**Meet the editor**

Dr. Grigorashvili Yury Evgenievich was born in 1947 in the town of Ussuriysk, in the Far East region of Russia. In 1972, he graduated from the Moscow Institute of Electronic Technology with his Master's degree in Electronic. In 1974, Dr. Grigorashvili received his Ph.D. in Electronic Engineering from the same university. After that, he worked as the lead engineer at the plant, specializing in

electronic chip manufacturing. From 1978 to 1988, Dr. Grigorashvili had held a position of senior research associate. Since then, he has taken the role of the director of superconducting microelectronics laboratory at the Moscow National Research University of Electronic Technology (MIET). As a result of completed projects over the years he has published more than 150 papers.

Contents

**Preface IX** 

Y.C. Tao

G.K. Padam

Chapter 7 **Josephson Vortices in** 

Chapter 8 **Shapiro Steps in** 

Kazuto Hirata

Michihide Kitamura

Chapter 1 **Role of Antiferromagnetic** 

Sung-Ho Suck Salk, Jae-Hyeon Eom,

Chapter 2 **Pairing Symmetry and Multiple Energy Gaps in** 

Liang-Jian Zou and Feng Lu

Chapter 3 **Josephson Current in a Ferromagnetic** 

Chapter 4 **Investigating Superconductivity with** 

Chapter 5 **Non Resonant Microwave Absorption (NRMA)** 

Chapter 6 **Superconductivity in Nanoscale Systems 105**  Meenakshi Singh, Yi Sun and Jian Wang

**High Tc Superconductors 137** 

**BSCCO Intrinsic Josephson Junctions 157** 

Seung Joon Shin, Jae-Gon Eom and Sung-Sik Lee

**Multi-Orbital Iron-Pnictide Superconductors 25** 

**Semiconductor/Semiconductor/Ferromagnetic** 

**Semiconductor Junction with Superconducting Contacts 45** 

**Electron Paramagnetic Resonance (EPR) Spectrometer 63**  D. Shaltiel, H.A. Krug von Nidda, B.Ya. Shapiro, A. Loidl, T. Tamegai, T. Kurz, B. Bogoslavsky, B. Rosenstein and I. Shapiro

**Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson (ESJ) Fluxons 83** 

**Fluctuations in High Temperature Superconductivity 1** 

## Contents

## **Preface XI**


X Contents


J.A. Veira, M.R. Osorio and F. Vidal

## Preface

Superconductivity is a very interesting phenomenon found in nature. Initially, the superconductivity was characterized by an abrupt drop to zero in the material's electrical resistance. An electric current, induced by the magnetic field in a superconducting loop, can flow indefinitely without any loss. Further research constantly modifies our concepts and understanding of the superconductivity as a more complex phenomenon.

The statement about the zero value of electrical resistance is valid only for direct electric current. In an AC electric field, the superconductor's resistance is different from zero even at the below critical temperatures. This is due to the fact that the sample has not only the superconducting current carriers but also normal current carriers. Normal electrons are the cause of heat allocation in a superconductor when it receives irradiation by high-frequency electromagnetic field. The losses rise with the increase in frequency and become noticeable in the microwave range of frequencies.

The difference between a superconductor and a perfect conductor (resistance of which is also equal to zero) is shown in the Meissner effect. Upon cooling of an ideal conductor in a constant magnetic field to a below critical temperature, the magnetic field remains unchanged in its volume. Under similar conditions the magnetic field found inside the superconductor would be absent after cooling. Even if the external magnetic field remains unchanged, a superconductor has arising electric currents on its surface that compensate for the field present inside the sample. However let's remember that generally the external magnetic field is pushed out only from a partial volume of a superconductor. It depends on the strength of the magnetic field, and the sample's shape and material. In the majority of known superconductors the magnetic field located in limited space, penetrates into the superconductor deeper in the form of a thin thread. The volume of this thread has no superconducting properties. The superconducting material is located outside this zone. Constant electric current finds its way out without any heat loss and the material appears to have zero electrical resistance. Ohmic resistance occurs in sufficiently strong magnetic fields only, when zones with the material in normal state increase and completely cover the path for the movement of the superconducting carriers.

#### X Preface

Considerable difference between the superconductor and conductor is found and proven by two Josephson effects. The first effect occurs in a superconductor that is divided by a thin dielectric layer. If the current's value passing through the structure is below critical, the dielectric voltage drop is equal to zero. In the metal-based structures divided by a thin dielectric, the current-voltage characteristics of the tunneling current have no sites with zero voltage found on the dielectric. The second effect occurs when the direct current with the magnitude above critical flows through the superconducting structure. In this case, the voltage drop will occur on the dielectric and the structure will begin to emit electromagnetic waves.

Preface XI

The book gives an overview of major problems encountered in this field of study. A list of countries, in which researchers work, demonstrates global interest to the problem of superconductivity. Thus, this book would be a great reference for further

First chapter of the book presents the theoretical results of the research done on superconductivity. The following chapters describe the manufacturing technology of the superconducting materials and structures as well as studied results of their properties. In the end, the book presents the devices currently designed for commercial applications. Besides the superconducting elements, these devices have the cooling systems of various design, electronic control units, information storage and processing units, and electrical elements that run required functions of the device. Technical realization of such devices demands great financial commitment by companies and other investors. Yet, the effect of these investments into the technosphere will pay off in the future, marking new heights in imminent progress

Most of the material presented in this book is the result of authors' own research that has been carried out over a long period of time. A number of chapters thoroughly describe the fundamental electrical and structural properties of the superconductors as well as the methods researching those properties. The sourcebook comprehensively covers the advanced techniques and concepts of superconductivity. It's intended for a

National Research University, Moscow Institute of Electronic Technology, Moscow,

**Dr. Yury Grigorashvili** 

Russia

research and development of ideas and applications.

and development.

wide range of readers.

For a long time all discovered superconducting materials had a very low critical temperature. The liquid helium was used to cool the superconducting materials. In 1976, the record for the low-temperature superconductors was achieved by finding the Nb3Ge (Tc = 23,2K) compound. Eminent changes also happened in 1986 when the superconductor – La2-xBaxCuO4. (Тс = 30 К) was synthesized. It took a year for other researchers to prove the existence of the superconductor with Тс > 30 К. In January 1987, the discovery of superconductivity at temperature of 90 К was announced (Y1 Ba2 Cu3 O7-x). One year later, bismuth-based high temperature superconductor was revealed (Тс = 110 К) and in a month, Tl2 Ba2 Ca2 Cu3 O10. (Тс = 125 К) was discovered as well. After the year 2002, there were invented both MgB2 (40 K) superconductors and iron-based superconductors (56 К). Today the record is held by HgBa2Ca2Cu3O10-x (135 К) compound.

Needed progress forward is impossible without thorough understanding of all the features found in superconductivity on both the macro and micro levels. The result of the research taken in this direction may lead to a discovery of a material, which would have high critical temperature and current. The findings of such materials could bring us closer to a breakthrough in the creation of highly efficient electric motors and power transmission lines, levitating transportation, and frictionless bearings. Currently known superconducting materials either have the critical temperature below 77 K, or are very fragile, preventing us from receiving an acceptable economical cost-effectiveness.

Yet, the pace of modern research is extraordinary. Today we already have and use the superconducting SQUID's that have the highest sensitivity to the magnetic field with minimal magnetosensitive zone. We use the Josephson electronic devices in the primary voltage standards. Among the infrared sensors, the superconducting receivers have maximum sensitivity and performance. The wires used in both systems for fusion devices and colliders are also superconductive. It's possible to assume that the quantum computer will be built on the basis of elements of superconducting spintronics. For the high-temperature superconductors the cooling process has already been considerably developed. On today's market we have both the acoustic refrigerators and the refrigerators made based on the thermoelectric effects, which are opening up new horizons for commercial applications of the superconducting devices.

The book gives an overview of major problems encountered in this field of study. A list of countries, in which researchers work, demonstrates global interest to the problem of superconductivity. Thus, this book would be a great reference for further research and development of ideas and applications.

X Preface

Considerable difference between the superconductor and conductor is found and proven by two Josephson effects. The first effect occurs in a superconductor that is divided by a thin dielectric layer. If the current's value passing through the structure is below critical, the dielectric voltage drop is equal to zero. In the metal-based structures divided by a thin dielectric, the current-voltage characteristics of the tunneling current have no sites with zero voltage found on the dielectric. The second effect occurs when the direct current with the magnitude above critical flows through the superconducting structure. In this case, the voltage drop will occur on the dielectric

For a long time all discovered superconducting materials had a very low critical temperature. The liquid helium was used to cool the superconducting materials. In 1976, the record for the low-temperature superconductors was achieved by finding the Nb3Ge (Tc = 23,2K) compound. Eminent changes also happened in 1986 when the superconductor – La2-xBaxCuO4. (Тс = 30 К) was synthesized. It took a year for other researchers to prove the existence of the superconductor with Тс > 30 К. In January 1987, the discovery of superconductivity at temperature of 90 К was announced (Y1 Ba2 Cu3 O7-x). One year later, bismuth-based high temperature superconductor was revealed (Тс = 110 К) and in a month, Tl2 Ba2 Ca2 Cu3 O10. (Тс = 125 К) was discovered as well. After the year 2002, there were invented both MgB2 (40 K) superconductors and iron-based superconductors (56 К). Today the record is held by

Needed progress forward is impossible without thorough understanding of all the features found in superconductivity on both the macro and micro levels. The result of the research taken in this direction may lead to a discovery of a material, which would have high critical temperature and current. The findings of such materials could bring us closer to a breakthrough in the creation of highly efficient electric motors and power transmission lines, levitating transportation, and frictionless bearings. Currently known superconducting materials either have the critical temperature below 77 K, or are very fragile, preventing us from receiving an acceptable economical

Yet, the pace of modern research is extraordinary. Today we already have and use the superconducting SQUID's that have the highest sensitivity to the magnetic field with minimal magnetosensitive zone. We use the Josephson electronic devices in the primary voltage standards. Among the infrared sensors, the superconducting receivers have maximum sensitivity and performance. The wires used in both systems for fusion devices and colliders are also superconductive. It's possible to assume that the quantum computer will be built on the basis of elements of superconducting spintronics. For the high-temperature superconductors the cooling process has already been considerably developed. On today's market we have both the acoustic refrigerators and the refrigerators made based on the thermoelectric effects, which are opening up new horizons for commercial applications of the superconducting devices.

and the structure will begin to emit electromagnetic waves.

HgBa2Ca2Cu3O10-x (135 К) compound.

cost-effectiveness.

First chapter of the book presents the theoretical results of the research done on superconductivity. The following chapters describe the manufacturing technology of the superconducting materials and structures as well as studied results of their properties. In the end, the book presents the devices currently designed for commercial applications. Besides the superconducting elements, these devices have the cooling systems of various design, electronic control units, information storage and processing units, and electrical elements that run required functions of the device. Technical realization of such devices demands great financial commitment by companies and other investors. Yet, the effect of these investments into the technosphere will pay off in the future, marking new heights in imminent progress and development.

Most of the material presented in this book is the result of authors' own research that has been carried out over a long period of time. A number of chapters thoroughly describe the fundamental electrical and structural properties of the superconductors as well as the methods researching those properties. The sourcebook comprehensively covers the advanced techniques and concepts of superconductivity. It's intended for a wide range of readers.

**Dr. Yury Grigorashvili** 

National Research University, Moscow Institute of Electronic Technology, Moscow, Russia

**1. Introduction**

temperature, T∗.

The two-dimensional layers of cuprate oxides are known to be the systems of strongly repulsive (correlated) electrons as the Mott insulators which have revealed various novel physical properties uniquely different from the conventional low temperature superconductors. They show the antiferromagnetic (AF) infinite-range or long-range order (AFLRO) at and near half-filling. As hole doping increases, the AFLRO diminishes and the short-range (finite-range) AF order takes over with the emergence of d-wave superconductivity. The two-dimensional systems of strongly correlated electrons involved with strong repulsive interactions may favor the spin singlet pairing order (or correlations) of d-wave symmetry over that of s-wave symmetry. Here the spin singlet paring correlations are concerned with the AF spin fluctuations of the shortest possible correlation length among the AF spin fluctuations of all possible correlation lengths which appear in the region of hole doping away from half-filling. In this region of hole doping the cuprate oxides exhibit the novel structure of the high T*<sup>C</sup>* phase diagram characterized by the dome-shaped superconducting transition temperature, T*<sup>C</sup>* and the monotonously decreasing pseudogap

**Role of Antiferromagnetic Fluctuations in** 

*1Department of Physics, Pohang University of Science and Technology, Pohang* 

*4Department of Physics and Astronomy, McMaster University, Hamilton, ON* 

**High Temperature Superconductivity** 

Sung-Ho Suck Salk1,2,3, Jae-Hyeon Eom1, Seung Joon Shin1,

Jae-Gon Eom1 and Sung-Sik Lee4

*Science and Technology, Sungnam-si 3Korea Institute of Advanced Study, Seoul* 

> *1,2,3Korea 4Canada*

**1**

*2Natural Science Division, The Korean Academy of* 

Earlier. slave-boson approaches of the t-J Hamiltionian were proposed by researchers in the field[1–3] in an attempt to reproduce the observed high T*<sup>C</sup>* phase diagram. They were successful in reproducing the monotonously decreasing pseudogap temperature T∗ in agreement with observation. T∗ is shown to be the spin gap temperature at which the spin singlet pairing order or the spin (spinon) pairing correlations begins to open. However, their treatment of single-holon bose condensation has led to a linear increase of the bose condensation temperature T*<sup>C</sup>* rather than the observed dome-shaped T*C*[2, 3]. Later we introduced a slave-boson approach which allows the double-holon bose condensation[4] and failed to reproduce the dome-shaped T*C*, also yielding the linearly increasing trend of *TC*.

## **Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity**

Sung-Ho Suck Salk1,2,3, Jae-Hyeon Eom1, Seung Joon Shin1, Jae-Gon Eom1 and Sung-Sik Lee4 *1Department of Physics, Pohang University of Science and Technology, Pohang 2Natural Science Division, The Korean Academy of Science and Technology, Sungnam-si 3Korea Institute of Advanced Study, Seoul 4Department of Physics and Astronomy, McMaster University, Hamilton, ON 1,2,3Korea 4Canada* 

#### **1. Introduction**

The two-dimensional layers of cuprate oxides are known to be the systems of strongly repulsive (correlated) electrons as the Mott insulators which have revealed various novel physical properties uniquely different from the conventional low temperature superconductors. They show the antiferromagnetic (AF) infinite-range or long-range order (AFLRO) at and near half-filling. As hole doping increases, the AFLRO diminishes and the short-range (finite-range) AF order takes over with the emergence of d-wave superconductivity. The two-dimensional systems of strongly correlated electrons involved with strong repulsive interactions may favor the spin singlet pairing order (or correlations) of d-wave symmetry over that of s-wave symmetry. Here the spin singlet paring correlations are concerned with the AF spin fluctuations of the shortest possible correlation length among the AF spin fluctuations of all possible correlation lengths which appear in the region of hole doping away from half-filling. In this region of hole doping the cuprate oxides exhibit the novel structure of the high T*<sup>C</sup>* phase diagram characterized by the dome-shaped superconducting transition temperature, T*<sup>C</sup>* and the monotonously decreasing pseudogap temperature, T∗.

Earlier. slave-boson approaches of the t-J Hamiltionian were proposed by researchers in the field[1–3] in an attempt to reproduce the observed high T*<sup>C</sup>* phase diagram. They were successful in reproducing the monotonously decreasing pseudogap temperature T∗ in agreement with observation. T∗ is shown to be the spin gap temperature at which the spin singlet pairing order or the spin (spinon) pairing correlations begins to open. However, their treatment of single-holon bose condensation has led to a linear increase of the bose condensation temperature T*<sup>C</sup>* rather than the observed dome-shaped T*C*[2, 3]. Later we introduced a slave-boson approach which allows the double-holon bose condensation[4] and failed to reproduce the dome-shaped T*C*, also yielding the linearly increasing trend of *TC*.

diagonal hopping *t*

hopping integral; *t*

holon (*b*), that is, *ci<sup>σ</sup>* = *fiσb*†

*Ht*−*t*�

Hamiltonian is given by

with

coupling constant.

� term into the *t* − *J* Hamiltonian. It is given by,

(**S***<sup>i</sup>* · **<sup>S</sup>***<sup>j</sup>* <sup>−</sup> <sup>1</sup>

Here ∑<*i*,*j*<sup>&</sup>gt; denotes summation over the nearest neighbor sites i and j, ∑<*i*,*j*>, the summation

operator with the constraint of no double occupancy at each site i. t is the nearest neighbor

We take the slave-boson representation of electron operator as a composite of spinon (*f*) and

in the above equation, the resulting U(1) slave-boson representation of the above *t* − *t*� − *J*

Here *λ<sup>i</sup>* is the Lagrange multiplier field which enforces the single occupancy constraint.

Taking proper Hubbard-Stratonovich transformations and associated algebras by closely following our recently proposed slave-boson theory[5] (see Appendix A for details), one

)*fi<sup>σ</sup>*

*<sup>i</sup><sup>σ</sup> fj<sup>σ</sup>* + *c*.*c*.

*ij*(*fi*<sup>↓</sup> *fj*<sup>↑</sup> − *fi*<sup>↑</sup> *fj*↓) + *c*.*c*.

 *χ*∗ *ij <sup>f</sup>* †

<sup>2</sup> <sup>∑</sup>*αβ <sup>c</sup>*†

*<sup>j</sup> bi* + *c*.*c*.) − *t*

*<sup>i</sup>σc*˜*j<sup>σ</sup>* + *c*.*c*) − *t*

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 3

4

� ∑ <*i*,*j*>� (*c*˜ †

†

, the next-nearest neighbor hopping integral and J, the Heisenberg

� ∑ <*i*,*j*>� ,*σ* (*f* † *<sup>i</sup><sup>σ</sup> fjσb*†

*<sup>i</sup>* with the single occupancy constraint at each site i. Following

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

L*eff* = L<sup>0</sup> + L*<sup>f</sup>* + L*<sup>b</sup>* (3)

*<sup>i</sup> bi* − 1). (2)

*<sup>i</sup>σc*˜*j<sup>σ</sup>* + *c*.*c*)

*ninj*), . (1)

*<sup>i</sup>σ*), the electron annihilation(creation)

*<sup>i</sup>ασαβci<sup>β</sup>* with *σαβ*, the Pauli spin matrix

*<sup>j</sup> bi* + *c*.*c*.)

<sup>2</sup> + *x*2), (4)

, (5)

<*i*,*j*> (*c*˜ †

+*J* ∑ <*i*,*j*>

*Ht*−*t*�−*<sup>J</sup>* <sup>=</sup> <sup>−</sup>*<sup>t</sup>* ∑

over the next-nearest neighbor (diagonal) sites and *c*˜*iσ*(*c*˜

�

a rigorous slave-boson treatment of **S***<sup>i</sup>* = <sup>1</sup>

<sup>−</sup>*<sup>J</sup>* <sup>=</sup> <sup>−</sup>*<sup>t</sup>* ∑

− *J* <sup>2</sup> ∑ <*i*,*j*>

+*i*∑ *i <sup>λ</sup>i*(∑*<sup>σ</sup> f* † *<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>+</sup> *<sup>b</sup>*†

obtains the following effective Lagrangian,

<sup>L</sup><sup>0</sup> <sup>=</sup> *Jx*

<sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| <sup>2</sup> + 1 2 |*χij*| <sup>2</sup> + 1 4 + *J* <sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| <sup>2</sup>(|Δ*<sup>b</sup> ij*|

L*<sup>f</sup>* = ∑ *i*,*σ f* † *<sup>i</sup>σ*(*∂τ* <sup>−</sup> *<sup>μ</sup><sup>f</sup>*

> − *Jx* <sup>4</sup> ∑ <*i*,*j*>,*σ*

−*t* � *x* ∑ <*i*,*j*>� ,*σ f* † *<sup>i</sup><sup>σ</sup> fj<sup>σ</sup>* + *c*.*c*.

− *Jx* <sup>2</sup> ∑ <*i*,*j*> Δ*<sup>f</sup>* <sup>∗</sup>

<*i*,*j*>,*σ*

(*f* † *<sup>i</sup><sup>σ</sup> fjσb*†

*bibjb*† *j b*† *<sup>i</sup>* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

Soon after this study we[5] proposed an improved slave-boson theory which fundamentally differs from these approaches in that a term involving coupling between the spin and charge degrees of freedom or simply spin-charge coupling appears in our rigorous slave-boson treatment of the t-J Hamiltonian. The resulting effective mean field Lagrangian reveals coupling between the spin (spinon) paring order, Δ*<sup>f</sup>* and the charge (holon) pairing order, Δ*b*. As a consequence the Cooper pairing order is satisfactorily seen to be a composite of these two order parameters, Δ*<sup>f</sup>* and Δ*<sup>b</sup>* to allow for the bose condensation of the Cooper pairs rather than the single-holon bose condensation or the double-holon bose condensation. Accordingly this theory has led to successful reproductions of not only the monotonously decreasing spin gap temperature but also the long-waited dome-shaped structure of the superconducting transition temperature in the phase diagram. Further other important physical observations such as the boomerang behavior of superfluid weight, the peak-dip-hump structure of optical conductivity and both the temperature and doping dependence of spectral functions are reproduced in agreement with observations[6].

For the sake of self-containment we will first review our earlier proposed slave-boson theory[5] of the t-J Hamiltonian which reveals the spin-charge coupling mentioned above. Earlier it was shown by others that inclusion of the *t*� term in the t-J Hamiltonian leads to satisfactory descriptions of the electronic structure of high T*<sup>C</sup>* cuprates[7–11] and the enhancement of pairing correlation resulting in an increasing trend of T*<sup>C</sup>* in the overdoped region in the phase diagram for the choice of *t* � /*t* < 0, e.g., *t* � /*t* = −0.3[12, 13]. It is, thus, of great interest to see how its inclusion affects the entire structure of the phase diagram which includes the pseudogap temperature. At present there has been no study which addresses the role of the diagonal hopping *t*� on the spin gap temperature, T∗. Such study is needed to find whether there exists any relation between T∗ and T*<sup>C</sup>* or the spin gap phase and the superconducting phase. In this regard we would like to draw attention to the fact that the observed phase diagrams of high T*<sup>C</sup>* cuprate samples (e.g., LSCO and BSCCO samples) reveal that higher the T∗, higher the T*<sup>C</sup>* as earlier discussed by Oda et al.[14] This suggests that the two energy or temperature scales, T∗ and T*<sup>C</sup>* are no longer independent of each other. Thus one of our main objectives is to study how the pseudogap or spin gap temperature, T∗ and the superconducting transition temperature, *TC* are correlated and show that such correlation arises owing to the presence of the short-range antiferromagnetic (AF) spin fluctuations of the shortest possible correlation length involved with the spin pairing correlations. For a concerted, self-consistent study, we use a predicted phase diagram to calculate magnetic susceptibility and discuss two important observations made by the inelastic neutron scattering (INS) measurements, namely the temperature dependence of the magnetic resonance peak[17] and the linear scaling behavior between the magnetic peak resonance energy, E*res* and the superconducting transition temperature[18]. From this study we show that the short-range AF spin fluctuations are directly responsible for the magnetic susceptibility observed by the INS measurements mentioned here.

#### **2. Theory: U(1) slave boson representation of the t-J Hamiltonian**

In the present study we limit ourselves to the derivation of the U(1) slave boson representation of the t-J Hamiltonian. We refer details of its derivation to Appendix A. In Appendix B a brief exposure of the SU(2) approach is made in association with the U(1) representation. Here only a rudimentary description is presented by introducing the next-nearest neighbor hopping or diagonal hopping *t* � term into the *t* − *J* Hamiltonian. It is given by,

$$H\_{l-l'-l} = -t\sum\_{} \left(\tilde{c}\_{i\sigma}^{\dagger}\tilde{c}\_{j\sigma} + c.c\right) - t'\sum\_{} \left(\tilde{c}\_{i\sigma}^{\dagger}\tilde{c}\_{j\sigma} + c.c\right)$$

$$+J\sum\_{} (\mathbf{S}\_{i\uparrow}\cdot\mathbf{S}\_{j} - \frac{1}{4}n\_{i}n\_{j})\_{\prime}.\tag{1}$$

Here ∑<*i*,*j*<sup>&</sup>gt; denotes summation over the nearest neighbor sites i and j, ∑<*i*,*j*>, the summation over the next-nearest neighbor (diagonal) sites and *c*˜*iσ*(*c*˜ † *<sup>i</sup>σ*), the electron annihilation(creation) operator with the constraint of no double occupancy at each site i. t is the nearest neighbor hopping integral; *t* � , the next-nearest neighbor hopping integral and J, the Heisenberg coupling constant.

We take the slave-boson representation of electron operator as a composite of spinon (*f*) and holon (*b*), that is, *ci<sup>σ</sup>* = *fiσb*† *<sup>i</sup>* with the single occupancy constraint at each site i. Following a rigorous slave-boson treatment of **S***<sup>i</sup>* = <sup>1</sup> <sup>2</sup> <sup>∑</sup>*αβ <sup>c</sup>*† *<sup>i</sup>ασαβci<sup>β</sup>* with *σαβ*, the Pauli spin matrix in the above equation, the resulting U(1) slave-boson representation of the above *t* − *t*� − *J* Hamiltonian is given by

$$H\_{l-l'-l} = -t\sum\_{,\sigma} (f\_{i\sigma}^{\dagger}f\_{j\sigma}b\_{\slash}^{\dagger}b\_{\choose h} + c.c.) - t'\sum\_{,l\sigma} (f\_{i\sigma}^{\dagger}f\_{j\sigma}b\_{\slash}^{\dagger}b\_{\choose h} + c.c.)$$

$$-\frac{I}{2}\sum\_{}b\_{i}b\_{\slash}b\_{\slash}^{\dagger}b\_{i}^{\dagger}(f\_{i\downarrow}^{\dagger}f\_{j\uparrow}^{\dagger} - f\_{i\uparrow}^{\dagger}f\_{j\downarrow}^{\dagger})(f\_{j\uparrow}f\_{i\downarrow} - f\_{j\downarrow}f\_{i\uparrow})$$

$$+i\sum\_{i}\lambda\_{i}(\sum\_{\sigma}f\_{i\sigma}^{\dagger}f\_{i\sigma} + b\_{i}^{\dagger}b\_{i} - 1). \tag{2}$$

Here *λ<sup>i</sup>* is the Lagrange multiplier field which enforces the single occupancy constraint.

Taking proper Hubbard-Stratonovich transformations and associated algebras by closely following our recently proposed slave-boson theory[5] (see Appendix A for details), one obtains the following effective Lagrangian,

$$
\mathcal{L}\_{eff} = \mathcal{L}\_0 + \mathcal{L}\_f + \mathcal{L}\_b \tag{3}
$$

with

2 Will-be-set-by-IN-TECH

Soon after this study we[5] proposed an improved slave-boson theory which fundamentally differs from these approaches in that a term involving coupling between the spin and charge degrees of freedom or simply spin-charge coupling appears in our rigorous slave-boson treatment of the t-J Hamiltonian. The resulting effective mean field Lagrangian reveals coupling between the spin (spinon) paring order, Δ*<sup>f</sup>* and the charge (holon) pairing order, Δ*b*. As a consequence the Cooper pairing order is satisfactorily seen to be a composite of these two order parameters, Δ*<sup>f</sup>* and Δ*<sup>b</sup>* to allow for the bose condensation of the Cooper pairs rather than the single-holon bose condensation or the double-holon bose condensation. Accordingly this theory has led to successful reproductions of not only the monotonously decreasing spin gap temperature but also the long-waited dome-shaped structure of the superconducting transition temperature in the phase diagram. Further other important physical observations such as the boomerang behavior of superfluid weight, the peak-dip-hump structure of optical conductivity and both the temperature and doping dependence of spectral functions are

For the sake of self-containment we will first review our earlier proposed slave-boson theory[5] of the t-J Hamiltonian which reveals the spin-charge coupling mentioned above. Earlier it was shown by others that inclusion of the *t*� term in the t-J Hamiltonian leads to satisfactory descriptions of the electronic structure of high T*<sup>C</sup>* cuprates[7–11] and the enhancement of pairing correlation resulting in an increasing trend of T*<sup>C</sup>* in the overdoped

is, thus, of great interest to see how its inclusion affects the entire structure of the phase diagram which includes the pseudogap temperature. At present there has been no study which addresses the role of the diagonal hopping *t*� on the spin gap temperature, T∗. Such study is needed to find whether there exists any relation between T∗ and T*<sup>C</sup>* or the spin gap phase and the superconducting phase. In this regard we would like to draw attention to the fact that the observed phase diagrams of high T*<sup>C</sup>* cuprate samples (e.g., LSCO and BSCCO samples) reveal that higher the T∗, higher the T*<sup>C</sup>* as earlier discussed by Oda et al.[14] This suggests that the two energy or temperature scales, T∗ and T*<sup>C</sup>* are no longer independent of each other. Thus one of our main objectives is to study how the pseudogap or spin gap temperature, T∗ and the superconducting transition temperature, *TC* are correlated and show that such correlation arises owing to the presence of the short-range antiferromagnetic (AF) spin fluctuations of the shortest possible correlation length involved with the spin pairing correlations. For a concerted, self-consistent study, we use a predicted phase diagram to calculate magnetic susceptibility and discuss two important observations made by the inelastic neutron scattering (INS) measurements, namely the temperature dependence of the magnetic resonance peak[17] and the linear scaling behavior between the magnetic peak resonance energy, E*res* and the superconducting transition temperature[18]. From this study we show that the short-range AF spin fluctuations are directly responsible for the magnetic

�

/*t* < 0, e.g., *t*

�

/*t* = −0.3[12, 13]. It

reproduced in agreement with observations[6].

region in the phase diagram for the choice of *t*

susceptibility observed by the INS measurements mentioned here.

**2. Theory: U(1) slave boson representation of the t-J Hamiltonian**

In the present study we limit ourselves to the derivation of the U(1) slave boson representation of the t-J Hamiltonian. We refer details of its derivation to Appendix A. In Appendix B a brief exposure of the SU(2) approach is made in association with the U(1) representation. Here only a rudimentary description is presented by introducing the next-nearest neighbor hopping or

$$\mathcal{L}\_0 = \frac{I\_\mathbf{x}}{2} \sum\_{} \left\{ |\Delta\_{ij}^f|^2 + \frac{1}{2} |\chi\_{ij}|^2 + \frac{1}{4} \right\} + \frac{J}{2} \sum\_{} |\Delta\_{ij}^f|^2 (|\Delta\_{ij}^b|^2 + \mathbf{x}^2),\tag{4}$$

$$\begin{split} \mathcal{L}\_{f} &= \sum\_{i,\tau} f\_{i\tau}^{\dagger} (\mathfrak{d}\_{\tau} - \mu^{f}) f\_{i\tau} \\ &- \frac{I\_{\text{X}}}{\mathbf{4}} \sum\_{,\sigma} \left\{ \chi\_{ij}^{\*} f\_{i\sigma}^{\dagger} f\_{j\sigma} + c.c. \right\} \\ &- t^{\dagger} \mathbf{x} \sum\_{,\tau} \left\{ f\_{i\sigma}^{\dagger} f\_{j\tau} + c.c. \right\} \\ &- \frac{I\_{\text{X}}}{\mathbf{2}} \sum\_{} \left\{ \Delta f\_{ij}^{\dagger} (f\_{i\downarrow} f\_{j\uparrow} - f\_{i\uparrow} f\_{j\downarrow}) + c.c. \right\}, \end{split} \tag{5}$$

 **0**

 **0 0.1 0.2 0.3**

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 5

Fig. 1. (color online). Variation of phase diagram with *t*� for Heisenberg coupling strength,

The contribution of the next-nearest neighbor hopping or the diagonal hopping is readily understood from the inspection of Eq.8 by noting that the value of cos *kx* cos *ky* is negative at the hot spot (*π*, 0), zero at the cold spot (*π*/2, *π*/2) and positive at (0, 0). From this we see

expected to lead to the enhancement (depression) of AF spin (spinon) pairing correlations or the spin singlet pairing order of d-wave symmetry compared to the case of *t*� = 0), that is, no diagonal hoping. The charge (holon) pairing of s-wave symmetry will be enhanced at the

Here we explore the role of the next-nearest neighbor hopping, i.e., the diagonal hopping on both the pseudogap temperatureT∗ and the superconducting transition temperature, T*<sup>C</sup>* and the cause of correlation between these two temperature scales or relatedly the spin gap phase and the d-wave superconducting phase. Earlier the negative value of *t*� was shown to match well the observed Fermi surface of the hole doped cuprate oxides while its positive value

value of the Heisenberg coupling constant, *J* = 0.3*t*. In the underdoped region both T∗ and T*<sup>C</sup>* are predicted to remain largely unchanged despite the considerable change of *t*�

� = 0.3*t*), we examine the dependence of the phase diagram on *t*� for

that stabilization (destabilization) of the spinon energy at the hot spot with *t*� < 0 (*t*

**3. Role of next-nearest neighbor hopping on the structure of phase diagram**

matches that of the electron doped cuprate oxides[7] as mentioned above.

Choosing the two different cases of the diagonal hopping, one for *t*� < 0 (e.g., *t*

Fig.1 shows the computed results of the phase diagram with the variation of *t*�

x

**t'/t=0.3 t'/t=0 t'/t=-0.3**

J/t=0.3

� > 0) is

� = −0.3*t*) and

/*t* at a fixed

/*t*

 **0.02**

 **0.04**

T/t

*J* = 0.3*t*.

nodal points.

the other for *t*

� > 0 (e.g., *t*

the hole doped cuprate oxides.

 **0.06**

 **0.08**

for the spin spinon sector and

$$\mathcal{L}\_b = \sum\_i b\_i^\dagger (\partial\_\mathbf{r} - \mu^b) b\_i - t \sum\_{} \left\{ \chi\_{ij}^\* b\_i^\dagger b\_j + c.c. \right\} $$

$$ -t'(1-x) \sum\_{} \left\{ b\_i^\dagger b\_j + c.c. \right\} $$

$$ -\frac{I}{2} \sum\_{} |\Delta\_{ij}^\dagger|^2 \left\{ \Delta\_{ij}^{b\_i} b\_i b\_j + c.c. \right\}, \tag{6} $$

for the charge (holon) sector. Here *μf*(*μb*) is the spinon(holon) chemical potential. *χij* is the hopping order parameter and Δ*<sup>f</sup> ij* <sup>=</sup><sup>&</sup>lt; (*fi*<sup>↑</sup> *fj*<sup>↓</sup> <sup>−</sup> *fi*<sup>↓</sup> *fj*↑) <sup>&</sup>gt; (Δ*<sup>b</sup> ij* =< *bibj* >), the spinon (holon) pairing order parameter; *<sup>x</sup>*, the hole doping concentration and *Jx* <sup>=</sup> *<sup>J</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)2. The last term of Eq. 6 reveals the presence of coupling between the spin (spinon) and charge (holon) degrees of freedom, i.e., simply termed as spin-charge coupling, as seen in the form of the product of the spin (spinon) single pairing order, Δ*<sup>f</sup>* and the charge (holon) pairing order, Δ*b*. Thanks to this coupling the Cooper pairing order Δ is, now, properly represented as a composite of these two order parameters, Δ*<sup>f</sup>* and Δ*b*. We point out that the holon (charge) sector, Eq. 6 is coupled with the spinon (spin) sector, Eq. 5 owing to the presence of coupling between the spinon paring order Δ*<sup>f</sup>* and the holon pairing order Δ*<sup>b</sup>* as shown in the last term of Eq. 6.

The resulting free energy (see derivation in Appendix A) is given by,

$$\begin{split} F &= -\frac{1}{\beta} \ln Z \\ &= J\_X N \left( 2|\Delta^f|^2 + \frac{1}{2} |\chi|^2 \right) + J N |\Delta^f|^2 |\Delta^b|^2 \\ &\quad - (1 - x) N \mu\_{eff}^f + N x \mu\_{eff}^b + \frac{1}{2} \sum\_{\mathbf{k}} E\_{\mathbf{k}\mathbf{s}}^b \\ &\quad - 2N k\_B T \ln 2 - 2 k\_B T \sum\_{\mathbf{k}} \cosh \frac{\beta E\_{\mathbf{k}\mathbf{s}}^f}{2} \\ &\quad + k\_B T \sum\_{\mathbf{k}} \ln \left( 1 - e^{-\beta E\_{\mathbf{k}\mathbf{s}}^b} \right), \end{split} \tag{7}$$

where <sup>Z</sup> <sup>=</sup> <sup>D</sup> *<sup>f</sup>* <sup>D</sup>*b*D*χ*DΔ*f*DΔ*b*D*λe*<sup>−</sup> *<sup>β</sup>* <sup>0</sup> *<sup>d</sup>τ*L*eff* is the partition function; Δ*<sup>f</sup>* =< (*fi*<sup>↑</sup> *fj*<sup>↓</sup> <sup>−</sup> *fi*<sup>↓</sup> *fj*↑) <sup>&</sup>gt;(Δ*<sup>b</sup>* <sup>=</sup><sup>&</sup>lt; *bibj* <sup>&</sup>gt;), the spinon(holon) pairing order parameter; *<sup>E</sup><sup>f</sup>* **<sup>k</sup>** <sup>=</sup> (*� f* **<sup>k</sup>** <sup>−</sup> *<sup>μ</sup><sup>f</sup> eff*) + <sup>Δ</sup>*<sup>f</sup>* **k** 2 (*E<sup>b</sup>* **<sup>k</sup>** = (*�<sup>b</sup>* **<sup>k</sup>** <sup>−</sup> *<sup>μ</sup><sup>b</sup> eff*) <sup>−</sup> <sup>Δ</sup>*<sup>b</sup>* **k** 2 ), the spinon(holon) quasiparticle energy; *x*, the hole doping rate;*Jx* <sup>=</sup> *<sup>J</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> and N, the total number of sites in a square lattice. Here the spinon and holon energies, *� <sup>f</sup>* **<sup>k</sup>** and *�<sup>b</sup>* **<sup>k</sup>** are respectively,

$$
\epsilon\_\mathbf{k}^f = -\frac{I\_\mathbf{x}}{2}\chi(\cos k\_\mathbf{x} + \cos k\_\mathbf{y}) - 4t'\mathbf{x}\cos k\_\mathbf{x}\cos k\_\mathbf{y}
$$

$$
\epsilon\_\mathbf{k}^b = -2t\chi(\cos k\_\mathbf{x} + \cos k\_\mathbf{y}) - 4t'\mathbf{x}\cos k\_\mathbf{x}\cos k\_\mathbf{y}.\tag{8}
$$

4 Will-be-set-by-IN-TECH

 *b*† *<sup>i</sup> bj* + *c*.*c*.

for the charge (holon) sector. Here *μf*(*μb*) is the spinon(holon) chemical potential. *χij* is the

*ij* <sup>=</sup><sup>&</sup>lt; (*fi*<sup>↑</sup> *fj*<sup>↓</sup> <sup>−</sup> *fi*<sup>↓</sup> *fj*↑) <sup>&</sup>gt; (Δ*<sup>b</sup>*

pairing order parameter; *<sup>x</sup>*, the hole doping concentration and *Jx* <sup>=</sup> *<sup>J</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)2. The last term of Eq. 6 reveals the presence of coupling between the spin (spinon) and charge (holon) degrees of freedom, i.e., simply termed as spin-charge coupling, as seen in the form of the product of the spin (spinon) single pairing order, Δ*<sup>f</sup>* and the charge (holon) pairing order, Δ*b*. Thanks to this coupling the Cooper pairing order Δ is, now, properly represented as a composite of these two order parameters, Δ*<sup>f</sup>* and Δ*b*. We point out that the holon (charge) sector, Eq. 6 is coupled with the spinon (spin) sector, Eq. 5 owing to the presence of coupling between the spinon paring order Δ*<sup>f</sup>* and the holon pairing order Δ*<sup>b</sup>* as shown in the last term of Eq. 6.

*eff* <sup>+</sup> *Nxμ<sup>b</sup>*

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*βE<sup>b</sup>* **k***s* 

(*fi*<sup>↑</sup> *fj*<sup>↓</sup> <sup>−</sup> *fi*<sup>↓</sup> *fj*↑) <sup>&</sup>gt;(Δ*<sup>b</sup>* <sup>=</sup><sup>&</sup>lt; *bibj* <sup>&</sup>gt;), the spinon(holon) pairing order parameter; *<sup>E</sup><sup>f</sup>* **<sup>k</sup>** <sup>=</sup>

the hole doping rate;*Jx* <sup>=</sup> *<sup>J</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> and N, the total number of sites in a square lattice. Here

**<sup>k</sup>** are respectively,

�

�

*x* cos *kx* cos *ky*

*eff*) <sup>−</sup> <sup>Δ</sup>*<sup>b</sup>* **k** 2

<sup>2</sup> *<sup>χ</sup>*(cos *kx* <sup>+</sup> cos *ky*) <sup>−</sup> <sup>4</sup>*<sup>t</sup>*

**<sup>k</sup>** = −2*tχ*(cos *kx* + cos *ky*) − 4*t*

<*i*,*j*> *χ*∗ *ijb*† *<sup>i</sup> bj* + *c*.*c*.

*bibj* + *c*.*c*.

<sup>+</sup> *JN*|Δ*<sup>f</sup>*

cosh *<sup>β</sup>E<sup>f</sup>* **k***s* 2

*eff* + 1 <sup>2</sup> ∑ **k** *Eb* **k***s*

**k**


, (6)

*ij* =< *bibj* >), the spinon (holon)

, (7)

<sup>0</sup> *<sup>d</sup>τ*L*eff* is the partition function; Δ*<sup>f</sup>* =<

), the spinon(holon) quasiparticle energy; *x*,

*x* cos *kx* cos *ky*. (8)

*<sup>i</sup>* (*∂τ* <sup>−</sup> *<sup>μ</sup>b*)*bi* <sup>−</sup> *<sup>t</sup>* ∑

(<sup>1</sup> − *<sup>x</sup>*) ∑ <*i*,*j*>�

for the spin spinon sector and

hopping order parameter and Δ*<sup>f</sup>*

L*<sup>b</sup>* = ∑ *i b*†

> −*t* �

− *J* <sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| 2 Δ*b ij* ∗

The resulting free energy (see derivation in Appendix A) is given by,

*<sup>β</sup>* ln <sup>Z</sup>

 <sup>2</sup>|Δ*<sup>f</sup>* | <sup>2</sup> + 1 2 |*χ*| 2 

<sup>−</sup>(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)*Nμ<sup>f</sup>*

+*kBT*∑ **k** ln 

**<sup>k</sup>** and *�<sup>b</sup>*

where <sup>Z</sup> <sup>=</sup> <sup>D</sup> *<sup>f</sup>* <sup>D</sup>*b*D*χ*DΔ*f*DΔ*b*D*λe*<sup>−</sup> *<sup>β</sup>*

*� f* **<sup>k</sup>** <sup>=</sup> <sup>−</sup> *Jx*

*�b*

 (*�<sup>b</sup>* **<sup>k</sup>** <sup>−</sup> *<sup>μ</sup><sup>b</sup>*

(*� f* **<sup>k</sup>** <sup>−</sup> *<sup>μ</sup><sup>f</sup>*

*eff*) + <sup>Δ</sup>*<sup>f</sup>* **k** 2 (*E<sup>b</sup>* **<sup>k</sup>** =

the spinon and holon energies, *� <sup>f</sup>*

−2*NkBT* ln 2 − <sup>2</sup>*kBT* ∑

*<sup>F</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup>

= *JxN*

Fig. 1. (color online). Variation of phase diagram with *t*� for Heisenberg coupling strength, *J* = 0.3*t*.

The contribution of the next-nearest neighbor hopping or the diagonal hopping is readily understood from the inspection of Eq.8 by noting that the value of cos *kx* cos *ky* is negative at the hot spot (*π*, 0), zero at the cold spot (*π*/2, *π*/2) and positive at (0, 0). From this we see that stabilization (destabilization) of the spinon energy at the hot spot with *t*� < 0 (*t* � > 0) is expected to lead to the enhancement (depression) of AF spin (spinon) pairing correlations or the spin singlet pairing order of d-wave symmetry compared to the case of *t*� = 0), that is, no diagonal hoping. The charge (holon) pairing of s-wave symmetry will be enhanced at the nodal points.

#### **3. Role of next-nearest neighbor hopping on the structure of phase diagram**

Here we explore the role of the next-nearest neighbor hopping, i.e., the diagonal hopping on both the pseudogap temperatureT∗ and the superconducting transition temperature, T*<sup>C</sup>* and the cause of correlation between these two temperature scales or relatedly the spin gap phase and the d-wave superconducting phase. Earlier the negative value of *t*� was shown to match well the observed Fermi surface of the hole doped cuprate oxides while its positive value matches that of the electron doped cuprate oxides[7] as mentioned above.

Choosing the two different cases of the diagonal hopping, one for *t*� < 0 (e.g., *t* � = −0.3*t*) and the other for *t* � > 0 (e.g., *t* � = 0.3*t*), we examine the dependence of the phase diagram on *t*� for the hole doped cuprate oxides.

Fig.1 shows the computed results of the phase diagram with the variation of *t*� /*t* at a fixed value of the Heisenberg coupling constant, *J* = 0.3*t*. In the underdoped region both T∗ and T*<sup>C</sup>* are predicted to remain largely unchanged despite the considerable change of *t*� /*t*

 **0**

 **0 0.1 0.2 0.3 0.4**

**J/t=0.2 J/t=0.3 J/t=0.4**

t'/t=-0.3

x

Fig. 2. (color online). Variaton of phase diagram with Heisenberg coupling strength J for

**4. Magnetic susceptibility based on the U(1) slave-boson representation**

To put it otherwise, the short-range AF spin fluctuations play a key role of causing such inseparable relation between the two temperature scales or relatedly the spin gap phase and the superconducting phase. This finding is consistent with the observed phase diagrams with different cuprate samples which shows higher the T∗, higher the T*C*[14] as mentioned earlier. It is then assured that the superconducting phase transition will not arise in the absence of the spin gap phase below T∗ which is initiated by the short-range AF spin fluctuations involved

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 7

The observed high *Tc* phase diagrams of cuprate oxides are characterized by the pseudogap or spin gap phase which exists below the monotonously decreasing T∗ and the d-wave superconducting phase below the dome shaped T*C*[15, 16]. From their inelastic neutron scattering measurements (INS) of the temperature dependence of magnetic resonance peaks for YB2Cu3O6+*<sup>x</sup>* (YBCO) Dai et al. [17] reported that the magnetic resonance begins to appear at the pseudogap temperature *T*∗ as its onset temperature and continues to exist with an increasing trend of the resonance peak height in the underdoped region as temperature is lowered and that near the optimal doping *T* tends to get closer to *Tc*. On the other hand, He et al. observed from their INS measurements of the doping dependence of the resonance peak energy, *Eres* for Bi2Sr2CaCu2O8+*<sup>δ</sup>* (BSCCO) that in the underdoped region *Eres* increases with increasing hole concentration *x* up to optimal doping *x*0, and that in the overdoped cuprates it decreases with increasing *x*, by exhibiting a linear scaling behavior of *Eres* with *Tc* at all hole concentrations[18]. Most recently Stock et al.[19] observed that spin waves decay above the pseudogap of a heavily underdoped YBCO. Using a time-of-flight neutron spectroscopy for the studies of dynamic spin correlations or spin fluctuations in the overdoped La2−*x*Sr*x*CuO4

 **0.02**

with the spin paring correlations

 **0.04**

 **0.06**

T/t

*t*�

/*t* = −0.3.

 **0.08**

 **0.1**

from a positive value (*t* � /*t* = 0.3) to a negative (*t* � /*t* = −0.3) one. On the other hand, in the overdoped region both T∗ and T*<sup>C</sup>* are seen to simultaneously increase (decrease) for the case of *t*� /*t* < 0 (*t* � /*t* > 0) with reference to that of *t* � /*t* = 0. The predicted superconducting transition temperature at optimal doping concentration did not change appreciably despite the considerable variation of *t* � /*t* as shown in the figure. The simultaneous increase (decrease) of T∗ and T*<sup>C</sup>* with *t* � /*t* = −0.3 (*t*� /*t* = 0.3) indicates that the two temperature scales, T∗ and T*<sup>C</sup>* or the spin gap phase and the superconducting phase are interrelated. To see the cause of such interplay, below we probe the role of the short-range AF spin fluctuations or the spin paring correlations on the determination of the phase diagram.

For the case of *t* � < 0 (*t* � > 0) the spinon energy at the hot spot(*π*, 0) is lowered (raised) with reference that of *t* � = 0, i.e., no diagonal hopping, as can be readily understood from Eq. 8. Thus the spin (spinon) pairing correlation at the hot spot for *t*� < 0 is energetically more stable than the case for *t*� > 0. It is to be recalled that the spin gap temperature is the temperature at which the spin singlet paring order or (correlations) of d-wave symmetry or the spin pairing correlations emerges. The spin paring correlations will be less prone to change in the underdoped region compared to the case of the overdoped region. This is because owing to lower hole concentrations in this region, chances of electron hopping from site to site are reduced and, consequently, the existing short-range AF order is not easily perturbed. Thus the spin paring correlations or the short-range antiferromagnetic spin fluctuations is expected to remain more robust in the underdoped region compared to the case of the overdoped region. Indeed, the predicted T<sup>∗</sup> and T*<sup>C</sup>* is shown to be sensitive to the variation of *t* � , preferentially in the overdoped region. This is displayed in Fig. 1.

It is reminded that the Cooper pairing order can be seen as the composite of the spin (spinon) pairing order Δ*<sup>f</sup>* and the charge (holon) pairing order Δ*b*, which results from the presence of the spin-charge coupling shown in the last term of Eq. 6. As a result of the coupling between the two orders, the superconducting phase transition with its onset temperature, T*<sup>C</sup>* may arise owing the short-range AF spin fluctuations involved with the formation of the spin pairing order (correlations) which initiates the onset of the spin gap temperature T∗. To put it otherwise, owing to the spin-charge coupling both T∗ and T*<sup>C</sup>* are simultaneously affected or correlated. Indeed, such simultaneous change is seen to appear by exhibiting the simultaneous increase (decrease) of both T<sup>∗</sup> and T*<sup>C</sup>* with *t*� /*t* < 0 (*t* � /*t* > 0) as J increases. Such trend is seen in Fig. 2. Our findings of both the enhancement of the spin pairing correlations and the increasing trend of the superconducting transition temperature for *t*� = −0.3*t* which appear in the overdoped region agree well with the variational Monte Carlo, mean-field calculations of Lee and coworkers [13]. However, unlike our present study they did not show a study of the spin gap temperature concerned with the role of the spin paring correlations.

For further verification from a different angle we closely examine the predicted structural dependence of the phase diagram on J in Fig.2. Both *T*∗ and *TC* are predicted to simultaneously increase with J as shown in Fig. 2. Needless to say, spin pairing correlations should increase with J. This will, in turn, cause the simultaneous increase of both the spin gap temperature and the superconducting transition temperature with increasing J. Such simultaneous increase with J is predicted as shown in the figure. This clearly demonstrates that the the superconducting transition temperature and the pseudogap temperature or relatedly the spin gap phase and the d-wave superconducting phase are correlated via the spin pairing correlations or the AF spin fluctuations of the shortest possible correlation length.

6 Will-be-set-by-IN-TECH

overdoped region both T∗ and T*<sup>C</sup>* are seen to simultaneously increase (decrease) for the case

transition temperature at optimal doping concentration did not change appreciably despite

T*<sup>C</sup>* or the spin gap phase and the superconducting phase are interrelated. To see the cause of such interplay, below we probe the role of the short-range AF spin fluctuations or the spin

Eq. 8. Thus the spin (spinon) pairing correlation at the hot spot for *t*� < 0 is energetically more stable than the case for *t*� > 0. It is to be recalled that the spin gap temperature is the temperature at which the spin singlet paring order or (correlations) of d-wave symmetry or the spin pairing correlations emerges. The spin paring correlations will be less prone to change in the underdoped region compared to the case of the overdoped region. This is because owing to lower hole concentrations in this region, chances of electron hopping from site to site are reduced and, consequently, the existing short-range AF order is not easily perturbed. Thus the spin paring correlations or the short-range antiferromagnetic spin fluctuations is expected to remain more robust in the underdoped region compared to the case of the overdoped region.

It is reminded that the Cooper pairing order can be seen as the composite of the spin (spinon) pairing order Δ*<sup>f</sup>* and the charge (holon) pairing order Δ*b*, which results from the presence of the spin-charge coupling shown in the last term of Eq. 6. As a result of the coupling between the two orders, the superconducting phase transition with its onset temperature, T*<sup>C</sup>* may arise owing the short-range AF spin fluctuations involved with the formation of the spin pairing order (correlations) which initiates the onset of the spin gap temperature T∗. To put it otherwise, owing to the spin-charge coupling both T∗ and T*<sup>C</sup>* are simultaneously affected or correlated. Indeed, such simultaneous change is seen to appear by exhibiting the

trend is seen in Fig. 2. Our findings of both the enhancement of the spin pairing correlations and the increasing trend of the superconducting transition temperature for *t*� = −0.3*t* which appear in the overdoped region agree well with the variational Monte Carlo, mean-field calculations of Lee and coworkers [13]. However, unlike our present study they did not show a study of the spin gap temperature concerned with the role of the spin paring correlations. For further verification from a different angle we closely examine the predicted structural dependence of the phase diagram on J in Fig.2. Both *T*∗ and *TC* are predicted to simultaneously increase with J as shown in Fig. 2. Needless to say, spin pairing correlations should increase with J. This will, in turn, cause the simultaneous increase of both the spin gap temperature and the superconducting transition temperature with increasing J. Such simultaneous increase with J is predicted as shown in the figure. This clearly demonstrates that the the superconducting transition temperature and the pseudogap temperature or relatedly the spin gap phase and the d-wave superconducting phase are correlated via the spin pairing correlations or the AF spin fluctuations of the shortest possible correlation length.

Indeed, the predicted T<sup>∗</sup> and T*<sup>C</sup>* is shown to be sensitive to the variation of *t*

�

�

/*t* = −0.3) one. On the other hand, in the

/*t* = 0. The predicted superconducting

�

/*t* > 0) as J increases. Such

, preferentially

/*t* as shown in the figure. The simultaneous increase (decrease)

� > 0) the spinon energy at the hot spot(*π*, 0) is lowered (raised)

/*t* < 0 (*t* �

� = 0, i.e., no diagonal hopping, as can be readily understood from

/*t* = 0.3) indicates that the two temperature scales, T∗ and

/*t* = 0.3) to a negative (*t*

/*t* > 0) with reference to that of *t*

�

paring correlations on the determination of the phase diagram.

/*t* = −0.3 (*t*�

in the overdoped region. This is displayed in Fig. 1.

simultaneous increase (decrease) of both T<sup>∗</sup> and T*<sup>C</sup>* with *t*�

from a positive value (*t*

�

the considerable variation of *t*

�

� < 0 (*t*

/*t* < 0 (*t*

of T∗ and T*<sup>C</sup>* with *t*

For the case of *t*

with reference that of *t*

of *t*�

�

Fig. 2. (color online). Variaton of phase diagram with Heisenberg coupling strength J for *t*� /*t* = −0.3.

To put it otherwise, the short-range AF spin fluctuations play a key role of causing such inseparable relation between the two temperature scales or relatedly the spin gap phase and the superconducting phase. This finding is consistent with the observed phase diagrams with different cuprate samples which shows higher the T∗, higher the T*C*[14] as mentioned earlier. It is then assured that the superconducting phase transition will not arise in the absence of the spin gap phase below T∗ which is initiated by the short-range AF spin fluctuations involved with the spin paring correlations

#### **4. Magnetic susceptibility based on the U(1) slave-boson representation**

The observed high *Tc* phase diagrams of cuprate oxides are characterized by the pseudogap or spin gap phase which exists below the monotonously decreasing T∗ and the d-wave superconducting phase below the dome shaped T*C*[15, 16]. From their inelastic neutron scattering measurements (INS) of the temperature dependence of magnetic resonance peaks for YB2Cu3O6+*<sup>x</sup>* (YBCO) Dai et al. [17] reported that the magnetic resonance begins to appear at the pseudogap temperature *T*∗ as its onset temperature and continues to exist with an increasing trend of the resonance peak height in the underdoped region as temperature is lowered and that near the optimal doping *T* tends to get closer to *Tc*. On the other hand, He et al. observed from their INS measurements of the doping dependence of the resonance peak energy, *Eres* for Bi2Sr2CaCu2O8+*<sup>δ</sup>* (BSCCO) that in the underdoped region *Eres* increases with increasing hole concentration *x* up to optimal doping *x*0, and that in the overdoped cuprates it decreases with increasing *x*, by exhibiting a linear scaling behavior of *Eres* with *Tc* at all hole concentrations[18]. Most recently Stock et al.[19] observed that spin waves decay above the pseudogap of a heavily underdoped YBCO. Using a time-of-flight neutron spectroscopy for the studies of dynamic spin correlations or spin fluctuations in the overdoped La2−*x*Sr*x*CuO4

where *J*(*q*) = 2*J*(cos *qx* + cos *qy*) and *χ*0(*q*, *ω*) is the irreducible magnetic susceptibility given

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 9

*<sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*) + <sup>Δ</sup>*<sup>f</sup>*

*<sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*) <sup>−</sup> <sup>Δ</sup>*<sup>f</sup>*

*<sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*) + <sup>Δ</sup>*<sup>f</sup>*

expression of the effective Lagrangian Eq. 3, interplay between the two sectors, one for the spinon (spin) sector and the other for holon (charge) sector, namely Eq. 5 and Eq. 6 appears owing to the presence of coupling between the spinon pairing order and holon pairing order as shown in the last term of Eq. 6. Thus it should be noted that the effect of coupling between the two order parameter is embedded in the expression of the above irreducible magnetic

able to obtain the generic feature of the dome shaped superconducting transition temperature and the monotonously decreasing pseudogap temperature in the phase diagram[5] in agreement with observaions[15, 16]. Now with the inclusion of the diagonal hopping term, such generic feature is, still, well predicted in the computed result of the phase diagram as shown in Fig. 1 and Fig. 2. As a concerted study we use the predicted phase diagram shown in Fig. 3 to calculate the magnetic spin susceptibility of present interest. As in our earlier study of the magnetic susceptibility[21, 22], we take the negative value[7, 13] of the next-nearest neighbor hopping integral with the choice of *t*� = −0.45 (to conform with the

In Fig. 4 we display the variation of magnetic susceptibility at (*π*, *π*) with temperature T and transfer energy E at a fixed hole doping, *x* = 0.05. The magnetic resonance peak is shown to

) <sup>−</sup> <sup>Δ</sup>*<sup>f</sup>*

*<sup>k</sup>* <sup>−</sup> *<sup>μ</sup><sup>f</sup>*

*k*+*q* Δ*f k* 

> *k*+*q* Δ*f k*

> *k*+*q*Δ*<sup>f</sup> k*

> *k*+*q*Δ*<sup>f</sup> k*

*<sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*)<sup>2</sup> + (Δ*<sup>f</sup>*

*<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup>*

*<sup>w</sup>* <sup>−</sup> (*E<sup>f</sup>*

*n <sup>f</sup>*(*E<sup>f</sup>*

*<sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup>*

*<sup>w</sup>* <sup>−</sup> (−*E<sup>f</sup>*

*<sup>w</sup>* <sup>−</sup> (−*E<sup>f</sup>*

*<sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup>*

*coskxcosky*; the spinon chemical potential, *μ<sup>f</sup>* ; the

*<sup>k</sup>* ) = 1/(*eβE<sup>f</sup>*

*<sup>w</sup>* <sup>−</sup> (*E<sup>f</sup>*

*<sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup>*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>E</sup><sup>f</sup>*

*k* )

*k* )

*<sup>k</sup>* )) + *iη*

*k* )

*k* )

 ,

(15)

*<sup>k</sup>* )) + *iη*

*<sup>k</sup>* + 1). In the complete

� term we were

*<sup>k</sup>* ) + *iη*

*<sup>k</sup>* ) + *iη*

*<sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup>*

*<sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup>*

*<sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup>*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> (−*E<sup>f</sup>*

*<sup>k</sup>* )<sup>2</sup> with the effective bare spinon

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>E</sup><sup>f</sup>*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> (−*E<sup>f</sup>*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup>*

2*E<sup>f</sup> k*+*qE<sup>f</sup> k*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup>*

2*E<sup>f</sup> k*+*qE<sup>f</sup> k*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup>*

2*E<sup>f</sup> k*+*q Ef k*

*<sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup>*

2*E<sup>f</sup> k*+*q Ef k*

> *<sup>k</sup>* = (*� f*

*<sup>k</sup>* <sup>=</sup> *Jp*Δ*<sup>f</sup> <sup>ϕ</sup><sup>k</sup>* with *<sup>ϕ</sup><sup>k</sup>* <sup>=</sup> *coskx* <sup>−</sup> *cosky* and *<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup>*

susceptibility, Eq. 15, including the effect of the nearest neighbor hopping.

Earlier, with the neglect of the next-nearest neighbor (or diagonal) hopping *t*

study of Brinckmann and Lee[23]) in the *t* − *t*� − *J* Hamiltonian of interest[21] .

*<sup>k</sup>* <sup>−</sup> (*� <sup>f</sup>*

*<sup>k</sup>* + (*� <sup>f</sup>*

<sup>2</sup> (*coskx* <sup>+</sup> *cosky*) <sup>−</sup> <sup>4</sup>*xt*�

**5. Computed results of magnetic susceptibility**

by

*<sup>χ</sup>*0(*<sup>q</sup>*, *<sup>w</sup>*) = (<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup>

 *Ef k*+*q Ef <sup>k</sup>* + (*� <sup>f</sup>*

+ *Ef k*+*q Ef <sup>k</sup>* <sup>−</sup> (*� <sup>f</sup>*

+ *Ef k*+*qE<sup>f</sup>*

+ *Ef k*+*qE<sup>f</sup>*

*<sup>k</sup>* <sup>=</sup> <sup>−</sup> *Jx<sup>χ</sup>*

energy, *� <sup>f</sup>*

spinon gap, Δ*<sup>f</sup>*

where the quasi-spinon energy is *E<sup>f</sup>*

<sup>4</sup>*<sup>N</sup>* ∑ *k*

(LSCO) sample, Wakimoto et al.[20] showed from their study of the doping dependence of antiferromagnetic spin excitations that the excitations decrease with hole doping above the optimal doping of La2−*x*Sr*x*CuO4 (LSCO) and disappear at *x* = 0.3. Here we discuss the magnetic susceptibility[21, 22] at the wave vector *Q*� = (*π*, *π*) in association with our computed phase diagram and focus our attention to the observed linear scaling behavior of magnetic resonance peak energy *Eres* with the superconducting transition temperature *Tc*. For self-containment we first discuss the U(1) slave-boson representation of irreducible magnetic susceptibility for our calculations of magnetic susceptibility.

Allowing external magnetic field�*h*, we introduce into the effective Lagrangian <sup>L</sup>*eff* above the Zeeman coupling term,

$$
\hat{H}^{e\mathbf{x}} = -\sum\_{i} \vec{S}\_{i} \cdot \vec{h}\_{i\nu} \tag{9}
$$

where in the U(1) slave boson representation,

�

$$\begin{split} \vec{S}\_{i} \cdot \vec{h}\_{i} &= \frac{1}{2} \sum\_{k=1}^{3} (c\_{i\alpha}^{\dagger} \sigma\_{\alpha\beta}^{k} c\_{i\beta}) h\_{i}^{k} \\ &= \frac{1}{2} \sum\_{k=1}^{3} (f\_{i\alpha}^{\dagger} b\_{i} \sigma\_{\alpha\beta}^{k} b\_{i}^{\dagger} f\_{i\beta}) h\_{i}^{k} \\ &= \frac{1}{2} \sum\_{k=1}^{3} (1 + b\_{i}^{\dagger} b\_{i}) (f\_{i\alpha}^{\dagger} \sigma\_{\alpha\beta}^{k} f\_{i\beta}) h\_{i}^{k}, \\\\ &\simeq \frac{1}{2} \sum\_{k=1}^{3} < (1 + b\_{i}^{\dagger} b\_{i}) > (f\_{i\alpha}^{\dagger} \sigma\_{\alpha\beta}^{k} f\_{i\beta}) h\_{i}^{k}, \end{split} \tag{10}$$

The associated free energy is formally,

$$F = -\frac{1}{\beta} \text{Im}Z,\tag{11}$$

where *β* = 1/*kT* and the partition function,

$$\mathcal{Z} = \int \mathcal{D}f \mathcal{D}b \mathcal{D}\chi \mathcal{D}\Delta^f \mathcal{D}\Delta^b \mathcal{D}\lambda \varepsilon^{-\int\_0^{\oint} d\tau \mathcal{L}\_{eff}}.\tag{12}$$

Converting the magnetic (spin) susceptibility,

$$\chi\_{kk'}(\vec{r}\_i - \vec{r}\_{j\prime}\,\pi - \pi^{\prime}) = -\beta\hbar \left. \frac{\delta^2 F[\vec{h}]}{\delta h\_i^k(\tau)\delta h\_j^{k\prime}(\tau^{\prime})} \right|\_{\vec{h}=0}.\tag{13}$$

into its four momentum space (�*q*, *ω*)expression and allowing isotropic response to the applied magnetic field, the RPA form of magnetic susceptibility is obtained to be,[23]

$$\chi(\vec{q}, w) = \frac{\chi^0(\vec{q}, w)}{1 + f(\vec{q}) \chi^0(\vec{q}, w)'} \tag{14}$$

8 Will-be-set-by-IN-TECH

(LSCO) sample, Wakimoto et al.[20] showed from their study of the doping dependence of antiferromagnetic spin excitations that the excitations decrease with hole doping above the optimal doping of La2−*x*Sr*x*CuO4 (LSCO) and disappear at *x* = 0.3. Here we discuss the magnetic susceptibility[21, 22] at the wave vector *Q*� = (*π*, *π*) in association with our computed phase diagram and focus our attention to the observed linear scaling behavior of magnetic resonance peak energy *Eres* with the superconducting transition temperature *Tc*. For self-containment we first discuss the U(1) slave-boson representation of irreducible magnetic

Allowing external magnetic field�*h*, we introduce into the effective Lagrangian <sup>L</sup>*eff* above the

*i* �

*αβciβ*)*h<sup>k</sup> i*

> *<sup>i</sup> bi*)(*<sup>f</sup>* † *iασ<sup>k</sup>*

*αβ fiβ*)*h<sup>k</sup> i* ,

*iασ<sup>k</sup>*

*αβ fiβ*)*h<sup>k</sup>*

*<sup>β</sup> lnZ*, (11)

<sup>0</sup> *<sup>d</sup>τ*L*eff* . (12)

, (14)

. (13)

*<sup>i</sup>* , (10)

*<sup>i</sup> bi*) <sup>&</sup>gt; (*<sup>f</sup>* †

<sup>D</sup>Δ*b*D*λe*<sup>−</sup> *<sup>β</sup>*

�*h*]

 �*h*=0

) = <sup>−</sup>*βh*¯ *<sup>δ</sup>*2*F*[

into its four momentum space (�*q*, *ω*)expression and allowing isotropic response to the applied

*<sup>χ</sup>*(�*q*, *<sup>w</sup>*) = *<sup>χ</sup>*0(�*q*, *<sup>w</sup>*)

*δh<sup>k</sup> <sup>i</sup>* (*τ*)*δhk*� *<sup>j</sup>* (*τ*� )

1 + *J*(�*q*)*χ*0(�*q*, *w*)

*Si* ·�*hi*, (9)

*<sup>H</sup>*<sup>ˆ</sup> *ex* <sup>=</sup> <sup>−</sup> ∑

susceptibility for our calculations of magnetic susceptibility.

where in the U(1) slave boson representation,

The associated free energy is formally,

where *β* = 1/*kT* and the partition function,

Converting the magnetic (spin) susceptibility,

Z = 

*χkk*�(�*ri* −�*rj*, *τ* − *τ*

� *Si* ·�*hi* <sup>=</sup> <sup>1</sup>

2

<sup>=</sup> <sup>1</sup> 2

<sup>=</sup> <sup>1</sup> 2

� 1 2

3 ∑ *k*=1 (*c*† *iασ<sup>k</sup>*

3 ∑ *k*=1 (*f* † *iαbiσ<sup>k</sup> αβb*† *<sup>i</sup> fiβ*)*h<sup>k</sup> i*

3 ∑ *k*=1

3 ∑ *k*=1

(1 + *b*†

< (1 + *b*†

*<sup>F</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup>

<sup>D</sup> *<sup>f</sup>* <sup>D</sup>*b*D*χ*DΔ*<sup>f</sup>*

�

magnetic field, the RPA form of magnetic susceptibility is obtained to be,[23]

Zeeman coupling term,

where *J*(*q*) = 2*J*(cos *qx* + cos *qy*) and *χ*0(*q*, *ω*) is the irreducible magnetic susceptibility given by

*<sup>χ</sup>*0(*<sup>q</sup>*, *<sup>w</sup>*) = (<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> <sup>4</sup>*<sup>N</sup>* ∑ *k Ef k*+*q Ef <sup>k</sup>* + (*� <sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup> <sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*) + <sup>Δ</sup>*<sup>f</sup> k*+*q* Δ*f k* 2*E<sup>f</sup> k*+*qE<sup>f</sup> k <sup>n</sup> <sup>f</sup>*(*E<sup>f</sup> <sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup> k* ) *<sup>w</sup>* <sup>−</sup> (*E<sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>E</sup><sup>f</sup> <sup>k</sup>* ) + *iη* + *Ef k*+*q Ef <sup>k</sup>* <sup>−</sup> (*� <sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup> <sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*) <sup>−</sup> <sup>Δ</sup>*<sup>f</sup> k*+*q* Δ*f k* 2*E<sup>f</sup> k*+*qE<sup>f</sup> k n <sup>f</sup>*(*E<sup>f</sup> <sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup> k* ) *<sup>w</sup>* <sup>−</sup> (*E<sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> (−*E<sup>f</sup> <sup>k</sup>* )) + *iη* + *Ef k*+*qE<sup>f</sup> <sup>k</sup>* <sup>−</sup> (*� <sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup><sup>f</sup>* )(*� <sup>f</sup> <sup>k</sup>* <sup>−</sup> *<sup>μ</sup><sup>f</sup>* ) <sup>−</sup> <sup>Δ</sup>*<sup>f</sup> k*+*q*Δ*<sup>f</sup> k* 2*E<sup>f</sup> k*+*q Ef k <sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup> <sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup> k* ) *<sup>w</sup>* <sup>−</sup> (−*E<sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>E</sup><sup>f</sup> <sup>k</sup>* ) + *iη* + *Ef k*+*qE<sup>f</sup> <sup>k</sup>* + (*� <sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> *<sup>μ</sup>f*)(*� <sup>f</sup> <sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*) + <sup>Δ</sup>*<sup>f</sup> k*+*q*Δ*<sup>f</sup> k* 2*E<sup>f</sup> k*+*q Ef k <sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup> <sup>k</sup>*+*q*) <sup>−</sup> *<sup>n</sup> <sup>f</sup>*(−*E<sup>f</sup> k* ) *<sup>w</sup>* <sup>−</sup> (−*E<sup>f</sup> <sup>k</sup>*+*<sup>q</sup>* <sup>−</sup> (−*E<sup>f</sup> <sup>k</sup>* )) + *iη* , (15)

where the quasi-spinon energy is *E<sup>f</sup> <sup>k</sup>* = (*� f <sup>k</sup>* <sup>−</sup> *<sup>μ</sup>f*)<sup>2</sup> + (Δ*<sup>f</sup> <sup>k</sup>* )<sup>2</sup> with the effective bare spinon energy, *� <sup>f</sup> <sup>k</sup>* <sup>=</sup> <sup>−</sup> *Jx<sup>χ</sup>* <sup>2</sup> (*coskx* <sup>+</sup> *cosky*) <sup>−</sup> <sup>4</sup>*xt*� *coskxcosky*; the spinon chemical potential, *μ<sup>f</sup>* ; the spinon gap, Δ*<sup>f</sup> <sup>k</sup>* <sup>=</sup> *Jp*Δ*<sup>f</sup> <sup>ϕ</sup><sup>k</sup>* with *<sup>ϕ</sup><sup>k</sup>* <sup>=</sup> *coskx* <sup>−</sup> *cosky* and *<sup>n</sup> <sup>f</sup>*(*E<sup>f</sup> <sup>k</sup>* ) = 1/(*eβE<sup>f</sup> <sup>k</sup>* + 1). In the complete expression of the effective Lagrangian Eq. 3, interplay between the two sectors, one for the spinon (spin) sector and the other for holon (charge) sector, namely Eq. 5 and Eq. 6 appears owing to the presence of coupling between the spinon pairing order and holon pairing order as shown in the last term of Eq. 6. Thus it should be noted that the effect of coupling between the two order parameter is embedded in the expression of the above irreducible magnetic susceptibility, Eq. 15, including the effect of the nearest neighbor hopping.

#### **5. Computed results of magnetic susceptibility**

Earlier, with the neglect of the next-nearest neighbor (or diagonal) hopping *t* � term we were able to obtain the generic feature of the dome shaped superconducting transition temperature and the monotonously decreasing pseudogap temperature in the phase diagram[5] in agreement with observaions[15, 16]. Now with the inclusion of the diagonal hopping term, such generic feature is, still, well predicted in the computed result of the phase diagram as shown in Fig. 1 and Fig. 2. As a concerted study we use the predicted phase diagram shown in Fig. 3 to calculate the magnetic spin susceptibility of present interest. As in our earlier study of the magnetic susceptibility[21, 22], we take the negative value[7, 13] of the next-nearest neighbor hopping integral with the choice of *t*� = −0.45 (to conform with the study of Brinckmann and Lee[23]) in the *t* − *t*� − *J* Hamiltonian of interest[21] .

In Fig. 4 we display the variation of magnetic susceptibility at (*π*, *π*) with temperature T and transfer energy E at a fixed hole doping, *x* = 0.05. The magnetic resonance peak is shown to

Fig. 5. (color online) Resonance peak energy (*Eres*/*J* in reduced unit) vs. superconducting

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 11

In this study we applied the recently proposed slave-boson theory[5] in which the spin (spinon) paring order and the charge (holon) pairing order are coupled to result in the generic feature of the dome-shaped superconducting transition temperature and the monotonously decreasing spin gap temperature in the phase diagram. From the present study with the inclusion of the diagonal hopping *t*� term we also found that such generic feature still holds, as shown in Fig. 1 through Fig. 3. Further we showed that there exists correlation (or interplay) between the two different temperature scales, T∗ and T*C*, resulting in the increasing T*<sup>C</sup>* with increasing T∗. Relatedly, it can be said that the superconducting phase is correlated with the spin gap phase. We find that such correlation between the two phases is attributed to the short-range AF spin fluctuations involved with spin pairing correlations. The simultaneous increase of the superconducting transition temperature with the spin gap temperature with incrasing J is shown to be consistent with the observed phase diagrams for high T*<sup>C</sup>* cuprate samples (e.g., LSCO and BSCCO samples)[15] which shows that the higher T∗ samples always accompany higher T*C*. In addition, to achieve a self-consistent, concerted study we used the predicted phase diagram to study the magnetic susceptibility. Specifically, resorting to the computed phase diagram shown in Fig. 3 we found that both the temperature dependence of the magnetic resonace peak and the linear scaling behavior of the magnetic (spin) resonance peak energy *Eres* with the superconducting transition temperature *Tc* agree with the INS measurements[17, 18]. We showed that this linear scaling behavior is attributed to the short-range AF spin fluctuations. Although not discussed here, such linear relation is found to be invariant with the Heisenberg coupling constant[22], implying high *TC* cuprate sample independence. In short, based on the above concerted studies of both the phase diagram and the magnetic susceptibility we find that the short-range (spin dimer) AF spin fluctuations of the shortest possible correlation length involved with the spin pairing correlations are responsible for high T*<sup>C</sup>* superconductivity. We argue that this finding is supported by the reproducibility of both the dome-shaped superconducting transition temperature, T*<sup>C</sup>* in the phase diagram and the linear scaling behavior between *Eres* and T*C*, in both of which the T*<sup>C</sup>*

/*t* = −0.45.

transition temperature (*Tc*/*t* in reduced unit) with *t*�

**6. Summary**

Fig. 3. (color online) Phase diagram with *t* � /*t* = −0.45, *J*/*t* = 0.5. Both temperature *T* and hole concentration *x* are in reduced units, *T*/*t* and *x*/*x*<sup>0</sup> respectively.

Fig. 4. (color online) Imaginary part of magnetic susceptibility vs. temperature and resonance energy at a fixed hole doping, *x* = 0.05 in the underdoped region

decrease with increasing temperature and disappears at the onset temperature *T*∗. It shows a steady decrease of the resonance peak peak height with increasing temperature and eventual disappearance at *T*∗ in agreement with observation[17]. This indicates that the short-range AF spin fluctuations involved with the spin pairing correlations or the spin (spinon) singlet pairing order disappears at the onset temperature, T∗.

He et al.[18] showed from their INS measurements of Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (BSCCO) that in the underdoped cuprates the magnetic (spin) resonance peak energy *Eres* (or *ωres*) increases with *Tc* showing a linear scaling behavior between the two energy scales, *Eres* and *Tc*, i.e., *Eres*/*Tc* � *const*. In Fig. 5 we show that the predicted *Eres* with *Jeff* = *αJ* (where *α* = 0.4[23]) monotonously increases with increasing *Tc*, yielding a linear scaling behavior of *Eres*/*Tc* � *const*. This predicted linear scaling behavior is in agreement with the observations made by He et al.[18]. We note some quantitative differences between the observed value (around 5) and the predicted one (around 3).

Fig. 5. (color online) Resonance peak energy (*Eres*/*J* in reduced unit) vs. superconducting transition temperature (*Tc*/*t* in reduced unit) with *t*� /*t* = −0.45.

#### **6. Summary**

10 Will-be-set-by-IN-TECH

�

Fig. 4. (color online) Imaginary part of magnetic susceptibility vs. temperature and resonance energy at a fixed hole doping, *x* = 0.05 in the underdoped region

pairing order disappears at the onset temperature, T∗.

and the predicted one (around 3).

decrease with increasing temperature and disappears at the onset temperature *T*∗. It shows a steady decrease of the resonance peak peak height with increasing temperature and eventual disappearance at *T*∗ in agreement with observation[17]. This indicates that the short-range AF spin fluctuations involved with the spin pairing correlations or the spin (spinon) singlet

He et al.[18] showed from their INS measurements of Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (BSCCO) that in the underdoped cuprates the magnetic (spin) resonance peak energy *Eres* (or *ωres*) increases with *Tc* showing a linear scaling behavior between the two energy scales, *Eres* and *Tc*, i.e., *Eres*/*Tc* � *const*. In Fig. 5 we show that the predicted *Eres* with *Jeff* = *αJ* (where *α* = 0.4[23]) monotonously increases with increasing *Tc*, yielding a linear scaling behavior of *Eres*/*Tc* � *const*. This predicted linear scaling behavior is in agreement with the observations made by He et al.[18]. We note some quantitative differences between the observed value (around 5)

hole concentration *x* are in reduced units, *T*/*t* and *x*/*x*<sup>0</sup> respectively.

/*t* = −0.45, *J*/*t* = 0.5. Both temperature *T* and

Fig. 3. (color online) Phase diagram with *t*

In this study we applied the recently proposed slave-boson theory[5] in which the spin (spinon) paring order and the charge (holon) pairing order are coupled to result in the generic feature of the dome-shaped superconducting transition temperature and the monotonously decreasing spin gap temperature in the phase diagram. From the present study with the inclusion of the diagonal hopping *t*� term we also found that such generic feature still holds, as shown in Fig. 1 through Fig. 3. Further we showed that there exists correlation (or interplay) between the two different temperature scales, T∗ and T*C*, resulting in the increasing T*<sup>C</sup>* with increasing T∗. Relatedly, it can be said that the superconducting phase is correlated with the spin gap phase. We find that such correlation between the two phases is attributed to the short-range AF spin fluctuations involved with spin pairing correlations. The simultaneous increase of the superconducting transition temperature with the spin gap temperature with incrasing J is shown to be consistent with the observed phase diagrams for high T*<sup>C</sup>* cuprate samples (e.g., LSCO and BSCCO samples)[15] which shows that the higher T∗ samples always accompany higher T*C*. In addition, to achieve a self-consistent, concerted study we used the predicted phase diagram to study the magnetic susceptibility. Specifically, resorting to the computed phase diagram shown in Fig. 3 we found that both the temperature dependence of the magnetic resonace peak and the linear scaling behavior of the magnetic (spin) resonance peak energy *Eres* with the superconducting transition temperature *Tc* agree with the INS measurements[17, 18]. We showed that this linear scaling behavior is attributed to the short-range AF spin fluctuations. Although not discussed here, such linear relation is found to be invariant with the Heisenberg coupling constant[22], implying high *TC* cuprate sample independence. In short, based on the above concerted studies of both the phase diagram and the magnetic susceptibility we find that the short-range (spin dimer) AF spin fluctuations of the shortest possible correlation length involved with the spin pairing correlations are responsible for high T*<sup>C</sup>* superconductivity. We argue that this finding is supported by the reproducibility of both the dome-shaped superconducting transition temperature, T*<sup>C</sup>* in the phase diagram and the linear scaling behavior between *Eres* and T*C*, in both of which the T*<sup>C</sup>*

with the Heisenberg interaction term,

*HJ* <sup>=</sup> <sup>−</sup> *<sup>J</sup>*

<sup>2</sup> ∑ <*i*,*j*> *bibjb*† *j b*† *<sup>i</sup>* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

site with spin 0. Thus physical spin-charge separation is not allowed.

*Ht*−*<sup>J</sup>* in the large *<sup>U</sup>* limit. The Coulomb repulsion, *Uni*↑*ni*<sup>↓</sup> <sup>=</sup> *<sup>U</sup>*

by the four holon (boson) operator *bibjb*†

from the intersite charge coupling term *<sup>J</sup>*

in the Heisenberg interaction term.

*HJ* <sup>=</sup> *<sup>J</sup>* <sup>∑</sup><*i*,*j*>(**S***<sup>i</sup>* · **<sup>S</sup>***<sup>j</sup>* <sup>−</sup> <sup>1</sup>

The Heisenberg interaction term, Eq.(A4) shows coupling between the charge and spin degrees of freedom. Physics involved with the charge degree of freedom is manifested

in its slave-boson representation. It is to be noted that the Hubbard Hamiltonian contains repulsive interaction *U* between charged particles and is mapped into the t-J Hamiltonian

obviously manifests the presence of both the charge (the first term) and spin (the second term) degrees of freedom. Thus, under mapping the charge part of contribution naturally appears

Let us now take another look at the importance of the charge contribution. In general, uncertainty principle between the number density (amplitude ) and the phase of a boson order parameter applies. As an example, arbitrarily large fluctuations of the number density fix the phase, or arbitrarily large phase fluctuations fix the number density of the boson. The

*j b*†

The first term represents hopping of a spinon from site *j* to site *i* and of a holon (positively charged boson) from site *i* to site *j*. In the slave-boson representation a charged fermion (electron or hole) is taken as a composite particle of a 'spinon' and a 'holon'. They can conveniently serve as book-keeping labels to discern physical properties or objects involved with the charge or spin degree of freedom (e.g., spin gap phase, spin singlet pairs, hole pairs, ...). With the single occupancy constraint, electron is allowed to hop from a singly occupied copper site *i* only to a vacant copper site *j*. A site of single occupancy in the *CuO*<sup>2</sup> plane of high *Tc* cuprates physically represents an electrically neutral site (net charge 0) with an electron of spin 1/2 and the vacant site, a site of positive charge +*e* with net spin 0. In the slave-boson representation, hopping of an electron (a composite of spinon and holon) from a singly occupied copper site (neutral site) *j* to an empty site (positively charged site with +*e*) *i* results in the annihilation of a spinon (a fermion of charge 0 and spin 1/2) and the creation of a positively charged holon (a boson of charge +*e* and spin 0) at site *j* while at the copper site *i* a composite of a spinon (fermion of charge 0 and spin 1/2) and a negatively charged holon is created. It is of note that as a result of electron hopping the newly occupied copper site *i* in the *CuO*<sup>2</sup> plane can, also, be labeled as 'spinon' since this is an electrically neutral (charge 0) site with an electron of spin 1/2 and the vacant site *j*, 'holon' since this is a positively charged site with a single charge +*e* and the net spin 0 as mentioned above. Thus in practical sense, there is no distinction between the two different cases above. At times, we will call the singly occupied site as 'spinon' and the vacant (empty) site as 'holon' as long as there is no confusion. This is because any site occupied by a spinon is identified as an electrically neutral site occupied by a single electron with spin 1/2 and the site with a positive holon is a positively charged vacant

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 13

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑). (A4)

*<sup>i</sup>* in the Heisenberg interaction term. Judging

<sup>4</sup> (*ni*<sup>↑</sup> <sup>+</sup> *ni*↓)<sup>2</sup> <sup>−</sup> *<sup>U</sup>*

<sup>4</sup> (*ni*<sup>↑</sup> <sup>−</sup> *ni*↓)<sup>2</sup>

<sup>4</sup> *ninj* present in the Heisenberg interaction term

<sup>4</sup> *ninj*), it is obvious that this charge contribution can not be neglected

and thus the superconducting phase transition is shown to occur as a result of the short-range AF spin fluctuations in association with the spin-charge coupling.

#### **7. Acknowledgement**

One(SHSS) of us greatly acknowledges the Korean Ministry of Education (HakJin Program) and Pohang University of Science and Technology for financial supports at the initial stage of the present work. He thanks the Korean Academy of Science and Technology for continuous encouragements in science research. We are grateful to Professor Ki-Seok Kim at Pohang University of Science and Technology for his earlier contribution.

#### **8. Appendix A: Heisenberg interaction term in the U(1) slave-boson representation**

The t-J Hamiltonian of interest is given by,

$$H = -t\sum\_{} \left(c\_{i\sigma}^{\dagger}c\_{j\sigma} + c.c.\right) + J \sum\_{} \left(\mathbf{S}\_i \cdot \mathbf{S}\_j - \frac{1}{4}n\_in\_j\right)$$

$$-\mu \sum\_i c\_{i\sigma}^{\dagger}c\_{i\sigma} \tag{A1}$$

and the Heisenberg interaction term is rewritten

$$\begin{split} H\_{I} &= J \sum\_{} (\mathbf{S}\_{i} \cdot \mathbf{S}\_{j} - \frac{1}{4} n\_{i} n\_{j}) \\ &= -\frac{J}{2} \sum\_{} (c\_{i\downarrow}^{\dagger} c\_{j\uparrow}^{\dagger} - c\_{i\uparrow}^{\dagger} c\_{j\downarrow}^{\dagger})(c\_{j\uparrow} c\_{i\downarrow} - c\_{j\downarrow} c\_{i\uparrow}). \end{split} \tag{A2}$$

Here *t* is the hopping energy and **S***i*, the electron spin operator at site *i*, **S***<sup>i</sup>* = <sup>1</sup> 2 *c*† *<sup>i</sup>ασαβci<sup>β</sup>* with *σαβ*, the Pauli spin matrix element. *ni* is the electron number operator at site *i*, *ni* = *c*† *<sup>i</sup>σciσ*. *μ* is the chemical potential.

In the U(1) slave-boson representation[1, 2, 24, 25], with single occupancy constraint at site *i* the electron annihilation operator *ci<sup>σ</sup>* is taken as a composite operator of the spinon (neutrally charged fermion) annihilation operator *fi<sup>σ</sup>* and the holon (positively charged boson) creation operator *b*† *<sup>i</sup>* , and thus, *ci<sup>σ</sup>* <sup>=</sup> *fiσb*† *<sup>i</sup>* . Rigorously speaking, it should be noted that the expression *ci<sup>σ</sup>* = *b*† *<sup>i</sup> fi<sup>σ</sup>* is not precise since these operators belong to different Hilbert spaces and thus the equality sign here should be taken only as a symbol for mapping. Using *ci<sup>σ</sup>* = *fiσb*† *i* and introducing the Lagrange multiplier term (the last term in Eq.(A3)) to enforce single occupancy constraint, the t-J Hamiltonian is rewritten,

$$H = -t\sum\_{} \left( (f\_{i\sigma}^{\dagger}b\_i)(b\_j^{\dagger}f\_{j\sigma}) + c.c. \right) + H\_I$$

$$-\mu \sum\_i f\_{i\sigma}^{\dagger}b\_i f\_{i\sigma} b\_i^{\dagger}$$

$$-i\sum\_i \lambda\_i (b\_i^{\dagger}b\_i + f\_{i\sigma}^{\dagger}f\_{i\sigma} - 1) \tag{A3}$$

with the Heisenberg interaction term,

12 Will-be-set-by-IN-TECH

and thus the superconducting phase transition is shown to occur as a result of the short-range

One(SHSS) of us greatly acknowledges the Korean Ministry of Education (HakJin Program) and Pohang University of Science and Technology for financial supports at the initial stage of the present work. He thanks the Korean Academy of Science and Technology for continuous encouragements in science research. We are grateful to Professor Ki-Seok Kim at Pohang

**8. Appendix A: Heisenberg interaction term in the U(1) slave-boson representation**

*<sup>i</sup>σcj<sup>σ</sup>* + *c*.*c*.) + *J* ∑

4 *ninj*)

In the U(1) slave-boson representation[1, 2, 24, 25], with single occupancy constraint at site *i* the electron annihilation operator *ci<sup>σ</sup>* is taken as a composite operator of the spinon (neutrally charged fermion) annihilation operator *fi<sup>σ</sup>* and the holon (positively charged boson) creation

the equality sign here should be taken only as a symbol for mapping. Using *ci<sup>σ</sup>* = *fiσb*†

and introducing the Lagrange multiplier term (the last term in Eq.(A3)) to enforce single

*<sup>i</sup> fi<sup>σ</sup>* is not precise since these operators belong to different Hilbert spaces and thus

*<sup>j</sup> fjσ*) + *c*.*c*.

(**S***<sup>i</sup>* · **<sup>S</sup>***<sup>j</sup>* <sup>−</sup> <sup>1</sup>

Here *t* is the hopping energy and **S***i*, the electron spin operator at site *i*, **S***<sup>i</sup>* = <sup>1</sup>

*σαβ*, the Pauli spin matrix element. *ni* is the electron number operator at site *i*, *ni* = *c*†

<*i*,*j*>

(**S***<sup>i</sup>* · **<sup>S</sup>***<sup>j</sup>* <sup>−</sup> <sup>1</sup>

*<sup>i</sup>σci<sup>σ</sup>* (A1)

*<sup>i</sup>* . Rigorously speaking, it should be noted that the expression

 + *HJ*

*<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* − 1) (A3)

4 *ninj*)

*<sup>j</sup>*↓)(*cj*↑*ci*<sup>↓</sup> <sup>−</sup> *cj*↓*ci*↑). (A2)

2 *c*†

*<sup>i</sup>ασαβci<sup>β</sup>* with

*<sup>i</sup>σciσ*. *μ* is

*i*

AF spin fluctuations in association with the spin-charge coupling.

University of Science and Technology for his earlier contribution.

<*i*,*j*> (*c*†

−*μ*∑ *i c*†

*HJ* = *J* ∑

<sup>=</sup> <sup>−</sup> *<sup>J</sup>* <sup>2</sup> ∑ <*i*,*j*> (*c*† *i*↓*c*† *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>c</sup>*† *i*↑*c*†

<*i*,*j*>

The t-J Hamiltonian of interest is given by,

*<sup>H</sup>* = −*<sup>t</sup>* ∑

and the Heisenberg interaction term is rewritten

*<sup>i</sup>* , and thus, *ci<sup>σ</sup>* <sup>=</sup> *fiσb*†

occupancy constraint, the t-J Hamiltonian is rewritten,

*<sup>H</sup>* = −*<sup>t</sup>* ∑

−*μ*∑ *i f* † *<sup>i</sup>σbi fiσb*† *i*

−*i*∑ *i*

<*i*,*j*>

*λi*(*b*†

 (*f* † *iσbi*)(*b*†

*<sup>i</sup> bi* <sup>+</sup> *<sup>f</sup>* †

**7. Acknowledgement**

the chemical potential.

operator *b*†

*ci<sup>σ</sup>* = *b*†

$$H\_{\!\!\!\!\!\!} = -\frac{J}{2} \sum\_{} b\_i b\_j b\_j^\dagger b\_i^\dagger (f\_{i\downarrow}^\dagger f\_{j\uparrow}^\dagger - f\_{i\uparrow}^\dagger f\_{j\downarrow}^\dagger)(f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}).\tag{A4}$$

The first term represents hopping of a spinon from site *j* to site *i* and of a holon (positively charged boson) from site *i* to site *j*. In the slave-boson representation a charged fermion (electron or hole) is taken as a composite particle of a 'spinon' and a 'holon'. They can conveniently serve as book-keeping labels to discern physical properties or objects involved with the charge or spin degree of freedom (e.g., spin gap phase, spin singlet pairs, hole pairs, ...). With the single occupancy constraint, electron is allowed to hop from a singly occupied copper site *i* only to a vacant copper site *j*. A site of single occupancy in the *CuO*<sup>2</sup> plane of high *Tc* cuprates physically represents an electrically neutral site (net charge 0) with an electron of spin 1/2 and the vacant site, a site of positive charge +*e* with net spin 0. In the slave-boson representation, hopping of an electron (a composite of spinon and holon) from a singly occupied copper site (neutral site) *j* to an empty site (positively charged site with +*e*) *i* results in the annihilation of a spinon (a fermion of charge 0 and spin 1/2) and the creation of a positively charged holon (a boson of charge +*e* and spin 0) at site *j* while at the copper site *i* a composite of a spinon (fermion of charge 0 and spin 1/2) and a negatively charged holon is created. It is of note that as a result of electron hopping the newly occupied copper site *i* in the *CuO*<sup>2</sup> plane can, also, be labeled as 'spinon' since this is an electrically neutral (charge 0) site with an electron of spin 1/2 and the vacant site *j*, 'holon' since this is a positively charged site with a single charge +*e* and the net spin 0 as mentioned above. Thus in practical sense, there is no distinction between the two different cases above. At times, we will call the singly occupied site as 'spinon' and the vacant (empty) site as 'holon' as long as there is no confusion. This is because any site occupied by a spinon is identified as an electrically neutral site occupied by a single electron with spin 1/2 and the site with a positive holon is a positively charged vacant site with spin 0. Thus physical spin-charge separation is not allowed.

The Heisenberg interaction term, Eq.(A4) shows coupling between the charge and spin degrees of freedom. Physics involved with the charge degree of freedom is manifested by the four holon (boson) operator *bibjb*† *j b*† *<sup>i</sup>* in the Heisenberg interaction term. Judging from the intersite charge coupling term *<sup>J</sup>* <sup>4</sup> *ninj* present in the Heisenberg interaction term *HJ* <sup>=</sup> *<sup>J</sup>* <sup>∑</sup><*i*,*j*>(**S***<sup>i</sup>* · **<sup>S</sup>***<sup>j</sup>* <sup>−</sup> <sup>1</sup> <sup>4</sup> *ninj*), it is obvious that this charge contribution can not be neglected in its slave-boson representation. It is to be noted that the Hubbard Hamiltonian contains repulsive interaction *U* between charged particles and is mapped into the t-J Hamiltonian *Ht*−*<sup>J</sup>* in the large *<sup>U</sup>* limit. The Coulomb repulsion, *Uni*↑*ni*<sup>↓</sup> <sup>=</sup> *<sup>U</sup>* <sup>4</sup> (*ni*<sup>↑</sup> <sup>+</sup> *ni*↓)<sup>2</sup> <sup>−</sup> *<sup>U</sup>* <sup>4</sup> (*ni*<sup>↑</sup> <sup>−</sup> *ni*↓)<sup>2</sup> obviously manifests the presence of both the charge (the first term) and spin (the second term) degrees of freedom. Thus, under mapping the charge part of contribution naturally appears in the Heisenberg interaction term.

Let us now take another look at the importance of the charge contribution. In general, uncertainty principle between the number density (amplitude ) and the phase of a boson order parameter applies. As an example, arbitrarily large fluctuations of the number density fix the phase, or arbitrarily large phase fluctuations fix the number density of the boson. The

**8.1 U(1) mean field Hamiltonian**

term) in Eq.(A7) is rewritten,

− *J* 2 (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

intersite spin (spinon) interaction and is rewritten,

− *J* 2 *bibjb*† *j b*† *i* (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

<sup>=</sup> <sup>−</sup> *Jp* <sup>2</sup> (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

*<sup>i</sup> bi* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*†

1+ < *b*†

− *Jp* <sup>2</sup> (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

<sup>=</sup> *Jp* 4 <sup>3</sup> ∑ *k*=1 (*f* † *iασ<sup>k</sup>*

<sup>=</sup> <sup>−</sup> *<sup>J</sup>*

<sup>2</sup> <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup>*

*ij*<sup>|</sup> <sup>&</sup>gt;<sup>2</sup>

*<sup>j</sup> bj* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*†

hole doping concentration[27]. The fourth term in Eq.(A7) is written,

*<sup>i</sup> bi* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*αβ fiβ*)(*<sup>f</sup>* †

3 ∑ *k*=0 (*f* †

<sup>2</sup> (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

*vD* <sup>=</sup> <sup>−</sup> *Jp* 8

*vE* <sup>=</sup> <sup>−</sup> *Jp* 4 (*f* † *<sup>i</sup><sup>σ</sup> fjσ*)(*<sup>f</sup>* †

*vP* <sup>=</sup> <sup>−</sup> *Jp*

Here *σ*<sup>0</sup> is the unit matrix and *σ*1,2,3, the Pauli spin matrices.

*jγσ<sup>k</sup>*

*<sup>j</sup>* ] = *δij* for boson, the intersite charge (holon) interaction term (the second

 *bibjb*† *j b*† *i*

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑), (A9)

*<sup>j</sup> bj* <sup>&</sup>gt;) or *Jp* <sup>=</sup> *<sup>J</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> with *<sup>x</sup>*, the uniform

, (A8)

(A10)

*<sup>j</sup> bj* <sup>+</sup> *<sup>b</sup>*† *i b*† *<sup>j</sup> bjbi* 

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>i</sup> bi* <sup>+</sup> *<sup>b</sup>*†

*ij* = *fj*<sup>↑</sup> *fi*<sup>↓</sup> − *fj*<sup>↓</sup> *fi*↑, the spinon pairing field. The third term in Eq.(A7) represents the

*<sup>i</sup> bib*†

*<sup>j</sup> bj* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*†

The intersite spinon interaction term in Eq.(A9) is decomposed into the direct, exchange and

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

with *σ*<sup>0</sup> = *I*, the identity matrix and *σ*1,2,3, the Pauli spin matrices, where *vD*, *vE* and *vP* are the spinon interaction terms of the direct, exchange and pairing channels respectively ,

*<sup>i</sup> <sup>σ</sup><sup>k</sup> fi*)(*<sup>f</sup>* †

*<sup>j</sup><sup>σ</sup> fiσ*) − *ni*

*γδ fjδ*) <sup>−</sup> (*<sup>f</sup>* †

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>i</sup> bib*† *<sup>j</sup> bj* > <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> > .

*iασ*<sup>0</sup>

= *vD* + *vE* + *vP* (A11)

*αβ fiβ*)(*<sup>f</sup>* †

*iγσ*<sup>0</sup> *γδ fjδ*) 

*<sup>j</sup> <sup>σ</sup><sup>k</sup> fj*), (A12)

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑). (A14)

, (A13)

1 + *b*†

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 15

Noting that [*bi*, *b*†

where *Jp* = *J*(1+ < *b*†

pairing channels[25],

*J* 2 *bibjb*† *j b*† *i* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

= *J* 2 

with Δ*<sup>f</sup>*

conventional BCS superconductors of long coherence length meet the former classification, and thus the phase fluctuations of the Cooper pair order parameter are minimal. For charged bosons, e.g., the Cooper pairs, the number density fluctuations refer to charge density fluctuations. For short coherence length superconductors such as the high *Tc* cuprate systems of present interest, local charge density fluctuations exist and cause large phase fluctuations. Thus, both the charge and phase fluctuations need to be taken into account to fully exploit the quantum fluctuations .

Let us now consider the importance of the charge and spin fluctuations. In generally, coupling between physical quantities *A* and *B* is decomposed into terms involving fluctuations of *A*, i.e., (*A*− < *A* >) and *B*, i.e., (*B*− < *B* >), separately uncorrelated mean field contribution of < *A* > and < *B* > and correlation between fluctuations of *A* and *B*, that is, (*A*− < *A* >) and (*B*− < *B* >) respectively; *AB* = (*A*− < *A* >) < *B* > +(*B*− < *B* >) < *A* > + < *A* >< *<sup>B</sup>* <sup>&</sup>gt; +(*A*<sup>−</sup> <sup>&</sup>lt; *<sup>A</sup>* <sup>&</sup>gt;)(*B*<sup>−</sup> <sup>&</sup>lt; *<sup>B</sup>* <sup>&</sup>gt;). Setting *<sup>A</sup>* <sup>=</sup> *bibjb*† *j b*† *<sup>i</sup>* for charge (holon) contribution and *B* = (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) for spin (spinon) contribution, the Heisenberg coupling term, Eq(A4) can be decomposed into terms involving coupling between the charge and spin fluctuations separately, the mean field contributions and coupling (correlation) between fluctuations (charge and spin fluctuations). Using such decomposition of the Heisenberg interaction term for Eq.(A3), we write the partition function,

*Z* = *DbDf*↑*D f*↓*Dλe*−*S*[*b*, *<sup>f</sup>*,*λ*] , (A5)

where

$$S[b\_{\prime}f,\lambda] = \int\_{0}^{\oint} d\tau \left[ \sum\_{i} b\_{i}^{\dagger} \partial\_{\tau} b\_{i} + \sum\_{i} f\_{i\sigma}^{\dagger} \partial\_{\tau} f\_{i\sigma} + H\_{t-I}^{II(1)} \right] \tag{A6}$$

with *β* = <sup>1</sup> *kBT* , the inverse temperature and *<sup>H</sup>U*(1) *<sup>t</sup>*−*<sup>J</sup>* , the U(1) symmetry preserved Hamiltonian,

*HU*(1) *<sup>t</sup>*−*<sup>J</sup>* <sup>=</sup> <sup>−</sup>*<sup>t</sup>* ∑ <*i*,*j*> (*f* † *<sup>i</sup><sup>σ</sup> fjσb*† *<sup>j</sup> bi* + *c*.*c*.) − *J* <sup>2</sup> ∑ <*i*,*j*> (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) *bibjb*† *j b*† *i* + *bibjb*† *j b*† *i* (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) − *bibjb*† *j b*† *i* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) + *bibjb*† *j b*† *i* − *bibjb*† *j b*† *i* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) − (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) −*μ*∑ *i f* † *<sup>i</sup><sup>σ</sup> fiσ*(<sup>1</sup> <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup> bi*) − *<sup>i</sup>*∑ *i λi*(*f* † *<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup> bi* − 1). (A7)

#### **8.1 U(1) mean field Hamiltonian**

14 Will-be-set-by-IN-TECH

conventional BCS superconductors of long coherence length meet the former classification, and thus the phase fluctuations of the Cooper pair order parameter are minimal. For charged bosons, e.g., the Cooper pairs, the number density fluctuations refer to charge density fluctuations. For short coherence length superconductors such as the high *Tc* cuprate systems of present interest, local charge density fluctuations exist and cause large phase fluctuations. Thus, both the charge and phase fluctuations need to be taken into account to fully exploit the

Let us now consider the importance of the charge and spin fluctuations. In generally, coupling between physical quantities *A* and *B* is decomposed into terms involving fluctuations of *A*, i.e., (*A*− < *A* >) and *B*, i.e., (*B*− < *B* >), separately uncorrelated mean field contribution of < *A* > and < *B* > and correlation between fluctuations of *A* and *B*, that is, (*A*− < *A* >) and (*B*− < *B* >) respectively; *AB* = (*A*− < *A* >) < *B* > +(*B*− < *B* >) < *A* > + < *A* ><

term, Eq(A4) can be decomposed into terms involving coupling between the charge and spin fluctuations separately, the mean field contributions and coupling (correlation) between fluctuations (charge and spin fluctuations). Using such decomposition of the Heisenberg

*DbDf*↑*D f*↓*Dλe*−*S*[*b*, *<sup>f</sup>*,*λ*]

*<sup>i</sup> ∂τbi* + ∑ *i f* †

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>+</sup> *<sup>b</sup>*†

*λi*(*f* †

*j b*†

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑) for spin (spinon) contribution, the Heisenberg coupling

*<sup>i</sup>σ∂τ fi<sup>σ</sup>* <sup>+</sup> *<sup>H</sup>U*(1)

 *bibjb*† *j b*† *i*

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>i</sup> bi* − 1). (A7)

*t*−*J* 

*<sup>t</sup>*−*<sup>J</sup>* , the U(1) symmetry preserved Hamiltonian,

*<sup>i</sup>* for charge (holon) contribution and

, (A5)

(A6)

quantum fluctuations .

*B* = (*f* †

where

with *β* = <sup>1</sup>

*HU*(1)

*<sup>t</sup>*−*<sup>J</sup>* <sup>=</sup> <sup>−</sup>*<sup>t</sup>* ∑

− *J* <sup>2</sup> ∑ <*i*,*j*>

+ *bibjb*† *j b*† *i* (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

− *bibjb*† *j b*† *i* (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

+ *bibjb*† *j b*† *i* − *bibjb*† *j b*† *i* (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

− (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

−*μ*∑ *i f* †

*<sup>i</sup><sup>σ</sup> fiσ*(<sup>1</sup> <sup>+</sup> *<sup>b</sup>*†

*i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

*<sup>B</sup>* <sup>&</sup>gt; +(*A*<sup>−</sup> <sup>&</sup>lt; *<sup>A</sup>* <sup>&</sup>gt;)(*B*<sup>−</sup> <sup>&</sup>lt; *<sup>B</sup>* <sup>&</sup>gt;). Setting *<sup>A</sup>* <sup>=</sup> *bibjb*†

interaction term for Eq.(A3), we write the partition function,

*S*[*b*, *f* , *λ*] =

<*i*,*j*>

*kBT* , the inverse temperature and *<sup>H</sup>U*(1)

(*f* † *<sup>i</sup><sup>σ</sup> fjσb*†

> (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

*Z* = 

> *β* 0 *dτ* ∑ *i b*†

> > *<sup>j</sup> bi* + *c*.*c*.)

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>i</sup> bi*) − *<sup>i</sup>*∑ *i*

Noting that [*bi*, *b*† *<sup>j</sup>* ] = *δij* for boson, the intersite charge (holon) interaction term (the second term) in Eq.(A7) is rewritten,

$$\begin{aligned} &-\frac{1}{2}\Big\langle \left( f\_{i\downarrow}^{\dagger} f\_{j\uparrow}^{\dagger} - f\_{i\uparrow}^{\dagger} f\_{j\downarrow}^{\dagger} \right) (f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}) \right\rangle b\_{i} b\_{j} b\_{j}^{\dagger} b\_{i}^{\dagger} \\ &= -\frac{J}{2} < |\Delta\_{ij}^{f}| > \,^{2} \left( 1 + b\_{i}^{\dagger} b\_{i} + b\_{j}^{\dagger} b\_{j} + b\_{i}^{\dagger} b\_{j}^{\dagger} b\_{j} b\_{i} \right) . \end{aligned} \tag{A8}$$

with Δ*<sup>f</sup> ij* = *fj*<sup>↑</sup> *fi*<sup>↓</sup> − *fj*<sup>↓</sup> *fi*↑, the spinon pairing field. The third term in Eq.(A7) represents the intersite spin (spinon) interaction and is rewritten,

$$-\frac{I}{2}\left\langle b\_i b\_j b\_j^\dagger b\_i^\dagger \right\rangle (f\_{i\downarrow}^\dagger f\_{j\uparrow}^\dagger - f\_{i\uparrow}^\dagger f\_{j\downarrow}^\dagger)(f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow})$$

$$=-\frac{I\_p}{2} (f\_{i\downarrow}^\dagger f\_{j\uparrow}^\dagger - f\_{i\uparrow}^\dagger f\_{j\downarrow}^\dagger)(f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}),\tag{A9}$$

where *Jp* = *J*(1+ < *b*† *<sup>i</sup> bi* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*† *<sup>j</sup> bj* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*† *<sup>i</sup> bib*† *<sup>j</sup> bj* <sup>&</sup>gt;) or *Jp* <sup>=</sup> *<sup>J</sup>*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> with *<sup>x</sup>*, the uniform hole doping concentration[27]. The fourth term in Eq.(A7) is written,

$$\begin{split} & \frac{1}{2} \left\langle b\_{i} b\_{j} b\_{j}^{\dagger} b\_{i}^{\dagger} \right\rangle \left\langle (f\_{i\downarrow}^{\dagger} f\_{j\uparrow}^{\dagger} - f\_{i\uparrow}^{\dagger} f\_{j\downarrow}^{\dagger})(f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}) \right\rangle \\ &= \frac{1}{2} \left( 1 + < b\_{i}^{\dagger} b\_{i} > + < b\_{j}^{\dagger} b\_{j} > + < b\_{i}^{\dagger} b\_{i} b\_{j}^{\dagger} b\_{j} > \right) < |\Delta\_{ij}^{f}|^{2} > \,. \tag{A10} \end{split} \tag{A10}$$

The intersite spinon interaction term in Eq.(A9) is decomposed into the direct, exchange and pairing channels[25],

$$\begin{aligned} &-\frac{I\_p}{2} (f\_{i\downarrow}^\dagger f\_{j\uparrow}^\dagger - f\_{i\uparrow}^\dagger f\_{j\downarrow}^\dagger)(f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}) \\ &= \frac{I\_p}{4} \left[ \sum\_{k=1}^3 (f\_{i\alpha}^\dagger \sigma\_{a\beta}^k f\_{i\beta})(f\_{j\gamma}^\dagger \sigma\_{\gamma\delta}^k f\_{j\delta}) - (f\_{i\alpha}^\dagger \sigma\_{a\beta}^0 f\_{i\beta})(f\_{i\gamma}^\dagger \sigma\_{\gamma\delta}^0 f\_{j\delta}) \right] \\ &= v\_D + v\_E + v\_P \end{aligned} \tag{A11}$$

with *σ*<sup>0</sup> = *I*, the identity matrix and *σ*1,2,3, the Pauli spin matrices, where *vD*, *vE* and *vP* are the spinon interaction terms of the direct, exchange and pairing channels respectively ,

$$v\_D = -\frac{J\_p}{8} \sum\_{k=0}^{3} (f\_i^\dagger \sigma^k f\_i)(f\_j^\dagger \sigma^k f\_j),\tag{A12}$$

$$v\_E = -\frac{I\_p}{4} \left( (f\_{i\sigma}^\dagger f\_{j\sigma})(f\_{j\sigma}^\dagger f\_{i\sigma}) - n\_i \right) \tag{A13}$$

$$v\_P = -\frac{I\_p}{2} (f\_{i\downarrow}^\dagger f\_{j\uparrow}^\dagger - f\_{i\uparrow}^\dagger f\_{j\downarrow}^\dagger)(f\_{\uparrow\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}).\tag{A14}$$

Here *σ*<sup>0</sup> is the unit matrix and *σ*1,2,3, the Pauli spin matrices.

transformations corresponding to these channels and saddle point approximation leads to the

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 17

*<sup>i</sup> bi* >< *<sup>b</sup>*†

*<sup>i</sup> bi*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*<sup>i</sup> bi*) − *<sup>i</sup>*∑ *i*

*<sup>i</sup> bj* <sup>&</sup>gt;, <sup>Δ</sup>*<sup>f</sup>*

for the contribution of the direct spinon interaction term (the fourth term). The expression

*HMF* = *H*Δ,*<sup>χ</sup>* + *H<sup>b</sup>* + *H <sup>f</sup>*

where *H*Δ,*<sup>χ</sup>* represents the the saddle point energy involved with the spinon pairing order

*<sup>j</sup> bj* >

*<sup>i</sup> bi* > + *b*†

*λi*(*f* †

*<sup>i</sup>* <sup>&</sup>gt;<sup>=</sup> 0 for *<sup>l</sup>* <sup>=</sup> 1, 2, 3, *<sup>ρ</sup>*<sup>0</sup>

*<sup>i</sup> bj*) in the fifth term of Eq.(A19) represents the exchange interaction channel. The

, the holon pairing order parameter Δ*<sup>b</sup>* and the hopping order parameter *χ*,

*<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>+</sup> *<sup>b</sup>*†

*ij* <sup>=</sup><sup>&</sup>lt; *fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*<sup>↑</sup> <sup>&</sup>gt; and *<sup>ρ</sup><sup>k</sup>*

*<sup>i</sup>* <sup>=</sup> <sup>1</sup>

<sup>2</sup> <sup>&</sup>lt; *<sup>f</sup>* †

4*t Jp b*†

*ij* (*bjbi*) − *c*.*c*

*<sup>i</sup> bj*) − *c*.*c*.

*<sup>j</sup>*↓) <sup>−</sup> *<sup>c</sup>*.*c*.

*<sup>j</sup> bj*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*<sup>j</sup> bj* > 

*<sup>i</sup> bi* − 1), (A19)

*<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>&</sup>gt;<sup>=</sup> <sup>1</sup>

2

, (A21)

, (A20)

*<sup>i</sup>* <sup>=</sup><sup>&</sup>lt; <sup>1</sup> 2 *f* † *<sup>i</sup> <sup>σ</sup><sup>k</sup> fi* <sup>&</sup>gt;

<sup>2</sup> for *l* = 0

*<sup>J</sup>* [25, 26]. The

effective Hamiltonian,

where Δ*<sup>b</sup>*

(*b*† *<sup>j</sup> bi*)(*b*†

We note that *ρ<sup>l</sup>*

parameter Δ*<sup>f</sup>*

*Heff* <sup>=</sup> *Jp*

<sup>4</sup> ∑ <*i*,*j*>

+ *J* <sup>2</sup> ∑ <*i*,*j*>

+ *Jp* <sup>2</sup> ∑ <*i*,*j*>

+ *Jp* <sup>2</sup> ∑ <*i*,*j*>

+ 4*t* 2 *Jp* ∑ <*i*,*j*> (*b*† *<sup>i</sup> bj*)(*b*† *<sup>j</sup> bi*)

+ *J* <sup>2</sup> ∑ <*i*,*j*>

− *J* <sup>2</sup> ∑ <*i*,*j*>

−*μ*∑ *i f* †

*ij* <sup>=</sup><sup>&</sup>lt; *bibj* <sup>&</sup>gt;, *<sup>χ</sup>ij* <sup>=</sup><sup>&</sup>lt; *<sup>f</sup>* †

<sup>2</sup> <sup>&</sup>lt; *<sup>f</sup>* †

*<sup>i</sup>* <sup>=</sup> <sup>1</sup>

resulting effective Hamiltonian is

are proper saddle points.

 |*χij*|

> |Δ*f ij*| 2 |Δ*b ij*| <sup>2</sup> <sup>−</sup> <sup>Δ</sup>*b*<sup>∗</sup>

 |Δ*f ij*| <sup>2</sup> <sup>−</sup> <sup>Δ</sup>*<sup>f</sup> ij*(*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

3 ∑ *l*=0 (*ρl j* )<sup>2</sup> <sup>−</sup> *<sup>ρ</sup><sup>l</sup> j* (*f* † *<sup>i</sup> <sup>σ</sup><sup>l</sup> fi*) <sup>+</sup> *Jp* <sup>2</sup> ∑ *i* (*f* † *<sup>i</sup><sup>σ</sup> fiσ*)



*<sup>i</sup><sup>σ</sup> fiσ*(<sup>1</sup> <sup>+</sup> *<sup>b</sup>*†

*<sup>i</sup><sup>σ</sup> fj<sup>σ</sup>* <sup>+</sup> <sup>4</sup>*<sup>t</sup>*

*<sup>i</sup> <sup>σ</sup><sup>l</sup> fi* <sup>&</sup>gt;=<sup>&</sup>lt; *<sup>S</sup><sup>l</sup>*

*<sup>H</sup>*Δ,*<sup>χ</sup>* <sup>=</sup> *<sup>J</sup>* ∑

<*i*,*j*>

+ *Jp* <sup>2</sup> ∑ <*i*,*j*>

 1 2 |Δ*f ij*| <sup>2</sup>|Δ*<sup>b</sup> ij*| <sup>2</sup> + 1 2 |Δ*f ij*| <sup>2</sup>*x*<sup>2</sup> 

> |Δ*f ij*| <sup>2</sup> + 1 2 |*χij*| <sup>2</sup> + 1 4

*Jp <sup>b</sup>*†

exchange channel will be ignored owing to a large cost in energy, *<sup>U</sup>* <sup>≈</sup> <sup>4</sup>*<sup>t</sup>*

<sup>2</sup> <sup>−</sup> *<sup>χ</sup>*<sup>∗</sup> *ij*(*<sup>f</sup>* † *<sup>i</sup><sup>σ</sup> fj<sup>σ</sup>* +

Combining Eq.(A8) and Eq.(A10), we have

$$\begin{split} & -\frac{J}{2} < |\boldsymbol{\Delta}\_{ij}^{\dagger}|^{2} > \left( 1 + b\_{i}^{\dagger}b\_{i} + b\_{j}^{\dagger}b\_{j} + b\_{i}^{\dagger}b\_{j}^{\dagger}b\_{j}b\_{i} \right) \\ & + \frac{J}{2} < |\boldsymbol{\Delta}\_{ij}^{\dagger}|^{2} > \left( 1 + < b\_{i}^{\dagger}b\_{i} > + < b\_{j}^{\dagger}b\_{j} > + < b\_{i}^{\dagger}b\_{i}b\_{j}^{\dagger}b\_{j} > \right) \\ & = -\frac{J}{2} < |\boldsymbol{\Delta}\_{ij}^{\dagger}|^{2} > b\_{i}^{\dagger}b\_{i}^{\dagger}b\_{j}b\_{i} + \frac{J}{2} < |\boldsymbol{\Delta}\_{ij}^{\dagger}|^{2} > < b\_{i}^{\dagger}b\_{i}b\_{j}^{\dagger}b\_{j} > \\ & -\frac{J}{2} < |\boldsymbol{\Delta}\_{ij}^{\dagger}|^{2} > \left[ \left( b\_{i}^{\dagger}b\_{i} - < b\_{i}^{\dagger}b\_{i} > \right) + \left( b\_{j}^{\dagger}b\_{j} - < b\_{j}^{\dagger}b\_{j} > \right) \right]. \end{split} \tag{A15}$$

Collecting the decomposed terms Eq.(A8) through Eq.(A10) in association with Eqs.(A11) through Eq.(A15), we write

$$\begin{split} H\_{I} &= -\frac{I}{2} \sum\_{} |\Delta\_{ij}^{f}|^{2} b\_{i}^{\dagger} b\_{j}^{\dagger} b\_{j} b\_{i} \\ &- J\_{P} \sum\_{} \left[ \frac{1}{2} (f\_{i\downarrow}^{\dagger} f\_{j\uparrow}^{\dagger} - f\_{i\uparrow}^{\dagger} f\_{j\downarrow}^{\dagger}) (f\_{\uparrow\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}) \\ &+ \frac{1}{4} ((f\_{i\sigma}^{\dagger} f\_{j\sigma}) (f\_{j\sigma}^{\dagger} f\_{i\sigma}) - n\_{i}) \\ &+ \frac{1}{8} \sum\_{k=0}^{3} (f\_{i\uparrow}^{\dagger} \sigma^{k} f\_{i\uparrow}) (f\_{j\uparrow}^{\dagger} \sigma^{k} f\_{j\uparrow}) \right] \\ &+ \frac{I}{2} \sum\_{} |\Delta\_{ij}^{f}|^{2} < b\_{i}^{\dagger} b\_{i} > < b\_{j}^{\dagger} b\_{j} > \\ &- \frac{I}{2} \sum\_{} |\Delta\_{ij}^{f}|^{2} \left[ \left( b\_{i}^{\dagger} b\_{i\uparrow} - < b\_{i}^{\dagger} b\_{i} > \right) + \left( b\_{j}^{\dagger} b\_{j} - < b\_{j}^{\dagger} b\_{j} > \right) \right] . \end{split} \tag{A16}$$

where we considered <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> <sup>&</sup>gt;<sup>=</sup> <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> and ignored the fifth term in Eq.(A7).

Hubbard Stratonovich transformation for the holon pairing term (the second term of Eq.(A16)) leads to

$$e^{\sum\_{}\frac{1}{2}|\Delta\_{ij}^{\dagger}|^{2}b\_{l}^{\dagger}b\_{j}^{\dagger}b\_{l}b\_{j}} \propto \int \prod\_{}d\Delta\_{ij}^{b\*}d\Delta\_{ij}^{b}e^{-\sum\_{}\frac{1}{2}|\Delta\_{ij}^{\dagger}|^{2}\left[|\Delta\_{ij}^{b}|^{2}-\Delta\_{ij}^{b\*}(b\_{l}b\_{j})-\Delta\_{ij}^{b}(b\_{j}^{\dagger}b\_{l}^{\dagger})\right]\_{s} \tag{A17}$$

and the saddle point approximation yields,

$$\Delta H\_P^b = \sum\_{} \frac{J}{2} |\Delta\_{ij}^f|^2 \left[ |\Delta\_{ij}^{0b}|^2 - \Delta\_{ij}^{0b\*} (b\_j b\_i) - \Delta\_{ij}^{0b} (b\_j^\dagger b\_i^\dagger) \right],\tag{A18}$$

where Δ0*<sup>b</sup> ij* <sup>=</sup><sup>&</sup>lt; *bibj* <sup>&</sup>gt; is the saddle point for the holon pairing order parameter <sup>Δ</sup>*<sup>b</sup> ij*. Since confusion is not likely to occur, we will use the notation Δ*<sup>b</sup> ij* in place of <sup>Δ</sup>0*<sup>b</sup> ij* for the saddle point. As are shown in Eqs.(A12) through (A14) the spinon interaction term is decomposed into the direct, exchange and pairing channels respectively. Proper Hubbard-Stratonovich 16 Will-be-set-by-IN-TECH

*<sup>j</sup> bj* <sup>+</sup> *<sup>b</sup>*† *i b*† *<sup>j</sup> bjbi* 

<sup>2</sup> <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*|

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>j</sup> bj* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*†

<sup>2</sup> >< *b*†

*<sup>i</sup> bib*† *<sup>j</sup> bj* > 

*<sup>j</sup> bj* > 

. (A15)

, (A16)

, (A17)

*ij*. Since

, (A18)

*ij* for the saddle

*<sup>i</sup> bib*† *<sup>j</sup> bj* >

*<sup>j</sup> bj*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*<sup>j</sup> bj*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*ij* (*b*† *j b*† *i* ) 

<sup>2</sup> and ignored the fifth term in Eq.(A7).

*<sup>j</sup> bj* > 

*ij* (*bibj*)−Δ*<sup>b</sup>*

*ij* in place of <sup>Δ</sup>0*<sup>b</sup>*

*ij*(*b*† *j b*† *i* ) 

*<sup>i</sup> bi* <sup>&</sup>gt; <sup>+</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*<sup>i</sup> bi* > + *b*†

Collecting the decomposed terms Eq.(A8) through Eq.(A10) in association with Eqs.(A11)

*<sup>i</sup> bi* <sup>+</sup> *<sup>b</sup>*†

*<sup>j</sup> bjbi* <sup>+</sup> *<sup>J</sup>*

*<sup>i</sup> bi*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*†

1+ < *b*†

Combining Eq.(A8) and Eq.(A10), we have

− *J* <sup>2</sup> <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> > 1 + *b*†

+ *J* <sup>2</sup> <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> > 

− *J* <sup>2</sup> <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> > *b*†

through Eq.(A15), we write

<sup>=</sup> <sup>−</sup> *<sup>J</sup>*

<sup>2</sup> <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> > *b*† *i b*†

*HJ* <sup>=</sup> <sup>−</sup> *<sup>J</sup>*

−*Jp* ∑ <*i*,*j*>

> 3 ∑ *k*=0 (*f* †

+ 1 4 (*f* † *<sup>i</sup><sup>σ</sup> fjσ*)(*<sup>f</sup>* †

+ 1 8

+ *J* <sup>2</sup> ∑ <*i*,*j*>

− *J* <sup>2</sup> ∑ <*i*,*j*>

and the saddle point approximation yields,

*Hb <sup>P</sup>* = ∑ <*i*,*j*>

where we considered <sup>&</sup>lt; <sup>|</sup>Δ*<sup>f</sup>*

*e* <sup>∑</sup><*i*,*j*<sup>&</sup>gt; *<sup>J</sup>* <sup>2</sup> <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup>*b*† *i b*† *<sup>j</sup> bibj* ∝ ∏ <*i*,*j*>

leads to

where Δ0*<sup>b</sup>*

<sup>2</sup> ∑ <*i*,*j*>

> 1 2 (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †



*ij*|


*<sup>i</sup> <sup>σ</sup><sup>k</sup> fi*)(*<sup>f</sup>* †

<sup>2</sup> <sup>&</sup>gt;<sup>=</sup> <sup>|</sup>Δ*<sup>f</sup>*

*J* 2 |Δ*f ij*| 2 <sup>|</sup>Δ0*<sup>b</sup> ij* |

confusion is not likely to occur, we will use the notation Δ*<sup>b</sup>*

*ij*|

*d*Δ*b*<sup>∗</sup> *ij <sup>d</sup>*Δ*<sup>b</sup> ije*

*<sup>j</sup><sup>σ</sup> fiσ*) − *ni*

*<sup>j</sup> <sup>σ</sup><sup>k</sup> fj*) 

*<sup>i</sup> bi* >< *<sup>b</sup>*†

*<sup>i</sup> bi*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*†

*<sup>j</sup> bj* >

*<sup>i</sup> bi* > + *b*†

Hubbard Stratonovich transformation for the holon pairing term (the second term of Eq.(A16))

*ij* <sup>=</sup><sup>&</sup>lt; *bibj* <sup>&</sup>gt; is the saddle point for the holon pairing order parameter <sup>Δ</sup>*<sup>b</sup>*

point. As are shown in Eqs.(A12) through (A14) the spinon interaction term is decomposed into the direct, exchange and pairing channels respectively. Proper Hubbard-Stratonovich

<sup>−</sup> <sup>∑</sup><*i*,*j*<sup>&</sup>gt; *<sup>J</sup>*

<sup>2</sup> <sup>|</sup>Δ*<sup>f</sup> ij*| 2 |Δ*b ij*| <sup>2</sup>−Δ*b*<sup>∗</sup>

<sup>2</sup> <sup>−</sup> <sup>Δ</sup>0*b*<sup>∗</sup> *ij* (*bjbi*) <sup>−</sup> <sup>Δ</sup>0*<sup>b</sup>*

transformations corresponding to these channels and saddle point approximation leads to the effective Hamiltonian,

*Heff* <sup>=</sup> *Jp* <sup>4</sup> ∑ <*i*,*j*> |*χij*| <sup>2</sup> <sup>−</sup> *<sup>χ</sup>*<sup>∗</sup> *ij*(*<sup>f</sup>* † *<sup>i</sup><sup>σ</sup> fj<sup>σ</sup>* + 4*t Jp b*† *<sup>i</sup> bj*) − *c*.*c*. + *J* <sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| 2 |Δ*b ij*| <sup>2</sup> <sup>−</sup> <sup>Δ</sup>*b*<sup>∗</sup> *ij* (*bjbi*) − *c*.*c* + *Jp* <sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| <sup>2</sup> <sup>−</sup> <sup>Δ</sup>*<sup>f</sup> ij*(*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* † *<sup>j</sup>*↓) <sup>−</sup> *<sup>c</sup>*.*c*. + *Jp* <sup>2</sup> ∑ <*i*,*j*> 3 ∑ *l*=0 (*ρl j* )<sup>2</sup> <sup>−</sup> *<sup>ρ</sup><sup>l</sup> j* (*f* † *<sup>i</sup> <sup>σ</sup><sup>l</sup> fi*) <sup>+</sup> *Jp* <sup>2</sup> ∑ *i* (*f* † *<sup>i</sup><sup>σ</sup> fiσ*) + 4*t* 2 *Jp* ∑ <*i*,*j*> (*b*† *<sup>i</sup> bj*)(*b*† *<sup>j</sup> bi*) + *J* <sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| <sup>2</sup> < *b*† *<sup>i</sup> bi* >< *<sup>b</sup>*† *<sup>j</sup> bj* > − *J* <sup>2</sup> ∑ <*i*,*j*> |Δ*f ij*| 2 *<sup>b</sup>*† *<sup>i</sup> bi*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*† *<sup>i</sup> bi* > + *b*† *<sup>j</sup> bj*<sup>−</sup> <sup>&</sup>lt; *<sup>b</sup>*† *<sup>j</sup> bj* > −*μ*∑ *i f* † *<sup>i</sup><sup>σ</sup> fiσ*(<sup>1</sup> <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup> bi*) − *<sup>i</sup>*∑ *i λi*(*f* † *<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup> bi* − 1), (A19)

where Δ*<sup>b</sup> ij* <sup>=</sup><sup>&</sup>lt; *bibj* <sup>&</sup>gt;, *<sup>χ</sup>ij* <sup>=</sup><sup>&</sup>lt; *<sup>f</sup>* † *<sup>i</sup><sup>σ</sup> fj<sup>σ</sup>* <sup>+</sup> <sup>4</sup>*<sup>t</sup> Jp <sup>b</sup>*† *<sup>i</sup> bj* <sup>&</sup>gt;, <sup>Δ</sup>*<sup>f</sup> ij* <sup>=</sup><sup>&</sup>lt; *fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*<sup>↑</sup> <sup>&</sup>gt; and *<sup>ρ</sup><sup>k</sup> <sup>i</sup>* <sup>=</sup><sup>&</sup>lt; <sup>1</sup> 2 *f* † *<sup>i</sup> <sup>σ</sup><sup>k</sup> fi* <sup>&</sup>gt; are proper saddle points.

We note that *ρ<sup>l</sup> <sup>i</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>f</sup>* † *<sup>i</sup> <sup>σ</sup><sup>l</sup> fi* <sup>&</sup>gt;=<sup>&</sup>lt; *<sup>S</sup><sup>l</sup> <sup>i</sup>* <sup>&</sup>gt;<sup>=</sup> 0 for *<sup>l</sup>* <sup>=</sup> 1, 2, 3, *<sup>ρ</sup>*<sup>0</sup> *<sup>i</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> <sup>&</sup>lt; *<sup>f</sup>* † *<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>&</sup>gt;<sup>=</sup> <sup>1</sup> <sup>2</sup> for *l* = 0 for the contribution of the direct spinon interaction term (the fourth term). The expression (*b*† *<sup>j</sup> bi*)(*b*† *<sup>i</sup> bj*) in the fifth term of Eq.(A19) represents the exchange interaction channel. The exchange channel will be ignored owing to a large cost in energy, *<sup>U</sup>* <sup>≈</sup> <sup>4</sup>*<sup>t</sup>* 2 *<sup>J</sup>* [25, 26]. The resulting effective Hamiltonian is

$$H^{MF} = H^{\Delta, \chi} + H^b + H^f,\tag{A20}$$

where *H*Δ,*<sup>χ</sup>* represents the the saddle point energy involved with the spinon pairing order parameter Δ*<sup>f</sup>* , the holon pairing order parameter Δ*<sup>b</sup>* and the hopping order parameter *χ*,

$$\begin{split} H^{\Delta,\chi} = J \sum\_{} \left[ \frac{1}{2} |\Delta\_{ij}^f|^2 |\Delta\_{ij}^b|^2 + \frac{1}{2} |\Delta\_{ij}^f|^2 x^2 \right] \\ + \frac{J\_p}{2} \sum\_{} \left[ |\Delta\_{ij}^f|^2 + \frac{1}{2} |\chi\_{ij}|^2 + \frac{1}{4} \right], \end{split} \tag{A21}$$

where *E<sup>f</sup>*

energy,

*ks* is the quasispinon excitation energy,

with the spinon pairing energy (gap), Δ*<sup>f</sup>*

where the holon pairing energy, Δ*<sup>b</sup>*

respectively, and *βks*(*β*†

we calculate the total free energy.

*HMF <sup>U</sup>*(1) =

*ks* and *�<sup>b</sup>*

Rewriting Eq.(A24) as

0. *� <sup>f</sup>*

with

*Ef ks* = (*� f*

*Eb ks* = (*�<sup>b</sup>*

<sup>0</sup> <sup>=</sup> *<sup>J</sup>*Δ<sup>2</sup>

*γ*2

*<sup>ξ</sup>k*(*φ*) =

*� f ks* <sup>=</sup> *Jp*

*�b*

and lower energy bands of quasiparticles respectively. Here *<sup>α</sup>ks*↑(*α*†

with *<sup>γ</sup><sup>k</sup>* = (cos *kx* <sup>+</sup> cos *ky*) and *<sup>ϕ</sup><sup>k</sup>* = (cos *kx* <sup>−</sup> cos *ky*). <sup>∑</sup>�

� ∑ *k*,*s*=±1

*Hc* = *N J*Δ<sup>2</sup>

� ∑ *k*,*s*=±1

*Z* = exp(−*βHc*)

+

the partition function is derived to be,

 *Ef ks*(*α*†

*f* Δ2 *<sup>b</sup>* <sup>+</sup> *<sup>x</sup>*<sup>2</sup> + *N Jp* Δ2 *<sup>f</sup>* <sup>+</sup> *<sup>χ</sup>*<sup>2</sup> <sup>2</sup> <sup>+</sup> 1 4 

� ∏ *k*,*s*=±1

*Eb ks* <sup>+</sup> *<sup>μ</sup><sup>b</sup>* *ks* <sup>−</sup> *<sup>μ</sup>f*)<sup>2</sup> + (Δ*<sup>f</sup>*

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 19

*ks* <sup>−</sup> *<sup>μ</sup>b*)<sup>2</sup> <sup>−</sup> (Δ*<sup>b</sup>*

*<sup>k</sup>* cos<sup>2</sup> *<sup>φ</sup>* <sup>+</sup> *<sup>ϕ</sup>*<sup>2</sup>

momentum *k* in the half reduced Brillouin zone, and *s* = +1 and −1 represent the upper

the annihilation(creation) operators of spinon quasiparticles of spin up and spin down

sign (−Δ2) in the expression of the holon quasiparticle energy (*�* <sup>−</sup> *<sup>μ</sup>*)<sup>2</sup> <sup>−</sup> <sup>Δ</sup><sup>2</sup> arises as a consequence of the Bose Einstein statistics[28]. From the diagonalized Hamiltonian Eq.(A24),

*ks*↑*αks*<sup>↑</sup> <sup>−</sup> *<sup>α</sup>ks*↓*α*†

<sup>2</sup> <sup>−</sup> *Nxμ<sup>f</sup>* <sup>+</sup> *Nxμ<sup>b</sup>*

(2 cosh *<sup>β</sup>E<sup>f</sup>*

*ks* <sup>2</sup> )

<sup>2</sup>(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*

<sup>−</sup>*βE<sup>b</sup>*

<sup>0</sup> <sup>=</sup> *Jpξk*(*<sup>τ</sup> <sup>f</sup>*)Δ*<sup>f</sup>* , and *<sup>E</sup><sup>b</sup>*

*<sup>f</sup> <sup>ξ</sup>k*(*τb*)Δ*<sup>b</sup>* and with *<sup>φ</sup>* <sup>=</sup> *<sup>θ</sup>*, *<sup>τ</sup> <sup>f</sup>* or *<sup>τ</sup>b*,

*ks*), the annihilation(creation) operators of holon quasiparticles of spin

*ks*↓) + *<sup>E</sup><sup>b</sup>*

+*Hc* (A30)

*ksβ*† *ksβks*

*ks* are the kinetic energies for spinons and holons respectively. The minus

<sup>0</sup> )<sup>2</sup> (A25)

<sup>0</sup>)2, (A26)

*<sup>k</sup>* sin2 *<sup>φ</sup>*, , (A27)

<sup>2</sup> *<sup>s</sup>χξk*(*θ*), (A28)

*ks* = 2*tsχξk*(*θ*), (A29)

*ks* is the quasiholon excitation

denotes the summation over

*ks*↑) and *<sup>α</sup>ks*↓(*α*†

, (A31)

*ks* )<sup>−</sup>1. (A32)

*ks*↓) are

*H<sup>b</sup>* is the holon Hamiltonian,

$$H^b = -t\sum\_{} \left[\chi\_{ij}^\*(b\_i^\dagger b\_j) + c.c.\right]$$

$$-\sum\_{} \frac{I}{2} |\Delta\_{ij}^f|^2 \left[\Delta\_{ij}^{b\*}(b\_i b\_j) + c.c.\right]$$

$$-\sum\_i \mu\_i^b (b\_i^\dagger b\_i - x)\_i\tag{A22}$$

where *μ<sup>b</sup> <sup>i</sup>* <sup>=</sup> *<sup>i</sup>λ<sup>i</sup>* <sup>+</sup> *<sup>J</sup>* <sup>2</sup> <sup>∑</sup>*j*=*i*±*x*ˆ,*i*±*y*<sup>ˆ</sup> <sup>|</sup>Δ*<sup>f</sup> ij*| <sup>2</sup> and *H <sup>f</sup>* , the spinon Hamiltonian,

$$\begin{split} \boldsymbol{H}^{f} &= -\frac{I\_{p}}{4} \sum\_{} \left[ \chi\_{ij}^{\*} (f\_{i\sigma}^{\dagger} f\_{j\sigma}) + c.c. \right] \\ &- \frac{I\_{p}}{2} \sum\_{} \left[ \Delta\_{ij}^{f\*} (f\_{j\uparrow} f\_{i\downarrow} - f\_{j\downarrow} f\_{i\uparrow}) + c.c. \right] \\ &- \sum\_{i} \mu\_{i}^{f} \left( f\_{i\sigma}^{\dagger} f\_{i\sigma} - (1-\mathbf{x}) \right), \end{split} \tag{A23}$$

where *<sup>μ</sup><sup>f</sup> <sup>i</sup>* = *μ*(1 − *x*) + *iλi*.

As can be seen from Eqs.(A21) through (A23), Eq.(A20) reveals the importance of coupling between the spin and charge degrees of freedom, that is, coupling between the spinon pairing and holon pairing. Thus no spin-charge separation appears in the mean-field Hamiltonian above contrary to other mean field theories[1–3, 24, 26] which pay attention to the single-holon bose condensation. As can be seen from the second term in Eq.(A22) which represents holon pairing contribution it is expected that, owing to the coupling effect, bose condensation (or superconducting phase transition) will occur only in the presence of the non-vanishing spin singlet pairing order, Δ*<sup>f</sup>* owing to the coupling effects mentioned above. Indeed, in high *Tc* cuprates superconductivity is not observed above the pseudogap (spin gap) temperatures *T*∗ where the spin singlet pairing order disappears.

#### **8.2 U(1) free energy**

The diagonalized Hamiltonian for Eq.(A20) above is obtained to be (see Appendix A for detailed derivations),

$$H\_{\mathcal{U}(1)}^{\rm MF} = Nj \left[ \Delta\_f^2 \Delta\_b^2 + \Delta\_f^2 \mathbf{x}^2 \right] + Nj\_p \left[ \Delta\_f^2 + \frac{1}{2} \chi^2 + \frac{1}{4} \right]$$

$$\begin{split} \left. + \sum\_{k,s}^{\prime} E\_{ks}^{\dagger} (a\_{ks\uparrow}^{\dagger} a\_{ks\uparrow} - a\_{ks\downarrow}^{\dagger} a\_{ks\downarrow}) - N\mathbf{x}\mu^{\dagger} \\ + \sum\_{k,s=\pm 1}^{\prime} E\_{ks}^{b} \beta\_{ks}^{\dagger} \rho\_{ks} + \sum\_{k,s=\pm 1}^{\prime} \frac{1}{2} (E\_{ks}^{b} + \mu^{b}) + \mu^{b} N\mathbf{x}\_{\prime} \end{split} \tag{A24}$$

where *E<sup>f</sup> ks* is the quasispinon excitation energy,

$$E\_{\rm ks}^f = \sqrt{(\epsilon\_{\rm ks}^f - \mu^f)^2 + (\Delta\_0^f)^2} \tag{A25}$$

with the spinon pairing energy (gap), Δ*<sup>f</sup>* <sup>0</sup> <sup>=</sup> *Jpξk*(*<sup>τ</sup> <sup>f</sup>*)Δ*<sup>f</sup>* , and *<sup>E</sup><sup>b</sup> ks* is the quasiholon excitation energy,

$$E\_{\rm ks}^b = \sqrt{(\epsilon\_{\rm ks}^b - \mu^b)^2 - (\Delta\_0^b)^2} \,\,\,\,\tag{A26}$$

where the holon pairing energy, Δ*<sup>b</sup>* <sup>0</sup> <sup>=</sup> *<sup>J</sup>*Δ<sup>2</sup> *<sup>f</sup> <sup>ξ</sup>k*(*τb*)Δ*<sup>b</sup>* and with *<sup>φ</sup>* <sup>=</sup> *<sup>θ</sup>*, *<sup>τ</sup> <sup>f</sup>* or *<sup>τ</sup>b*,

$$\xi\_k(\phi) = \sqrt{\gamma\_k^2 \cos^2 \phi + \varphi\_k^2 \sin^2 \phi} \,\prime \,\prime \,\tag{A27}$$

$$
\epsilon\_{\rm ks}^f = \frac{\rm J\_p}{2} s \chi\_k^\pi(\theta),
\tag{A28}
$$

$$
\epsilon\_{\rm ks}^{b} = 2 \text{ts} \chi\_{k}^{\pi}(\theta) \,\tag{A29}
$$

with *<sup>γ</sup><sup>k</sup>* = (cos *kx* <sup>+</sup> cos *ky*) and *<sup>ϕ</sup><sup>k</sup>* = (cos *kx* <sup>−</sup> cos *ky*). <sup>∑</sup>� denotes the summation over momentum *k* in the half reduced Brillouin zone, and *s* = +1 and −1 represent the upper and lower energy bands of quasiparticles respectively. Here *<sup>α</sup>ks*↑(*α*† *ks*↑) and *<sup>α</sup>ks*↓(*α*† *ks*↓) are the annihilation(creation) operators of spinon quasiparticles of spin up and spin down respectively, and *βks*(*β*† *ks*), the annihilation(creation) operators of holon quasiparticles of spin 0. *� <sup>f</sup> ks* and *�<sup>b</sup> ks* are the kinetic energies for spinons and holons respectively. The minus sign (−Δ2) in the expression of the holon quasiparticle energy (*�* <sup>−</sup> *<sup>μ</sup>*)<sup>2</sup> <sup>−</sup> <sup>Δ</sup><sup>2</sup> arises as a consequence of the Bose Einstein statistics[28]. From the diagonalized Hamiltonian Eq.(A24), we calculate the total free energy.

Rewriting Eq.(A24) as

$$\begin{aligned} H\_{lI(1)}^{MF} &= \sum\_{k,s=\pm 1}^{'} \left[ E\_{ks}^{f} (a\_{ks\uparrow}^{\dagger} a\_{ks\uparrow} - a\_{ks\downarrow} a\_{ks\downarrow}^{\dagger}) + E\_{ks}^{b} \beta\_{ks}^{\dagger} \beta\_{ks} \right] \\ &+ H\_{\emptyset} \end{aligned} \tag{A30}$$

with

18 Will-be-set-by-IN-TECH

*<sup>i</sup> bj*) + *c*.*c*.

<sup>2</sup> and *H <sup>f</sup>* , the spinon Hamiltonian,

*<sup>i</sup><sup>σ</sup> fjσ*) + *c*.*c*.

*ij* (*fj*<sup>↑</sup> *fi*<sup>↓</sup> − *fj*<sup>↓</sup> *fi*↑) + *<sup>c</sup>*.*c*.

*<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* − (1 − *x*)

As can be seen from Eqs.(A21) through (A23), Eq.(A20) reveals the importance of coupling between the spin and charge degrees of freedom, that is, coupling between the spinon pairing and holon pairing. Thus no spin-charge separation appears in the mean-field Hamiltonian above contrary to other mean field theories[1–3, 24, 26] which pay attention to the single-holon bose condensation. As can be seen from the second term in Eq.(A22) which represents holon pairing contribution it is expected that, owing to the coupling effect, bose condensation (or superconducting phase transition) will occur only in the presence of the non-vanishing spin singlet pairing order, Δ*<sup>f</sup>* owing to the coupling effects mentioned above. Indeed, in high *Tc* cuprates superconductivity is not observed above the pseudogap (spin gap) temperatures *T*∗

The diagonalized Hamiltonian for Eq.(A20) above is obtained to be (see Appendix A for

� ∑ *k*,*s*=±1

*ks*↓*αks*↓) <sup>−</sup> *Nxμ<sup>f</sup>*

1 2 (*E<sup>b</sup>* *<sup>i</sup> bi* − *x*), (A22)

, (A23)

*ks* <sup>+</sup> *<sup>μ</sup>b*) + *<sup>μ</sup>bNx*, (A24)

*ij* (*bibj*) + *c*.*c*.

*<sup>H</sup><sup>b</sup>* <sup>=</sup> <sup>−</sup>*<sup>t</sup>* ∑

− ∑ <*i*,*j*>

− ∑ *i μb <sup>i</sup>* (*b*†

*ij*|

<sup>4</sup> ∑ <*i*,*j*>

 *χ*∗ *ij*(*<sup>f</sup>* †

 Δ*<sup>f</sup>* <sup>∗</sup>

*<sup>H</sup> <sup>f</sup>* <sup>=</sup> <sup>−</sup> *Jp*

<sup>−</sup> *Jp* <sup>2</sup> ∑ <*i*,*j*>

− ∑ *i μf i f* †

<sup>2</sup> <sup>∑</sup>*j*=*i*±*x*ˆ,*i*±*y*<sup>ˆ</sup> <sup>|</sup>Δ*<sup>f</sup>*

<*i*,*j*>

*J* 2 |Δ*f ij*| 2 Δ*b*<sup>∗</sup>

 *χ*∗ *ij*(*b*†

*H<sup>b</sup>* is the holon Hamiltonian,

*<sup>i</sup>* <sup>=</sup> *<sup>i</sup>λ<sup>i</sup>* <sup>+</sup> *<sup>J</sup>*

*<sup>i</sup>* = *μ*(1 − *x*) + *iλi*.

where the spin singlet pairing order disappears.

*HMF <sup>U</sup>*(1) = *N J*

+ � ∑ *k*,*s Ef ks*(*α*†

+

� ∑ *k*,*s*=±1

*Eb ksβ*† *ksβks* +

 Δ2 *f* Δ2 *<sup>b</sup>* <sup>+</sup> <sup>Δ</sup><sup>2</sup> *f x*2 + *N Jp* Δ2 *<sup>f</sup>* + 1 2 *χ*<sup>2</sup> + 1 4 

*ks*↑*αks*<sup>↑</sup> <sup>−</sup> *<sup>α</sup>*†

where *μ<sup>b</sup>*

where *<sup>μ</sup><sup>f</sup>*

**8.2 U(1) free energy**

detailed derivations),

$$H\_c = Nl\Delta\_f^2 \left(\Delta\_b^2 + \mathbf{x}^2\right) + Nl\_p \left(\Delta\_f^2 + \frac{\chi^2}{2} + \frac{1}{4}\right)$$

$$+ \sum\_{k,s=\pm 1}^{'} \frac{E\_{ks}^b + \mu^b}{2} - N\mathbf{x}\mu^f + N\mathbf{x}\mu^b.\tag{A31}$$

the partition function is derived to be,

$$Z = \exp(-\beta H\_{\mathcal{E}}) \prod\_{k,s=\pm 1}^{'} (2 \cosh \frac{\beta E\_{ks}^{f}}{2})^2 (1 - e^{-\beta E\_{ks}^{b}})^{-1}. \tag{A32}$$

For fixed numbers of spinon and holon at a given hole concentration, we obtain, for the

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 21

*� <sup>f</sup>*

*�b ks* <sup>−</sup> *<sup>μ</sup><sup>b</sup> Eb ks*

*ks* <sup>−</sup> *<sup>μ</sup><sup>f</sup> Ef ks*

<sup>0</sup> and the holon pairing energy (gap) <sup>Δ</sup>*<sup>b</sup>*

− *Nx* = 0, (A37)

<sup>0</sup> respectively

 *fi*↓ <sup>−</sup> *<sup>f</sup>* † *i*↑ =

 =

 *fi*<sup>↓</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *<sup>i</sup>* with

+ *Nx* = 0. (A38)

chemical potentials, *μ<sup>f</sup>* and *μb*,

*∂FU*(1) *∂μ<sup>f</sup>* <sup>=</sup>

*∂FU*(1) *∂μ<sup>b</sup>* <sup>=</sup> <sup>−</sup>

temperatures at which the spin gap Δ*<sup>f</sup>*

= *gi*

*i*↑ *f* †

*<sup>i</sup>* =

+*iλ*�� <sup>2</sup> , *a*

 *bi* 0

0(2) *<sup>i</sup>* <sup>=</sup> <sup>−</sup>*λ*�

 *fi*<sup>2</sup> <sup>−</sup> *<sup>f</sup>* † *i*1

constraints *f* †

for spinon, *h*<sup>0</sup>

*a* 0(1) *<sup>i</sup>* <sup>=</sup> *<sup>i</sup>λ*�  *bi* 0 

− *<sup>i</sup>*∑ *i*

**Appendix B: SU(2) action from the U(1) action**

for both the spinon and holon spinors with *fi*<sup>1</sup>

and *ci*<sup>↓</sup> <sup>=</sup> *<sup>b</sup>*†

*<sup>i</sup>*<sup>↓</sup> <sup>=</sup> 0 and *fi*<sup>↓</sup> *fi*<sup>↑</sup> <sup>=</sup> 0 to write

*<sup>i</sup><sup>σ</sup> fi<sup>σ</sup>* <sup>+</sup> *<sup>b</sup>*†

0(3)

in order to allow for SU(2) symmetry. Thus writing spinors *ψ*<sup>0</sup>

+*λ*�� <sup>2</sup> , *a*

*λi*(*f* †

begin to open.

and *bi*<sup>1</sup> *bi*2 

(*b*† *<sup>i</sup>*1, *<sup>b</sup>*† *i*2) *fi*<sup>1</sup> *f* † *i*2

(*b*† *<sup>i</sup>*1, *<sup>b</sup>*† *i*2)

� ∑ *k*,*s*=±1

+ *�b* � ∑ *k*,*s*=±1

tanh *<sup>β</sup>E<sup>f</sup> ks* 2

 1 *eβE<sup>b</sup> ks* − 1

*ks* <sup>−</sup> *<sup>μ</sup><sup>b</sup>* <sup>−</sup> *<sup>E</sup><sup>b</sup>*

2*E<sup>b</sup> ks*

, satisfying *ci*<sup>↑</sup> <sup>=</sup> *<sup>b</sup>*†

*<sup>i</sup> fi*<sup>↓</sup> = (*b*†

*<sup>i</sup> bi* − <sup>1</sup>) − *<sup>i</sup>*∑

*ks*

Using the five self-consistent equations of Eqs.(A34) through (A38), we determine *χ*, Δ*b*, Δ*<sup>f</sup>* , *μ<sup>f</sup>* and *μ<sup>b</sup>* at each doping and temperature. Both the pseudogap temperature *T*<sup>∗</sup> and the superconducting transition (bose condensation) temperature *Tc* are determined to be the

The t-J Hamiltonian is manifestly invariant under the local SU(2) transformation *gi* = *eiσ*·*<sup>θ</sup>*

*f* † *i*2

*<sup>i</sup> fi*<sup>↑</sup> = (*b*†

*<sup>i</sup>* , 0) *fi*↓ <sup>−</sup> *<sup>f</sup>* † *i*↑

> *i λ* � *i f* † *i*↑ *f* † *<sup>i</sup>*<sup>↓</sup> <sup>−</sup> *<sup>i</sup>*∑ *i λ* ��

= *gi*

*<sup>i</sup>* , 0) *fi*↑ *f* † *i*↓

[26]. We introduce additional Lagrange multiplier terms involved with the

for holon and the three-component Lagrangian multiplier field **a**<sup>0</sup>

*<sup>i</sup>*<sup>1</sup> =

*<sup>i</sup>* = *iλi*, the U(1) action in Eq.(A6) can be rewritten as

 *fi*<sup>↑</sup> *f* † *i*↓  *fi*↑ *f* † *i*↓  , *fi*<sup>2</sup> <sup>−</sup> *<sup>f</sup>* † *i*1

= (*b*†

= (*b*†

*<sup>i</sup>* , 0)*g*† *i gi fi*↓ <sup>−</sup> *<sup>f</sup>* † *i*↑

*<sup>i</sup>* , 0)*g*† *i gi fi*↑ *f* † *i*↓

= *gi*

*<sup>i</sup> fi*<sup>↓</sup> *fi*<sup>↑</sup> (B1)

and *ψ*<sup>0</sup> *<sup>i</sup>*<sup>2</sup> =

Using the above expression, the total free energy is given by

$$F\_{II(1)} = NI\lambda\_f^2 \left(\Delta\_b^2 + \mathbf{x}^2\right) + NI\_p \left(\Delta\_f^2 + \frac{\chi^2}{2} + \frac{1}{4}\right)$$

$$\begin{split} \left(-2k\_BT \sum\_{k,s=\pm 1}^{\prime} \ln(\cosh(\beta E\_{ks}^{\frac{\ell}{}}/2)) - N\mathbf{x}\mu^{\ell} - 2Nk\_BT\ln 2 \\ + k\_BT \sum\_{k,s=\pm 1}^{\prime} \ln(1 - e^{-\beta E\_{ks}^b}) + \sum\_{k,s=\pm 1}^{\prime} \frac{E\_{ks}^b + \mu^b}{2} + N\mathbf{x}\mu^b. \end{split} \tag{A33}$$

The set of uniform phase (*θ* = 0) for the hopping order parameter, d-wave symmetry (*τ <sup>f</sup>* = *π*/2) for the spinon pairing order parameter and s-wave symmetry (*τ<sup>b</sup>* = 0) for the holon pairing order parameter is found to yield a stable saddle point energy for both the underdoped and overdoped regions. There is another set of order parameters which yield the same energy as the above one; 2*π*-flux phase (*θ* = *π*/2) for the hopping order parameter, s-wave symmetry (*τ <sup>f</sup>* = 0) for the spinon pairing order parameter and d-wave symmetry (*τ<sup>b</sup>* = *π*/2) for the holon pairing order parameter. In both cases, the d-wave symmetry of the electron or hole (not holon) pairs occurs as a composite of the d-wave (s-wave) symmetry of spinon pairs and s-wave (d-wave) symmetry of holon pairs. Only at very low doping near half filling, the flux phase[25] becomes more stable. Thus, the phase of the order parameters of present interest are *θ* = 0, *τ <sup>f</sup>* = *π*/2 and *τ<sup>b</sup>* = 0. Then the d-wave symmetry of the electron or hole (not holon) pairs is a composite of the d-wave symmetry of spinon pairs and s-wave symmetry of holon pairs. Minimizing the free energy with respect to the amplitudes of the order parameters *χ*, Δ*<sup>b</sup>* and Δ*<sup>f</sup>* , we obtain the self-consistent equations for the order parameters,

$$\begin{split} \frac{\partial F\_{\mathrm{II}(1)}}{\partial \chi} &= Nl\_{p}\chi - \sum\_{\mathrm{ks}}^{\prime} \left( \tanh \frac{\beta E\_{\mathrm{ks}}^{f}}{2} \right) \left( \frac{\partial E\_{\mathrm{ks}}^{f}}{\partial \chi} \right) \\ &+ \sum\_{\mathrm{ks}}^{\prime} \left( \frac{1}{e^{\beta E\_{\mathrm{ks}}^{b}} - 1} + \frac{1}{2} \right) \left( \frac{\partial E\_{\mathrm{ks}}^{b}}{\partial \chi} \right) = 0, \end{split} \tag{A.34}$$

$$\begin{split} \frac{\partial F\_{\mathcal{U}(1)}}{\partial \Delta\_b} &= 2Nf \Delta\_f^2 \Delta\_b \\ &+ \sum\_{ks}^{\prime} \left( \frac{1}{e^{\beta E\_{ks}^b} - 1} + \frac{1}{2} \right) \left( \frac{\partial E\_{ks}^b}{\partial \Delta\_b} \right) = 0, \end{split} \tag{A35}$$

$$\begin{split} \frac{\partial F\_{II(1)}}{\partial \Delta\_{f}} &= 2Nl\_{p}\Delta\_{f} + 2N\Delta\_{f}(\Delta\_{b}^{2} + \mathbf{x}^{2}) \\ &- \sum\_{ks}^{\prime} \left( \tanh \frac{\beta E\_{ks}^{f}}{2} \right) \left( \frac{\partial E\_{ks}^{f}}{\partial \Delta\_{f}} \right) \\ &+ \sum\_{ks}^{\prime} \left( \frac{1}{e^{\beta E\_{ks}^{b}} - 1} + \frac{1}{2} \right) \left( \frac{\partial E\_{ks}^{b}}{\partial \Delta\_{f}} \right) = 0. \end{split} \tag{A36}$$

20 Will-be-set-by-IN-TECH

*ks* ) +

The set of uniform phase (*θ* = 0) for the hopping order parameter, d-wave symmetry (*τ <sup>f</sup>* = *π*/2) for the spinon pairing order parameter and s-wave symmetry (*τ<sup>b</sup>* = 0) for the holon pairing order parameter is found to yield a stable saddle point energy for both the underdoped and overdoped regions. There is another set of order parameters which yield the same energy as the above one; 2*π*-flux phase (*θ* = *π*/2) for the hopping order parameter, s-wave symmetry (*τ <sup>f</sup>* = 0) for the spinon pairing order parameter and d-wave symmetry (*τ<sup>b</sup>* = *π*/2) for the holon pairing order parameter. In both cases, the d-wave symmetry of the electron or hole (not holon) pairs occurs as a composite of the d-wave (s-wave) symmetry of spinon pairs and s-wave (d-wave) symmetry of holon pairs. Only at very low doping near half filling, the flux phase[25] becomes more stable. Thus, the phase of the order parameters of present interest are *θ* = 0, *τ <sup>f</sup>* = *π*/2 and *τ<sup>b</sup>* = 0. Then the d-wave symmetry of the electron or hole (not holon) pairs is a composite of the d-wave symmetry of spinon pairs and s-wave symmetry of holon pairs. Minimizing the free energy with respect to the amplitudes of the order parameters *χ*,

� ∑ *k*,*s*=±1

*ks*/2)) <sup>−</sup> *Nxμ<sup>f</sup>* <sup>−</sup> <sup>2</sup>*NkBT* ln 2

<sup>2</sup> <sup>+</sup> *Nxμ<sup>b</sup>*

. (A33)

*Eb ks* <sup>+</sup> *<sup>μ</sup><sup>b</sup>*

Using the above expression, the total free energy is given by

� ∑ *k*,*s*=±1

� ∑ *k*,*s*=±1 *f* Δ2 *<sup>b</sup>* <sup>+</sup> *<sup>x</sup>*<sup>2</sup> + *N Jp* Δ2 *<sup>f</sup>* <sup>+</sup> *<sup>χ</sup>*<sup>2</sup> <sup>2</sup> <sup>+</sup> 1 4 

ln(cosh(*βE<sup>f</sup>*

ln(<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−*βE<sup>b</sup>*

Δ*<sup>b</sup>* and Δ*<sup>f</sup>* , we obtain the self-consistent equations for the order parameters,

� ∑ *ks*

 1 *eβE<sup>b</sup> ks* − 1

*<sup>f</sup>* Δ*<sup>b</sup>*

 1 *eβE<sup>b</sup> ks* − 1

= <sup>2</sup>*N Jp*Δ*<sup>f</sup>* + <sup>2</sup>*N*Δ*f*(Δ<sup>2</sup>

 1 *eβE<sup>b</sup> ks* − 1

tanh *<sup>β</sup>E<sup>f</sup> ks* 2

tanh *<sup>β</sup>E<sup>f</sup> ks* 2

> + 1 2

> + 1 2

> + 1 2

 *<sup>∂</sup>E<sup>f</sup> ks ∂χ*

 *<sup>∂</sup>E<sup>b</sup> ks ∂χ*

 *<sup>∂</sup>E<sup>b</sup> ks ∂*Δ*<sup>b</sup>*

*<sup>b</sup>* <sup>+</sup> *<sup>x</sup>*2)

 *<sup>∂</sup>E<sup>b</sup> ks ∂*Δ*<sup>f</sup>*

 *<sup>∂</sup>E<sup>f</sup> ks ∂*Δ*<sup>f</sup>*

= 0, (A34)

= 0, (A35)

= 0. (A36)

*∂χ* <sup>=</sup> *N Jp<sup>χ</sup>* <sup>−</sup>

+ � ∑ *ks*

= 2*N J*Δ<sup>2</sup>

+ � ∑ *ks*

− � ∑ *ks*

+ � ∑ *ks*

*∂FU*(1)

*∂FU*(1) *∂*Δ*<sup>b</sup>*

*∂FU*(1) *∂*Δ*<sup>f</sup>*

*FU*(1) <sup>=</sup> *N J*Δ<sup>2</sup>

−2*kBT*

+*kBT*

For fixed numbers of spinon and holon at a given hole concentration, we obtain, for the chemical potentials, *μ<sup>f</sup>* and *μb*,

$$
\frac{\partial F\_{\mathcal{U}(1)}}{\partial \mu^f} = \sum\_{k,s=\pm 1}^{\prime} \left( \tanh \frac{\beta E\_{\text{ks}}^f}{2} \right) \left( \frac{\epsilon\_{\text{ks}}^f - \mu^f}{E\_{\text{ks}}^f} \right) - N\mathbf{x} = \mathbf{0}, \tag{A37}
$$

$$
\frac{\partial F\_{\mathcal{U}(1)}}{\partial \mu^b} = -\sum\_{k,s=\pm 1}^{\prime} \left[ \frac{1}{e^{\beta E\_{\text{ks}}^b} - 1} \frac{\epsilon\_{\text{ks}}^b - \mu^b}{E\_{\text{ks}}^b} \right.
$$

$$
+ \frac{\epsilon\_{\text{ks}}^b - \mu^b - E\_{\text{ks}}^b}{2E\_{\text{ks}}^b} \Big] + N\mathbf{x} = 0. \tag{A38}
$$

Using the five self-consistent equations of Eqs.(A34) through (A38), we determine *χ*, Δ*b*, Δ*<sup>f</sup>* , *μ<sup>f</sup>* and *μ<sup>b</sup>* at each doping and temperature. Both the pseudogap temperature *T*<sup>∗</sup> and the superconducting transition (bose condensation) temperature *Tc* are determined to be the temperatures at which the spin gap Δ*<sup>f</sup>* <sup>0</sup> and the holon pairing energy (gap) <sup>Δ</sup>*<sup>b</sup>* <sup>0</sup> respectively begin to open.

*ks*

#### **Appendix B: SU(2) action from the U(1) action**

The t-J Hamiltonian is manifestly invariant under the local SU(2) transformation *gi* = *eiσ*·*<sup>θ</sup>* for both the spinon and holon spinors with *fi*<sup>1</sup> *f* † *i*2 = *gi fi*↑ *f* † *i*↓ , *fi*<sup>2</sup> <sup>−</sup> *<sup>f</sup>* † *i*1 = *gi fi*↓ <sup>−</sup> *<sup>f</sup>* † *i*↑ and *bi*<sup>1</sup> *bi*2 = *gi bi* 0 , satisfying *ci*<sup>↑</sup> <sup>=</sup> *<sup>b</sup>*† *<sup>i</sup> fi*<sup>↑</sup> = (*b*† *<sup>i</sup>* , 0) *fi*↑ *f* † *i*↓ = (*b*† *<sup>i</sup>* , 0)*g*† *i gi fi*↑ *f* † *i*↓ = (*b*† *<sup>i</sup>*1, *<sup>b</sup>*† *i*2) *fi*<sup>1</sup> *f* † *i*2 and *ci*<sup>↓</sup> <sup>=</sup> *<sup>b</sup>*† *<sup>i</sup> fi*<sup>↓</sup> = (*b*† *<sup>i</sup>* , 0) *fi*↓ <sup>−</sup> *<sup>f</sup>* † *i*↑ = (*b*† *<sup>i</sup>* , 0)*g*† *i gi fi*↓ <sup>−</sup> *<sup>f</sup>* † *i*↑ = (*b*† *<sup>i</sup>*1, *<sup>b</sup>*† *i*2) *fi*<sup>2</sup> <sup>−</sup> *<sup>f</sup>* † *i*1 [26]. We introduce additional Lagrange multiplier terms involved with the constraints *f* † *i*↑ *f* † *<sup>i</sup>*<sup>↓</sup> <sup>=</sup> 0 and *fi*<sup>↓</sup> *fi*<sup>↑</sup> <sup>=</sup> 0 to write

$$-i\sum\_{i} \lambda\_i (f\_{i\sigma}^{\dagger} f\_{i\sigma} + b\_i^{\dagger} b\_i - 1) - i \sum\_{i} \lambda\_i^{'} f\_{i\uparrow}^{\dagger} f\_{i\downarrow}^{\dagger} - i \sum\_{i} \lambda\_i^{''} f\_{i\downarrow} f\_{i\uparrow} \tag{B1}$$

in order to allow for SU(2) symmetry. Thus writing spinors *ψ*<sup>0</sup> *<sup>i</sup>*<sup>1</sup> = *fi*<sup>↑</sup> *f* † *i*↓ and *ψ*<sup>0</sup> *<sup>i</sup>*<sup>2</sup> = *fi*<sup>↓</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ for spinon, *h*<sup>0</sup> *<sup>i</sup>* = *bi* 0 for holon and the three-component Lagrangian multiplier field **a**<sup>0</sup> *<sup>i</sup>* with *a* 0(1) *<sup>i</sup>* <sup>=</sup> *<sup>i</sup>λ*� +*iλ*�� <sup>2</sup> , *a* 0(2) *<sup>i</sup>* <sup>=</sup> <sup>−</sup>*λ*� +*λ*�� <sup>2</sup> , *a* 0(3) *<sup>i</sup>* = *iλi*, the U(1) action in Eq.(A6) can be rewritten as

where

*HSU*(2) *<sup>t</sup>*−*<sup>J</sup>* <sup>=</sup> <sup>−</sup> *<sup>t</sup>*

<sup>2</sup> ∑ <*i*,*j*>

− *J* <sup>2</sup> ∑ <*i*,*j*>

+ (1 + *h*†

− (1 + *h*†

+ (1 + *h*†

 (*f* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* †

−*μ*∑ *i*

− ∑ *i <sup>i</sup>λ*(1) *<sup>i</sup>* (*<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*†

+*iλ*(3) *<sup>i</sup>* (*<sup>f</sup>* †

cond-mat/0212436.

(2004).

106402 (2005); references there-in.

[9] A. Himeda and M. Ogata, Phys. Rev. Lett. 85, 4345 (2000).

**9. References**

(<sup>1</sup> <sup>−</sup> *<sup>h</sup>*† *<sup>i</sup> hi*)

 (*f* † *<sup>i</sup><sup>α</sup> fjα*)(*b*†

+(*fi*<sup>2</sup> *fj*<sup>1</sup> <sup>−</sup> *fi*<sup>1</sup> *fj*2)(*b*†

(*<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* †

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

*<sup>i</sup>*<sup>1</sup> *fi*<sup>1</sup> <sup>−</sup> *fi*<sup>2</sup> *<sup>f</sup>* †

*<sup>j</sup>*1*bi*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*†

*<sup>j</sup>*1*bi*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*†

*<sup>j</sup> hj*) (*f* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* †

*<sup>j</sup> hj*)

*<sup>j</sup> hj*) − (1 + *h*†

*<sup>i</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*†

[1] G. Kotliar and J. Liu, Phys. Rev. B 38, 5142 (1988); references there-in.

Soc. 37, 545 (2000); S.-S. Lee and S.-H. S. Salk, cond-mat/0304293.

*<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1) −

*<sup>i</sup>*1*bi*2) + *<sup>i</sup>λ*(2)

[2] Y. Suzumura, Y. Hasegawa and H. Fukuyama, J. Phys. Soc. Jpn. 57, 2768 (1988). [3] M.U. Ubbens and P. A. Lee, Phys. Rev. B 49, 6853 (1994); references therein.

[4] T.-H. Gimm, S.-S. Lee, S.-P. Hong and S.-H. Suck Salk, Phys. Rev. B 60, 6324 (1999). [5] S.-S. Lee and Sung-Ho Suck Salk, Phys. Rev. B 64, 052501 (2001); S.-S. Lee and Sung-Ho Suck Salk, Phys. Rev. B 66, 054427 (2002); S.-S. Lee and Sung-Ho Suck Salk, J. Kor. Phys.

[6] S.-S. Lee and S.-H. S. Salk, Phys. Rev. B 71, 134518 (2005); J.-H. Eom and S.-H. S. Salk, Phys. Rev. B 72, 064508 (2005); J.-H. Eom, S.-S. Lee, K.-S. Kim and S.-H. S. Salk, Phys. Rev. B 70, 024522 (2004); S.-S. Lee, J.-H. Eom, K.-S. Kim and S.-H. Suck Salk, Phys. Rev. B 66, 064520 (2002); S.-S, Lee and S.-H. S. Salk, Phys. Rev. B 66, 054427 (2002); S.-S. Lee and S.-H. S. Salk, Phys. Rev. B 64, 052501 (2001); Physica C. 353, 130(2001); J. Kor. Phys. Soc. 37, 545 (2000); Int. J. Mod. Phys. B 13, 3455 (1999); cond-mat/0410229;

[7] M. Civelli M. Capone, S. S. Kancharla, O. Parcollet, and G. Kotliar, Phys. Rev. Lett. 95,

[8] K. Tanaka, T. Yoshida, A. Fujimori, D. H. Lu, Z.-X. Shen, X.-J. Shou, H. Eisaki, Z. Hussain, S. Uchida, Y. Aiura. K. Ono, T. Sugaya, T. Mizuno. and I. Terasaki, Phys. Rev. B 70 092503

*<sup>i</sup>*1*bi*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*†

(*<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* †

*<sup>i</sup>*2*bj*2) + *c*.*c*.

Role of Antiferromagnetic Fluctuations in High Temperature Superconductivity 23

*<sup>i</sup>*1*bj*2) + *c*.*c*.

*<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1)

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

*<sup>i</sup>* (*fi*<sup>2</sup> *fi*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*†

*<sup>i</sup>*2*bi*2) 

 (*f* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* †

 (1 + *h*†

*<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1)

*<sup>j</sup> hj*) <sup>×</sup>

*<sup>i</sup>*2*bi*1)

*<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1)

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

*<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1)

. (B6)

*<sup>j</sup> hj*)

$$\mathcal{S}\_{\mathrm{II}(1)}[b, f, a] = \int\_0^\beta d\tau \left[ \sum\_i h^{0\dagger}\_{\bar{i}} \partial\_\tau h^0\_i + \frac{1}{2} \sum\_{i,\mu} \psi^{0\dagger}\_{i\bar{\mu}} \partial\_\tau \psi^0\_{i\bar{\mu}} - t \sum\_{} \left( (\psi^{0\dagger}\_{\bar{i}a} h^0\_i)(h^{0\dagger}\_{\bar{j}} \psi^0\_{j\bar{\mu}}) + c.c.\right) \right. \\ \left. + \frac{1}{2} \sum\_{} \left( -\frac{1}{2} \sum\_{i,\mu} \psi^{0\dagger}\_{i\bar{\mu}} \psi^0\_{j\bar{\mu}} - \frac{1}{2} \sum\_{i,\mu} \psi^0\_{i\bar{\mu}} \psi^0\_{i\bar{\mu}} \right) \right. \\ \left. + c.c.\right] \\ = \frac{1}{2} \sum\_i \psi^\dagger\_{i\bar{\sigma}} f\_{i\bar{\sigma}} (h^\dagger\_i b\_i + 1) - \sum\_i \mathbf{a}^0\_i \cdot \left( \frac{1}{2} \psi^{0\dagger}\_{i\bar{\mu}} \sigma \psi^0\_{i\bar{\mu}} + h^0\_i \tau \sigma h^0\_i \right) \Big]. \tag{B2}$$

Here the fourth term is the Heisenberg interaction term, *HJ* <sup>=</sup> <sup>−</sup> *<sup>J</sup>* <sup>2</sup> <sup>∑</sup><*i*,*j*<sup>&</sup>gt; *bibjb*† *j b*† *<sup>i</sup>* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> − *f* † *i*↑ *f* † *<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑).

We rewrite the spinon part of the Heisenberg interaction,

$$-\frac{I}{2}(f\_{i\downarrow}^{\sf t}f\_{j\uparrow}^{\sf t}-f\_{i\uparrow}^{\sf t}f\_{j\downarrow}^{\sf t})(f\_{j\uparrow}f\_{i\downarrow}-f\_{j\downarrow}f\_{i\uparrow})$$

$$=\frac{I}{4}\left[\sum\_{k=1}^{3}(f\_{ia}^{\sf t}\sigma\_{a\beta\}f\_{i\beta})(f\_{j\uparrow}^{\sf t}\sigma\_{\gamma\delta}^{k}f\_{j\delta})-(f\_{ia}^{\sf t}f\_{ia})(f\_{j\beta}^{\sf t}f\_{j\beta})\right]$$

$$=\frac{I}{4}\left[\frac{1}{4}\left(tr\Psi\_{i}^{0\sf t}\Psi\_{i}^{0}(\sigma^{k})^{T}\right)\left(tr\Psi\_{j}^{0\sf t}\Psi\_{j}^{0}(\sigma^{k})^{T}\right)-(f\_{ia}^{\sf t}f\_{ia})(f\_{j\beta}^{\sf t}f\_{j\beta})\right],\tag{B3}$$

where Ψ<sup>0</sup> *i* ≡ *fi*<sup>↑</sup> *fi*<sup>↓</sup> *f* † *<sup>i</sup>*<sup>↓</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ and (*f* † *iασ<sup>k</sup> αβ fiβ*) = <sup>1</sup> <sup>2</sup> *tr* Ψ0† *<sup>i</sup>* <sup>Ψ</sup><sup>0</sup> *<sup>i</sup>* (*σk*)*<sup>T</sup>* [**?** ]. Here (*σk*)*<sup>T</sup>* denotes the transpose of the Pauli matrices for *k* = 1, 2, 3.

Realizing *hi* = *gih*<sup>0</sup> *<sup>i</sup>* and Ψ*<sup>i</sup>* = *fi*<sup>1</sup> *fi*<sup>2</sup> *f* † *<sup>i</sup>*<sup>2</sup> <sup>−</sup> *<sup>f</sup>* † *i*1 = *gi fi*<sup>↑</sup> *fi*<sup>↓</sup> *f* † *<sup>i</sup>*<sup>↓</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ , and using Eq.(B3), the SU(2) symmetric Heisenberg interaction term is given by

$$H\_{I}^{SU(2)} = \frac{I}{4} \sum\_{} (1 + h\_i^\dagger h\_i)(1 + h\_j^\dagger h\_j) \left[ \frac{1}{4} \left( tr \mathbf{Y}\_i^\dagger \mathbf{Y}\_i (\sigma^k)^T \right) \left( tr \mathbf{Y}\_j^\dagger \mathbf{Y}\_j (\sigma^k)^T \right) - (f\_{ia}^\dagger f\_{ia})(f\_{j\beta}^\dagger f\_{j\beta}) \right]$$

$$= -\frac{I}{2} \sum\_{} (1 + h\_i^\dagger h\_i)(1 + h\_j^\dagger h\_j)(f\_{i2}^\dagger f\_{j1}^\dagger - f\_{i1}^\dagger f\_{j2}^\dagger)(f\_{j1} f\_{i2} - f\_{j2} f\_{i1}). \tag{B4}$$

Taking decomposition of the Heisenberg interaction term above into terms involving charge and spin fluctuations separately, uncorrelated mean field contributions and correlated fluctuations, i.e., correlations between charge and spin fluctuations as in the U(1) case, the SU(2) action is rewritten,

$$S[b\_{\rm av} f\_{\rm av} \lambda\_i] = \int\_0^\beta d\tau \left[ \sum\_{i,a=1,2} (b\_{\rm ia}^\dagger \partial\_\tau b\_{\rm ia} + f\_{\rm ia}^\dagger \partial\_\tau f\_{\rm ia}) + H\_{t-f}^{SU(2)} \right],\tag{B5}$$

where

22 Will-be-set-by-IN-TECH

*<sup>i</sup> bi* + <sup>1</sup>) − ∑

Here the fourth term is the Heisenberg interaction term, *HJ* <sup>=</sup> <sup>−</sup> *<sup>J</sup>*

*αβ fiβ*)(*<sup>f</sup>* †

 *fi*<sup>1</sup> *fi*<sup>2</sup> *f* † *<sup>i</sup>*<sup>2</sup> <sup>−</sup> *<sup>f</sup>* † *i*1

> *<sup>j</sup> hj*) 1 4 *tr*Ψ†

> > *<sup>j</sup> hj*)(*<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* †

*jγσ<sup>k</sup>*

 *tr*Ψ0† *<sup>j</sup>* <sup>Ψ</sup><sup>0</sup> *j*(*σk*)*<sup>T</sup>* <sup>−</sup> (*<sup>f</sup>* †

*αβ fiβ*) = <sup>1</sup>

 = *gi*

We rewrite the spinon part of the Heisenberg interaction,

*i* **a**0 *<sup>i</sup>* · ( 1 2 *ψ*0† *<sup>i</sup><sup>α</sup> σψ*<sup>0</sup>

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*γδ fjδ*) <sup>−</sup> (*<sup>f</sup>* †

2 *tr* Ψ0† *<sup>i</sup>* <sup>Ψ</sup><sup>0</sup> *<sup>i</sup>* (*σk*)*<sup>T</sup>* 

*<sup>i</sup><sup>α</sup> fiα*)(*<sup>f</sup>* †

 *fi*<sup>↑</sup> *fi*<sup>↓</sup> *f* † *<sup>i</sup>*<sup>↓</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑

> *tr*Ψ†

*<sup>i</sup>* <sup>Ψ</sup>*i*(*σk*)*<sup>T</sup>*

Taking decomposition of the Heisenberg interaction term above into terms involving charge and spin fluctuations separately, uncorrelated mean field contributions and correlated fluctuations, i.e., correlations between charge and spin fluctuations as in the U(1) case, the

(*b*†

*<sup>i</sup>α∂τbi<sup>α</sup>* <sup>+</sup> *<sup>f</sup>* †

*<sup>j</sup><sup>β</sup> fjβ*) 

*<sup>i</sup><sup>α</sup> fiα*)(*<sup>f</sup>* †

*<sup>j</sup>* <sup>Ψ</sup>*j*(*σk*)*<sup>T</sup>*

*<sup>i</sup>α∂τ fiα*) + *<sup>H</sup>SU*(2)

*t*−*J*

 <sup>−</sup> (*<sup>f</sup>* †

*<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1). (B4)

*<sup>j</sup><sup>β</sup> fjβ*) 

[**?** ]. Here (*σk*)*<sup>T</sup>* denotes the

, and using Eq.(B3), the SU(2)

*<sup>i</sup><sup>α</sup> fiα*)(*<sup>f</sup>* †

, (B5)

*<sup>j</sup><sup>β</sup> fjβ*) 

*<sup>i</sup><sup>α</sup>* − *<sup>t</sup>* ∑ <*i*,*j*>  (*ψ*0† *<sup>i</sup><sup>α</sup> <sup>h</sup>*<sup>0</sup> *<sup>i</sup>* )(*h*0† *<sup>j</sup> <sup>ψ</sup>*<sup>0</sup>

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑)

*<sup>i</sup><sup>α</sup>* <sup>+</sup> *<sup>h</sup>*0† *<sup>i</sup> <sup>σ</sup>h*<sup>0</sup> *i* )  *<sup>j</sup>α*) + *c*.*c*.

. (B2)

*j b*† *<sup>i</sup>* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> −

, (B3)

<sup>2</sup> <sup>∑</sup><*i*,*j*<sup>&</sup>gt; *bibjb*†

*SU*(1)[*b*, *f* , *a*] =

*<sup>j</sup>*↓)(*fj*<sup>↑</sup> *fi*<sup>↓</sup> <sup>−</sup> *fj*<sup>↓</sup> *fi*↑).

*f* † *i*↑ *f* †

where Ψ<sup>0</sup>

*HSU*(2) *<sup>J</sup>* <sup>=</sup> *<sup>J</sup>*

*i* ≡

Realizing *hi* = *gih*<sup>0</sup>

<sup>4</sup> ∑ <*i*,*j*>

<sup>=</sup> <sup>−</sup> *<sup>J</sup>* <sup>2</sup> ∑ <*i*,*j*>

SU(2) action is rewritten,

 *β* 0 *dτ* ∑ *i h*0† *<sup>i</sup> ∂τh*<sup>0</sup> *<sup>i</sup>* + 1 <sup>2</sup> ∑ *i*,*α ψ*0† *<sup>i</sup><sup>α</sup> ∂τψ*<sup>0</sup>

− *J* <sup>2</sup> ∑ <*i*,*j*> *h*0 *iαh*<sup>0</sup> *jβh*0† *<sup>j</sup><sup>β</sup> <sup>h</sup>*0† *<sup>i</sup><sup>α</sup>* (*<sup>f</sup>* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

−*μ*∑ *i f* † *<sup>i</sup><sup>σ</sup> fiσ*(*b*†

− *J* 2 (*f* † *i*↓ *f* † *<sup>j</sup>*<sup>↑</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *f* †

= *J* 4 <sup>3</sup> ∑ *k*=1 (*f* † *iασ<sup>k</sup>*

= *J* 4 1 4 *tr*Ψ0† *<sup>i</sup>* <sup>Ψ</sup><sup>0</sup> *<sup>i</sup>* (*σk*)*<sup>T</sup>*

 *fi*<sup>↑</sup> *fi*<sup>↓</sup> *f* † *<sup>i</sup>*<sup>↓</sup> <sup>−</sup> *<sup>f</sup>* † *i*↑ *<sup>i</sup>* and Ψ*<sup>i</sup>* =

symmetric Heisenberg interaction term is given by

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

*<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*†

 *β* 0 *dτ* ∑ *i*,*α*=1,2

transpose of the Pauli matrices for *k* = 1, 2, 3.

(1 + *h*†

(1 + *h*†

*S*[*bα*, *fα*, *λi*] =

and (*f* † *iασ<sup>k</sup>*

*HSU*(2) *<sup>t</sup>*−*<sup>J</sup>* <sup>=</sup> <sup>−</sup> *<sup>t</sup>* <sup>2</sup> ∑ <*i*,*j*> (*f* † *<sup>i</sup><sup>α</sup> fjα*)(*b*† *<sup>j</sup>*1*bi*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*† *<sup>i</sup>*2*bj*2) + *c*.*c*. +(*fi*<sup>2</sup> *fj*<sup>1</sup> <sup>−</sup> *fi*<sup>1</sup> *fj*2)(*b*† *<sup>j</sup>*1*bi*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup>*1*bj*2) + *c*.*c*. − *J* <sup>2</sup> ∑ <*i*,*j*> (*<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1) (1 + *h*† *<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*† *<sup>j</sup> hj*) + (1 + *h*† *<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*† *<sup>j</sup> hj*) (*f* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1) − (1 + *h*† *<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*† *<sup>j</sup> hj*) (*<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1) + (1 + *h*† *<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*† *<sup>j</sup> hj*) − (1 + *h*† *<sup>i</sup> hi*)(<sup>1</sup> <sup>+</sup> *<sup>h</sup>*† *<sup>j</sup> hj*) <sup>×</sup> (*f* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1) − (*f* † *<sup>i</sup>*<sup>2</sup> *<sup>f</sup>* † *<sup>j</sup>*<sup>1</sup> <sup>−</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>j</sup>*2)(*fj*<sup>1</sup> *fi*<sup>2</sup> − *fj*<sup>2</sup> *fi*1) −*μ*∑ *i* (<sup>1</sup> <sup>−</sup> *<sup>h</sup>*† *<sup>i</sup> hi*) − ∑ *i <sup>i</sup>λ*(1) *<sup>i</sup>* (*<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup>*1*bi*2) + *<sup>i</sup>λ*(2) *<sup>i</sup>* (*fi*<sup>2</sup> *fi*<sup>1</sup> <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup>*2*bi*1) +*iλ*(3) *<sup>i</sup>* (*<sup>f</sup>* † *<sup>i</sup>*<sup>1</sup> *fi*<sup>1</sup> <sup>−</sup> *fi*<sup>2</sup> *<sup>f</sup>* † *<sup>i</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*† *<sup>i</sup>*1*bi*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*† *<sup>i</sup>*2*bi*2) . (B6)

#### **9. References**


**0**

**2**

Liang-Jian Zou and Feng Lu

*the Chinese Academy of Sciences People's Republic of China*

**Pairing Symmetry and Multiple Energy Gaps in**

Since the new high-Tc superconducting family based on iron pnictides was discovered Kamihara et al. (2008); Chen et al. (2008a;b), and the critical temperature was lifted to 56 K under high pressure Wu et al. (2009) which is considerably larger than the McMillan Limit McMillan (1968), the superconductive pairing mechanism and properties have attracted great interests experimentally and theoretically. These newly-found superconductors also provide a potential application prospect in two respects: on the one hand, simple components and rich resource of the FeAs-based compounds show the most possibility of large-scale applications; on the other hand, extremely large upper critical fields H*c*2Wen et al. (2008) in FeAs superconductors imply realistic applications in the near future. To find higher T*c* FeAs superconductors in further experiments and to improve their critical currrent density, it is essentially important to theoretically understand various normal and superconducting properties of iron pnictides, especially the superconducting pairing symmetry and its microscopic pairing mechanism. Once the pairing symmetry is known,

Soon after the finding of the superconductivity in LaFeAsO, the first-principles electronic structure calculations Mazin et al. (2008); Boeri et al. (2008); Cao et al. (2008); Ma et al. (2010) showed that Fe 3*d* orbits contribute the major spectral weight near the Fermi surface: the Fermi surface of LaFeAsO consists of two hole-type circles around the Γ point and two electron-type co-centered ellipses around the *M* point Zhang et al. (2009), indicating that the multi-orbital character is important in the FeAs superconductors. This is in a sharp contrast to the single-orbital character of high-T*<sup>C</sup>* cuprate. In these multi-orbital systems, one of the central problems is the superconducting pairing mechanism and its pairing symmetry. In fact, shortly after the determination of crystal structures of F-doped LaFeAsO, several authors proposed the constraint of the superconducting pairing symmetry of iron pnictides by applying the symmetric operations of the crystal point group on the pairing wavefunctions Mazin et al. (2008). However, such an analysis does not consider the Fermi surface topologies and the pairing potential of the superconductivity, hence can not solely determine the pairing

Meanwhile, based on microscopic spin fluctuation interactions in LaFeAsO, some authors proposed various superconducting pairing symmetries range from spin singlet *d*-wave Mu et al. (2008); Millo et al. (2008), *s*-wave Wang et al. (2009a;b); Nomura et al. (2008), the

many superconducting properties could be qualitatively understood.

symmetry in superconducting LaFeAsO1−*x*F*<sup>x</sup>* Lin et al. (2008).

**1. Introduction**

**Multi-Orbital Iron-Pnictide Superconductors**

*Key Laboratory of Materials Physics, Institute of Solid State Physics,*


## **Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors**

Liang-Jian Zou and Feng Lu

*Key Laboratory of Materials Physics, Institute of Solid State Physics, the Chinese Academy of Sciences People's Republic of China*

### **1. Introduction**

24 Will-be-set-by-IN-TECH

24 Superconductors – Properties, Technology, and Applications

[10] F. Ronning, C. Kim, D. L. Feng, D. S. Marshall, A. G. Loeser, L. L. Miller, J. N. Eckstein,

[13] C. T. Shih, T. K. Lee, T. Eder, C.-Y. Mou, and Y. C. Chen, Phys. Rev. Lett. 92, 227002

[14] M. Oda, R. Dipasupil, N. Momono, and M. Ido, J. Phys. Soc. Jpn. 69, 983 (2000); T. Nakano, N. Momono, M. Oda and M. Ido, J. Phys. Soc. Jpn. 67, 2622 (1998). [15] M. Oda, R. Dipasupil, N. Momono, M. Ido, J. Phys. Soc. Jpn. 69, 983 (2000)

[17] P. Dai, H. A. Mook, S. M. Hayden, G. Appeli, T. G. Perring, R. D. Hunt and F. Dogan,

[18] H. He, Y. Sidis, P. Bourges, G. D. Gu, A. Ivanov, N. Koshizuka, B. Liang, C. T. Lin, L. P. Regnault, E. Schoenherr and B. Keimer, Phys. Rev. Lett. 86, 1610 (2001); references

[19] C. Stock, R. A. Cowley, W. J. L. Buyers, R. Coldea, C. Broholm, C. D. Frost, R. J. Birgeneau, R. Liang, D. Bonn and W. N. Hardy, Phys. Rev. B 75, 172510 (2007). [20] S. Wakimoto, K. Yamada, J. M. Tranquada, C. D. Frost, R. J. Birgeneau and H. Zhang,

[23] J. Brinckmann and P. A. Lee, Phys. Rev. Lett. 82, 2915 (1999); J. Brinckmann and P. A.

[26] a) X. G. Wen and P. A. Lee, Phys. Rev. Lett. 76, 503 (1996); b) X. G. Wen and P. A. Lee, Phys. Rev. Lett. 80, 2193 (1998); c) P. A. Lee, N. Nagaosa, T.-K. Ng and X.-G. Wen, Phys.

charge <sup>−</sup>2*<sup>e</sup>* but not the holon pair of positive charge <sup>+</sup>2*<sup>e</sup>* at intersites *<sup>i</sup>* and *<sup>j</sup>*.

to the electron occupation number operator at site *i* and from the consideration of uniform electron removal and thus hole doping concentration *x*, we readily note that

zero, that is, when the two intersites or one of the two sites are vacant; *bib*†

[28] P. Nozières and D. Saint James, J. Physique, 43, 1133 (1982); references therein.

 *bib*† *<sup>i</sup> bjb*† *j* ≤ 1.

*<sup>i</sup>* represents the occupation number operator of the holon pair of negative

= 1 arises where an electron pair occupied at intersites *i* and *j*. Otherwise it is

*<sup>i</sup>* is equivalent

[21] J.-G. Eom, J.-H. Eom and Sung-Ho Suck Salk, Int. J. Mod. Phys. B, 21 3132 (2007).

[22] Seung-Joon Shin and Sung-Ho Salk, submitted to Phys. Rev. Lett..

[24] P. A. Lee and N. Nagaosa, Phys. Rev. B 46, 5621 (1992). [25] M. U. Ubbens and P. A. Lee, Phys. Rev. B 46, 8434 (1992).

= (<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)<sup>2</sup> with 0 <sup>≤</sup>

I.Bozovic, Z.-X. Shen, Science, 282, 2067 (1998).

[16] T. Timusk, B. Stattt, Rep. Prog. Phys. 62, 61 (1999).

Science 284, 1344 (1999); references there-in.

Phys. Rev. Lett. 98, 247003 (2007).

Lee, Phys. Rev. B 65, 014502 (2001).

Rev. B 57, 6003 (1998).

*j b*†

[27] Here *bibjb*†

*bibjb*† *j b*† *i* 

 *bib*† *<sup>i</sup> bjb*† *j* 

(2004); references there-in.

there-in.

[11] E. Gagliano, S. Bacci, and E. Dagotto, Phys. Rev. B 42 6222 (1990). [12] P. Prelovsek and A. Ram ˇ sak, Phys. Rev. B 72 012510 (2005). ˇ

> Since the new high-Tc superconducting family based on iron pnictides was discovered Kamihara et al. (2008); Chen et al. (2008a;b), and the critical temperature was lifted to 56 K under high pressure Wu et al. (2009) which is considerably larger than the McMillan Limit McMillan (1968), the superconductive pairing mechanism and properties have attracted great interests experimentally and theoretically. These newly-found superconductors also provide a potential application prospect in two respects: on the one hand, simple components and rich resource of the FeAs-based compounds show the most possibility of large-scale applications; on the other hand, extremely large upper critical fields H*c*2Wen et al. (2008) in FeAs superconductors imply realistic applications in the near future. To find higher T*c* FeAs superconductors in further experiments and to improve their critical currrent density, it is essentially important to theoretically understand various normal and superconducting properties of iron pnictides, especially the superconducting pairing symmetry and its microscopic pairing mechanism. Once the pairing symmetry is known, many superconducting properties could be qualitatively understood.

> Soon after the finding of the superconductivity in LaFeAsO, the first-principles electronic structure calculations Mazin et al. (2008); Boeri et al. (2008); Cao et al. (2008); Ma et al. (2010) showed that Fe 3*d* orbits contribute the major spectral weight near the Fermi surface: the Fermi surface of LaFeAsO consists of two hole-type circles around the Γ point and two electron-type co-centered ellipses around the *M* point Zhang et al. (2009), indicating that the multi-orbital character is important in the FeAs superconductors. This is in a sharp contrast to the single-orbital character of high-T*<sup>C</sup>* cuprate. In these multi-orbital systems, one of the central problems is the superconducting pairing mechanism and its pairing symmetry. In fact, shortly after the determination of crystal structures of F-doped LaFeAsO, several authors proposed the constraint of the superconducting pairing symmetry of iron pnictides by applying the symmetric operations of the crystal point group on the pairing wavefunctions Mazin et al. (2008). However, such an analysis does not consider the Fermi surface topologies and the pairing potential of the superconductivity, hence can not solely determine the pairing symmetry in superconducting LaFeAsO1−*x*F*<sup>x</sup>* Lin et al. (2008).

> Meanwhile, based on microscopic spin fluctuation interactions in LaFeAsO, some authors proposed various superconducting pairing symmetries range from spin singlet *d*-wave Mu et al. (2008); Millo et al. (2008), *s*-wave Wang et al. (2009a;b); Nomura et al. (2008), the

naturally thought to contribute the pairing glue of the superconducting electron pairs, due to the antiferromagnetic ground state in updoped FeAs compounds. Considering the multi-band and electron correlation characters, a minimal model for describing the low-energy physics of the FeAs-based superconductors is the two-orbital t-J model and its extension. Based on the band structures results and theoretical analysis, the twofold-degenerate d*xz*/d*yz* orbits are essential for the ironpnictide superconductors. We firstly depict such physical processes with

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 27

*<sup>i</sup>ασc*ˆ*jβσ* + ∑

�*ij*�*αβσ*

*<sup>i</sup>ασ* creates a *dxz* (*α*=1) or *dyz* (*α*=2) electron with orbit *α* and spin *σ* at site R*i*. t and

denotes the hopping integrals of the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) sites, respectively. U, U� and J*<sup>H</sup>* are the intra-orbital, inter-orbital Coulomb interactions

Eq.(1); meanwhile, even in the weak correlation regime in the atomic limit Manousakis et al.

strictly. Thus, we can describe the low-energy processes in iron pnictides with the two-orbital

on a quasi-two-dimensional square lattice. This Hamiltonian consists of the tight-binding

*εkxz* = −2(*t*<sup>1</sup> cos *kx* + *t*<sup>2</sup> cos *ky* + 2*t*<sup>3</sup> cos *kx* cos *ky*), *εkyz* = −2(*t*<sup>2</sup> cos *kx* + *t*<sup>1</sup> cos *ky* + 2*t*<sup>3</sup> cos *kx* cos *ky*),

The intra-orbital components of the nearest-neighbour (NN) hopping integrals t*αβ*

are measured in units of | *t*<sup>1</sup> |. The carrier concentration is equal to 0.18, which is a typical

doping concentration in the iron-based superconductors Dubroka et al. (2008).

� <sup>+</sup> *HJ*−*<sup>J</sup>*

*<sup>H</sup>* <sup>=</sup> *Ht*−*<sup>t</sup>*

*<sup>k</sup>*1*σck*<sup>1</sup>*<sup>σ</sup>* + (*εkyz* <sup>−</sup> *<sup>μ</sup>*)*c*†

*<sup>i</sup>α*↓*c*ˆ*iα*<sup>↓</sup> <sup>+</sup> *<sup>U</sup>*� ∑

*t* � *αβ ij c*ˆ † *<sup>i</sup>ασc*ˆ*jβσ*

*iσσ*� *c*ˆ † *<sup>i</sup>*1*σc*ˆ*i*1*σc*ˆ † *<sup>i</sup>*2*σ*� *c*ˆ*i*2*σ*�

<sup>↓</sup> (1)


� , (2)

*<sup>k</sup>*1*σck*<sup>2</sup>*<sup>σ</sup>* <sup>+</sup> *<sup>c</sup>*†

model can be derived from

model, although not

*<sup>k</sup>*2*σck*<sup>1</sup>*σ*)]

*ij* are


� . The kinetic energy term reads,

*<sup>k</sup>*2*σck*<sup>2</sup>*<sup>σ</sup>* <sup>+</sup> *<sup>ε</sup>kxy*(*c*†

*<sup>x</sup>* =t2 = 1.3. The components of next-nearest-neighbour (NNN) hopping

*ij* , are *t*<sup>3</sup> = *t*<sup>4</sup> = −0.85 Raghu et al. (2008). Throughout this paper, all the energies

the two-orbital Hubbard model,

where *c*†

and the Hund's coupling.

model,

kinetic energy *Ht*−*<sup>t</sup>*

*Ht*−*<sup>t</sup>*

with the notations

*<sup>x</sup>* =t1=-1, and t22

integrals, t�*αβ*

� = ∑ *kσ*

[(*εkxz* <sup>−</sup> *<sup>μ</sup>*)*c*†

t �

t-t� -J-J�

t 11 *<sup>H</sup>*<sup>ˆ</sup> <sup>=</sup> ∑ <*ij*>*αβσ*

> +*U* ∑ *iα c*ˆ † *<sup>i</sup>α*↑*c*ˆ*iα*↑*c*<sup>ˆ</sup> †

−*JH* ∑ *iσσ*� *c*ˆ † *<sup>i</sup>*1*σc*ˆ*i*1*σ*� *c*ˆ † *<sup>i</sup>*2*σ*�*c*ˆ*i*2*<sup>σ</sup>*

+*JH* ∑ *iα*�=*α*� *c*ˆ † *<sup>i</sup>α*↑*c*ˆ*iα*� ↑*c*ˆ † *<sup>i</sup>α*↓*c*ˆ*iα*�

In the strongly correlated regime, it is well known that the t-t�

(2008), the two-orbital Hubbard model in Eq.(1) can derive to the t-t�

� and the interaction part *HJ*−*<sup>J</sup>*

*εkxy* = −4*t*<sup>4</sup> sin *kx* sin *ky*,

*t αβ ij c*ˆ †

mixture of *Sx*<sup>2</sup>*y*<sup>2</sup> -wave and *dx*<sup>2</sup>−*y*<sup>2</sup> -wave Seo et al. (2008) to spin triplet *<sup>p</sup>*-wave Dai et al. (2008); Lee et al. (2008). These suggestions on the pairing symmetry raised critical and hot debates in the literature. A few authors focused on the microscopic origin of the superconducting pairing according to the antiferromagnetic spin fluctuation and the Fermi surface nesting topology through the characteristic wavevector **Q**=(*π*, 0) of the antiferromagnetic spin fluctuations. They proposed the s+−-wave pairing symmetry Mazin et al. (2008); Kuroki et al. (2008), *i.e.* the phase of the superconducting order parameters of the inner Fermi surface around the Γ point is antiphase to that of the Fermi surface around the M point. The s+−-wave symmetry of the superconducting order parameters seems to receive sufficient support in theory and experiment Mazin et al. (2008), and is consistent with the nesting picture of electron-type and hole-type Fermi surfaces in FeAs-based superconductors. However, the most recently found K*x*Fe2−*<sup>y</sup>*Se2 compounds clearly rule out the presence of a hole-type Fermi surface around the Γ point Xiang et al. (2011), suggesting that an alternative pairing symmetry is possible. Actually, from the researching history of the high-T*c* cuperates, we have known that a pairing mechanism based on the Fermi surface nesting is rather delicate, since any finite electron-electron interaction, which usually occurs in high-T*c* cuprates and iron pnictides, will destroy the perfect nesting of the Fermi surfaces. These disagreements and debates in the experimental data and theoretical results on the superconducting pairing symmetry of iron pnictides appeal for more efforts to unveil the mysterious nature of the superconducting iron pnictides.

On the other hand, the effect of the electron correlation in iron pnictides should be taken into account, since the bad metallic behavior and the existence of antiferromagnetic spin moments suggest that the iron pnictide is close to a metal insulator transition Haule et al. (2008). In this Chapter, starting with the minimal two-orbital t-t� -J-J� model Manousakis et al. (2008); Raghu et al. (2008), we develop a mean-field theory of the multi-orbital superconductors for the weak, intermediate, and strong correlation regimes, respectively. Taking a concrete t-t� -J-J� model which has the same topology as the Fermi surface and the band structures of LaFeAsO, we obtain the superconducting phase diagram, the quasiparticle spectra in normal state and superconducting phase, and the ARPES manifestation of the superconducting energy gaps. Our theory is applicable not only for FeAs superconductors, but also for ruthenate and heavy fermion, and other spin-fluctuation mediated multi-orbital superconductors. For realistic iron pnictides, we show that the pairing symmetry *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> -wave is stable in the reasonable parameters region; two superconducting gaps and their weak anisotropy and nodeless qualitatively agree with the observations in ARPES experiments. However, a quantitative comparison between theory and experiment shows a more elaborate theoretical model is necessary.

The rest of this Chapter is arranged as follows: in Sec.II we present the theory and methods for multi-orbital superconductivity; in Sec.III and IV we show the numerical results on the pairing symmetry of multi-orbital iron-pnictide superconductors, and the orbital dependence of superconducting energy gaps; Sec.V is devoted to the comparison between our theory and experimental observations, and finally we make a concluding remarks in Sec.VI.

#### **2. Theory and methods of multi-orbital superconductivity**

For the iron-pnictide compounds, the electron-phonon coupling seems to be irrelevant to the superconducting pairing origin Boeri et al. (2008), the antiferromagnetic spin fluctuation is naturally thought to contribute the pairing glue of the superconducting electron pairs, due to the antiferromagnetic ground state in updoped FeAs compounds. Considering the multi-band and electron correlation characters, a minimal model for describing the low-energy physics of the FeAs-based superconductors is the two-orbital t-J model and its extension. Based on the band structures results and theoretical analysis, the twofold-degenerate d*xz*/d*yz* orbits are essential for the ironpnictide superconductors. We firstly depict such physical processes with the two-orbital Hubbard model,

$$\begin{split} \hat{H} &= \sum\_{\alpha\beta\sigma\prime} t\_{ij}^{\alpha\beta} \hat{c}\_{i\alpha\sigma}^{\dagger} \hat{c}\_{j\beta\sigma} + \sum\_{\alpha\beta\sigma\prime} t\_{ij}^{\alpha\beta} \hat{c}\_{i\alpha\sigma}^{\dagger} \hat{c}\_{j\beta\sigma} \\ &+ \mathcal{U} \sum\_{i\alpha} \hat{c}\_{i\alpha\uparrow}^{\dagger} \hat{c}\_{i\alpha\uparrow} \hat{c}\_{i\alpha\downarrow}^{\dagger} \hat{c}\_{i\alpha\downarrow} + \mathcal{U}' \sum\_{i\sigma\prime\prime} \hat{c}\_{i1\sigma}^{\dagger} \hat{c}\_{i1\sigma} \hat{c}\_{i2\sigma\prime}^{\dagger} \hat{c}\_{i2\sigma\prime} \\ &- \mathcal{H} \sum\_{i\sigma\prime\prime} \hat{c}\_{i1\sigma}^{\dagger} \hat{c}\_{i1\sigma\prime} \hat{c}\_{i2\sigma\prime}^{\dagger} \hat{c}\_{i2\sigma} \\ &+ \mathcal{H} \sum\_{i\alpha\uparrow\not\equiv \alpha\prime} \hat{c}\_{i\alpha\uparrow}^{\dagger} \hat{c}\_{i\alpha\uparrow} \hat{c}\_{i\alpha\downarrow}^{\dagger} \hat{c}\_{i\alpha\downarrow} \end{split} \tag{1}$$

where *c*† *<sup>i</sup>ασ* creates a *dxz* (*α*=1) or *dyz* (*α*=2) electron with orbit *α* and spin *σ* at site R*i*. t and t � denotes the hopping integrals of the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) sites, respectively. U, U� and J*<sup>H</sup>* are the intra-orbital, inter-orbital Coulomb interactions and the Hund's coupling.

In the strongly correlated regime, it is well known that the t-t� -J-J� model can be derived from Eq.(1); meanwhile, even in the weak correlation regime in the atomic limit Manousakis et al. (2008), the two-orbital Hubbard model in Eq.(1) can derive to the t-t� -J-J� model, although not strictly. Thus, we can describe the low-energy processes in iron pnictides with the two-orbital t-t� -J-J� model,

$$H = H\_{t-t'} + H\_{I-I'} \tag{2}$$

on a quasi-two-dimensional square lattice. This Hamiltonian consists of the tight-binding kinetic energy *Ht*−*<sup>t</sup>* � and the interaction part *HJ*−*<sup>J</sup>* � . The kinetic energy term reads,

$$H\_{\mathfrak{t}-\mathfrak{t}'} = \sum\_{k\sigma} [ (\varepsilon\_{k\mathbf{x}\mathbf{z}} - \mu) c\_{k1\sigma}^{\dagger} c\_{k1\sigma} + (\varepsilon\_{k\mathbf{y}\mathbf{z}} - \mu) c\_{k2\sigma}^{\dagger} c\_{k2\sigma} + \varepsilon\_{k\mathbf{x}\mathbf{y}} (c\_{k1\sigma}^{\dagger} c\_{k2\sigma} + c\_{k2\sigma}^{\dagger} c\_{k1\sigma}) ]$$

with the notations

2 Will-be-set-by-IN-TECH

mixture of *Sx*<sup>2</sup>*y*<sup>2</sup> -wave and *dx*<sup>2</sup>−*y*<sup>2</sup> -wave Seo et al. (2008) to spin triplet *<sup>p</sup>*-wave Dai et al. (2008); Lee et al. (2008). These suggestions on the pairing symmetry raised critical and hot debates in the literature. A few authors focused on the microscopic origin of the superconducting pairing according to the antiferromagnetic spin fluctuation and the Fermi surface nesting topology through the characteristic wavevector **Q**=(*π*, 0) of the antiferromagnetic spin fluctuations. They proposed the s+−-wave pairing symmetry Mazin et al. (2008); Kuroki et al. (2008), *i.e.* the phase of the superconducting order parameters of the inner Fermi surface around the Γ point is antiphase to that of the Fermi surface around the M point. The s+−-wave symmetry of the superconducting order parameters seems to receive sufficient support in theory and experiment Mazin et al. (2008), and is consistent with the nesting picture of electron-type and hole-type Fermi surfaces in FeAs-based superconductors. However, the most recently found K*x*Fe2−*<sup>y</sup>*Se2 compounds clearly rule out the presence of a hole-type Fermi surface around the Γ point Xiang et al. (2011), suggesting that an alternative pairing symmetry is possible. Actually, from the researching history of the high-T*c* cuperates, we have known that a pairing mechanism based on the Fermi surface nesting is rather delicate, since any finite electron-electron interaction, which usually occurs in high-T*c* cuprates and iron pnictides, will destroy the perfect nesting of the Fermi surfaces. These disagreements and debates in the experimental data and theoretical results on the superconducting pairing symmetry of iron pnictides appeal for more efforts to unveil the mysterious nature of the superconducting

On the other hand, the effect of the electron correlation in iron pnictides should be taken into account, since the bad metallic behavior and the existence of antiferromagnetic spin moments suggest that the iron pnictide is close to a metal insulator transition Haule et al. (2008). In

Raghu et al. (2008), we develop a mean-field theory of the multi-orbital superconductors for the weak, intermediate, and strong correlation regimes, respectively. Taking a concrete t-t�

model which has the same topology as the Fermi surface and the band structures of LaFeAsO, we obtain the superconducting phase diagram, the quasiparticle spectra in normal state and superconducting phase, and the ARPES manifestation of the superconducting energy gaps. Our theory is applicable not only for FeAs superconductors, but also for ruthenate and heavy fermion, and other spin-fluctuation mediated multi-orbital superconductors. For realistic iron pnictides, we show that the pairing symmetry *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> -wave is stable in the reasonable parameters region; two superconducting gaps and their weak anisotropy and nodeless qualitatively agree with the observations in ARPES experiments. However, a quantitative comparison between theory and experiment shows a more elaborate theoretical

The rest of this Chapter is arranged as follows: in Sec.II we present the theory and methods for multi-orbital superconductivity; in Sec.III and IV we show the numerical results on the pairing symmetry of multi-orbital iron-pnictide superconductors, and the orbital dependence of superconducting energy gaps; Sec.V is devoted to the comparison between our theory and

For the iron-pnictide compounds, the electron-phonon coupling seems to be irrelevant to the superconducting pairing origin Boeri et al. (2008), the antiferromagnetic spin fluctuation is

experimental observations, and finally we make a concluding remarks in Sec.VI.

**2. Theory and methods of multi-orbital superconductivity**


model Manousakis et al. (2008);


this Chapter, starting with the minimal two-orbital t-t�

iron pnictides.

model is necessary.

$$\begin{aligned} \varepsilon\_{kxz} &= -2(t\_1 \cos k\_x + t\_2 \cos k\_y + 2t\_3 \cos k\_x \cos k\_y), \\ \varepsilon\_{kyz} &= -2(t\_2 \cos k\_x + t\_1 \cos k\_y + 2t\_3 \cos k\_x \cos k\_y), \\ \varepsilon\_{kxy} &= -4t\_4 \sin k\_x \sin k\_y. \end{aligned}$$

The intra-orbital components of the nearest-neighbour (NN) hopping integrals t*αβ ij* are t 11 *<sup>x</sup>* =t1=-1, and t22 *<sup>x</sup>* =t2 = 1.3. The components of next-nearest-neighbour (NNN) hopping integrals, t�*αβ ij* , are *t*<sup>3</sup> = *t*<sup>4</sup> = −0.85 Raghu et al. (2008). Throughout this paper, all the energies are measured in units of | *t*<sup>1</sup> |. The carrier concentration is equal to 0.18, which is a typical doping concentration in the iron-based superconductors Dubroka et al. (2008).

*HMF* <sup>=</sup> <sup>∑</sup>*<sup>k</sup>* <sup>Ψ</sup>(*k*)†*A*(*k*)Ψ(*k*)+*const*. Here <sup>Ψ</sup>(*k*)=(*ck*,1,↑, *<sup>c</sup>*†

⎛

*�k*1<sup>↑</sup> − *<sup>μ</sup>* <sup>Δ</sup><sup>∗</sup>

⎜⎜⎜⎝

*<sup>ε</sup>k*1*<sup>σ</sup>* <sup>=</sup> *<sup>ε</sup>kxz* <sup>−</sup> <sup>2</sup>*J*(*P*<sup>1</sup>

*<sup>ε</sup>k*2*<sup>σ</sup>* <sup>=</sup> *<sup>ε</sup>kyz* <sup>−</sup> <sup>2</sup>*J*(*P*<sup>2</sup>

*<sup>ε</sup>k*<sup>12</sup> <sup>=</sup> *<sup>ε</sup>kxy* <sup>−</sup> <sup>2</sup>*J*(*P*<sup>1</sup>

−4*J* � (*P*<sup>3</sup>

<sup>Δ</sup>*α*(*k*) = <sup>−</sup>4(Δ1*<sup>α</sup>*

<sup>−</sup>4(Δ2*<sup>α</sup>*

<sup>=</sup> <sup>−</sup>4Δ1*<sup>α</sup>*

cos(*ky*)) <sup>+</sup> *ξα*(cos(*kx* <sup>+</sup> *ky*) <sup>±</sup> *<sup>η</sup>*2*<sup>α</sup>* cos(*kx* <sup>−</sup> *ky*)). Where *ξα* <sup>=</sup> <sup>Δ</sup>2*<sup>α</sup>*

*<sup>x</sup>*+*y*.

*<sup>x</sup>*−*y*/Δ2*<sup>α</sup>*

is

*<sup>η</sup>*1*α*=<sup>|</sup> <sup>Δ</sup>1*<sup>α</sup>*

respect to Δ1*<sup>α</sup>*

*<sup>y</sup>* /Δ1*<sup>α</sup>*

*<sup>x</sup>* <sup>|</sup> and *<sup>η</sup>*2*α*=<sup>|</sup> <sup>Δ</sup>2*<sup>α</sup>*

*<sup>x</sup>*−*<sup>y</sup>* to <sup>Δ</sup>2*<sup>α</sup>*

*<sup>x</sup>* or Δ2*<sup>α</sup>*

*dxy*-wave symmetry at *η* = 1.

−4(*J* + *J* �

−4(*J* + *J* �

The modified intra-orbital and inter-orbital kinetic energy reads,

*A*(*k*) =

−*k*,1,↓, *ck*,2,↑, *<sup>c</sup>*†

<sup>2</sup>(*k*)

� *P*1

� *P*2

*xy* cos(*kx* − *ky*)).

*<sup>x</sup>*−*<sup>y</sup>* cos(*kx* − *ky*))

*x*+*y*/Δ1*<sup>α</sup>*

*<sup>x</sup>* ; the anisotropic factors,

*<sup>y</sup>* with

+*ξα*(cos(*kx* + *ky*) ± *η*2*<sup>α</sup>* cos(*kx* − *ky*))] (7)

*<sup>x</sup>*+*<sup>y</sup>* <sup>|</sup>, are positive. <sup>±</sup> denotes the relative phase of <sup>Δ</sup>1*<sup>α</sup>*

<sup>1</sup>(*k*) *�k*<sup>12</sup> 0

*<sup>y</sup>* cos *ky*) − 4*J*

*<sup>y</sup>* cos *ky*) − 4*J*

*xy* cos *ky*)

*<sup>y</sup>* cos *ky*)

<sup>Δ</sup>1(*k*) −*�k*1<sup>↓</sup> + *<sup>μ</sup>* <sup>0</sup> −*�k*<sup>12</sup> *�k*<sup>12</sup> <sup>0</sup> *�k*2<sup>↑</sup> − *<sup>μ</sup>* <sup>Δ</sup><sup>∗</sup>

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 29

and *const* is the collection of all the constant energy terms from the mean-field decoupling.

)(< *n*1*<sup>σ</sup>* > + < *n*2*<sup>σ</sup>* >)

)(< *n*1*<sup>σ</sup>* > + < *n*2*<sup>σ</sup>* >)

The superconducting order parameter Δ*α*(*k*) of each orbital channel in the momentum space

*<sup>x</sup>*+*<sup>y</sup>* cos(*kx* + *ky*) + <sup>Δ</sup>2*<sup>α</sup>*

*<sup>x</sup>* [(cos(*kx*) ± *η*1*<sup>α</sup>* cos(*ky*))

Thus the superconducting pairing symmetry of the *α*-orbit is determined by (cos(*kx*) ± *η*1*<sup>α</sup>*

To characterize the complicated superconducting order parameters in different parameter regions, we define the *Sx*<sup>2</sup>+*ηy*<sup>2</sup> -wave or *dx*<sup>2</sup>−*ηy*<sup>2</sup> -wave as the pairing symmetry when <sup>Δ</sup>*<sup>α</sup>* <sup>∝</sup> *cos*(*kx*) + *ηcos*(*ky*) or Δ*<sup>α</sup>* ∝ *cos*(*kx*) − *ηcos*(*ky*). It reduces to the conventional *Sx*<sup>2</sup>+*y*<sup>2</sup> -wave or *dx*<sup>2</sup>−*y*<sup>2</sup> -wave symmetry at *<sup>η</sup>* <sup>=</sup> 1. We also define the *<sup>S</sup>ηx*2*y*<sup>2</sup> -wave or *<sup>d</sup>ηxy*-wave as the superconducting pairing symmetry when Δ*<sup>α</sup>* ∝ *cos*(*kx* + *ky*) + *ηcos*(*kx* − *ky*) or Δ*<sup>α</sup>* ∝ *cos*(*kx* + *ky*) − *ηcos*(*kx* − *ky*)). In this situation, it reduces to the familiar *Sx*<sup>2</sup>*y*<sup>2</sup> -wave or

Diagonalizing the matrix *A*(*k*) by an unitary transformation U(k), *U*(*k*)†*A*(*k*)*U*(*k*), and minimizing the free energy of the system with respect to these parameters in Eq.(5-7), one

*<sup>x</sup>* cos *kx* + *<sup>P</sup>*<sup>1</sup>

*<sup>x</sup>* cos *kx* + *<sup>P</sup>*<sup>2</sup>

*xy* cos *kx* + *<sup>P</sup>*<sup>2</sup>

*xy* cos(*kx* + *ky*) + *<sup>P</sup>*<sup>4</sup>

*<sup>x</sup>* cos *kx* + <sup>Δ</sup>1*<sup>α</sup>*

<sup>0</sup> −*�k*<sup>12</sup> <sup>Δ</sup>2(*k*) −*�k*2<sup>↓</sup> + *<sup>μ</sup>*

−*k*,2,↓),

⎟⎟⎟⎠ , (5)

(6)

⎞

<sup>3</sup> cos *kx* cos *ky*

<sup>3</sup> cos *kx* cos *ky*

The interaction term reads,

$$\mathrm{i}\ H\_{\mathrm{I}-\mathrm{f}'} = \mathrm{I} \sum\_{\mathrm{a}\mathfrak{F}} (\vec{\mathcal{S}}\_{\mathrm{ia}} \cdot \vec{\mathcal{S}}\_{\mathrm{j}\mathfrak{F}} - \frac{1}{\mathbf{4}} n\_{\mathrm{ia}} \cdot n\_{\mathrm{j}\mathfrak{F}}) + \mathrm{f}' \sum\_{\mathrm{a}\mathfrak{F}} (\vec{\mathcal{S}}\_{\mathrm{ia}} \cdot \vec{\mathcal{S}}\_{\mathrm{j}\mathfrak{F}} - \frac{1}{\mathbf{4}} n\_{\mathrm{ia}} \cdot n\_{\mathrm{j}\mathfrak{F}}) \tag{3}$$

which contains a NN and a NNN antiferromagnetic spin couplings. Here *J* and *J* � are the NN and the NNN spin coupling strengths, respectively. � *Si<sup>α</sup>* is the spin operator of the electron in the *α*-orbit at R*<sup>i</sup>* and *ni<sup>α</sup>* is the particle number operator. *α*, *β*(=1,2) are orbital index.

To explore the essence of the iron-pnictide superconductors and other multi-orbital superconductors, we discuss the t-t� -J-J� model in three different correlation regimes:

#### **2.1 Weak correlation regime**

When the kinetic energy of the d*xz*- and d*yz*-electrons is much larger than the Coulomb interaction, we adopt the conventional mean-field decoupling approach to study the superconducting pairing symmetry and its orbital dependence. This ansatz is applicable for many FeAs-based and other multi-orbital superconductors with metallic ground states.

Notice that the d*xz* and d*yz* orbits are spatially anisotropic, in other words, the intra-orbital hopping integral along the x-direction is not equal to that along the y-direction for each orbital, as one can see from | *t*<sup>1</sup> |�=| *t*<sup>2</sup> |. Due to the asymmetry of the different directions in different orbits, the amplitude of the superconducting gap of the local pairing along the x-direction may be not equal to that along the y-direction in each orbit. Thus, the single orbital d-wave or s-wave superconducting order parameter, in which the superconducting energy gap has 4-fold symmetry of rotational invariance in the xy plane, is not suitable for describing the pairing symmetry of the intra-orbital superconducting order parameters in this multi-orbital system. Considering all of the possible kinetic correlations and the superconducting pairings for the NN and NNN sites along different directions, we introduce the following order parameters,

$$P\_{x'y'}^{\mathfrak{a}} = <\epsilon\_{i\alpha\sigma}^{\dagger} c\_{j\alpha\sigma} > \prime \quad (j = i \pm \mathfrak{x}/\mathfrak{j})$$

$$P\_{xy}^{1/2} = <\epsilon\_{i1\sigma}^{\dagger} c\_{j2\sigma} > \prime \quad (j = i \pm \mathfrak{x}/\mathfrak{j})$$

$$P\_3^{\mathfrak{a}} = <\epsilon\_{i\alpha\sigma}^{\dagger} c\_{j\alpha\sigma} > \prime \quad (j = i \pm (\mathfrak{x} \pm \mathfrak{j}))$$

$$P\_{xy}^{3/4} = <\epsilon\_{i1\sigma}^{\dagger} c\_{j2\sigma} > \prime \quad (j = i \pm (\mathfrak{x} \pm \mathfrak{j}))$$

$$\Delta\_{x'y}^{1\mathfrak{a}} = j <\epsilon\_{i\alpha\uparrow}^{\dagger} c\_{j\alpha\downarrow}^{\dagger} > \prime \quad (j = i \pm \mathfrak{x}/\mathfrak{j})$$

$$\Delta\_{x\pm y}^{2\mathfrak{a}} = j' <\epsilon\_{i\alpha\uparrow}^{\dagger} c\_{j\alpha\downarrow}^{\dagger} > \prime \quad (j = i \pm (\mathfrak{x} \pm \mathfrak{j})). \tag{4}$$

Here *P<sup>α</sup> <sup>x</sup>*/*<sup>y</sup>* and *<sup>P</sup>*1/2 *xy* (*P<sup>α</sup>* <sup>3</sup> and *<sup>P</sup>*3/4 *xy* ) are the kinetic average of the NN (NNN) intra-orbital and inter-orbital hopping integrals. These terms could be decoupled within the framework of the mean-field approximationSeo et al. (2008). Δ1*<sup>α</sup> <sup>x</sup>*/*<sup>y</sup>* (Δ2*<sup>α</sup> <sup>x</sup>*±*y*) is the mean-field amplitude of the local NN (NNN) pairing order parameter in the *α*-orbit. The inter-orbital pairing parameter < *c*† *<sup>i</sup>*1↑*c*† *<sup>j</sup>*2<sup>↓</sup> <sup>&</sup>gt; is very small, hence is neglected Seo et al. (2008).

With these parameters, one can decouple the interaction terms in Eq.(3) within the framework of the self-consistent mean-field approximation, and obtain the mean-field Hamiltonian,

*HMF* <sup>=</sup> <sup>∑</sup>*<sup>k</sup>* <sup>Ψ</sup>(*k*)†*A*(*k*)Ψ(*k*)+*const*. Here <sup>Ψ</sup>(*k*)=(*ck*,1,↑, *<sup>c</sup>*† −*k*,1,↓, *ck*,2,↑, *<sup>c</sup>*† −*k*,2,↓),

4 Will-be-set-by-IN-TECH

*ni<sup>α</sup>* · *njβ*) + *J*

To explore the essence of the iron-pnictide superconductors and other multi-orbital

When the kinetic energy of the d*xz*- and d*yz*-electrons is much larger than the Coulomb interaction, we adopt the conventional mean-field decoupling approach to study the superconducting pairing symmetry and its orbital dependence. This ansatz is applicable for many FeAs-based and other multi-orbital superconductors with metallic ground states.

Notice that the d*xz* and d*yz* orbits are spatially anisotropic, in other words, the intra-orbital hopping integral along the x-direction is not equal to that along the y-direction for each orbital, as one can see from | *t*<sup>1</sup> |�=| *t*<sup>2</sup> |. Due to the asymmetry of the different directions in different orbits, the amplitude of the superconducting gap of the local pairing along the x-direction may be not equal to that along the y-direction in each orbit. Thus, the single orbital d-wave or s-wave superconducting order parameter, in which the superconducting energy gap has 4-fold symmetry of rotational invariance in the xy plane, is not suitable for describing the pairing symmetry of the intra-orbital superconducting order parameters in this multi-orbital system. Considering all of the possible kinetic correlations and the superconducting pairings for the NN and NNN sites along different directions, we introduce the following order

*<sup>i</sup>ασcjασ* >, (*j* = *i* ± *x*ˆ/*y*ˆ)

*<sup>i</sup>*1*σcj*<sup>2</sup>*<sup>σ</sup>* >, (*j* = *i* ± *x*ˆ/*y*ˆ)

*<sup>i</sup>ασcjασ* >, (*j* = *i* ± (*x*ˆ ± *y*ˆ))

*<sup>i</sup>*1*σcj*<sup>2</sup>*<sup>σ</sup>* >, (*j* = *i* ± (*x*ˆ ± *y*ˆ))

*<sup>j</sup>α*<sup>↓</sup> <sup>&</sup>gt;, (*<sup>j</sup>* <sup>=</sup> *<sup>i</sup>* <sup>±</sup> *<sup>x</sup>*ˆ/*y*ˆ)

*<sup>x</sup>*/*<sup>y</sup>* (Δ2*<sup>α</sup>*

*<sup>j</sup>α*<sup>↓</sup> <sup>&</sup>gt;, (*<sup>j</sup>* <sup>=</sup> *<sup>i</sup>* <sup>±</sup> (*x*<sup>ˆ</sup> <sup>±</sup> *<sup>y</sup>*ˆ)). (4)

*<sup>x</sup>*±*y*) is the mean-field amplitude of the

*xy* ) are the kinetic average of the NN (NNN) intra-orbital and

�

∑ <<*ij*>>*αβ*

(� *Si<sup>α</sup>* · �

model in three different correlation regimes:

*Sj<sup>β</sup>* <sup>−</sup> <sup>1</sup> 4

*Si<sup>α</sup>* is the spin operator of the electron in

*ni<sup>α</sup>* · *njβ*) (3)

are the NN

�

The interaction term reads,

� = *J* ∑ <*ij*>*αβ*

superconductors, we discuss the t-t�

**2.1 Weak correlation regime**

parameters,

Here *P<sup>α</sup>*

< *c*† *<sup>i</sup>*1↑*c*†

*<sup>x</sup>*/*<sup>y</sup>* and *<sup>P</sup>*1/2

*xy* (*P<sup>α</sup>*

(� *Si<sup>α</sup>* · �

and the NNN spin coupling strengths, respectively. �

*Pα*

*P*1/2 *xy* = < *<sup>c</sup>*†

*P*3/4 *xy* = < *<sup>c</sup>*†

Δ1*<sup>α</sup>*

Δ2*<sup>α</sup> <sup>x</sup>*±*<sup>y</sup>* <sup>=</sup> *<sup>J</sup>* � < *c*† *<sup>i</sup>α*↑*c*†

<sup>3</sup> and *<sup>P</sup>*3/4

the mean-field approximationSeo et al. (2008). Δ1*<sup>α</sup>*

*Pα* <sup>3</sup> <sup>=</sup> <sup>&</sup>lt; *<sup>c</sup>*†

*<sup>x</sup>*/*<sup>y</sup>* <sup>=</sup> <sup>&</sup>lt; *<sup>c</sup>*†

*<sup>x</sup>*/*<sup>y</sup>* <sup>=</sup> *<sup>J</sup>* <sup>&</sup>lt; *<sup>c</sup>*†

*<sup>j</sup>*2<sup>↓</sup> <sup>&</sup>gt; is very small, hence is neglected Seo et al. (2008).

*<sup>i</sup>α*↑*c*†

inter-orbital hopping integrals. These terms could be decoupled within the framework of

local NN (NNN) pairing order parameter in the *α*-orbit. The inter-orbital pairing parameter

With these parameters, one can decouple the interaction terms in Eq.(3) within the framework of the self-consistent mean-field approximation, and obtain the mean-field Hamiltonian,

*Sj<sup>β</sup>* <sup>−</sup> <sup>1</sup> 4

which contains a NN and a NNN antiferromagnetic spin couplings. Here *J* and *J*


the *α*-orbit at R*<sup>i</sup>* and *ni<sup>α</sup>* is the particle number operator. *α*, *β*(=1,2) are orbital index.

*HJ*−*<sup>J</sup>*

$$A(k) = \begin{pmatrix} \varepsilon\_{k1\uparrow} - \mu & \Delta\_1^\*(k) & \varepsilon\_{k12} & 0\\ \Delta\_1(k) & -\varepsilon\_{k1\downarrow} + \mu & 0 & -\varepsilon\_{k12} \\ \varepsilon\_{k12} & 0 & \varepsilon\_{k2\uparrow} - \mu & \Delta\_2^\*(k) \\ 0 & -\varepsilon\_{k12} & \Delta\_2(k) & -\varepsilon\_{k2\downarrow} + \mu \end{pmatrix} \tag{5}$$

and *const* is the collection of all the constant energy terms from the mean-field decoupling. The modified intra-orbital and inter-orbital kinetic energy reads,

$$\begin{aligned} \varepsilon\_{k1\sigma} &= \varepsilon\_{kxz} - 2f(P\_x^1 \cos k\_x + P\_y^1 \cos k\_y) - 4f'P\_3^1 \cos k\_x \cos k\_y \\ &- 4(f+f')(+) \\ \varepsilon\_{k2\sigma} &= \varepsilon\_{kyz} - 2f(P\_x^2 \cos k\_x + P\_y^2 \cos k\_y) - 4f'P\_3^2 \cos k\_x \cos k\_y \end{aligned}$$

$$\begin{aligned} -4(f+f')(+) \\ \varepsilon\_{k12} &= \varepsilon\_{kxy} - 2f(P\_{xy}^1 \cos k\_x + P\_{xy}^2 \cos k\_y) \\ &- 4f'(P\_{xy}^3 \cos(k\_x + k\_y) + P\_{xy}^4 \cos(k\_x - k\_y)). \end{aligned} \tag{6}$$

The superconducting order parameter Δ*α*(*k*) of each orbital channel in the momentum space is

$$\begin{aligned} \Delta\_{\mathbb{R}}(k) &= -4(\Delta\_{\text{x}}^{1a}\cos k\_{\text{x}} + \Delta\_{\text{y}}^{1a}\cos k\_{\text{y}}) \\ &- 4(\Delta\_{\text{x}+\text{y}}^{2a}\cos(k\_{\text{x}} + k\_{\text{y}}) + \Delta\_{\text{x}-\text{y}}^{2a}\cos(k\_{\text{x}} - k\_{\text{y}})) \\ &= -4\Delta\_{\text{x}}^{1a}[(\cos(k\_{\text{x}}) \pm \eta\_{1a}\cos(k\_{\text{y}})) \\ &+ \xi\_{\text{z}}(\cos(k\_{\text{x}} + k\_{\text{y}}) \pm \eta\_{2a}\cos(k\_{\text{x}} - k\_{\text{y}}))] \end{aligned} \tag{7}$$

Thus the superconducting pairing symmetry of the *α*-orbit is determined by (cos(*kx*) ± *η*1*<sup>α</sup>* cos(*ky*)) <sup>+</sup> *ξα*(cos(*kx* <sup>+</sup> *ky*) <sup>±</sup> *<sup>η</sup>*2*<sup>α</sup>* cos(*kx* <sup>−</sup> *ky*)). Where *ξα* <sup>=</sup> <sup>Δ</sup>2*<sup>α</sup> x*+*y*/Δ1*<sup>α</sup> <sup>x</sup>* ; the anisotropic factors, *<sup>η</sup>*1*α*=<sup>|</sup> <sup>Δ</sup>1*<sup>α</sup> <sup>y</sup>* /Δ1*<sup>α</sup> <sup>x</sup>* <sup>|</sup> and *<sup>η</sup>*2*α*=<sup>|</sup> <sup>Δ</sup>2*<sup>α</sup> <sup>x</sup>*−*y*/Δ2*<sup>α</sup> <sup>x</sup>*+*<sup>y</sup>* <sup>|</sup>, are positive. <sup>±</sup> denotes the relative phase of <sup>Δ</sup>1*<sup>α</sup> <sup>y</sup>* with respect to Δ1*<sup>α</sup> <sup>x</sup>* or Δ2*<sup>α</sup> <sup>x</sup>*−*<sup>y</sup>* to <sup>Δ</sup>2*<sup>α</sup> <sup>x</sup>*+*y*.

To characterize the complicated superconducting order parameters in different parameter regions, we define the *Sx*<sup>2</sup>+*ηy*<sup>2</sup> -wave or *dx*<sup>2</sup>−*ηy*<sup>2</sup> -wave as the pairing symmetry when <sup>Δ</sup>*<sup>α</sup>* <sup>∝</sup> *cos*(*kx*) + *ηcos*(*ky*) or Δ*<sup>α</sup>* ∝ *cos*(*kx*) − *ηcos*(*ky*). It reduces to the conventional *Sx*<sup>2</sup>+*y*<sup>2</sup> -wave or *dx*<sup>2</sup>−*y*<sup>2</sup> -wave symmetry at *<sup>η</sup>* <sup>=</sup> 1. We also define the *<sup>S</sup>ηx*2*y*<sup>2</sup> -wave or *<sup>d</sup>ηxy*-wave as the superconducting pairing symmetry when Δ*<sup>α</sup>* ∝ *cos*(*kx* + *ky*) + *ηcos*(*kx* − *ky*) or Δ*<sup>α</sup>* ∝ *cos*(*kx* + *ky*) − *ηcos*(*kx* − *ky*)). In this situation, it reduces to the familiar *Sx*<sup>2</sup>*y*<sup>2</sup> -wave or *dxy*-wave symmetry at *η* = 1.

Diagonalizing the matrix *A*(*k*) by an unitary transformation U(k), *U*(*k*)†*A*(*k*)*U*(*k*), and minimizing the free energy of the system with respect to these parameters in Eq.(5-7), one

Providing 1/T1*<sup>N</sup>* in the normal state satisfies the Korringa law, the spin lattice relaxation rate

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 31

When the kinetic energy of the conduction bands becomes small and is comparable with the Coulomb interaction, we need to consider the electronic correlation effect, as one sees in FeTe1−*x*Se*<sup>x</sup>* and other superconductors. We utilize the Kotliar-Ruckenstein's slave boson approach for some FeAs-based and ruthenate superconductor with intermediate magnetic moments. To reflect the multi-orbital character of iron pnictides, we need to extend the single-orbital Kotliar-Ruckenstein's slave boson approach Kotliar et al. (1988) to the two-orbital Hubbard models for various configurations. In the multiorbital Hubbard model, a few of auxiliary boson field operators representing the possibilities of various electron occupations are introduced, such as *e*, *p*, *d*, *b*,*t*, *q*, which denote the possibilities of none, single, double, triplicate, quaternity occupations. With these auxiliary boson fields, an original

*<sup>i</sup><sup>α</sup> piασ* + ∑

*<sup>i</sup>ασ* is the new slaved fermion operator and *Qiασ* is an auxiliary particle number operator Kotliar et al. (1988). Projecting the original fermion operators into these boson field and fermion field operators, one could not only obtain an effective Hamiltonian, but also get the groundstate energy in the saddle point approximation with the normalization condition and the fermion number constraints Kotliar et al. (1988). Here we employ a generalized Lagrange multiplier method to enforce these constraint conditions, thus the interorbital hoppings and crystal field splitting can be treated on the same foot. The fermion occupation number is constrained with the penalty function method. To enforce the normalization condition, we

*σ*� *d*† *iσασ*� *β*

(<sup>1</sup> <sup>−</sup> *Qiασ*)<sup>−</sup> <sup>1</sup>

*ασσ*� <sup>+</sup> <sup>∑</sup>*ασ*

*t* 2

*piβσ*� + *t* † *<sup>i</sup>ασbi<sup>β</sup>*

<sup>2</sup> *f* †

*<sup>i</sup>ασ* (10)

*ασ* + *<sup>q</sup>*<sup>2</sup> (11)


model

1/*T*1*<sup>s</sup>* becomes Xiang (2007): 1/*T*1*<sup>s</sup>* ∝ (*kBT*) · *T*1*N*/*T*1*s*.

**2.2 Intermediate correlation regime**

fermion operator can be expressed as:

Here *f* †

*c*†

have a boundary constrained condition:

**2.3 Strong correlation regime**

moments in the parent phases.

*<sup>i</sup>ασ* <sup>=</sup> *<sup>Q</sup>*<sup>−</sup> <sup>1</sup>

+ ∑ *σ*� *t* † *<sup>i</sup>βσ*�*diσ*¯ *ασ*� *β* + *q*† *<sup>i</sup> tiασ*¯ 

<sup>1</sup> <sup>≥</sup> <sup>∑</sup>*ασ*

*p*2 *ασ* <sup>+</sup> <sup>∑</sup>*<sup>α</sup>* *b*2 *<sup>α</sup>* + ∑ *ασσ*� *d*2

With these projections to the boson states, one can easily obtain an effective t-t�

properties in the intermediate correlated iron pnictides, such as FeSe/FeTe, etc.

subjected to the normalization and fermion number constraints. Following the similar steps above in the weak correlation regime to decouple the spin exchange terms, one could obtain the self-consistent equations similar to Eq.(8) to get various supercoundcting groundstate

Once the Coulomb interaction is so large that double occupation is excluded on each site, we use the Barnes-Coleman's slave boson approach to discuss the pairing symmetry and orbital-dependent superconducting energy gaps, which is applicable for some FeAs-based superconductors and heavy fermion superconductors with significantly large magnetic

2 *iασ p*† *<sup>i</sup>ασei* <sup>+</sup> *<sup>b</sup>*†

obtains the self-consistent equations,

*<sup>n</sup>*(1/2)<sup>↑</sup> <sup>=</sup> <sup>1</sup> *<sup>N</sup>* ∑ *k*,*α U*∗ (1/3)*α*(*k*)*U*(1/3)*α*(*k*)*f*(*Eα*(*k*)) *<sup>n</sup>*(1/2)<sup>↓</sup> <sup>=</sup> <sup>1</sup> *<sup>N</sup>* ∑ *k*,*α U*∗ (2/4)*α*(*k*)*U*(2/4)*α*(*k*)(1 − *f*(*Eα*(*k*))) *P*1/2 *<sup>x</sup>*/*<sup>y</sup>* <sup>=</sup> <sup>1</sup> *<sup>N</sup>* ∑ *k*,*α coskx*/*yU*<sup>∗</sup> (1/3)*α*(*k*)*U*(1/3)*α*(*k*)*f*(*Eα*(*k*)) *P*1/2 <sup>3</sup> <sup>=</sup> <sup>1</sup> *<sup>N</sup>* ∑ *k*,*α coskxcoskyU*∗ (1/3)*α*(*k*)*U*(1/3)*α*(*k*)*f*(*Eα*(*k*)) *P*1/2 *xy* <sup>=</sup> <sup>1</sup> *<sup>N</sup>* ∑ *k*,*α coskx*/*yU*<sup>∗</sup> <sup>1</sup>*α*(*k*)*U*3*α*(*k*)*f*(*Eα*(*k*)) *P*3/4 *xy* <sup>=</sup> <sup>1</sup> *<sup>N</sup>* ∑ *k*,*α cos*(*kx* + / − *ky*)*U*<sup>∗</sup> <sup>1</sup>*α*(*k*)*U*3*α*(*k*)*f*(*Eα*(*k*)) Δ1(1/2) *<sup>x</sup>*/*<sup>y</sup>* <sup>=</sup> *<sup>J</sup> <sup>N</sup>* ∑ *k*,*α coskx*/*yU*<sup>∗</sup> (1/3)*α*(*k*)*U*(2/4)*α*(*k*)*f*(*Eα*(*k*)) Δ2(1/2) *<sup>x</sup>*±*<sup>y</sup>* <sup>=</sup> *<sup>J</sup>* � *<sup>N</sup>* ∑ *k*,*α cos*(*kx* ± *ky*)*U*<sup>∗</sup> (1/3)*α*(*k*)*U*(2/4)*α*(*k*)*f*(*Eα*(*k*)) (8)

where *Eα*(*k*) is the Bogoliubov quasiparticle eigenvalues obtained from H*MF*, and *f*(*E*) is the Fermi-Dirac distribution function, f(E) = 1/(1 + *exp*(*E*/*kBT*)). U*αβ*(k) denotes the (*α*,*β*) element of the 4x4 unitary matrix U(k).

With these self-consistent equations, we could obtain not only the groundstate phase diagram, but also the temperature dependence of the Fermi surfaces in normal state and the quasiparticle spectra in the normal and superconducting states. In fact, the intra-orbital hopping integral of the *dyz* orbit is symmetric with that of the *dyz* orbit under a coordinate transformation (x,y,z) ← (y,x,z). Due to this symmetry, the superconducting order parameters Δ2(k) can be obtained from Δ1(k) under the coordinate transformation. Therefore, we mainly focus on the properties of the superconducting order parameters Δ1(k) in the first orbit *dxz*. Nevertheless, the global superconducting pairing order parameter of the two-orbital t-t� -J-J� model should be rotationally symmetric in the *xy*-plane, as we can see from the Hamiltonian Eq.(2).

Within the present scenario, we could obtain not only the groundstate phase diagram, but also the quasiparticle spectra in the normal and the superconducting states. The temperature dependence of the Fermi surface in normal state and that of the spin-lattice relaxation rate in the superconducting state can also be obtained. Among these quantities, the spin-lattice relaxation rate in the NMR experiment is expressed as Matano (2008):

$$\begin{split} \frac{T\_{1N}}{T\_{1s}} &= \frac{2}{k\_B T} \int \int \mathcal{N}\_s(E) \mathcal{N}\_s(E') f(E) \\ &\times [1 - f(E') \delta(E - E') dE dE'] \end{split} \tag{9}$$

Providing 1/T1*<sup>N</sup>* in the normal state satisfies the Korringa law, the spin lattice relaxation rate 1/*T*1*<sup>s</sup>* becomes Xiang (2007): 1/*T*1*<sup>s</sup>* ∝ (*kBT*) · *T*1*N*/*T*1*s*.

#### **2.2 Intermediate correlation regime**

6 Will-be-set-by-IN-TECH

(1/3)*α*(*k*)*U*(1/3)*α*(*k*)*f*(*Eα*(*k*))

(2/4)*α*(*k*)*U*(2/4)*α*(*k*)(1 − *f*(*Eα*(*k*)))

<sup>1</sup>*α*(*k*)*U*3*α*(*k*)*f*(*Eα*(*k*))

(1/3)*α*(*k*)*U*(1/3)*α*(*k*)*f*(*Eα*(*k*))

(1/3)*α*(*k*)*U*(2/4)*α*(*k*)*f*(*Eα*(*k*))

(1/3)*α*(*k*)*U*(1/3)*α*(*k*)*f*(*Eα*(*k*))

<sup>1</sup>*α*(*k*)*U*3*α*(*k*)*f*(*Eα*(*k*))

(1/3)*α*(*k*)*U*(2/4)*α*(*k*)*f*(*Eα*(*k*))

(8)


obtains the self-consistent equations,

*<sup>n</sup>*(1/2)<sup>↑</sup> <sup>=</sup> <sup>1</sup>

*<sup>n</sup>*(1/2)<sup>↓</sup> <sup>=</sup> <sup>1</sup>

*P*1/2 *<sup>x</sup>*/*<sup>y</sup>* <sup>=</sup> <sup>1</sup>

*P*1/2 <sup>3</sup> <sup>=</sup> <sup>1</sup>

*P*1/2 *xy* <sup>=</sup> <sup>1</sup>

*P*3/4 *xy* <sup>=</sup> <sup>1</sup>

Δ1(1/2) *<sup>x</sup>*/*<sup>y</sup>* <sup>=</sup> *<sup>J</sup>*

Δ2(1/2) *<sup>x</sup>*±*<sup>y</sup>* <sup>=</sup> *<sup>J</sup>*

element of the 4x4 unitary matrix U(k).

Eq.(2).

*<sup>N</sup>* ∑ *k*,*α U*∗

*<sup>N</sup>* ∑ *k*,*α U*∗

*<sup>N</sup>* ∑ *k*,*α*

*<sup>N</sup>* ∑ *k*,*α*

*<sup>N</sup>* ∑ *k*,*α*

*<sup>N</sup>* ∑ *k*,*α*

*<sup>N</sup>* ∑ *k*,*α*

� *<sup>N</sup>* ∑ *k*,*α* *coskx*/*yU*<sup>∗</sup>

*coskx*/*yU*<sup>∗</sup>

*coskx*/*yU*<sup>∗</sup>

*cos*(*kx* ± *ky*)*U*<sup>∗</sup>

where *Eα*(*k*) is the Bogoliubov quasiparticle eigenvalues obtained from H*MF*, and *f*(*E*) is the Fermi-Dirac distribution function, f(E) = 1/(1 + *exp*(*E*/*kBT*)). U*αβ*(k) denotes the (*α*,*β*)

With these self-consistent equations, we could obtain not only the groundstate phase diagram, but also the temperature dependence of the Fermi surfaces in normal state and the quasiparticle spectra in the normal and superconducting states. In fact, the intra-orbital hopping integral of the *dyz* orbit is symmetric with that of the *dyz* orbit under a coordinate transformation (x,y,z) ← (y,x,z). Due to this symmetry, the superconducting order parameters Δ2(k) can be obtained from Δ1(k) under the coordinate transformation. Therefore, we mainly focus on the properties of the superconducting order parameters Δ1(k) in the first orbit *dxz*. Nevertheless, the global superconducting pairing order parameter of the two-orbital t-t�

model should be rotationally symmetric in the *xy*-plane, as we can see from the Hamiltonian

Within the present scenario, we could obtain not only the groundstate phase diagram, but also the quasiparticle spectra in the normal and the superconducting states. The temperature dependence of the Fermi surface in normal state and that of the spin-lattice relaxation rate in the superconducting state can also be obtained. Among these quantities, the spin-lattice

×[1 − *f*(*E*�

*Ns*(*E*)*Ns*(*E*�

)*δ*(*E* − *E*�

)*f*(*E*)

] (9)

)*dEdE*�

relaxation rate in the NMR experiment is expressed as Matano (2008):

<sup>=</sup> <sup>2</sup> *kBT*

*T*1*<sup>N</sup> T*1*<sup>s</sup>*

*coskxcoskyU*∗

*cos*(*kx* + / − *ky*)*U*<sup>∗</sup>

When the kinetic energy of the conduction bands becomes small and is comparable with the Coulomb interaction, we need to consider the electronic correlation effect, as one sees in FeTe1−*x*Se*<sup>x</sup>* and other superconductors. We utilize the Kotliar-Ruckenstein's slave boson approach for some FeAs-based and ruthenate superconductor with intermediate magnetic moments. To reflect the multi-orbital character of iron pnictides, we need to extend the single-orbital Kotliar-Ruckenstein's slave boson approach Kotliar et al. (1988) to the two-orbital Hubbard models for various configurations. In the multiorbital Hubbard model, a few of auxiliary boson field operators representing the possibilities of various electron occupations are introduced, such as *e*, *p*, *d*, *b*,*t*, *q*, which denote the possibilities of none, single, double, triplicate, quaternity occupations. With these auxiliary boson fields, an original fermion operator can be expressed as:

$$c\_{i\alpha\sigma}^{\dagger} = Q\_{i\alpha\sigma}^{-\frac{1}{2}} \left( p\_{i\alpha\sigma}^{\dagger} e\_i + b\_{i\alpha}^{\dagger} p\_{i\alpha\sigma} + \sum\_{\sigma'} d\_{i\sigma,\sigma'\_{\beta}}^{\dagger} p\_{i\beta\sigma'} + t\_{i\alpha\sigma}^{\dagger} b\_{i\beta} \right)$$

$$+ \sum\_{\sigma'} t\_{i\beta\sigma'}^{\dagger} d\_{i\sigma,\sigma'\_{\beta}} + q\_{i\cdot}^{\dagger} t\_{i\alpha\sigma} \right) (1 - Q\_{i\alpha\sigma})^{-\frac{1}{2}} f\_{i\alpha\sigma}^{\dagger} \tag{10}$$

Here *f* † *<sup>i</sup>ασ* is the new slaved fermion operator and *Qiασ* is an auxiliary particle number operator Kotliar et al. (1988). Projecting the original fermion operators into these boson field and fermion field operators, one could not only obtain an effective Hamiltonian, but also get the groundstate energy in the saddle point approximation with the normalization condition and the fermion number constraints Kotliar et al. (1988). Here we employ a generalized Lagrange multiplier method to enforce these constraint conditions, thus the interorbital hoppings and crystal field splitting can be treated on the same foot. The fermion occupation number is constrained with the penalty function method. To enforce the normalization condition, we have a boundary constrained condition:

$$1 \ge \sum\_{a\sigma} p\_{a\sigma}^2 + \sum\_a b\_a^2 + \sum\_{a\sigma\sigma'} d\_{a\sigma\sigma'}^2 + \sum\_{a\sigma} t\_{a\sigma}^2 + q^2 \tag{11}$$

With these projections to the boson states, one can easily obtain an effective t-t� -J-J� model subjected to the normalization and fermion number constraints. Following the similar steps above in the weak correlation regime to decouple the spin exchange terms, one could obtain the self-consistent equations similar to Eq.(8) to get various supercoundcting groundstate properties in the intermediate correlated iron pnictides, such as FeSe/FeTe, etc.

#### **2.3 Strong correlation regime**

Once the Coulomb interaction is so large that double occupation is excluded on each site, we use the Barnes-Coleman's slave boson approach to discuss the pairing symmetry and orbital-dependent superconducting energy gaps, which is applicable for some FeAs-based superconductors and heavy fermion superconductors with significantly large magnetic moments in the parent phases.

Fig. 1. (a) Superconducting phase diagram of the t-t�

J and a small NNN spin coupling J�

respectively.

(2008) and our ground states vs the NNN spin coupling J�

by N in Fig.1a. Obviously, when the superexchange coupling J and J�


concentration x=0.18. N denotes the normal state, the other four phases are superconductive with different pairing symmetries. (b) The energy difference ΔE between Seo *et al's* Seo et al.

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 33

provide the pairing glue, the kinetic energy is dominant, and the electrons stay in the normal state, which is analogous to the single-orbital t-J model Kotliar et al. (1988). Among the four superconducting phases mediated through the spin fluctuations, a large NN spin coupling

superconducting phase with the gap Δ1(k) ∝ *cos*(*kx*) + *ηcos*(*ky*), where the pairing symmetry is the combination of the *Sx*<sup>2</sup>+*y*<sup>2</sup> -wave and the *dx*<sup>2</sup>−*y*<sup>2</sup> -wave components, as seen the pink region in Fig.1a. The *Sx*<sup>2</sup>+*ηy*<sup>2</sup> symmetry arises from the major contribution of the NN spin

*Sx*<sup>2</sup>*y*<sup>2</sup> superconducting phase with the symmetry Δ1(k) ∝ *cos*(*kx* + *ky*) + *cos*(*kx* − *ky*), as seen in the blue region in Fig.1a, which is mainly attributed to the NNN spin coupling. In this situation, Δ1(k) is almost isotropic in the *xy*-plane due to the isotropy of the dominant

�

, or J>>J

coupling J term. The NNN spin coupling contributes very little to Δ1(k) due to J>>J

On the other hand, small NN spin coupling J and large NNN spin coupling J�

model for the d*xz*-orbit at the carrier

are too small to

� .

favor the

at different J, J=1,2 and 3,

, favor the *Sx*<sup>2</sup>+*ηy*<sup>2</sup> (here and below *η*1=*η*)

Within the slave-boson representation Barnes (1976); Coleman (1984), Eq.(4) is rewritten in terms of the projected fermion operators *f* † *im<sup>σ</sup>* and *fimσ*, as well as the slave boson operators at each site, which rule out the double and multiple fermion occupancies. The constrained Hilbert space (*S*) of each site *i* is

$$S\_i = \{ |\ 1, \uparrow\rangle, |\ 1, \downarrow\rangle, |\ 2, \uparrow\rangle, |\ 2, \downarrow\rangle, |\ 0, 0>\} \tag{12}$$

including the single-occupied states of the spin-up and spin-down in 1-orbit, and those in 2-orbit, together with the vacancy state, respectively. The present constrained spin-orbital formulation resembles to the 4-fold degenerate state of pseudo-angular momentum *j*=3/2 proposed by Barnes Barnes (1976) and Coleman Coleman (1984), if we define <sup>|</sup> *<sup>f</sup>* <sup>0</sup> <sup>&</sup>gt;=<sup>|</sup> 0, 0 <sup>&</sup>gt;, <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, <sup>−</sup>3/2 <sup>&</sup>gt; <sup>=</sup><sup>|</sup> 1, <sup>↑</sup>>, <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, <sup>−</sup>1/2 <sup>&</sup>gt;=<sup>|</sup> 1, <sup>↓</sup>>, <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, 1/2 <sup>&</sup>gt;=<sup>|</sup> 2, <sup>↑</sup>>, and <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, 3/2 <sup>&</sup>gt;=<sup>|</sup> 2, <sup>↓</sup>>. In this representation, *Pc*† *imσ<sup>P</sup>* <sup>=</sup> *<sup>f</sup>* † *imσbi*. The boson operator *<sup>b</sup>*† *i* creates an empty occupation state at the *i*th site, and the fermion operator *f* † *imσ*(*fimσ*) creates (annihilates) a slaved electron at site *i* with the orbit *m* and spin *σ*(=↑, ↓). After projecting the original fermion representation into the present boson representation, one could obtain an effective t-t� -J-J� model subjected to these constraints. Following the similar steps above in Sec.2.1 to decouple the spin exchange terms, one could readily solve the superconducting groundstate properties in strongly correlated K*x*Fe2−*<sup>y</sup>*Se2 compound.

#### **3. Pairing symmetry of multi-orbital iron-pnictide superconductors**

We present the pairing symmetry of the two-orbital t-t'-J-J' models for weak-correlated FeAs-based superconductors. The main numerical results in the weak correlation situation are addressed as follows. Also one could easily obtain the numerical results of the intermediate and strong correlation situations.

#### **3.1 Stability of unusual superconducting pairing symmetry**

First of all, we determine the stable ground state of the present two-orbital *t* − *t* � − *J* − *J*� model with the electron filling n=1-*δ* in a square lattice through comparing the groundstate energies of various pairing-symmetric superconducting states: the isotropic *s* wave, anisotropic <sup>s</sup>*x*<sup>2</sup>−*y*<sup>2</sup> wave, s*x*2*y*<sup>2</sup> wave, d*x*<sup>2</sup>−*y*<sup>2</sup> wave, and d*x*2*y*<sup>2</sup> wave, etc. By minimizing the groundstate energies of various candidates and finding a stablest state, we obtain phase diagrams of the systems for various parameters, such as the hopping integrals *tab*, the doping concentration *n*, the exchange parameters *J* and *J*� and so on. Our numerical results show that in the superconducting phase of iron pnictides, the energy of the weakly anisotropic and nodeless *dx*<sup>2</sup>−*y*<sup>2</sup> <sup>+</sup> *sx*<sup>2</sup>*y*<sup>2</sup> -wave-like superconducting state is lower than those of the *<sup>s</sup>*-waves and *<sup>d</sup>*-wave states for most of the situations we investigated.

#### **3.2 Phase diagram of superconducting pairing symmetry**

In this subsection, we first obtain the phase diagrams and mark the pairing symmetry of each stable superconducting phase in the J-J' and t-J planes, and locate the most possible position of the pairing symmetry of iron-pnictide superconductors.

The J� -J phase diagram of the t-t� -J-J� model at carrier concentration x=0.18 is shown in Fig.1a. Different from Seo *et al.*'s phase diagram Seo et al. (2008), we obtain five stable phases in the present model. The first one is a normal phase in the small J and J� region, denoted 8 Will-be-set-by-IN-TECH

Within the slave-boson representation Barnes (1976); Coleman (1984), Eq.(4) is rewritten in

at each site, which rule out the double and multiple fermion occupancies. The constrained

including the single-occupied states of the spin-up and spin-down in 1-orbit, and those in 2-orbit, together with the vacancy state, respectively. The present constrained spin-orbital formulation resembles to the 4-fold degenerate state of pseudo-angular momentum *j*=3/2 proposed by Barnes Barnes (1976) and Coleman Coleman (1984), if we define <sup>|</sup> *<sup>f</sup>* <sup>0</sup> <sup>&</sup>gt;=<sup>|</sup> 0, 0 <sup>&</sup>gt;, <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, <sup>−</sup>3/2 <sup>&</sup>gt; <sup>=</sup><sup>|</sup> 1, <sup>↑</sup>>, <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, <sup>−</sup>1/2 <sup>&</sup>gt;=<sup>|</sup> 1, <sup>↓</sup>>, <sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, 1/2 <sup>&</sup>gt;=<sup>|</sup> 2, <sup>↑</sup>>, and

(annihilates) a slaved electron at site *i* with the orbit *m* and spin *σ*(=↑, ↓). After projecting the original fermion representation into the present boson representation, one could obtain

in Sec.2.1 to decouple the spin exchange terms, one could readily solve the superconducting

We present the pairing symmetry of the two-orbital t-t'-J-J' models for weak-correlated FeAs-based superconductors. The main numerical results in the weak correlation situation are addressed as follows. Also one could easily obtain the numerical results of the intermediate

with the electron filling n=1-*δ* in a square lattice through comparing the groundstate energies of various pairing-symmetric superconducting states: the isotropic *s* wave, anisotropic <sup>s</sup>*x*<sup>2</sup>−*y*<sup>2</sup> wave, s*x*2*y*<sup>2</sup> wave, d*x*<sup>2</sup>−*y*<sup>2</sup> wave, and d*x*2*y*<sup>2</sup> wave, etc. By minimizing the groundstate energies of various candidates and finding a stablest state, we obtain phase diagrams of the systems for various parameters, such as the hopping integrals *tab*, the doping concentration *n*, the exchange parameters *J* and *J*� and so on. Our numerical results show that in the superconducting phase of iron pnictides, the energy of the weakly anisotropic and nodeless *dx*<sup>2</sup>−*y*<sup>2</sup> <sup>+</sup> *sx*<sup>2</sup>*y*<sup>2</sup> -wave-like superconducting state is lower than those of the *<sup>s</sup>*-waves and *<sup>d</sup>*-wave

In this subsection, we first obtain the phase diagrams and mark the pairing symmetry of each stable superconducting phase in the J-J' and t-J planes, and locate the most possible position

Different from Seo *et al.*'s phase diagram Seo et al. (2008), we obtain five stable phases in

model at carrier concentration x=0.18 is shown in Fig.1a.

creates an empty occupation state at the *i*th site, and the fermion operator *f* †

groundstate properties in strongly correlated K*x*Fe2−*<sup>y</sup>*Se2 compound.

**3.1 Stability of unusual superconducting pairing symmetry**

states for most of the situations we investigated.


**3.2 Phase diagram of superconducting pairing symmetry**

of the pairing symmetry of iron-pnictide superconductors.


the present model. The first one is a normal phase in the small J and J�

**3. Pairing symmetry of multi-orbital iron-pnictide superconductors**

First of all, we determine the stable ground state of the present two-orbital *t* − *t*

*im<sup>σ</sup>* and *fimσ*, as well as the slave boson operators

*imσbi*. The boson operator *<sup>b</sup>*†

*imσ*(*fimσ*) creates

� − *J* − *J*� model

region, denoted

*i*

*Si* = {| 1, ↑>, | 1, ↓>, | 2, ↑>, | 2, ↓>, | 0, 0 >} (12)

*imσ<sup>P</sup>* <sup>=</sup> *<sup>f</sup>* †

model subjected to these constraints. Following the similar steps above

terms of the projected fermion operators *f* †

<sup>|</sup> *<sup>f</sup>* <sup>1</sup> : 3/2, 3/2 <sup>&</sup>gt;=<sup>|</sup> 2, <sup>↓</sup>>. In this representation, *Pc*†

Hilbert space (*S*) of each site *i* is


and strong correlation situations.

an effective t-t�

The J�

Fig. 1. (a) Superconducting phase diagram of the t-t� -J-J� model for the d*xz*-orbit at the carrier concentration x=0.18. N denotes the normal state, the other four phases are superconductive with different pairing symmetries. (b) The energy difference ΔE between Seo *et al's* Seo et al. (2008) and our ground states vs the NNN spin coupling J� at different J, J=1,2 and 3, respectively.

by N in Fig.1a. Obviously, when the superexchange coupling J and J� are too small to provide the pairing glue, the kinetic energy is dominant, and the electrons stay in the normal state, which is analogous to the single-orbital t-J model Kotliar et al. (1988). Among the four superconducting phases mediated through the spin fluctuations, a large NN spin coupling J and a small NNN spin coupling J� , or J>>J � , favor the *Sx*<sup>2</sup>+*ηy*<sup>2</sup> (here and below *η*1=*η*) superconducting phase with the gap Δ1(k) ∝ *cos*(*kx*) + *ηcos*(*ky*), where the pairing symmetry is the combination of the *Sx*<sup>2</sup>+*y*<sup>2</sup> -wave and the *dx*<sup>2</sup>−*y*<sup>2</sup> -wave components, as seen the pink region in Fig.1a. The *Sx*<sup>2</sup>+*ηy*<sup>2</sup> symmetry arises from the major contribution of the NN spin coupling J term. The NNN spin coupling contributes very little to Δ1(k) due to J>>J � .

On the other hand, small NN spin coupling J and large NNN spin coupling J� favor the *Sx*<sup>2</sup>*y*<sup>2</sup> superconducting phase with the symmetry Δ1(k) ∝ *cos*(*kx* + *ky*) + *cos*(*kx* − *ky*), as seen in the blue region in Fig.1a, which is mainly attributed to the NNN spin coupling. In this situation, Δ1(k) is almost isotropic in the *xy*-plane due to the isotropy of the dominant

Fig. 2. The Fermi surface topology in the large Brillouin zone at the different temperatures: T=0.15 (black), 0.2 (red) and 0.7 (green). Dashed square outlines the reduced Brillouin Zone.

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 35

electron-like Fermi sheet may play a more important role in the low-energy processes in finite temperatures. This behavior arises from that the electronic thermal excitations increase with the lift of temperature, leading to the chemical potential decreasing with increasing T. Thus the electron-like Fermi surface decreases and the hole-like Fermi surface increases with increasing

Fig.3 shows the temperature dependence of the superconducting energy gaps on the two hole-like Fermi sheets along the *θ* = 0*<sup>o</sup>* direction in the polar coordinate system for the two sets of parameters. With the increasing temperature, the two energy gaps decrease monotonously and vanish simultaneously, as observed in the ARPES experiments Ding et al. (2008). Obviously, the superconducting-state to normal-state transition is a second-order phase transition. For the case I, the magnitude of the energy gap on the small Fermi srface (*α*1) is larger than that on the large Fermi surface (*α*2) in the Γ point, in agreement with the ARPES results Ding et al. (2008); Zhao et al. (2008). In contrast, for the case II, the magnitude of the gap on the small fermi surface (*α*1) is smaller than that on large Fermi surface (*α*2) in the Γ point, which disagrees with the experiment Ding et al. (2008); Zhao et al. (2008). These indicate that the first set of parameters in Case I is more suitable for describing the FeAs

From the present theoretical results in Fig.3, we find that in Case I, the ratios of the energy gaps to the transition temperature are 2Δ1/*kBTc*=3.6 for the large gap, and 2Δ2/*kBTc*=2.9 for the small gap, respectively. The ratio of the large energy gap around the small Fermi sheet to the small one around the large Fermi sheet gives rise to Δ1/Δ2=1.25. These theoretical

= 0.7; the other parameters are the same as these in

The theoretical parameters: *J* = 0.3, *J*

the Fig.1.

superconductors.

T.

NNN hopping integrals in the *xy*-plane. The superconducting order parameter becomes complicated when J and J� compete with each other. As seen in Fig.1a, the pairing symmetry of the superconducting phase in the green region of Fig.1a is the combination of the *Sx*<sup>2</sup>+*ηy*<sup>2</sup> and the *Sx*<sup>2</sup>*y*<sup>2</sup> components, and the symmetry of the superconducting phase in the yellow region of Fig.1a is the combination of the *dx*<sup>2</sup>−*ηy*<sup>2</sup> and the *Sx*<sup>2</sup>*y*<sup>2</sup> components.

It is interesting to ask in which region the realistic parameters of the iron pnictides fall. From the first-principles calculations, Ma *et*. *al*. suggested that J ≈ J � <sup>≈</sup> 0.05 *eV*/*S*<sup>2</sup> Ma et al. (2010), where *S* is the spin of each Fe ion. When the hopping integral |*t*1| ≈0.1 ∼0.5 eV, such a set of parameters falls in the yellow region in Fig.1a, implying that the FeAs superconductors should have the *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> pairing symmetry, and the anisotropic factor *<sup>η</sup>* is not equal to 1. Also, some other authors suggested other parameters for the FeAs superconductors, for example, Seo *et al*.Seo et al. (2008) proposed that J=0.25 and J� =0.5; and Si *et al*.Si et al. (2008) thought that J > J � /2. This shows that further effort is needed to obtain more accurate interaction parameters in iron pnictides.

We notice that Seo *et al.'s* J-J� parameters also falls in the yellow region in Fig.1a, *i.e.* the pairing symmetry is *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> type, rather than the *<sup>δ</sup><sup>s</sup> Sx*<sup>2</sup>*y*<sup>2</sup> <sup>±</sup> *<sup>δ</sup>ddx*<sup>2</sup>−*y*<sup>2</sup> type with *<sup>η</sup>* <sup>=</sup> 1 (here *<sup>δ</sup><sup>s</sup>* and *<sup>δ</sup><sup>d</sup>* are the weights of the *Sx*<sup>2</sup>*y*<sup>2</sup> wave and the *dx*<sup>2</sup>−*y*<sup>2</sup> wave component, respectively). In Fig.1b, we compare the groundstate energy difference between theirs and ours, and find that the groundstate energy in the present superconducting phase, E*η*, is lower than the E*<sup>d</sup>* in Seo *et al*.'s paper Seo et al. (2008). Fig.1b shows the J� dependence of the groundstate energy difference, ΔE=E*d*-E*η*, between the two superconducting phases at different J, here E*<sup>d</sup>* and <sup>E</sup>*<sup>η</sup>* are the energies of the *<sup>δ</sup>sSx*<sup>2</sup>*y*<sup>2</sup> ± *<sup>δ</sup>ddx*<sup>2</sup>−*y*<sup>2</sup> symmetric phase and of the symmetric phase in Fig. 1a, respectively. It is clear that in wide J-J� range, the *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> phase is always more stable than the *<sup>δ</sup><sup>s</sup> Sx*<sup>2</sup>*y*<sup>2</sup> <sup>±</sup> *<sup>δ</sup><sup>d</sup> dx*<sup>2</sup>−*y*<sup>2</sup> phase. Thus *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> -wave is most likely the superconducting pairing symmetry in iron pnictide superconductors.

#### **3.3 T-dependence of fermi surface and superconducting energy gaps**

In the present situation with a weakly broken orbital symmetry, we find that two superconducting energy gaps synchronously approach zero as T is lifted to T*c*. To concretely discuss the properties of the superconducting state and the normal state, and compare the theory with the ARPES experimental results, in what follows, we focus on two sets of typical superexchange coupling parameters, Case I: J=0.3 and J� =0.7, *i.e.*, the NN spin coupling is weaker than the NNN coupling; and Case II: J=0.7 and J� =0.3, *i.e.* the NN spin coupling is stronger than the NNN coupling. In both of the situations, the parameters fall in the yellow region in Fig.1a, so the superconducting pairing symmetries are *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> -wave. We present the temperature evolution of the Fermi surfaces in the normal state in Fig.2 for the case I with J=0.3 and J� =0.7. The Fermi surface topology for the case II with J=0.7 and J� =0.3 is almost identical to Fig.2, hence is not plotted. From Fig.2, one sees that in the large Brillouin zone (BZ) associated with the present t-t� -J-J� model with one Fe atom per unit cell, there exist two hole-like Fermi sheets (*α*<sup>1</sup> and *α*2) around the Γ point, and two electron-like Fermi sheets (*β*<sup>1</sup> and *β*2) around the *M* point. This is in agreement with the ARPES experiment Ding et al. (2008) and consistent with the first-principle electronic structures calculations Mazin et al. (2008); Boeri et al. (2008); Cao et al. (2008); Singh et al. (2008); Lebègue et al. (2007); Ma et al. (2008). Interestingly, the hole-like Fermi sheets expand a little with increasing temperature; in contrast, the electron-like Fermi sheets shrink considerably. This indicates that the 10 Will-be-set-by-IN-TECH

NNN hopping integrals in the *xy*-plane. The superconducting order parameter becomes

the superconducting phase in the green region of Fig.1a is the combination of the *Sx*<sup>2</sup>+*ηy*<sup>2</sup> and the *Sx*<sup>2</sup>*y*<sup>2</sup> components, and the symmetry of the superconducting phase in the yellow region

It is interesting to ask in which region the realistic parameters of the iron pnictides fall. From

where *S* is the spin of each Fe ion. When the hopping integral |*t*1| ≈0.1 ∼0.5 eV, such a set of parameters falls in the yellow region in Fig.1a, implying that the FeAs superconductors should have the *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> pairing symmetry, and the anisotropic factor *<sup>η</sup>* is not equal to 1. Also, some other authors suggested other parameters for the FeAs superconductors,

symmetry is *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> type, rather than the *<sup>δ</sup><sup>s</sup> Sx*<sup>2</sup>*y*<sup>2</sup> <sup>±</sup> *<sup>δ</sup>ddx*<sup>2</sup>−*y*<sup>2</sup> type with *<sup>η</sup>* <sup>=</sup> 1 (here *<sup>δ</sup><sup>s</sup>* and *<sup>δ</sup><sup>d</sup>* are the weights of the *Sx*<sup>2</sup>*y*<sup>2</sup> wave and the *dx*<sup>2</sup>−*y*<sup>2</sup> wave component, respectively). In Fig.1b, we compare the groundstate energy difference between theirs and ours, and find that the groundstate energy in the present superconducting phase, E*η*, is lower than the E*<sup>d</sup>* in

difference, ΔE=E*d*-E*η*, between the two superconducting phases at different J, here E*<sup>d</sup>* and <sup>E</sup>*<sup>η</sup>* are the energies of the *<sup>δ</sup>sSx*<sup>2</sup>*y*<sup>2</sup> ± *<sup>δ</sup>ddx*<sup>2</sup>−*y*<sup>2</sup> symmetric phase and of the symmetric phase in

more stable than the *<sup>δ</sup><sup>s</sup> Sx*<sup>2</sup>*y*<sup>2</sup> <sup>±</sup> *<sup>δ</sup><sup>d</sup> dx*<sup>2</sup>−*y*<sup>2</sup> phase. Thus *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> -wave is most likely the

In the present situation with a weakly broken orbital symmetry, we find that two superconducting energy gaps synchronously approach zero as T is lifted to T*c*. To concretely discuss the properties of the superconducting state and the normal state, and compare the theory with the ARPES experimental results, in what follows, we focus on two sets of typical

stronger than the NNN coupling. In both of the situations, the parameters fall in the yellow region in Fig.1a, so the superconducting pairing symmetries are *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> -wave. We present the temperature evolution of the Fermi surfaces in the normal state in Fig.2 for the

almost identical to Fig.2, hence is not plotted. From Fig.2, one sees that in the large Brillouin

two hole-like Fermi sheets (*α*<sup>1</sup> and *α*2) around the Γ point, and two electron-like Fermi sheets (*β*<sup>1</sup> and *β*2) around the *M* point. This is in agreement with the ARPES experiment Ding et al. (2008) and consistent with the first-principle electronic structures calculations Mazin et al. (2008); Boeri et al. (2008); Cao et al. (2008); Singh et al. (2008); Lebègue et al. (2007); Ma et al. (2008). Interestingly, the hole-like Fermi sheets expand a little with increasing temperature; in contrast, the electron-like Fermi sheets shrink considerably. This indicates that the


=0.7. The Fermi surface topology for the case II with J=0.7 and J�

of Fig.1a is the combination of the *dx*<sup>2</sup>−*ηy*<sup>2</sup> and the *Sx*<sup>2</sup>*y*<sup>2</sup> components.

for example, Seo *et al*.Seo et al. (2008) proposed that J=0.25 and J�

superconducting pairing symmetry in iron pnictide superconductors.

**3.3 T-dependence of fermi surface and superconducting energy gaps**

superexchange coupling parameters, Case I: J=0.3 and J�

weaker than the NNN coupling; and Case II: J=0.7 and J�

�

Seo *et al*.'s paper Seo et al. (2008). Fig.1b shows the J�

Fig. 1a, respectively. It is clear that in wide J-J�

interaction parameters in iron pnictides.

the first-principles calculations, Ma *et*. *al*. suggested that J ≈ J

compete with each other. As seen in Fig.1a, the pairing symmetry of

�

/2. This shows that further effort is needed to obtain more accurate

parameters also falls in the yellow region in Fig.1a, *i.e.* the pairing

<sup>≈</sup> 0.05 *eV*/*S*<sup>2</sup> Ma et al. (2010),

=0.5; and Si *et al*.Si et al.

dependence of the groundstate energy

=0.7, *i.e.*, the NN spin coupling is

=0.3, *i.e.* the NN spin coupling is

model with one Fe atom per unit cell, there exist

=0.3 is

range, the *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> phase is always

complicated when J and J�

(2008) thought that J > J

case I with J=0.3 and J�

zone (BZ) associated with the present t-t�

We notice that Seo *et al.'s* J-J�

Fig. 2. The Fermi surface topology in the large Brillouin zone at the different temperatures: T=0.15 (black), 0.2 (red) and 0.7 (green). Dashed square outlines the reduced Brillouin Zone. The theoretical parameters: *J* = 0.3, *J* = 0.7; the other parameters are the same as these in the Fig.1.

electron-like Fermi sheet may play a more important role in the low-energy processes in finite temperatures. This behavior arises from that the electronic thermal excitations increase with the lift of temperature, leading to the chemical potential decreasing with increasing T. Thus the electron-like Fermi surface decreases and the hole-like Fermi surface increases with increasing T.

Fig.3 shows the temperature dependence of the superconducting energy gaps on the two hole-like Fermi sheets along the *θ* = 0*<sup>o</sup>* direction in the polar coordinate system for the two sets of parameters. With the increasing temperature, the two energy gaps decrease monotonously and vanish simultaneously, as observed in the ARPES experiments Ding et al. (2008). Obviously, the superconducting-state to normal-state transition is a second-order phase transition. For the case I, the magnitude of the energy gap on the small Fermi srface (*α*1) is larger than that on the large Fermi surface (*α*2) in the Γ point, in agreement with the ARPES results Ding et al. (2008); Zhao et al. (2008). In contrast, for the case II, the magnitude of the gap on the small fermi surface (*α*1) is smaller than that on large Fermi surface (*α*2) in the Γ point, which disagrees with the experiment Ding et al. (2008); Zhao et al. (2008). These indicate that the first set of parameters in Case I is more suitable for describing the FeAs superconductors.

From the present theoretical results in Fig.3, we find that in Case I, the ratios of the energy gaps to the transition temperature are 2Δ1/*kBTc*=3.6 for the large gap, and 2Δ2/*kBTc*=2.9 for the small gap, respectively. The ratio of the large energy gap around the small Fermi sheet to the small one around the large Fermi sheet gives rise to Δ1/Δ2=1.25. These theoretical

Fig. 4. The angle dependence of the superconducting energy gaps near the small hole Fermi surface (*α*1) and the large hole Fermi surface (*α*2) around the Γ point in the polar coordinate

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 37

�

both cases, two distinct gaps open on the hole-like Fermi sheets, *α*<sup>1</sup> and *α*2, as seen in Fig.2. The presence of two different energy gaps demonstrates the nature of a multi-gap superconductor

structure exhibits a nearly isotropic symmetry with invisible anisotropy, as seen in Fig.4. A large energy gap opens on the small hole Fermi sheet (*α*1), and a small energy gap opens on the large hole Fermi sheet (*α*2). For the case II, the angular dependence of the energy gaps is visible, exhibiting weak spatial anisotropy. The oscillation amplitude is about 16%. However, the amplitudes of the superconducting energy gaps on the different Fermi surfaces, *α*<sup>1</sup> and *α*2, are contrast to these in Case I, i.e., a small energy gap opens on the small Fermi surface sheet

One finds that in Case I, the anisotropy of the superconducting energy gaps is very weak,

In Case II, the superconducting energy gaps with about 16% anisotropy is in agreement with the ARPES experiment by Kondo *et al*. Kondo et al. (2008). Noticing that in Case II, such spatial anisotropy is still under the resolution of the ARPES experiment, hence does not

It is the *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> -wave pairing symmetry that leads to weakly anisotropy and nodeless superconducting energy gap structures. Although the *dx*<sup>2</sup>−*ηy*<sup>2</sup> -wave pairing has nodes in the line cos *kx* − *η* cos *ky* = 0, and the *Sx*<sup>2</sup>*y*<sup>2</sup> -wave pairing has nodes in the lines *kx* = ±*π*/2 or *ky* <sup>=</sup> *<sup>π</sup>*/2. The mixed superconducting pairing symmetry, *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> , diminishes the

model. Our results show that for the case I, the superconducting energy gap

= 0.3 (red and blue circles).

�

�

� = 0.7

*s* Ding et al. (2008) ARPES data.

*s* Ding et al. (2008) observation.

system. The theoretical parameters are the same as these in Fig. 3, Case I, *J* = 0.3, *J*

(*α*1), and a large energy gap opens on the large hole fermi surface sheet (*α*2).

consistent with Zhao *et al*.'s Zhao et al. (2008) and Ding *et al*.

conflict with Zhao *et al*.'s Zhao et al. (2008) and Ding *et al*.

(wine and green circles); and Case II, *J* = 0.7, *J*

in the t-t�


Fig. 3. Temperature dependence of the superconducting energy gaps near the small hole Fermi surface (*α*1) and the large hole Fermi surface (*α*2) around the Γ point along the *θ* = 0*<sup>o</sup>* direction in the polar coordinate system. The doping concentration x=0.18. Theoretical parameters: Case I, *J* = 0.3, *J* � = 0.7 (wine and green circles); and Case II, *J* = 0.7, *J* � = 0.3 (red and blue circles).

results significantly deviate from the ARPES experimental data Ding et al. (2008). In Case II, Δ1/Δ2=2, in agreement with Ref.Ding et al. (2008), however, the ratios of these two gaps with respect to T*c*, 2Δ*α*/*kBTc*, also strongly disagree with Ref.Ding et al. (2008). These facts demonstrate that there exist some essential shortages in the present t-t� -J-J� model or in the self-consistent field method. One also notices that for Case II, the decline of the two superconducting energy gaps with the increasing temperature is not smooth, which comes from the fact that the different local pairing order parameters, Δ1*<sup>α</sup> <sup>x</sup>*/*<sup>y</sup>* and <sup>Δ</sup>2*<sup>α</sup> <sup>x</sup>*±*y*, interplay with each other, reflecting the anisotropic pairing symmetry in Case II.

#### **3.4 Angle dependence of superconducting energy gaps**

In this section we present the dependence of each superconducting energy gap with the d+s-wave pairing on the orientational angle, and show that the anisotropy of the superconducting energy gaps crucially depends on the inter-orbital hopping and the ratio of J'/J.

The ARPES experiment provides direct information about the quasiparticle spectra in normal state and the pairing symmetry of the superconducting energy gaps in the superconducting state. Here we present our theoretical results of the angle resolved energy gaps of the two orbits, and compare them with experimental observation. Fig. 4 shows the superconducting energy gap characters of the t-t� -J-J� model for the two sets of the parameters in Case I and II. In 12 Will-be-set-by-IN-TECH

Fig. 3. Temperature dependence of the superconducting energy gaps near the small hole Fermi surface (*α*1) and the large hole Fermi surface (*α*2) around the Γ point along the *θ* = 0*<sup>o</sup>* direction in the polar coordinate system. The doping concentration x=0.18. Theoretical

results significantly deviate from the ARPES experimental data Ding et al. (2008). In Case II, Δ1/Δ2=2, in agreement with Ref.Ding et al. (2008), however, the ratios of these two gaps with respect to T*c*, 2Δ*α*/*kBTc*, also strongly disagree with Ref.Ding et al. (2008). These facts

the self-consistent field method. One also notices that for Case II, the decline of the two superconducting energy gaps with the increasing temperature is not smooth, which comes

In this section we present the dependence of each superconducting energy gap with the d+s-wave pairing on the orientational angle, and show that the anisotropy of the superconducting energy gaps crucially depends on the inter-orbital hopping and the ratio

The ARPES experiment provides direct information about the quasiparticle spectra in normal state and the pairing symmetry of the superconducting energy gaps in the superconducting state. Here we present our theoretical results of the angle resolved energy gaps of the two orbits, and compare them with experimental observation. Fig. 4 shows the superconducting

demonstrate that there exist some essential shortages in the present t-t�

from the fact that the different local pairing order parameters, Δ1*<sup>α</sup>*

each other, reflecting the anisotropic pairing symmetry in Case II.


**3.4 Angle dependence of superconducting energy gaps**

= 0.7 (wine and green circles); and Case II, *J* = 0.7, *J*

� = 0.3

model or in

*<sup>x</sup>*±*y*, interplay with


*<sup>x</sup>*/*<sup>y</sup>* and <sup>Δ</sup>2*<sup>α</sup>*

model for the two sets of the parameters in Case I and II. In

parameters: Case I, *J* = 0.3, *J*

energy gap characters of the t-t�

(red and blue circles).

of J'/J.

�

Fig. 4. The angle dependence of the superconducting energy gaps near the small hole Fermi surface (*α*1) and the large hole Fermi surface (*α*2) around the Γ point in the polar coordinate system. The theoretical parameters are the same as these in Fig. 3, Case I, *J* = 0.3, *J* � = 0.7 (wine and green circles); and Case II, *J* = 0.7, *J* � = 0.3 (red and blue circles).

both cases, two distinct gaps open on the hole-like Fermi sheets, *α*<sup>1</sup> and *α*2, as seen in Fig.2. The presence of two different energy gaps demonstrates the nature of a multi-gap superconductor in the t-t� -J-J� model. Our results show that for the case I, the superconducting energy gap structure exhibits a nearly isotropic symmetry with invisible anisotropy, as seen in Fig.4. A large energy gap opens on the small hole Fermi sheet (*α*1), and a small energy gap opens on the large hole Fermi sheet (*α*2). For the case II, the angular dependence of the energy gaps is visible, exhibiting weak spatial anisotropy. The oscillation amplitude is about 16%. However, the amplitudes of the superconducting energy gaps on the different Fermi surfaces, *α*<sup>1</sup> and *α*2, are contrast to these in Case I, i.e., a small energy gap opens on the small Fermi surface sheet (*α*1), and a large energy gap opens on the large hole fermi surface sheet (*α*2).

One finds that in Case I, the anisotropy of the superconducting energy gaps is very weak, consistent with Zhao *et al*.'s Zhao et al. (2008) and Ding *et al*. � *s* Ding et al. (2008) ARPES data. In Case II, the superconducting energy gaps with about 16% anisotropy is in agreement with the ARPES experiment by Kondo *et al*. Kondo et al. (2008). Noticing that in Case II, such spatial anisotropy is still under the resolution of the ARPES experiment, hence does not conflict with Zhao *et al*.'s Zhao et al. (2008) and Ding *et al*. � *s* Ding et al. (2008) observation. It is the *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> -wave pairing symmetry that leads to weakly anisotropy and nodeless superconducting energy gap structures. Although the *dx*<sup>2</sup>−*ηy*<sup>2</sup> -wave pairing has nodes in the line cos *kx* − *η* cos *ky* = 0, and the *Sx*<sup>2</sup>*y*<sup>2</sup> -wave pairing has nodes in the lines *kx* = ±*π*/2 or *ky* <sup>=</sup> *<sup>π</sup>*/2. The mixed superconducting pairing symmetry, *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> , diminishes the

(2008). Such a behavior deviating from the *T*<sup>3</sup> law may be attributed from the multi-gap character of this system, and such a drop reflects the different energy gaps in different orbits. Surely, more meticulous studies are needed in the near future. We also notice that the extended *s*<sup>±</sup> energy gaps found by Parker *etal*. also can give the same NMR relaxation rate in superconducting pnictides Parker et al. (2008). Parish *et al*. Parish et al. (2008) suggested that the deviation of the *T*<sup>3</sup> law in the spin-lattice relaxation arises from the inter-band

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 39

From the preceding discussions, we find that many unusual properties in the normal state and the superconductive phase of newly discovered FeAs compounds could be qualitatively

good approximation to describe iron pnictide superconductors. Within this scenario, the mixing pairing symmetry with *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> -wave contributes to the weakly anisotropic and nodeless energy gaps. Such a pairing symmetry assembles the characters of usual *d*-wave and *s*-wave, hence shares the properties of the usual *d*-wave superconductors, like cuprates, and the *s*-wave superconductor, such as MgB2. Nevertheless, to quantitatively compare the theoretical results with the experimental observation, more subtle band structures of the

state which is the same as the pairing symmetry obtained by Chubukov *et al*Chubukov et al. (2008). It may seem strange that this intermediate coupling theory based upon the proximity to a Mott transition has essentially the same pairing solutions as the Fermi-liquid analysis of Ref.Mazin et al. (2009). But it is not surprising at all because the fermiology and the spin fluctuation wave vector (the structure of magnetic excitations in the reciprocal space) predetermine this symmetry, as is suggested by Mazin et al. (2009). There is, however, an important difference between our results and those of Chubukov *et al* Chubukov et al. (2008). In their case, pairing mechanism is due to the increase in the intra-band pairing hopping term,

not necessarily due to spin-fluctuations that is the pairing mechanism in our analysis.

expect that the more elaborate tight-binding parameters and anisotropic coupling J-J�

Also, one should keep in mind that a completely quantitative comparison between the theory

topology structure of the Fermi surfaces of the FeAs superconductors, but does not contain all the details of the Fermi surfaces and the band structures in iron pnictide compounds. On the

will improve the present results in future studies. Also the present constrained mean-field

In some FeAs-based superconductors, the weakly anisotropic orbital symmetry makes it very difficult to distinguish which orbitals are involved in the formation of the superconducting state. To further uncover the orbital dependence of the superconducting energy gaps, we study the superconducting properties of a highly anisotropic two-orbital t-J model in the strong correlation regime. We study how the phase diagram evolves with the band asymmetric factor *R* = *t*22/*t*11, and the numerical result is shown in Fig. 6 in the

model, showing that to some extent, this model is a



, *Sx*<sup>2</sup>*y*<sup>2</sup> is a dominating pairing

model only describes the

model. Hence, we

model

, might be a strong asymmetry

**4. Comparison with other theories and experimental observations**


It is of interest that for sufficiently large NNN spin coupling J�

and experiment is still difficult, since the present two-orbital t-t�

other hand, in the realistic material, the spin couplings, *J* and J�

approximation needs to be further improved.

Yin et al. (2008), which is not taken into account in the present t-t�

contribution.

t-t� -J-J�

interpreted in the two-orbital t-t�

model are expected.

nodes, so the system exhibits weakly anisotropic and nodeless *s*-wave-like energy gaps on the Fermi surface sheets.

#### **3.5 Spin-lattice relaxation rate in NMR**

In this subsection one can also obtain the theoretical spin-lattice relaxation data under different temperatures, especially the temperature-dependence of the Knight shift in iron-pnictide superconductors. We attribute the unusual T-dependence of the Knight shift to the multi-gap character.

Fig. 5. Temperature dependence of the spin lattice relaxation rate in the t-t� -J-J� model. The red arrow indicates the superconducting critical temperature T*c*. The red line is the T<sup>3</sup> law for comparison. Inset shows the detail near T*c*. The theoretical parameters are the same as case II in Fig.2.

Although many experimental measurements, such as the Andreev reflection Chen et al. (2008), the exponential temperature dependence of the penetration depths and the ARPES Ding et al. (2008); Zhao et al. (2008), observed the nodeless gap function in the superconducting phase of ReFeAsO1−*x*F*<sup>x</sup>* and Ba1−*x*K*x*Fe2As2 compounds, the line nodes in the superconducting energy gap was also suggested by the NMR experiment Matano (2008). The two characters in the NMR experiment supported the line nodes: lack of the coherence peak and the *T*<sup>3</sup> behavior in the nuclear spin-lattice relaxation rate, 1/*T*1. Using the gap function obtained in this paper, we calculate the spin lattice relaxation rate 1/T*s*, and the numerical result is shown in Fig. 5. We also plot the *T*<sup>3</sup> law (the red line) for a comparison. It is found that over a wide temperature range, the spin lattice relaxation rate in the present model can be fitted by the *T*<sup>3</sup> law, in agreement with the observation of the NMR experiments Matano (2008).

A small coherence peak appears around the critical transition temperature, as clearly seen in the inset of Fig.5. Experimentally, such a small coherence peak may be easily suppressed by the impurity effect or the antiferromagnetic spin fluctuations, similar to the situations in cuprates. This leads to the missing of the Hebel-Slichter coherence peak in the NMR experiment in iron pnictide SC. With decreasing temperature, one finds a drop in the spin lattice relaxation rate, 1/*T*1*s*, consistent with the observation of the NMR experiments Matano 14 Will-be-set-by-IN-TECH

nodes, so the system exhibits weakly anisotropic and nodeless *s*-wave-like energy gaps on the

In this subsection one can also obtain the theoretical spin-lattice relaxation data under different temperatures, especially the temperature-dependence of the Knight shift in iron-pnictide superconductors. We attribute the unusual T-dependence of the Knight shift

Fig. 5. Temperature dependence of the spin lattice relaxation rate in the t-t�

red arrow indicates the superconducting critical temperature T*c*. The red line is the T<sup>3</sup> law for comparison. Inset shows the detail near T*c*. The theoretical parameters are the same as

Although many experimental measurements, such as the Andreev reflection Chen et al. (2008), the exponential temperature dependence of the penetration depths and the ARPES Ding et al. (2008); Zhao et al. (2008), observed the nodeless gap function in the superconducting phase of ReFeAsO1−*x*F*<sup>x</sup>* and Ba1−*x*K*x*Fe2As2 compounds, the line nodes in the superconducting energy gap was also suggested by the NMR experiment Matano (2008). The two characters in the NMR experiment supported the line nodes: lack of the coherence peak and the *T*<sup>3</sup> behavior in the nuclear spin-lattice relaxation rate, 1/*T*1. Using the gap function obtained in this paper, we calculate the spin lattice relaxation rate 1/T*s*, and the numerical result is shown in Fig. 5. We also plot the *T*<sup>3</sup> law (the red line) for a comparison. It is found that over a wide temperature range, the spin lattice relaxation rate in the present model can be fitted by the *T*<sup>3</sup> law, in agreement with the observation of the NMR experiments

A small coherence peak appears around the critical transition temperature, as clearly seen in the inset of Fig.5. Experimentally, such a small coherence peak may be easily suppressed by the impurity effect or the antiferromagnetic spin fluctuations, similar to the situations in cuprates. This leads to the missing of the Hebel-Slichter coherence peak in the NMR experiment in iron pnictide SC. With decreasing temperature, one finds a drop in the spin lattice relaxation rate, 1/*T*1*s*, consistent with the observation of the NMR experiments Matano


model. The

Fermi surface sheets.

to the multi-gap character.

case II in Fig.2.

Matano (2008).

**3.5 Spin-lattice relaxation rate in NMR**

(2008). Such a behavior deviating from the *T*<sup>3</sup> law may be attributed from the multi-gap character of this system, and such a drop reflects the different energy gaps in different orbits. Surely, more meticulous studies are needed in the near future. We also notice that the extended *s*<sup>±</sup> energy gaps found by Parker *etal*. also can give the same NMR relaxation rate in superconducting pnictides Parker et al. (2008). Parish *et al*. Parish et al. (2008) suggested that the deviation of the *T*<sup>3</sup> law in the spin-lattice relaxation arises from the inter-band contribution.

## **4. Comparison with other theories and experimental observations**

From the preceding discussions, we find that many unusual properties in the normal state and the superconductive phase of newly discovered FeAs compounds could be qualitatively interpreted in the two-orbital t-t� -J-J� model, showing that to some extent, this model is a good approximation to describe iron pnictide superconductors. Within this scenario, the mixing pairing symmetry with *dx*<sup>2</sup>−*ηy*<sup>2</sup> <sup>+</sup> *Sx*<sup>2</sup>*y*<sup>2</sup> -wave contributes to the weakly anisotropic and nodeless energy gaps. Such a pairing symmetry assembles the characters of usual *d*-wave and *s*-wave, hence shares the properties of the usual *d*-wave superconductors, like cuprates, and the *s*-wave superconductor, such as MgB2. Nevertheless, to quantitatively compare the theoretical results with the experimental observation, more subtle band structures of the t-t� -J-J� model are expected.

It is of interest that for sufficiently large NNN spin coupling J� , *Sx*<sup>2</sup>*y*<sup>2</sup> is a dominating pairing state which is the same as the pairing symmetry obtained by Chubukov *et al*Chubukov et al. (2008). It may seem strange that this intermediate coupling theory based upon the proximity to a Mott transition has essentially the same pairing solutions as the Fermi-liquid analysis of Ref.Mazin et al. (2009). But it is not surprising at all because the fermiology and the spin fluctuation wave vector (the structure of magnetic excitations in the reciprocal space) predetermine this symmetry, as is suggested by Mazin et al. (2009). There is, however, an important difference between our results and those of Chubukov *et al* Chubukov et al. (2008). In their case, pairing mechanism is due to the increase in the intra-band pairing hopping term, not necessarily due to spin-fluctuations that is the pairing mechanism in our analysis.

Also, one should keep in mind that a completely quantitative comparison between the theory and experiment is still difficult, since the present two-orbital t-t� -J-J� model only describes the topology structure of the Fermi surfaces of the FeAs superconductors, but does not contain all the details of the Fermi surfaces and the band structures in iron pnictide compounds. On the other hand, in the realistic material, the spin couplings, *J* and J� , might be a strong asymmetry Yin et al. (2008), which is not taken into account in the present t-t� -J-J� model. Hence, we expect that the more elaborate tight-binding parameters and anisotropic coupling J-J� model will improve the present results in future studies. Also the present constrained mean-field approximation needs to be further improved.

In some FeAs-based superconductors, the weakly anisotropic orbital symmetry makes it very difficult to distinguish which orbitals are involved in the formation of the superconducting state. To further uncover the orbital dependence of the superconducting energy gaps, we study the superconducting properties of a highly anisotropic two-orbital t-J model in the strong correlation regime. We study how the phase diagram evolves with the band asymmetric factor *R* = *t*22/*t*11, and the numerical result is shown in Fig. 6 in the

the orbital dependent superconducting regime, and the orbital dependent superconducting phase is more robust with the deviation of R from the unity, as we see in Fig.6b. As one expects, when the hopping integral ratio R is larger than the unity, the behavior of Δ<sup>1</sup> is inter-changed with that of Δ2. The properties in the system with R are analogous to those with 1/R in the

Pairing Symmetry and Multiple Energy Gaps in Multi-Orbital Iron-Pnictide Superconductors 41

We notice the profound difference of the superconducting properties between cuprates and iron pnictides. Comparing with the copper-based superconductors with a 4-fold rotational symmetry, the inequivalence between the *x*- and *y*-direction of the orbit d*xz*/*yz* in iron pnictides results in the anisotropic factor, *η*, and leads to a distinct pairing symmetry. In the present iron pnictide superconductors, the NNN spin coupling contributes an important role to the *Sx*<sup>2</sup>*y*<sup>2</sup> pairing symmetry. Further, the multi-orbital character also contributes two weakly anisotropic and nodeless energy gaps, significantly different from the single energy

Strong next-nearest-neighbour coupling and inter-orbital hopping in iron-pnictide superconductors favor a weak anisotropic and nodeless d+s wave symmetry. From

In summary, our results have shown that many properties observed in iron-based superconductors could be comprehensively understood in the present model qualitatively. In the reasonable physical parameters region of LaFeAsO1−*x*F*x*, the pairing symmetry of the model is nearly isotropic and nodeless *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> -wave, mainly originating from the Fermi surface topology and the spin fluctuation in these systems, which is in agreement with the

Kamihara Y., Watanabe T., Hirano M. and Hosono H. (2008). Iron-based layered

Chen X. H., Wu T., Wu G., Liu R. H., Chen H., and Fang D. F. (2008). Superconductivity at 43

Chen G. F., Li Z., Li G., Zhou J., Wu D., Dong J., Hu W. Z., Zheng P., Chen Z. J., Luo J. L., and

Wu G., Xie Y. L., Chen H., Zhong M., Liu R. H., Shi B. C., Li Q. J., Wang X. F., Wu T., Yan Y. J.,

McMillan W. L. (1968). Transition Temperature of Strong-Coupled Superconductors. *Phys.*

superconductor La[O1−*x*F*x*]FeAs(x=0.05-0.12) with Tc=26 K, *J. Am. Chem. Soc.*, Vol.

Wang N. L. (2008). Superconducting properties of Fe-based layered superconductor

Ying J. J., Chen X. H. (2009). Superconductivity at 56 K in samarium-doped SrFeAsF.

*<sup>x</sup>*±*y*. Thus, when the NN interaction J is dominant in the system,

is considerably larger than J, the local superconducting order

*<sup>x</sup>*/*<sup>y</sup>* become a dominant term in Eq.(5); and

*<sup>x</sup>*/*<sup>y</sup>* and the

Eq.(3), one could see that the NN interaction J favors the order parameters Δ1*<sup>α</sup>*

observation of ARPES and the NMR experiments in ironpnictide superconductors.

K in SmFeAsO1−*x*F*x*. *Nature*, Vol. 453, 761-762, ISSN 0028-0836.

LaO0.9F0.1FeAs. *Phys. Rev. Lett.*, Vol. 101, 057007, ISSN 0031-9007.

*J. Phys.: Condens. Matter*, Vol. 21, 142203, ISSN 0953-8984.

absence of the crystalline field splitting.

**5. Remarks and conclusions**

gap in the cuprate superconductors.

favors Δ2*<sup>α</sup>*

the local superconducting order parameters Δ1*<sup>α</sup>*

130, 3296-3297, ISSN 0002-7863.

*Rev.*, Vol. 167, 331, ISSN 0031-899X.

*<sup>x</sup>*±*<sup>y</sup>* become dominant in Eq.(5).

NNN interaction J�

parameters Δ2*<sup>α</sup>*

**6. References**

when the NNN interaction J�

strongly correlation regime. Note that we consider only the nearest-neighbor hopping on a square lattice. It is found that at *n* = 1.98, the difference between Δ<sup>1</sup> and Δ<sup>2</sup> increases with *R* deviating from the unity. The superconducting order parameters exhibit different behaviors: Δ<sup>2</sup> monotonously increases and almost saturates as R < *Rc* ≈ 0.6; however, Δ<sup>1</sup> monotonously decreases and vanishes at *Rc*, indicating the appearance of an orbital dependent superconducting phase, where the superconducting gap in orbit 2 exponentially approaches zero, and the energy gap in orbit remains finite. As the doping concentration increases to *n* = 1.95 in Fig.6b, the two superconducting order parameters behave similarly to Fig.6a. Finally, the TGSC-intermediate superconducting phase transition occurs at *Rc* ≈ 0.7. Obviously, with the decrease of *R*, the bandwidth of 2-orbit considerably shrinks and the pairing coupling of the orbit-2 electrons significantly deviates from that of 1-orbit. Thus, the orbital-dependent intermediate superconducting phase easily occurs when the symmetry of the orbital hopping is broken.

With the increase of the hopping integral asymmetry, the bandwidth of the 2-orbit becomes narrower and narrower, and more and more orbit-2 electrons transfer to orbit-1, hence the amplitude of the superconducting order parameter of 2-orbit gradually decreases to zero. In the same time, the superconducting order parameter of 1-orbit increases. The system enters

Fig. 6. (Color online) Dependence of the superconducting order parameters Δ*m* (m=1,2) on the level splitting *E*<sup>Δ</sup> for (a) *R* = 1 and *δ* = 0.02, and (b) *R* = 0.8 and *δ* = 0.02, respectively. Here TGSC and OSSC denote the two-gap and orbital dependent superconducting phases, respectively

the orbital dependent superconducting regime, and the orbital dependent superconducting phase is more robust with the deviation of R from the unity, as we see in Fig.6b. As one expects, when the hopping integral ratio R is larger than the unity, the behavior of Δ<sup>1</sup> is inter-changed with that of Δ2. The properties in the system with R are analogous to those with 1/R in the absence of the crystalline field splitting.

### **5. Remarks and conclusions**

16 Will-be-set-by-IN-TECH

strongly correlation regime. Note that we consider only the nearest-neighbor hopping on a square lattice. It is found that at *n* = 1.98, the difference between Δ<sup>1</sup> and Δ<sup>2</sup> increases with *R* deviating from the unity. The superconducting order parameters exhibit different behaviors: Δ<sup>2</sup> monotonously increases and almost saturates as R < *Rc* ≈ 0.6; however, Δ<sup>1</sup> monotonously decreases and vanishes at *Rc*, indicating the appearance of an orbital dependent superconducting phase, where the superconducting gap in orbit 2 exponentially approaches zero, and the energy gap in orbit remains finite. As the doping concentration increases to *n* = 1.95 in Fig.6b, the two superconducting order parameters behave similarly to Fig.6a. Finally, the TGSC-intermediate superconducting phase transition occurs at *Rc* ≈ 0.7. Obviously, with the decrease of *R*, the bandwidth of 2-orbit considerably shrinks and the pairing coupling of the orbit-2 electrons significantly deviates from that of 1-orbit. Thus, the orbital-dependent intermediate superconducting phase easily occurs when the symmetry of

With the increase of the hopping integral asymmetry, the bandwidth of the 2-orbit becomes narrower and narrower, and more and more orbit-2 electrons transfer to orbit-1, hence the amplitude of the superconducting order parameter of 2-orbit gradually decreases to zero. In the same time, the superconducting order parameter of 1-orbit increases. The system enters

Fig. 6. (Color online) Dependence of the superconducting order parameters Δ*m* (m=1,2) on the level splitting *E*<sup>Δ</sup> for (a) *R* = 1 and *δ* = 0.02, and (b) *R* = 0.8 and *δ* = 0.02, respectively. Here TGSC and OSSC denote the two-gap and orbital dependent superconducting phases,

the orbital hopping is broken.

respectively

We notice the profound difference of the superconducting properties between cuprates and iron pnictides. Comparing with the copper-based superconductors with a 4-fold rotational symmetry, the inequivalence between the *x*- and *y*-direction of the orbit d*xz*/*yz* in iron pnictides results in the anisotropic factor, *η*, and leads to a distinct pairing symmetry. In the present iron pnictide superconductors, the NNN spin coupling contributes an important role to the *Sx*<sup>2</sup>*y*<sup>2</sup> pairing symmetry. Further, the multi-orbital character also contributes two weakly anisotropic and nodeless energy gaps, significantly different from the single energy gap in the cuprate superconductors.

Strong next-nearest-neighbour coupling and inter-orbital hopping in iron-pnictide superconductors favor a weak anisotropic and nodeless d+s wave symmetry. From Eq.(3), one could see that the NN interaction J favors the order parameters Δ1*<sup>α</sup> <sup>x</sup>*/*<sup>y</sup>* and the NNN interaction J� favors Δ2*<sup>α</sup> <sup>x</sup>*±*y*. Thus, when the NN interaction J is dominant in the system, the local superconducting order parameters Δ1*<sup>α</sup> <sup>x</sup>*/*<sup>y</sup>* become a dominant term in Eq.(5); and when the NNN interaction J� is considerably larger than J, the local superconducting order parameters Δ2*<sup>α</sup> <sup>x</sup>*±*<sup>y</sup>* become dominant in Eq.(5).

In summary, our results have shown that many properties observed in iron-based superconductors could be comprehensively understood in the present model qualitatively. In the reasonable physical parameters region of LaFeAsO1−*x*F*x*, the pairing symmetry of the model is nearly isotropic and nodeless *dx*<sup>2</sup>−*ηy*2+*Sx*<sup>2</sup>*y*<sup>2</sup> -wave, mainly originating from the Fermi surface topology and the spin fluctuation in these systems, which is in agreement with the observation of ARPES and the NMR experiments in ironpnictide superconductors.

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**1. Introduction**

semiconductors (FS), and so on.

technological relevance. [5–8]

temperature (*TC*) SCs have a *dx*<sup>2</sup>

Superconducting spintronics [1–3] is one of the most attractive subjects of spintronics. A variety of spin-injecting tunnelling junctions composed of ferromagnet and superconductor (SC) with different wave pairing symmetries have been explored theoretically and experimentally, where the ferromagnet includes ferromagnetic metals (FM), ferromagnetic

**Josephson Current in a Ferromagnetic** 

**Semiconductor Junction with** 

*Department of Physics, Nanjing Normal University, Nanjing* 

**Superconducting Contacts** 

Y.C. Tao

**3**

*People's Republic of China* 

**Semiconductor/Semiconductor/Ferromagnetic** 

Recently, the FS arouses great interests due to several important material properties, for example, carrier type (i.e., *n* or *p*), carrier densities, and ferromagnetism, can be easily controlled in it. [4] The origin of ferromagnetism in the FS can be explained by applying a picture that uniform itinerant carrier spin mediates a long-range ferromagnetic order between the ions Mn+<sup>2</sup> with spin. *p*-type III-V GaMnAs is a prototype FS with a large spin polarization even for small Mn concentrations [4, 5] and recent experiments demonstrate that the Curie temperature of this material can be as high as 150 K, [4, 5] indicating promise for possible

On the other hand, current can flow without dissipation through a thin insulating layer or weak link separating two superconductors (SC). A phase difference *φ* between the SCs appears on account of the quantum character of the superflow and the current-phase relationship of the link is periodic function of *φ*. For a tunnel barrier, it is given by *I* = *IC* sin(*φ*) as first found by Josephson [9], where *IC* is the critical Josephson current. When the two SCs coupled by a thin ferromagnetic layer [10–29], *IC* can change sign. Referring to the Josephson current-phase relationship, the change is corresponding to a *π*-phase shift of *φ*. And thus Josephson junctions can present a negative coupling, which are usually called *π* junctions. The transitions from positive to negative coupling (the so-called transition from 0 to *π* junction) by increasing the FS layer thickness observed by experiments are qualitatively in agreement with those of theoretical calculations. It has been widely accepted that high-critical

gap of such a pairing state manifests a sign change at some directions of the Fermi wave


*a*−*x*<sup>2</sup> *b*


## **Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic Semiconductor Junction with Superconducting Contacts**

Y.C. Tao *Department of Physics, Nanjing Normal University, Nanjing People's Republic of China* 

#### **1. Introduction**

20 Will-be-set-by-IN-TECH

44 Superconductors – Properties, Technology, and Applications

Parker D., Dolgov O. V., Korshunov M. M., Golubov A. A., and Mazin I. I. (2008). Extended

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Chubukov A. V., Efremov D. V., and Eremin I. (2008). Magnetism, superconductivity, and

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ISSN 1098-0121.

1098-0121.

l´

Yin Z. P., Leb´

s+<sup>−</sup> scenario for the nuclear spin-lattice relaxation rate in superconducting pnictides.

cos(kx)cos(ky) superconductivity in the iron pnictides. *Phys. Rev. B*, Vol. 78, 144514,

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lgue S., Han M. J., Neal B. P., Savrasov S. Y., and Pickett W. E. (2008). Electron-Hole Symmetry and Magnetic Coupling in Antiferromagnetic LaFeAsO.

> Superconducting spintronics [1–3] is one of the most attractive subjects of spintronics. A variety of spin-injecting tunnelling junctions composed of ferromagnet and superconductor (SC) with different wave pairing symmetries have been explored theoretically and experimentally, where the ferromagnet includes ferromagnetic metals (FM), ferromagnetic semiconductors (FS), and so on.

> Recently, the FS arouses great interests due to several important material properties, for example, carrier type (i.e., *n* or *p*), carrier densities, and ferromagnetism, can be easily controlled in it. [4] The origin of ferromagnetism in the FS can be explained by applying a picture that uniform itinerant carrier spin mediates a long-range ferromagnetic order between the ions Mn+<sup>2</sup> with spin. *p*-type III-V GaMnAs is a prototype FS with a large spin polarization even for small Mn concentrations [4, 5] and recent experiments demonstrate that the Curie temperature of this material can be as high as 150 K, [4, 5] indicating promise for possible technological relevance. [5–8]

> On the other hand, current can flow without dissipation through a thin insulating layer or weak link separating two superconductors (SC). A phase difference *φ* between the SCs appears on account of the quantum character of the superflow and the current-phase relationship of the link is periodic function of *φ*. For a tunnel barrier, it is given by *I* = *IC* sin(*φ*) as first found by Josephson [9], where *IC* is the critical Josephson current. When the two SCs coupled by a thin ferromagnetic layer [10–29], *IC* can change sign. Referring to the Josephson current-phase relationship, the change is corresponding to a *π*-phase shift of *φ*. And thus Josephson junctions can present a negative coupling, which are usually called *π* junctions. The transitions from positive to negative coupling (the so-called transition from 0 to *π* junction) by increasing the FS layer thickness observed by experiments are qualitatively in agreement with those of theoretical calculations. It has been widely accepted that high-critical temperature (*TC*) SCs have a *dx*<sup>2</sup> *a*−*x*<sup>2</sup> *b* -wave pairing symmetry, [30–32] in which the energy gap of such a pairing state manifests a sign change at some directions of the Fermi wave

Superconducting Contacts 3

<sup>47</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

At finite temperatures, the spin splitting energy Γ different for different kind of holes, is given by <sup>Γ</sup> <sup>=</sup> *Jpd* <sup>&</sup>lt; *<sup>S</sup><sup>z</sup>* <sup>&</sup>gt;, where *Jpd* stands for the *<sup>p</sup>* <sup>−</sup> *<sup>d</sup>* exchange coupling strength between itinerant holes and Mn2<sup>+</sup> ion impurity spins connected with the densities of Mn impurity and heavy or light holes, and < *S<sup>z</sup>* > implies the thermal average of the impurity Mn ion spins. To model the subband spin splitting and the relative spin-polarized degree *P*, we define the parameter *P* ≡ Γ/*EF*. The SC is assumed *s*-wave pairing and described by a BCS-like

one-electron energy relative to *EF* and Δ the energy gap. The energy gap Δ is determined by

The quasiparticle wave functions, in the Blonder-Tinkham-Klapwijk model, BDG

where *ω<sup>D</sup>* refers to Debye frequency and Δ<sup>0</sup> denotes energy gap at zero temperature.

0 ∗  *�*2

> 2 1 + *eβζ<sup>k</sup>*

> > *u<sup>σ</sup> vσ* = *E uσ vσ*

(*x*) = *U*�

+ *aeik*−*<sup>x</sup>*

 *veiφL*/2 *ue*−*iφL*/2

*d�<sup>k</sup> ζk*

(r) + *ησh*(r)

<sup>0</sup> (r) except for the effective mass replaced by the mass of electron *me*, Δ(*x*) is pair potential, *E* is the quasiparticle energy relative to *EF*, *σ* is the spin opposite to *σ* with ↑ and ↓, *ησ* = 1 for *σ* =↑, and *ησ* = −1 for *σ* =↓. In Eq. (3), as a result of that the spin-flip process has been neglected in the SC near interfaces in Refs. 34 and 35, the spin-dependent (four-component) BDG equation may be decoupled into two sets of two-component equations: *uσ*, *v<sup>σ</sup>* describing the spin-*σ* electronlike quasiparticle (ELQ) and

A SC/FS/SM/FS/SC Josephson tunnel junction as in Fig. 1 is considered [34, 35]. The left and right electrodes are made up of the same SCs: they are separated from a FS/SM/FS trilayer with the same FS GaMnAs layers of thickness *b* and SM GaAs of thickness *a*. Either the left and right FS/SC interfaces or SM/FS interfaces, can be approximately described by four *δ*-type

assumed to be symmetrical due to the same SCs and the same FS's, where *U* and *U*� depend on the product of the barrier height and width. For the ferromagnetic (F) alignment of the magnetizations of the two bilayers, h(−*b* − *a*/2 ≤ *z* ≤ −*a*/2) = h(*a*/2 ≤ *z* ≤ *b* + *a*/2), while for the antiferromagnetic (A) alignment, h(−*b* − *a*/2 ≤ *z* ≤ −*a*/2) = −h(*a*/2 ≤ *z* ≤ *b* + *a*/2). For the injection of an ELQ from the left SC with Andreev reflection as a HLQ (*a*) and normal

> *ueiφL*/2 *ve*−*iφL*/2

*<sup>k</sup>* <sup>+</sup> <sup>Δ</sup><sup>2</sup> with *�<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*2/2*me* <sup>−</sup> *EF* the

, (2)

<sup>0</sup> (r) has the same expression

*δ*(*z* ± *a*/2), respectively and

, (4)

, (3)

Hamiltonian, in which the excitation energy is *ζ<sup>k</sup>* =

Semiconductor Junction with Superconducting Contacts

ln <sup>Δ</sup><sup>0</sup> Δ = *h*¯ *ω<sup>D</sup>* 0

<sup>0</sup> (r) − *ησh*(r) Δ(*x*)

<sup>Δ</sup>∗(*x*) <sup>−</sup>*HSC*

where *u<sup>σ</sup>* and *v<sup>σ</sup>* are the two-component wavefunction, *HSC*

the self-consistent equation [34, 35]

*HSC*

spin-*σ* holelike quasiparticle (HLQ), respectively.

barrier potentials *V*(*x*) = *Uδ*[*z* ± (*b* + *a*/2)] and *V*�

reflection (*b*), the wave function are given by [34, 35]

*ik*+*<sup>x</sup>* + *be*−*ik*+*x*]

*ψSC*

*<sup>L</sup>* (*x*)=[*e*

**3. FS Josephson tunnelling junctions**

**3.1 Model and theory**

equation [36]

as *HFS*

vector. This feature is much different from that for a conventional *s*-wave SC. As a result of this sign change, it was predicted [33] that a sizable areal density of midgap states exists on a {110} surface of a *d*-wave SC. These midgap states have exerted the influence on the Josephson current of high *TC* SC junctions. More recently, the relative Josephson tunnelling junctions composed of FS and SC with *s*-wave pairing symmetry have being explored and some valuable results have been obtained,[34, 35] however, the works focused on the systems are scarce. Particularly, to our knowledge, we seldom see theoretical and experimental works taking into account the unconventional pairing symmetry, which is relevant to the high *TC* superconducting junctions composed of the FS. Therefore, it seems to be timely to review the *Superconducting Spintronics* based on the Josephson junctions consisting of FS and SC with s-wave pairing symmetry and consider the outlook for the future work.

In this chapter, the Josephson current flowing through an FS/semiconductor (SM) /FS trilayer connected to two superconductor (SC) electrodes is studied. We extend a general expression for dc Josephson current to the structure, in which the quasiparticle wave function, satisfying Bogoliubov-de Gennes (BDG) equation, is first obtained and then the temperature Green's function formalism is used. The Josephson current in the junction is find to be strongly dependent on not only the kinds of holes in the FS's and scattering potential strengths at the interfaces but also the relative orientations of the effective exchange field of the two FS's. We also show that in the coherent regime, the oscillatory dependences of the critical Josephson current on the FS and SM layer thicknesses and Josephson current on the macroscopic phase difference for the heavy and light holes in different orientations of the effective exchange field of the two FS's, are much different due to the different mismatches in the effective mass and Fermi velocity between the FS and SC, which is connected with the transition from positive (0) to negative (*π*) coupling. The effect can be exploited for the design of devices such as superconducting quantum interference devices with improved accuracy and convenience. In Sec. II, we present the theoretical approach. In Sec. III, FS Josephson tunneling junctions are studied. The summary and outlook are given in Sec. IV. Finally, the list of used symbols is presented in Sec. V.

#### **2. Theoretical approach: BDG equation**

Applying Bogoliubov-de Gennes (BDG) equation, [36] *Superconducting Spintronics* built on the Josephson junctions composed of GaMnAs FS and *s*-wave SC are studied, in which the AR ,[34, 35] hole types, mismatches in the effective mass and Fermi velocity of holes between the FS and SC, and strengths of potential scattering at the interface are taken into account. In the next parts of the chapter, the layers are taken to be the *y* − *z* plane and to be stacked along the *x*-direction, and the FS/SC interface, for example at *x* = *a*, is described by a *δ*-type potential *V*(*x*) = *Uδ*(*x* − *a*) with *U* dependent on the product of the barrier height and width.

As in FMs, a spin-polarized subband model with the spins parallel and antiparallel to the local magnetization is extended to the heavy and light hole bands. The hole Hamiltonian in the FS is simply given by

$$H\_{FS}(\boldsymbol{r}) = H\_0(\boldsymbol{r}) - h(\boldsymbol{r}) \cdot \boldsymbol{\sigma}. \tag{1}$$

Here *<sup>H</sup>*0(r) = <sup>−</sup>*h*¯ <sup>2</sup>∇*r*/2*mp* <sup>+</sup> *<sup>V</sup>*(r) is the kinetic energy with *mp* the mass of a hole plus the usual static potential, which is also the hole Hamiltonian of the SM GaAs, h(r) is the effective exchange field with the magnitude equal to Γ/2, and σ the conventional Pauli spin operator. At finite temperatures, the spin splitting energy Γ different for different kind of holes, is given by <sup>Γ</sup> <sup>=</sup> *Jpd* <sup>&</sup>lt; *<sup>S</sup><sup>z</sup>* <sup>&</sup>gt;, where *Jpd* stands for the *<sup>p</sup>* <sup>−</sup> *<sup>d</sup>* exchange coupling strength between itinerant holes and Mn2<sup>+</sup> ion impurity spins connected with the densities of Mn impurity and heavy or light holes, and < *S<sup>z</sup>* > implies the thermal average of the impurity Mn ion spins. To model the subband spin splitting and the relative spin-polarized degree *P*, we define the parameter *P* ≡ Γ/*EF*. The SC is assumed *s*-wave pairing and described by a BCS-like Hamiltonian, in which the excitation energy is *ζ<sup>k</sup>* = *�*2 *<sup>k</sup>* <sup>+</sup> <sup>Δ</sup><sup>2</sup> with *�<sup>k</sup>* <sup>=</sup> *<sup>h</sup>*¯ <sup>2</sup>*k*2/2*me* <sup>−</sup> *EF* the one-electron energy relative to *EF* and Δ the energy gap. The energy gap Δ is determined by the self-consistent equation [34, 35]

$$\ln\left(\frac{\Delta\_0}{\Delta}\right) = \int\_0^{\hbar\omega\_D} \frac{d\varepsilon\_k}{\zeta\_k} \frac{2}{1 + e^{\beta\zeta\_k}},\tag{2}$$

where *ω<sup>D</sup>* refers to Debye frequency and Δ<sup>0</sup> denotes energy gap at zero temperature.

The quasiparticle wave functions, in the Blonder-Tinkham-Klapwijk model, BDG equation [36]

$$
\begin{bmatrix} H\_0^{\text{SC}}(\mathbf{r}) - \eta\_{\mathcal{T}}h(\mathbf{r}) & \Delta(\mathbf{x}) \\ \Delta^\*(\mathbf{x}) & -H\_0^{\text{SC}^\*}(\mathbf{r}) + \eta\_{\overline{\sigma}}h(\mathbf{r}) \end{bmatrix} \begin{bmatrix} u\_{\sigma} \\ v\_{\overline{\sigma}} \end{bmatrix} = E \begin{bmatrix} u\_{\sigma} \\ v\_{\overline{\sigma}} \end{bmatrix} \tag{3}
$$

where *u<sup>σ</sup>* and *v<sup>σ</sup>* are the two-component wavefunction, *HSC* <sup>0</sup> (r) has the same expression as *HFS* <sup>0</sup> (r) except for the effective mass replaced by the mass of electron *me*, Δ(*x*) is pair potential, *E* is the quasiparticle energy relative to *EF*, *σ* is the spin opposite to *σ* with ↑ and ↓, *ησ* = 1 for *σ* =↑, and *ησ* = −1 for *σ* =↓. In Eq. (3), as a result of that the spin-flip process has been neglected in the SC near interfaces in Refs. 34 and 35, the spin-dependent (four-component) BDG equation may be decoupled into two sets of two-component equations: *uσ*, *v<sup>σ</sup>* describing the spin-*σ* electronlike quasiparticle (ELQ) and spin-*σ* holelike quasiparticle (HLQ), respectively.

#### **3. FS Josephson tunnelling junctions**

#### **3.1 Model and theory**

2 Will-be-set-by-IN-TECH

vector. This feature is much different from that for a conventional *s*-wave SC. As a result of this sign change, it was predicted [33] that a sizable areal density of midgap states exists on a {110} surface of a *d*-wave SC. These midgap states have exerted the influence on the Josephson current of high *TC* SC junctions. More recently, the relative Josephson tunnelling junctions composed of FS and SC with *s*-wave pairing symmetry have being explored and some valuable results have been obtained,[34, 35] however, the works focused on the systems are scarce. Particularly, to our knowledge, we seldom see theoretical and experimental works taking into account the unconventional pairing symmetry, which is relevant to the high *TC* superconducting junctions composed of the FS. Therefore, it seems to be timely to review the *Superconducting Spintronics* based on the Josephson junctions consisting of FS and SC with

In this chapter, the Josephson current flowing through an FS/semiconductor (SM) /FS trilayer connected to two superconductor (SC) electrodes is studied. We extend a general expression for dc Josephson current to the structure, in which the quasiparticle wave function, satisfying Bogoliubov-de Gennes (BDG) equation, is first obtained and then the temperature Green's function formalism is used. The Josephson current in the junction is find to be strongly dependent on not only the kinds of holes in the FS's and scattering potential strengths at the interfaces but also the relative orientations of the effective exchange field of the two FS's. We also show that in the coherent regime, the oscillatory dependences of the critical Josephson current on the FS and SM layer thicknesses and Josephson current on the macroscopic phase difference for the heavy and light holes in different orientations of the effective exchange field of the two FS's, are much different due to the different mismatches in the effective mass and Fermi velocity between the FS and SC, which is connected with the transition from positive (0) to negative (*π*) coupling. The effect can be exploited for the design of devices such as superconducting quantum interference devices with improved accuracy and convenience. In Sec. II, we present the theoretical approach. In Sec. III, FS Josephson tunneling junctions are studied. The summary and outlook are given in Sec. IV. Finally, the list of used symbols is

Applying Bogoliubov-de Gennes (BDG) equation, [36] *Superconducting Spintronics* built on the Josephson junctions composed of GaMnAs FS and *s*-wave SC are studied, in which the AR ,[34, 35] hole types, mismatches in the effective mass and Fermi velocity of holes between the FS and SC, and strengths of potential scattering at the interface are taken into account. In the next parts of the chapter, the layers are taken to be the *y* − *z* plane and to be stacked along the *x*-direction, and the FS/SC interface, for example at *x* = *a*, is described by a *δ*-type potential *V*(*x*) = *Uδ*(*x* − *a*) with *U* dependent on the product of the barrier height and width. As in FMs, a spin-polarized subband model with the spins parallel and antiparallel to the local magnetization is extended to the heavy and light hole bands. The hole Hamiltonian in the FS

Here *<sup>H</sup>*0(r) = <sup>−</sup>*h*¯ <sup>2</sup>∇*r*/2*mp* <sup>+</sup> *<sup>V</sup>*(r) is the kinetic energy with *mp* the mass of a hole plus the usual static potential, which is also the hole Hamiltonian of the SM GaAs, h(r) is the effective exchange field with the magnitude equal to Γ/2, and σ the conventional Pauli spin operator.

*HFS*(r) = *H*0(r) − h(r) · σ. (1)

s-wave pairing symmetry and consider the outlook for the future work.

presented in Sec. V.

is simply given by

**2. Theoretical approach: BDG equation**

A SC/FS/SM/FS/SC Josephson tunnel junction as in Fig. 1 is considered [34, 35]. The left and right electrodes are made up of the same SCs: they are separated from a FS/SM/FS trilayer with the same FS GaMnAs layers of thickness *b* and SM GaAs of thickness *a*. Either the left and right FS/SC interfaces or SM/FS interfaces, can be approximately described by four *δ*-type barrier potentials *V*(*x*) = *Uδ*[*z* ± (*b* + *a*/2)] and *V*� (*x*) = *U*� *δ*(*z* ± *a*/2), respectively and assumed to be symmetrical due to the same SCs and the same FS's, where *U* and *U*� depend on the product of the barrier height and width. For the ferromagnetic (F) alignment of the magnetizations of the two bilayers, h(−*b* − *a*/2 ≤ *z* ≤ −*a*/2) = h(*a*/2 ≤ *z* ≤ *b* + *a*/2), while for the antiferromagnetic (A) alignment, h(−*b* − *a*/2 ≤ *z* ≤ −*a*/2) = −h(*a*/2 ≤ *z* ≤ *b* + *a*/2). For the injection of an ELQ from the left SC with Andreev reflection as a HLQ (*a*) and normal reflection (*b*), the wave function are given by [34, 35]

$$\psi\_{L}^{\rm SC}(\mathbf{x}) = [e^{i\mathbf{k}^+ \cdot \mathbf{x}} + be^{-i\mathbf{k}^+ \cdot \mathbf{x}}] \begin{pmatrix} ue^{i\boldsymbol{\phi}\_{\mathbb{L}}/2} \\ ue^{-i\boldsymbol{\phi}\_{\mathbb{L}}/2} \end{pmatrix} + ae^{i\mathbf{k}^- \cdot \mathbf{x}} \begin{pmatrix} ve^{i\boldsymbol{\phi}\_{\mathbb{L}}/2} \\ ue^{-i\boldsymbol{\phi}\_{\mathbb{L}}/2} \end{pmatrix},\tag{4}$$

Superconducting Contacts 5

<sup>49</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

*E* → *iωn*, the Matsubara frequencies are *ω<sup>n</sup>* = *πkBT*(2*n* + 1) with *n* = 0, ±1, ±2,..., and

*<sup>d</sup><sup>θ</sup>* sin *<sup>θ</sup>* cos *<sup>θ</sup>* <sup>∑</sup>*<sup>ω</sup>n*,*<sup>σ</sup>*

In Fig. 2 are illustrated the numerical results for *IC* as a function of the FS thickness *b* with

the F and A alignments. [34] The damped oscillations of the critical Josephson currents *IC* for the heavy and light holes are all found to be as a function of *b*, which is also the characteristic feature of the ballistic SC/FM/SC junctions and always related to the crossovers between 0 and *π* states [24]. Microscopically, Andreev reflections play an important role in the transfer of Cooper pairs through the FS. An electronlike quasiparticle in FS with energy lower than the superconducting energy gap Δ, cannot enter into the SC, but is reflected at the FS/SC interface as a hole and reflected back as an electron at the opposite SC/FS interface. The constructive interference of the electronlike and holelike excitations gives rise to Andreev bound states in FS, which carry the supercurrent. However, in the FS, the spin splitting of spin band exerts an influence on the Andreev reflection when the spin of the quasiparticles is reversed by the Andreev reflection, which means that the exchange energy is gained or lost by a quasiparticle Andreev-reflected at the FS/SC interface. It follows that the superconducting pair amplitude induced in FS by the proximity to the SC is spatially modulated and the phase increases linearly with the distance from the SC/FS interface. Since the Josephson critical current *IC* is proportional to the pair amplitude in the FS, *IC* follows the sign of the order parameter in the FS. Whether it is either positive or negative is determined by the phase accumulating in the FS. Furthermore, it is found that as shown in Fig. 2, at *Z* = 0 and *Z*� = 0, for the heavy holes, the general tendency of variations of the oscillation amplitude with increasing *b* is to first decrease, then increase, and last gradually reduce to zero in the F and A alignments, while for the light holes, gradually decrease to zero all the time. We also find that for either the heavy or light holes, the behaviors in the F and A alignments are very similar in spite of the values

*<sup>F</sup>* at different scattering potential strengths *Z* and *Z*� for the heavy and light holes in

. However, for *Z* and *Z*� unequal to zero, it is noticeable that the peak values for

heavy holes all have bigger difference between the F and A alignments, whereas for the light holes, no much bigger difference, which is the same as that for *Z* = 0 and *Z*� = 0. Besides,

kinds of holes in the F and A alignments, however, the number of periodicity is decreased for

*<sup>n</sup>* + <sup>Δ</sup>2. As for the same *<sup>E</sup>*, *<sup>σ</sup>*, and *<sup>θ</sup>* in the approximation with <sup>Ω</sup>/*EF* � 1 and

− *a*� *n k*− *n*

*<sup>n</sup>* are obtained from *k*+, *k*−, *a*, and *a*� by the analytic continuation

*<sup>F</sup>* )<sup>2</sup> with *<sup>S</sup>* the area of junction. In accordance with Eq. (10), we can

, (9)

(*φ*) are simply connected by

[*an*(*φ*) − *an*(−*φ*)], (10)

, the peak values all decrease for the two

1 2Ω*<sup>n</sup>* (*k*<sup>+</sup> *<sup>n</sup>* + *k*<sup>−</sup> *n* ) *an k*+ *n*

*kBT* <sup>∑</sup>*<sup>ω</sup><sup>n</sup>*

*E*/*EF* � 1, the amplitudes of two Andreev reflections *a*(*φ*) and *a*�

(*φ*) = *<sup>a</sup>*(−*φ*). Performing integration over *<sup>k</sup>*�, we can deduce [34, 35]

*π* /2

0

formalism [37]

*<sup>n</sup>* , *k*<sup>−</sup>

where *R* = 2*π*2*h*¯ /*Se*2(*kSC*

**3.2 Calculations and results**

*<sup>n</sup>* , *an*, and *a*�

where *k*<sup>+</sup>

*a*�

<sup>Ω</sup>*<sup>n</sup>* <sup>=</sup> *<sup>ω</sup>*<sup>2</sup>

*a* = 40/*kSC*

of *Z* and *Z*�

*<sup>I</sup>* <sup>=</sup> *<sup>e</sup>*<sup>Δ</sup> 2¯*<sup>h</sup>* ∑ *σ*,*k*�

Semiconductor Junction with Superconducting Contacts

*<sup>I</sup>* <sup>=</sup> <sup>2</sup>*πkBT*<sup>Δ</sup> *eRkSC F*

calculate the Josephson currents in the F and A alignments.

with the increase of the barrier strengths *Z* and *Z*�

Fig. 1. The SC/FS/SM/FS/SC system (from Ref. 34).

for *x* ≤ −*b* − *a*/2,

$$\psi\_1(\mathbf{x}) = \left[c\_{11}e^{iq\_{\sigma}^{L+} \mathbf{x}} + c\_{12}e^{-iq\_{\sigma}^{L+} \mathbf{x}}\right] \begin{pmatrix} 1\\0 \end{pmatrix} + \left[c\_{13}e^{iq\_{\overline{\sigma}}^{L-} \mathbf{x}} + c\_{14}e^{-iq\_{\overline{\sigma}}^{L-} \mathbf{x}}\right] \begin{pmatrix} 0\\1 \end{pmatrix} \tag{5}$$

for −*b* − *a*/2 ≤ *x* ≤ −*a*/2,

$$\psi\_2(z) = \left[c\_{21}e^{iq' \cdot \mathbf{x}} + c\_{22}e^{-iq' \cdot \mathbf{x}}\right] \begin{pmatrix} 1\\0 \end{pmatrix} + \left[c\_{23}e^{iq' \cdot \mathbf{x}} + c\_{24}e^{-iq' \cdot \mathbf{x}}\right] \begin{pmatrix} 0\\1 \end{pmatrix} \tag{6}$$

for −*a*/2 ≤ *x* ≤ *a*/2,

$$\psi\_3(\mathbf{x}) = \left[c\_{31}e^{i q\_{\sigma}^{R+} \mathbf{x}} + c\_{32}e^{-i q\_{\sigma}^{R+} \mathbf{x}}\right] \begin{pmatrix} 1 \\ 0 \end{pmatrix} + \left[c\_{33}e^{i q\_{\overline{\sigma}}^{R-} \mathbf{x}} + c\_{34}e^{-i q\_{\overline{\sigma}}^{R-} \mathbf{x}}\right] \begin{pmatrix} 0 \\ 1 \end{pmatrix} \tag{7}$$

for *a*/2 ≤ *x* ≤ *a*/2 + *b*,

$$\psi\_R^{\rm SC}(\mathbf{x}) = c e^{i k^+ \mathbf{x}} \begin{pmatrix} \imath \iota e^{i \phi\_R/2} \\ \imath e^{-i \phi\_R/2} \end{pmatrix} + d e^{-i k^- \mathbf{x}} \begin{pmatrix} \imath e^{i \phi\_R/2} \\ \imath e^{-i \phi\_R/2} \end{pmatrix} . \tag{8}$$

for *x* ≥ *a*/2 + *b*, where *q <sup>L</sup>*(*R*)<sup>±</sup> *<sup>σ</sup>* <sup>=</sup> (2*mp*/¯*h*2)(*EF* <sup>±</sup> *<sup>E</sup>* <sup>+</sup> *ησh*0) <sup>−</sup> *<sup>k</sup>*<sup>2</sup> �, *<sup>q</sup>*�± <sup>=</sup> (2*mp*/¯*h*2)(*EF* <sup>±</sup> *<sup>E</sup>*) <sup>−</sup> *<sup>k</sup>*<sup>2</sup> �, *<sup>k</sup>*<sup>±</sup> <sup>=</sup> (2*me*/¯*h*2)(*EF* <sup>±</sup> <sup>Ω</sup>) <sup>−</sup> *<sup>k</sup>*<sup>2</sup> �, *<sup>u</sup>* <sup>=</sup> (<sup>1</sup> <sup>+</sup> <sup>Ω</sup>/*E*)/2, and *<sup>v</sup>* <sup>=</sup> (<sup>1</sup> <sup>−</sup> <sup>Ω</sup>/*E*)/2 with <sup>Ω</sup> <sup>=</sup> <sup>√</sup>*E*<sup>2</sup> <sup>−</sup> <sup>Δ</sup><sup>2</sup> and the parallel component of the wave vector *<sup>k</sup>*� which is conserved. In Eqs. (4)-(8), all coefficients will be determined by the usual matching conditions of the wave functions.

From Eqs. (4)-(8), the transmission and reflection coefficients can be obtained. [34, 35] In the derivation, two dimensionless parameters *Z* = 2*meU*/(*h*¯ <sup>2</sup>*kSC <sup>F</sup>* ) and *Z*� = 2*meU*� /(*h*¯ <sup>2</sup>*kSC <sup>F</sup>* ), and *χ* are respectively introduced to describe the barrier strengths at the interfaces and ratio of the masses *mp* and *me*, the phase difference between the SCs *<sup>φ</sup>* <sup>=</sup> *<sup>φ</sup><sup>L</sup>* <sup>−</sup> *<sup>φ</sup>R*, <sup>|</sup>*k*�| <sup>=</sup> *<sup>k</sup>SC <sup>F</sup>* sin *θ*, where the corresponding Fermi vector in the SC is *kSC <sup>F</sup>* <sup>=</sup> <sup>√</sup>2*meEF*/¯*h*. Analogously, one can easily obtain the Andreev reflection (*a*� ) as an ELQ for the injection of a HLQ into the right FS from the left SC.

After having obtained the coefficients *a* and *a*� , we can express the dc Josephson current in terms of the Andreev reflections amplitudes by using the temperature Green's function formalism [37]

4 Will-be-set-by-IN-TECH


+ [*c*13*e*

+ [*c*23*e*

+ [*c*33*e*

+ *de*−*ik*−*<sup>x</sup>*

�, *<sup>q</sup>*�± <sup>=</sup>

which is conserved. In Eqs. (4)-(8), all coefficients will be determined by the usual matching

From Eqs. (4)-(8), the transmission and reflection coefficients can be obtained. [34, 35] In the

*χ* are respectively introduced to describe the barrier strengths at the interfaces and ratio of the

in terms of the Andreev reflections amplitudes by using the temperature Green's function

(2*me*/¯*h*2)(*EF* <sup>±</sup> <sup>Ω</sup>) <sup>−</sup> *<sup>k</sup>*<sup>2</sup> �, *<sup>u</sup>* <sup>=</sup> (<sup>1</sup> <sup>+</sup> <sup>Ω</sup>/*E*)/2, and *<sup>v</sup>* <sup>=</sup> (<sup>1</sup> <sup>−</sup> <sup>Ω</sup>/*E*)/2 with <sup>Ω</sup> <sup>=</sup> <sup>√</sup>*E*<sup>2</sup> <sup>−</sup> <sup>Δ</sup><sup>2</sup> and the parallel component of the wave vector *<sup>k</sup>*�

*iqR*<sup>−</sup> *<sup>σ</sup> <sup>x</sup>* + *c*34*e*

*iqL*<sup>−</sup> *<sup>σ</sup> <sup>x</sup>* + *c*14*e*

*iq*�−*<sup>x</sup>* + *c*24*e*

 *veiφR*/2 *ue*−*iφR*/2

(2*mp*/¯*h*2)(*EF* <sup>±</sup> *<sup>E</sup>* <sup>+</sup> *ησh*0) <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*<sup>F</sup>* ) and *Z*� = 2*meU*�

*<sup>F</sup>* <sup>=</sup> <sup>√</sup>2*meEF*/¯*h*. Analogously, one can easily

, we can express the dc Josephson current

) as an ELQ for the injection of a HLQ into the right FS from

<sup>−</sup>*iqL*<sup>−</sup> *<sup>σ</sup> <sup>x</sup>*] 0 1 

<sup>−</sup>*iq*�−*x*]

−*iqR*<sup>−</sup> *σ x*] 0 1 

 0 1  (5)

(6)

(7)

, (8)

/(*h*¯ <sup>2</sup>*kSC*

*<sup>F</sup>* sin *θ*, where

*<sup>F</sup>* ), and

SC FS SM FS SC

Fig. 1. The SC/FS/SM/FS/SC system (from Ref. 34).

*iqL*<sup>+</sup> *<sup>σ</sup> <sup>x</sup>* + *c*12*e*

*iqR*<sup>+</sup> *<sup>σ</sup> <sup>x</sup>* + *c*32*e*

*<sup>R</sup>* (*x*) = *ceik*+*<sup>x</sup>*

�, *<sup>k</sup>*<sup>±</sup> <sup>=</sup>

derivation, two dimensionless parameters *Z* = 2*meU*/(*h*¯ <sup>2</sup>*kSC*

the corresponding Fermi vector in the SC is *kSC*

After having obtained the coefficients *a* and *a*�

*iq*�+*<sup>x</sup>* <sup>+</sup> *<sup>c</sup>*22*<sup>e</sup>*

<sup>−</sup>*iqL*<sup>+</sup> *<sup>σ</sup> <sup>x</sup>*] 1 0 

<sup>−</sup>*iq*�+*x*]

<sup>−</sup>*iqR*<sup>+</sup> *<sup>σ</sup> <sup>x</sup>*] 1 0 

 *ueiφR*/2 *ve*−*iφR*/2

*<sup>L</sup>*(*R*)<sup>±</sup> *<sup>σ</sup>* <sup>=</sup>

masses *mp* and *me*, the phase difference between the SCs *<sup>φ</sup>* <sup>=</sup> *<sup>φ</sup><sup>L</sup>* <sup>−</sup> *<sup>φ</sup>R*, <sup>|</sup>*k*�| <sup>=</sup> *<sup>k</sup>SC*

 1 0 

*ψ*1(*x*)=[*c*11*e*

*ψ*2(*z*)=[*c*21*e*

*ψ*3(*x*)=[*c*31*e*

for *x* ≥ *a*/2 + *b*, where *q*

conditions of the wave functions.

obtain the Andreev reflection (*a*�

the left SC.

(2*mp*/¯*h*2)(*EF* <sup>±</sup> *<sup>E</sup>*) <sup>−</sup> *<sup>k</sup>*<sup>2</sup>

*ψSC*

for *x* ≤ −*b* − *a*/2,

for −*b* − *a*/2 ≤ *x* ≤ −*a*/2,

for −*a*/2 ≤ *x* ≤ *a*/2,

for *a*/2 ≤ *x* ≤ *a*/2 + *b*,

$$I = \frac{e\Delta}{2\hbar} \sum\_{\sigma, k\_{\parallel}} k\_B T \sum\_{\omega\_n} \frac{1}{2\Omega\_n} (k\_n^+ + k\_n^-) \left(\frac{a\_n}{k\_n^+} - \frac{a\_n'}{k\_n^-}\right),\tag{9}$$

where *k*<sup>+</sup> *<sup>n</sup>* , *k*<sup>−</sup> *<sup>n</sup>* , *an*, and *a*� *<sup>n</sup>* are obtained from *k*+, *k*−, *a*, and *a*� by the analytic continuation *E* → *iωn*, the Matsubara frequencies are *ω<sup>n</sup>* = *πkBT*(2*n* + 1) with *n* = 0, ±1, ±2,..., and <sup>Ω</sup>*<sup>n</sup>* <sup>=</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>n</sup>* + <sup>Δ</sup>2. As for the same *<sup>E</sup>*, *<sup>σ</sup>*, and *<sup>θ</sup>* in the approximation with <sup>Ω</sup>/*EF* � 1 and *E*/*EF* � 1, the amplitudes of two Andreev reflections *a*(*φ*) and *a*� (*φ*) are simply connected by *a*� (*φ*) = *<sup>a</sup>*(−*φ*). Performing integration over *<sup>k</sup>*�, we can deduce [34, 35]

$$I = \frac{2\pi k\_B T \Delta}{e \mathcal{R} k\_F^{\text{SC}}} \int\_0^{\pi/2} d\theta \sin \theta \cos \theta \sum\_{\omega\_h \sigma} [a\_\hbar(\phi) - a\_\hbar(-\phi)],\tag{10}$$

where *R* = 2*π*2*h*¯ /*Se*2(*kSC <sup>F</sup>* )<sup>2</sup> with *<sup>S</sup>* the area of junction. In accordance with Eq. (10), we can calculate the Josephson currents in the F and A alignments.

#### **3.2 Calculations and results**

In Fig. 2 are illustrated the numerical results for *IC* as a function of the FS thickness *b* with *a* = 40/*kSC <sup>F</sup>* at different scattering potential strengths *Z* and *Z*� for the heavy and light holes in the F and A alignments. [34] The damped oscillations of the critical Josephson currents *IC* for the heavy and light holes are all found to be as a function of *b*, which is also the characteristic feature of the ballistic SC/FM/SC junctions and always related to the crossovers between 0 and *π* states [24]. Microscopically, Andreev reflections play an important role in the transfer of Cooper pairs through the FS. An electronlike quasiparticle in FS with energy lower than the superconducting energy gap Δ, cannot enter into the SC, but is reflected at the FS/SC interface as a hole and reflected back as an electron at the opposite SC/FS interface. The constructive interference of the electronlike and holelike excitations gives rise to Andreev bound states in FS, which carry the supercurrent. However, in the FS, the spin splitting of spin band exerts an influence on the Andreev reflection when the spin of the quasiparticles is reversed by the Andreev reflection, which means that the exchange energy is gained or lost by a quasiparticle Andreev-reflected at the FS/SC interface. It follows that the superconducting pair amplitude induced in FS by the proximity to the SC is spatially modulated and the phase increases linearly with the distance from the SC/FS interface. Since the Josephson critical current *IC* is proportional to the pair amplitude in the FS, *IC* follows the sign of the order parameter in the FS. Whether it is either positive or negative is determined by the phase accumulating in the FS. Furthermore, it is found that as shown in Fig. 2, at *Z* = 0 and *Z*� = 0, for the heavy holes, the general tendency of variations of the oscillation amplitude with increasing *b* is to first decrease, then increase, and last gradually reduce to zero in the F and A alignments, while for the light holes, gradually decrease to zero all the time. We also find that for either the heavy or light holes, the behaviors in the F and A alignments are very similar in spite of the values of *Z* and *Z*� . However, for *Z* and *Z*� unequal to zero, it is noticeable that the peak values for heavy holes all have bigger difference between the F and A alignments, whereas for the light holes, no much bigger difference, which is the same as that for *Z* = 0 and *Z*� = 0. Besides, with the increase of the barrier strengths *Z* and *Z*� , the peak values all decrease for the two kinds of holes in the F and A alignments, however, the number of periodicity is decreased for

Superconducting Contacts 7

<sup>51</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

A, [34] including the product of *b* and wave vector *qL*(*R*)<sup>±</sup> and varying periodically. However, the Fermi vector *qF* for the heavy holes is much bigger than that for the light because of effective mass bigger than that for the light, as a result, the number of the periodicity for the heavy is also slightly bigger than that for the light as shown in Fig. 2 at *Z* = 0 and *Z*� = 0. More importantly, when *b* is smaller, *exp*−*i*[(*k*±±*q*�±)*a*/2 have the bigger effect on the peak value for the heavy holes than that for the light holes, so the behavior of the periodicity for the two holes are different at the smaller *b*. In fact, the effect of the coefficients of the above two terms corresponds to a physical phase for each equation, therefore, when *Z* and *Z*� are unequal to *zero* and vary, the phase shift appears and the peaks all shift with the *b*. The phase shifts for heavy and light holes are different on account of different mismatches in the Fermi velocity between the FS and SC, resultantly, the behaviors for the variations of the number of

current, physically, for the heavy or light holes, the exchange energy is not easily or lost by a quasiparticle Andreev-reflected at the the FS/SC interface with bigger *z*, therefore, Josephson current can not easily pass the junction and thus decreases with increasing the *Z* and *Z*�

Fig. 3 shows the SM thickness *a* dependence of Josephson critical current *IC* for the heavy and

features as in Fig. 2 except that the behaviors and values of the critical Josephson currents *IC*

In Figs. 4 (5) are illustrated the phases of the Josephson current *I*(*φ*) of the heavy (light) holes for different *Z* and *Z*� in the F and A alignments. [34] It is shown that, for the dependence of the Josephson currents on the phase, if the junction is the 0 or *π* state or transition from the 0 to *π* state is determined by not only the magnetic alignment but also kinds of the carriers in the FS. No matter what *P*, *Z* and *Z*� are, for the heavy hole, the junctions in the F and A alignments are only the 0 state, otherwise these are *π* state, while for the light holes, the situations are thoroughly different from those for the heavy. In the F alignment, regardless of *P*, the minimum values of *Z* and *Z*� at the same time when there is a transition of the junction changing from the 0 to *π* state are respectively 1.5 and 1.7 or so although the transition is not too obvious for the bigger *P*. If one of *Z* and *Z*� is increased, the other is at least the above value and *P* is only decreased for the existence of the transition. In the A alignment, ii spite of *P*, the minimum values of *Z* and *Z*� at the same time when there exists a transition of the junction changing from the 0 to *π* state are basically the same as those in the F. However, in the A alignment, with the increase of *Z* , *Z*� can be decreased in small magnitude and *P* is only enhanced for the existence of the transition, yet both of *Z* and *Z*� can be increased in the meantime and can be bigger for the bigger *P*. At the half metallic limit, they are respectively approximate to 2.2 and 2.1. It can be concluded from these results that the different mismatches in the effective mass and Fermi velocity between the FS and SC and the relative orientations of the effective exchange field of the two FS's play an important role in transition of the junction changing from the 0 to *π* state. The interpretation can be given as follows. In Eqs. (A4)-(A7) of Appendix A, [34] there are two terms, including

. For the effect of *Z* and *Z*� on the critical Josephson

. If the junction is the 0 or *π* state or transition from

. [34] We find that there are same

.

the periodicity are different with *Z* and *Z*�

Semiconductor Junction with Superconducting Contacts

*exp*−*i*[(*K*±−*qL*<sup>±</sup>

*<sup>σ</sup>* )*b*−*φ*] and *exp*−*i*[(*K*±+*qL*<sup>±</sup>

*<sup>σ</sup>* )*b*−*φ*]

the 0 to *π* state is mainly dependent on the contribution of the two terms. For the heavy holes, due to the effective mass bigger than that in the SC, Fermi vectors in the F and A alignments are not much smaller compared with those in the SC, therefore the sum of the two terms can not easily change the sign with *φ* increased, therefore, there only 0 or *π* state.

light holes in the F and A alignments at different *Z* and *Z*�

are slightly different, which can be similarly easily interpreted as in Fig. 2.

Fig. 2. Dependences of *IC* on the FS thickness *b* at different interface scattering potential strengths (*Z*, *Z*� ) with (0, 0) (solid line) and (2, 2) (dotted line). Here, *EF* = 100.0meV, Δ<sup>0</sup> = 1.4meV, *kBT* = 0.05Δ0, *me* = 1.0, *mp* = 0.45*me* for the heavy holes in the F (a) and A (b) alignments, and *mp* = 0.08 for the light holes in the F (c) and A (d) (from Ref. 34).

the heavy holes while that for the light is increased. The occurrence of periodic behavior can be explained by the two terms *exp*−*i*[(*k*±±*q*�±)*a*/2+(*k*±±*qL*<sup>±</sup> *<sup>σ</sup>* )*b*−*φ*] in Eqs. (A4)-(A13) of Appendix 6 Will-be-set-by-IN-TECH

(Z,Z'

(Z,Z'

(d)

)=(0,0)

(Z,Z'

)=(0,0)

(b)

)=(2,2)

0 20 40 60

2b

*<sup>σ</sup>* )*b*−*φ*] in Eqs. (A4)-(A13) of Appendix

(Z,Z'

)=(2,2)


0.020


strengths (*Z*, *Z*�



0.000

eRI

C/πΔ0

0.005

0.010

0.015


(Z,Z'

(Z,Z'

(c)

)=(0,0)

(Z,Z'

(a)

)=(0,0)

)=(2,2)

0 20 40 60

2b

be explained by the two terms *exp*−*i*[(*k*±±*q*�±)*a*/2+(*k*±±*qL*<sup>±</sup>

(Z,Z'

)=(2,2)

Fig. 2. Dependences of *IC* on the FS thickness *b* at different interface scattering potential

the heavy holes while that for the light is increased. The occurrence of periodic behavior can

alignments, and *mp* = 0.08 for the light holes in the F (c) and A (d) (from Ref. 34).

) with (0, 0) (solid line) and (2, 2) (dotted line). Here, *EF* = 100.0meV, Δ<sup>0</sup> = 1.4meV, *kBT* = 0.05Δ0, *me* = 1.0, *mp* = 0.45*me* for the heavy holes in the F (a) and A (b)


eRI

C/πΔ0

0.000

0.005

0.010

0.015

0.020

A, [34] including the product of *b* and wave vector *qL*(*R*)<sup>±</sup> and varying periodically. However, the Fermi vector *qF* for the heavy holes is much bigger than that for the light because of effective mass bigger than that for the light, as a result, the number of the periodicity for the heavy is also slightly bigger than that for the light as shown in Fig. 2 at *Z* = 0 and *Z*� = 0. More importantly, when *b* is smaller, *exp*−*i*[(*k*±±*q*�±)*a*/2 have the bigger effect on the peak value for the heavy holes than that for the light holes, so the behavior of the periodicity for the two holes are different at the smaller *b*. In fact, the effect of the coefficients of the above two terms corresponds to a physical phase for each equation, therefore, when *Z* and *Z*� are unequal to *zero* and vary, the phase shift appears and the peaks all shift with the *b*. The phase shifts for heavy and light holes are different on account of different mismatches in the Fermi velocity between the FS and SC, resultantly, the behaviors for the variations of the number of the periodicity are different with *Z* and *Z*� . For the effect of *Z* and *Z*� on the critical Josephson current, physically, for the heavy or light holes, the exchange energy is not easily or lost by a quasiparticle Andreev-reflected at the the FS/SC interface with bigger *z*, therefore, Josephson current can not easily pass the junction and thus decreases with increasing the *Z* and *Z*� .

Fig. 3 shows the SM thickness *a* dependence of Josephson critical current *IC* for the heavy and light holes in the F and A alignments at different *Z* and *Z*� . [34] We find that there are same features as in Fig. 2 except that the behaviors and values of the critical Josephson currents *IC* are slightly different, which can be similarly easily interpreted as in Fig. 2.

In Figs. 4 (5) are illustrated the phases of the Josephson current *I*(*φ*) of the heavy (light) holes for different *Z* and *Z*� in the F and A alignments. [34] It is shown that, for the dependence of the Josephson currents on the phase, if the junction is the 0 or *π* state or transition from the 0 to *π* state is determined by not only the magnetic alignment but also kinds of the carriers in the FS. No matter what *P*, *Z* and *Z*� are, for the heavy hole, the junctions in the F and A alignments are only the 0 state, otherwise these are *π* state, while for the light holes, the situations are thoroughly different from those for the heavy. In the F alignment, regardless of *P*, the minimum values of *Z* and *Z*� at the same time when there is a transition of the junction changing from the 0 to *π* state are respectively 1.5 and 1.7 or so although the transition is not too obvious for the bigger *P*. If one of *Z* and *Z*� is increased, the other is at least the above value and *P* is only decreased for the existence of the transition. In the A alignment, ii spite of *P*, the minimum values of *Z* and *Z*� at the same time when there exists a transition of the junction changing from the 0 to *π* state are basically the same as those in the F. However, in the A alignment, with the increase of *Z* , *Z*� can be decreased in small magnitude and *P* is only enhanced for the existence of the transition, yet both of *Z* and *Z*� can be increased in the meantime and can be bigger for the bigger *P*. At the half metallic limit, they are respectively approximate to 2.2 and 2.1. It can be concluded from these results that the different mismatches in the effective mass and Fermi velocity between the FS and SC and the relative orientations of the effective exchange field of the two FS's play an important role in transition of the junction changing from the 0 to *π* state. The interpretation can be given as follows. In Eqs. (A4)-(A7) of Appendix A, [34] there are two terms, including *exp*−*i*[(*K*±−*qL*<sup>±</sup> *<sup>σ</sup>* )*b*−*φ*] and *exp*−*i*[(*K*±+*qL*<sup>±</sup> *<sup>σ</sup>* )*b*−*φ*] . If the junction is the 0 or *π* state or transition from the 0 to *π* state is mainly dependent on the contribution of the two terms. For the heavy holes, due to the effective mass bigger than that in the SC, Fermi vectors in the F and A alignments are not much smaller compared with those in the SC, therefore the sum of the two terms can not easily change the sign with *φ* increased, therefore, there only 0 or *π* state.

Superconducting Contacts 9

<sup>53</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

easily change the sign, particularly for the minority holes. It follows that the general tendency is that smaller *P* more easily gives rise to the sign variation of the sum of the two terms. In the A alignment, a majority (minority) spin in the right FS will be regard as a minority (majority) spin in the right FS, therefore the increase of *P* is favorable for the sign change of the sum of the two terms. Furthermore, the barrier strengths in coefficients of the two terms in Eqs. (A4)-(A7) also exert an important influence on the sign of the value. When the two barrier strengths *Z* and *Z*� or one of them is much bigger or smaller, one of the two coefficients is

In Fig. 6, the normalized critical currents *IC* at different scattering potential strengths *Z* and *Z*� in the F alignment for the heavy and light holes are shown as a function of *P*� [35], where *P*� is define as *P*� ≡ Γ/Δ<sup>0</sup> = *PEF*/Δ<sup>0</sup> [28]. It is found that the critical Josephson current *IC* for

*Z*� unequal to zero, the situations are very different. As *Z*� = 0, for the heavy hole, *IC* always increases with *P*� at the small *Z* and decreases at the big *Z*, however, as *Z* = 0, at some *Z*�

*IC* are found, at *Z*� = 0, to decrease all the time with *P*� regardless of *Z*, while at *Z* = 0, there

In Fig. 7, are illustrated the *P*� dependences of Josephson critical current *IC* for the heavy and

behaviors and values of the critical Josephson currents *IC* with *P*� increased are remarkably different from those in Fig. 6, which are obviously due to the existence of spin splitting energy in the FS's and resultant different mismatches in the Fermi vectors between the FS and SC for the F and A alignments. The critical Josephson current in the A alignment *IC* for the heavy

Fig. 8 shows the *Z* (*Z*� ) dependence of Josephson critical current *IC* for different *Z*� (*Z* ) in the F alignment for the heavy and light holes. [35] The critical Josephson currents *IC* for the heavy

two peaks, however, at *Z*� = 0, firstly increase with the increase of *Z*, whereas at *Z* = 0, firstly

behaviors of *IC* with *Z*� and *Z* are also very similar, and *IC* basically decreases, then increases, and lastly decreases to zero with increasing *Z*� or *Z*, which indicates that there is only a peak. In Fig. 9, the variations of Josephson critical current *IC* for the heavy and light holes with

) at different *Z*� (*Z*) in the A alignment are plotted. [35]It is found that although there are the same features as those in Fig. 8, the values of the critical Josephson currents *IC* with increasing *Z* or *Z*� are a little different, this can be also ascribed to the same reason as in the explanation of the same and different properties between Figs. 6 and 7. Here, the differences

*<sup>σ</sup>* )*b*−*φ*] and *exp*−*i*[(*K*±+*qL*<sup>±</sup>

, while for the light, decreases. For *Z* or

. [35] It is shown that there exist some

. Simultaneously, it is also found that some

, while for the light, basically has no change.

) enhanced, finally reduce to zero and there are

. For the light hole, the critical Josephson

. At *Z* and *Z*� unequal to zero, the varied

) increased. In the meantime, we

, while as *Z* = 0, *IC* firstly

) = 0, always increases with *P*� for the light holes

) = 0, for the heavy hole, *IC* increases

, decreases. For the light hole, the critical Josephson currents

*<sup>σ</sup>* )*b*−*φ*] can

, *IC*

than those for the heavy and in the SC, and thus *exp*−*i*[(*K*±−*qL*<sup>±</sup>

Semiconductor Junction with Superconducting Contacts

the heavy holes, at *Z* = 0 and *Z*� = 0, increases with *P*�

, and at other *Z*�

light holes in the A alignment at different *Z* and *Z*�

at some *Z*� with *P*� and decreases at other *Z*�

holes, at *Z* = 0 and *Z*� = 0, decreases with *P*�

holes, is found to exhibit oscillations with *Z*(*Z*�

decreases and then increases with increasing *Z*�

increases and then decreases with the increase of *Z*�

currents *IC* is also found to display oscillations with *Z*(*Z*�

find that as *Z*� = 0, *IC* firstly decreases with the enhancement of *Z*�

In addition, *IC* in the A alignment, at *Z*(*Z*�

(*Z*) .

features similar to those in Fig. 6, for example, as *Z*(*Z*�

increases with *P*�

regardless of *Z*�

*Z* (*Z*�

zero, as result, the increase of *φ* can not cause the change of sign of the sum.

are the same features of *IC* with variation of *P*� as those for heavy holes.

Fig. 3. Dependences of *IC* on the SM thickness *a* at *b* = 20/*kSC <sup>F</sup>* with the same situations as in Fig. 2 (from Ref. 34).

However, as concerned about the light holes, for the F alignment in which the spin of the majority (minority) holes is still that of the majority (minority) holes in the right FS, since the effective mass is much smaller than that for the heavy hole, the Fermi vector is much smaller 8 Will-be-set-by-IN-TECH

)=(0,0) (Z,Z'

)=(0,0)

(b)

(Z,Z'

(Z,Z'

)=(0,0)

(d)

0 20 40 60

a

*<sup>F</sup>* with the same situations as in

(Z,Z'

)=(2,2)

)=(2,2)


0.020


Fig. 2 (from Ref. 34).



0.000

eRIC/πΔ0

0.005

0.010

0.015


(Z,Z'

(Z,Z'

)=(2,2)

(c)

(Z,Z'

)=(0,0)

0 20 40 60

a

Fig. 3. Dependences of *IC* on the SM thickness *a* at *b* = 20/*kSC*

(Z,Z'

)=(2,2)

However, as concerned about the light holes, for the F alignment in which the spin of the majority (minority) holes is still that of the majority (minority) holes in the right FS, since the effective mass is much smaller than that for the heavy hole, the Fermi vector is much smaller

(a)


eRIC/πΔ0

0.000

0.005

0.010

0.015

than those for the heavy and in the SC, and thus *exp*−*i*[(*K*±−*qL*<sup>±</sup> *<sup>σ</sup>* )*b*−*φ*] and *exp*−*i*[(*K*±+*qL*<sup>±</sup> *<sup>σ</sup>* )*b*−*φ*] can easily change the sign, particularly for the minority holes. It follows that the general tendency is that smaller *P* more easily gives rise to the sign variation of the sum of the two terms. In the A alignment, a majority (minority) spin in the right FS will be regard as a minority (majority) spin in the right FS, therefore the increase of *P* is favorable for the sign change of the sum of the two terms. Furthermore, the barrier strengths in coefficients of the two terms in Eqs. (A4)-(A7) also exert an important influence on the sign of the value. When the two barrier strengths *Z* and *Z*� or one of them is much bigger or smaller, one of the two coefficients is zero, as result, the increase of *φ* can not cause the change of sign of the sum.

In Fig. 6, the normalized critical currents *IC* at different scattering potential strengths *Z* and *Z*� in the F alignment for the heavy and light holes are shown as a function of *P*� [35], where *P*� is define as *P*� ≡ Γ/Δ<sup>0</sup> = *PEF*/Δ<sup>0</sup> [28]. It is found that the critical Josephson current *IC* for the heavy holes, at *Z* = 0 and *Z*� = 0, increases with *P*� , while for the light, decreases. For *Z* or *Z*� unequal to zero, the situations are very different. As *Z*� = 0, for the heavy hole, *IC* always increases with *P*� at the small *Z* and decreases at the big *Z*, however, as *Z* = 0, at some *Z*� , *IC* increases with *P*� , and at other *Z*� , decreases. For the light hole, the critical Josephson currents *IC* are found, at *Z*� = 0, to decrease all the time with *P*� regardless of *Z*, while at *Z* = 0, there are the same features of *IC* with variation of *P*� as those for heavy holes.

In Fig. 7, are illustrated the *P*� dependences of Josephson critical current *IC* for the heavy and light holes in the A alignment at different *Z* and *Z*� . [35] It is shown that there exist some features similar to those in Fig. 6, for example, as *Z*(*Z*� ) = 0, for the heavy hole, *IC* increases at some *Z*� with *P*� and decreases at other *Z*� . Simultaneously, it is also found that some behaviors and values of the critical Josephson currents *IC* with *P*� increased are remarkably different from those in Fig. 6, which are obviously due to the existence of spin splitting energy in the FS's and resultant different mismatches in the Fermi vectors between the FS and SC for the F and A alignments. The critical Josephson current in the A alignment *IC* for the heavy holes, at *Z* = 0 and *Z*� = 0, decreases with *P*� , while for the light, basically has no change. In addition, *IC* in the A alignment, at *Z*(*Z*� ) = 0, always increases with *P*� for the light holes regardless of *Z*� (*Z*) .

Fig. 8 shows the *Z* (*Z*� ) dependence of Josephson critical current *IC* for different *Z*� (*Z* ) in the F alignment for the heavy and light holes. [35] The critical Josephson currents *IC* for the heavy holes, is found to exhibit oscillations with *Z*(*Z*� ) enhanced, finally reduce to zero and there are two peaks, however, at *Z*� = 0, firstly increase with the increase of *Z*, whereas at *Z* = 0, firstly decreases and then increases with increasing *Z*� . For the light hole, the critical Josephson currents *IC* is also found to display oscillations with *Z*(*Z*� ) increased. In the meantime, we find that as *Z*� = 0, *IC* firstly decreases with the enhancement of *Z*� , while as *Z* = 0, *IC* firstly increases and then decreases with the increase of *Z*� . At *Z* and *Z*� unequal to zero, the varied behaviors of *IC* with *Z*� and *Z* are also very similar, and *IC* basically decreases, then increases, and lastly decreases to zero with increasing *Z*� or *Z*, which indicates that there is only a peak.

In Fig. 9, the variations of Josephson critical current *IC* for the heavy and light holes with *Z* (*Z*� ) at different *Z*� (*Z*) in the A alignment are plotted. [35]It is found that although there are the same features as those in Fig. 8, the values of the critical Josephson currents *IC* with increasing *Z* or *Z*� are a little different, this can be also ascribed to the same reason as in the explanation of the same and different properties between Figs. 6 and 7. Here, the differences

Superconducting Contacts 11

<sup>55</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic




Last, I want to briefly mention that if at finite voltage, there also can exist the dc current which exhibits the so-called subharmonic structure [38, 39], i.e., current singularities at voltages *V* = 2Δ/*en* with *n* =, 1 2,..., where Δ the energy gap in the SC. Therefore, in such situation, the decoherence due to the subgap currents would have to be taken into account as pointed by

P=0.6

P=0.85

0.0 0.2 0.4 0.6 0.8 1.0

φ/π

Fig. 5. The same dependence for the light holes as in Fig. 4 (from Ref. 34).

P=0.0 A

P=0.6

P=0.85

0.0 0.2 0.4 0.6 0.8 1.0

φ/π

P=0.0 F




Ref. [39].

eRIC/πΔ0

(Z,Z' )=(0,0) (Z,Z' )=(0,2) (Z,Z' )=(2,0) (Z,Z'

(Z,Z' )=(2,2)

)=(1.6,1.8)

Semiconductor Junction with Superconducting Contacts

Fig. 4. The phases of the Josephson current *I*(*φ*) for the heavy holes at different (*Z*, *Z* ) in the F and A alignments. Here, *a* = 40/*kSC <sup>F</sup>* , *<sup>b</sup>* <sup>=</sup> 20/*kSC <sup>F</sup>* , *Z* and *Z* are given in the inset of Fig. 4, and the other parameters are the same as in Fig. 2 (from Ref. 34).

of the results between the heavy and light holes in Fig. 9, together with Figs. 6-8, can be all owing to the fact that there are the different mismatches in the effective mass and Fermi velocity between the FS and SC for the two kinds of holes.

10 Will-be-set-by-IN-TECH




P=0.6

P=0.85

Fig. 4. The phases of the Josephson current *I*(*φ*) for the heavy holes at different (*Z*, *Z*

*<sup>F</sup>* , *<sup>b</sup>* <sup>=</sup> 20/*kSC*

of the results between the heavy and light holes in Fig. 9, together with Figs. 6-8, can be all owing to the fact that there are the different mismatches in the effective mass and Fermi

0.0 0.2 0.4 0.6 0.8 1.0

φ/π

and the other parameters are the same as in Fig. 2 (from Ref. 34).

velocity between the FS and SC for the two kinds of holes.

F and A alignments. Here, *a* = 40/*kSC*

A P=0.0

0.0010 P=0.6

0.0 0.2 0.4 0.6 0.8 1.0

φ/π

*<sup>F</sup>* , *Z* and *Z* are given in the inset of Fig. 4,

P=0.85

) in the

F P=0.0


0.0010



0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0000

0.0002

0.0004

eRIC/πΔ0

0.0006

0.0008

eRIC/πΔ0

(Z,Z' )=(0,0) (Z,Z' )=(0,2) (Z,Z' )=(2,0) (Z,Z' )=(1.6,1.8)

(Z,Z' )=(2,2)

Fig. 5. The same dependence for the light holes as in Fig. 4 (from Ref. 34).

Last, I want to briefly mention that if at finite voltage, there also can exist the dc current which exhibits the so-called subharmonic structure [38, 39], i.e., current singularities at voltages *V* = 2Δ/*en* with *n* =, 1 2,..., where Δ the energy gap in the SC. Therefore, in such situation, the decoherence due to the subgap currents would have to be taken into account as pointed by Ref. [39].

Superconducting Contacts 13

<sup>57</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

0.0002

0.0016

Fig. 7. The same dependence as in Fig. 6 in the case of the A alignment (from Ref. 35).

which is connected with the transition from positive (0) to negative (*π*) coupling. The effect can be exploited for the design of devices such as superconducting quantum interference

Up now, we hardly find theoretical and experimental works taking into account the unconventional pairing symmetry, which is relevant to the high *TC* superconducting junctions comprising the FS. Due to the interplay between the various pairing symmetry in the unconventional SC and hole types in the FS, there will exhibit substantial physical content, which will have not only great intrinsic scientific importance but also potential device applications in the future. We will focus our attention on the *superconducting Spintronics* based on the junctions consisting of FS and SC with unconventional pairing symmetry in the

0.0020

0.0024

0.0028

0.0032

0.0036

0.0040

0.0044

0.0004

0.0006

0.0008

0.0010

0.0012

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/ =2.0

(b)

(d)

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/ =2.0

0.0 0.2 0.4 0.6 0.8 1.0

P/

Z=0.0

Z=0.0

0.0014

Z/ =0.0

0.0 0.2 0.4 0.6 0.8 1.0

P/

devices with improved accuracy and convenience.

Z=0.0 Z=0.5 Z=1.0 Z=2.0

(c)

Z/ =0.0

(a)

Semiconductor Junction with Superconducting Contacts

Z=0.0 Z=0.5 Z=1.0 Z=2.0

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045

future work.

eRIC/πΔ0

eRIC/πΔ0

Fig. 6. Dependence of the normalized critical current *IC* on *P* in the F alignment at different scattering potential strengths: *Z* (a) and *Z* (b) for the heavy holes; *Z* (c) and *Z* (d) for for the light. Here. the parameters are the same as in Fig. 4 (from Ref. 35).

#### **4. Summary and outlook**

In this chapter, we study the Josephson current flowing through an FS/semiconductor (SM) /FS trilayer connected to two superconductor (SC) electrodes. It is shown that the Josephson current in the junction is strongly dependent on not only the kinds of holes in the FS's and scattering potential strengths at the interfaces but also the relative orientations of the effective exchange field of the two FS's. We also find that in the coherent regime, the oscillatory dependences of the critical Josephson current on the FS and SM layer thicknesses and Josephson current on the macroscopic phase difference for the heavy and light holes in different orientations of the effective exchange field of the two FS's, are much different due to the different mismatches in the effective mass and Fermi velocity between the FS and SC, 12 Will-be-set-by-IN-TECH

0.0002

0.0045

0.0015

Fig. 6. Dependence of the normalized critical current *IC* on *P* in the F alignment at different scattering potential strengths: *Z* (a) and *Z* (b) for the heavy holes; *Z* (c) and *Z* (d) for for the

In this chapter, we study the Josephson current flowing through an FS/semiconductor (SM) /FS trilayer connected to two superconductor (SC) electrodes. It is shown that the Josephson current in the junction is strongly dependent on not only the kinds of holes in the FS's and scattering potential strengths at the interfaces but also the relative orientations of the effective exchange field of the two FS's. We also find that in the coherent regime, the oscillatory dependences of the critical Josephson current on the FS and SM layer thicknesses and Josephson current on the macroscopic phase difference for the heavy and light holes in different orientations of the effective exchange field of the two FS's, are much different due to the different mismatches in the effective mass and Fermi velocity between the FS and SC,

0.0020

0.0025

0.0030

0.0035

0.0040

0.0004

0.0006

0.0008

0.0010

Z=0.0

Z=0.0

(d)

(b)

0.0 0.2 0.4 0.6 0.8 1.0

P/

0.0012

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/ =2.0

> Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/ =2.0

0.0014

Z/ =0.0

> Z/ =0.0

0.0 0.2 0.4 0.6 0.8 1.0

P/

light. Here. the parameters are the same as in Fig. 4 (from Ref. 35).

Z=0.0 Z=0.5 Z=1.0 Z=2.0

> Z=0.0 Z=0.5 Z=1.0 Z=2.0

(c)

(a)

0.0000

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045

**4. Summary and outlook**

eRIC/πΔ0

0.0002

0.0004

eRIC/πΔ0

0.0006

0.0008

Fig. 7. The same dependence as in Fig. 6 in the case of the A alignment (from Ref. 35).

which is connected with the transition from positive (0) to negative (*π*) coupling. The effect can be exploited for the design of devices such as superconducting quantum interference devices with improved accuracy and convenience.

Up now, we hardly find theoretical and experimental works taking into account the unconventional pairing symmetry, which is relevant to the high *TC* superconducting junctions comprising the FS. Due to the interplay between the various pairing symmetry in the unconventional SC and hole types in the FS, there will exhibit substantial physical content, which will have not only great intrinsic scientific importance but also potential device applications in the future. We will focus our attention on the *superconducting Spintronics* based on the junctions consisting of FS and SC with unconventional pairing symmetry in the future work.

Superconducting Contacts 15

<sup>59</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

0.000

0.005

0.000

Fig. 9. Dependence of *IC* on *Z* and *Z* at *P* = 0.9 with the same situations as in Fig. 7 (from

0.001

0.002

0.003

0.004

0.001

0.002

0.003

Z=0.0 Z=0.5 Z=1.0 Z=2.0

02468

(d)

Z/

Z=0.0 Z=0.5 Z=1.0 Z=2.0

(b)

0.004

0.005

0.000

0.005

0.000

Ref. 35).

0.001

0.002

eRIC/πΔ0

0.003

0.004

0.001

0.002

0.003

eRIC/πΔ0

0.004

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/ =2.0

02468

Z

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/ =2.0

(C)

(a)

Semiconductor Junction with Superconducting Contacts

0.005

0.006

Fig. 8. Dependence of *IC* on *Z* and *Z* at *P* = 0.9 with the same situations as in Fig. 6 (from Ref. 35).

14 Will-be-set-by-IN-TECH

0.0000

0.005

0.000

Fig. 8. Dependence of *IC* on *Z* and *Z* at *P* = 0.9 with the same situations as in Fig. 6 (from

0.001

0.002

0.003

0.004

0.0005

0.0010

0.0015

0.0020

Z=0.0 Z=0.5 Z=1.0 Z=2.0

(b)

02468

Z/

Z=0.0 Z=0.5 Z=1.0 Z=2.0

(d)

0.0025

0.0030

0.0035

0.0040

0.000

0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045

Ref. 35).

eRIC/πΔ0

0.001

0.002

eRI

C/πΔ0

0.003

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z/

02468

Z

Z/ =0.0 Z/ =0.5 Z/ =1.0 Z=2.0

(c)

(a)

=2.0

0.004

0.005

Fig. 9. Dependence of *IC* on *Z* and *Z* at *P* = 0.9 with the same situations as in Fig. 7 (from Ref. 35).

Superconducting Contacts 17

<sup>61</sup> Josephson Current in a Ferromagnetic Semiconductor/Semiconductor/Ferromagnetic

[1] B. M. Andersen. Y. S. Barash, S. Graser, and P.J. Hirschfeld, Phys. Rev. B 77, 054501 (2008).

[4] T. Jungwirth, J. Sinova, J. Mašek, J. Kuˇcera, and A.H. MacDonald, Rev. Mod. Phys. 78, 809 (2006); H. Saito, S. Yuasa, and K. Ando, Phys. Rev. Lett 95, 086604 (2005). [5] H. Ohno, A. Shen. F. Matsukura, A. Oiwa, A. Endo, S. Katsumoto, and Y. Iye, Appl. Phys. Lett 69, 363 (1996); H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietal, Y. Ohno, and K. Ohtani, Nature (London) 408, 994 (2000); K.C. Ku, S.J. Potashnik, R.F. Wang, S.H. Chun, P. Schiffer, N. Samarth, M.J. Seong, A. Mascarenhas, E. Johnston-Halperin, R.C. Myers, A.C. Gossard, and D.D. Awschalom, Appl. Phys. Lett.

[6] H. Ohno, Science 281, (1991); J. Magn. Magn. Matter. 200, 110 (1999); T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Ferrand, Science 287, 019 (2000); J. Konig, H.-H. Lin, and A.H. MacDonald, Phys. Rev. Lett. 84, 5682 (2000); J. Konig, T. Jungwirth, and A.H.

[10] V.V. Ryazanov, V.A. Oboznov, A. Yu. Rusanov, A.V. Veretennikov, A.A. Golubov, and J.

[11] V.V. Ryazanov, V.A. Oboznov, A.V. Veretennikov, A. Yu. Rusanov Phys. Rev. B. 65,

[13] J.M.E. Geers, M.B.S. Hesselberth, J. Aarts, and A.A.Golubov, Phys. Rev. B 64, 094506

[12] T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Phys. Rev. Lett 86, 304 (2001).

[2] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). ˇ

*uσ*, *v<sup>σ</sup>* ··· **the two-component wavefunction**

*a*, *b*, *c*, *d* ··· **transmission and reflection coefficients**

Semiconductor Junction with Superconducting Contacts

*<sup>L</sup>*(*R*)<sup>±</sup> *<sup>σ</sup>* ··· **the wave vectors in the left (right) FS**

*<sup>k</sup>*� ··· **the parallel component of the wave vector**

[3] A.I. Buzdin, Rev. Mod. Phys. 77, 935 (2005)

MacDonald, Phys. Rev. B 64, 184423 (2001).

[9] B.D. Josephson, Phys. Lett. 1, 251 (1962).

Aarts, Phys. Rev. Lett. 86, 2427 (2001).

[7] Y.C. Tao, J.G. Hu, and H. Liu, J. Appl. Phys. 96, 498 (2004) [8] Y.C. Tao and J.G. Hu, Int. J. Mod. Phys B 16, 2247 (2004).

*Z*/*Z*� ··· **two dimensionless parameters**

*ψ*(*x*)··· **the wave function**

*q*�± ··· **the wave vectors in the SM** *k*<sup>±</sup> ··· **the wave vectors in the SCs**

*kF* ··· **the Fermi wave vector**

*ω* ··· **the Matsubara frequencies**

82, 2302 (2003).

020501 (2001).

(2001).

*kB* ··· **boltzmann factor**

*T* ··· **tempreture**

**6. References**

*q*


*uσ*, *v<sup>σ</sup>* ··· **the two-component wavefunction**

*ψ*(*x*)··· **the wave function**

16 Will-be-set-by-IN-TECH

**5. Symbol list**

**SC** ··· **superconductor**

**SM** ··· **semiconductor**

**FS** ···**ferromagnetic semiconductor**

**ELQ** ··· **electronlike quasiparticle HLQ** ··· **holelike quasiparticle**

*φ* ··· **a phase difference between the SCs**

*n* ··· **n-type semiconductor** *p* ··· **p-type semiconductor**

*I* ··· **the dc Josephson current**

*U* ··· **the barrier height** *δ*(*x*)··· *δ***-type function**

*h*¯ ··· **Planck constant**

*IC* ··· **the critical Josephson current**

*V*(*x*)/*V*(r)··· **the usual static potential**

*HFS*(r)··· **the hole Hamiltonian in the FS**

*mp*/*me* ··· **the mass of a hole/electron** h(r)··· **the effective exchange field**

Γ ··· **the spin splitting energy**

*EF* ··· **the fermi-energy**

*ζ<sup>k</sup>* ··· **the excitation energy** *�<sup>k</sup>* ··· **the one-electron energy**

Δ(*x*)··· **the pair potential**

σ ··· **the conventional Pauli spin operator**

*Jpd* ··· **the p-d exchange coupling strength**

*p* ··· **the relative spin-polarized degree**

Δ<sup>0</sup> ··· **energy gap at zero temperature**

Δ ··· **energy gap at non-zero temperature**

<sup>&</sup>lt; *<sup>S</sup><sup>z</sup>* <sup>&</sup>gt; ··· **the thermal average of the impurity Mn ion spins**

*H*0(r)··· **the hole Hamiltonian of the SM GaAs**

**BDG** ··· **Bogoliubov-de Gennes (BDG) equation**

*a*, *b*, *c*, *d* ··· **transmission and reflection coefficients**

*q <sup>L</sup>*(*R*)<sup>±</sup> *<sup>σ</sup>* ··· **the wave vectors in the left (right) FS**

*q*�± ··· **the wave vectors in the SM**


#### **6. References**


**4** 

*Israel* 

1D. Shaltiel et al\*

*The Hebrew University of Jerusalem* 

**Investigating Superconductivity with Electron** 

**Paramagnetic Resonance (EPR) Spectrometer** 

Shortly after the discovery of high *Tc* superconductors there was an influx of publications where EPR technique was used to investigate their properties (Srinivasu et al., 2001). As no substantial information was obtained it was soon almost abandoned. The fact that EPR technique provided a large number of experimental data in the superconducting regime, without significantly contributing to the deeper understanding of high *Tc* superconductors was puzzling. Meanwhile, our continuous effort to further investigate superconductivity using EPR technique yielded important experimental and theoretical results related to properties of high *Tc* superconductors. Furthermore, some of these results are unique as they

The work shows that an electron paramagnetic resonance spectrometer is a powerful tool to investigate properties of superconductors. As superconducting systems are not conventional paramagnetic substances, the EPR signals do not originate from the usual resonant absorption, namely magnetic dipolar transitions between spin levels. Thus, it is essential in analyzing the experimental results, to explore the mechanism that induces the EPR signals. Indeed, by detailed experimental and theoretical studies of the EPR spectra of different high *Tc* superconductors, it has been possible to apprehend the mechanism that induces the observed microwave absorption. Based on these results, new unknown properties of superconductors were obtained. As the EPR spectrometer is the central experimental tool in the present work, we start by describing the EPR spectrometer in details, to facilitate the reader understanding its various functions. This step is necessary, as it turned out that among experts in superconductivity many are not familiar with the elements of the EPR spectrometer, how and what it measures. We will then continue by presenting experimental

H.A. Krug von Nidda2, B.Ya. Shapiro3, A. Loidl2, T. Tamegai4, T. Kurz2, B. Bogoslavsky1,

*5National Chiao Tung University, Department of Electrophysics, Hsinchu, Taiwan, R.O.C. 6Ariel University Center, Israel*

*2Experimental Physics V, Center for Electronic Correlations and Magnetism,* 

were not obtained with conventional experimental techniques.

**1. Introduction** 

 \*

B. Rosenstein5,6 and I. Shapiro3

*1The Hebrew University of Jerusalem, Israel* 

*University of Augsburg, Augsburg, Germany 3Bar-Ilan University, Ramat-Gan, Israel 4The University of Tokyo, Hongo, Japan* 


## **Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer**

1D. Shaltiel et al\* *The Hebrew University of Jerusalem Israel* 

## **1. Introduction**

18 Will-be-set-by-IN-TECH

62 Superconductors – Properties, Technology, and Applications

[14] O. Bourgeois, P. Gandit, J. Lesueur, A. Sulpice, X. Grison, and J. Chaussy, Eur, Phys. J. B

[15] T. Kontos, M. Aprili, J. Lesueur, F. Genet*e*ˆt, B. Stephanidis, and R. Boursier, Phys. Rev.

[20] N.M. Chtchelkatchev, W. Belzig, Yu. V. Nazarov, and C. Bruder, Pis'ma Zh. Éksp. Teor.

[21] A.A. Golubov, M. Yu. Kupriyanov, and Ya. V. Fominov, Pis'ma Zh. Éksp. Teor. Fiz. 75,

[24] Z. Radovi´c, L. Dobrosavljevi´c-Gruji´c, and B. Vujiˇci´c, Phys. Rev. B 63, 214512 (2001); M. Božovi´c and Z. Radovi´c, Phys. Rev. B 66, 134524 (2002); Z. Radovi´c, N. Lazarides, and

[25] K. Halterman and O.T. Valls, Phys. Rev. B 65, 014509 (2002); M. Zareyan, W. Belzig, and

[29] J.S. Jang, D. Davidovi´c, D.H. Reih, and C.L. Chien, Phys. Rev. Lett. 74, 314 (1995); Y. Obi,

[36] P.G. de Gennes, *Superconductivity o f Metal and Alloys* (Benjamin, New York, 1996).

[38] W.M. van Huffelen, T.M. Klapwijk, D.R. Heslinga, M.J. de Boer, and N. van der Post,

21, 75 (2001).

Lett. 89, 137007 (2002).

[16] M. Fogelström, Phys. Rev. B 62, 11812 (2000).

Fiz. 74, 357 (2001) [JEPT Lett. 74, 323 (2001)].

N. Flytzanis, Phys. Rev. B 68, 014501 (2001).

Yu. V. Nazarov, Phys. Rev. Lett. 86, 308 (2001).

[30] D.J. van Harlingen, Rev. Mod. Phys. 67, 515 (1995).

[33] C.R. Hu, Phys. Rev. Lett. 72, 1526 (1994). [34] Y.C. Tao, J. Appl. Phys. 100, 063901 (2006) [35] Y.C. Tao, Eur. Phys. J. B. 56, 83 (2007).

Phys. Rev. B 47, 5170-5189 (1993).

[26] A. Brinkman and A.A. Golubov, Phys. Rev. B 61, 11297 (2000). [27] I. Baladie and A. Buzdin, Phys. Rev. B 64, 224514 (2001).

709 (2002) [JEPT Lett. 75, 588 (2002)].

[17] Yu. S. Barash and I. V. Bobkova, Phys. Rev. B 65, 144502 (2002).

[18] F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys. Rev. B 64, 134506 (2001). [19] V.N. Krivoruchko and E. A. Koshina, Phys. Rev. B 64, 172511 (2001).

[22] Yu. S. Barash, I. V. Bobkova, and T. Kopp, Phys. Rev. B 66, 140503 (2003). [23] A. Yu. Zyuzin, B. Spivak, and M. Hruška, Europhys. Lett. 62, 97 (2003).

[28] F.S. Bergeret, A.F. Volkov, and K.B. Efetov, Phys. Rev. B 65, 134505 ( 2002).

M. Ikebe, T. Kubo, and H. Fujimori, Physica C 317-318, 149 (1999).

[31] J.X. Zhu, B. Friedman and C.S. Ting, Phys. Rev. B 59, 9558 (1999). [32] J.X. Zhu and C.S. Ting, and A.V. Balatsky Phys. Rev. B 66,

[37] A. Furusaki and M. Tsukada, Solid State Commun. 78, 299 (1991).

[39] M.J. Storcz and F.K. Wilhelm, Appl. Phys. Lett. 83, 2589 (2003).

Shortly after the discovery of high *Tc* superconductors there was an influx of publications where EPR technique was used to investigate their properties (Srinivasu et al., 2001). As no substantial information was obtained it was soon almost abandoned. The fact that EPR technique provided a large number of experimental data in the superconducting regime, without significantly contributing to the deeper understanding of high *Tc* superconductors was puzzling. Meanwhile, our continuous effort to further investigate superconductivity using EPR technique yielded important experimental and theoretical results related to properties of high *Tc* superconductors. Furthermore, some of these results are unique as they were not obtained with conventional experimental techniques.

The work shows that an electron paramagnetic resonance spectrometer is a powerful tool to investigate properties of superconductors. As superconducting systems are not conventional paramagnetic substances, the EPR signals do not originate from the usual resonant absorption, namely magnetic dipolar transitions between spin levels. Thus, it is essential in analyzing the experimental results, to explore the mechanism that induces the EPR signals. Indeed, by detailed experimental and theoretical studies of the EPR spectra of different high *Tc* superconductors, it has been possible to apprehend the mechanism that induces the observed microwave absorption. Based on these results, new unknown properties of superconductors were obtained. As the EPR spectrometer is the central experimental tool in the present work, we start by describing the EPR spectrometer in details, to facilitate the reader understanding its various functions. This step is necessary, as it turned out that among experts in superconductivity many are not familiar with the elements of the EPR spectrometer, how and what it measures. We will then continue by presenting experimental

<sup>\*</sup> H.A. Krug von Nidda2, B.Ya. Shapiro3, A. Loidl2, T. Tamegai4, T. Kurz2, B. Bogoslavsky1,

B. Rosenstein5,6 and I. Shapiro3

*<sup>1</sup>The Hebrew University of Jerusalem, Israel* 

*<sup>2</sup>Experimental Physics V, Center for Electronic Correlations and Magnetism,* 

*University of Augsburg, Augsburg, Germany* 

*<sup>3</sup>Bar-Ilan University, Ramat-Gan, Israel* 

*<sup>4</sup>The University of Tokyo, Hongo, Japan* 

*<sup>5</sup>National Chiao Tung University, Department of Electrophysics, Hsinchu, Taiwan, R.O.C. 6Ariel University Center, Israel*

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 65

measurements were performed in steps of the variables involved, and at each step the lockin detector phase had to be adjusted accordingly. This procedure results in an extremely cumbersome measurement, where the possibilities offered by the modern EPR Bruker spectrometer ELEXSYS E500 are extremely useful. A feature of this spectrometer is its possibility to measure the signal intensity in steps of two variables. It allows obtaining the measured signal intensity at each step, where the lock-in phase is varied from zero to 1800. Repeating the measurement, by changing, in steps, the magnitude of the variable involved (such as magnetic field, temperature, angle of the oriented crystal) gives the measured signal intensity as function of the variable at the full 1800 range of the lock-in phase. Then the measurements were analyzed using a proper computer program. It allowed obtaining two curves, the measured signal amplitude, and the corresponding signal phase, as a function of the variable involved such as: DC magnetic field, AC magnetic field, sample orientation angle and temperature. Thus, the actual "calculated" signal was derived from

the measured signal amplitude and its corresponding signal phase (see below).

Fig. 2 presents such a result, for a Bi2Sr2CaCu2O8+δ (Bi2212) single crystal, Zero Field Cooled **(ZFC)** to 4K with the DC magnetic field applied parallel to the a-b plane. The upper curve shows the measured signal amplitude, and the lower curve shows its corresponding signal phase (with respect to the lock-in AC phase) as function of DC magnetic field. The measured signal amplitude exhibits two maxima followed by an exponential decay towards zero at high fields. The signal phase is close to zero at low fields and drops steeply towards -1800 at high fields. It indicates that the signal is in phase and out of phase with the AC field at low and high fields, respectively. The details of the phase shift, i.e. deviations (±20°) from 0° at low fields and from 180° at high fields have not been investigated so far and need further attention. As the allover phase shift from low to high fields is evidently less than 180° it

Fig. 3 presents EPR results similar to those shown in Fig. 2, for temperatures, 80K, 30K and 4K. It indicates that the measured signal amplitude and signal phase have similar functional behavior as in Fig. 2. It implies that the EPR responses as function of field are similar in the

Fig. 1. Block diagram of the EPR spectrometer.

temperature range from *Tc* down to 4K.

cannot be explained just by a constant shift of the calibration.

results obtained in high *Tc* superconductors subjected to various factors such as DC and AC magnetic fields, temperature, sample orientation and to different sample treatments such as zero-field-cooling or field-cooling. This overview will allow a systematic classification of the measurements and will provide the theoretical background (Shaltiel et al., 2008) necessary to analyze these experiments and to understand the inherent physics. The presentation is based on publications by the authors over a period from just after the discovery of high *Tc* superconductors to date.

## **2. Experimental**

An EPR spectrometer, Fig. 1**,** consists of a microwave source that feeds a cavity where a sample is installed. The sample is exposed to three magnetic fields; a DC field, HDC, a collinear low frequency AC field, hAC, and a transverse microwave field, Hrf. The microwave and the AC frequencies used in the present investigation were 9.3 GHz and 100 kHz, respectively. At zero DC field the coupling of the waveguide to the cavity is adjusted critically in such a way that the reflection from the cavity is zero. Regarding a usual EPR experiment on a paramagnetic sample, the variation of the DC field results in a detuning from the critical adjustment on passing the resonance condition for magnetic dipolar transitions between the Zeeman levels (=HDC, with microwave frequency and gyromagnetic ratio ) and in turn yields microwave reflection from the cavity. The reflected microwave power is rectified by a detector, i.e. a microwave diode, yielding a signal proportional to the absolute value of the reflected power P at HDC and modulated by the AC frequency (note that hAC<<HDC). This signal is further fed into a lock-in-detector **(LID)** which samples the signal by convolution with a periodic signal of the same AC frequency over a large number of AC periods, thus yielding amplitude and phase while suppressing statistical noise. Provided that the width of the resonance absorption dependent on HDC is larger than the amplitude of the AC modulation field hAC, the LID output submits a signal proportional to the field derivative of the paramagnetic resonance signal dP/dH.

An accepted assumption in analyzing experimental results obtained by an EPR spectrometer is that the role of the AC field is to introduce variations in the magnitude of the DC magnetic field; the AC field is usually referred to as the modulating magnetic field. However, as will be shown below, the AC field does not just result in a modulation of the microwave signal dependent of the modulated static field like in a usual EPR experiment, but in superconductors it may also induce an additional mechanism of microwave power dissipation.

Therefore, one should be perceptive to the different dissipation process, when analyzing the EPR results in the superconducting phase. Here it is important to note that a complication in the measuring process has been observed. It is related to variations in the signal phase that may develop through the measuring process. The reference AC phase of the LID is adjusted for usual EPR measurements to yield the optimum paramagnetic resonance signal. However in case of the microwave dissipation observed in anisotropic superconductors this adjustment turns out to be not appropriate. Variations in the signal phase were observed when varying the magnitude of the variables involved such as DC magnetic field, temperature, sample orientation and others. This effect necessitates a procedure where the lock-in detector phase has to be adjusted, throughout the measuring process, to be in-phase with the AC phase. To overcome this problem, conducted during the measuring process, the measurements were performed in steps of the variables involved, and at each step the lockin detector phase had to be adjusted accordingly. This procedure results in an extremely cumbersome measurement, where the possibilities offered by the modern EPR Bruker spectrometer ELEXSYS E500 are extremely useful. A feature of this spectrometer is its possibility to measure the signal intensity in steps of two variables. It allows obtaining the measured signal intensity at each step, where the lock-in phase is varied from zero to 1800. Repeating the measurement, by changing, in steps, the magnitude of the variable involved (such as magnetic field, temperature, angle of the oriented crystal) gives the measured signal intensity as function of the variable at the full 1800 range of the lock-in phase. Then the measurements were analyzed using a proper computer program. It allowed obtaining two curves, the measured signal amplitude, and the corresponding signal phase, as a function of the variable involved such as: DC magnetic field, AC magnetic field, sample orientation angle and temperature. Thus, the actual "calculated" signal was derived from the measured signal amplitude and its corresponding signal phase (see below).

Fig. 1. Block diagram of the EPR spectrometer.

64 Superconductors – Properties, Technology, and Applications

results obtained in high *Tc* superconductors subjected to various factors such as DC and AC magnetic fields, temperature, sample orientation and to different sample treatments such as zero-field-cooling or field-cooling. This overview will allow a systematic classification of the measurements and will provide the theoretical background (Shaltiel et al., 2008) necessary to analyze these experiments and to understand the inherent physics. The presentation is based on publications by the authors over a period from just after the discovery of high *Tc*

An EPR spectrometer, Fig. 1**,** consists of a microwave source that feeds a cavity where a sample is installed. The sample is exposed to three magnetic fields; a DC field, HDC, a collinear low frequency AC field, hAC, and a transverse microwave field, Hrf. The microwave and the AC frequencies used in the present investigation were 9.3 GHz and 100 kHz, respectively. At zero DC field the coupling of the waveguide to the cavity is adjusted critically in such a way that the reflection from the cavity is zero. Regarding a usual EPR experiment on a paramagnetic sample, the variation of the DC field results in a detuning from the critical adjustment on passing the resonance condition for magnetic dipolar transitions between the Zeeman levels (=HDC, with microwave frequency and gyromagnetic ratio ) and in turn yields microwave reflection from the cavity. The reflected microwave power is rectified by a detector, i.e. a microwave diode, yielding a signal proportional to the absolute value of the reflected power P at HDC and modulated by the AC frequency (note that hAC<<HDC). This signal is further fed into a lock-in-detector **(LID)** which samples the signal by convolution with a periodic signal of the same AC frequency over a large number of AC periods, thus yielding amplitude and phase while suppressing statistical noise. Provided that the width of the resonance absorption dependent on HDC is larger than the amplitude of the AC modulation field hAC, the LID output submits a signal

proportional to the field derivative of the paramagnetic resonance signal dP/dH.

An accepted assumption in analyzing experimental results obtained by an EPR spectrometer is that the role of the AC field is to introduce variations in the magnitude of the DC magnetic field; the AC field is usually referred to as the modulating magnetic field. However, as will be shown below, the AC field does not just result in a modulation of the microwave signal dependent of the modulated static field like in a usual EPR experiment, but in superconductors it may also induce an additional mechanism of microwave power

Therefore, one should be perceptive to the different dissipation process, when analyzing the EPR results in the superconducting phase. Here it is important to note that a complication in the measuring process has been observed. It is related to variations in the signal phase that may develop through the measuring process. The reference AC phase of the LID is adjusted for usual EPR measurements to yield the optimum paramagnetic resonance signal. However in case of the microwave dissipation observed in anisotropic superconductors this adjustment turns out to be not appropriate. Variations in the signal phase were observed when varying the magnitude of the variables involved such as DC magnetic field, temperature, sample orientation and others. This effect necessitates a procedure where the lock-in detector phase has to be adjusted, throughout the measuring process, to be in-phase with the AC phase. To overcome this problem, conducted during the measuring process, the

superconductors to date.

**2. Experimental** 

dissipation.

Fig. 2 presents such a result, for a Bi2Sr2CaCu2O8+δ (Bi2212) single crystal, Zero Field Cooled **(ZFC)** to 4K with the DC magnetic field applied parallel to the a-b plane. The upper curve shows the measured signal amplitude, and the lower curve shows its corresponding signal phase (with respect to the lock-in AC phase) as function of DC magnetic field. The measured signal amplitude exhibits two maxima followed by an exponential decay towards zero at high fields. The signal phase is close to zero at low fields and drops steeply towards -1800 at high fields. It indicates that the signal is in phase and out of phase with the AC field at low and high fields, respectively. The details of the phase shift, i.e. deviations (±20°) from 0° at low fields and from 180° at high fields have not been investigated so far and need further attention. As the allover phase shift from low to high fields is evidently less than 180° it cannot be explained just by a constant shift of the calibration.

Fig. 3 presents EPR results similar to those shown in Fig. 2, for temperatures, 80K, 30K and 4K. It indicates that the measured signal amplitude and signal phase have similar functional behavior as in Fig. 2. It implies that the EPR responses as function of field are similar in the temperature range from *Tc* down to 4K.

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 67

0 100 200 300 400 500 600 700 800

Bi2212 ZFC HDCIIa-b hAC=12 gauss T=4K

Exponential

Magnetic Field (gauss)

Fig. 5 presents the measured signal amplitude and the corresponding signal phase as function of magnetic field applied parallel to the a-b plane for a Bi2Sr2Ca2Cu3O10+x (Bi2223) single crystal ZFC to 4K. It shows a response similar to the observed signal in Fig. 2 for Bi2212. As both Bi2212 and Bi2223 are high-anisotropy superconductors, we deduce that a high anisotropy is a necessary condition to observe similar results in the high-*Tc* superconductors. This conjecture is also confirmed by our measurements in the high-*Tc*, low-anisotropy YBCO (YBa2Cu3O7-δ) superconductor: no significant EPR signal was observed in optimally dopes YBCO, when the DC magnetic field is applied either parallel or perpendicular to the a-b plane, while sizable signals appeared for overdoped, i.e. more anisotropic YBCO with the magnetic field applied within the a-b plane. These recent results are intended to be published elsewhere. Fig. 6 presents the actual calculated signal as a function of field in Bi2223 derived from the measured signal amplitude and from its corresponding signal phase of Fig. 5. It shows a similar DC-field dependence like that observed in Fig. 4 for Bi2212. It confirms again the earlier conclusion that high-anisotropy is

Fig. 4. Actual calculated signal in the Bi2212 crystal as function of field parallel to the conduction plane at 4K, derived by multiplying the measured signal amplitude with the

a necessary condition for obtaining a similar behavior in different superconductors.

0 500 1000 1500 2000 2500

Bi2223 HDCIIa-b ZFC T=4K hAC=12 gauss

Magnetic Field (gauss)

Fig. 5. AC induced microwave dissipation signal amplitude with the corresponding phase value of a Bi2223 single crystal, ZFC to 4K, as function of magnetic field aligned parallel to the a-b plane. The temperature behavior of the signals in Bi2223 is similar to that observed



Phase (degrees)

in Bi2212.

Signal (arb. units)

cosine of its corresponding phase using the values shown in Fig. 2.

0

20000 40000 60000

Signal (arb. units)

Fig. 2. AC induced microwave dissipation signal amplitude (a) with the corresponding phase value (b) of a Bi2212 single crystal, ZFC to 4K, as function of DC magnetic field aligned parallel to the a-b plane.

Fig. 3. AC induced microwave dissipation signal amplitudes with their corresponding phase values, of the same Bi2212 single crystal shown in Fig. 2, ZFC to 80K, 30K, and 4K, as function of DC magnetic field aligned parallel to the a-b plane.

Understanding the mechanism that induces the signal, the origin of the spectra, and its corresponding signal-phase as function of field, is an important key in studying superconducting properties with the EPR spectrometer.

Fig. 4 shows the actual calculated signal as function of field in Bi2212, derived by multiplying the measured signal amplitude of Fig. 2 with the cosine of its phase. It shows a sharp increase in the intensity at very low magnetic field followed by a steep decrease with change of sign to negative values and a final increase that asymptotically tends to zero. The theoretical considerations presented in (Shaltiel et al., 2008) showed that this corresponds to the field derivative of the dissipated microwave power due to shaking of JV away from the zero crossing, but is subject to strong nonlinearities close to the zero crossing.

100 200 300 400 500

Bi2212 HDCIIa-b ZFC T=4K hAC=12 gauss

80K

80K

4K

30K

100 200 300 400 500

Magnetic Field (gauss) Fig. 3. AC induced microwave dissipation signal amplitudes with their corresponding phase

Understanding the mechanism that induces the signal, the origin of the spectra, and its corresponding signal-phase as function of field, is an important key in studying

Fig. 4 shows the actual calculated signal as function of field in Bi2212, derived by multiplying the measured signal amplitude of Fig. 2 with the cosine of its phase. It shows a sharp increase in the intensity at very low magnetic field followed by a steep decrease with change of sign to negative values and a final increase that asymptotically tends to zero. The theoretical considerations presented in (Shaltiel et al., 2008) showed that this corresponds to the field derivative of the dissipated microwave power due to shaking of JV away from the

values, of the same Bi2212 single crystal shown in Fig. 2, ZFC to 80K, 30K, and 4K, as

zero crossing, but is subject to strong nonlinearities close to the zero crossing.

30K

4K

Magnetic Field (gauss)

Fig. 2. AC induced microwave dissipation signal amplitude (a) with the corresponding phase value (b) of a Bi2212 single crystal, ZFC to 4K, as function of DC magnetic field

> > -160 -120 -80 -40

Phase (degrees)

function of DC magnetic field aligned parallel to the a-b plane.

superconducting properties with the EPR spectrometer.

signal (arb. units)

b

Bi2212 HDCIIa-b ZFC T=4K hAC=12 gauss a


Phase (degrees)

aligned parallel to the a-b plane.

Signal (arb. uni.)

Fig. 4. Actual calculated signal in the Bi2212 crystal as function of field parallel to the conduction plane at 4K, derived by multiplying the measured signal amplitude with the cosine of its corresponding phase using the values shown in Fig. 2.

Fig. 5 presents the measured signal amplitude and the corresponding signal phase as function of magnetic field applied parallel to the a-b plane for a Bi2Sr2Ca2Cu3O10+x (Bi2223) single crystal ZFC to 4K. It shows a response similar to the observed signal in Fig. 2 for Bi2212. As both Bi2212 and Bi2223 are high-anisotropy superconductors, we deduce that a high anisotropy is a necessary condition to observe similar results in the high-*Tc* superconductors. This conjecture is also confirmed by our measurements in the high-*Tc*, low-anisotropy YBCO (YBa2Cu3O7-δ) superconductor: no significant EPR signal was observed in optimally dopes YBCO, when the DC magnetic field is applied either parallel or perpendicular to the a-b plane, while sizable signals appeared for overdoped, i.e. more anisotropic YBCO with the magnetic field applied within the a-b plane. These recent results are intended to be published elsewhere. Fig. 6 presents the actual calculated signal as a function of field in Bi2223 derived from the measured signal amplitude and from its corresponding signal phase of Fig. 5. It shows a similar DC-field dependence like that observed in Fig. 4 for Bi2212. It confirms again the earlier conclusion that high-anisotropy is a necessary condition for obtaining a similar behavior in different superconductors.

Fig. 5. AC induced microwave dissipation signal amplitude with the corresponding phase value of a Bi2223 single crystal, ZFC to 4K, as function of magnetic field aligned parallel to the a-b plane. The temperature behavior of the signals in Bi2223 is similar to that observed in Bi2212.

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 69

Similar results are observed for a Bi2223 superconductor as shown in **Fig. 8**. They were obtained by zero-field cooling the crystal down to 30K, followed by application of a DC field of 30 gauss, 500 gauss or 5000 gauss prior to sweeping the AC field. Here again, as with the Bi2212 crystal, the Bi2223 signal amplitude, obtained when the low magnetic field of 30 gauss was applied, increases strongly and linearly with increasing AC field, whilst the signal amplitude observed, when 500 gauss was applied, is low and remains constant with increasing AC field. The signal obtained, when the much stronger 5000 gauss cooling field was applied, shows negligibly small signal amplitude for all AC field values. It confirms the previous conclusion regarding the similarity of EPR response in different high anisotropy

> HDC=30 gauss HDC=500 gauss HDC=5000 gauss

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

b

a

AC Magnetic Field (gauss p. to p.)

Fig. 8. Signal amplitude (a) and signal phase (b) in Bi2223 vs. AC magnetic field, obtained by zero-field cooling the crystal to 30K, followed by applying a DC field of 30 gauss or 500

Fig. 9 presents an extremely important result vital to understand the physics, and to comprehend the mechanism of the AC field induced microwave dissipation results in the highly anisotropic superconductors. The figure shows an unexpected effect where the signal intensity as function of temperature in the high anisotropy Bi2212 crystal for the DC magnetic field aligned parallel to the c-axis is zero from Tc to 4K, whilst at DC magnetic field parallel to the a-b plane, the signal intensity increases sharply already just below Tc and continues to increase sharply towards lower temperatures. The small signal observed close to Tc in Fig. 9 for the DC magnetic field applied parallel to the c-axis results from the thermally activated flux flow resistivity (FFR) (Shaltiel et al., 2001)**.** (See Fig. 6 of Shaltiel et

A reasonable conclusion to the extreme difference in the superconducting response under the parallel and the perpendicular DC magnetic field, is derived from Fig. 10. It shows that, when a magnetic field is parallel to the a-b plane, only Josephson vortices (JV) are formed and, while applying a magnetic field parallel to the c-axis, only Abricosov vortices, i.e.

superconductors.

al., 2001)

Bi-2223 HDCIIa-b ZFC T=30K


gauss or 5000 gauss prior to sweeping the AC field.

**3. Understanding the physics of the AC-field-induced signals** 

Phase (degrees)

signal (arb. units)

Fig. 6. Actual calculated signal in the Bi2223 crystal as function of field parallel to the conduction plane at 4K, derived by multiplying the measured signal amplitude with the cosine of its corresponding phase using the values shown in Fig. 3.

The experimental results presented in Figs. 2-6 show the EPR signal as a function of DC magnetic field using a constant AC field. An additional picture is obtained by interchanging the roles of the variables, namely the DC field is kept constant and the EPR signal is measured as function of AC field. Such a result is illustrated in Fig. 7**;** it shows the signal amplitude and signal phase as function of AC magnetic field for a Bi2212 crystal. The measurements were performed by zero-field cooling the crystal down to 30K, followed by applying a DC field of 20 gauss or 150 gauss, prior to sweeping the AC field. The figure shows striking differences in the signal amplitude and in the signal phase as function of AC field when either of the two DC fields is applied. The 20 gauss signal amplitude increases strongly and linearly with increasing AC field. In contrast, the 150 gauss signal amplitude after a slight increase - remains low and constant. At low AC field the two signals are out of phase with respect to the AC phase. At high AC field the signals are in phase and out of phase respectively when either the low or the high DC field are applied.

Fig. 7. Signal amplitude (a) and signal phase (b) in Bi2212 vs. AC magnetic field, obtained by zero-field cooling the crystal to 30K, followed by applying a DC field of 20 gauss or 150 gauss prior to sweeping the AC field.

Bi2223 ZFC HDCIIa-b hAC=12 gauss T=4K

0 100 200 300 400 500

Magnetic Field (gauss)

The experimental results presented in Figs. 2-6 show the EPR signal as a function of DC magnetic field using a constant AC field. An additional picture is obtained by interchanging the roles of the variables, namely the DC field is kept constant and the EPR signal is measured as function of AC field. Such a result is illustrated in Fig. 7**;** it shows the signal amplitude and signal phase as function of AC magnetic field for a Bi2212 crystal. The measurements were performed by zero-field cooling the crystal down to 30K, followed by applying a DC field of 20 gauss or 150 gauss, prior to sweeping the AC field. The figure shows striking differences in the signal amplitude and in the signal phase as function of AC field when either of the two DC fields is applied. The 20 gauss signal amplitude increases strongly and linearly with increasing AC field. In contrast, the 150 gauss signal amplitude after a slight increase - remains low and constant. At low AC field the two signals are out of phase with respect to the AC phase. At high AC field the signals are in phase and out of

0 3 6 9 12 15 18 21 24 27 30

**b**

**a**

Bi-2212 Optimally doped HDCIIa-b ZFC T=30K

AC Magnetic Field (p to p, gauss)

Fig. 7. Signal amplitude (a) and signal phase (b) in Bi2212 vs. AC magnetic field, obtained by zero-field cooling the crystal to 30K, followed by applying a DC field of 20 gauss or 150

Fig. 6. Actual calculated signal in the Bi2223 crystal as function of field parallel to the conduction plane at 4K, derived by multiplying the measured signal amplitude with the

cosine of its corresponding phase using the values shown in Fig. 3.

phase respectively when either the low or the high DC field are applied.

HDC= 20gauss HDC=150gauss

> -180 -150 -120 -90 -60 -30

Phase (degrees)

gauss prior to sweeping the AC field.

Signal (arb. units)

Signal (arb. units)

Similar results are observed for a Bi2223 superconductor as shown in **Fig. 8**. They were obtained by zero-field cooling the crystal down to 30K, followed by application of a DC field of 30 gauss, 500 gauss or 5000 gauss prior to sweeping the AC field. Here again, as with the Bi2212 crystal, the Bi2223 signal amplitude, obtained when the low magnetic field of 30 gauss was applied, increases strongly and linearly with increasing AC field, whilst the signal amplitude observed, when 500 gauss was applied, is low and remains constant with increasing AC field. The signal obtained, when the much stronger 5000 gauss cooling field was applied, shows negligibly small signal amplitude for all AC field values. It confirms the previous conclusion regarding the similarity of EPR response in different high anisotropy superconductors.

Fig. 8. Signal amplitude (a) and signal phase (b) in Bi2223 vs. AC magnetic field, obtained by zero-field cooling the crystal to 30K, followed by applying a DC field of 30 gauss or 500 gauss or 5000 gauss prior to sweeping the AC field.

## **3. Understanding the physics of the AC-field-induced signals**

Fig. 9 presents an extremely important result vital to understand the physics, and to comprehend the mechanism of the AC field induced microwave dissipation results in the highly anisotropic superconductors. The figure shows an unexpected effect where the signal intensity as function of temperature in the high anisotropy Bi2212 crystal for the DC magnetic field aligned parallel to the c-axis is zero from Tc to 4K, whilst at DC magnetic field parallel to the a-b plane, the signal intensity increases sharply already just below Tc and continues to increase sharply towards lower temperatures. The small signal observed close to Tc in Fig. 9 for the DC magnetic field applied parallel to the c-axis results from the thermally activated flux flow resistivity (FFR) (Shaltiel et al., 2001)**.** (See Fig. 6 of Shaltiel et al., 2001)

A reasonable conclusion to the extreme difference in the superconducting response under the parallel and the perpendicular DC magnetic field, is derived from Fig. 10. It shows that, when a magnetic field is parallel to the a-b plane, only Josephson vortices (JV) are formed and, while applying a magnetic field parallel to the c-axis, only Abricosov vortices, i.e.

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 71

This conclusion is corroborated by the fact that microwave dissipation results similar to those shown in Fig. 7, were observed also in Bi2223 which like in Bi2212 forms JV and PV for the magnetic field applied parallel and perpendicular to the a-b plane, respectively. Moreover, we recall that, unlike in Bi2212 or in Bi2223, no significant microwave dissipation was observed in optimally doped YBa2Cu3O7-δ for any direction of the external DC magnetic field besides the signal at Tc from the thermally activated flux flow resistivity. As only PV are formed in optimally doped YBa2Cu3O7-δ for all magnetic field orientations, this again confirms the conclusion that no microwave dissipation is induced by the AC magnetic field,

A mechanism that explains the induced microwave dissipation by the AC field, when interacting with JV, is derived from soliton propagation in Josephson transition lines observed in itinerant Josephson junction (IJJ) mesas (Hechtfischer et al., 1997)**.** Such a mesa is given in Fig. 111 : A DC bias current applied parallel to the a-b plane induces a Lorentz force on the JV **(fluxons)** that reside between the conduction layers of the superconductor. Extremely high JV velocities, that may reach the speed of light, are obtained when the DC current is applied (Ustinov, 1998). The JV motion induces electromagnetic oscillations and,

Mesa structure

Fig. 11. Mesa structure of a BSCCO crystal. The bias current applies a Lorentz force on the

Fig. 12 shows that the AC magnetic field applied parallel to the a-b plane induces oscillating Eddy currents in the conduction plane in a Bi2212 compound. As with the current in IJJ mesas, the Eddy currents give rise to a Lorentz force on the JV whose oscillating intensity induces oscillating interaction with the microwave field that generates the EPR microwave losses. This explains why EPR signals are observed at H||a-b plane but none at H||c-axis, as JV are formed only when the magnetic field is parallel to the conduction plane and are not formed when the field is parallel to the c axis. A straight forward conclusion in this analysis is that EPR dissipation signals in high anisotropy superconductors are observed only due to the motion of

<sup>1</sup> Permission to publish this figure (taken from Hechtfischer et al., 1997) was granted by APS, that keeps

JV that are present between the conduction planes.

its copyrights).

when interacting with the microwave field, microwave dissipation may result.

when interacting with PV.

pancake vortices (PV), are formed. This indicates that the AC magnetic field induces microwave dissipation, when interacting with JV, but none when interacting with PV.

Fig. 9. Signal amplitude as function of temperature in the high anisotropy Bi2212 crystal for DC magnetic field aligned parallel to the a-b plane or to the c-axis. Note the large different response for the two orientations. The small signal observed close to Tc for magnetic field parallel to the c-axis results from thermally activated flux flow resistivity.

Vortex structure for magnetic field parallel or perpendicular to the a-b plane

Fig. 10. The figure shows that applying a magnetic field parallel to the a-b plane forms only JV, whilst, applying a magnetic field parallel to the c-axis forms only PV.

pancake vortices (PV), are formed. This indicates that the AC magnetic field induces microwave dissipation, when interacting with JV, but none when interacting with PV.

HDCIIa-b

0 10 20 30 40 50 60 70 80 90

HDCIIc

Temperature (K)

Fig. 9. Signal amplitude as function of temperature in the high anisotropy Bi2212 crystal for DC magnetic field aligned parallel to the a-b plane or to the c-axis. Note the large different response for the two orientations. The small signal observed close to Tc for magnetic field

Vortex structure for magnetic field parallel or

Fig. 10. The figure shows that applying a magnetic field parallel to the a-b plane forms only

perpendicular to the a-b plane

JV, whilst, applying a magnetic field parallel to the c-axis forms only PV.

0

2000

4000

Signal (arb. units)

6000

Bi2212 optimally doped HDC=20 gauss

hAC=10 gauss peak to peak

parallel to the c-axis results from thermally activated flux flow resistivity.

8000

This conclusion is corroborated by the fact that microwave dissipation results similar to those shown in Fig. 7, were observed also in Bi2223 which like in Bi2212 forms JV and PV for the magnetic field applied parallel and perpendicular to the a-b plane, respectively. Moreover, we recall that, unlike in Bi2212 or in Bi2223, no significant microwave dissipation was observed in optimally doped YBa2Cu3O7-δ for any direction of the external DC magnetic field besides the signal at Tc from the thermally activated flux flow resistivity. As only PV are formed in optimally doped YBa2Cu3O7-δ for all magnetic field orientations, this again confirms the conclusion that no microwave dissipation is induced by the AC magnetic field, when interacting with PV.

A mechanism that explains the induced microwave dissipation by the AC field, when interacting with JV, is derived from soliton propagation in Josephson transition lines observed in itinerant Josephson junction (IJJ) mesas (Hechtfischer et al., 1997)**.** Such a mesa is given in Fig. 111 : A DC bias current applied parallel to the a-b plane induces a Lorentz force on the JV **(fluxons)** that reside between the conduction layers of the superconductor. Extremely high JV velocities, that may reach the speed of light, are obtained when the DC current is applied (Ustinov, 1998). The JV motion induces electromagnetic oscillations and, when interacting with the microwave field, microwave dissipation may result.

## Mesa structure

Fig. 11. Mesa structure of a BSCCO crystal. The bias current applies a Lorentz force on the JV that are present between the conduction planes.

Fig. 12 shows that the AC magnetic field applied parallel to the a-b plane induces oscillating Eddy currents in the conduction plane in a Bi2212 compound. As with the current in IJJ mesas, the Eddy currents give rise to a Lorentz force on the JV whose oscillating intensity induces oscillating interaction with the microwave field that generates the EPR microwave losses. This explains why EPR signals are observed at H||a-b plane but none at H||c-axis, as JV are formed only when the magnetic field is parallel to the conduction plane and are not formed when the field is parallel to the c axis. A straight forward conclusion in this analysis is that EPR dissipation signals in high anisotropy superconductors are observed only due to the motion of

<sup>1</sup> Permission to publish this figure (taken from Hechtfischer et al., 1997) was granted by APS, that keeps its copyrights).

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 73

Fig. 13 presents the calculated signal of Bi2212 and Bi2223 as a function of magnetic field applied parallel to the a-b plane at *T*=4K**.** It is obtained by multiplying the measured signal amplitudes shown in Figs. 2 & 3, with their corresponding phase. It indicates the presence of three regions: A low field region, from zero to around 30 gauss, where the signal intensity increases almost linearly with field up to a maximum value; a high field region, where the signal intensity is zero; and an intermediate transition region. At low JV densities, the JV form the unpinned "dilute lattice region" due to negligible interaction between the JV. As the JV density increases linearly with the magnetic field, the Lorentz force on the JV induces a signal in the low-field region proportional to the DC field, as observed. At higher fields the inter JV interaction is strong and the JV form the "dense lattice region" where the JV are strongly pinned. The intensity of the AC field is not strong enough to induce motion of the JV in this region; therefore the signal intensity is zero. The inbetween intermediate region is characterized by a sharp decrease in the signal intensity and by an exponential decrease towards zero with increasing field. The field widths of the intermediate regions observed in Bi2212 and Bi2223 are different. This indicates a different distribution of the JV within the two compounds. The evolution of the JV system in the intermediate region has not been

**4. Study of superconducting properties with EPR spectrometer** 

**4.1 Interaction between Josephson vortices and Abricosov vortices** 

The theory which discusses the EPR experimental results was published earlier (Shaltiel et al., 2008). It shows that pinning of Josephson vortices effects strongly on their behavior. This section presents EPR results obtained in high anisotropic superconductors, which document the strength of this method in the investigation of the properties of Josephson vortices. It will also discuss the possibility to study small and nano-scale superconducting systems with

We have shown that the absence or presence of JV pinning is vital when investigating superconducting properties with the EPR technique. For example, the interaction of pancake vortices (PV), also known as Abrikosov vortices (AV), with JV induces pinning. Pinning of JV can be studied by varying the density of the Josephson vortices and Abricosov vortices in a controlled way and measuring the effect on the EPR signal. This can be achieved by tilting the orientation of a Bi2212 crystal, exposed to a given magnetic field HDC, from magnetic field parallel to the a-b plane, where only JV are formed, towards the c-axis where only AV are formed. The density of the JV and PV as a function of the tilting angle, *θ*, of the magnetic field (*θ* being zero at magnetic field parallel to the a-b plane) varies as cos*θ* and sin*θ*, respectively. Thus, the signal amplitude being proportional to the JV density should vary as cos*θ*, provided the JV remain unpinned. Deviation from cos*θ* dependence would result from pinning of JV induced via the JV-AV interaction. This is demonstrated in **Fig. 14** for HDC=20

The figure shows that the signal amplitude decreases close to cos *θ* at *θ* values, 00 to 400. But at higher angles the signal amplitude decreases faster than cos *θ*. The increase in deviation indicates an increase in pinning above 400 due to the increase of the JV-AV interaction. As the pinning strength increases with angle, at angles below 400 the JV-AV interaction is not strong enough to induce an observable effect. The intensity of the interaction depends on Hi.

reported elsewhere.

EPR.

gauss.

JV. Therefore JV pinning plays an essential role on the signal shape. As JV pinning depends on the variables such as temperature and sample orientation, the analysis of the signal as a function of the variables involved enables the microscopic study of JV pinning.

Fig. 12. Schematic picture that interprets the formation of AC-induced oscillating Eddy currents. The currents apply oscillating Lorentz forces on the Josephson vortices.

Fig. 13. Calculated signal in Bi2212 (a) and in Bi2223 (b) as a function of magnetic field parallel to the a-b plane at T=4K. It is obtained by multiplying the signal amplitude, with its corresponding phase. It shows formation of the dilute and dense region at low and high magnetic field respectively and an in-between intermediate region, in both compounds.

JV. Therefore JV pinning plays an essential role on the signal shape. As JV pinning depends on the variables such as temperature and sample orientation, the analysis of the signal as a

Fig. 12. Schematic picture that interprets the formation of AC-induced oscillating Eddy currents. The currents apply oscillating Lorentz forces on the Josephson vortices.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400

Magnetic Field (gauss)

Exponential fit

Fig. 13. Calculated signal in Bi2212 (a) and in Bi2223 (b) as a function of magnetic field parallel to the a-b plane at T=4K. It is obtained by multiplying the signal amplitude, with its corresponding phase. It shows formation of the dilute and dense region at low and high magnetic field respectively and an in-between intermediate region, in both compounds.

Exponential fit

Intermediate lattice region

Dense lattice region Intermediate lattice region

Dilute lattice region

Dilute lattice region

Dense lattice region

Bi2223

HDCIIa-b; ZFC hac=12gauss p. to p. T=4K a

Bi2212

b



Signal (arb. units)

Signal (arb. units)

function of the variables involved enables the microscopic study of JV pinning.

Fig. 13 presents the calculated signal of Bi2212 and Bi2223 as a function of magnetic field applied parallel to the a-b plane at *T*=4K**.** It is obtained by multiplying the measured signal amplitudes shown in Figs. 2 & 3, with their corresponding phase. It indicates the presence of three regions: A low field region, from zero to around 30 gauss, where the signal intensity increases almost linearly with field up to a maximum value; a high field region, where the signal intensity is zero; and an intermediate transition region. At low JV densities, the JV form the unpinned "dilute lattice region" due to negligible interaction between the JV. As the JV density increases linearly with the magnetic field, the Lorentz force on the JV induces a signal in the low-field region proportional to the DC field, as observed. At higher fields the inter JV interaction is strong and the JV form the "dense lattice region" where the JV are strongly pinned. The intensity of the AC field is not strong enough to induce motion of the JV in this region; therefore the signal intensity is zero. The inbetween intermediate region is characterized by a sharp decrease in the signal intensity and by an exponential decrease towards zero with increasing field. The field widths of the intermediate regions observed in Bi2212 and Bi2223 are different. This indicates a different distribution of the JV within the two compounds. The evolution of the JV system in the intermediate region has not been reported elsewhere.

## **4. Study of superconducting properties with EPR spectrometer**

The theory which discusses the EPR experimental results was published earlier (Shaltiel et al., 2008). It shows that pinning of Josephson vortices effects strongly on their behavior. This section presents EPR results obtained in high anisotropic superconductors, which document the strength of this method in the investigation of the properties of Josephson vortices. It will also discuss the possibility to study small and nano-scale superconducting systems with EPR.

## **4.1 Interaction between Josephson vortices and Abricosov vortices**

We have shown that the absence or presence of JV pinning is vital when investigating superconducting properties with the EPR technique. For example, the interaction of pancake vortices (PV), also known as Abrikosov vortices (AV), with JV induces pinning. Pinning of JV can be studied by varying the density of the Josephson vortices and Abricosov vortices in a controlled way and measuring the effect on the EPR signal. This can be achieved by tilting the orientation of a Bi2212 crystal, exposed to a given magnetic field HDC, from magnetic field parallel to the a-b plane, where only JV are formed, towards the c-axis where only AV are formed. The density of the JV and PV as a function of the tilting angle, *θ*, of the magnetic field (*θ* being zero at magnetic field parallel to the a-b plane) varies as cos*θ* and sin*θ*, respectively. Thus, the signal amplitude being proportional to the JV density should vary as cos*θ*, provided the JV remain unpinned. Deviation from cos*θ* dependence would result from pinning of JV induced via the JV-AV interaction. This is demonstrated in **Fig. 14** for HDC=20 gauss.

The figure shows that the signal amplitude decreases close to cos *θ* at *θ* values, 00 to 400. But at higher angles the signal amplitude decreases faster than cos *θ*. The increase in deviation indicates an increase in pinning above 400 due to the increase of the JV-AV interaction. As the pinning strength increases with angle, at angles below 400 the JV-AV interaction is not strong enough to induce an observable effect. The intensity of the interaction depends on Hi.

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 75

**4.2 Comparison of EPR signals, derived from either the field cooling or from the zero** 

The maximum signal amplitude as function of temperature, derived from measurements of signal amplitude vs. magnetic field applied parallel to the a-b conduction planes in Bi2212, for zero field cooled (ZFC) protocol and for 500 gauss field cooled (FC) protocol is shown in Fig. 16. The maximum signal amplitudes were obtained from the corresponding maxima in the amplitude values similar to that shown in Fig 2. The figure shows that in both procedures, the maximum signal amplitudes have similar values from Tc down to 40K. Below 40K the maximum signal amplitude strongly increases for the ZFC measurement and sharply decreases for the FC measurement. The sharp decrease of the signal below 40K indicates that field cooling introduces pinning of JV. The increase of the ZFC signal indicates

0 10 20 30 40 50 60 70 80 90

500 gauss Field Cooled

temperature

Temperature (K)

Fig. 16. Maximum signal amplitude as function of temperature in Bi2212 obtained when the

A memory signal is observed in the high anisotropy superconductors, Bi2212 and Bi2223. It is obtained by field cooling the superconductors in a magnetic field Hm, applied parallel to the conduction planes down to 4K, the lowest available temperature of the spectrometer. After setting the DC field to zero and then measuring the microwave dissipation on increasing DC field, a signal is observed slightly above Hm, indicating a **memory effect**. Memory signals are shown in Fig. 17 for Bi2212; with Hm values that range from 1500 gauss to 17000 gauss. (17000

A detailed theoretical analysis of the memory effect was presented in an earlier publication (Shaltiel et al., 2010)**.** It indicates that by adiabatic field cooling from *Tc* to a low temperature, the JV explore the available phase space that nucleates into the deepest valley of the

sample is field cooled at either zero magnetic field or at 500 gauss magnetic field.

gauss was the maximum available field with the operating spectrometer).

Zero Field Cooled

**field cooling protocol using identical measurements**

0

10000

20000

30000

Signal (arb. units)

**4.3 Memory effect** 

40000

50000

60000

the absence, or only a very small pinning effect, of the JV below 40K.

Increasing HDC increases both JV and AV densities for all tilting angles. Therefore, the deviation from cos *θ* at higher HDC should start at a lower tilting angle. This is observed in Fig. 15 for HDC strengths of 20 gauss, 50 gauss, and 90 gauss where the deviation angle starts at 400, 200, and 50, respectively, indicating that the higher HDC the lower the angle where the deviation starts. The systematic analysis of the signal amplitude as a function of the tilting angle for different HDC values can be used to study the JV-AV interaction.

Fig. 14. Bi2212 signal amplitude as function of tilted angle *θ* in an applied field of 20 gauss.

Fig. 15. Bi2212 signal ampltude as function of tilted angle *θ* in applied fields of 20 gauss, 50 gauss and 90 gauss.

### **4.2 Comparison of EPR signals, derived from either the field cooling or from the zero field cooling protocol using identical measurements**

The maximum signal amplitude as function of temperature, derived from measurements of signal amplitude vs. magnetic field applied parallel to the a-b conduction planes in Bi2212, for zero field cooled (ZFC) protocol and for 500 gauss field cooled (FC) protocol is shown in Fig. 16. The maximum signal amplitudes were obtained from the corresponding maxima in the amplitude values similar to that shown in Fig 2. The figure shows that in both procedures, the maximum signal amplitudes have similar values from Tc down to 40K. Below 40K the maximum signal amplitude strongly increases for the ZFC measurement and sharply decreases for the FC measurement. The sharp decrease of the signal below 40K indicates that field cooling introduces pinning of JV. The increase of the ZFC signal indicates the absence, or only a very small pinning effect, of the JV below 40K.

Fig. 16. Maximum signal amplitude as function of temperature in Bi2212 obtained when the sample is field cooled at either zero magnetic field or at 500 gauss magnetic field.

## **4.3 Memory effect**

74 Superconductors – Properties, Technology, and Applications

Increasing HDC increases both JV and AV densities for all tilting angles. Therefore, the deviation from cos *θ* at higher HDC should start at a lower tilting angle. This is observed in Fig. 15 for HDC strengths of 20 gauss, 50 gauss, and 90 gauss where the deviation angle starts at 400, 200, and 50, respectively, indicating that the higher HDC the lower the angle where the deviation starts. The systematic analysis of the signal amplitude as a function of the tilting


Tilted Angle off a-b (Degrees)

50 gauss

Fig. 14. Bi2212 signal amplitude as function of tilted angle *θ* in an applied field of 20 gauss.

200

Bi2212 HDC=50 gauss hAC=12 gauss T=40K

Tilted Angle off a-b plane (Degrees)

Fig. 15. Bi2212 signal ampltude as function of tilted angle *θ* in applied fields of 20 gauss, 50


Signal

40 degrees

Ixcos


50

90 gauss

Bi2212 HDC=20 gauss hAC=12 gauss T=40K

angle for different HDC values can be used to study the JV-AV interaction.

0


40<sup>0</sup>

20 gauss

0

2000

gauss and 90 gauss.

4000

6000

Signal (arb. units)

8000

10000

12000

2000

4000

6000

Signal (arb. units)

8000

10000

12000

14000

A memory signal is observed in the high anisotropy superconductors, Bi2212 and Bi2223. It is obtained by field cooling the superconductors in a magnetic field Hm, applied parallel to the conduction planes down to 4K, the lowest available temperature of the spectrometer. After setting the DC field to zero and then measuring the microwave dissipation on increasing DC field, a signal is observed slightly above Hm, indicating a **memory effect**. Memory signals are shown in Fig. 17 for Bi2212; with Hm values that range from 1500 gauss to 17000 gauss. (17000 gauss was the maximum available field with the operating spectrometer).

A detailed theoretical analysis of the memory effect was presented in an earlier publication (Shaltiel et al., 2010)**.** It indicates that by adiabatic field cooling from *Tc* to a low temperature, the JV explore the available phase space that nucleates into the deepest valley of the

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 77

field region. To check this conjecture, the following procedure to measure the magnetization in a Bi2212 crystal was carried out. The magnetization was measured in a commercial SQUID (superconducting quantum interference device) magnetometer MPMS5 (Quantum Design) working in a temperature range from 1.8 < *T* < 400K and in magnetic fields up to 50 Kgauss. The Bi2212 single crystal was aligned within the a-b plane. To obtain the field dependence of the ZFC or FC magnetization at 4K, the sample was cooled down to 4K from above *Tc* at the field indicated at the corresponding hysteresis loop, of 0 Kgauss, 5 Kgauss and 10 Kgauss. Then the field was set to zero and the magnetization measured with increasing field up to 50 Kgauss, then in decreasing field down to -50 Kgauss and again up to the highest field to close the hysteresis loop. The magnetization results are shown in **Figs. 18** a and b. Indeed, as expected, no magnetization is observed at zero magnetic field when a zero magnetic field is applied. In contrast, a certain magnetization is observed at zero magnetic field when the sample is field cooled at 5 Kgauss. An even larger magnetization occurs at a cooling field of 10 Kgauss. A magnetization ratio of about 2 is obtained from Figs. 18 at H=0 in these field cooling processes of 5 Kgauss and 10 Kgauss, as expected.


Magnetic Field (kOe)

indicates the formation of JV remnant magnetization.

and loose the equilibrium state with increasing temperature.

Mmol = 824.824 g/mol m = 0.38 mg

**a**

The memory results presented so far, did not involve the effect of temperature or time on the microwave-dissipaton signals. A limited experimental data related to time effects was achived as follows. After obtaining a Hm memory signal, the magnetic field was decreased to zero and kept at this field for 2 hours. Then the field was increased to a value above Hm to disclose the memory signal. The observed signal intensity and signal shape were similar to those observed earlier. It indicates that, if a time effect is present, it should be much longer than two hours.

The dependence of the maximum signal amplitude of a memory signal on the temperature was obtained as follows: A Bi2212 crystal aligned parallel to the a-b plane is cooled in a Hm field to 4K. After decreasing the field to zero, it was increased to a field that yields the maximum memory signal. Keeping the field constant the signal amplitude is measured by increasing temperature from 4K to 30K. The result is shown in Fig. 19. The figure illustrates that the amplitude of the memory signal remains constant in the temperature range 4K to 10K and decreases at higher temperature. This indicates that the JV remain frozen up to 10K

Figs. 18a and 18b. Magnetization as function of magnetic field in a Bi2212 crystal, field cooled at DC fields of 0, 5 and 10 Kgauss. Note that non zero magnetization is observed at zero magnetic fields when the compound is field cooled at 5 Kgauss or 10 Kgauss. It


M (emu/mol)


Magnetic Field (kOe)

Mmol = 824.824 g/mol m = 0.38 mg **Bi2 Sr2 CaCuO8**

 cool 0 Kgauss cool 5 Kgauss cool 10 Kgauss T = 4 K

**b**

**Bi2 Sr2 CaCuO8**

 cool 0 Kgauss cool 5 Kgauss cool 10 Kgauss T = 4 K


0

4000

M (emu/mol)

8000

landslide potential and are confined inside deep pinning landslide-potential minima in a quasi equilibrium glassy state. As the field is subsequently increased the fluxons are immobile and prevent penetration of new JV sites and cannot absorb microwave energy up to Hm. No signal was observed in this field region. When the magnetic field is increased above Hm new unpinned vortices start to penetrate.

Fig. 17. Memory signal observed at magnetic fields, for Hm values 1.5 kgauss to 17 kgauss. Note that the memory signals appear at field values slightly above Hm.

Further analysis of the Bi2212 memory results presented in Fig. 17 demonstrates additional interesting properties. They indicate that the memory signals hold the following features, common to all Hm values: (a) the signals exhibit the same shape and retain similar intensities, and similar widths for all Hm values. (b) The field position Hfp of each memory signal occurs at a field slightly above Hm with similar Hfp-Hm value. (c) A narrow singlepeak, small-intensity signal is observed at H=0 for all Hm values that decays strongly towards zero. d) In the field region from above H=0 to H=Hm, only a very weak signal is observed indicating the presence of a very small amount of unpinned Josephson vortices along this field region. Analyzing the memory features described in (a) to (d), implies the following additional properties: The signal shape of the memory signals described in (a) indicates similar features as those obtained in the signal shape of Fig. 2 presented earlier. The signal in Fig. 2 was obtained by zero field cooling the crystal (with the field oriented parallel to the a-b plane) followed by increasing the intensity of the magnetic field. In this process the AC field interacts with the JV that penetrate the crystal from H=0 and above, inducing the observed signal. The similarity between the signal shapes of Fig. 2 with the memory signals indicates that the same mechanism induces the memory signals in both cases. Thus as in Fig. 2, unpinned JV start to penetrate the crystal at magnetic fields above Hm, generating the memory effect. This conclusion is in agreement with the theoretical analysis given in reference (Shaltiel et al., 2010).

The analysis presented above in the field region where the memory signal is formed implies unexpected and intriguing consequences. It indicates that JV are present in the magnetic field region from zero to Hm. It predicts the formation of JV remanent magnetization in this

landslide potential and are confined inside deep pinning landslide-potential minima in a quasi equilibrium glassy state. As the field is subsequently increased the fluxons are immobile and prevent penetration of new JV sites and cannot absorb microwave energy up to Hm. No signal was observed in this field region. When the magnetic field is increased

FC 1.5 Kgauss

FC 3 Kgauss

200 400 600

200 400 600

200 400 600

Fig. 17. Memory signal observed at magnetic fields, for Hm values 1.5 kgauss to 17 kgauss.

Further analysis of the Bi2212 memory results presented in Fig. 17 demonstrates additional interesting properties. They indicate that the memory signals hold the following features, common to all Hm values: (a) the signals exhibit the same shape and retain similar intensities, and similar widths for all Hm values. (b) The field position Hfp of each memory signal occurs at a field slightly above Hm with similar Hfp-Hm value. (c) A narrow singlepeak, small-intensity signal is observed at H=0 for all Hm values that decays strongly towards zero. d) In the field region from above H=0 to H=Hm, only a very weak signal is observed indicating the presence of a very small amount of unpinned Josephson vortices along this field region. Analyzing the memory features described in (a) to (d), implies the following additional properties: The signal shape of the memory signals described in (a) indicates similar features as those obtained in the signal shape of Fig. 2 presented earlier. The signal in Fig. 2 was obtained by zero field cooling the crystal (with the field oriented parallel to the a-b plane) followed by increasing the intensity of the magnetic field. In this process the AC field interacts with the JV that penetrate the crystal from H=0 and above, inducing the observed signal. The similarity between the signal shapes of Fig. 2 with the memory signals indicates that the same mechanism induces the memory signals in both cases. Thus as in Fig. 2, unpinned JV start to penetrate the crystal at magnetic fields above Hm, generating the memory effect. This conclusion is in agreement with the theoretical

The analysis presented above in the field region where the memory signal is formed implies unexpected and intriguing consequences. It indicates that JV are present in the magnetic field region from zero to Hm. It predicts the formation of JV remanent magnetization in this

2 4 6 8 10 0

FC 15 Kgauss

3 6 9 12 15 18 0

FC 17 Kgauss

3 6 9 12 15 18 0

Magnetic Field (Kgauss)

FC 8 Kgauss

above Hm new unpinned vortices start to penetrate.

200 400 600

analysis given in reference (Shaltiel et al., 2010).

200 400 600

Signal (Arb. units)

<sup>1234567</sup> 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

FC 5 Kgauss

<sup>12345</sup> 0

Magnetic Field (Kgauss)

Note that the memory signals appear at field values slightly above Hm.

field region. To check this conjecture, the following procedure to measure the magnetization in a Bi2212 crystal was carried out. The magnetization was measured in a commercial SQUID (superconducting quantum interference device) magnetometer MPMS5 (Quantum Design) working in a temperature range from 1.8 < *T* < 400K and in magnetic fields up to 50 Kgauss. The Bi2212 single crystal was aligned within the a-b plane. To obtain the field dependence of the ZFC or FC magnetization at 4K, the sample was cooled down to 4K from above *Tc* at the field indicated at the corresponding hysteresis loop, of 0 Kgauss, 5 Kgauss and 10 Kgauss. Then the field was set to zero and the magnetization measured with increasing field up to 50 Kgauss, then in decreasing field down to -50 Kgauss and again up to the highest field to close the hysteresis loop. The magnetization results are shown in **Figs. 18** a and b. Indeed, as expected, no magnetization is observed at zero magnetic field when a zero magnetic field is applied. In contrast, a certain magnetization is observed at zero magnetic field when the sample is field cooled at 5 Kgauss. An even larger magnetization occurs at a cooling field of 10 Kgauss. A magnetization ratio of about 2 is obtained from Figs. 18 at H=0 in these field cooling processes of 5 Kgauss and 10 Kgauss, as expected.

Figs. 18a and 18b. Magnetization as function of magnetic field in a Bi2212 crystal, field cooled at DC fields of 0, 5 and 10 Kgauss. Note that non zero magnetization is observed at zero magnetic fields when the compound is field cooled at 5 Kgauss or 10 Kgauss. It indicates the formation of JV remnant magnetization.

The memory results presented so far, did not involve the effect of temperature or time on the microwave-dissipaton signals. A limited experimental data related to time effects was achived as follows. After obtaining a Hm memory signal, the magnetic field was decreased to zero and kept at this field for 2 hours. Then the field was increased to a value above Hm to disclose the memory signal. The observed signal intensity and signal shape were similar to those observed earlier. It indicates that, if a time effect is present, it should be much longer than two hours.

The dependence of the maximum signal amplitude of a memory signal on the temperature was obtained as follows: A Bi2212 crystal aligned parallel to the a-b plane is cooled in a Hm field to 4K. After decreasing the field to zero, it was increased to a field that yields the maximum memory signal. Keeping the field constant the signal amplitude is measured by increasing temperature from 4K to 30K. The result is shown in Fig. 19. The figure illustrates that the amplitude of the memory signal remains constant in the temperature range 4K to 10K and decreases at higher temperature. This indicates that the JV remain frozen up to 10K and loose the equilibrium state with increasing temperature.

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 79

**4.5 Evaluating the possibility to study small or nano-scale superconducting systems** 

The microwave-dissipation signal of a high *Tc* Bi2212 superconductor, shown in Fig. 21, is obtained from Fig. 2*.* The sample size was 10x10x0.1 mm3. Its maximum intensity is of the order of 10,000 arbitrary units. The remarkably high intensity signal promises the possibility to investigate nano-scale superconductors and sizable small samples obtained by reducing

> Bi2212 ZFC HDCIIa-b

T=4K

hAC=12 kgauss

0 100 200 300 400 500

Fig. 21. Signal amplitude as function of magnetic field in a 10x10x0.1mm3 Bi2212 crystal. The remarkable high intensity signal promises investigation of EPR properties in nano crystals.

Extremely thin superconductors with a limited number of layers are required when studying sample size effects on their properties. EPR technique can be considered in investigating their properties, by comparing results obtained in small layered nano crystals of similar structure with those obtained in crystals with a larger layer number. In samples with a small number of conducting layers the following EPR measurements are anticipated: a) Differences in the interaction between JV and AV vortices due to the small number of neighboring conducting layers. b) Results presented so far have shown that the observed properties depend on sample treatments, such as field cooling or zero field cooling procedure. Therefore the effect of cooling on thin samples may show different response that would depend on the sample thickness. c) The magnetic field memory effect, being by itself a very interesting effect, is explained using the hierarchical model of glassy state, commonly used in spin glass theory (Shaltiel et al., 2010). As the glassy state may depend on the thickness of the sample, it would be interesting to investigate the memory effect in small layer samples. Nano superconducting samples needed to investigate their properties are readily obtained using cleavage methods. In contrast the preparation method of thin superconductors for investigating IJJ properties, when regular methods are applied, is cumbersome, the reason being that conducting elements should be installed on the mesas' surface. Here we quote such a sample preparation in mesas that has been used to study teraherz-wave emission from Intrinsic Josephson Junctions in high *Tc* superconductors (Kadowaki et al., 2008): "The experiments were performed with the conventional methods commonly used in performing similar measurements and include

Magnetic Field (gauss)

**with EPR spectrometers** 

the sample thickness to a few superconducting layers.

0

1000

2000

3000

Signal (arb. units)

4000

5000

6000

Fig. 19. Dependence of the maximum memory signal ampitude on temperature, derived from the memory signal after field cooling the sample to 4K and measuring the signal ampitude at the DC field of maxium memory signal on increasing the temperature continuously to higher values.

#### **4.4 JV phase diagram**

The magnetic field of the JV phase diagram, as function of temperature, derived from the results of the memory effect, is shown in Fig. 20. It indicates the presence of a glass state at the lowest temperature followed by a cross over state, an Anderson glass state, and a liquid state at higher temperatures.

Fig. 20. Magnetic field-temperature phase diagram of JV in Bi2212.

Bi2212 HDCII a-b HAC=12 gauss

Cooling field HDC =3000 gauss

Liquid

Anderson

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

Temperature (K)

Fig. 19. Dependence of the maximum memory signal ampitude on temperature, derived from the memory signal after field cooling the sample to 4K and measuring the signal ampitude at the DC field of maxium memory signal on increasing the temperature

The magnetic field of the JV phase diagram, as function of temperature, derived from the results of the memory effect, is shown in Fig. 20. It indicates the presence of a glass state at the lowest temperature followed by a cross over state, an Anderson glass state, and a liquid

glass Glass

Cross over

0 10 20 30 40 50 60 70 80 90

Temperature (K)

0

Fig. 20. Magnetic field-temperature phase diagram of JV in Bi2212.

Magnetic Field (Kgauss)

100

200

300

Maximum memory Signal (arb. units)

continuously to higher values.

state at higher temperatures.

**4.4 JV phase diagram** 

400

500

600

#### **4.5 Evaluating the possibility to study small or nano-scale superconducting systems with EPR spectrometers**

The microwave-dissipation signal of a high *Tc* Bi2212 superconductor, shown in Fig. 21, is obtained from Fig. 2*.* The sample size was 10x10x0.1 mm3. Its maximum intensity is of the order of 10,000 arbitrary units. The remarkably high intensity signal promises the possibility to investigate nano-scale superconductors and sizable small samples obtained by reducing the sample thickness to a few superconducting layers.

Fig. 21. Signal amplitude as function of magnetic field in a 10x10x0.1mm3 Bi2212 crystal. The remarkable high intensity signal promises investigation of EPR properties in nano crystals.

Extremely thin superconductors with a limited number of layers are required when studying sample size effects on their properties. EPR technique can be considered in investigating their properties, by comparing results obtained in small layered nano crystals of similar structure with those obtained in crystals with a larger layer number. In samples with a small number of conducting layers the following EPR measurements are anticipated: a) Differences in the interaction between JV and AV vortices due to the small number of neighboring conducting layers. b) Results presented so far have shown that the observed properties depend on sample treatments, such as field cooling or zero field cooling procedure. Therefore the effect of cooling on thin samples may show different response that would depend on the sample thickness. c) The magnetic field memory effect, being by itself a very interesting effect, is explained using the hierarchical model of glassy state, commonly used in spin glass theory (Shaltiel et al., 2010). As the glassy state may depend on the thickness of the sample, it would be interesting to investigate the memory effect in small layer samples. Nano superconducting samples needed to investigate their properties are readily obtained using cleavage methods. In contrast the preparation method of thin superconductors for investigating IJJ properties, when regular methods are applied, is cumbersome, the reason being that conducting elements should be installed on the mesas' surface. Here we quote such a sample preparation in mesas that has been used to study teraherz-wave emission from Intrinsic Josephson Junctions in high *Tc* superconductors (Kadowaki et al., 2008): "The experiments were performed with the conventional methods commonly used in performing similar measurements and include

Investigating Superconductivity with Electron Paramagnetic Resonance (EPR) Spectrometer 81

Fig. 22. Schematic, three dimensional presentation of the resistivity as function of magnetic field and temperature in a superconductor. Application of an AC field induces variation in the resistivity. The figure shows that an EPR signal is observed at the superconducting

The title of the present article: "Investigating the superconductivity with EPR (Electron Paramagnetic Resonance) spectrometer", also known as ESR (Electron Spin Resonance) spectrometer, is to a certain sense misleading but was intentionally chosen to put attention to an unconventional possibility to use the EPR technique. It gives the impression that the spectrometer investigates properties related to paramagnetic states and transitions between spin states in superconductors. As spin states and paramagnetic properties have not been an important subject in investigating properties in superconductors, various physicists ignore publications related to EPR technique. However, studying superconductors with EPR spectrometers generates a rich spectrum of experimental results of different nature. Hence, questions appeared regarding the mechanism that gives rise to the EPR signals and what do the results indicate. The present work provides answers to both questions. Investigating mesas shows that applying current in Josephson transmission lines (JTLs) induces motion of the fluxons that reside between the conduction planes and that the fluxons interact with the environment (Ustinov, 1998). Fig. 12 shows that in the case of EPR technique the AC modulation field induces Eddy currents that apply a Lorentz force that induces the motion of fluxons in the high anisotropy superconductors Bi2212 and Bi2223. Their motion interacts with the microwaves and induces the observed signal. This is the mechanism that provides the experimental results. The detailed theoretical derivation in the framework of Josephson phase electrodynamics (Shaltiel et al., 2008) enables to model the observed dependence of the microwave absorption on the DC magnetic field and the strong increase of the signal with increasing anisotropy. The analysis of the results yields information related to the properties of Josephson vortices in high anisotropy superconductors, which are otherwise not or only difficult to access. Here we recall two prominent achievements: (a) by means of

transition temperature.

**5. Conclusion** 

soldering of conducting wires on the sample's surface. Preparation of the mesas needs skilful techniques as can be deduced from the following description. "The samples were prepared from a piece of single crystal grown by the traveling floating zone method. A cleaved thin single crystal with a thickness of several μm was glued on a sapphire substrate by U-vanish (polyamide resin), and a silver thin layer and a gold film are sputtered on the cleaved sample surface. Then, the sample mesa and two contact pads with a desired size were patterned by the photolithography and Argon ion milling techniques. After removing unnecessary resist, CaF2 is evaporated on the part of mesa for the electrical isolation purpose, and Au is finally deposited for the contact pad. This Au is again patterned by photolithography and Argon milling techniques again." In contrast to this difficult and cumbersome sample-preparation method in preparing thin mesas, preparation of samples to be used in investigating nano samples with the EPR technique can be described as follows: "A cleaved thin single crystal of conventional size should be glued on a sapphire with proper dimensions to fit the sample holder of the spectrometer". The contrast between the extremely complicated preparation method, where the so called "conventional measurement method" is used, and the simplicity in sample preparation in performing EPR nano measurements is striking. The EPR method should therefore be preferred in studying nano superconductors, when similar properties are to be investigated. Microwave emission from Intrinsic Josephson Junctions in high *Tc* superconductors can readily be investigated using a device where an AC modulating technique is applied, and whose features are similar to those of the EPR spectrometer. With such a device the microwave output will be modulated at AC frequency.

#### **4.6 Searching for superconducting compounds, with EPR spectrometers**

Fig. 22**,** derived from Fig. 1 of Shaltiel et al., 1991, plots, in schematic representation, the resistivity R(T,H) in cooling a superconductor from above to below *Tc* at different DC magnetic fields. The figure shows that an ESR signal is obtained when sweeping the temperature from above to below *Tc*. The signal is induced by the AC field. It results from variations of the sample resistivity during the cooling process that affects the quality factor Q of the cavity of the spectrometer. Thus, EPR spectrometers can be used to search for superconducting materials, to determine their *Tc*, to study resistivity in superconductors, and to investigate the effects of various paramenets on the sample resistivity, without applying wiring contacts.

#### **4.7 Determination of the anisotropy in superconductors**

A description of a procedure that measures the anisotropy in superconductors using an EPR spectrometer was reported earlier (Shaltiel et al., 1992)**.** The anisotropy was derived by analyzing EPR results of the signal-peak intensity, obtained when varying the temperature across *Tc*, as function of the orientation-angle *θ* of the magnetic field with respect to the crystal axis (Shaltiel et al., 1992)**.** The detailed description of the measuring procedure including analysis of the results is presented in the above article**.** The results derived from these measurements, agree with a theory proposed by Tinkham, where the superconducting anisotropy constant is involved. Values of = 5 and = 40-60 were obtained for YBCO and Bi2212, respectively (Shaltiel et al., 1992). The results described in Shaltiel et al., 1992, which show the dependence of the signal intensity as function of the orientation angle, are a guide that can be used to orient a superconducting crystal with respect to an externally applied magnetic field. The orientation of the superconducting crystal through all the measurements presented in this work were performed by this procedure.

Fig. 22. Schematic, three dimensional presentation of the resistivity as function of magnetic field and temperature in a superconductor. Application of an AC field induces variation in the resistivity. The figure shows that an EPR signal is observed at the superconducting transition temperature.

## **5. Conclusion**

80 Superconductors – Properties, Technology, and Applications

soldering of conducting wires on the sample's surface. Preparation of the mesas needs skilful techniques as can be deduced from the following description. "The samples were prepared from a piece of single crystal grown by the traveling floating zone method. A cleaved thin single crystal with a thickness of several μm was glued on a sapphire substrate by U-vanish (polyamide resin), and a silver thin layer and a gold film are sputtered on the cleaved sample surface. Then, the sample mesa and two contact pads with a desired size were patterned by the photolithography and Argon ion milling techniques. After removing unnecessary resist, CaF2 is evaporated on the part of mesa for the electrical isolation purpose, and Au is finally deposited for the contact pad. This Au is again patterned by photolithography and Argon milling techniques again." In contrast to this difficult and cumbersome sample-preparation method in preparing thin mesas, preparation of samples to be used in investigating nano samples with the EPR technique can be described as follows: "A cleaved thin single crystal of conventional size should be glued on a sapphire with proper dimensions to fit the sample holder of the spectrometer". The contrast between the extremely complicated preparation method, where the so called "conventional measurement method" is used, and the simplicity in sample preparation in performing EPR nano measurements is striking. The EPR method should therefore be preferred in studying nano superconductors, when similar properties are to be investigated. Microwave emission from Intrinsic Josephson Junctions in high *Tc* superconductors can readily be investigated using a device where an AC modulating technique is applied, and whose features are similar to those of the EPR spectrometer. With

such a device the microwave output will be modulated at AC frequency.

**4.7 Determination of the anisotropy in superconductors** 

presented in this work were performed by this procedure.

**4.6 Searching for superconducting compounds, with EPR spectrometers** 

Fig. 22**,** derived from Fig. 1 of Shaltiel et al., 1991, plots, in schematic representation, the resistivity R(T,H) in cooling a superconductor from above to below *Tc* at different DC magnetic fields. The figure shows that an ESR signal is obtained when sweeping the temperature from above to below *Tc*. The signal is induced by the AC field. It results from variations of the sample resistivity during the cooling process that affects the quality factor Q of the cavity of the spectrometer. Thus, EPR spectrometers can be used to search for superconducting materials, to determine their *Tc*, to study resistivity in superconductors, and to investigate the effects of various paramenets on the sample resistivity, without applying wiring contacts.

A description of a procedure that measures the anisotropy in superconductors using an EPR spectrometer was reported earlier (Shaltiel et al., 1992)**.** The anisotropy was derived by analyzing EPR results of the signal-peak intensity, obtained when varying the temperature across *Tc*, as function of the orientation-angle *θ* of the magnetic field with respect to the crystal axis (Shaltiel et al., 1992)**.** The detailed description of the measuring procedure including analysis of the results is presented in the above article**.** The results derived from these measurements, agree with a theory proposed by Tinkham, where the superconducting anisotropy constant is involved. Values of = 5 and = 40-60 were obtained for YBCO and Bi2212, respectively (Shaltiel et al., 1992). The results described in Shaltiel et al., 1992, which show the dependence of the signal intensity as function of the orientation angle, are a guide that can be used to orient a superconducting crystal with respect to an externally applied magnetic field. The orientation of the superconducting crystal through all the measurements The title of the present article: "Investigating the superconductivity with EPR (Electron Paramagnetic Resonance) spectrometer", also known as ESR (Electron Spin Resonance) spectrometer, is to a certain sense misleading but was intentionally chosen to put attention to an unconventional possibility to use the EPR technique. It gives the impression that the spectrometer investigates properties related to paramagnetic states and transitions between spin states in superconductors. As spin states and paramagnetic properties have not been an important subject in investigating properties in superconductors, various physicists ignore publications related to EPR technique. However, studying superconductors with EPR spectrometers generates a rich spectrum of experimental results of different nature. Hence, questions appeared regarding the mechanism that gives rise to the EPR signals and what do the results indicate. The present work provides answers to both questions. Investigating mesas shows that applying current in Josephson transmission lines (JTLs) induces motion of the fluxons that reside between the conduction planes and that the fluxons interact with the environment (Ustinov, 1998). Fig. 12 shows that in the case of EPR technique the AC modulation field induces Eddy currents that apply a Lorentz force that induces the motion of fluxons in the high anisotropy superconductors Bi2212 and Bi2223. Their motion interacts with the microwaves and induces the observed signal. This is the mechanism that provides the experimental results. The detailed theoretical derivation in the framework of Josephson phase electrodynamics (Shaltiel et al., 2008) enables to model the observed dependence of the microwave absorption on the DC magnetic field and the strong increase of the signal with increasing anisotropy. The analysis of the results yields information related to the properties of Josephson vortices in high anisotropy superconductors, which are otherwise not or only difficult to access. Here we recall two prominent achievements: (a) by means of

**5** 

G.K. Padam

*India* 

**Non Resonant Microwave Absorption (NRMA)** 

**Electromagnetic Interactions (EMI) and Energy** 

*Materials Physics and Engineering Division, National Physical Laboratory, New Delhi,* 

High temperature superconductors (HTS) due to their higher operating temperatures, 2 dimensional (2D) layered structure and very short coherence length (ξ) are known to show some interesting properties especially magnetic field dependent microwave (MW) absorption effects at very low magnetic fields of the order of zero gauss; Kadam et al.1987; Blazy et al.1987, 1988, 1989;) while carrying out energy /absorption studies at MW frequencies. One striking example of this is the appearance of very strong non-resonant microwave absorption (NRMA) signal centered around zero magnetic field in the superconducting state at near below Tc in cuprate based HTS :YBCO, reported by Bhat et al.(1987) using conventional electron spin resonance (ESR) spectrometer. This was followed by reports on other cuprate based HTS (Portis et al.,1987, Dulcic et al.1987,1988,1989,1990; Pakulis & G.V.Chandershekar,1989; Sugawara et al., 2000;Hashmizume et al.2001; Padam et al.,1999) and also on a different variety of superconductors like: borocarbides (Kadam et al.,1994), oxyflorides (Kadam et al.,1995) alkali fullerides (Roberts,1998-II), granular conventional low-Tc typeI [Pb, Al, Sn, In etc.]/type II [Nb, Nb3Sn, Nb3Ge/PbMoS6O8 etc.] (Kheifetsand et al., 1990; Blazey et al.1988,1989; Suss et al,1989;Kataev et al.,1991;Baranowsky et al.1991;Moorjani et al.1990; Bele et al.1994; Blazey et al 1987;Veinger et al. 1995),MgB2(S.Sarangi & S.V.Bhat, Joshi et al.,2002) including newly discovered iron pniticide (Panarina et al. 2010; Pascher et al.2010 etc. ).Before coming to the actual problem, what follows is a brief introduction of the

It is to note that the NRMA response using ESR technique was reported for the first time in 1970s by Indovina et al. in the case of granular thin films of conventional type I superconductor: Pb. Similar results were then reported in other type I superconductors: Sn (Kim et al., 1972) and Al (Muller et al., 1980) at magnetic fields varying from few hundred gauss to many thousand gauss. These reports strongly pointed out that the NRMA signal appears only if Josephson junctions (JJs) are present and their thickness (t) is ≥ ξ and it does

**1. Introduction** 

NRMA studies done before the advent of HTS.

**Anomalies in High Temperature** 

**Superconductors (HTS) Relevance of** 

**Stabilized Josephson (ESJ) Fluxons** 

field cooling and zero-field cooling experiments a memory effect was discovered and a corresponding phase diagram of the Josephson-vortex system was derived. (b) From angular dependent measurements pinning of the Josephson vortices by Abricosov vortices was established. Hence, the EPR spectrometer is the experimental device that measures these properties. We note that it was originally designed to investigate different properties than those probed presently. Experimentally the spectrometer investigates properties of superconductors, where Josephson transmission lines are present. If such an instrument did not exist, it would have to be designed to enable the investigation of such properties.

## **6. References**


## **Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson (ESJ) Fluxons**

G.K. Padam

*Materials Physics and Engineering Division, National Physical Laboratory, New Delhi, India* 

## **1. Introduction**

82 Superconductors – Properties, Technology, and Applications

field cooling and zero-field cooling experiments a memory effect was discovered and a corresponding phase diagram of the Josephson-vortex system was derived. (b) From angular dependent measurements pinning of the Josephson vortices by Abricosov vortices was established. Hence, the EPR spectrometer is the experimental device that measures these properties. We note that it was originally designed to investigate different properties than those probed presently. Experimentally the spectrometer investigates properties of superconductors, where Josephson transmission lines are present. If such an instrument did

Hechtfischer, G.; Kleiner, R.; Schlenga, K.; Walkenhorst, W. & Müller, P. (1997). Collective

Kadowaki, K.; Yamaguchi, H.; Kawamata, K.; Yamamoto, T.; Minami, H.; Kakeya, I.; Welp,

Shaltiel, D.; Bill, H.; Grayevsky, A.; Junod, A.; Lovy, D.; Sadowski, W. & Walker, E. (1991).

Hc2(θ)/Hc2. *Physical Review B*, Vol.43, No.16, (June 1991), pp. 13594-13597. Shaltiel, D.; Ginodman, V.; Golosovsky, M.; Katz, U.; Boasson, H.; Gerhouser, W. & Fischer,

Shaltiel, D.; Krug von Nidda, H.-A.; Loidl, A.; Rosenstein, B.; Shapiro, B.Ya.; Shapiro, I.;

Shaltiel, D.; Krug von Nidda, H.-A.; Shapiro, B. Ya.; Rosenstein, B.; Loidl, A.; Bogoslavsky,

Shaltiel, D.; Krug von Nidda, H.-A.; Rosenstein, B.; Shapiro, B.Ya.; Golosovsky, M.; Shapiro,

Srinivasu, V.V.; Ken-ichi, I.; Hashizume, A.; Sreedevi, V.; Kohmoto, H.; Endo, T.; da Silva,

Ustinov, A.V. (1998). Solitons in Josephson Junctions. *Physica D*, Vol.123, (November 1998),

technique, *Phys. Rev. B* 77, (June 2008), pp. 214522-1-214522-5.

*and Technology*, Vol.23, No.7, (July 2010), pp. 75001-75006.

*B*, Vol.77, (January 2008), pp. 014508-1-014508-7.

Vol.14, No.1,(February 2001), pp. 41-46.

pp. 315-329.

Motion of Josephson Vortices in Intrinsic Josephson Junctions in Bi2Sr2CaCu2O8+y.

U.; Ozyuzer, L.; Koshelev, A.; Kurter, C.; Gray, K.E. & Kwok, W.-K. (2008). Direct Observation of Tetrahertz Electromagnetic Waves Emitted from Intrinsic Josephson Junctions in Single Crystalline Bi2Sr2CaCu2O8+δ. *Physica C*, Vol.468, (April 2008), pp.

Microwave Absorption across *Tc*: Determination of the Angular Dependence

P. (1992). Investigation of the Magnetic Field-induced Microwave Absorption across Tc in Single Crystals of YBa2Cu3O7-δ and BiSrCaCuO Anisotropy and Magnetic Field Dependence. *Physica C*, Vol.202, (November 1992), pp. 303-320. Shaltiel, D.; Bezalel, M.; Revaz, B.; Walker, E.; Tamegai, T. & Ooi, S. (2001). Dynamic

Response of Microwave Absorption Triggered by Magnetic Modulation in High Anisotropy Bi2Sr2CaCu2Ox Crystals. *Physica C*, Vol.349, (January 2001), pp. 139-149.

Tamegai, T. & Bogoslavsky, B. (2008). Sharp Increase of Microwave Absorption Due to Shaking of Josephson Vortices in BiSCCO Superconductor. *Physical Review* 

B.; Shapiro, I. & Tamegai, T. (2008a) Interaction between Josephson and pancake vortices investigated by the induced microwave dissipation by ac magnetic field

I.; Loidl, A.; Bogoslavsky, B.; Fujii, T.; Watanabe, T. & Tamegai, T. (2010). Field Cooling Memory Effect in Bi2212 and Bi2223 Single Crystals. *Superconductor Science* 

R.R.; Kopelevich, Y.; Moehlecke, S.; Masui, T. & Hayashi, K. (2001). Nonresonant Microwave Absorption in Bi2212 Single Crystal: Second Peak and Microwave Power Dependence. *Journal of Superconductivity: Incorporating Novel Magnetism*,

not exist, it would have to be designed to enable the investigation of such properties.

*Physical Review B*, Vol.55, No.21, (June 1997), pp. 14638-14644.

**6. References** 

634-639.

High temperature superconductors (HTS) due to their higher operating temperatures, 2 dimensional (2D) layered structure and very short coherence length (ξ) are known to show some interesting properties especially magnetic field dependent microwave (MW) absorption effects at very low magnetic fields of the order of zero gauss; Kadam et al.1987; Blazy et al.1987, 1988, 1989;) while carrying out energy /absorption studies at MW frequencies. One striking example of this is the appearance of very strong non-resonant microwave absorption (NRMA) signal centered around zero magnetic field in the superconducting state at near below Tc in cuprate based HTS :YBCO, reported by Bhat et al.(1987) using conventional electron spin resonance (ESR) spectrometer. This was followed by reports on other cuprate based HTS (Portis et al.,1987, Dulcic et al.1987,1988,1989,1990; Pakulis & G.V.Chandershekar,1989; Sugawara et al., 2000;Hashmizume et al.2001; Padam et al.,1999) and also on a different variety of superconductors like: borocarbides (Kadam et al.,1994), oxyflorides (Kadam et al.,1995) alkali fullerides (Roberts,1998-II), granular conventional low-Tc typeI [Pb, Al, Sn, In etc.]/type II [Nb, Nb3Sn, Nb3Ge/PbMoS6O8 etc.] (Kheifetsand et al., 1990; Blazey et al.1988,1989; Suss et al,1989;Kataev et al.,1991;Baranowsky et al.1991;Moorjani et al.1990; Bele et al.1994; Blazey et al 1987;Veinger et al. 1995),MgB2(S.Sarangi & S.V.Bhat, Joshi et al.,2002) including newly discovered iron pniticide (Panarina et al. 2010; Pascher et al.2010 etc. ).Before coming to the actual problem, what follows is a brief introduction of the NRMA studies done before the advent of HTS.

It is to note that the NRMA response using ESR technique was reported for the first time in 1970s by Indovina et al. in the case of granular thin films of conventional type I superconductor: Pb. Similar results were then reported in other type I superconductors: Sn (Kim et al., 1972) and Al (Muller et al., 1980) at magnetic fields varying from few hundred gauss to many thousand gauss. These reports strongly pointed out that the NRMA signal appears only if Josephson junctions (JJs) are present and their thickness (t) is ≥ ξ and it does

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 85

anomalies are still far from clear understanding and thus needs modifications.

for explaining the anomalous magnetic hysteresis as seen by NRMA.

hysteresis and their comparison with DC magnetization hysteresis.

hysteresis.

and give some conclusions.

more interesting and intriguing.

signal amplitude and magnetic anomaly respectively.

manifestation of the JJinter / JJintra with markedly different temperature dependences originate in such JJs? Secondly, how could the anomalous nature of NRMA hysteresis in contrast to the normal nature of DC magnetic hysteresis under similar conditions of temperature and magnetic field is generally observed? A critical analysis of the existing understanding to explain these observed features reveals that the full explanation of these

Recently (Padam et al. 2006, 2010) these two aspects have been taken seriously and analyzed in a different perspective, i.e. (i) by including the role of electromagnetic interactions (EMI) for explaining the reason for undulatory temperature dependence of the NRMA signal amplitude, and (ii) by giving a special reference to energy stabilized Josephson fluxons (ESJ)

The main objective of this article is to further elucidate these two problems and the importance of NRMA technique in understanding their origin in a broader and deeper perspective with inclusion of studies made on recently discovered iron pniticides. The first problem has been analyzed in more generalized way by including some more data (i.e. temperature dependence under magnetic field) so that the confusion related with the different explanations for the origin of the temperature dependence of the NRMA signal can be minimized. Similarly, the second problem based on the existing status of NRMA

The samples of BSCCO based HTS used in the present study are (Bi, Pb)2Sr2Ca2Cu3O10+x which were prepared from spray dried route in addition to the earlier studied Bi2Sr2Ca1Cu2O8+x samples with some more data and were characterized. The analysis of the results leads us to emphasize the role of EMI to explain a temperature-dependent JJs structure and to assert the role of ESJ fluxons to explain the anomalous nature of the

With this objective, in the section 2 of this article, the experimental details of the sample preparation and the characterization are given. In section 3, first results of work already done related with different temperature dependences of NRMA signal amplitude and the explanations given so far (3.1) are presented, then report results of present NRMA studies on sintered pellets of BSCCO samples (3.2) followed by discussion asserting the role of EMI (3.3). Similarly, section 4, will be first devoted to a brief summary of the results reported earlier on the nature of magnetic hysteresis in granular superconductors while carrying out DC magnetization and NRMA measurements (4.1) and the explanations for its origin, then report the present results (4.2) followed by discussion and emphasizing the role of energy stabilized Josephson fluxons (ESJ) in subsection 4.3. Finally, in section 5, we will summarize

Based on this exhaustive study, it is shown in this article that the inclusion of EMI and ESJ can further confirm their roles in the observed different temperature dependent NRMA

In this article, the additional NRMA data showing the anomalous nature of hysteresis in some other variety of superconductors including the recently discovered iron superconductors which further confirms/supports the earlier reports makes the scenario

not appear in JJs free superconductors and/or superconductors having very thin JJs. These JJs are the non-superconducting regions separating (i) microscopic metal superconducting grains and/or (ii) superconducting regions within the grain. The main mechanism of MW power absorption was assigned to: resistance of JJs (Kim et al.,1972; Muller et al.1980) and/or viscous motion of fluxons penetrated at JJs (Kim et al., 1972; Gittleman & B.Rosenblum,1966) It is to be pointed out that in both these models, JJs of intergranular type (here after called as JJinter are considered.

Despite these reports on the observation of NRMA revealing an important and direct information related with manifestation of JJsinter in granular conventional superconductors which cannot be rendered by resistivity and specific heat measurements, not much literature on NRMA studies was available at that time. It was until the advent of HTS which are intrinsically granular in nature and showed a very strong NRMA signal at nearly zero magnetic field, NRMA became a valuable, informative and indispensable tool for examining all kinds of granular superconductors existing till today from conventional (both type I and type II) to recently discovered iron pniticide etc. particularly for the detection, variety and behavior of JJs and to know the mechanism of MW absorption.

Further, to explain the origin of NRMA signal in HTS (Portis et al., 1988) and Dulcic et al.1889) extended the previous models for conventional granular superconductors in which JJinter play the dominant role (Indovina et al.,1970;Kim et al.1972; Muller et al.1980;Gittleman & B.Rosenblum, 1966).This was done by including the contribution from intragranular JJs (JJintra) also. As is known that unlike in the granular conventional superconductors where, the effect of JJintra will be suppressed due to large ξ, in HTS due to very short ξ, the effect of JJintra can be easily observed. Although these models considered contribution from both the JJs (JJinter and JJintra), however, they associated to different mechanism of absorption. That is, Portis et al. associated NRMA to damped fluxon (intergranular fluxons at JJinter / intragranular fluxons at JJsintra) motion, whereas Dulcic et al. associated to JJinter and JJintra resistance to the induced boundary currents (Josephson currents).Further, it was pointed out by Portis et al. (1988) "although we associate the microwave absorption with damped fluxon motion, it may be equivalent to associate the absorption with Josephson currents as others have done".

The continuous growth in this field of NRMA studies, along with the extreme sensitivity and expedition of the NRMA technique apart from its ability to detect the phase transition to the superconducting state, it has opened a new window (i) to unambiguously discover/detect new superconducting phases present even in quantity as low as of the order of sub micrograms, (ii) to distinguish multiple superconducting phases in a single matrix, (iii) to tell something about the quality of the superconductor, (iv) to study a new type of device for example: a single chip comprising of a combination of insulating, semiconducting, and superconducting layers, (v) to study the phase diagram of vortex matter through temperature dependent NRMA signal amplitude and to (vi) reveal important information on the anomalous nature of magnetic hysteresis etc. This provides new insights to both the fundamental and technological aspects of HTS.

There are two very important observed features which need further elucidation as their origins are still not very clear though there is a consensus on the origin of NRMA – a manifestation of JJs. Firstly, how could the amplitude of the NRMA signal which is a

not appear in JJs free superconductors and/or superconductors having very thin JJs. These JJs are the non-superconducting regions separating (i) microscopic metal superconducting grains and/or (ii) superconducting regions within the grain. The main mechanism of MW power absorption was assigned to: resistance of JJs (Kim et al.,1972; Muller et al.1980) and/or viscous motion of fluxons penetrated at JJs (Kim et al., 1972; Gittleman & B.Rosenblum,1966) It is to be pointed out that in both these models, JJs of intergranular type

Despite these reports on the observation of NRMA revealing an important and direct information related with manifestation of JJsinter in granular conventional superconductors which cannot be rendered by resistivity and specific heat measurements, not much literature on NRMA studies was available at that time. It was until the advent of HTS which are intrinsically granular in nature and showed a very strong NRMA signal at nearly zero magnetic field, NRMA became a valuable, informative and indispensable tool for examining all kinds of granular superconductors existing till today from conventional (both type I and type II) to recently discovered iron pniticide etc. particularly for the detection, variety and

Further, to explain the origin of NRMA signal in HTS (Portis et al., 1988) and Dulcic et al.1889) extended the previous models for conventional granular superconductors in which JJinter play the dominant role (Indovina et al.,1970;Kim et al.1972; Muller et al.1980;Gittleman & B.Rosenblum, 1966).This was done by including the contribution from intragranular JJs (JJintra) also. As is known that unlike in the granular conventional superconductors where, the effect of JJintra will be suppressed due to large ξ, in HTS due to very short ξ, the effect of JJintra can be easily observed. Although these models considered contribution from both the JJs (JJinter and JJintra), however, they associated to different mechanism of absorption. That is, Portis et al. associated NRMA to damped fluxon (intergranular fluxons at JJinter / intragranular fluxons at JJsintra) motion, whereas Dulcic et al. associated to JJinter and JJintra resistance to the induced boundary currents (Josephson currents).Further, it was pointed out by Portis et al. (1988) "although we associate the microwave absorption with damped fluxon motion, it may be equivalent to associate the absorption with Josephson currents as others

The continuous growth in this field of NRMA studies, along with the extreme sensitivity and expedition of the NRMA technique apart from its ability to detect the phase transition to the superconducting state, it has opened a new window (i) to unambiguously discover/detect new superconducting phases present even in quantity as low as of the order of sub micrograms, (ii) to distinguish multiple superconducting phases in a single matrix, (iii) to tell something about the quality of the superconductor, (iv) to study a new type of device for example: a single chip comprising of a combination of insulating, semiconducting, and superconducting layers, (v) to study the phase diagram of vortex matter through temperature dependent NRMA signal amplitude and to (vi) reveal important information on the anomalous nature of magnetic hysteresis etc. This provides

There are two very important observed features which need further elucidation as their origins are still not very clear though there is a consensus on the origin of NRMA – a manifestation of JJs. Firstly, how could the amplitude of the NRMA signal which is a

new insights to both the fundamental and technological aspects of HTS.

(here after called as JJinter are considered.

have done".

behavior of JJs and to know the mechanism of MW absorption.

manifestation of the JJinter / JJintra with markedly different temperature dependences originate in such JJs? Secondly, how could the anomalous nature of NRMA hysteresis in contrast to the normal nature of DC magnetic hysteresis under similar conditions of temperature and magnetic field is generally observed? A critical analysis of the existing understanding to explain these observed features reveals that the full explanation of these anomalies are still far from clear understanding and thus needs modifications.

Recently (Padam et al. 2006, 2010) these two aspects have been taken seriously and analyzed in a different perspective, i.e. (i) by including the role of electromagnetic interactions (EMI) for explaining the reason for undulatory temperature dependence of the NRMA signal amplitude, and (ii) by giving a special reference to energy stabilized Josephson fluxons (ESJ) for explaining the anomalous magnetic hysteresis as seen by NRMA.

The main objective of this article is to further elucidate these two problems and the importance of NRMA technique in understanding their origin in a broader and deeper perspective with inclusion of studies made on recently discovered iron pniticides. The first problem has been analyzed in more generalized way by including some more data (i.e. temperature dependence under magnetic field) so that the confusion related with the different explanations for the origin of the temperature dependence of the NRMA signal can be minimized. Similarly, the second problem based on the existing status of NRMA hysteresis and their comparison with DC magnetization hysteresis.

The samples of BSCCO based HTS used in the present study are (Bi, Pb)2Sr2Ca2Cu3O10+x which were prepared from spray dried route in addition to the earlier studied Bi2Sr2Ca1Cu2O8+x samples with some more data and were characterized. The analysis of the results leads us to emphasize the role of EMI to explain a temperature-dependent JJs structure and to assert the role of ESJ fluxons to explain the anomalous nature of the hysteresis.

With this objective, in the section 2 of this article, the experimental details of the sample preparation and the characterization are given. In section 3, first results of work already done related with different temperature dependences of NRMA signal amplitude and the explanations given so far (3.1) are presented, then report results of present NRMA studies on sintered pellets of BSCCO samples (3.2) followed by discussion asserting the role of EMI (3.3). Similarly, section 4, will be first devoted to a brief summary of the results reported earlier on the nature of magnetic hysteresis in granular superconductors while carrying out DC magnetization and NRMA measurements (4.1) and the explanations for its origin, then report the present results (4.2) followed by discussion and emphasizing the role of energy stabilized Josephson fluxons (ESJ) in subsection 4.3. Finally, in section 5, we will summarize and give some conclusions.

Based on this exhaustive study, it is shown in this article that the inclusion of EMI and ESJ can further confirm their roles in the observed different temperature dependent NRMA signal amplitude and magnetic anomaly respectively.

In this article, the additional NRMA data showing the anomalous nature of hysteresis in some other variety of superconductors including the recently discovered iron superconductors which further confirms/supports the earlier reports makes the scenario more interesting and intriguing.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 87

Fig. 1b. XRD pattern of powder of Bi-2212 (5wt%Ag) sintered pellet.

present experiments were single phase.

resistance

Room temperature X-ray powder diffraction (Fig.1a) of the material 0, 5 and 10 wt% Ag doped Bi1.84Pb 0.4Sr2Ca2.2Cu3O10+y samples, showed all these products consisted of (Bi, Pb)- 2223 phase. However, on addition of Ag, the samples was found to improve in terms of crystallinity as revealed from the sharper peaks and directionally oriented as revealed from the increased 00ℓ reflections. This indicated these contents of Ag addition improved the quality of the samples. Similarly, room temperature powder X-ray diffraction (Fig.1b) for the Bi2Sr2CaCu2O8+x (5wt% Ag) samples established that the products consisted of (Bi-2212). Furthermore, the absence of lines associated with secondary phases like Ca2PbO4, CaO, CuO etc. in the Bi-2212 and in the (Bi, Pb)-2223 samples indicated that the materials used in the

A scanning electron microscopic studies of three (Bi, Pb)-2223 (0wt%, 5wt% and 10wt% Ag) samples revealed that in pure samples, there is a broad distribution of grain size and in Ag doped samples, there is a uniform grain size distribution with better oriented well connected plate like grains (not shown). Similarly, Bi2Sr2CaCu2O8+x (5wt% Ag) revealed a reasonably good microstructure with well aligned and well connected grains (not shown). Superconducting transition temperature (Tc) was measured by both the electrical resistivity and ac susceptibility technique for these samples and the results are shown in (Fig.2). For the (Bi, Pb)-2223 samples Tc increases from 105 to 117K on increasing Ag content from 0 to 10wt%. The normal state DC resistance is lowest for the 5wt% doped sample and is highest for the 10wt% sample. In 5wt% Ag doped sample, it appears that most of the silver goes to the grain boundaries and hence is available for conduction. Therefore, minimum normal state resistance is observed in this sample. However, for the 10wt% Ag doped sample, most of the silver ions go into the lattice; therefore less silver is available for normal electrical conduction, hence responsible for highest normal state

## **2. Experimental details**

The bulk pellets (diameter=12mm, thickness= 2mm) of Bi2Sr2CaCu2O8+x: Ag (5wt %) [Bi-2212] were synthesized by the matrix reaction method and the details are described earlier (Padam et al., 1999)

Whereas, the bulk pellet samples of Bi1.84Pb 0.4Sr2Ca2.2Cu3O10+y: Ag (0 to 10wt %) [(Bi, Pb)- 2223] were synthesized by using spray drying method. For this, high purity powders of Bi (NO3)3.5H2O, Pb (NO3)2, Sr (NO3)2, Ca (NO3)2.4H2O and Cu (NO3)2.3H2O were used. These powders were taken in a nominal composition of 1.84:0.4:2:2.2:3 and dissolved separately in double distilled water. Each of the individual solutions was continuously stirred with mild heating to clear all solutions and a few drops concentrated nitric acid were added, whenever needed, to fasten the process. Then all the solutions of individual powders were mixed in a beaker and labeled 'pure solution'. Different silver nitrate solutions were also made in double distilled water to obtain a silver content that corresponds to the composition of (Bi,Pb)-2223 by weight percentages of 0,1,5 and 10. They were mixed, one by one, with the same amount of 'pure solution'. These final solutions were then kept separately in a beaker and spray dried separately to prepare the desired powders (0-10wt% Ag) of the above mentioned compositions. The powders so obtained were calcined separately first at 8000C and then at 8200C for one hour with one intermediate grinding. Then pellets having 12mm diameter and 1mm thickness, were made of each 0-10wt% Ag doped (Bi,Pb)-2223 calcined powders and sintered at 8350C in air for 100 hours with one intermediate grinding.

Since the properties of the non- resonant microwave absorption depend to some extent on the phase purity and microstructure of the sample etc, which may vary from sample to sample even made under identical conditions, therefore, to avoid any ambiguity all the characterization studies: NRMA, X-ray diffraction (XRD), scanning electron microscope (SEM), EDAX, electrical and ac susceptibility for (Tc) were done on a single sample.

Fig. 1a. XRD patterns of (0-10wt%Ag) added (Bi, Pb)-2223 samples showing improvement in grain alignment on increasing Ag content.

The bulk pellets (diameter=12mm, thickness= 2mm) of Bi2Sr2CaCu2O8+x: Ag (5wt %) [Bi-2212] were synthesized by the matrix reaction method and the details are described earlier

Whereas, the bulk pellet samples of Bi1.84Pb 0.4Sr2Ca2.2Cu3O10+y: Ag (0 to 10wt %) [(Bi, Pb)- 2223] were synthesized by using spray drying method. For this, high purity powders of Bi (NO3)3.5H2O, Pb (NO3)2, Sr (NO3)2, Ca (NO3)2.4H2O and Cu (NO3)2.3H2O were used. These powders were taken in a nominal composition of 1.84:0.4:2:2.2:3 and dissolved separately in double distilled water. Each of the individual solutions was continuously stirred with mild heating to clear all solutions and a few drops concentrated nitric acid were added, whenever needed, to fasten the process. Then all the solutions of individual powders were mixed in a beaker and labeled 'pure solution'. Different silver nitrate solutions were also made in double distilled water to obtain a silver content that corresponds to the composition of (Bi,Pb)-2223 by weight percentages of 0,1,5 and 10. They were mixed, one by one, with the same amount of 'pure solution'. These final solutions were then kept separately in a beaker and spray dried separately to prepare the desired powders (0-10wt% Ag) of the above mentioned compositions. The powders so obtained were calcined separately first at 8000C and then at 8200C for one hour with one intermediate grinding. Then pellets having 12mm diameter and 1mm thickness, were made of each 0-10wt% Ag doped (Bi,Pb)-2223 calcined

powders and sintered at 8350C in air for 100 hours with one intermediate grinding.

(SEM), EDAX, electrical and ac susceptibility for (Tc) were done on a single sample.

Since the properties of the non- resonant microwave absorption depend to some extent on the phase purity and microstructure of the sample etc, which may vary from sample to sample even made under identical conditions, therefore, to avoid any ambiguity all the characterization studies: NRMA, X-ray diffraction (XRD), scanning electron microscope

Fig. 1a. XRD patterns of (0-10wt%Ag) added (Bi, Pb)-2223 samples showing improvement in

grain alignment on increasing Ag content.

**2. Experimental details** 

(Padam et al., 1999)

Fig. 1b. XRD pattern of powder of Bi-2212 (5wt%Ag) sintered pellet.

Room temperature X-ray powder diffraction (Fig.1a) of the material 0, 5 and 10 wt% Ag doped Bi1.84Pb 0.4Sr2Ca2.2Cu3O10+y samples, showed all these products consisted of (Bi, Pb)- 2223 phase. However, on addition of Ag, the samples was found to improve in terms of crystallinity as revealed from the sharper peaks and directionally oriented as revealed from the increased 00ℓ reflections. This indicated these contents of Ag addition improved the quality of the samples. Similarly, room temperature powder X-ray diffraction (Fig.1b) for the Bi2Sr2CaCu2O8+x (5wt% Ag) samples established that the products consisted of (Bi-2212). Furthermore, the absence of lines associated with secondary phases like Ca2PbO4, CaO, CuO etc. in the Bi-2212 and in the (Bi, Pb)-2223 samples indicated that the materials used in the present experiments were single phase.

A scanning electron microscopic studies of three (Bi, Pb)-2223 (0wt%, 5wt% and 10wt% Ag) samples revealed that in pure samples, there is a broad distribution of grain size and in Ag doped samples, there is a uniform grain size distribution with better oriented well connected plate like grains (not shown). Similarly, Bi2Sr2CaCu2O8+x (5wt% Ag) revealed a reasonably good microstructure with well aligned and well connected grains (not shown). Superconducting transition temperature (Tc) was measured by both the electrical resistivity and ac susceptibility technique for these samples and the results are shown in (Fig.2). For the (Bi, Pb)-2223 samples Tc increases from 105 to 117K on increasing Ag content from 0 to 10wt%. The normal state DC resistance is lowest for the 5wt% doped sample and is highest for the 10wt% sample. In 5wt% Ag doped sample, it appears that most of the silver goes to the grain boundaries and hence is available for conduction. Therefore, minimum normal state resistance is observed in this sample. However, for the 10wt% Ag doped sample, most of the silver ions go into the lattice; therefore less silver is available for normal electrical conduction, hence responsible for highest normal state resistance

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 89

binary phases) which is showing Cu2+ is also depicted.

and confirmed that the signal shapes are not distorted.

dependent NRMA signal amplitude behavior.

absorption (P) changes.

(100 mW, x1.25x103)

The quality test was also confirmed by measuring ESR of the samples at room temperature and observed no ESR signal due to most expected Cu2+ ions in all the samples was observed (Fig.4). For comparison, ESR spectra of (Bi, Pb)-2223 under sintered sample (containing

Fig. 4. EPR of (Bi, PB)-2223 and Bi-2212 samples showing absence of any signal to Cu2+ ions

All these results showing nearly single phase nature of these samples with good microstructures and sharp superconducting transitions suggested that both the samples are of a reasonably good quality. For NRMA studies, an electron paramagnetic resonance (EPR) spectrometer Bruker ER 200D X-band (9.47GHz) equipped with an Oxford ESR900 continuous helium gas flow temperature variation accessory was used. Details are given in reference 7. The measurements were done at modulation frequency at 100 KHz for different values of temperature (5-300K) by cooling the samples in zero field and then sweeping the static magnetic field. To avoid any possibility of experimental conditions (modulation amplitude, MW power, magnetic field scan range etc.) dependent shifting, shape, phase reversal and hysteresis or other artifacts of the signal, all the measurements were done under identical conditions for each study. That is, moderate modulation amplitude of typically 4G and the microwave power of 20mW were kept the same. Similarly, the magnetic field scan ranges were also kept the same for both the forward and reverse scans: the field was varied between the field was varied between -50 (+ 50) to +50 (-50) G. Some of the measurements were also done for high field scans from -50 to + 1050G. It was checked

To see the actual variation of the NRMA signal amplitude as a function of temperature, the NRMA signal amplitude is recorded by the temperature scanning method. The NRMA signal amplitude was monitored by warming the samples from 5K to 300K. For the Bi-2212 sample, these measurements were also done at four different fixed magnetic field values of 0, 50 and 100G for each scan in order to see the effect of magnetic field on the temperature

Signals were recorded in the derivative form [i.e. field derivative (H) of microwave power absorbed (P): dP/dH] resulting from magnetic field modulation of typically 4G peak-topeak at 100 KHz and phase-sensitive detection technique when Hmw (magnetic field component of microwaves) is perpendicularly to DC magnetic field HDC. This provides information on extremely small temperature and magnetic field dependent microwave

Fig. 2a. AC susceptibility dependence on temperature for silver added (0-10wt%) samples.

Fig. 2b. Temperature dependence of DC resistance of silver added (0-10wt%) samples. For the Bi-2212 sample, Tc of 96K was obtained (Fig .3).

Fig. 3. Resistance and Susceptibility vs. Temperature characteristics of the Bi-2212(5wt%Ag) sample.

Fig. 2a. AC susceptibility dependence on temperature for silver added (0-10wt%) samples.

Fig. 2b. Temperature dependence of DC resistance of silver added (0-10wt%) samples.

Fig. 3. Resistance and Susceptibility vs. Temperature characteristics of the Bi-2212(5wt%Ag)

For the Bi-2212 sample, Tc of 96K was obtained (Fig .3).

sample.

The quality test was also confirmed by measuring ESR of the samples at room temperature and observed no ESR signal due to most expected Cu2+ ions in all the samples was observed (Fig.4). For comparison, ESR spectra of (Bi, Pb)-2223 under sintered sample (containing binary phases) which is showing Cu2+ is also depicted.

Fig. 4. EPR of (Bi, PB)-2223 and Bi-2212 samples showing absence of any signal to Cu2+ ions (100 mW, x1.25x103)

All these results showing nearly single phase nature of these samples with good microstructures and sharp superconducting transitions suggested that both the samples are of a reasonably good quality. For NRMA studies, an electron paramagnetic resonance (EPR) spectrometer Bruker ER 200D X-band (9.47GHz) equipped with an Oxford ESR900 continuous helium gas flow temperature variation accessory was used. Details are given in reference 7. The measurements were done at modulation frequency at 100 KHz for different values of temperature (5-300K) by cooling the samples in zero field and then sweeping the static magnetic field. To avoid any possibility of experimental conditions (modulation amplitude, MW power, magnetic field scan range etc.) dependent shifting, shape, phase reversal and hysteresis or other artifacts of the signal, all the measurements were done under identical conditions for each study. That is, moderate modulation amplitude of typically 4G and the microwave power of 20mW were kept the same. Similarly, the magnetic field scan ranges were also kept the same for both the forward and reverse scans: the field was varied between the field was varied between -50 (+ 50) to +50 (-50) G. Some of the measurements were also done for high field scans from -50 to + 1050G. It was checked and confirmed that the signal shapes are not distorted.

To see the actual variation of the NRMA signal amplitude as a function of temperature, the NRMA signal amplitude is recorded by the temperature scanning method. The NRMA signal amplitude was monitored by warming the samples from 5K to 300K. For the Bi-2212 sample, these measurements were also done at four different fixed magnetic field values of 0, 50 and 100G for each scan in order to see the effect of magnetic field on the temperature dependent NRMA signal amplitude behavior.

Signals were recorded in the derivative form [i.e. field derivative (H) of microwave power absorbed (P): dP/dH] resulting from magnetic field modulation of typically 4G peak-topeak at 100 KHz and phase-sensitive detection technique when Hmw (magnetic field component of microwaves) is perpendicularly to DC magnetic field HDC. This provides information on extremely small temperature and magnetic field dependent microwave absorption (P) changes.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 91

various approaches and explanations has been given.

**3.1.1 Peak type temperature dependence** 

(Portis et al., 1988, Dulcic et al., 1989)

not be ruled out.

larger variety of JJs and are more of thick type like JJinter.

It is interesting to note that these different behavior are independent of the sample form (i.e. whether polycrystalline, thin film or single crystal) and types of superconductors. However, to realize, the full understanding of these different behavior it is necessary to clarify a considerable amount of confusion in the literature regarding the different origins of the temperature dependence of the NRMA signal amplitude by different researchers given so far which probably is due to some missing parameter. In the following, a brief account of

Single peak type temperature dependence in HTS which is a characteristic feature of a sample having single superconducting phase has been well studied and its origin generally has been attributed to the different coupling strength of different JJs. So, throughout the temperature range in the superconducting state, some junctions decoupled, flluxons (JFs) penetrated and form a flux line lattice (FLL) which is controlled by Josephson interactions (JI) and dissipate power. That is, on increasing temperature, coupling strength of JJs and strength of (JI) among JFs decreases this result into an increase in number density and mobility of the fluxons penetrated within JJs. Consequently, an initial rise in the NRMA signal amplitude reaching maxima in the superconducting region at a few degrees below Tc, until most of the junctions are broken and showed weak superconductivity, hence an exponential fall in the superconducting-tonormal transition region. Above this temperature, all the JJs are broken and the material becomes normal, where the signal disappears. Details can be seen in earlier references

Further, width and position of the peak have been related to a different variety of JJs. This different variety of JJs are: (i) JJinter which dominates in sintered pellets and/or dense powders can be due to misorientation of anisotropic grains, oxygen depleted grain boundaries or secondary phases at the grain boundaries etc. and are generally thick, and (ii) JJintra which dominates in isolated grains can be due to twin boundaries, stacking faults or point defects etc. and are generally thin. Narrow peaks near below Tc have been assigned to a sample having a small variety of JJs which are generally thin type (i.e.JJintra). On the other hand, broader peaks at temperatures lower than Tc have been cited as samples having a

It is to be noted that in contradiction to the above explanation given by majority of the researchers regarding the origin of peak type behavior assigned to temperature dependent coupling strength of different kinds of JJs, however, there are some researchers for example Moorjani et al.(1990) who still have a different opinion. That is, according to them: (i) the peak response is due to the intrinsic superconducting transition, (ii) if any NRMA signal (i.e. rising base line) appears below this peak then that is due to the presence of different JJinter. Drawback of this report is that it considers only the role of JJsinter and neglects that of JJintra even in HTS which are known to have inevitable presence of JJintra. Considering the role of JJintra e.g. due to the formation of a thin oxide layer on their Nb samples as reported by other researchers, can be one of explanations for the peak behavior in their Nb samples. Another possible reason of formation of superconducting droplets near Tc giving peak response can

## **3. Results and explanations on temperature dependent behavior of NRMA signal amplitude**

#### **3.1 Temperature dependent behavior of NRMA signal amplitude in different superconductors reported by other researchers**

Fig. 5 shows temperature dependence of NRMA signal amplitude reported in for different superconductors: cuprates (Blazey et al., 1987; Moorjani et al.1987; Glarum et al.1988) organic superconductors (Haddon et al., 1991) conventional superconductors14,17 ( Suss et al., 1989; Moorjani et al., 1987)etc. It is evident that markedly different variations in the NRMA signal amplitude in all these superconductors are seen on increasing temperature i.e. going from superconducting state to normal state. These different behavior mostly observed are: peak, undulatory and monotonous fall:

Fig. 5. Temperature dependence of NRMA signal amplitude showing different behavior irrespective of class of granular superconductor (data from literature).


It is interesting to note that these different behavior are independent of the sample form (i.e. whether polycrystalline, thin film or single crystal) and types of superconductors. However, to realize, the full understanding of these different behavior it is necessary to clarify a considerable amount of confusion in the literature regarding the different origins of the temperature dependence of the NRMA signal amplitude by different researchers given so far which probably is due to some missing parameter. In the following, a brief account of various approaches and explanations has been given.

## **3.1.1 Peak type temperature dependence**

90 Superconductors – Properties, Technology, and Applications

Fig. 5 shows temperature dependence of NRMA signal amplitude reported in for different superconductors: cuprates (Blazey et al., 1987; Moorjani et al.1987; Glarum et al.1988) organic superconductors (Haddon et al., 1991) conventional superconductors14,17 ( Suss et al., 1989; Moorjani et al., 1987)etc. It is evident that markedly different variations in the NRMA signal amplitude in all these superconductors are seen on increasing temperature i.e. going from superconducting state to normal state. These different behavior mostly observed are:

Fig. 5. Temperature dependence of NRMA signal amplitude showing different behavior

i. Observation of peak/peaks in temperature vs. NRMA signal amplitude i.e. on increasing temperature, an initial exponential rise in the amplitude reaching a maximum with a subsequent exponential fall near below Tc (Fig.5a), or there can be

ii. Undulatory i.e. an initial fall then a rise reaching maxima followed by a final fall (Fig.5c)

irrespective of class of granular superconductor (data from literature).

more than one peak as shown in Fig.5b.

iii. A monotonous fall (Fig.5d) on increasing temperature.

on increasing temperature.

**3. Results and explanations on temperature dependent behavior of NRMA** 

**3.1 Temperature dependent behavior of NRMA signal amplitude in different** 

**superconductors reported by other researchers** 

peak, undulatory and monotonous fall:

**signal amplitude** 

Single peak type temperature dependence in HTS which is a characteristic feature of a sample having single superconducting phase has been well studied and its origin generally has been attributed to the different coupling strength of different JJs. So, throughout the temperature range in the superconducting state, some junctions decoupled, flluxons (JFs) penetrated and form a flux line lattice (FLL) which is controlled by Josephson interactions (JI) and dissipate power. That is, on increasing temperature, coupling strength of JJs and strength of (JI) among JFs decreases this result into an increase in number density and mobility of the fluxons penetrated within JJs. Consequently, an initial rise in the NRMA signal amplitude reaching maxima in the superconducting region at a few degrees below Tc, until most of the junctions are broken and showed weak superconductivity, hence an exponential fall in the superconducting-tonormal transition region. Above this temperature, all the JJs are broken and the material becomes normal, where the signal disappears. Details can be seen in earlier references (Portis et al., 1988, Dulcic et al., 1989)

Further, width and position of the peak have been related to a different variety of JJs. This different variety of JJs are: (i) JJinter which dominates in sintered pellets and/or dense powders can be due to misorientation of anisotropic grains, oxygen depleted grain boundaries or secondary phases at the grain boundaries etc. and are generally thick, and (ii) JJintra which dominates in isolated grains can be due to twin boundaries, stacking faults or point defects etc. and are generally thin. Narrow peaks near below Tc have been assigned to a sample having a small variety of JJs which are generally thin type (i.e.JJintra). On the other hand, broader peaks at temperatures lower than Tc have been cited as samples having a larger variety of JJs and are more of thick type like JJinter.

It is to be noted that in contradiction to the above explanation given by majority of the researchers regarding the origin of peak type behavior assigned to temperature dependent coupling strength of different kinds of JJs, however, there are some researchers for example Moorjani et al.(1990) who still have a different opinion. That is, according to them: (i) the peak response is due to the intrinsic superconducting transition, (ii) if any NRMA signal (i.e. rising base line) appears below this peak then that is due to the presence of different JJinter. Drawback of this report is that it considers only the role of JJsinter and neglects that of JJintra even in HTS which are known to have inevitable presence of JJintra. Considering the role of JJintra e.g. due to the formation of a thin oxide layer on their Nb samples as reported by other researchers, can be one of explanations for the peak behavior in their Nb samples. Another possible reason of formation of superconducting droplets near Tc giving peak response can not be ruled out.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 93

samples showing a peak behavior indicating that all the samples have a different variety of JJs. Since these results are similar to as reported by others1-7,10,15, therefore, its origin can also be explained on the bases of temperature dependent coupling strength of JJs / strength of JI.

Fig. 6. Temperature of NRMA signal amplitude of Ag added (Bi, Pb)-2223 sample showing narrowing and shifting of the peaks towards higher temperature on increasing Ag content.

However, width and position of the peak was found to depend upon the amount of Ag. That is, narrowest peak at the highest temperature of 135K for the 10wt% Ag doped sample, whereas the broadest peak at the lowest temperature of 118K for the pure sample was observed. This narrowing and shifting towards higher temperature of the peak on increasing Ag content may be due to the improvement in the quality of the sample in terms of decreasing number probably of JJinter. These results are supported by the sharper XRD peaks with enhanced (ooℓ) reflections (Fig.2) and sharper superconducting transition and increased Tc (Fig.1) on increasing Ag content. Similar results related with observation of narrow peaks at high temperatures for good quality samples and broader peaks at lower temperatures for poor quality samples have been reported by others ( Bhat et al.,1987; Portis et al.,1988; Dulcic et al.,1989; Pakulis & G.V.Chandershekar, 1989; Sugawara et al.,2000;

Fig.7a depicts an undualtory temperature dependence of the NRMA signal amplitude for the Bi-2212 (5wt %) bulk samples kept at constant magnetic field of 0G and varying the temperature from 15K to 105K rather than the peak behavior as shown by all the three (Bi,Pb)-2223 samples. This Bi-2212 sample showed showing three distinct regions, i.e. an initial decreasing curve ABC (15-71.8), narrow flat CD (71.8-76.5K) and then a rise DE (76.5-

This undulatory behavior may be discussed first in the frame work of the explanation used for peak behavior. That is, on increasing temperature, strength of JI decreases (due to increasing decoupling of JJs) which results into an increase in number density and mobility of JF. This leads to an initial rise in the NRMA signal amplitude reaching a maxima at a few degrees below Tc, until most of the JJs are broken and showed weak superconductivity, hence an exponential fall in the superconducting - to - normal region. Above this temperature, all the JJs are broken and the material becomes normal, where the NRMA

82.5K) with maxima at around 82.5K followed by subsequent exponential fall EF.

Kataev et al., 1991,Padam et al.,1999).

**3.2.2 Bi-2212 samples** 

signal disappears.

Like single peak which is a characteristic feature of a single superconducting phase, multiple peak behavior is a signature of multiple superconducting phases with different Tc (Kataev et al., 1991; Moorjani et al., 1987) in the same sample. Like the origin of single peak behavior assigned to temperature dependent coupling strength of different JJs and strength of JI in a single superconducting phase, the origin of multiple peaks is attributed to temperature dependent coupling strength of different JJs present in each superconducting phase.

## **3.1.2 Undulatory and monotonous type temperature dependence**

With regard to the origin of undulatory and monotonous fall type behaviors, following different explanations have been given.


It is clear that as compared to the origin of peak type response, origins of the undulatory and monotonous type behaviors are uncertain. That is, whether it is due to some impurities, imperfections, multiple superconducting / non-superconducting phases or variations in flux dynamics etc. still not clear. Further, all these reports consider the role of JI interactions only but none of them give reason for the initial fall in the signal amplitude on increasing temperature.

All these observations irrespective of the class of superconductors make this problem intriguing and motivate to investigate this discrepant data more carefully.

Recently in our earlier work (Padam et al., 2006) an attempt has been made to analyze this problem by considering the effect of electromagnetic interactions (EMI) in addition to JI and found that involving EMI can satisfactorily explain particularly the undulatory behavior.

Before going into further details, lets us see the temperature dependence of the NRMA signal amplitude in the present samples.

## **3.2 NRMA signal amplitude vs. temperature behaviors in present BSCCO samples – Relevance of EMI**

## **3.2.1 (Bi, Pb) samples**

Fig.6 depicts the peak behavior for Ag (0, 5 and10 wt %) added (Bi, Pb)2Sr2Ca2Cu3O10+x [here after called as (Bi, Pb)-2223] bulk samples the temperature dependence of the signal amplitude for of constant magnetic field of 0 G by varying the temperature (45-160K). In these experiments equal amounts of each sample and identical spectrometer operating conditions were used to facilitate comparison of the resulting spectra. It is clear that all the samples showing a peak behavior indicating that all the samples have a different variety of JJs. Since these results are similar to as reported by others1-7,10,15, therefore, its origin can also be explained on the bases of temperature dependent coupling strength of JJs / strength of JI.

Fig. 6. Temperature of NRMA signal amplitude of Ag added (Bi, Pb)-2223 sample showing narrowing and shifting of the peaks towards higher temperature on increasing Ag content.

However, width and position of the peak was found to depend upon the amount of Ag. That is, narrowest peak at the highest temperature of 135K for the 10wt% Ag doped sample, whereas the broadest peak at the lowest temperature of 118K for the pure sample was observed. This narrowing and shifting towards higher temperature of the peak on increasing Ag content may be due to the improvement in the quality of the sample in terms of decreasing number probably of JJinter. These results are supported by the sharper XRD peaks with enhanced (ooℓ) reflections (Fig.2) and sharper superconducting transition and increased Tc (Fig.1) on increasing Ag content. Similar results related with observation of narrow peaks at high temperatures for good quality samples and broader peaks at lower temperatures for poor quality samples have been reported by others ( Bhat et al.,1987; Portis et al.,1988; Dulcic et al.,1989; Pakulis & G.V.Chandershekar, 1989; Sugawara et al.,2000; Kataev et al., 1991,Padam et al.,1999).

## **3.2.2 Bi-2212 samples**

92 Superconductors – Properties, Technology, and Applications

Like single peak which is a characteristic feature of a single superconducting phase, multiple peak behavior is a signature of multiple superconducting phases with different Tc (Kataev et al., 1991; Moorjani et al., 1987) in the same sample. Like the origin of single peak behavior assigned to temperature dependent coupling strength of different JJs and strength of JI in a single superconducting phase, the origin of multiple peaks is attributed to temperature

With regard to the origin of undulatory and monotonous fall type behaviors, following

a. For the undulatory type behavior, different sources (Dulcic et al., 1990; Moorjani et al., 1990; Bohandy et al. 1988) have been reported to responsible. For example: (i) impurities, (ii) high density of defects and dominance of a variety of thick JJs etc.,(iii) to a mixture of phases with a distribution of superconducting transition temperatures and

b. Similarly, for a monotonous fall has been attributed to (i) impurities( Glarum et al., 1988) (ii) a deteriorated Tc superconducting phase (Sastry et al.,1987), (iii) presence of imperfections such as stacking faults ( Bohandy et al.,1991)and (iv) a decrease in the

It is clear that as compared to the origin of peak type response, origins of the undulatory and monotonous type behaviors are uncertain. That is, whether it is due to some impurities, imperfections, multiple superconducting / non-superconducting phases or variations in flux dynamics etc. still not clear. Further, all these reports consider the role of JI interactions only but none of them give reason for the initial fall in the signal amplitude on increasing

All these observations irrespective of the class of superconductors make this problem

Recently in our earlier work (Padam et al., 2006) an attempt has been made to analyze this problem by considering the effect of electromagnetic interactions (EMI) in addition to JI and found that involving EMI can satisfactorily explain particularly the undulatory

Before going into further details, lets us see the temperature dependence of the NRMA

**3.2 NRMA signal amplitude vs. temperature behaviors in present BSCCO samples –** 

Fig.6 depicts the peak behavior for Ag (0, 5 and10 wt %) added (Bi, Pb)2Sr2Ca2Cu3O10+x [here after called as (Bi, Pb)-2223] bulk samples the temperature dependence of the signal amplitude for of constant magnetic field of 0 G by varying the temperature (45-160K). In these experiments equal amounts of each sample and identical spectrometer operating conditions were used to facilitate comparison of the resulting spectra. It is clear that all the

viscosity of the fluxons on increasing temperature (Haddon et al., 1991).

intriguing and motivate to investigate this discrepant data more carefully.

dependent coupling strength of different JJs present in each superconducting phase.

**3.1.2 Undulatory and monotonous type temperature dependence** 

different explanations have been given.

signal amplitude in the present samples.

temperature.

behavior.

**Relevance of EMI** 

**3.2.1 (Bi, Pb) samples** 

lower modulation frequency/amplitude.

Fig.7a depicts an undualtory temperature dependence of the NRMA signal amplitude for the Bi-2212 (5wt %) bulk samples kept at constant magnetic field of 0G and varying the temperature from 15K to 105K rather than the peak behavior as shown by all the three (Bi,Pb)-2223 samples. This Bi-2212 sample showed showing three distinct regions, i.e. an initial decreasing curve ABC (15-71.8), narrow flat CD (71.8-76.5K) and then a rise DE (76.5- 82.5K) with maxima at around 82.5K followed by subsequent exponential fall EF.

This undulatory behavior may be discussed first in the frame work of the explanation used for peak behavior. That is, on increasing temperature, strength of JI decreases (due to increasing decoupling of JJs) which results into an increase in number density and mobility of JF. This leads to an initial rise in the NRMA signal amplitude reaching a maxima at a few degrees below Tc, until most of the JJs are broken and showed weak superconductivity, hence an exponential fall in the superconducting - to - normal region. Above this temperature, all the JJs are broken and the material becomes normal, where the NRMA signal disappears.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 95

From the above one can say that with the inclusion of EMI the undulatory particularly the initial fall portion has been explained which otherwise was unexplainable with taking JI alone. Based upon this explanation of inclusion of EMI in addition to JI, the undulatory temperature dependence of NRMA signal amplitude as reported by others in superconducting sample having thick JJinter can also be explained. To confirm, the measurements were done by keeping the sample at a constant magnetic field as is known that the magnetic field suppresses JJinter response. The results obtained for two fields of 50G and 100G are shown in Figs.7b, 7c. Interestingly, for both the fields, the undulatory behavior which was obtained at 0G disappeared and instead a peak response obtained and the peak gets narrowed on increasing magnetic field. This clearly suggested that the initial fall is related with the presence of thicker JJinter which play a dominating role at low temperatures.

Fig. 7. NRMA signal amplitude vs. temperature curve for Bi-2212 (5wt%Ag) sample in OG,

Based upon this explanation, all the results reported in literature showing undulatory can also be explained on the basis of presence of thick JJs which dominate at low temperature and lead to rising strength of EMI initially, hence an initial fall in the NRMA signal

Similarly, the monotonous fall in the NRMA signal amplitude might also be due to the dominance of thicker JJinter (which can be due to impurities, impurity phases, defects etc.) which allows fluxons to enter at low temperatures and increase the probability of EMI among fluxons, thereby decreasing the mobility of the fluxons hence the NRMA signal

From the above one can say that if a superconductor has more of multitude of thick JJinter, then the strength EMI overpowers that of JI and generally show undulatory or monotonous

In the next section the second equally important results related with anomalous nature of

It would be appropriate to give a brief account of earlier reports related particularly with the nature of NRMA hysteresis in comparison to that of the DC magnetization method and its universality in different granular superconducting systems including the present results

**4. Results and explanations on anomalous nature of magnetic hysteresis** 

10G and 50G.

amplitude on increasing temperature.

amplitude with increasing temperature.

response on increasing temperature.

before any elaborative discussion related to its origin.

NRMA hysteresis are given.

**behavior of NRMA** 

From this it is very clear that, the undulatory behavior of an initial fall in the present samples can not be explained by the temperature dependent strength of JI alone as it will lead to an initial rise with increasing temperature.

Other explanations which relates with impurities and to a mixture of phases with a distribution of superconducting transition temperatures are unlikely as the Bi-2212 samples are single phase as revealed from the observations of: (i) sharp superconducting transition from resistivity/AC susceptibility data, (ii) no reflections, corresponding to the Bi-2201 structure within the accuracy of 3% from even the powder XRD measurements and (iii) no impurity signal from EPR data (Fig.4).

Similarly another explanation which relates to the presence of high density of defects and dominance of a variety of thick JJs etc. considering only the JI is also ruled out for the obvious reason that JI alone can only explain the initial rise but not the initial fall as in the present case. This indicated that there is a need to look for some other alternative explanation and/or to include the role of some other interaction in addition to JI.

Recently the present authors (Padam et al.,2006)have given an alternative explanation for the origin of undulatory behavior by including in addition to JI, the role of electromagnetic interactions (EMI) which are not only significant but are also dominant over most of the low magnetic field and low temperature regimes, where generally the NRMA measurements are carried out. The combined effect of JI and EMI controls the FLL behavior and play an important role in absorption of microwaves. Based upon the inclusion of EMI in addition of JI, the undulatory behavior in the present Bi-2212 (5wt %) sample has been explained as follows:

Appearance of NRMA signal at 15K, suggested the presence of thicker JJs (most likely JJinter ) which allow fluxons to penetrate at such low temperatures. Since density of fluxons is low at 15K, therefore there is a large separation among them. This leads to a weak attractive EMI. However, at this temperature, JI among fluxons is still finite. Thus combined effect of finite JI but feeble attractive EMI results into a loose packing of fluxons to form, a quasi-3D dilute liquid like FLL (i.e. disordered phase) in which the fluxons are mobile. An increase in temperature causes breaking of more number of JJinter and allows more number of fluxons to penetrate, thus reduces the inter-fluxon distance. This in turn decreases strength of JI but increases that of the EM attractions among the fluxons. Thus are able to suppress their even increasing thermally activated fluxon motion as a result of raising temperature. This trend continues up to 71.8K and results in the decrease of MWA, hence, a decrease in the signal amplitude.

On further increasing temperature, due to continuation of the above trend, probably, the EM attraction becomes which minimize the thermally activated fluxon motion and results in weak temperature dependence of the amplitude in the narrow temperature range (71.8- 76.5K). In this range, a phase transition from quasi-3D dilute liquid like FLL to a quasi-3D solid like FLL (ordered phase) probably also occur.

However, above 76.5K fluxon density increases to an extent that the inter-fluxon EM repulsion overpowers inter-fluxon EM attraction and the system now becomes more sensitive to increasing thermal fluctuations. Due to dominance of both the EM repulsion and increasing thermal fluctuations, fluxon motion is more prevented and the FLL starts melting and transits into a quasi-2D dense liquid. This leads to an increase in MWA, hence a rise in the amplitude with a peak at around 82.5K, where it melts and leads to a fall in MWA in 82.5-105K.

From this it is very clear that, the undulatory behavior of an initial fall in the present samples can not be explained by the temperature dependent strength of JI alone as it will

Other explanations which relates with impurities and to a mixture of phases with a distribution of superconducting transition temperatures are unlikely as the Bi-2212 samples are single phase as revealed from the observations of: (i) sharp superconducting transition from resistivity/AC susceptibility data, (ii) no reflections, corresponding to the Bi-2201 structure within the accuracy of 3% from even the powder XRD measurements and (iii) no

Similarly another explanation which relates to the presence of high density of defects and dominance of a variety of thick JJs etc. considering only the JI is also ruled out for the obvious reason that JI alone can only explain the initial rise but not the initial fall as in the present case. This indicated that there is a need to look for some other alternative

Recently the present authors (Padam et al.,2006)have given an alternative explanation for the origin of undulatory behavior by including in addition to JI, the role of electromagnetic interactions (EMI) which are not only significant but are also dominant over most of the low magnetic field and low temperature regimes, where generally the NRMA measurements are carried out. The combined effect of JI and EMI controls the FLL behavior and play an important role in absorption of microwaves. Based upon the inclusion of EMI in addition of JI, the undulatory behavior in the present Bi-2212 (5wt %) sample has been explained as follows: Appearance of NRMA signal at 15K, suggested the presence of thicker JJs (most likely JJinter ) which allow fluxons to penetrate at such low temperatures. Since density of fluxons is low at 15K, therefore there is a large separation among them. This leads to a weak attractive EMI. However, at this temperature, JI among fluxons is still finite. Thus combined effect of finite JI but feeble attractive EMI results into a loose packing of fluxons to form, a quasi-3D dilute liquid like FLL (i.e. disordered phase) in which the fluxons are mobile. An increase in temperature causes breaking of more number of JJinter and allows more number of fluxons to penetrate, thus reduces the inter-fluxon distance. This in turn decreases strength of JI but increases that of the EM attractions among the fluxons. Thus are able to suppress their even increasing thermally activated fluxon motion as a result of raising temperature. This trend continues up to 71.8K and

On further increasing temperature, due to continuation of the above trend, probably, the EM attraction becomes which minimize the thermally activated fluxon motion and results in weak temperature dependence of the amplitude in the narrow temperature range (71.8- 76.5K). In this range, a phase transition from quasi-3D dilute liquid like FLL to a quasi-3D

However, above 76.5K fluxon density increases to an extent that the inter-fluxon EM repulsion overpowers inter-fluxon EM attraction and the system now becomes more sensitive to increasing thermal fluctuations. Due to dominance of both the EM repulsion and increasing thermal fluctuations, fluxon motion is more prevented and the FLL starts melting and transits into a quasi-2D dense liquid. This leads to an increase in MWA, hence a rise in the amplitude

with a peak at around 82.5K, where it melts and leads to a fall in MWA in 82.5-105K.

explanation and/or to include the role of some other interaction in addition to JI.

results in the decrease of MWA, hence, a decrease in the signal amplitude.

solid like FLL (ordered phase) probably also occur.

lead to an initial rise with increasing temperature.

impurity signal from EPR data (Fig.4).

From the above one can say that with the inclusion of EMI the undulatory particularly the initial fall portion has been explained which otherwise was unexplainable with taking JI alone. Based upon this explanation of inclusion of EMI in addition to JI, the undulatory temperature dependence of NRMA signal amplitude as reported by others in superconducting sample having thick JJinter can also be explained. To confirm, the measurements were done by keeping the sample at a constant magnetic field as is known that the magnetic field suppresses JJinter response. The results obtained for two fields of 50G and 100G are shown in Figs.7b, 7c. Interestingly, for both the fields, the undulatory behavior which was obtained at 0G disappeared and instead a peak response obtained and the peak gets narrowed on increasing magnetic field. This clearly suggested that the initial fall is related with the presence of thicker JJinter which play a dominating role at low temperatures.

Fig. 7. NRMA signal amplitude vs. temperature curve for Bi-2212 (5wt%Ag) sample in OG, 10G and 50G.

Based upon this explanation, all the results reported in literature showing undulatory can also be explained on the basis of presence of thick JJs which dominate at low temperature and lead to rising strength of EMI initially, hence an initial fall in the NRMA signal amplitude on increasing temperature.

Similarly, the monotonous fall in the NRMA signal amplitude might also be due to the dominance of thicker JJinter (which can be due to impurities, impurity phases, defects etc.) which allows fluxons to enter at low temperatures and increase the probability of EMI among fluxons, thereby decreasing the mobility of the fluxons hence the NRMA signal amplitude with increasing temperature.

From the above one can say that if a superconductor has more of multitude of thick JJinter, then the strength EMI overpowers that of JI and generally show undulatory or monotonous response on increasing temperature.

In the next section the second equally important results related with anomalous nature of NRMA hysteresis are given.

## **4. Results and explanations on anomalous nature of magnetic hysteresis behavior of NRMA**

It would be appropriate to give a brief account of earlier reports related particularly with the nature of NRMA hysteresis in comparison to that of the DC magnetization method and its universality in different granular superconducting systems including the present results before any elaborative discussion related to its origin.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 97

Fig. 9. NRMA hysteresis in different superconductors (data from literature) showing

dP/dH vs. H curves of NRMA signals recorded for the present Bi-2223 samples in the high field scans for both forward (-50 to +1025G ) directions at temperatures in the range of 5.5 to 135K were taken and the results for 90K are shown in Fig.10a. Two signals: one the usual near zero field signal and another a hump like feature (encircled) which varies in amplitude and width with increasing temperature along with the former signal have been observed. Similar observations of zero field signal along with this kind of feature (encircled) have been reported earlier by some researchers including present authors (Padam et al. 2010) and are also shown in Fig. 10b for Bi-2212 single crystal (rf) ( Baginskii et al. 1994, rf absorption studies) and in Fig.10c for Tl-2212 single crystal (NRMA studies) (Joshi et al. 2000, MW absorption studies)The former well known zero field signal is due to viscous motion of intergranular (IGBJ)/intragranular (IG) fluxons. Whereas the latter hump like feature is due to fluxons generated due to tilt motion of flux lattice (Brandt, 1991; Kleiner et al. 1992) It is these fluxons which have been considered to be responsible for the general anomalous magnetic hysteresis and are named as ESJ in our earlier

Similar results were obtained in Bi-2212 samples. Fig.11 depicts low field forward (-50 to +50G) and reverse field (+50 to -50G) magnetic sweeps of the NRMA signals in the both the Bi-2223 and Bi-2212 samples showing hysteresis at LT (5.5K, 20K) and at HT (110K, 80K). It can be seen that the decreasing field curve lies below the increasing field curve for both the samples not only at HT but also at LT. This indicated that the nature of NRMA magnetic hysteresis is anomalous in the present samples even at low temperatures as low

Thus all the above data reported earlier and in the present work confirms the anomalous nature of hysteresis in different granular superconductors including recently discovered

iron pnictides under LT/LH conditions as seen by the NRMA method.

anomalous hysteresis irrespective of class of granular superconductor.

**4.3 NRMA hysteresis in the present BSCCO samples** 

publication (Padam et al. 2010)

5.5K.

#### **4.1 A comparative data on the nature of DC and NRMA magnetic hysteresis reported by other researchers**

Fig.8 shows a comparison of set of data available in literature in single crystals of YBCO (Dulcic et al. 1990; Pakulis & G.V.Chandershekar, 1989) and Bi-2212 (Pradhan et al., 1995) of these two measurements under almost similar experimental conditions of temperature, magnetic field scan and orientation of the magnetic field with respect to crystal axis. It is worth noting that for both the YBCO and Bi-2212 systems, the decreasing H curve lying above the increasing H curve observed in DC magnetization, the decreasing H curve lies below the increasing H curve nature of hysteresis shown by NRMA method. This indicated that in contrast to the normal nature of hysteresis observed by DC method, the nature of hysteresis shown by NRMA is anomalous under similar experimental conditions of temperature, magnetic field.

Fig. 8. Comparison of NRMA and DC magnetic hysteresis in YBCO/Bi-2212 single crystals showing the anomalous hysteresis as seen in NRMA in contrast to normal in the DC magnetization (data from literature).

### **4.2 Magnetic hysteresis of NRMA in different superconductors from unconventional to conventional observed by other researchers**

Fig.9 exhibits the results of NRMA hysteresis reported earlier in different superconducting systems like: unconventional intrinsically granular cuprates (Blazey et al.,1988),borocarbides (Kadam et al.,1994) fullerene ( Zakhidov et al.1991) MgB2( Joshi et al.,2002) Iron pniticides (Panarina et al. 2010)and granular conventional type I/type II superconductors ( Baranowsky et al., 1991; Mahel & S.Benacka,1992)) It further confirms that irrespective of the system of the superconductor, in all the NRMA curves, the decreasing H lies below the increasing H curve at temperatures much below Tc and in low magnetic field regions. This clearly indicated anomalous hysteresis nature at LT/LH in different superconducting systems from unconventional to conventional while carrying out NRMA measurements.

Fig. 9. NRMA hysteresis in different superconductors (data from literature) showing anomalous hysteresis irrespective of class of granular superconductor.

## **4.3 NRMA hysteresis in the present BSCCO samples**

96 Superconductors – Properties, Technology, and Applications

**4.1 A comparative data on the nature of DC and NRMA magnetic hysteresis reported** 

Fig.8 shows a comparison of set of data available in literature in single crystals of YBCO (Dulcic et al. 1990; Pakulis & G.V.Chandershekar, 1989) and Bi-2212 (Pradhan et al., 1995) of these two measurements under almost similar experimental conditions of temperature, magnetic field scan and orientation of the magnetic field with respect to crystal axis. It is worth noting that for both the YBCO and Bi-2212 systems, the decreasing H curve lying above the increasing H curve observed in DC magnetization, the decreasing H curve lies below the increasing H curve nature of hysteresis shown by NRMA method. This indicated that in contrast to the normal nature of hysteresis observed by DC method, the nature of hysteresis shown by NRMA is anomalous under similar experimental conditions of

Fig. 8. Comparison of NRMA and DC magnetic hysteresis in YBCO/Bi-2212 single crystals showing the anomalous hysteresis as seen in NRMA in contrast to normal in the DC

**4.2 Magnetic hysteresis of NRMA in different superconductors from unconventional** 

Fig.9 exhibits the results of NRMA hysteresis reported earlier in different superconducting systems like: unconventional intrinsically granular cuprates (Blazey et al.,1988),borocarbides (Kadam et al.,1994) fullerene ( Zakhidov et al.1991) MgB2( Joshi et al.,2002) Iron pniticides (Panarina et al. 2010)and granular conventional type I/type II superconductors ( Baranowsky et al., 1991; Mahel & S.Benacka,1992)) It further confirms that irrespective of the system of the superconductor, in all the NRMA curves, the decreasing H lies below the increasing H curve at temperatures much below Tc and in low magnetic field regions. This clearly indicated anomalous hysteresis nature at LT/LH in different superconducting systems from unconventional to conventional while carrying

**by other researchers** 

temperature, magnetic field.

magnetization (data from literature).

out NRMA measurements.

**to conventional observed by other researchers** 

dP/dH vs. H curves of NRMA signals recorded for the present Bi-2223 samples in the high field scans for both forward (-50 to +1025G ) directions at temperatures in the range of 5.5 to 135K were taken and the results for 90K are shown in Fig.10a. Two signals: one the usual near zero field signal and another a hump like feature (encircled) which varies in amplitude and width with increasing temperature along with the former signal have been observed. Similar observations of zero field signal along with this kind of feature (encircled) have been reported earlier by some researchers including present authors (Padam et al. 2010) and are also shown in Fig. 10b for Bi-2212 single crystal (rf) ( Baginskii et al. 1994, rf absorption studies) and in Fig.10c for Tl-2212 single crystal (NRMA studies) (Joshi et al. 2000, MW absorption studies)The former well known zero field signal is due to viscous motion of intergranular (IGBJ)/intragranular (IG) fluxons. Whereas the latter hump like feature is due to fluxons generated due to tilt motion of flux lattice (Brandt, 1991; Kleiner et al. 1992) It is these fluxons which have been considered to be responsible for the general anomalous magnetic hysteresis and are named as ESJ in our earlier publication (Padam et al. 2010)

Similar results were obtained in Bi-2212 samples. Fig.11 depicts low field forward (-50 to +50G) and reverse field (+50 to -50G) magnetic sweeps of the NRMA signals in the both the Bi-2223 and Bi-2212 samples showing hysteresis at LT (5.5K, 20K) and at HT (110K, 80K). It can be seen that the decreasing field curve lies below the increasing field curve for both the samples not only at HT but also at LT. This indicated that the nature of NRMA magnetic hysteresis is anomalous in the present samples even at low temperatures as low 5.5K.

Thus all the above data reported earlier and in the present work confirms the anomalous nature of hysteresis in different granular superconductors including recently discovered iron pnictides under LT/LH conditions as seen by the NRMA method.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 99

Fig. 11. NRMA hysteresis in present BSCCO sintered pellets of LT and at HT.

J.Stankowski, 1991).

**4.4 Origin of NRMA anomalous magnetic hysteresis and the relevance of ESJ fluxons**  Before elaborating more on the origin of anomalous hysteresis, let us give a brief account of the development of the understanding related with the observations of different nature of hysteresis depending upon the (i) class of superconductors (i.e. conventional or unconventional), (ii) experimental conditions of temperature (T) and magnetic field (H) and (iii) the sensitivity of the method (DC magnetization methods: DC magnetic field (HDC) variation of (i) flux density (B), (ii) microwave surface resistance (Rs) and (iii) transport critical current density [Jc] (Ji et al. 1993)and of modulated magnetic field variation of non-resonant microwave power absorption (P) NRMA technique( Czyzak &

That is, (i) normal nature of hysteresis in conventional superconductors at all temperatures and/or magnetic field by any measurement method, (ii) normal hysteresis at low temperatures (LT) and/or low magnetic field (LH) and anomalous hysteresis at high magnetic field (HH) and/or high temperatures (HT) by in granular conventional / unconventional superconductors by DC magnetization and microwave methods and (iii) anomalous hysteresis not only at HH /HT but also at LH/LT in granular conventional

The normal nature of hysteresis under any conditions conventional superconductors has been very well explained by the Beans model (Beans, 1962) which considers the role of

/unconventional superconductors while carrying out NRMA measurements.

Fig. 10. dP/dH as a function of magnetic field showing signal (encircled) due to ESJ fluxon

Fig. 10. dP/dH as a function of magnetic field showing signal (encircled) due to ESJ fluxon

Fig. 11. NRMA hysteresis in present BSCCO sintered pellets of LT and at HT.

## **4.4 Origin of NRMA anomalous magnetic hysteresis and the relevance of ESJ fluxons**

Before elaborating more on the origin of anomalous hysteresis, let us give a brief account of the development of the understanding related with the observations of different nature of hysteresis depending upon the (i) class of superconductors (i.e. conventional or unconventional), (ii) experimental conditions of temperature (T) and magnetic field (H) and (iii) the sensitivity of the method (DC magnetization methods: DC magnetic field (HDC) variation of (i) flux density (B), (ii) microwave surface resistance (Rs) and (iii) transport critical current density [Jc] (Ji et al. 1993)and of modulated magnetic field variation of non-resonant microwave power absorption (P) NRMA technique( Czyzak & J.Stankowski, 1991).

That is, (i) normal nature of hysteresis in conventional superconductors at all temperatures and/or magnetic field by any measurement method, (ii) normal hysteresis at low temperatures (LT) and/or low magnetic field (LH) and anomalous hysteresis at high magnetic field (HH) and/or high temperatures (HT) by in granular conventional / unconventional superconductors by DC magnetization and microwave methods and (iii) anomalous hysteresis not only at HH /HT but also at LH/LT in granular conventional /unconventional superconductors while carrying out NRMA measurements.

The normal nature of hysteresis under any conditions conventional superconductors has been very well explained by the Beans model (Beans, 1962) which considers the role of

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 101

techniques unable to do so.

IGJ, IG and ESJ etc.

superconductors can be explained.

**5. Summary and conclusions** 

parameters/interactions not envisaged before.

the ESR spectrometer (Khachaturyan et al. 1987).

Before proceeding further, it is essential to essential to highlight the experimental features of NRMA method, which makes it unique. For example, its ability (i) in creating a situation for enhanced ESJ fluxons,(ii) in detecting even minute changes in the fluxon density and (iii) in distinguishing between different types of fluxons,where all other DC magnetization

In this context it is known that NRMA technique uses ESR spectrometer, where DC magnetic field (HDC) is modulated with a modulation field (Hm) superimposed parallel on HDC. Due to this Hm of magnetically modulated hysteresis are observed unlike direct field hysteresis measurements. This modulation field in addition to HDC (used in B vs.H and Jc vs. H methods) and HDC+Hmw (both used in Rs vs. H method) gives a few advantages. For example, it provides extra energy (due to a combined effect of three fields: HDC+Hmw+Hm) required to overcome the strength of the Josephson junctions; as a result, IGBJ/IGJ fluxons enter at much lower temperatures/low magnetic fields. Another is a stronger Lorentz force, particularly due to Hm induced oscillatory currents, which tilt the IGBJ/IGJ fluxons, and probably becomes a volume source of generation of ESJ fluxons. Further, a combination of the modulation field in conjunction with phase sensitive makes ESR an extremely sensitive and powerful tool to detect any feeble change and to distinguish between different types of fluxons (Portis et al.1988; Dulcic et al.1897, 1988, 1989). Furthermore, the sensitivity of the NRMA method, which uses ESR, is even five orders of magnitude greater than the limit of

All these parameters make ESR technique an extra sensitive and unique in comparison of other techniques. Therefore, any change however minute and at times seemingly intractable, related with density/distribution/motion and different types of fluxons depending upon temperature/magnetic field/sample form etc. can be easily detected in NRMA. Furthermore, NRMA is the only method (Portis et al.1988; Dulcic et al.1897, 1988, 1989) which is able to distinguish between above mentioned different types of fluxons, i.e.IGBJ,

Using this explanation, the observed anomalous nature of hysteresis in the present samples (which also shows the signal due to ESJ fluxons) and reported in a variety of

The present NRMA studies made on Bi-2223 **/** Bi-2212 samples and similar studies by other researchers in a variety of superconductors from conventional to unconventional have been reported. Main focus has been given on two most important aspects. First is the different temperature dependences of the NRMA signal amplitude observed by us and by others. The second one is on the observation of anomalous magnetic hysteresis in the present samples and in reports of others. These studies showing different types of behaviors: peak type undulatory and monotonous temperature dependent signal amplitude behavior and the anomalous magnetic hysteresis at all temperatures below Tc have been analyzed and discussed by taking into consideration of

intragranular Abrikisov/Pancake fluxons (IG). The anomalous nature of hysteresis at HT/HH in granular conventional/ unconventional superconductors has been well explained by Ji et al model by modifying Beans model on considering the role of intergranular Josephson (IGBJ) fluxons in addition to IG.

It is surprising that despite the voluminous literature on the NRMA hysteresis; only a few groups (Sugawara et al. 2000; Padam et al. 1991; Czyzak & J.Stankowski, 1991; Mahel et al., 1992; Ramachandran et al. 1994) have mentioned about its anomalous nature and related it to Ji et al model and its variant by Mahel et al. and Ramachandran et al. but they have their own drawbacks as will be clear below.

The present/ others results on the presence of anomalous magnetic hysteresis can be discussed in the frame work of the most successful model by Ji et al.which explain HT/HH anomalous hysteresis in granular superconductors. According to this model, there are two kinds of fluxons: intergranular (IGBJ) and intragranular (IJ) in granular superconductors. Further, at HT/HH, due to the higher number density and higher mobility of the IGBJ fluxons to those of the IJ fluxons, the magnetization during increasing field curve is lower than that during decreasing field curve and gives anomalous hysteresis. Although this model successfully explains the anomalous hysteresis at HT/HH, however the observed anomalous hysteresis at LT/LH in the present samples and reported by others can not be explained.

Further, these results also cannot be explained even by using variant of Ji et al. model by Mahel et al. and Ramachandran et al. particularly used for explaining NRMA anomalous hysteresis. This is because, the observation of anomalous magnetic hysteresis in the present samples as well as reported by others is not limited by the conditions of imposed by these models. For example, the former model includes the effect of modulation amplitude threshold field on the distribution of both the intergranular Josephson (IGBJ) and intragranular (IJ) fluxons). However, this model is applicable only for a limited modulation amplitude range. Similarly, the latter model which includes the effect of modulation amplitude threshold field on the distribution of flux suffers from its stringent requirement of threshold field value.

In our recent work (Padam et al., 2010) an attempt has been made to solve this problem by using another variant of Ji et al. model by including the role of Energy Stabilized Josephson (ESJ) fluxons in addition to IGB/IG fluxons and highlighting the features of the extremely sensitive NRMA method which makes it unique in creating and detecting a situation for the appearance of anomalous hysteresis. It was found that by combining the role of ESJ fluxons with the uniqueness of the NRMA technique, it is possible to explain the observation of LT/LH anomalous hysteresis too.

It is known that apart from IGBJ fluxons and IJ fluxons, there exists another kind of Josephson fluxons which are generated as a result of thermal/Lorentz forces induced tilt motion of flux lattice (Brandt, 1991; Kleiner et al. 1992) It is these fluxons are ESJ fluxons which not only increase the total number of more mobile fluxons at LT and/ LH, etc but also are detected due to the extra sensitivity of the NRMA method (Khachaturyan et al., 1987), thereby revealing the anomalous hysteresis.

Before proceeding further, it is essential to essential to highlight the experimental features of NRMA method, which makes it unique. For example, its ability (i) in creating a situation for enhanced ESJ fluxons,(ii) in detecting even minute changes in the fluxon density and (iii) in distinguishing between different types of fluxons,where all other DC magnetization techniques unable to do so.

In this context it is known that NRMA technique uses ESR spectrometer, where DC magnetic field (HDC) is modulated with a modulation field (Hm) superimposed parallel on HDC. Due to this Hm of magnetically modulated hysteresis are observed unlike direct field hysteresis measurements. This modulation field in addition to HDC (used in B vs.H and Jc vs. H methods) and HDC+Hmw (both used in Rs vs. H method) gives a few advantages. For example, it provides extra energy (due to a combined effect of three fields: HDC+Hmw+Hm) required to overcome the strength of the Josephson junctions; as a result, IGBJ/IGJ fluxons enter at much lower temperatures/low magnetic fields. Another is a stronger Lorentz force, particularly due to Hm induced oscillatory currents, which tilt the IGBJ/IGJ fluxons, and probably becomes a volume source of generation of ESJ fluxons. Further, a combination of the modulation field in conjunction with phase sensitive makes ESR an extremely sensitive and powerful tool to detect any feeble change and to distinguish between different types of fluxons (Portis et al.1988; Dulcic et al.1897, 1988, 1989). Furthermore, the sensitivity of the NRMA method, which uses ESR, is even five orders of magnitude greater than the limit of the ESR spectrometer (Khachaturyan et al. 1987).

All these parameters make ESR technique an extra sensitive and unique in comparison of other techniques. Therefore, any change however minute and at times seemingly intractable, related with density/distribution/motion and different types of fluxons depending upon temperature/magnetic field/sample form etc. can be easily detected in NRMA. Furthermore, NRMA is the only method (Portis et al.1988; Dulcic et al.1897, 1988, 1989) which is able to distinguish between above mentioned different types of fluxons, i.e.IGBJ, IGJ, IG and ESJ etc.

Using this explanation, the observed anomalous nature of hysteresis in the present samples (which also shows the signal due to ESJ fluxons) and reported in a variety of superconductors can be explained.

## **5. Summary and conclusions**

100 Superconductors – Properties, Technology, and Applications

intragranular Abrikisov/Pancake fluxons (IG). The anomalous nature of hysteresis at HT/HH in granular conventional/ unconventional superconductors has been well explained by Ji et al model by modifying Beans model on considering the role of

It is surprising that despite the voluminous literature on the NRMA hysteresis; only a few groups (Sugawara et al. 2000; Padam et al. 1991; Czyzak & J.Stankowski, 1991; Mahel et al., 1992; Ramachandran et al. 1994) have mentioned about its anomalous nature and related it to Ji et al model and its variant by Mahel et al. and Ramachandran et al. but they have their

The present/ others results on the presence of anomalous magnetic hysteresis can be discussed in the frame work of the most successful model by Ji et al.which explain HT/HH anomalous hysteresis in granular superconductors. According to this model, there are two kinds of fluxons: intergranular (IGBJ) and intragranular (IJ) in granular superconductors. Further, at HT/HH, due to the higher number density and higher mobility of the IGBJ fluxons to those of the IJ fluxons, the magnetization during increasing field curve is lower than that during decreasing field curve and gives anomalous hysteresis. Although this model successfully explains the anomalous hysteresis at HT/HH, however the observed anomalous hysteresis at LT/LH in the present samples and reported by others can not be

Further, these results also cannot be explained even by using variant of Ji et al. model by Mahel et al. and Ramachandran et al. particularly used for explaining NRMA anomalous hysteresis. This is because, the observation of anomalous magnetic hysteresis in the present samples as well as reported by others is not limited by the conditions of imposed by these models. For example, the former model includes the effect of modulation amplitude threshold field on the distribution of both the intergranular Josephson (IGBJ) and intragranular (IJ) fluxons). However, this model is applicable only for a limited modulation amplitude range. Similarly, the latter model which includes the effect of modulation amplitude threshold field on the distribution of flux suffers from its stringent requirement

In our recent work (Padam et al., 2010) an attempt has been made to solve this problem by using another variant of Ji et al. model by including the role of Energy Stabilized Josephson (ESJ) fluxons in addition to IGB/IG fluxons and highlighting the features of the extremely sensitive NRMA method which makes it unique in creating and detecting a situation for the appearance of anomalous hysteresis. It was found that by combining the role of ESJ fluxons with the uniqueness of the NRMA technique, it is possible to explain the observation of

It is known that apart from IGBJ fluxons and IJ fluxons, there exists another kind of Josephson fluxons which are generated as a result of thermal/Lorentz forces induced tilt motion of flux lattice (Brandt, 1991; Kleiner et al. 1992) It is these fluxons are ESJ fluxons which not only increase the total number of more mobile fluxons at LT and/ LH, etc but also are detected due to the extra sensitivity of the NRMA method (Khachaturyan et al., 1987),

intergranular Josephson (IGBJ) fluxons in addition to IG.

own drawbacks as will be clear below.

explained.

of threshold field value.

LT/LH anomalous hysteresis too.

thereby revealing the anomalous hysteresis.

The present NRMA studies made on Bi-2223 **/** Bi-2212 samples and similar studies by other researchers in a variety of superconductors from conventional to unconventional have been reported. Main focus has been given on two most important aspects. First is the different temperature dependences of the NRMA signal amplitude observed by us and by others. The second one is on the observation of anomalous magnetic hysteresis in the present samples and in reports of others. These studies showing different types of behaviors: peak type undulatory and monotonous temperature dependent signal amplitude behavior and the anomalous magnetic hysteresis at all temperatures below Tc have been analyzed and discussed by taking into consideration of parameters/interactions not envisaged before.

Non Resonant Microwave Absorption (NRMA) Anomalies in High Temperature Superconductors (HTS) Relevance of Electromagnetic Interactions (EMI) and Energy Stabilized Josephson… 103

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From the above studies and discussion, it comes out that the undulatory type/ monotonous temperature dependent behavior of NRMA signal amplitude in a variety of granular superconductors can be very well explained if the role of EMI is included with JI.

In addition, the presence of additional fluxons (ESJ) in the present Bi-2223 / Bi-2212 samples and in other granular superconductors at all temperatures which probably formed due to a combined effect of additional fields like oscillatory MW field and modulation field are responsible for the anomalous hysteresis while carrying out NRMA measurements.

## **6. Acknowledgements**

The author is grateful to Prof.R.C.Budani, Director, and National Physical Laboratory for his constant encouragement during the course of this work. The author is highly indebted to Prof.S.V.Bhat (IISC, Banglore) and Prof.M.R.Tripathy (Dept.Of Electrical and Communications Engng. ASET, Noida) for NRMA experiments. The author is highly grateful to Dr.S.N.Ekbote for his guidance and valuable suggestions.

## **7. References**


From the above studies and discussion, it comes out that the undulatory type/ monotonous temperature dependent behavior of NRMA signal amplitude in a variety of granular

In addition, the presence of additional fluxons (ESJ) in the present Bi-2223 / Bi-2212 samples and in other granular superconductors at all temperatures which probably formed due to a combined effect of additional fields like oscillatory MW field and modulation field are

The author is grateful to Prof.R.C.Budani, Director, and National Physical Laboratory for his constant encouragement during the course of this work. The author is highly indebted to Prof.S.V.Bhat (IISC, Banglore) and Prof.M.R.Tripathy (Dept.Of Electrical and Communications Engng. ASET, Noida) for NRMA experiments. The author is highly

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Physica C 357-360,481.


**6** 

*1China 2USA* 

**Superconductivity in Nanoscale Systems** 

*1International Center for Quantum Materials & State Key Laboratory for Mesoscopic* 

*2The Center for Nanoscale Science & Department of Physics, The Pennsylvania State* 

According to the BCS theory, charge transport in superconductors is carried out by paired electrons named 'Cooper pairs'. The distance between a pair of electrons is called the superconducting coherence length (ξ). If superconductivity is made possible by electrons 'holding hands' to form Cooper pairs, what happens to the superconductivity when one or more dimensions of the sample are smaller than the size of the cooper pair (ξ)? The answer to this question is of both fundamental interest and practical importance. The fundamental interest stems from the fact that in these reduced dimensionality systems, fluctuations play a very important role, even at low temperatures. In addition, because of the large surface to volume ratio, surface effects become important. Therefore, many novel phenomena not seen in bulk manifest in reduced dimensionality systems. From a practical standpoint, reduced dimensionality systems are becoming important because of rapid miniaturization of electronic circuits. With electronic devices getting smaller by the day engineers have to design smaller transistors. For comparison, the first chip had about 2,200 transistors on it whereas today, hundreds of millions of transistors can fit on a single chip. There are already computer chips that have nanometer sized transistors and future transistors will have to be even smaller. This makes understanding electronics in general and

superconductivity in particular in reduced dimensionality samples very relevant.

Let us begin by introducing the specific scales involved and systems of interest. A superconductor with one, two or three dimensions smaller than ξ is in the quasi-twodimensional (2D), quasi-one-dimensional (1D) or quasi zero dimensional (0D) regime respectively. For most conventional superconductors, ξ is of the order of microns. Therefore, systems falling in the 2D, 1D or 0D category are nanoscale systems. According to Hohenberg-Mermin-Wagner theorem (Hohenberg, 1967; Mermin & Wagner 1966), in these reduced dimensionality systems, fluctuations should destroy superconducting order even at low temperature. In 2D samples, the Berezinski-Kosterlitz-Thousless transition occurs, enabling superconducting order to exist at low temperature. However, the existence, limits

**1. Introduction** 

 \*

Corresponding author

Meenakshi Singh2, Yi Sun1 and Jian Wang1,2\*

*Physics, School of Physics, Peking University, Beijing,* 

*University, University Park,* 

Veinger, A.I., Zabrodskii A.G. and Tisnek, T.V. (1995).Supercond.Sci.Technol. 8,368. Zakhidov, A.A., Imaeda, K., Ugawa, A., Yakushi, K., Inokuchi, H., Iqbal, Z., Baughman, R.H., Ramakrishana B.L.and Achiba, Y. (1991) Physica C 185,411.

## **Superconductivity in Nanoscale Systems**

Meenakshi Singh2, Yi Sun1 and Jian Wang1,2\*

*1International Center for Quantum Materials & State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing, 2The Center for Nanoscale Science & Department of Physics, The Pennsylvania State University, University Park, 1China 2USA* 

## **1. Introduction**

104 Superconductors – Properties, Technology, and Applications

Zakhidov, A.A., Imaeda, K., Ugawa, A., Yakushi, K., Inokuchi, H., Iqbal, Z., Baughman,

Veinger, A.I., Zabrodskii A.G. and Tisnek, T.V. (1995).Supercond.Sci.Technol. 8,368.

R.H., Ramakrishana B.L.and Achiba, Y. (1991) Physica C 185,411.

According to the BCS theory, charge transport in superconductors is carried out by paired electrons named 'Cooper pairs'. The distance between a pair of electrons is called the superconducting coherence length (ξ). If superconductivity is made possible by electrons 'holding hands' to form Cooper pairs, what happens to the superconductivity when one or more dimensions of the sample are smaller than the size of the cooper pair (ξ)? The answer to this question is of both fundamental interest and practical importance. The fundamental interest stems from the fact that in these reduced dimensionality systems, fluctuations play a very important role, even at low temperatures. In addition, because of the large surface to volume ratio, surface effects become important. Therefore, many novel phenomena not seen in bulk manifest in reduced dimensionality systems. From a practical standpoint, reduced dimensionality systems are becoming important because of rapid miniaturization of electronic circuits. With electronic devices getting smaller by the day engineers have to design smaller transistors. For comparison, the first chip had about 2,200 transistors on it whereas today, hundreds of millions of transistors can fit on a single chip. There are already computer chips that have nanometer sized transistors and future transistors will have to be even smaller. This makes understanding electronics in general and superconductivity in particular in reduced dimensionality samples very relevant.

Let us begin by introducing the specific scales involved and systems of interest. A superconductor with one, two or three dimensions smaller than ξ is in the quasi-twodimensional (2D), quasi-one-dimensional (1D) or quasi zero dimensional (0D) regime respectively. For most conventional superconductors, ξ is of the order of microns. Therefore, systems falling in the 2D, 1D or 0D category are nanoscale systems. According to Hohenberg-Mermin-Wagner theorem (Hohenberg, 1967; Mermin & Wagner 1966), in these reduced dimensionality systems, fluctuations should destroy superconducting order even at low temperature. In 2D samples, the Berezinski-Kosterlitz-Thousless transition occurs, enabling superconducting order to exist at low temperature. However, the existence, limits

<sup>\*</sup> Corresponding author

Superconductivity in Nanoscale Systems 107

Fig. 1. A schematic of the setup used to electrodeposit nanowires into the pores of a trackeetched polycarbonate or anodized oxide (AAO) membrane. The inset shows scanning electron micrographs of (a) a cross section of an AAO membrane and a (b) longitudnal section of an AAO membrane. The regularly arranged pores meant for deposition of

For restrictive template based electrodeposition, commercially available track etched polycarbonate or anodic aluminum oxide (AAO) filtration membranes are used as templates. The pores in these membranes are created by irradiating the membrane with energetic heavy ions and anodizing in acid respectively (Lee, 2006). The resultant pores are parallel to each other and perpendicular to a cross section of the membrane (insets of figure 1). The length and diameter of the nanowire can be controlled by changing the thickness and the pore size of the membrane. The typical pore density varies from 108 to 1011 pores/cm2 which even with low filling fractions yields a large number of nanowires. A schematic of the electrochemical cell typically used is shown in figure 1. For use in electrodeposition, silver or gold is thermally evaporated on one surface of the membrane closing off the pores on that side. This evaporated layer acts as the cathode for electrodeposition. The membrane is then immersed in an electrolyte containing cations of the material(s) to be electrodeposited. A counter electrode and sometimes a reference electrode are put in the circuit and a potential is applied. Since the membranes are insulating, the metal evaporated on one surface acts as the cathode through which the current flows. As the open side of the template is exposed to the electrolyte, positively charged ions move towards the metal at the bottom of the pore. When they reach the metal, they accept electrons and get reduced to their metallic form. When the pores are filled with material, the deposition is halted. The membrane can be dissolved away and the nanowires are released for further use. Variations of this process are used to deposit multilayered nanowires and nanotubes. Some metals like gold can be deposited without an applied potential in an electroless process (Menon & Martin, 1995). In another variation, the pores of the membrane are wet by a precursor followed by thermal decomposition leading to formation of either nanowires or nanotubes in the pore (Nielsch et

nanowires can be seen.

al., 2005).

and nature of superconductivity in 1D remain a subject of interest. Fortunately, the advent of advanced synthesis techniques like electron beam lithography has enabled scientists to probe the nature of superconductivity in quasi-1D samples. In these systems, superconductivity exhibits many novel phenomena that shed light on the nature of superconductivity. In addition, the aforementioned miniaturization of electronic devices is opening up new horizons for applications of 1D nanowires, and need for understanding superconductivity at this scale. These quasi-1D systems therefore form the basis of the studies described in this chapter.

In this chapter, we hope to provide a comprehensive picture of superconductivity in nanoscale systems starting from synthesis techniques currently in use to examples of the new physics these systems are helping in uncovering. The systems that we focus on are quasi-1D nanowires and nanobelts. In the synthesis section, we will give a brief overview of the common techniques currently in use for making nanoscale samples, especially nanowires. The synthesis and manipulation of these samples can be extremely challenging. Many of the more advanced techniques are continuously evolving and have merited chapters and books on their own. We will include references to some of these more detailed descriptions of the techniques. In the section 3, we will discuss some unexpected discoveries in quasi-1D nanowires and nanobelts. Some of the novel phenomena discussed include macroscopic quantum phenomena, extremely long range proximity effects, the counter-intuitive anti-proximity effect and superconductivity in new materials amongst others. We hope to provide a glimpse of the measurements performed as well as some physical insight into the mechanism behind these phenomena. Finally and most importantly, we will discuss the potential of these new discoveries to be used in technology. The potential for application has not been the least of the reasons why superconductivity continues to capture the imagination of scientists and engineers alike even 100 years after its discovery.

## **2. Synthesis**

The first controlled fabrication of nanowires started in the late 1980s. Early experiments used a top-down approach where a large piece of the material was taken and whittled down using an electron beam to the required size (Knoedler, 1990). Most approaches to nanowire synthesis used today are however bottom-up: where the nanowire is assembled from smaller particles.

## **2.1 Electrochemical synthesis**

Electrochemical synthesis is an inexpensive and versatile method that yields high quality nanowires in a large number. Its limitation is that it is only useful for materials that can be electrochemically deposited from solution (like metals) and its application to semiconductor and non-elemental nanowires is possible but limited. It is a bottom-up approach in which wires are assembled atom by atom from an electrolytic solution with or without an applied potential. Electrochemical synthesis can further roughly be categorized into restrictive template based electrodeposition and active template assisted electrodeposition (Bera, 2004).

and nature of superconductivity in 1D remain a subject of interest. Fortunately, the advent of advanced synthesis techniques like electron beam lithography has enabled scientists to probe the nature of superconductivity in quasi-1D samples. In these systems, superconductivity exhibits many novel phenomena that shed light on the nature of superconductivity. In addition, the aforementioned miniaturization of electronic devices is opening up new horizons for applications of 1D nanowires, and need for understanding superconductivity at this scale. These quasi-1D systems therefore form the basis of the

In this chapter, we hope to provide a comprehensive picture of superconductivity in nanoscale systems starting from synthesis techniques currently in use to examples of the new physics these systems are helping in uncovering. The systems that we focus on are quasi-1D nanowires and nanobelts. In the synthesis section, we will give a brief overview of the common techniques currently in use for making nanoscale samples, especially nanowires. The synthesis and manipulation of these samples can be extremely challenging. Many of the more advanced techniques are continuously evolving and have merited chapters and books on their own. We will include references to some of these more detailed descriptions of the techniques. In the section 3, we will discuss some unexpected discoveries in quasi-1D nanowires and nanobelts. Some of the novel phenomena discussed include macroscopic quantum phenomena, extremely long range proximity effects, the counter-intuitive anti-proximity effect and superconductivity in new materials amongst others. We hope to provide a glimpse of the measurements performed as well as some physical insight into the mechanism behind these phenomena. Finally and most importantly, we will discuss the potential of these new discoveries to be used in technology. The potential for application has not been the least of the reasons why superconductivity continues to capture the imagination of scientists and engineers alike even 100 years after its

The first controlled fabrication of nanowires started in the late 1980s. Early experiments used a top-down approach where a large piece of the material was taken and whittled down using an electron beam to the required size (Knoedler, 1990). Most approaches to nanowire synthesis used today are however bottom-up: where the nanowire is assembled from

Electrochemical synthesis is an inexpensive and versatile method that yields high quality nanowires in a large number. Its limitation is that it is only useful for materials that can be electrochemically deposited from solution (like metals) and its application to semiconductor and non-elemental nanowires is possible but limited. It is a bottom-up approach in which wires are assembled atom by atom from an electrolytic solution with or without an applied potential. Electrochemical synthesis can further roughly be categorized into restrictive template based electrodeposition and active template assisted

studies described in this chapter.

discovery.

**2. Synthesis** 

smaller particles.

**2.1 Electrochemical synthesis** 

electrodeposition (Bera, 2004).

Fig. 1. A schematic of the setup used to electrodeposit nanowires into the pores of a trackeetched polycarbonate or anodized oxide (AAO) membrane. The inset shows scanning electron micrographs of (a) a cross section of an AAO membrane and a (b) longitudnal section of an AAO membrane. The regularly arranged pores meant for deposition of nanowires can be seen.

For restrictive template based electrodeposition, commercially available track etched polycarbonate or anodic aluminum oxide (AAO) filtration membranes are used as templates. The pores in these membranes are created by irradiating the membrane with energetic heavy ions and anodizing in acid respectively (Lee, 2006). The resultant pores are parallel to each other and perpendicular to a cross section of the membrane (insets of figure 1). The length and diameter of the nanowire can be controlled by changing the thickness and the pore size of the membrane. The typical pore density varies from 108 to 1011 pores/cm2 which even with low filling fractions yields a large number of nanowires. A schematic of the electrochemical cell typically used is shown in figure 1. For use in electrodeposition, silver or gold is thermally evaporated on one surface of the membrane closing off the pores on that side. This evaporated layer acts as the cathode for electrodeposition. The membrane is then immersed in an electrolyte containing cations of the material(s) to be electrodeposited. A counter electrode and sometimes a reference electrode are put in the circuit and a potential is applied. Since the membranes are insulating, the metal evaporated on one surface acts as the cathode through which the current flows. As the open side of the template is exposed to the electrolyte, positively charged ions move towards the metal at the bottom of the pore. When they reach the metal, they accept electrons and get reduced to their metallic form. When the pores are filled with material, the deposition is halted. The membrane can be dissolved away and the nanowires are released for further use. Variations of this process are used to deposit multilayered nanowires and nanotubes. Some metals like gold can be deposited without an applied potential in an electroless process (Menon & Martin, 1995). In another variation, the pores of the membrane are wet by a precursor followed by thermal decomposition leading to formation of either nanowires or nanotubes in the pore (Nielsch et al., 2005).

Superconductivity in Nanoscale Systems 109

Fig. 2. A schematic of the lithography process for making nanostructures (a) a layer of resist is spin-coated on the substrate, (b) the resist is exposed to either UV radiation in case of photolithography or an electron beam in case of electron-beam lithography through a mask which has the required pattern cut out from it, (c) the substrate with the resist is soaked in a developer which removes the parts of the resist that were exposed to radiation earlier (in case of pisitive resist) or parts that were not exposed to radiation earlier (in case of negative resist), (d) material is evaporated onto the substrate with the resist acting as a mask, (e) the substrate is soaked in a solvent that dissolves the remaining resist and also the layer of evaporated material on the resist. This 'lift-off' process only leaves behind the material

The technique of Chemical Vapor Deposition (CVD) is mostly used for the synthesis of thin films. The process is very versatile and allows for growth of amorphous, epitaxial, monocrystalline and polycrystalline films. The CVD equipment consists of a tube-like, chamber in which the substrate for synthesis is placed, a mechanism to heat the substrate, a gas handling system which is used to introduce vapors into the chamber and an exhaust

For the synthesis, a precursor dissolved in a solvent or in the form of an emulsion is vaporized and introduced into the chamber where some of the precursor mixture deposits on the substrate. The substrate is then heated causing the precursor mixture to decompose and the solvent to evaporate away. A layer of the required precursor is thus left on the substrate. If a metal organic compound is used as the precursor, the method is referred to as Metal-Organic Chemical Vapor Deposition (MOCVD). A slight variation of this

evaporated directly on the substrate in step (d).

**2.3 Chemical vapor deposition**

from which the waste vapors escape.

The formation of nanostructures in the active template-based synthesis results from preferential electrodeposition of atoms or molecules at hole and defect sites on the electrode surface. Once the initial layer of atoms is crystallized on the defect site, subsequent atoms also preferentially deposit at the site leading to formation of different nanostructures. The preparation of the electrode surface plays a major role in determining the morphology and structure of the electrodeposited nanostructure. Other nanosynthesis techniques like photolithography have been used in conjunction with this technique. Photolithography is used to create nanostructures on the electrode surface into which the required material is then electrodeposited (Yang et al., 2008).

## **2.2 Lithography**

Lithography translates to 'stone writing' and originally referred to the art of writing text on stone. Modern lithographic techniques find a wide variety of uses in the printing process. Our foci of interest for this section are, however, micro-lithographic processes like photolithography and electron beam lithography (EBL) with which most top down nanofabrication processes start. Photo-lithography is the more popular of the two processes and is used for manufacturing of semiconductor microchips. EBL, although more expensive, offers significantly improved resolution, sometimes going down to a few nanometers. Figure 2 shows a schematic of the lithography process.

The substrate (most often doped silicon coated with an insulating layer of silicon oxide or silicon nitride) is spin coated with a photo resist or e-beam resist. The choice of resist depends on the resolution required, spacing of features and thickness of metal to be evaporated post lithography. The resist is baked for hardening. In case of photo-lithography a hard mask is used with the required pattern made on it. In case of EBL, a 'soft' mask drawn using computer software like L-Edit is used. The areas within the pattern are exposed to either UV light for photolithography or an electron beam for EBL. The exposure time and beam-current used depend on the resolution required and the feature size. The exposure of the resist causes it to undergo a chemical reaction. The substrate is then soaked in a developer. In case of positive resist (as depicted in figure 2), the parts of the resist exposed to the beam and chemically altered are dissolved by the developer. For negative resist, the parts not exposed to the beam are dissolved away. The substrate is now left with resist everywhere except for places where the pattern should be. The required material is now evaporated on the substrate. This material fills the blanks left by the dissolved resist and also covers up the remaining resist. The substrate is then taken for the 'lift-off' process where it is dipped in a solvent that dissolves away the remaining resist. The material covering the resist is stripped away with the resist, and only the material in the pattern remains on the substrate. Lithography has the advantage of being an extremely flexible technique that can be used to create a large variety of shapes and sizes of nanostructures. It is also a technique most likely to become scalable to uses in chip design and manufacture in the future. The disadvantage is that lithography, especially EBL tends to be very expensive and the resulting nanostructures are most often granular (this depends on the evaporation conditions). A more in-depth description of lithography can be found in (Madou, 2002).

The formation of nanostructures in the active template-based synthesis results from preferential electrodeposition of atoms or molecules at hole and defect sites on the electrode surface. Once the initial layer of atoms is crystallized on the defect site, subsequent atoms also preferentially deposit at the site leading to formation of different nanostructures. The preparation of the electrode surface plays a major role in determining the morphology and structure of the electrodeposited nanostructure. Other nanosynthesis techniques like photolithography have been used in conjunction with this technique. Photolithography is used to create nanostructures on the electrode surface into which the required material is

Lithography translates to 'stone writing' and originally referred to the art of writing text on stone. Modern lithographic techniques find a wide variety of uses in the printing process. Our foci of interest for this section are, however, micro-lithographic processes like photolithography and electron beam lithography (EBL) with which most top down nanofabrication processes start. Photo-lithography is the more popular of the two processes and is used for manufacturing of semiconductor microchips. EBL, although more expensive, offers significantly improved resolution, sometimes going down to a few nanometers.

The substrate (most often doped silicon coated with an insulating layer of silicon oxide or silicon nitride) is spin coated with a photo resist or e-beam resist. The choice of resist depends on the resolution required, spacing of features and thickness of metal to be evaporated post lithography. The resist is baked for hardening. In case of photo-lithography a hard mask is used with the required pattern made on it. In case of EBL, a 'soft' mask drawn using computer software like L-Edit is used. The areas within the pattern are exposed to either UV light for photolithography or an electron beam for EBL. The exposure time and beam-current used depend on the resolution required and the feature size. The exposure of the resist causes it to undergo a chemical reaction. The substrate is then soaked in a developer. In case of positive resist (as depicted in figure 2), the parts of the resist exposed to the beam and chemically altered are dissolved by the developer. For negative resist, the parts not exposed to the beam are dissolved away. The substrate is now left with resist everywhere except for places where the pattern should be. The required material is now evaporated on the substrate. This material fills the blanks left by the dissolved resist and also covers up the remaining resist. The substrate is then taken for the 'lift-off' process where it is dipped in a solvent that dissolves away the remaining resist. The material covering the resist is stripped away with the resist, and only the material in the pattern remains on the substrate. Lithography has the advantage of being an extremely flexible technique that can be used to create a large variety of shapes and sizes of nanostructures. It is also a technique most likely to become scalable to uses in chip design and manufacture in the future. The disadvantage is that lithography, especially EBL tends to be very expensive and the resulting nanostructures are most often granular (this depends on the evaporation conditions). A more in-depth description of lithography can

then electrodeposited (Yang et al., 2008).

Figure 2 shows a schematic of the lithography process.

**2.2 Lithography** 

be found in (Madou, 2002).

Fig. 2. A schematic of the lithography process for making nanostructures (a) a layer of resist is spin-coated on the substrate, (b) the resist is exposed to either UV radiation in case of photolithography or an electron beam in case of electron-beam lithography through a mask which has the required pattern cut out from it, (c) the substrate with the resist is soaked in a developer which removes the parts of the resist that were exposed to radiation earlier (in case of pisitive resist) or parts that were not exposed to radiation earlier (in case of negative resist), (d) material is evaporated onto the substrate with the resist acting as a mask, (e) the substrate is soaked in a solvent that dissolves the remaining resist and also the layer of evaporated material on the resist. This 'lift-off' process only leaves behind the material evaporated directly on the substrate in step (d).

## **2.3 Chemical vapor deposition**

The technique of Chemical Vapor Deposition (CVD) is mostly used for the synthesis of thin films. The process is very versatile and allows for growth of amorphous, epitaxial, monocrystalline and polycrystalline films. The CVD equipment consists of a tube-like, chamber in which the substrate for synthesis is placed, a mechanism to heat the substrate, a gas handling system which is used to introduce vapors into the chamber and an exhaust from which the waste vapors escape.

For the synthesis, a precursor dissolved in a solvent or in the form of an emulsion is vaporized and introduced into the chamber where some of the precursor mixture deposits on the substrate. The substrate is then heated causing the precursor mixture to decompose and the solvent to evaporate away. A layer of the required precursor is thus left on the substrate. If a metal organic compound is used as the precursor, the method is referred to as Metal-Organic Chemical Vapor Deposition (MOCVD). A slight variation of this

Superconductivity in Nanoscale Systems 111

In addition to these commonly used techniques, several other ingenious methods have been used for nanostructure synthesis. These include DNA (Stoltenberg & Woolley, 2004) and carbon nanotube template nanowire synthesis (Bezryadin et al., 2000) and microbial

When a material transits from a normal to a superconducting state, electrons with opposite spins pair up. As mentioned in the introduction, this pair is called a 'Cooper pair'. Once part of a Cooper pair, the two electrons can be described by a single quantum mechanical wave function߰ ൌ ȁ߰ȁ݁ఝ. The spatial extent of this wavefunction, or the 'size' of a cooper pair, is known as the coherence length (ξ) and φ is the phase of the wavefunction. If the superconducting state is made possible by electrons 'holding hands' to form Cooper pairs, what happens to the superconductivity when the diameter of the nanowire the Cooper pairs are supposed to go through is smaller than ξ? Do the Cooper pairs break up and is superconductivity destroyed in these quasi-1D systems? This dilemma lies at the root of why superconductivity in quasi-1D systems is different. It is also responsible for the plethora of novel phenomena seen on studying superconductivity in quasi-1D wires as

In the superconducting state, the phase φ is spatially coherent. This means that if the phase at any one point is known, the phase at any other point can be predicted. Fluctuations (thermal and quantum) lead to loss of phase coherence in superconducting samples from time to time. The region where the coherence is lost becomes temporarily normal (φ becomes ill defined). The spatial extent of such a region is given by ξ. For 3D or 2D samples, these normal regions without phase coherence do not affect the transport measurement as the charge carriers can bypass them. In a quasi-1D sample however, since the diameter (d) < ξ, the normal region encompasses the entire cross section of the wire and cannot be bypassed. Therefore, phase slips result in a loss of superconductivity in quasi-1D systems.

**3. Novel phenomena – What they imply and their potential applications** 

Fig. 4. A molecular beam epitaxy chamber.

described below.

nanowires (Reguera et al., 2005) amongst others.

**3.1 Phase slips – Thermally activated and quantum**

Mathematically, this can be understood using Josephson relation:

technique is also used for nanowire and nanotube growth under the name of Vapour Liquid Solid method (VLS). This method was developed in 1964 for the growth of Silicon whiskers (Wagner & Ellis, 1964). In the first step, a layer of a solid impurity such as gold is thermally evaporated on the substrate. The substrate is then heated so that liquid droplets of an alloy of the impurity and the substrate form on the surface of the substrate. In the next step, like the CVD method, a vaporized solution of the desired material is introduced into the chamber. The liquid droplets formed on the substrate in the earlier step act as catalysts in the reaction to deposit atoms of the required material on the substrate. By a continuation of this process, a nanowire on the substrate is formed. A schematic of this process is shown in figure 3. This technique has recently been used for successful synthesis of topological insulator nanoribbons (Peng et al., 2010). Using lithography in conjunction with this technique also allows for ordering the nanowire growth. Recently a variation of the CVD method has been used to grow very large aspect ratio (length:diameter) nanowires in microstructured optical fibers (Sazio et al., 2006). The advantage of this method is its adaptability to a wide range of materials. The disadvantages mostly involve safety and contamination as poisonous and pyrophoric (ignite in contact with air) precursors are used.

Fig. 3. A schematic of the chemical vapor deposition process modified to the vapor-liquidsolid method for nanowire growth.

#### **2.4 Molecular beam epitaxy**

Molecular Beam Epitaxy (MBE) is a refined form of thermal evaporation. In MBE, directed neutral beams of atoms or molecules of a substance being evaporated get deposited on a substrate. The major difference from thermal evaporation is that the deposition is extremely slow, allowing for atomic layer by atomic layer epitaxial growth. The samples grown by MBE are high quality single crystals. The very slow growth rate requires the evaporation to be done in ultra-high vacuum to keep the contamination levels in check. The apparatus to fulfill these requirements is considerably complicated (schematics of a typical MBE chamber are shown in figure 4). In addition, the substrates for sample growth also have to be of very high quality. This technique generates extremely smooth and thin samples with a high degree of compositional control and crystal perfection. The payoff is its high cost. The method is largely used in the synthesis of thin films but, can also have applications to nanowire growth (section 3.6). Further details of this method of sample synthesis can be found in (Arthur, 2002).

technique is also used for nanowire and nanotube growth under the name of Vapour Liquid Solid method (VLS). This method was developed in 1964 for the growth of Silicon whiskers (Wagner & Ellis, 1964). In the first step, a layer of a solid impurity such as gold is thermally evaporated on the substrate. The substrate is then heated so that liquid droplets of an alloy of the impurity and the substrate form on the surface of the substrate. In the next step, like the CVD method, a vaporized solution of the desired material is introduced into the chamber. The liquid droplets formed on the substrate in the earlier step act as catalysts in the reaction to deposit atoms of the required material on the substrate. By a continuation of this process, a nanowire on the substrate is formed. A schematic of this process is shown in figure 3. This technique has recently been used for successful synthesis of topological insulator nanoribbons (Peng et al., 2010). Using lithography in conjunction with this technique also allows for ordering the nanowire growth. Recently a variation of the CVD method has been used to grow very large aspect ratio (length:diameter) nanowires in microstructured optical fibers (Sazio et al., 2006). The advantage of this method is its adaptability to a wide range of materials. The disadvantages mostly involve safety and contamination as poisonous and pyrophoric

Fig. 3. A schematic of the chemical vapor deposition process modified to the vapor-liquid-

Molecular Beam Epitaxy (MBE) is a refined form of thermal evaporation. In MBE, directed neutral beams of atoms or molecules of a substance being evaporated get deposited on a substrate. The major difference from thermal evaporation is that the deposition is extremely slow, allowing for atomic layer by atomic layer epitaxial growth. The samples grown by MBE are high quality single crystals. The very slow growth rate requires the evaporation to be done in ultra-high vacuum to keep the contamination levels in check. The apparatus to fulfill these requirements is considerably complicated (schematics of a typical MBE chamber are shown in figure 4). In addition, the substrates for sample growth also have to be of very high quality. This technique generates extremely smooth and thin samples with a high degree of compositional control and crystal perfection. The payoff is its high cost. The method is largely used in the synthesis of thin films but, can also have applications to nanowire growth (section 3.6). Further details of this method of sample synthesis can be

(ignite in contact with air) precursors are used.

solid method for nanowire growth.

**2.4 Molecular beam epitaxy**

found in (Arthur, 2002).

Fig. 4. A molecular beam epitaxy chamber.

In addition to these commonly used techniques, several other ingenious methods have been used for nanostructure synthesis. These include DNA (Stoltenberg & Woolley, 2004) and carbon nanotube template nanowire synthesis (Bezryadin et al., 2000) and microbial nanowires (Reguera et al., 2005) amongst others.

## **3. Novel phenomena – What they imply and their potential applications**

When a material transits from a normal to a superconducting state, electrons with opposite spins pair up. As mentioned in the introduction, this pair is called a 'Cooper pair'. Once part of a Cooper pair, the two electrons can be described by a single quantum mechanical wave function߰ ൌ ȁ߰ȁ݁ఝ. The spatial extent of this wavefunction, or the 'size' of a cooper pair, is known as the coherence length (ξ) and φ is the phase of the wavefunction. If the superconducting state is made possible by electrons 'holding hands' to form Cooper pairs, what happens to the superconductivity when the diameter of the nanowire the Cooper pairs are supposed to go through is smaller than ξ? Do the Cooper pairs break up and is superconductivity destroyed in these quasi-1D systems? This dilemma lies at the root of why superconductivity in quasi-1D systems is different. It is also responsible for the plethora of novel phenomena seen on studying superconductivity in quasi-1D wires as described below.

#### **3.1 Phase slips – Thermally activated and quantum**

In the superconducting state, the phase φ is spatially coherent. This means that if the phase at any one point is known, the phase at any other point can be predicted. Fluctuations (thermal and quantum) lead to loss of phase coherence in superconducting samples from time to time. The region where the coherence is lost becomes temporarily normal (φ becomes ill defined). The spatial extent of such a region is given by ξ. For 3D or 2D samples, these normal regions without phase coherence do not affect the transport measurement as the charge carriers can bypass them. In a quasi-1D sample however, since the diameter (d) < ξ, the normal region encompasses the entire cross section of the wire and cannot be bypassed. Therefore, phase slips result in a loss of superconductivity in quasi-1D systems. Mathematically, this can be understood using Josephson relation:

Superconductivity in Nanoscale Systems 113

where ΔF is the free energy barrier between the two states and the proportionality constant is related to the attempt frequency and the temperature. For low excitation currents

coherence length. As the temperature decreases, the size of the free energy barrier relative to kBT increases, leading to a rapid reduction in the thermal-activation rate. At very low temperature this rate becomes negligible. TAPS is therefore, only important close to Tc, and is associated with the rounding of the superconducting transition seen in the SnNW. However, it cannot explain the resistive tail that persists to low temperatures. To explain the resistive tail, some other process by which the phase can relax has to be taken into account. QPS, or the tunneling through the energy barrier (figure 6) is the dominant process at low temperatures (Giordano, 1988, 1991; Giordano & Schuler 1990). A heuristic argument (Giordano, 1994) suggests that the resistance given by the QPS will follow the same form as

���

�� � where σ is the cross sectional area, Hc is the critical field and ξ is the

��� � (2)

��� � instead of kBT. Here, τGL is the

� (3)

������ � �� � . Therefore,

����� � �

���� � �

�������

where is possible numerical factor. QPS has fascinated experimentalists and theorists alike for almost 2 decades now because of its tremendous implications (Altomare et al., 2006; Lau et al., 2001). The observed resistance tail is an example of macroscopic quantum tunneling because a macroscopic, measurable quantity (resistance) is changing in response to a quantum tunneling event. It has been suggested that QPS in nanowires can be used to realize a superconducting quantum bit that can have much lower 1/f noise than existing quantum bits using Josephson junctions (Mooij & Harmans, 2005). These quantum bits not using Josephson junctions can form a new basis for superconducting quantum computing.

Fig. 6. A representation of the (a) the thermally activated phase slip wherer thermal energy allows the system to go over the energy barrier separating two superconducting states, (b) quantum phase slips that allow the system to tunnel from one superconducting state to the other even at low temperatures. The two superconducting states in question have the phase difference between the two ends of the superconductor () different by 2.

ℏ

�� = √2���

��

RTAPS except for the energy scale being given by ℏ� = ℏ

Ginzburg Landau relaxation time given by ��� = �ℏ

$$V = \frac{d(\Delta \phi)}{dt} \tag{1}$$

Here, V is the voltage across the system, t is the time, φ is the phase, and Δφ is the phase difference across the system. In a phase slip event, the phase at some point in the sample becomes ill defined. When it recovers, the value of Δφ is changed by 2π causing a finite voltage V to appear across the system and hence, a loss in superconductivity. So, in a phase slip event, the system essentially goes from a superconducting state with some value of Δφ to a superconducting state with a value of Δφ different from the previous value by 2π. To go from one state to another, the system needs to cross a potential barrier (figure 6). The energy to cross the barrier can be provided by thermal energy (thermally activated phase slips) or by quantum fluctuations (quantum phase slips). Extensive experiments attempting to see both kinds of phase slips have been done.

Fig. 5. (Tian et al., 2003) Resistance vs. Temperature measurements on tin nanowire arrays embedded in track-etched polycarbonate membranes and squeezed between bulk tin electrodes. The schematics of the measurements are shown in the inset. The measurements shown here are performed on three different arrays which the wires are 40 nm, 70 nm and 100 nm in diameter. For the 40 nm diameter tin nanowires rounding of the superconducting transitions and a long resistance tail can be seen.

Transport measurements of resistance (R) as a function of temperature (T) of quasi-1D samples differ from bulk samples in two ways. The first is a rounding of the superconducting transition and the second is the presence of a superconducting tail in the quasi-1D samples. These two features can be seen to appear in figure 5. The figure shows R vs. T measurements performed on single crystalline tin nanowire (SnNW) arrays embedded in polycarbonate membranes contacted with bulk tin electrodes (Tian et al., 2003). A schematic of the measurement is shown in the inset of figure 5. Measurements are performed on 3 different sets of nanowires of diameters 100 nm, 70 nm and 40 nm. The transition for the 100 nm and 70 nm diameter nanowires is sharp and the resistance falls from the normal state value to zero in a temperature interval of 90 mK. The transition for the 40 nm nanowire on the other hand, is rounded and also has a long resistive tail stretching from 1.8 K to 3.7 K, a range of 1.9 K. These two features are attributed to thermally activated (TAPS) and quantum phase slips (QPS) respectively.

In TAPS, the system crosses the energy barrier between the superconducting states whose values of Δφ differ by 2π with the help of thermodynamic fluctuations (figure 6). The resistance due to TAPS below the critical temperature of the superconducting transition (Tc) can be written as (Langer & Ambegaokar, 1967; McCumber & Halperin, 1970):

ܸ ൌ ௗሺοఝሻ

Here, V is the voltage across the system, t is the time, φ is the phase, and Δφ is the phase difference across the system. In a phase slip event, the phase at some point in the sample becomes ill defined. When it recovers, the value of Δφ is changed by 2π causing a finite voltage V to appear across the system and hence, a loss in superconductivity. So, in a phase slip event, the system essentially goes from a superconducting state with some value of Δφ to a superconducting state with a value of Δφ different from the previous value by 2π. To go from one state to another, the system needs to cross a potential barrier (figure 6). The energy to cross the barrier can be provided by thermal energy (thermally activated phase slips) or by quantum fluctuations (quantum phase slips). Extensive experiments attempting

Fig. 5. (Tian et al., 2003) Resistance vs. Temperature measurements on tin nanowire arrays embedded in track-etched polycarbonate membranes and squeezed between bulk tin electrodes. The schematics of the measurements are shown in the inset. The measurements shown here are performed on three different arrays which the wires are 40 nm, 70 nm and 100 nm in diameter. For the 40 nm diameter tin nanowires rounding of the superconducting

Transport measurements of resistance (R) as a function of temperature (T) of quasi-1D samples differ from bulk samples in two ways. The first is a rounding of the superconducting transition and the second is the presence of a superconducting tail in the quasi-1D samples. These two features can be seen to appear in figure 5. The figure shows R vs. T measurements performed on single crystalline tin nanowire (SnNW) arrays embedded in polycarbonate membranes contacted with bulk tin electrodes (Tian et al., 2003). A schematic of the measurement is shown in the inset of figure 5. Measurements are performed on 3 different sets of nanowires of diameters 100 nm, 70 nm and 40 nm. The transition for the 100 nm and 70 nm diameter nanowires is sharp and the resistance falls from the normal state value to zero in a temperature interval of 90 mK. The transition for the 40 nm nanowire on the other hand, is rounded and also has a long resistive tail stretching from 1.8 K to 3.7 K, a range of 1.9 K. These two features are attributed to thermally activated (TAPS) and quantum phase slips (QPS) respectively.

In TAPS, the system crosses the energy barrier between the superconducting states whose values of Δφ differ by 2π with the help of thermodynamic fluctuations (figure 6). The resistance due to TAPS below the critical temperature of the superconducting transition (Tc)

can be written as (Langer & Ambegaokar, 1967; McCumber & Halperin, 1970):

to see both kinds of phase slips have been done.

transitions and a long resistance tail can be seen.

ௗ௧ (1)

$$R\_{TAPS} \propto e^{-\Delta F/\_{k\_B T}}\tag{2}$$

where ΔF is the free energy barrier between the two states and the proportionality constant is related to the attempt frequency and the temperature. For low excitation currents �� = √2��� �� �� � where σ is the cross sectional area, Hc is the critical field and ξ is the coherence length. As the temperature decreases, the size of the free energy barrier relative to kBT increases, leading to a rapid reduction in the thermal-activation rate. At very low temperature this rate becomes negligible. TAPS is therefore, only important close to Tc, and is associated with the rounding of the superconducting transition seen in the SnNW. However, it cannot explain the resistive tail that persists to low temperatures. To explain the resistive tail, some other process by which the phase can relax has to be taken into account. QPS, or the tunneling through the energy barrier (figure 6) is the dominant process at low temperatures (Giordano, 1988, 1991; Giordano & Schuler 1990). A heuristic argument (Giordano, 1994) suggests that the resistance given by the QPS will follow the same form as RTAPS except for the energy scale being given by ℏ� = ℏ ��� � instead of kBT. Here, τGL is the Ginzburg Landau relaxation time given by ��� = �ℏ ������ � �� � . Therefore,

$$R\_{QPS} \propto e^{-a\Delta F \tau\_{GL}}/\_{\hbar} \tag{3}$$

where is possible numerical factor. QPS has fascinated experimentalists and theorists alike for almost 2 decades now because of its tremendous implications (Altomare et al., 2006; Lau et al., 2001). The observed resistance tail is an example of macroscopic quantum tunneling because a macroscopic, measurable quantity (resistance) is changing in response to a quantum tunneling event. It has been suggested that QPS in nanowires can be used to realize a superconducting quantum bit that can have much lower 1/f noise than existing quantum bits using Josephson junctions (Mooij & Harmans, 2005). These quantum bits not using Josephson junctions can form a new basis for superconducting quantum computing.

Fig. 6. A representation of the (a) the thermally activated phase slip wherer thermal energy allows the system to go over the energy barrier separating two superconducting states, (b) quantum phase slips that allow the system to tunnel from one superconducting state to the other even at low temperatures. The two superconducting states in question have the phase difference between the two ends of the superconductor () different by 2.

Superconductivity in Nanoscale Systems 115

The occurrence of this long-range proximity effect was also reported in ordinary, hard ferromagnetic crystalline nanowires in standard 4-probe measurements (Wang et al., 2010a). The Co and Ni nanowires (CoNW and NiNW) were synthesized using template based electrodeposition into the pores of a track-etched polycarbonate membrane. The CoNW were single crystalline while the NiNW were polycrystalline. The nanowires were released from the membrane and dispersed on a Si/SiN substrate and contacted by a focused ion beam (FIB) system with superconducting tungsten (W) electrodes. The R vs. T measurements at different applied fields (H) on a 40 nm diameter, 1.5 μm long CoNW are shown in figure 7 (the inset shows the measurement geometry). A large resistance drop (to 11% of the normal state value) at the Tc of the W electrode suggests a very long range (~ 500 nm) proximity effect in the CoNW. The range of the proximity effect is smaller (~ 375 nm) in thicker (80 nm) diameter nanowires. The expected singlet coherence length in Co nanowires

and D = (1/3)vF *l* is the electron diffusion constant (here, vF is the Co Fermi velocity ∼108 cm s−1 and *l* is the mean freepath). The experimentally measured resistivities yield a singlet coherence length of 3.3 nm for the sample shown in figure 7, which is much smaller than the observed coherence length of 500 nm. The superconductivity in this system also may be triplet in nature. It has been suggested that the region between the W electrodes and the CoNW is damaged by the FIB process and may thus be largely inhomogeneous and act as a seeding ground for triplet pairing. Another explanation may be that the surface of the CoNW and NiNW is not ferromagnetic and the superconducting proximity effect is only on the non-ferromagnetic surface of the nanowires. In (Konschelle et al., 2010) it is suggested that a phase sensitive measurement, experimentally testing the Josephson current-phase

Other than the extremely long-range proximity effect, another striking feature visible in figure 7 is the very large resistance peak (25% of the normal-state resistance) found between 5 K and 4.5 K, just above the temperature of the superconducting resistance drop. This 'critical peak' is preceded by a small dip on the high-temperature side of the peak. Measurements in both warming up and cooling down scans show an absence of hysteretic behavior. Both the peak and the superconducting drop are stable on cycling the sample to room temperature. This peak does not conform to charge imbalance peaks typically seen in superconducting mesoscopic systems since a very large magnetic field is required to destroy it. Charge imbalance peaks on the other hand are destroyed by very small applied fields (Santhanam et al., 1991). Another mechanism used to explain resistance peaks in Fe-In junctions is spin accumulation (Chiang et al., 2007). In that case, however, the absolute peak value (∆R) was many orders of magnitude smaller (10-8Ω), as was the relative change from the normal-state resistance (∆R/R ~ 0.05%). Furthermore, spin accumulation would assume that the induced superconductivity is singlet in nature, which seems inconsistent with the long range proximity effect. A satisfactory explanation of this peak is yet to be found.

The long range proximity effect in ferromagnets offers the possibility of combination of the zero-resistance supercurrents of superconductors with the spin alignment of ferromagnets. This has tremendous implications for spintronics (Eschrig, 2011). Superconducting spintronic devices offer elements of non-locality (if the superconducting coherence length is larger than the nanoscale device), coherence (electrons in a cooper pair are coherently

relationship can be used to discriminate between the two scenarios.

�������� � (Keizer et al., 2006), where TCurie is the Curie temperature

is given by �� = ���

#### **3.2 Long range proximity effect in ferromagnetic nanowires**

In a singlet (s-wave) superconductor, the spins of the electrons making a Cooper pair are antiparallel (the spin part of the wavefunction is given by |↑↓� � | ↓↑�). The spins of electrons in a ferromagnet, on the other hand, are aligned parallel to each other. This incompatible spin order between singlet superconductors and ferromagnets makes for new and interesting physics when the two are placed in contact with each other. When a superconductor is placed in contact with a normal metal, the Cooper pairs in the superconductor tunnel into the normal metal, making it partially superconducting. This is known as the superconducting proximity effect. The distance to which Cooper pairs survive in the non-superconducting material is the range of the proximity effect. This range, for a bulk normal metal can be ~ 1μm. For a bulk ferromagnet however, the incompatible spin order ensures that the singlet Cooper pair cannot survive beyond a few nanometers. This expectation has been confirmed in macroscopic (Fe, Ni)–In junctions (Chiang et al., 2007) and submicrometre Ni–Al structures (Aumentado & Chandrasekhar, 2001) where the spatial range of the proximity effect is limited to ∼ 1 nm. In mesosopic ferromagnet-superconductor hybrid structures, a surprisingly long range proximity effect has been found (Bergeret et al., 2005; Giroud et al., 1998; Keizer et al., 2006; Pena, V. et al., 2004; Petrashov et al., 1999; Sosnin et al., 2006). In particular, a recent experiment detected super-current in half-metallic ferromagnet CrO2 thin film sandwiched between two singlet superconducting electrodes separated by 1 µm (Keizer et al., 2006). To account for the long range proximity in these systems, the superconductivity was attributed to the spin triplet (spin wavefunction given by one of |↑↑�, |↓↓� �� |↑↓� � | ↓↑� which do not preclude parallel spins) rather than the spin singlet state. The formation of the spin triplet state requires the presence of a region of noncollinear magnetization. This region, in the case of an experiment on a Co thin film was provided by a thin weakly superconducting CuNi or PdNi layer between the superconducting Nb and the ferromagnetic Co film (Khaire et al., 2010).

Fig. 7. (Wang et al., 2010a) Resistance vs. Temperature measurements for a 40 nm diameter, 1.5 μm long cobalt nanowire at different applied fields. The applied fields are perpendicular to the wire. The inset shows an scanning electron micrograph of the cobalt wire contacted with 4 focused ion beam deposited superconducting tungsten electrodes.

In a singlet (s-wave) superconductor, the spins of the electrons making a Cooper pair are antiparallel (the spin part of the wavefunction is given by |↑↓� � | ↓↑�). The spins of electrons in a ferromagnet, on the other hand, are aligned parallel to each other. This incompatible spin order between singlet superconductors and ferromagnets makes for new and interesting physics when the two are placed in contact with each other. When a superconductor is placed in contact with a normal metal, the Cooper pairs in the superconductor tunnel into the normal metal, making it partially superconducting. This is known as the superconducting proximity effect. The distance to which Cooper pairs survive in the non-superconducting material is the range of the proximity effect. This range, for a bulk normal metal can be ~ 1μm. For a bulk ferromagnet however, the incompatible spin order ensures that the singlet Cooper pair cannot survive beyond a few nanometers. This expectation has been confirmed in macroscopic (Fe, Ni)–In junctions (Chiang et al., 2007) and submicrometre Ni–Al structures (Aumentado & Chandrasekhar, 2001) where the spatial range of the proximity effect is limited to ∼ 1 nm. In mesosopic ferromagnet-superconductor hybrid structures, a surprisingly long range proximity effect has been found (Bergeret et al., 2005; Giroud et al., 1998; Keizer et al., 2006; Pena, V. et al., 2004; Petrashov et al., 1999; Sosnin et al., 2006). In particular, a recent experiment detected super-current in half-metallic ferromagnet CrO2 thin film sandwiched between two singlet superconducting electrodes separated by 1 µm (Keizer et al., 2006). To account for the long range proximity in these systems, the superconductivity was attributed to the spin triplet (spin wavefunction given by one of |↑↑�, |↓↓� �� |↑↓� � | ↓↑� which do not preclude parallel spins) rather than the spin singlet state. The formation of the spin triplet state requires the presence of a region of noncollinear magnetization. This region, in the case of an experiment on a Co thin film was provided by a thin weakly superconducting CuNi or PdNi layer between the

**3.2 Long range proximity effect in ferromagnetic nanowires** 

superconducting Nb and the ferromagnetic Co film (Khaire et al., 2010).

Fig. 7. (Wang et al., 2010a) Resistance vs. Temperature measurements for a 40 nm diameter, 1.5 μm long cobalt nanowire at different applied fields. The applied fields are perpendicular to the wire. The inset shows an scanning electron micrograph of the cobalt wire contacted

with 4 focused ion beam deposited superconducting tungsten electrodes.

The occurrence of this long-range proximity effect was also reported in ordinary, hard ferromagnetic crystalline nanowires in standard 4-probe measurements (Wang et al., 2010a). The Co and Ni nanowires (CoNW and NiNW) were synthesized using template based electrodeposition into the pores of a track-etched polycarbonate membrane. The CoNW were single crystalline while the NiNW were polycrystalline. The nanowires were released from the membrane and dispersed on a Si/SiN substrate and contacted by a focused ion beam (FIB) system with superconducting tungsten (W) electrodes. The R vs. T measurements at different applied fields (H) on a 40 nm diameter, 1.5 μm long CoNW are shown in figure 7 (the inset shows the measurement geometry). A large resistance drop (to 11% of the normal state value) at the Tc of the W electrode suggests a very long range (~ 500 nm) proximity effect in the CoNW. The range of the proximity effect is smaller (~ 375 nm) in thicker (80 nm) diameter nanowires. The expected singlet coherence length in Co nanowires is given by �� = ��� �������� � (Keizer et al., 2006), where TCurie is the Curie temperature and D = (1/3)vF *l* is the electron diffusion constant (here, vF is the Co Fermi velocity ∼108 cm s−1 and *l* is the mean freepath). The experimentally measured resistivities yield a singlet coherence length of 3.3 nm for the sample shown in figure 7, which is much smaller than the observed coherence length of 500 nm. The superconductivity in this system also may be triplet in nature. It has been suggested that the region between the W electrodes and the CoNW is damaged by the FIB process and may thus be largely inhomogeneous and act as a seeding ground for triplet pairing. Another explanation may be that the surface of the

CoNW and NiNW is not ferromagnetic and the superconducting proximity effect is only on the non-ferromagnetic surface of the nanowires. In (Konschelle et al., 2010) it is suggested that a phase sensitive measurement, experimentally testing the Josephson current-phase relationship can be used to discriminate between the two scenarios.

Other than the extremely long-range proximity effect, another striking feature visible in figure 7 is the very large resistance peak (25% of the normal-state resistance) found between 5 K and 4.5 K, just above the temperature of the superconducting resistance drop. This 'critical peak' is preceded by a small dip on the high-temperature side of the peak. Measurements in both warming up and cooling down scans show an absence of hysteretic behavior. Both the peak and the superconducting drop are stable on cycling the sample to room temperature. This peak does not conform to charge imbalance peaks typically seen in superconducting mesoscopic systems since a very large magnetic field is required to destroy it. Charge imbalance peaks on the other hand are destroyed by very small applied fields (Santhanam et al., 1991). Another mechanism used to explain resistance peaks in Fe-In junctions is spin accumulation (Chiang et al., 2007). In that case, however, the absolute peak value (∆R) was many orders of magnitude smaller (10-8Ω), as was the relative change from the normal-state resistance (∆R/R ~ 0.05%). Furthermore, spin accumulation would assume that the induced superconductivity is singlet in nature, which seems inconsistent with the long range proximity effect. A satisfactory explanation of this peak is yet to be found.

The long range proximity effect in ferromagnets offers the possibility of combination of the zero-resistance supercurrents of superconductors with the spin alignment of ferromagnets. This has tremendous implications for spintronics (Eschrig, 2011). Superconducting spintronic devices offer elements of non-locality (if the superconducting coherence length is larger than the nanoscale device), coherence (electrons in a cooper pair are coherently

Superconductivity in Nanoscale Systems 117

served to definitively decouple the APE seen in the 70 nm wire from negative magnetoresistance caused by a weak applied magnetic field seen in other experiments (Herzog et al., 1998; Rogachev et al., 2006; Xiong et al., 1997). The phenomenon also shows a strong dependence on the diameter and length of the sample and the nature of the measuring electrodes. It is not seen in 200 nm AlNW and is very weak in 50 µm long AlNW.

It is also weaker in ZnNW with Pb electrodes compared to ZnNW with In electrodes.

Fig. 8. (Tian et al., 2005) An ehancement in the critical current (Ic) of a zinc nanowire array squeezed between superconducting lead electrodes is seen when the magnetic field is increased toward the critical field of the bulk superconducting lead electrodes contacting the

A theoretical model applicable to this system was given by Fu et al. (Fu et al., 2006) generalizing the resistively shunted Josephson junction to include superconducting nanowires. In this model the nanowire array connected to the two bulk superconducting electrodes is modeled as a nanowire in series with two resistors and in parallel with a capacitor. When the electrodes are normal, the resistors in series provide a medium for dissipation of quantum phase slips, thus stabilizing the superconductivity. When the electrodes are superconducting, this dissipation path disappears and the wire becomes normal. A similar explanation was proposed to understand the results of the experiments on granular Zn nanowires. Although the model agrees with the results qualitatively, it does not explain how the strength of the effect depends on the contacting electrode in AlNWs or ZnNWs. In addition to the material of the electrodes, other important parameters in these experiments are the 'characteristic' diameter and the length of the nanowires that define the presence or absence of APE. It is difficult to obtain quantitative predictions for these quantities from this model. A more quantitative mechanism using the time independent Ginzburg-Landau equations was proposed by Vodolazov (Vodolazov, 2007)*.* The qualitative behavior of the system is reproduced at temperatures close to Tc and the authors expect traces of this behavior to exist at low temperatures also. The mechanism of Vodolazov uses the fact that the diameters of the wires are smaller than the superconducting coherence length to model them as a 1D system. The coherence length for the AlNWs in these experiments can be estimated using the value of ξ0.*l* (where *l* is the mean free path) as 4×10-16 Ω m2, the dirty limit coherence length ξ(0) = 0.855(ξ0*l*)0.5 and 1600 nm as the value for ξ0. ξ(0) is estimated to be ~ 50 nm. In comparison, APE was seen 70 and 80 nm diameter AlNW but, not in 200 nm diameter nanowires. In ZnNWs, the ξ(0) was estimated to be ~ 150 nm and the APE was seen in 40 nm diameter nanowires but

Pb of the electrodes.

nanowires. The critical current peaks at the critical field Hc

coupled even if they go in separate electrodes) and entanglement (stemming from coherence), all of which are essential for quantum computing. The availability of triplet supercurrents may also revolutionize the field of spintronics with new spin valves, pumps, switches, transistors and filters.

#### **3.3 Anti-proximity effect**

When a superconductor is placed in contact with a normal metal, signs of superconductivity appear in the normal metal. This 'proximity effect' is a much documented and well-studied phenomenon. However, a number of recent experiments have reported an unexpected 'antiproximity effect' (APE) in zinc nanowires (ZnNW) contacted with bulk superconducting electrodes (Chen et al., 2009, 2011; Singh et al., 2011; Tian et al., 2005, 2006). In the original experiment, (Tian et al., 2005) ZnNW 6 µm in length embedded in track-etched polycarbonate membranes were squeezed between superconducting bulk electrodes for two-electrode transport measurements (the schematics of the measurement are similar to those shown in the inset of figure 5). In this configuration, the resistance of the long nanowires dominates the overall series resistance, and the contact resistance is minimized as in a 4-electrode measurement. In 70 nm diameter nanowires contacted with bulk tin (Sn) electrodes, the superconducting transitions of Sn at 3.7 K and of the ZnNW at 1 K were both seen, consistent with expectation. However in 40 nm diameter nanowires, the transition for Sn was seen but the superconducting transition for Zn was absent. When the Sn electrodes were driven normal by a magnetic field of 300 Oe, the superconducting transition of Zn reappeared. The APE in the 40 nm Zn wires was replicated with indium (In) and lead (Pb) electrodes (Tian et al., 2006a). The experiment with In electrodes showed that the APE in the 40 nm Zn wires was switched off precisely when the magnetic field was increased above the critical field of In (Tian et al., 2005). With Pb electrodes, the strength of APE was weaker and there was no obvious suppression of superconductivity in the ZnNWs in resistance measurements as a function of temperature or magnetic field. The APE however, could be detected in the critical current (Ic) of the Zn wires. Specifically the Ic of a Pb/Zn/Pb sample showed a dramatic increase when the magnetic field was increased towards the critical field (Hc Pb) of Pb. Only at fields higher than Hc Pb did the Ic of the ZnNWs show the 'normal' behavior, namely a decrease with increasing field (figure 8). Similar Ic behavior vs. magnetic field was also confirmed in aluminum nanowire (AlNW) arrays embedded in anodic aluminum oxide membrane contacted with In electrodes (Singh et al., 2011) and in four electrode measurements on e-beam assisted evaporated granular ZnNWs (Chen et al., 2009, 2011). The nanowires and the electrodes for this experiment were evaporated in a single step using an e-beam lithographically fabricated mask. The authors attributed the increase in Ic to the creation of quasi-particles as the bulk electrodes are driven toward the normal state by the external magnetic field. In all these experiments, a magnetic field was used to see the APE. It is therefore natural to question whether the APE is caused by the magnetic field. To answer this question an experiment was performed where two single crystalline AlNW grown using electrodeposition (Singh et al., 2009) of the same diameter (70 nm) and length (2.5 µm) were contacted with superconducting W and normal platinum (Pt) electrodes using the FIB process. The Ic of the wire contacted with the superconducting electrode was found to be significantly smaller than the Ic of the wire contacted with the normal electrode (figure 9) indicating suppression of superconductivity of the AlNW because of the bulk superconducting electrode. As there were no applied magnetic fields in this experiment, this

coupled even if they go in separate electrodes) and entanglement (stemming from coherence), all of which are essential for quantum computing. The availability of triplet supercurrents may also revolutionize the field of spintronics with new spin valves, pumps,

When a superconductor is placed in contact with a normal metal, signs of superconductivity appear in the normal metal. This 'proximity effect' is a much documented and well-studied phenomenon. However, a number of recent experiments have reported an unexpected 'antiproximity effect' (APE) in zinc nanowires (ZnNW) contacted with bulk superconducting electrodes (Chen et al., 2009, 2011; Singh et al., 2011; Tian et al., 2005, 2006). In the original experiment, (Tian et al., 2005) ZnNW 6 µm in length embedded in track-etched polycarbonate membranes were squeezed between superconducting bulk electrodes for two-electrode transport measurements (the schematics of the measurement are similar to those shown in the inset of figure 5). In this configuration, the resistance of the long nanowires dominates the overall series resistance, and the contact resistance is minimized as in a 4-electrode measurement. In 70 nm diameter nanowires contacted with bulk tin (Sn) electrodes, the superconducting transitions of Sn at 3.7 K and of the ZnNW at 1 K were both seen, consistent with expectation. However in 40 nm diameter nanowires, the transition for Sn was seen but the superconducting transition for Zn was absent. When the Sn electrodes were driven normal by a magnetic field of 300 Oe, the superconducting transition of Zn reappeared. The APE in the 40 nm Zn wires was replicated with indium (In) and lead (Pb) electrodes (Tian et al., 2006a). The experiment with In electrodes showed that the APE in the 40 nm Zn wires was switched off precisely when the magnetic field was increased above the critical field of In (Tian et al., 2005). With Pb electrodes, the strength of APE was weaker and there was no obvious suppression of superconductivity in the ZnNWs in resistance measurements as a function of temperature or magnetic field. The APE however, could be detected in the critical current (Ic) of the Zn wires. Specifically the Ic of a Pb/Zn/Pb sample showed a dramatic increase when the magnetic field was increased towards the critical field

behavior, namely a decrease with increasing field (figure 8). Similar Ic behavior vs. magnetic field was also confirmed in aluminum nanowire (AlNW) arrays embedded in anodic aluminum oxide membrane contacted with In electrodes (Singh et al., 2011) and in four electrode measurements on e-beam assisted evaporated granular ZnNWs (Chen et al., 2009, 2011). The nanowires and the electrodes for this experiment were evaporated in a single step using an e-beam lithographically fabricated mask. The authors attributed the increase in Ic to the creation of quasi-particles as the bulk electrodes are driven toward the normal state by the external magnetic field. In all these experiments, a magnetic field was used to see the APE. It is therefore natural to question whether the APE is caused by the magnetic field. To answer this question an experiment was performed where two single crystalline AlNW grown using electrodeposition (Singh et al., 2009) of the same diameter (70 nm) and length (2.5 µm) were contacted with superconducting W and normal platinum (Pt) electrodes using the FIB process. The Ic of the wire contacted with the superconducting electrode was found to be significantly smaller than the Ic of the wire contacted with the normal electrode (figure 9) indicating suppression of superconductivity of the AlNW because of the bulk superconducting electrode. As there were no applied magnetic fields in this experiment, this

Pb did the Ic of the ZnNWs show the 'normal'

switches, transistors and filters.

**3.3 Anti-proximity effect**

(Hc

Pb) of Pb. Only at fields higher than Hc

served to definitively decouple the APE seen in the 70 nm wire from negative magnetoresistance caused by a weak applied magnetic field seen in other experiments (Herzog et al., 1998; Rogachev et al., 2006; Xiong et al., 1997). The phenomenon also shows a strong dependence on the diameter and length of the sample and the nature of the measuring electrodes. It is not seen in 200 nm AlNW and is very weak in 50 µm long AlNW. It is also weaker in ZnNW with Pb electrodes compared to ZnNW with In electrodes.

Fig. 8. (Tian et al., 2005) An ehancement in the critical current (Ic) of a zinc nanowire array squeezed between superconducting lead electrodes is seen when the magnetic field is increased toward the critical field of the bulk superconducting lead electrodes contacting the nanowires. The critical current peaks at the critical field Hc Pb of the electrodes.

A theoretical model applicable to this system was given by Fu et al. (Fu et al., 2006) generalizing the resistively shunted Josephson junction to include superconducting nanowires. In this model the nanowire array connected to the two bulk superconducting electrodes is modeled as a nanowire in series with two resistors and in parallel with a capacitor. When the electrodes are normal, the resistors in series provide a medium for dissipation of quantum phase slips, thus stabilizing the superconductivity. When the electrodes are superconducting, this dissipation path disappears and the wire becomes normal. A similar explanation was proposed to understand the results of the experiments on granular Zn nanowires. Although the model agrees with the results qualitatively, it does not explain how the strength of the effect depends on the contacting electrode in AlNWs or ZnNWs. In addition to the material of the electrodes, other important parameters in these experiments are the 'characteristic' diameter and the length of the nanowires that define the presence or absence of APE. It is difficult to obtain quantitative predictions for these quantities from this model. A more quantitative mechanism using the time independent Ginzburg-Landau equations was proposed by Vodolazov (Vodolazov, 2007)*.* The qualitative behavior of the system is reproduced at temperatures close to Tc and the authors expect traces of this behavior to exist at low temperatures also. The mechanism of Vodolazov uses the fact that the diameters of the wires are smaller than the superconducting coherence length to model them as a 1D system. The coherence length for the AlNWs in these experiments can be estimated using the value of ξ0.*l* (where *l* is the mean free path) as 4×10-16 Ω m2, the dirty limit coherence length ξ(0) = 0.855(ξ0*l*)0.5 and 1600 nm as the value for ξ0. ξ(0) is estimated to be ~ 50 nm. In comparison, APE was seen 70 and 80 nm diameter AlNW but, not in 200 nm diameter nanowires. In ZnNWs, the ξ(0) was estimated to be ~ 150 nm and the APE was seen in 40 nm diameter nanowires but

Superconductivity in Nanoscale Systems 119

the 1.2 µm long wire occured in two steps at two distinct Tcs (figure 10). In the R vs. H measurements two distinct Hcs were also seen. The two-step drop in resistance is suggestive of two distinct transitions. The higher Tc and Hc are almost same as the corresponding values for the W electrodes. The results are interpreted with proximity induced superconductivity in the nanowire with a small superconducting "mini-gap" δ near the centre of the wire which is different from the gap elsewhere in the wire. The gap near the electrodes in the wire corresponds to the bulk W gap (∆). This "mini-gap" state is a novel

Fig. 10. (Wang et al., 2009a) Resistance vs temperature plots for a 70 nm diameter, 1.2 μm long single crystaline gold nanowire at different magnetic fields applied perpendicular to wire axis. The gold nanowire is contacted with superconducting tungsten electrodes using a focused ion beam system and the measurement shown has been done in the standard fourprobe configuration. The superconducting transition can be seen to occur in two steps at two

The R vs. H measurement of a 1.0 µm long nanowire and 1.2 µm nanowire are shown in figure 11 (a) and 11(b) respectively. A small resistance valley is more apparent in the 1.2 µm long nanowire. This resistance valley is associated with the mini-gap state. The details of the temperature evolution of the mini-gap in the R vs. H measurements can be seen in the magnified graph in figure 11 (c). The Hc of the mini-gap is ~ 3000 Oe in comparison to the larger transition Hc corresponding to the W electrode Hc of ~ 80000 Oe (figure 11(b)) . The curves have been offset for clarity. At 2.4 K, there is no sign of the mini-gap. At 2.5 K, the resistance suddenly jumps and then quickly drops back to zero at 2.5 kOe. At 2.6 K, this magneticfield-symmetric resistance fluctuation becomes more clearly developed and the baseline of the fluctuating resistance at fields above the mini-gap region is seen to be smoothly increasing with field. With increasing temperature the mini-gap valley continues to shrink in width, disappearing at T = 4 K. Evidence for the mini-gap in proximity structures was also revealed in recent scanning tunnelling microscopy studies (Gupta et al., 2004; Le Sueur et al., 2008; Vinet et al., 2001) and another transport measurement (Lucignano et al., 2010). In addition to the mini-gap state, resistance plateaus can be seen in the R vs. H measurements in the transition region (figure 11(a) and

superconducting state not seen in bulk samples.

distinct 'critical' temperatures.

11(b)).

not in 70 nm diameter nanowires (Tian et al., 2005). The model also predicts a weakening or absence of the APE in nanowires with length L > ΛQ (the charge imbalance length). The charge imbalance length of the Zn NWs was calculated to be 22 µm (Vodolazov, 2007) and that of AlNW to be 19 µm (Singh et al., 2011), and this was used to explain the relatively weak APE in long ~ 35 μm Zn and ~ 50 µm AlNWs. Lastly, this mechanism uses an external magnetic field to model the critical current enhancement seen in the APE and the APE has now been seen in the absence of an external magnetic field.

The APE is a counterintuitive phenomenon that challenges our understanding of superconductivity. The phenomenon is qualitatively consistent with the Vodolazov model, but the presence of the phenomenon without an applied field shows that further theoretical work is needed to gain a more complete and quantitative understanding of the APE. A complete picture of the mechanism behind this behavior will offer great insight into superconductivity in nanoscale systems and perhaps open doors to new applications.

Fig. 9. (Singh et al. 2011) Normalizes voltage vs. applied current for two 70 nm diameter, 2.5 μm long single aluminum nanowires at a temperature of 0.1 K. One of the nanowires is measured using normal focused ion beam deposited normal Pt electrodes and the other is measured using superconducting W electrodes. The inset shows a scanning electron micrograph of the aluminum nanowire contacted with the normal platinum electrodes. The nanowire contacted tungsten also looks the same under a scanning electron microscope.

### **3.4 Mini-gap state**

In superconducting nanowires contacted with superconducting electrodes, the counterintuitive APE is seen. What happens when a normal nanowire is contacted with superconducting electrodes? In this situation, in addition to the expected and well-studied phenomenon of the proximity effect, some surprising new phenomena are uncovered. In a recent experiment (Wang et al., 2009a), crystalline gold nanowires (AuNW) 70 nm in diameter were contacted with superconducting W electrodes using FIB. As expected, the superconducting proximity effect caused the short wires (1 µm and 1.2 µm long) to go completely superconducting whereas the longer nanowires (1.9 µm) retained some residual resistivity to the lowest measured temperatures. This gives the range of the proximity effect to be ~ 600 nm. A surprising feature however, was that the superconducting transition of

not in 70 nm diameter nanowires (Tian et al., 2005). The model also predicts a weakening or absence of the APE in nanowires with length L > ΛQ (the charge imbalance length). The charge imbalance length of the Zn NWs was calculated to be 22 µm (Vodolazov, 2007) and that of AlNW to be 19 µm (Singh et al., 2011), and this was used to explain the relatively weak APE in long ~ 35 μm Zn and ~ 50 µm AlNWs. Lastly, this mechanism uses an external magnetic field to model the critical current enhancement seen in the APE and the APE has now been seen in

The APE is a counterintuitive phenomenon that challenges our understanding of superconductivity. The phenomenon is qualitatively consistent with the Vodolazov model, but the presence of the phenomenon without an applied field shows that further theoretical work is needed to gain a more complete and quantitative understanding of the APE. A complete picture of the mechanism behind this behavior will offer great insight into superconductivity in nanoscale systems and perhaps open doors to new

Fig. 9. (Singh et al. 2011) Normalizes voltage vs. applied current for two 70 nm diameter, 2.5 μm long single aluminum nanowires at a temperature of 0.1 K. One of the nanowires is measured using normal focused ion beam deposited normal Pt electrodes and the other is measured using superconducting W electrodes. The inset shows a scanning electron micrograph of the aluminum nanowire contacted with the normal platinum electrodes. The nanowire contacted tungsten also looks the same under a scanning electron

In superconducting nanowires contacted with superconducting electrodes, the counterintuitive APE is seen. What happens when a normal nanowire is contacted with superconducting electrodes? In this situation, in addition to the expected and well-studied phenomenon of the proximity effect, some surprising new phenomena are uncovered. In a recent experiment (Wang et al., 2009a), crystalline gold nanowires (AuNW) 70 nm in diameter were contacted with superconducting W electrodes using FIB. As expected, the superconducting proximity effect caused the short wires (1 µm and 1.2 µm long) to go completely superconducting whereas the longer nanowires (1.9 µm) retained some residual resistivity to the lowest measured temperatures. This gives the range of the proximity effect to be ~ 600 nm. A surprising feature however, was that the superconducting transition of

the absence of an external magnetic field.

applications.

microscope.

**3.4 Mini-gap state** 

the 1.2 µm long wire occured in two steps at two distinct Tcs (figure 10). In the R vs. H measurements two distinct Hcs were also seen. The two-step drop in resistance is suggestive of two distinct transitions. The higher Tc and Hc are almost same as the corresponding values for the W electrodes. The results are interpreted with proximity induced superconductivity in the nanowire with a small superconducting "mini-gap" δ near the centre of the wire which is different from the gap elsewhere in the wire. The gap near the electrodes in the wire corresponds to the bulk W gap (∆). This "mini-gap" state is a novel superconducting state not seen in bulk samples.

Fig. 10. (Wang et al., 2009a) Resistance vs temperature plots for a 70 nm diameter, 1.2 μm long single crystaline gold nanowire at different magnetic fields applied perpendicular to wire axis. The gold nanowire is contacted with superconducting tungsten electrodes using a focused ion beam system and the measurement shown has been done in the standard fourprobe configuration. The superconducting transition can be seen to occur in two steps at two distinct 'critical' temperatures.

The R vs. H measurement of a 1.0 µm long nanowire and 1.2 µm nanowire are shown in figure 11 (a) and 11(b) respectively. A small resistance valley is more apparent in the 1.2 µm long nanowire. This resistance valley is associated with the mini-gap state. The details of the temperature evolution of the mini-gap in the R vs. H measurements can be seen in the magnified graph in figure 11 (c). The Hc of the mini-gap is ~ 3000 Oe in comparison to the larger transition Hc corresponding to the W electrode Hc of ~ 80000 Oe (figure 11(b)) . The curves have been offset for clarity. At 2.4 K, there is no sign of the mini-gap. At 2.5 K, the resistance suddenly jumps and then quickly drops back to zero at 2.5 kOe. At 2.6 K, this magneticfield-symmetric resistance fluctuation becomes more clearly developed and the baseline of the fluctuating resistance at fields above the mini-gap region is seen to be smoothly increasing with field. With increasing temperature the mini-gap valley continues to shrink in width, disappearing at T = 4 K. Evidence for the mini-gap in proximity structures was also revealed in recent scanning tunnelling microscopy studies (Gupta et al., 2004; Le Sueur et al., 2008; Vinet et al., 2001) and another transport measurement (Lucignano et al., 2010). In addition to the mini-gap state, resistance plateaus can be seen in the R vs. H measurements in the transition region (figure 11(a) and 11(b)).

Superconductivity in Nanoscale Systems 121

the nanowire. An interesting fact is that if the spatial extent of each vortex is taken to be limited by the diameter of the nanowire, the total number of vortices that will fit in the 1 µm nanowire is 1um/70nm = 15 which in fact corresponds to the number of peaks seen before

The implications of the range of new phenomena discovered in proximity induced superconducting nanowires are profound. First of all, the fact that AuNW up to 1.2 µm long act as a superconductor provides us a large number of 'new' superconductors that can be used in nanoscale circuitry. The 'mini-gap' state is a new superconducting state that opens new avenues for exploration of the physics of nanoscale superconductors. The ability to introduce and manoeuvre vortices one by one in the proximity induced superconducting system is a first. If scalable synthesis for such samples can be developed, the manoeuvrable

Fig. 12. (He & Wang, 2011) The differential magnetoresistance as a function of magnetic field of the 1.0 and 1.2 μm long gold nanowires measured at T = 3K. The dashed lines represent dR/d|H| = 0 for the two wires. There are small oscillations of the value of dR/d|H| around zero H, possibly due to the limited resolution in the R and H readings in low magnetic field. The clearly resolved peak at about 1.65 T was picked as the first peak of the 1.0 μm wire. The peak at the differential magnetoresistance shows uniform oscillations with B of 0.25 T. The number on the peak was theorized to match with the number of vortices in

Bi has an extremely small Fermi surface (FS) (just 10-5 of the Brillouin zone), low carrier

for electrons along the trigonal direction) (Liu & Allen, 1995) and very long electron mean free path (exceeding 2 *μ*m at room temperature) (Hartman, 1969). These properties distinguish Bi from other metals and make it particularly suitable for studying quantum phenomena. A most striking example of superconductivity behaving differently in nanoscale systems was observed in granular bismuth nanowires (BiNW) electrodeposited in track-etched polycarbonate membranes (Tian et al., 2006b). Bulk rhombohedral Bi is a semimetal with no superconducting transition observed down to 50 mK under ambient pressure. In granular BiNW however, 70 nm and 100 nm in diameter, superconductivity

\* < 0.003 free electron mass

vortices can be utilized as memory units in quantum computing.

the wire becomes normal.

the nanowire.

**3.5 New superconducting materials – Bismuth**

density (∼ 3 × 1017 /cm3 at 2 K), small effective carrier mass (*m*<sup>e</sup>

Fig. 11. (Wang et al., 2009a) (a) Magnetoresistance of the 1 μm gold nanowire, from bottom to top, at 2.0 K (gray), 2.5 K (black), 3.0 (red), 3.5 K (green), 4.0 K (blue), and 5.5 K (cyan). (c) Close-up view of (b) near zero magnetic field at different temperatures for the 1.2 m long nanowire. The curves are offset for clarity; except for 3.5K, the resistance in all plots is zero at zero field. The mini-gap state seen as the small resistance valley is apparent in the 1.2 μm long nanowire but, not in the 1 μm long nanowire.

In figure 12, plots of the numerical derivative dR/dH of the 1 µm and 1.2 µm nanowires are shown as a function of H at 3K (He & Wang, 2011). The applied field is perpendicular to the wire. This differential magnetoresistance shows uniform oscillations with increasing field with a period of ϕ0*/(*2*πr*2*)* (where, ϕ0 = *h/*2*e* is the superconducting flux quantum, *r* = 35 nm is the radius of the nanowire). The period of the differential resistance plateaus can be explained by theorizing that each step in resistance is caused by a phase slip. The phase slips are created by vortices moving across the wire that are generated by the perpendicular magnetic field. The extraordinary feature in this experiment is that the nature of the gap in the proximity induced superconductivity in the AuNW allows for generation of individual vortices one at a time along the nanowire. The gap in the middle of the wire is the smallest and is the first site for single vortex creation. As the field is increased the vortex moves to higher gap regions and a new vortex is created in the center of the wire. As a result, the resistance of the Au wire increases with increasing field step by step so that the generated vortices move continuously across the wire. This is very different from the well-known behavior of superconductors where when the field is increased beyond a critical value, vortices proliferate all at once and the superconductivity is destroyed. In figure 12, the numbers assigned to each peak are theorized to be associated with the number of vortices in

Fig. 11. (Wang et al., 2009a) (a) Magnetoresistance of the 1 μm gold nanowire, from bottom to top, at 2.0 K (gray), 2.5 K (black), 3.0 (red), 3.5 K (green), 4.0 K (blue), and 5.5 K (cyan). (c) Close-up view of (b) near zero magnetic field at different temperatures for the 1.2 m long nanowire. The curves are offset for clarity; except for 3.5K, the resistance in all plots is zero at zero field. The mini-gap state seen as the small resistance valley is apparent in the 1.2

In figure 12, plots of the numerical derivative dR/dH of the 1 µm and 1.2 µm nanowires are shown as a function of H at 3K (He & Wang, 2011). The applied field is perpendicular to the wire. This differential magnetoresistance shows uniform oscillations with increasing field with a period of ϕ0*/(*2*πr*2*)* (where, ϕ0 = *h/*2*e* is the superconducting flux quantum, *r* = 35 nm is the radius of the nanowire). The period of the differential resistance plateaus can be explained by theorizing that each step in resistance is caused by a phase slip. The phase slips are created by vortices moving across the wire that are generated by the perpendicular magnetic field. The extraordinary feature in this experiment is that the nature of the gap in the proximity induced superconductivity in the AuNW allows for generation of individual vortices one at a time along the nanowire. The gap in the middle of the wire is the smallest and is the first site for single vortex creation. As the field is increased the vortex moves to higher gap regions and a new vortex is created in the center of the wire. As a result, the resistance of the Au wire increases with increasing field step by step so that the generated vortices move continuously across the wire. This is very different from the well-known behavior of superconductors where when the field is increased beyond a critical value, vortices proliferate all at once and the superconductivity is destroyed. In figure 12, the numbers assigned to each peak are theorized to be associated with the number of vortices in

μm long nanowire but, not in the 1 μm long nanowire.

the nanowire. An interesting fact is that if the spatial extent of each vortex is taken to be limited by the diameter of the nanowire, the total number of vortices that will fit in the 1 µm nanowire is 1um/70nm = 15 which in fact corresponds to the number of peaks seen before the wire becomes normal.

The implications of the range of new phenomena discovered in proximity induced superconducting nanowires are profound. First of all, the fact that AuNW up to 1.2 µm long act as a superconductor provides us a large number of 'new' superconductors that can be used in nanoscale circuitry. The 'mini-gap' state is a new superconducting state that opens new avenues for exploration of the physics of nanoscale superconductors. The ability to introduce and manoeuvre vortices one by one in the proximity induced superconducting system is a first. If scalable synthesis for such samples can be developed, the manoeuvrable vortices can be utilized as memory units in quantum computing.

Fig. 12. (He & Wang, 2011) The differential magnetoresistance as a function of magnetic field of the 1.0 and 1.2 μm long gold nanowires measured at T = 3K. The dashed lines represent dR/d|H| = 0 for the two wires. There are small oscillations of the value of dR/d|H| around zero H, possibly due to the limited resolution in the R and H readings in low magnetic field. The clearly resolved peak at about 1.65 T was picked as the first peak of the 1.0 μm wire. The peak at the differential magnetoresistance shows uniform oscillations with B of 0.25 T. The number on the peak was theorized to match with the number of vortices in the nanowire.

#### **3.5 New superconducting materials – Bismuth**

Bi has an extremely small Fermi surface (FS) (just 10-5 of the Brillouin zone), low carrier density (∼ 3 × 1017 /cm3 at 2 K), small effective carrier mass (*m*<sup>e</sup> \* < 0.003 free electron mass for electrons along the trigonal direction) (Liu & Allen, 1995) and very long electron mean free path (exceeding 2 *μ*m at room temperature) (Hartman, 1969). These properties distinguish Bi from other metals and make it particularly suitable for studying quantum phenomena. A most striking example of superconductivity behaving differently in nanoscale systems was observed in granular bismuth nanowires (BiNW) electrodeposited in track-etched polycarbonate membranes (Tian et al., 2006b). Bulk rhombohedral Bi is a semimetal with no superconducting transition observed down to 50 mK under ambient pressure. In granular BiNW however, 70 nm and 100 nm in diameter, superconductivity

Superconductivity in Nanoscale Systems 123

Bi under high pressure is known to turn superconducting due to the formation of highpressure metallic polymorphs, namely, monoclinic Bi-II at 2.55 GPa (Brugger et al., 1967; Degtyareva et al., 2004; Nédellec et al., 1974), a complex tetragonal Bi-III at 2.7 GPa (Degtyareva et al., 2004; Haussermann et al., 2002), and a body centered cubic (bcc) Bi-V at 7.7 GPa. (Chen et al., 1969; Degtyareva et al., 2004). The transition temperatures of these polymorphic phases are respectively 3.9, 7.2, and 8.3 K (Brandt & Ginzburg, 1969). Tcs of 8.3 K and 7.2 K are also seen in the superconducting transition of the granular BiNW further strengthening the idea that small regions of high pressure Bi phase exist in the nanowire. As discussed in the next paragraph, crystalline BiNW also remarkably displayed

In a recent experiment (Tian et al., 2009), quantum oscillations of the resistance as a function of applied field (H) were found in crystalline BiNW in a standard four-probe measurement with FIB deposited normal Pt electrodes. The crystalline BiNW were superconducting with a transition temperature of 1.3 K. In parallel field, below the critical field (Hc ~ 7T), a long resistance tail was present (figure 13(a)). On subtracting a uniform background from the resistance, and plotting the remainder ΔR║, the oscillations are clearly seen (figure 13 (c)). The oscillations are periodic with H and the period

oscillations are reminiscent of the Little-Parks (LP) oscillations caused by fluxoid quantization in hollow superconducting cylinders when the thickness of the shell is comparable to ξ (Little & Parks, 1962; Parks & Little, 1964). The standard LP oscillations in a small hollow cylinder are characterized by the integer multiples of the superconducting ux quantum, Φ0, that is, H(πd2/4) ) = nΦ0, where d is the diameter of the hollow cylinder and n is an integer. Using this model, the period of the oscillations seen in BiNW, ∆H = 5.85 kOe results in d = 67.0 nm. The diameter of the nanowire determined by TEM is 79 nm with a ∼ 3.7 ± 0.5 nm oxidation layer on the surface leaving the nanowire with an effective diameter of ~ 72 nm. This is close to the diameter calculated using LP oscillations. The LP-like oscillations unambiguously suggest that the observed superconductivity in BiNW originates from the cylindrical shell between the inner core of

When the eld is aligned perpendicular to the wire axis, the residual resistance below 0.8 K still shows oscillations with varying H (figure 13(b)). On subtracting the residual resistance and plotting the remainder ΔR┴, (figure 13(d)), it is seen that these oscillations are periodic with 1/H rather than with H. The 1/H periodicity of the residual resistance oscillations is suggestive of the Shubnikov-de Haas (SdH) effect due to the Landau quantization of the conduction electrons in a metal. The SdH behavior in the superconducting state is not expected; therefore it is tempting to attribute it to the normal core. This explanation however, is not correct, because in the presence of a superconducting shell, the current will bypass the normal core and the normal core will not contribute to the resistance. In the paper (Tian et al., 2009) it is speculated that the SdH oscillations also originate from the superconducting shell. The finite residual resistance is a result of the phase slips in the shell. The normal region created in the shell because of these phase slips, behaves like a normal metal and shows SdH oscillations in the presence of a strong field. Both LP-like oscillations in parallel H and SdH-like oscillations in perpendicular H being observed in the same system suggest a completely unexpected and novel coexistence of superconducting and

2� � = 2 × 10������). These

superconductivity with a Tc of 1.3 K.

the Bi nanowire and its surface oxide.

corresponds to the superconducting quantum flux ( �� = ℎ

was found under ambient pressure in 18 out of 38 samples. Careful transmission electron microscopy (TEM) studies revealed that the samples that exhibit superconductivity had a uniform granular morphology with all the rhombohedral Bi grains (~10 nm) aligned with a preferred orientation along the entire length of the wire. For the non-superconducting granular Bi samples, the rhombohedral Bi grains showed random orientation and both wire diameter and the grain size also showed considerable variations along the length of the wire. The superconductivity in the BiNW was attributed to small regions of high pressure Bi phase formed at the boundaries of partially aligned grains. These high pressure regions with the strained lattice probably cannot be detected by X-ray diffractometry because the strain regions are only a few atomic layers thick and do not contribute much to the signal from the sample.

Fig. 13. (Tian et al., 2009) Resistance vs. applied magnetic field plots for a 79 nm diameter bismuth nanowire contacted with focused ion beam deposited normal platinum electrodes in a four-probe configuration at different temperatures for magnetic fields: (a) applied parallel to the wire axis; (b) applied perpendicular to the wire axis. (c) R||-H and (d) R-H plots, obtained by substracting smooth background from the data in Figures 13(a) and (b) respectively. Periodic oscillations of R were clearly seen in parallel H, and the dashed lines in panel (c) indicate the positions of fluxoid quantization as predicted by H(d2/4) = n0 with d = diameter of the nanowire = 67 nm. In pependicular H, the R oscillations are more complex and their periodicity is with 1/H.

was found under ambient pressure in 18 out of 38 samples. Careful transmission electron microscopy (TEM) studies revealed that the samples that exhibit superconductivity had a uniform granular morphology with all the rhombohedral Bi grains (~10 nm) aligned with a preferred orientation along the entire length of the wire. For the non-superconducting granular Bi samples, the rhombohedral Bi grains showed random orientation and both wire diameter and the grain size also showed considerable variations along the length of the wire. The superconductivity in the BiNW was attributed to small regions of high pressure Bi phase formed at the boundaries of partially aligned grains. These high pressure regions with the strained lattice probably cannot be detected by X-ray diffractometry because the strain regions are only a few atomic layers thick and do not contribute much to the signal from the

Fig. 13. (Tian et al., 2009) Resistance vs. applied magnetic field plots for a 79 nm diameter bismuth nanowire contacted with focused ion beam deposited normal platinum electrodes in a four-probe configuration at different temperatures for magnetic fields: (a) applied parallel to the wire axis; (b) applied perpendicular to the wire axis. (c) R||-H and (d) R-H plots, obtained by substracting smooth background from the data in Figures 13(a) and (b) respectively. Periodic oscillations of R were clearly seen in parallel H, and the dashed lines

/4) = n0 with

in panel (c) indicate the positions of fluxoid quantization as predicted by H(d2

complex and their periodicity is with 1/H.

d = diameter of the nanowire = 67 nm. In pependicular H, the R oscillations are more

sample.

Bi under high pressure is known to turn superconducting due to the formation of highpressure metallic polymorphs, namely, monoclinic Bi-II at 2.55 GPa (Brugger et al., 1967; Degtyareva et al., 2004; Nédellec et al., 1974), a complex tetragonal Bi-III at 2.7 GPa (Degtyareva et al., 2004; Haussermann et al., 2002), and a body centered cubic (bcc) Bi-V at 7.7 GPa. (Chen et al., 1969; Degtyareva et al., 2004). The transition temperatures of these polymorphic phases are respectively 3.9, 7.2, and 8.3 K (Brandt & Ginzburg, 1969). Tcs of 8.3 K and 7.2 K are also seen in the superconducting transition of the granular BiNW further strengthening the idea that small regions of high pressure Bi phase exist in the nanowire. As discussed in the next paragraph, crystalline BiNW also remarkably displayed superconductivity with a Tc of 1.3 K.

In a recent experiment (Tian et al., 2009), quantum oscillations of the resistance as a function of applied field (H) were found in crystalline BiNW in a standard four-probe measurement with FIB deposited normal Pt electrodes. The crystalline BiNW were superconducting with a transition temperature of 1.3 K. In parallel field, below the critical field (Hc ~ 7T), a long resistance tail was present (figure 13(a)). On subtracting a uniform background from the resistance, and plotting the remainder ΔR║, the oscillations are clearly seen (figure 13 (c)). The oscillations are periodic with H and the period corresponds to the superconducting quantum flux ( �� = ℎ 2� � = 2 × 10������). These oscillations are reminiscent of the Little-Parks (LP) oscillations caused by fluxoid quantization in hollow superconducting cylinders when the thickness of the shell is comparable to ξ (Little & Parks, 1962; Parks & Little, 1964). The standard LP oscillations in a small hollow cylinder are characterized by the integer multiples of the superconducting ux quantum, Φ0, that is, H(πd2/4) ) = nΦ0, where d is the diameter of the hollow cylinder and n is an integer. Using this model, the period of the oscillations seen in BiNW, ∆H = 5.85 kOe results in d = 67.0 nm. The diameter of the nanowire determined by TEM is 79 nm with a ∼ 3.7 ± 0.5 nm oxidation layer on the surface leaving the nanowire with an effective diameter of ~ 72 nm. This is close to the diameter calculated using LP oscillations. The LP-like oscillations unambiguously suggest that the observed superconductivity in BiNW originates from the cylindrical shell between the inner core of the Bi nanowire and its surface oxide.

When the eld is aligned perpendicular to the wire axis, the residual resistance below 0.8 K still shows oscillations with varying H (figure 13(b)). On subtracting the residual resistance and plotting the remainder ΔR┴, (figure 13(d)), it is seen that these oscillations are periodic with 1/H rather than with H. The 1/H periodicity of the residual resistance oscillations is suggestive of the Shubnikov-de Haas (SdH) effect due to the Landau quantization of the conduction electrons in a metal. The SdH behavior in the superconducting state is not expected; therefore it is tempting to attribute it to the normal core. This explanation however, is not correct, because in the presence of a superconducting shell, the current will bypass the normal core and the normal core will not contribute to the resistance. In the paper (Tian et al., 2009) it is speculated that the SdH oscillations also originate from the superconducting shell. The finite residual resistance is a result of the phase slips in the shell. The normal region created in the shell because of these phase slips, behaves like a normal metal and shows SdH oscillations in the presence of a strong field. Both LP-like oscillations in parallel H and SdH-like oscillations in perpendicular H being observed in the same system suggest a completely unexpected and novel coexistence of superconducting and

Superconductivity in Nanoscale Systems 125

surface, the films exhibit distinctive negative magnetoresistance with wide magnetoresistance terrace under the perpendicular magnetic field. Analysis of the experimental results revealed that the observed effect originates from the coexistence of two superconducting phases in the system: the percolation structures with lower critical field (Hc) and the flat 2D Pb islands with higher Hc. In the low temperature regime, electron tunneling dominates the transport property of the percolation structures. When a magnetic field is applied, the superconducting gap is suppressed. Reducing the gap leads to a significant decrease of the resistance. The unusual magnetoresistance effect in the coexistence phase is of both fundamental interest and has possible applications in fabrication of hybrid devices based on the traditional microelectronics and the emerging

The high quality Pb films can also be etched to quasi-one dimensional (quasi-1D) single crystal Pb superconductor nanobridge (several nanometers thick and 100 to 500 nanometers wide) by using an FIB system (the measurement geometry are shown in figure 14). The Pb nanostructures thus formed were measured in a physical property measurement system (PPMS) with standard four-electrode method. Many interesting phenomena with the different widths and thicknesses of the Pb nanobridges were observed by studying the transport properties, such as the enhanced superconductivity (TC) compared to the film with the same thickness (Wang et al., 2008a), magnetoresistance (MR) oscillations (Wang et al., 2008a, 2009c, 2010b), the broadening of the transition near TC because of thermally activated phase-slip (TAPS), multiple voltage steps, and power-law Voltage vs. Current relationship (Wang et al., 2009b). The properties and the unique structures provide a new approach for integrating superconducting circuits on single Si chips, for example nanoscale SQUID-like

Furthermore, atomically uniform single-crystal epitaxial Pb films, several nanometers thick, were fabricated on a Si substrate to form a sharp superconductor–semiconductor heterojunction (SSH) by FIB etching (Wang et al., 2008b) as shown in figures 15 (a) and (b). The transport measurements of the junction showed an unusual magnetoresistance effect when the Pb film was superconducting. The resistance of the SSH decreased sharply with increasing magnetic field and was very sensitive to the temperature below TC (figure 15 (c)). Such sensitivity and the SSH structure make the Pb–Si junction highly promising for mass production for use in developing a magnetic field controlled "on–off" device and new cryogenic temperature detectors on Si chips (Day et al., 2003; Irwin et al., 1996). The devices derived by superconductor

The giant magnetoresistance (GMR) effect was discovered in the late 1980s on Fe/Cr magnetic multilayer (Baibich et al., 1988) and Fe/Cr/Fe tri-layers (Binasch et al., 1989). It was seen that as the applied field was increased, the resistance of the multilayer structure decreased (6% for Fe/Cr/Fe tri-layers and 50% for Fe/Cr magnetic multilayer, respectively). This discovery was significant to the scientific community and caught widespread attention in the industry. The researchers of IBM recognized its potential and invented spin-valves (Tsang et al., 1994) and the commercial read head was created (Wolf et al., 2001). GMR's application in the read head of hard discs greatly increases the density of stored information. Since the introduction of GMR-type sensors as reading elements, storage capacities have increased approximately 100 times (Grunberg, 2008). Moreover, magnetic random access memory (MRAM), medical/biological sensors and various logic components

devices (Johansson et al., 2005), "on-off" functional and logical nanodevices.

films on semiconductor substrates have many promising applications in the future.

superconducting quantum electronics (Leggett, 2002).

metallic states in the surface shell of the Bi nanowire below Tc. The presence of superconductivity in ambient pressure BiNW is in itself a surprising result that brings to mind possibilities of other materials known to be non-superconducting in bulk showing superconductivity in the nanoscale. In addition, the fact that the superconductivity seems to be present only on the surface while the core continues to be normal, indicates that this superconductivity is an interface related phenomenon. The presence of SdH oscillations indicates metallic states on the surface. Metallic surface states are a prerequisite for the formation of a topological insulator (Fu et al., 2007). These new ndings suggest that this old material may have potential application in developing next-generation quantum computing devices.

### **3.6 Pb nanobelts**

Superconducting films on semiconductor substrates have attracted much attention since the derived superconductor-based electronics have been shown to be promising for future data processing and storage technologies. Current semiconductor technology is mainly based on silicon wafers. The superconductivity and applications of low dimensional Pb nanostructural superconductors have been extensively studied in recent decades (Wang et al., 2007, 2008a, 2008b, 2009b, 2009c, 2010b) because of the relatively higher transition temperature (*TC* ~ 7.2 K) of bulk Pb compared to other conventional superconductors. The structure formed by two dimensional (2D) superconducting Pb films on Si substrate provides an excellent research system and has a wide application field. Qi-Kun Xue et al. have carried out a series of systematic experiments in high quality 2D single crystalline Pb film systems by utilizing scanning tunneling microscope (STM) and have achieved many outstanding results (Bao et al., 2005; Guo et al., 2004; T. Zhang et al., 2010; Y. F. Zhang et al., 2005). The Pb films are synthesized by using an MBE chamber. Based on previous work (Jalochowski et al., 1995; Upton et al., 2004), Q. K. Xue's group made great achievements in fabricating crystalline Pb films by MBE. They have made ultrathin Pb films from one atomic layer to tens of layers on Si (111) substrates. The Si substrate was firstly heated at 400℃ for hours to get rid of the molecular gas absorbed at the surface, and then the substrate was flashed or annealed to ~1200℃ several times to form a clean surface. After that, the substrate was cooled down to ~ 95 K by liquid nitrogen during growth to achieve atomically flat single crystal Pb thin films over a macroscopic area. By means of STM, scanning tunneling spectroscopy (STS), and transport measurements various intriguing properties of the 2D Pb systems have been discovered. These include "magic film thickness", "preferential island heights", "oscillating superconductivity transition temperatures" and other novel behaviors because of the strong quantum confinement effect (Bao et al., 2005; Eom et al., 2006; Guo, et al., 2004; Herzog et al., 1996; Markovic et al., 1998; Parendo et al., 2004; Xiong et al., 1997).

High quality Pb films are not only important for fundamental research, but also offer possibilities of electronic device fabrication. From the view point of applications, Pb films and related nanodevices have been predominantly prepared on semiconductor substrates. In recent years, J. Wang and his collaborators performed experiments on nanoscale systems based on Pb films grown on Si substrates (Wang et al., 2007, 2008a, 2008b, 2009b, 2009c, 2010b). In the Pb thin film with a unique fractal-like morphology grown on Si (111)-7×7

metallic states in the surface shell of the Bi nanowire below Tc. The presence of superconductivity in ambient pressure BiNW is in itself a surprising result that brings to mind possibilities of other materials known to be non-superconducting in bulk showing superconductivity in the nanoscale. In addition, the fact that the superconductivity seems to be present only on the surface while the core continues to be normal, indicates that this superconductivity is an interface related phenomenon. The presence of SdH oscillations indicates metallic states on the surface. Metallic surface states are a prerequisite for the formation of a topological insulator (Fu et al., 2007). These new ndings suggest that this old material may have potential application in developing next-generation quantum computing

Superconducting films on semiconductor substrates have attracted much attention since the derived superconductor-based electronics have been shown to be promising for future data processing and storage technologies. Current semiconductor technology is mainly based on silicon wafers. The superconductivity and applications of low dimensional Pb nanostructural superconductors have been extensively studied in recent decades (Wang et al., 2007, 2008a, 2008b, 2009b, 2009c, 2010b) because of the relatively higher transition temperature (*TC* ~ 7.2 K) of bulk Pb compared to other conventional superconductors. The structure formed by two dimensional (2D) superconducting Pb films on Si substrate provides an excellent research system and has a wide application field. Qi-Kun Xue et al. have carried out a series of systematic experiments in high quality 2D single crystalline Pb film systems by utilizing scanning tunneling microscope (STM) and have achieved many outstanding results (Bao et al., 2005; Guo et al., 2004; T. Zhang et al., 2010; Y. F. Zhang et al., 2005). The Pb films are synthesized by using an MBE chamber. Based on previous work (Jalochowski et al., 1995; Upton et al., 2004), Q. K. Xue's group made great achievements in fabricating crystalline Pb films by MBE. They have made ultrathin Pb films from one atomic layer to tens of layers on Si (111) substrates. The Si substrate was firstly heated at 400℃ for hours to get rid of the molecular gas absorbed at the surface, and then the substrate was flashed or annealed to ~1200℃ several times to form a clean surface. After that, the substrate was cooled down to ~ 95 K by liquid nitrogen during growth to achieve atomically flat single crystal Pb thin films over a macroscopic area. By means of STM, scanning tunneling spectroscopy (STS), and transport measurements various intriguing properties of the 2D Pb systems have been discovered. These include "magic film thickness", "preferential island heights", "oscillating superconductivity transition temperatures" and other novel behaviors because of the strong quantum confinement effect (Bao et al., 2005; Eom et al., 2006; Guo, et al., 2004; Herzog et al., 1996; Markovic et al., 1998; Parendo et al., 2004;

High quality Pb films are not only important for fundamental research, but also offer possibilities of electronic device fabrication. From the view point of applications, Pb films and related nanodevices have been predominantly prepared on semiconductor substrates. In recent years, J. Wang and his collaborators performed experiments on nanoscale systems based on Pb films grown on Si substrates (Wang et al., 2007, 2008a, 2008b, 2009b, 2009c, 2010b). In the Pb thin film with a unique fractal-like morphology grown on Si (111)-7×7

devices.

**3.6 Pb nanobelts**

Xiong et al., 1997).

surface, the films exhibit distinctive negative magnetoresistance with wide magnetoresistance terrace under the perpendicular magnetic field. Analysis of the experimental results revealed that the observed effect originates from the coexistence of two superconducting phases in the system: the percolation structures with lower critical field (Hc) and the flat 2D Pb islands with higher Hc. In the low temperature regime, electron tunneling dominates the transport property of the percolation structures. When a magnetic field is applied, the superconducting gap is suppressed. Reducing the gap leads to a significant decrease of the resistance. The unusual magnetoresistance effect in the coexistence phase is of both fundamental interest and has possible applications in fabrication of hybrid devices based on the traditional microelectronics and the emerging superconducting quantum electronics (Leggett, 2002).

The high quality Pb films can also be etched to quasi-one dimensional (quasi-1D) single crystal Pb superconductor nanobridge (several nanometers thick and 100 to 500 nanometers wide) by using an FIB system (the measurement geometry are shown in figure 14). The Pb nanostructures thus formed were measured in a physical property measurement system (PPMS) with standard four-electrode method. Many interesting phenomena with the different widths and thicknesses of the Pb nanobridges were observed by studying the transport properties, such as the enhanced superconductivity (TC) compared to the film with the same thickness (Wang et al., 2008a), magnetoresistance (MR) oscillations (Wang et al., 2008a, 2009c, 2010b), the broadening of the transition near TC because of thermally activated phase-slip (TAPS), multiple voltage steps, and power-law Voltage vs. Current relationship (Wang et al., 2009b). The properties and the unique structures provide a new approach for integrating superconducting circuits on single Si chips, for example nanoscale SQUID-like devices (Johansson et al., 2005), "on-off" functional and logical nanodevices.

Furthermore, atomically uniform single-crystal epitaxial Pb films, several nanometers thick, were fabricated on a Si substrate to form a sharp superconductor–semiconductor heterojunction (SSH) by FIB etching (Wang et al., 2008b) as shown in figures 15 (a) and (b). The transport measurements of the junction showed an unusual magnetoresistance effect when the Pb film was superconducting. The resistance of the SSH decreased sharply with increasing magnetic field and was very sensitive to the temperature below TC (figure 15 (c)). Such sensitivity and the SSH structure make the Pb–Si junction highly promising for mass production for use in developing a magnetic field controlled "on–off" device and new cryogenic temperature detectors on Si chips (Day et al., 2003; Irwin et al., 1996). The devices derived by superconductor films on semiconductor substrates have many promising applications in the future.

The giant magnetoresistance (GMR) effect was discovered in the late 1980s on Fe/Cr magnetic multilayer (Baibich et al., 1988) and Fe/Cr/Fe tri-layers (Binasch et al., 1989). It was seen that as the applied field was increased, the resistance of the multilayer structure decreased (6% for Fe/Cr/Fe tri-layers and 50% for Fe/Cr magnetic multilayer, respectively). This discovery was significant to the scientific community and caught widespread attention in the industry. The researchers of IBM recognized its potential and invented spin-valves (Tsang et al., 1994) and the commercial read head was created (Wolf et al., 2001). GMR's application in the read head of hard discs greatly increases the density of stored information. Since the introduction of GMR-type sensors as reading elements, storage capacities have increased approximately 100 times (Grunberg, 2008). Moreover, magnetic random access memory (MRAM), medical/biological sensors and various logic components

Superconductivity in Nanoscale Systems 127

Fig. 15. (Wang et al., 2008b) (a) A scanning electron micrograph of the Pb film after a 2μm wide gap (the dark region) was fabricated. (b) The schematic graph for the transport measurement across the Pb/Si (111) heterojunctions. (c) Magnetoresistance of the

heterojunctions with a magnetic field perpendicular to the film at different temperatures.

**3.7 Josephson effect and superconducting quantum interference device (SQUID)** 

A Josephson junction is composed of two superconducting layers coupled by a weak link. The weak link can consist of a thin insulating barrier (S-I-S), a short section of nonsuperconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-c-S). The Josephson Effect is the phenomenon that the current can tunnel through the junction (or weak link) without any voltage applied (Josephson, 1962, 1974). It was observed by the British physicist B. D. Josephson, who was the first to predict the tunneling of superconducting Cooper pairs (before that, it was only known that normal electrons can tunnel through an insulating barrier). Josephson received the Nobel prize in 1973 for this finding. Josephson junctions have important applications in superconducting quantum interference devices (SQUID), single-electron transistors, superconducting qubits and rapid single flux quantum (RSFQ) digital electronics. SQUID is a very sensitive magnetometer used to measure extremely weak magnetic fields (Range, 2002), based on superconducting loops containing one or two Josephson junctions. For instance, the SQUID magnetometer reaches 7×10-11 Os/Hz1/2 which is higher than that of the best traditional magnetometer by one order. SQUID is used as a sensitive sensor that can detect the tiny magnetic flux changes through measuring the other visible parameters. It can

Although the physical mechanisms in traditional GMR and the giant negative magnetoresistive effect in SSH are completely different and the resistance changes in SSH only happens at low temperatures, it is reasonable to believe that the large magnetoresistance effect in SSH may also be extended and applied in technology and other fields such as magnetic switches, temperature or magnetic field sensors, high-density information storage and quantum computing circuits. It is to be remembered that Pb is a metal and is easily oxidized in atmosphere. Therefore, cover layers of Ge, Si, Au etc. need to

The vertical scale is normalized to the resistance at zero magnetic field.

be used to protect Pb based devices for applications.

be used in many fields.

have been invented by using the GMR effect. Meanwhile, GMR is often viewed as the founding stone of spintronics, where the electron spin is the key to physical functionality. Thus GMR of the magnetic multilayers opened the way to an efficient control of the motion of the electrons by acting on their spin through the orientation of a magnetization. In the SSH composed of Pb/Si system, described in the paragraph above, J. Wang and his collaborators observed a dramatic negative magnetoresistive effect (Wang et al., 2008b). Compared to the Fe/Cr magnetic multilayer in 1988 (Baibich et al., 1988), the resistance changes more with increasing the magnetic field, and the saturated critical field of the SSH is smaller than that of Fe/Cr multilayer by an order at the same temperature.

Fig. 14. (Wang et al., 2008b) (a) A scanning tunneling microscope image of the atomically smooth lead thin film on the Si(111) 77 substrate. (b) The schematic graph for the transport measurement across the lead nanobelt. (c) A SEM image of the lead nanobelt made by milling the lead film.

have been invented by using the GMR effect. Meanwhile, GMR is often viewed as the founding stone of spintronics, where the electron spin is the key to physical functionality. Thus GMR of the magnetic multilayers opened the way to an efficient control of the motion of the electrons by acting on their spin through the orientation of a magnetization. In the SSH composed of Pb/Si system, described in the paragraph above, J. Wang and his collaborators observed a dramatic negative magnetoresistive effect (Wang et al., 2008b). Compared to the Fe/Cr magnetic multilayer in 1988 (Baibich et al., 1988), the resistance changes more with increasing the magnetic field, and the saturated critical field of the SSH

is smaller than that of Fe/Cr multilayer by an order at the same temperature.

Fig. 14. (Wang et al., 2008b) (a) A scanning tunneling microscope image of the atomically smooth lead thin film on the Si(111) 77 substrate. (b) The schematic graph for the transport measurement across the lead nanobelt. (c) A SEM image of the lead nanobelt made by

milling the lead film.

Fig. 15. (Wang et al., 2008b) (a) A scanning electron micrograph of the Pb film after a 2μm wide gap (the dark region) was fabricated. (b) The schematic graph for the transport measurement across the Pb/Si (111) heterojunctions. (c) Magnetoresistance of the heterojunctions with a magnetic field perpendicular to the film at different temperatures. The vertical scale is normalized to the resistance at zero magnetic field.

Although the physical mechanisms in traditional GMR and the giant negative magnetoresistive effect in SSH are completely different and the resistance changes in SSH only happens at low temperatures, it is reasonable to believe that the large magnetoresistance effect in SSH may also be extended and applied in technology and other fields such as magnetic switches, temperature or magnetic field sensors, high-density information storage and quantum computing circuits. It is to be remembered that Pb is a metal and is easily oxidized in atmosphere. Therefore, cover layers of Ge, Si, Au etc. need to be used to protect Pb based devices for applications.

## **3.7 Josephson effect and superconducting quantum interference device (SQUID)**

A Josephson junction is composed of two superconducting layers coupled by a weak link. The weak link can consist of a thin insulating barrier (S-I-S), a short section of nonsuperconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-c-S). The Josephson Effect is the phenomenon that the current can tunnel through the junction (or weak link) without any voltage applied (Josephson, 1962, 1974). It was observed by the British physicist B. D. Josephson, who was the first to predict the tunneling of superconducting Cooper pairs (before that, it was only known that normal electrons can tunnel through an insulating barrier). Josephson received the Nobel prize in 1973 for this finding. Josephson junctions have important applications in superconducting quantum interference devices (SQUID), single-electron transistors, superconducting qubits and rapid single flux quantum (RSFQ) digital electronics. SQUID is a very sensitive magnetometer used to measure extremely weak magnetic fields (Range, 2002), based on superconducting loops containing one or two Josephson junctions. For instance, the SQUID magnetometer reaches 7×10-11 Os/Hz1/2 which is higher than that of the best traditional magnetometer by one order. SQUID is used as a sensitive sensor that can detect the tiny magnetic flux changes through measuring the other visible parameters. It can be used in many fields.

Superconductivity in Nanoscale Systems 129

physics from the effects caused due to defects and morphology. It is also possible to fabricate such devices for use in integrated circuits on Si substrates. The e-beam lithography technique used for gold, aluminum and lead nanobelts is especially suitable for this. Examples like nano-squid and single-electron transistors (Fulton & Dolan, 1987) enabling extremely sensitive magnetic field and charge measurements demonstrate the tremendous potential for applications of nanoscale superconducting devices. The factor limiting widespread use of these devices is still the low temperatures required and the prohibitive cost of the helium needed to reach those low temperatures. Much research has been done to achieve usable high Tc superconductors. However, till now, high temperature superconductors have realized applications only in very limited fields. Although their Tc is higher than conventional metallic superconductors, they are ceramic, making them inflexible and hard to machine compared to metallic superconductors. The low Tc of metallic superconductors, limiting their applications in devices, is not a problem in potential applications to space technology (ambient temperature in space ~ 3K and many metallic superconductors have Tc higher than this). Due to properties like high speed (response rate is 10-100 times faster than conventional devices), ultra-low noise (the thermal noise is much lower than conventional semiconductor devices) and low heat dissipation, superconductivity in aerospace applications such as space communications, radar guidance

In summary, the use of superconducting nanoscale components will produce denser and more rapid chips since the resistance is a major source of heat generation and charging time in integrated circuits. Studying and understanding superconductivity on this scale is of

We are grateful to Professors Moses H. W. Chan, Mingliang Tian, Qi-Kun Xue, Jainendra K. Jain, Thomas E. Mallouk, Nitin Samarth and Xu-Cun Ma for the collaboration in this project. This work was financially supported by the National Natural Science Foundation of China (No. 11174007) and National Basic Research Program (NBRP) of China (No. 2012CB921300), the Penn State MRSEC under NSF grant DMR-0820404, and China Postdoctoral Science

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etc. is widely used.

great interest and importance.

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pp. (054505), ISSN 0163-1829

**6. References** 

**5. Acknowledgement** 

The most common use of SQUID is probably in magnetic property measurement systems (MPMS). The SQUID is being used as a detector to perform MRI because of its extreme sensitivity. It is an ideal device for studying biological systems. In the clinical environment, the applications of SQUID are wide. For example, Magnetoencephalography (MEG) (Cohen, 1972), which is an equipment that measures and traces the extremely weak biomagnetic signals emitted by the brain. The SQUID is also used in cardiology for magnetic field imaging (MFI) (Brockmeier et al., 1994), which detects the magnetic field of the heart for diagnosis and risk stratification. In the other fields of biology, SQUID also plays a significant role such as recording the weak magnetic fields of the stomach, tracing the path of orally applied drugs in human body. Additionally, in geological exploration, the SQUIDsuperconducting magnetic magnetometer can be used to measure the geomagnetic field accurately and analyze the distribution to observe the valuable mineral deposits. For earthquake prediction, it can be used to detect the gravity changes on a floating superconducting ball. The changes in gravity change the relative position of the ball and the superconducting ring and cause distortion of the magnetic field. The changing magnetic field can be detected by SQUID-Gravimeter (Verkin et al., 1976). Furthermore, it is also promising as a potential component in quantum computers (Bouchiat et al., 1998) and in precision sensors. The applications of SQUID are many.

With the development of semiconductor industry, electronic devices are rapidly becoming smaller. Although the sensitivity of the SQUID increases with the area of the superconducting ring, nano-SQUID is also needed for detecting the weak magnetic signals of small structures and integrating the logic components of computers*.* The British "New Scientist" magazine reported (Troeman et al., 2007) that researchers in Holland had fabricated the smallest SQUID in the world. It is 180 nm in diameter and was made by Nb. To maintain superconductivity of the device, the whole system needs to operate at low temperatures. Pb also has TC higher than liquid helium temperature and is suitable for work in helium cooling conditions. In addition, the possibility of using focused ion beam and electron beam lithography techniques discussed in this chapter on silicon substrate offers the possibility of synthesizing millions or billions of SQUIDs devices in one chip.

## **4. Conclusion**

Superconductivity and its applications in science and technology is currently one of the most important research fields. In recent decades, the developments in this field have been extremely rapid and superconductors are extensively used in many applications. With the dimensions of electronic circuits reduced to nanometer scale, it becomes important to study the physics of superconductivity on this scale for potential nano-superconducting circuit elements. Many interesting properties are exhibited in superconducting nanowires with various morphologies, which offer numerous opportunities for applications.

The interesting physics observed in nanowires and nanobelts discussed in the chapter proves that they are an ideal platform for helping in understanding the fundamental nature of superconductivity. Furthermore, the plethora of synthesis techniques available, offer flexibility in morphology and performance control. The very high quality, defect free nanostructures like Pb films discussed in the chapter are ideal candidates for studying lowdimensional superconductivity because they enable the separation of true low-dimensional

The most common use of SQUID is probably in magnetic property measurement systems (MPMS). The SQUID is being used as a detector to perform MRI because of its extreme sensitivity. It is an ideal device for studying biological systems. In the clinical environment, the applications of SQUID are wide. For example, Magnetoencephalography (MEG) (Cohen, 1972), which is an equipment that measures and traces the extremely weak biomagnetic signals emitted by the brain. The SQUID is also used in cardiology for magnetic field imaging (MFI) (Brockmeier et al., 1994), which detects the magnetic field of the heart for diagnosis and risk stratification. In the other fields of biology, SQUID also plays a significant role such as recording the weak magnetic fields of the stomach, tracing the path of orally applied drugs in human body. Additionally, in geological exploration, the SQUIDsuperconducting magnetic magnetometer can be used to measure the geomagnetic field accurately and analyze the distribution to observe the valuable mineral deposits. For earthquake prediction, it can be used to detect the gravity changes on a floating superconducting ball. The changes in gravity change the relative position of the ball and the superconducting ring and cause distortion of the magnetic field. The changing magnetic field can be detected by SQUID-Gravimeter (Verkin et al., 1976). Furthermore, it is also promising as a potential component in quantum computers (Bouchiat et al., 1998) and in

With the development of semiconductor industry, electronic devices are rapidly becoming smaller. Although the sensitivity of the SQUID increases with the area of the superconducting ring, nano-SQUID is also needed for detecting the weak magnetic signals of small structures and integrating the logic components of computers*.* The British "New Scientist" magazine reported (Troeman et al., 2007) that researchers in Holland had fabricated the smallest SQUID in the world. It is 180 nm in diameter and was made by Nb. To maintain superconductivity of the device, the whole system needs to operate at low temperatures. Pb also has TC higher than liquid helium temperature and is suitable for work in helium cooling conditions. In addition, the possibility of using focused ion beam and electron beam lithography techniques discussed in this chapter on silicon substrate offers

Superconductivity and its applications in science and technology is currently one of the most important research fields. In recent decades, the developments in this field have been extremely rapid and superconductors are extensively used in many applications. With the dimensions of electronic circuits reduced to nanometer scale, it becomes important to study the physics of superconductivity on this scale for potential nano-superconducting circuit elements. Many interesting properties are exhibited in superconducting nanowires with

The interesting physics observed in nanowires and nanobelts discussed in the chapter proves that they are an ideal platform for helping in understanding the fundamental nature of superconductivity. Furthermore, the plethora of synthesis techniques available, offer flexibility in morphology and performance control. The very high quality, defect free nanostructures like Pb films discussed in the chapter are ideal candidates for studying lowdimensional superconductivity because they enable the separation of true low-dimensional

the possibility of synthesizing millions or billions of SQUIDs devices in one chip.

various morphologies, which offer numerous opportunities for applications.

precision sensors. The applications of SQUID are many.

**4. Conclusion** 

physics from the effects caused due to defects and morphology. It is also possible to fabricate such devices for use in integrated circuits on Si substrates. The e-beam lithography technique used for gold, aluminum and lead nanobelts is especially suitable for this. Examples like nano-squid and single-electron transistors (Fulton & Dolan, 1987) enabling extremely sensitive magnetic field and charge measurements demonstrate the tremendous potential for applications of nanoscale superconducting devices. The factor limiting widespread use of these devices is still the low temperatures required and the prohibitive cost of the helium needed to reach those low temperatures. Much research has been done to achieve usable high Tc superconductors. However, till now, high temperature superconductors have realized applications only in very limited fields. Although their Tc is higher than conventional metallic superconductors, they are ceramic, making them inflexible and hard to machine compared to metallic superconductors. The low Tc of metallic superconductors, limiting their applications in devices, is not a problem in potential applications to space technology (ambient temperature in space ~ 3K and many metallic superconductors have Tc higher than this). Due to properties like high speed (response rate is 10-100 times faster than conventional devices), ultra-low noise (the thermal noise is much lower than conventional semiconductor devices) and low heat dissipation, superconductivity in aerospace applications such as space communications, radar guidance etc. is widely used.

In summary, the use of superconducting nanoscale components will produce denser and more rapid chips since the resistance is a major source of heat generation and charging time in integrated circuits. Studying and understanding superconductivity on this scale is of great interest and importance.

## **5. Acknowledgement**

We are grateful to Professors Moses H. W. Chan, Mingliang Tian, Qi-Kun Xue, Jainendra K. Jain, Thomas E. Mallouk, Nitin Samarth and Xu-Cun Ma for the collaboration in this project. This work was financially supported by the National Natural Science Foundation of China (No. 11174007) and National Basic Research Program (NBRP) of China (No. 2012CB921300), the Penn State MRSEC under NSF grant DMR-0820404, and China Postdoctoral Science Foundation (No. 2011M500180).

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**7** 

*Japan* 

Kazuto Hirata

*National Institute for Materials Science, Tsukuba* 

cm2). This is a general view of magnetic flux

, and the applied magnetic field *H* is

from the center of vortex

**Josephson Vortices in High Tc Superconductors** 

When a magnetic field is applied to the type-II superconductors in a superconducting state, the magnetic field penetrates into the superconductors above the first critical field *H*c1, and is

(vortex) in the type-II superconductors as schematically drawn in Fig.1 (a). The wave function

core. In this case, vortex has a core of normal state. In a lower magnetic field, vortices behave as isolated vortices. When the magnetic field becomes higher, interaction between/among vortices, vortices form "Abrikosov" vortex lattice in triangular or square distribution, which has been observed by scanning tunneling microscope (Hess et al. 1994) and small angle neutron

Fig. 1. Schematic drawing of a magnetic flux (vortex) in type-II superconductor (a) and the

High-temperature superconductors (HTSCs) also belong to the type-II superconductors. However, vortex state is quite different from that in conventional superconductors. This has mainly its origin in the layered crystal structures in atomic scale. The crystal structure of HTSCs is consisted of the Cu-O superconducting layers and the nonsuperconducting ones iteratively as shown in Fig.1 (b). It causes a weak Josephson coupling between the Cu-O superconducting layers through the insulating layers in superconducting state. This leads intrinsically to the nanoscale Josephson junctions (JJs) in crystalline unit cells, and also

extends to the coherence length

scattering (Yethiraj et al, 1993) mainly in mettalic and intermetallic superconductors.

screened by the supercurrent in a range of the penetration depth of

**1. Introduction** 

of superconductivity

quantized in a unit of 0 (=*h*/2e=2.07x 10-7gauss.

crystal structure of Bi2Sr2CaCu2O8+ (b).

induces a strong anisotropy of physical properties in HTSCs.


## **Josephson Vortices in High Tc Superconductors**

## Kazuto Hirata

*National Institute for Materials Science, Tsukuba Japan* 

#### **1. Introduction**

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(2005). Band Structure and Oscillatory Electron-phonon Coupling of Pb Thin Films Determined by Atomic-layer-resolved Quantum-well States. *Physical Review Letters,*  When a magnetic field is applied to the type-II superconductors in a superconducting state, the magnetic field penetrates into the superconductors above the first critical field *H*c1, and is quantized in a unit of 0 (=*h*/2e=2.07x 10-7gauss. cm2). This is a general view of magnetic flux (vortex) in the type-II superconductors as schematically drawn in Fig.1 (a). The wave function of superconductivity extends to the coherence length , and the applied magnetic field *H* is screened by the supercurrent in a range of the penetration depth of from the center of vortex core. In this case, vortex has a core of normal state. In a lower magnetic field, vortices behave as isolated vortices. When the magnetic field becomes higher, interaction between/among vortices, vortices form "Abrikosov" vortex lattice in triangular or square distribution, which has been observed by scanning tunneling microscope (Hess et al. 1994) and small angle neutron scattering (Yethiraj et al, 1993) mainly in mettalic and intermetallic superconductors.

Fig. 1. Schematic drawing of a magnetic flux (vortex) in type-II superconductor (a) and the crystal structure of Bi2Sr2CaCu2O8+ (b).

High-temperature superconductors (HTSCs) also belong to the type-II superconductors. However, vortex state is quite different from that in conventional superconductors. This has mainly its origin in the layered crystal structures in atomic scale. The crystal structure of HTSCs is consisted of the Cu-O superconducting layers and the nonsuperconducting ones iteratively as shown in Fig.1 (b). It causes a weak Josephson coupling between the Cu-O superconducting layers through the insulating layers in superconducting state. This leads intrinsically to the nanoscale Josephson junctions (JJs) in crystalline unit cells, and also induces a strong anisotropy of physical properties in HTSCs.

Josephson Vortices in High Tc Superconductors 139

Tachiki 2004). However, JV system has not been understood well experimentally. The melting transition of the JV system (Kwok et al. 1994), the oscillatory melting temperature (Gordeev et al. 2000), and the vortex smectic phase have been found only in YBa2Cu3O7- (YBCO). In strongly anisotropic HTSCs such as Bi2Sr2CaCu2O8+ (Bi-2212), few experimental results have been reported on the magnetic phases of the JV system (Fuhrer

Most of the difficulties to study on JV system in experiments are considered due to a little change of physical parameters at the phase boundary. Shear modulus of JVs between the layers is very little and it is considered that a change in free energy is so small at the boundary to be observed by the experiments (Shilling et al. 1997). Recently we have found a new method to study the magnetic phases of JV system by using the periodic oscillations in flow resistance of JVs (Ooi et al. 2002), from which we can deduce the configurations of JVs. This phenomenon is caused by the intrinsic boundary effect of the sample, related to the configurations of JVs (Ooi et al. 2002; Koshelev 2002; Machida 2003). The magnetic phase, in which the periodic oscillations are observed, has been confirmed as a three-dimensional (3D) ordered vortex state. The periodic oscillations are observed independent to temperature and magnetic field, which is a universal nature of JV flowresistance. Then, it is useful for determining an absolute magentic field. In this chapter, experimental methods for observing the periodic oscillations and for fabricating the sample for the measurements are described in section 2, general view of vortex flow in section 3, magnetic phases in section 4, application of the periodic oscillations in section 5,

Single crystals of Bi-2212 were grown with a travelling solvent floating zone method (Mochiku et al., 1997). The crystals have been cut and cleaved into the platelets with a thickness of about 20m and an area of about 5mm x 3mm. Samples for the focused ion beam (FIB) structuring were obtained by dicing the platelets into small pieces of a bar-shape with an area of about 50m x 3mm. The superconducting transition temperature (*T*c) of the samples is about 85K. After four electrodes have been put on the bar by silver paste, IJJs have been fabricated at the center of the bar by using FIB as shown in Fig.3 (a). Detailed fabrication is described elsewhere (Kim et al. 1999). In the inset of Fig. 3 (a) and Fig.3 (b), a schematic drawing IJJ of the in-line symmetric type is shown used for four-probe measurements in the applied field *H* with ac (13 Hz:LR-700 AC resistance bridge) and dc current (Keithley 2400). Since the resistance of the IJJ part is 3 orders larger than that of the other part, we can measure the flow resistance of the IJJ part along the *c*-axis of the single

The samples have been aligned to the parallel field by measuring the flow resistance at constant field and current as a function of relative angle. When the field is close to parallel to the layers, JVs are in a lock-in state (Hechtfischer et al. 1997), and they flow collectively parallel to the layers with a Lorentz force, which causes flow voltage and hence the flow resistance obtained. The mid point of the center of the plateaus in the flow-resistance obtained from both direction of scanning angle has been taken as a tentatively aligned position to the superconducting layers with a resolution of 0.005°. Refinement of the aligned

et al. 1997; Mirkovic et al. 2001).

and finally conclusion in section 6.

crystal mainly as shown in Fig.3 (b).

**2. Experimental method** 

When the magnetic field is applied perpendicular to the superconducting layers of HTSCs, vortices in the superconducting layer distribute in a regular two-dimensional (2D) lattice of triangular or square as in the case of conventional superconductors. However, along the perpedicular direction to the layers, flux line is connected weakly to each other with pancake vortex in each layer. Vortices of HTSCs in the perpendicular field are called as pancake vortices (PVs). In the parallel field, based on the layered strucure of HTSCs, namely, intrinsic Josephson junctions (IJJs) (Kleiner et al., 1992; Rapp et al. 1996), vortices shown in Fig.2 (a) behave as those in the Josephson junctions artificially made of superconductor/ insulator/superconductor structure. In the JJs, Josephson vortex (JV) is formed in a characteristic state of vortex without normal state core (Bulaevskii & Clem 1991; Koshelev 2000). JV is contracted along the direction perpendicular to the layers and extends to the direction parallel to the layers, depending on the anisotropy. Details are discribed in section 3.1.

Fig. 2. Schematic drawing of distribution of Josephson vortices with a) isolated, b) dilute, and c) dense case.

Even in the perpendicular field to the superconducting layers, HTSCs show distinguished features of PVs in the vortex state. The first-order melting transition of PV-lattice (Zeldov et al. 1995), Bragg glass phase and vortex glass phase (Blatter et al. 1994) have been made clear experimentally and theoretically. In the parallel fields to the layers, magnetic field penetrates as JVs in a superconducting state of HTSCs. Theoretical studies have been made extensively from just after the discovery of HTSCs. Thermally-activated disordered vortex state, Kosterlitz-Thouless (K-T) transition in the smectic vortex state (Blatter et al. 1994; Balents & Nelson 1995), new vortex phases and a change of a tri-critical point in the dimension of the melting transition (Hu & Tachiki 2000)and the K-T type phase and the melting of 2D quasi-lattice have been predicted theoretically (Ivlev et al. 1990; Hu &

When the magnetic field is applied perpendicular to the superconducting layers of HTSCs, vortices in the superconducting layer distribute in a regular two-dimensional (2D) lattice of triangular or square as in the case of conventional superconductors. However, along the perpedicular direction to the layers, flux line is connected weakly to each other with pancake vortex in each layer. Vortices of HTSCs in the perpendicular field are called as pancake vortices (PVs). In the parallel field, based on the layered strucure of HTSCs, namely, intrinsic Josephson junctions (IJJs) (Kleiner et al., 1992; Rapp et al. 1996), vortices shown in Fig.2 (a) behave as those in the Josephson junctions artificially made of superconductor/ insulator/superconductor structure. In the JJs, Josephson vortex (JV) is formed in a characteristic state of vortex without normal state core (Bulaevskii & Clem 1991; Koshelev 2000). JV is contracted along the direction perpendicular to the layers and extends to the direction parallel to the layers, depending on the anisotropy. Details are discribed in section 3.1.

Fig. 2. Schematic drawing of distribution of Josephson vortices with a) isolated, b) dilute,

Even in the perpendicular field to the superconducting layers, HTSCs show distinguished features of PVs in the vortex state. The first-order melting transition of PV-lattice (Zeldov et al. 1995), Bragg glass phase and vortex glass phase (Blatter et al. 1994) have been made clear experimentally and theoretically. In the parallel fields to the layers, magnetic field penetrates as JVs in a superconducting state of HTSCs. Theoretical studies have been made extensively from just after the discovery of HTSCs. Thermally-activated disordered vortex state, Kosterlitz-Thouless (K-T) transition in the smectic vortex state (Blatter et al. 1994; Balents & Nelson 1995), new vortex phases and a change of a tri-critical point in the dimension of the melting transition (Hu & Tachiki 2000)and the K-T type phase and the melting of 2D quasi-lattice have been predicted theoretically (Ivlev et al. 1990; Hu &

and c) dense case.

Tachiki 2004). However, JV system has not been understood well experimentally. The melting transition of the JV system (Kwok et al. 1994), the oscillatory melting temperature (Gordeev et al. 2000), and the vortex smectic phase have been found only in YBa2Cu3O7- (YBCO). In strongly anisotropic HTSCs such as Bi2Sr2CaCu2O8+ (Bi-2212), few experimental results have been reported on the magnetic phases of the JV system (Fuhrer et al. 1997; Mirkovic et al. 2001).

Most of the difficulties to study on JV system in experiments are considered due to a little change of physical parameters at the phase boundary. Shear modulus of JVs between the layers is very little and it is considered that a change in free energy is so small at the boundary to be observed by the experiments (Shilling et al. 1997). Recently we have found a new method to study the magnetic phases of JV system by using the periodic oscillations in flow resistance of JVs (Ooi et al. 2002), from which we can deduce the configurations of JVs. This phenomenon is caused by the intrinsic boundary effect of the sample, related to the configurations of JVs (Ooi et al. 2002; Koshelev 2002; Machida 2003). The magnetic phase, in which the periodic oscillations are observed, has been confirmed as a three-dimensional (3D) ordered vortex state. The periodic oscillations are observed independent to temperature and magnetic field, which is a universal nature of JV flowresistance. Then, it is useful for determining an absolute magentic field. In this chapter, experimental methods for observing the periodic oscillations and for fabricating the sample for the measurements are described in section 2, general view of vortex flow in section 3, magnetic phases in section 4, application of the periodic oscillations in section 5, and finally conclusion in section 6.

## **2. Experimental method**

Single crystals of Bi-2212 were grown with a travelling solvent floating zone method (Mochiku et al., 1997). The crystals have been cut and cleaved into the platelets with a thickness of about 20m and an area of about 5mm x 3mm. Samples for the focused ion beam (FIB) structuring were obtained by dicing the platelets into small pieces of a bar-shape with an area of about 50m x 3mm. The superconducting transition temperature (*T*c) of the samples is about 85K. After four electrodes have been put on the bar by silver paste, IJJs have been fabricated at the center of the bar by using FIB as shown in Fig.3 (a). Detailed fabrication is described elsewhere (Kim et al. 1999). In the inset of Fig. 3 (a) and Fig.3 (b), a schematic drawing IJJ of the in-line symmetric type is shown used for four-probe measurements in the applied field *H* with ac (13 Hz:LR-700 AC resistance bridge) and dc current (Keithley 2400). Since the resistance of the IJJ part is 3 orders larger than that of the other part, we can measure the flow resistance of the IJJ part along the *c*-axis of the single crystal mainly as shown in Fig.3 (b).

The samples have been aligned to the parallel field by measuring the flow resistance at constant field and current as a function of relative angle. When the field is close to parallel to the layers, JVs are in a lock-in state (Hechtfischer et al. 1997), and they flow collectively parallel to the layers with a Lorentz force, which causes flow voltage and hence the flow resistance obtained. The mid point of the center of the plateaus in the flow-resistance obtained from both direction of scanning angle has been taken as a tentatively aligned position to the superconducting layers with a resolution of 0.005°. Refinement of the aligned

Josephson Vortices in High Tc Superconductors 141

between the superconducting layers through the insulating layer, which causes a complicated dissipation. On the other hand, the motion of JVs in HTSCs is quite different from that of PVs. JVs are located at the insulating layer as shown in Fig.2. They can easily move along the superconducting layers, but not along the perpendicular direction to the layers. This is so-called intrinsic pinning of JVs. For the applications of HTSCs to superconducting magnets, the intrinsic pinning mechanism is used to generate a magnetic

Flow of JVs has been studied in artificially-made Josephson junctions. In principle, JV flows very fast close to the light velocity because of a very low dissipation without normal state core (Fujimaki 1987; Bulaevskii 1991). However, there are many imperfections in real materials, which lead to the larger dissipation and reduce the velocity. In HTSCs, Josephson junctions are formed intrinsically, consisted of a stack of atomic scale layers as shown in Fig.1 (b). In strongly anisotropic superconductors such as HTSCs, JV core is

layers, as shown in Fig.2 (a). JV flow accompanies a movement of JV core, which causes a

to overlap and dense distribution of JVs form JV lattice as shown in Fig.2 (c). *B*cr in Bi-2212

= 500) is about 6 kOe with *s* = 1.5 nm. This value is relatively small compared to YBCO,

dependence to *B* and is proportional to the number of JVs. In the magnetic fields of *B*>*B*cr,

increasing magnetic fields, flow resistivity gradually deviates from linear dependency, and

Figure 4 shows a magnetic field dependence of JV flow-resistance of Bi-2212 measured at 70

estimated from the values in the last paragraph. Actually, the flow resistance increases linearly with magnetic field at lower magnetic fields, and shows a upward curvature at larger magnetic fields, which is the behaviour of the *in-plane* channel dominated flow resistivity (Koshelev 2000). At near the zero magnetic fields, the flow-resistance shows a downward curvature, which is considered as pinning effects of surface boundary of the

c4*s*4 and

c+0.27ab/

ab/c<<

c at the field *B* for strong in-plane dissipation. There are two kinds of

2). *B* is much larger than *B*cr. In Bi-2212, *B* is about 40 kOe for

*s*2, following to the definition of Koshelev 2000, where JVs start

about 10 and has a crossover magnetic field larger than a few

ab are the *c*-axis quasi-particle tunneling conductivity and

is the anisotropic parameter defined as

c)) and *s* is the interlayer spacing of superconducting

ab and the *c*-axis quasi-particle tunneling

Jff of JVs, we define the crossover

2)) for the magnetic field *B*<

2) and in-plane dissipation

ab/c<<2,

Jff shows linear

= 500,

c is the *c*-axis resistivity. With

c/ab

field along the insulating layers.

defined as Josephson length

(London penetration depth (

JV flow-resistivity

saturates to

ab/c>>

channel (

magnetic field as *B*cr=0/

which has a small value of

*B*cr (Koshelev 2000), where

ab = 50000 -1cm-1, and

resistivity

(

Jff~

Tesla.

**3.1 Flow of Josephson vortices in HTSCs** 

dissipation of the *in-plane* resistivity

c*B*2/(*B*2+*B*2), where *B*2=

c. Before discussing the flow resistivity

ab and 

Jff is expressed as

c and 

dissipation dominated; the c-axis dissipation channel (

*s*, where

Jff ~*s*2*B*/(0(

ab02/2<sup>2</sup>

K with dc current density of 60 A/cm2. Bi-2212 belongs to the resume of

the *in-plane* conductivity, respectively. In this magnetic field range,

c = 0.002 -1cm-1.

junctions and pinning centres intrinsically existed in the sample.

position to the field has been made to show the maximum flow-resistance. Details of the JV flow measurements are also described in Ref. (Ooi et al. 2002). Figure 3 (c) shows a typical curve of the flow resistance at 70 K. The flow resistance begins to oscillate periodically from below 10 kOe, reaches to its maximum at 20 kOe, and suddenly drops to zero at around 23 kOe. This drop is caused by the misalignment of the sample. When the magnetic component *H*c along the c-axis of the applied field *H* reaches to *H*c1, then, PVs penetrate to the sample, are pinned at pinning centers (crystal defects), cause JVs pinning with cooperative phenomena, and reduce the JV flow-resistance. Refinement of the aligned position to the field has been made to show the maximum of the flow-resistance as large as possible in magnetic field. Then, the resistance reaches to saturate with a finite value after the maximum with increasing the parallel field. This angle is set as an optimum value for evaluating a magnetic phase of JVs.

Fig. 3. (a) Secondary ion image of the measured sample after the FIB fabrication. Inset in (a) shows a schematic drawing of the sample geometry and applied magnetic field. (b) Schematic drawing of IJJs parts and electrodes configuration for the transport measurements. (c) Typical example of Josephson vortex flow-resistance against magnetic field.

In section 5, for the application of the periodic oscillations to measure a magnetic field precisely, pair of modulation coils has been set to the both side of the sample in Fig.3 (b) in addition to the main superconducting 8 Tesla split magnet. It produces about 200 Oe.

## **3. Flow resistance of Josephson vortices**

When vortices in superconductors are driven by the applied current perpendicular to the direction of magnetic field, the vortices flow by the Lorentz force in the direction perpendicular to the field and the current. Movement of the vortex core, which is in normal state, causes a dissipation and hence flow-resistance. The flow velocity is determined by the vortex states and also the superconducting media. Vortex states are related to the glass state, vortex-lattice state, for example. Superconductors have many kinds of imperfections; inhomogenity of the materials, defects (stacking fault, dislocations, etc), and impurities. These imperfections sometimes become pinning centers for vortices and the vortex states cause a flow-resistance (Tinkham 1975; Blatter 1994). In HTSCs, PVs flow as to the vortex motion of vortices in conventional superconductors, although they have a variety of vortex phases. As HTSCs have a high *T*c, the vortices are easily influenced by the thermal fluctuation. Furthermore, PVs are weakly connected

position to the field has been made to show the maximum flow-resistance. Details of the JV flow measurements are also described in Ref. (Ooi et al. 2002). Figure 3 (c) shows a typical curve of the flow resistance at 70 K. The flow resistance begins to oscillate periodically from below 10 kOe, reaches to its maximum at 20 kOe, and suddenly drops to zero at around 23 kOe. This drop is caused by the misalignment of the sample. When the magnetic component *H*c along the c-axis of the applied field *H* reaches to *H*c1, then, PVs penetrate to the sample, are pinned at pinning centers (crystal defects), cause JVs pinning with cooperative phenomena, and reduce the JV flow-resistance. Refinement of the aligned position to the field has been made to show the maximum of the flow-resistance as large as possible in magnetic field. Then, the resistance reaches to saturate with a finite value after the maximum with increasing the parallel field. This angle is set as an optimum value for

Fig. 3. (a) Secondary ion image of the measured sample after the FIB fabrication. Inset in (a) shows a schematic drawing of the sample geometry and applied magnetic field. (b) Schematic drawing of IJJs parts and electrodes configuration for the transport measurements. (c)

In section 5, for the application of the periodic oscillations to measure a magnetic field precisely, pair of modulation coils has been set to the both side of the sample in Fig.3 (b) in

When vortices in superconductors are driven by the applied current perpendicular to the direction of magnetic field, the vortices flow by the Lorentz force in the direction perpendicular to the field and the current. Movement of the vortex core, which is in normal state, causes a dissipation and hence flow-resistance. The flow velocity is determined by the vortex states and also the superconducting media. Vortex states are related to the glass state, vortex-lattice state, for example. Superconductors have many kinds of imperfections; inhomogenity of the materials, defects (stacking fault, dislocations, etc), and impurities. These imperfections sometimes become pinning centers for vortices and the vortex states cause a flow-resistance (Tinkham 1975; Blatter 1994). In HTSCs, PVs flow as to the vortex motion of vortices in conventional superconductors, although they have a variety of vortex phases. As HTSCs have a high *T*c, the vortices are easily influenced by the thermal fluctuation. Furthermore, PVs are weakly connected

addition to the main superconducting 8 Tesla split magnet. It produces about 200 Oe.

Typical example of Josephson vortex flow-resistance against magnetic field.

**3. Flow resistance of Josephson vortices** 

evaluating a magnetic phase of JVs.

between the superconducting layers through the insulating layer, which causes a complicated dissipation. On the other hand, the motion of JVs in HTSCs is quite different from that of PVs. JVs are located at the insulating layer as shown in Fig.2. They can easily move along the superconducting layers, but not along the perpendicular direction to the layers. This is so-called intrinsic pinning of JVs. For the applications of HTSCs to superconducting magnets, the intrinsic pinning mechanism is used to generate a magnetic field along the insulating layers.

### **3.1 Flow of Josephson vortices in HTSCs**

Flow of JVs has been studied in artificially-made Josephson junctions. In principle, JV flows very fast close to the light velocity because of a very low dissipation without normal state core (Fujimaki 1987; Bulaevskii 1991). However, there are many imperfections in real materials, which lead to the larger dissipation and reduce the velocity. In HTSCs, Josephson junctions are formed intrinsically, consisted of a stack of atomic scale layers as shown in Fig.1 (b). In strongly anisotropic superconductors such as HTSCs, JV core is defined as Josephson length *s*, where is the anisotropic parameter defined as c/ab (London penetration depth (ab and c)) and *s* is the interlayer spacing of superconducting layers, as shown in Fig.2 (a). JV flow accompanies a movement of JV core, which causes a dissipation of the *in-plane* resistivity ab and the *c*-axis quasi-particle tunneling resistivityc. Before discussing the flow resistivity Jff of JVs, we define the crossover magnetic field as *B*cr=0/*s*2, following to the definition of Koshelev 2000, where JVs start to overlap and dense distribution of JVs form JV lattice as shown in Fig.2 (c). *B*cr in Bi-2212 ( = 500) is about 6 kOe with *s* = 1.5 nm. This value is relatively small compared to YBCO, which has a small value of about 10 and has a crossover magnetic field larger than a few Tesla.

JV flow-resistivity Jff is expressed as Jff ~*s*2*B*/(0(c+0.27ab/2)) for the magnetic field *B*< *B*cr (Koshelev 2000), where c and ab are the *c*-axis quasi-particle tunneling conductivity and the *in-plane* conductivity, respectively. In this magnetic field range, Jff shows linear dependence to *B* and is proportional to the number of JVs. In the magnetic fields of *B*>*B*cr, Jff~c*B*2/(*B*2+*B*2), where *B*2=ab<sup>0</sup> 2/22c4*s*4 and c is the *c*-axis resistivity. With increasing magnetic fields, flow resistivity gradually deviates from linear dependency, and saturates to c at the field *B* for strong in-plane dissipation. There are two kinds of dissipation dominated; the c-axis dissipation channel (ab/c<<2) and in-plane dissipation channel (ab/c>>2). *B* is much larger than *B*cr. In Bi-2212, *B* is about 40 kOe for = 500, ab = 50000 -1cm-1, and c = 0.002 -1cm-1.

Figure 4 shows a magnetic field dependence of JV flow-resistance of Bi-2212 measured at 70 K with dc current density of 60 A/cm2. Bi-2212 belongs to the resume of ab/c<<2, estimated from the values in the last paragraph. Actually, the flow resistance increases linearly with magnetic field at lower magnetic fields, and shows a upward curvature at larger magnetic fields, which is the behaviour of the *in-plane* channel dominated flow resistivity (Koshelev 2000). At near the zero magnetic fields, the flow-resistance shows a downward curvature, which is considered as pinning effects of surface boundary of the junctions and pinning centres intrinsically existed in the sample.

Josephson Vortices in High Tc Superconductors 143

Figure 6 (a) shows a part of Fig.3. (c), plotted from 15 to 18 kOe. The oscillations show quite constant periodicity *H*p even in this magnetic field. The periodic oscillations are reproducebly observed in a wide temperature range from (*T*c-3) to 4.2 K, and in a wide magnetic field range from *B*cr to close to *B*, which will be discussed in determining magnetic phases of JVs. We have investigated the origin of the periodicity by measuring the flow resistance on several samples with different width *w*. The width *w* is defined as the sample size in the

Fig. 5. (a) Current-dependency of the flow-resistance in a small current range. (b) *I-V* characteristic in magnetic fields from 14 kOe to 16 kOe for every 20 Oe at 65K. Each curve is

Fig. 6. (a) Periodic oscillations in JV flow-resistance; enlarged part of Fig.3. (b). (b) Power spectrum of the periodic oscillations in fast Fourier transform on samples A, B and C. Inset shows width (*w*) dependency of the inverse of *H*p. The broken and solid lines represent

shifted by a 1 mV step.

1/*H*0(*w*) and its double, respectively.

perpendicular direction to the magnetic field sown in the inset of Fig.3(a).

Fig. 4. Magnetic field dependence of the JV flow-resistance in Bi-2212 measured at 70K with dc current density of 60 A/cm2. At lower magnetic fields, the flow-resistance shows linear magnetic field dependence, and saturates to a finite value at higher fields, which suggests the *in-plane* channel dominates in Bi-2212.

### **3.2 Periodic oscillations in flow resistance of Josephson vortices**

In Fig.4, the JV flow-resistance of Bi-2212 was measured with dc current of 0.15 mA and the junction size of width 17.6 m, length 14.0 m, and thickness 1 m. From the size of the sample, this current corresponds to the current density of about 60 A/cm2. Figure 5 (a) shows current density dependence of the flow resistance of the sample at 70 K with the angle from the superconducting layers of 0.010 degree. Decreasing the current density from 60 A/cm2, it can be clearly observed that the flow-resistance starts to oscillate at lower magnetic fields around 6-7 kOe below around 8 A/cm2 (0.020 mA), and the oscillations disappear at higher magnetic fields. At lower current density than 8 A/cm2, the oscillations become pronounced and the flow-resistance shows a maximum at around 40 kOe. The flow-resistance decreases from about 40 kOe, but still remains even at 70 kOe, which is the maximum magnetic field applied by the split superconducting magnet.

It is noted that the oscillations are observed only in sufficiently small current region. Figure 5 (b) shows an *I-V* characteristic of the junction size of width 18.0 m, length 16.5 m, and thickness 1 m at 65K. *I-V* curves are plotted in magnetic fields from 14 kOe to 16 kOe for every 20 Oe. Each curve is shifted by a 1 mV step. At the magnetic fields where the flow-resistance shows minimum, the nonlinearity of *I-V* curves becomes large and a kink structure is seen at around 30-50 A. The periodic nonlinearity corresponds to the periodic oscillation of the JV flow-resistance. In the current larger than 100 A, the nonlinearity disappears. The current 30 A corresponds to the current density of 9 A/cm2. Appearance of the periodic oscillations at 30 A (9 A/cm2) is quite well to coincide with the value of 8 A/cm2 in the last paragraph.

Fig. 4. Magnetic field dependence of the JV flow-resistance in Bi-2212 measured at 70K with dc current density of 60 A/cm2. At lower magnetic fields, the flow-resistance shows linear magnetic field dependence, and saturates to a finite value at higher fields, which suggests

In Fig.4, the JV flow-resistance of Bi-2212 was measured with dc current of 0.15 mA and the junction size of width 17.6 m, length 14.0 m, and thickness 1 m. From the size of the sample, this current corresponds to the current density of about 60 A/cm2. Figure 5 (a) shows current density dependence of the flow resistance of the sample at 70 K with the angle from the superconducting layers of 0.010 degree. Decreasing the current density from 60 A/cm2, it can be clearly observed that the flow-resistance starts to oscillate at lower magnetic fields around 6-7 kOe below around 8 A/cm2 (0.020 mA), and the oscillations disappear at higher magnetic fields. At lower current density than 8 A/cm2, the oscillations become pronounced and the flow-resistance shows a maximum at around 40 kOe. The flow-resistance decreases from about 40 kOe, but still remains even at 70 kOe, which is the maximum magnetic field applied by the split

It is noted that the oscillations are observed only in sufficiently small current region. Figure 5 (b) shows an *I-V* characteristic of the junction size of width 18.0 m, length 16.5 m, and thickness 1 m at 65K. *I-V* curves are plotted in magnetic fields from 14 kOe to 16 kOe for every 20 Oe. Each curve is shifted by a 1 mV step. At the magnetic fields where the flow-resistance shows minimum, the nonlinearity of *I-V* curves becomes large and a kink structure is seen at around 30-50 A. The periodic nonlinearity corresponds to the periodic oscillation of the JV flow-resistance. In the current larger than 100 A, the nonlinearity disappears. The current 30 A corresponds to the current density of 9 A/cm2. Appearance of the periodic oscillations at 30 A (9 A/cm2) is quite well to coincide with

**3.2 Periodic oscillations in flow resistance of Josephson vortices** 

the *in-plane* channel dominates in Bi-2212.

superconducting magnet.

the value of 8 A/cm2 in the last paragraph.

Figure 6 (a) shows a part of Fig.3. (c), plotted from 15 to 18 kOe. The oscillations show quite constant periodicity *H*p even in this magnetic field. The periodic oscillations are reproducebly observed in a wide temperature range from (*T*c-3) to 4.2 K, and in a wide magnetic field range from *B*cr to close to *B*, which will be discussed in determining magnetic phases of JVs. We have investigated the origin of the periodicity by measuring the flow resistance on several samples with different width *w*. The width *w* is defined as the sample size in the perpendicular direction to the magnetic field sown in the inset of Fig.3(a).

Fig. 5. (a) Current-dependency of the flow-resistance in a small current range. (b) *I-V* characteristic in magnetic fields from 14 kOe to 16 kOe for every 20 Oe at 65K. Each curve is shifted by a 1 mV step.

Fig. 6. (a) Periodic oscillations in JV flow-resistance; enlarged part of Fig.3. (b). (b) Power spectrum of the periodic oscillations in fast Fourier transform on samples A, B and C. Inset shows width (*w*) dependency of the inverse of *H*p. The broken and solid lines represent 1/*H*0(*w*) and its double, respectively.

Josephson Vortices in High Tc Superconductors 145

barrier to the flowing JV-lattice. The period *H*p is determined only by the effective width, which means that JV-lattice flows is strongly affected by the boundary (Ooi et al. 2004).

Fig. 8. (a) Beating in JV flow-resistance. The flow resistance was measured with changing the

the angle dependence of the flow resistance at 9 kOe. (b) Difference of two effective widths

The above mentioned interpretation can be applied to the *I-V* characteristics and the current density dependence of the oscillations in the magnetic fields qualitatively. When the surface potential is smaller than the energy of JV-system flowing with Lorentz force by the larger applied current, it is expected that the periodic oscillations would be smeared out. This is the speculation from the experimental results, but it was confirmed theoretically by analytical method (Koshelev 2002; Ikeda 2002) and numerical calculations

As the periodic oscillations can be observed in a lower current density, it is considered that as a ground state the JV-system is in a triangular lattice at intermediate magnetic fields. We

effect of the anisotropy on forming a triangular lattice (Yu S. 2004 & 2007). As this is related to the magnetic phase diagram of Bi-2212 described in the next section, we only mentioned here that the starting magnetic field *H*s of the periodic oscillations is inversely proportional

2004 & Hirata K. 2003). On contrary, it is useful to determine the anisotropic parameter from the JV flow resistance measurement directly. The periodic oscillations have a universal

. This is also one evidence to prove the theories and the formation of triangular lattie. Furthermore, the triangular lattice of JV system changes to the rectangular lattice with changing the sample size smaller and increasing the magnetuc fields (Yu S. 2004 & Hirata K. 2003). The boundary magnetic field *B*TR between triangular and rectangular lattice is defined

<sup>2</sup>*s*3 (Koshelev 2002), which is also well coincided with experiments (Yu S.

also have investigated the JV flow-resistance with changing the anisotropy

schematic drawing of the sample in an in-plane field is put in the inset to explain the

estimated from the periods of beatings. Solid curve shows *w*=|2*l*sin

, the beating effect becomes significant. Inset shows


to confirm the

*in-plane* angle

as a function of

(Machida 2003).

as *B*TR=*w*0/2

to  effective width and definition of *w*.

. With increasing angle

Figure 6 (b) shows the fast Fourier transform of the periodic oscillations on samples A (*w*=7.3 m), B (*w*=18.0 m) and C (*w*=31.0 m). The sharp fundamental peaks can be seen, which suggests that the periodicity *H*p is quite constant in a wide range of fields and over the samples. The small peaks correspond to the second harmonic waves, which are caused by a little distortion from the sinusoidal curves. In the inset of Fig.6 (b), the position of the peaks is plotted against sample width (*w*). All the samples show a linear dependence to width (*w*) with a slope of 2, but not 1.

Fig. 7. Schematic drawing of Josephson vortex configuration to explain the magnetic period of the flow-resistance. The period indicates the magnetic increment of 0/2 *ws*, which means that the Josephson vortices form a triangular lattice above *B*cr.

The value of the slope is very indicative for the interpretation of the periodic oscillations. When the JVs are supposed to be uniformly distributed in the junctions above *B*cr, with increasing magnetic fields, JVs penetrate into each junction. Then, the increment of magnetic field (*H*0(*w*)) to put one JV per one layer is 0/*ws*. Then, the slope 1/(0/*ws*) would be 1. However, the experiments show the value of 2. So, the slope is double of 1/(0/*ws*), and is rewritten as 2/(0/*ws*) =1/(0/2*ws*). This means that ,in a increment of magnetic field *H*p, one JV penetrates into every two-junctions.

The above experimental results can be explained by assuming that the lattice structure of JV system is triangular, and, at both sides of the sample, a surface barrier, Bean-Livingston barrier, for example, acts on the JVs' lattice. Figure 7 shows schematic drawings of the JV penetration into the junctions as (a) *H*p = *n*(*H*0(*w*)/2) and (b) *H*p = (*n*+1)(*H*0(*w*)/2), where *n* is the number of JV at certain magnetic field. Figures 7 (a) and (b) show the examples of JV system, in which the JVs are distributed in the ordered states, matched exactly with the size of the sample. In Fig.7. (a) of flowing JV-lattice system, JVs enter from the right side and exit from the left side simaultaneously. Then, the total potential in the JV-lattice becomes muximum. The average velosity of JV flow and hence the flow-resistance becomes minimum. With increasing the magnetic field from this sistuation, additional JVs are forced to enter into the ordered lattice by the applied current. The JV-lattice can then move more easily because the total potential decreases due to the mismatching between JV-lattice and the width of the sample. The flow resistance becomes larger than that of the matching situation. When the magnetic field exceeds (*n*+1/2)( *H*0(*w*) /2), the JV-lattice starts to form the next matching state as shown in Fig.7 (b). Therefore, it is plausible to consider that the oscillation period of the flow-resistance becomes half of *H*0(*w*). The effect of the boundary to the JV-lattice is crucial as shown in Fig.8. Difference in the two main effective width *w*=|2*l*sin| shown in the inset of Fig.8 (b) acts to determine the beating period as a surface

Figure 6 (b) shows the fast Fourier transform of the periodic oscillations on samples A (*w*=7.3 m), B (*w*=18.0 m) and C (*w*=31.0 m). The sharp fundamental peaks can be seen, which suggests that the periodicity *H*p is quite constant in a wide range of fields and over the samples. The small peaks correspond to the second harmonic waves, which are caused by a little distortion from the sinusoidal curves. In the inset of Fig.6 (b), the position of the peaks is plotted against sample width (*w*). All the samples show a linear dependence to

Fig. 7. Schematic drawing of Josephson vortex configuration to explain the magnetic period of the flow-resistance. The period indicates the magnetic increment of 0/2 *ws*, which means

The value of the slope is very indicative for the interpretation of the periodic oscillations. When the JVs are supposed to be uniformly distributed in the junctions above *B*cr, with increasing magnetic fields, JVs penetrate into each junction. Then, the increment of magnetic field (*H*0(*w*)) to put one JV per one layer is 0/*ws*. Then, the slope 1/(0/*ws*) would be 1. However, the experiments show the value of 2. So, the slope is double of 1/(0/*ws*), and is rewritten as 2/(0/*ws*) =1/(0/2*ws*). This means that ,in a increment of magnetic field *H*p,

The above experimental results can be explained by assuming that the lattice structure of JV system is triangular, and, at both sides of the sample, a surface barrier, Bean-Livingston barrier, for example, acts on the JVs' lattice. Figure 7 shows schematic drawings of the JV penetration into the junctions as (a) *H*p = *n*(*H*0(*w*)/2) and (b) *H*p = (*n*+1)(*H*0(*w*)/2), where *n* is the number of JV at certain magnetic field. Figures 7 (a) and (b) show the examples of JV system, in which the JVs are distributed in the ordered states, matched exactly with the size of the sample. In Fig.7. (a) of flowing JV-lattice system, JVs enter from the right side and exit from the left side simaultaneously. Then, the total potential in the JV-lattice becomes muximum. The average velosity of JV flow and hence the flow-resistance becomes minimum. With increasing the magnetic field from this sistuation, additional JVs are forced to enter into the ordered lattice by the applied current. The JV-lattice can then move more easily because the total potential decreases due to the mismatching between JV-lattice and the width of the sample. The flow resistance becomes larger than that of the matching situation. When the magnetic field exceeds (*n*+1/2)( *H*0(*w*) /2), the JV-lattice starts to form the next matching state as shown in Fig.7 (b). Therefore, it is plausible to consider that the oscillation period of the flow-resistance becomes half of *H*0(*w*). The effect of the boundary to the JV-lattice is crucial as shown in Fig.8. Difference in the two main effective width


that the Josephson vortices form a triangular lattice above *B*cr.

one JV penetrates into every two-junctions.

*w*=|2*l*sin

width (*w*) with a slope of 2, but not 1.

barrier to the flowing JV-lattice. The period *H*p is determined only by the effective width, which means that JV-lattice flows is strongly affected by the boundary (Ooi et al. 2004).

Fig. 8. (a) Beating in JV flow-resistance. The flow resistance was measured with changing the *in-plane* angle . With increasing angle , the beating effect becomes significant. Inset shows the angle dependence of the flow resistance at 9 kOe. (b) Difference of two effective widths as a function of estimated from the periods of beatings. Solid curve shows *w*=|2*l*sin|. A schematic drawing of the sample in an in-plane field is put in the inset to explain the effective width and definition of *w*.

The above mentioned interpretation can be applied to the *I-V* characteristics and the current density dependence of the oscillations in the magnetic fields qualitatively. When the surface potential is smaller than the energy of JV-system flowing with Lorentz force by the larger applied current, it is expected that the periodic oscillations would be smeared out. This is the speculation from the experimental results, but it was confirmed theoretically by analytical method (Koshelev 2002; Ikeda 2002) and numerical calculations (Machida 2003).

As the periodic oscillations can be observed in a lower current density, it is considered that as a ground state the JV-system is in a triangular lattice at intermediate magnetic fields. We also have investigated the JV flow-resistance with changing the anisotropy to confirm the effect of the anisotropy on forming a triangular lattice (Yu S. 2004 & 2007). As this is related to the magnetic phase diagram of Bi-2212 described in the next section, we only mentioned here that the starting magnetic field *H*s of the periodic oscillations is inversely proportional to . This is also one evidence to prove the theories and the formation of triangular lattie. Furthermore, the triangular lattice of JV system changes to the rectangular lattice with changing the sample size smaller and increasing the magnetuc fields (Yu S. 2004 & Hirata K. 2003). The boundary magnetic field *B*TR between triangular and rectangular lattice is defined as *B*TR=*w*0/2<sup>2</sup>*s*3 (Koshelev 2002), which is also well coincided with experiments (Yu S. 2004 & Hirata K. 2003). On contrary, it is useful to determine the anisotropic parameter from the JV flow resistance measurement directly. The periodic oscillations have a universal

Josephson Vortices in High Tc Superconductors 147

Fig. 9. Josephson-vortex flow resistance as a function of the parallel fields at the temperature close to *T*c. We can find a hump, continuously connected to the beginning of the oscillations

Figures 10 (a) and (b) show the flow resistance as a function of the parallel field at temperatures from 4.2 K to 80K on the same sample in Fig.9. Below 20 K, we can obseve a constant period *H*<sup>p</sup> (*H*<sup>p</sup> = 0/(2*ws*)) of the oscillations till 70 kOe. The flow resistance still increases in this temperature range. At intermediate temperatures from 30 K to 50K, it shows a maximum and decreases and becomes constant at higher magnetic fields. But at higher temperatures it increases again. The decrease in flow resistance at the intermediate temperatures is not caused by the misalignment of the sample as following. In the case of the misalignment about 0.03°, on which the results are shown in Fig.3 (c), the resistance shows a maximum around 20 kOe and a sharp drop to zero at about 23 kOe. Incorporation of the PVs with the *c*-axis component of the field, JVs are easily pinned to the pinning sites, because of the strong pinning of the PVs at lower temperatures in Bi-2212 and of considering the free energy of the pinned JVs to the PVs (Koshelev 1999). Decrease of the flow resistance after the maximum may be caused by a disordered state of JVs magnetically and thermally from the 3D-ordered state. Above 60 K, the pinning strength of the PVs in Bi-2212 is decreased, which can explain the increment of the flow resistance. Then, it is considered that, when the sample is close to the aligned configuration like this experiment, the finite flow resistance in the intermediate temperatures at higher magnetic fields is an intrinsic one in JV system. With increasing temperature and magnetic field, it is suggested that JV system changes from a triangular-lattice state to another one. The phase boundaries are drawn in Fig. 11 as a magnetic phase diagram of JV system. This may be considered as a magnetically induced transition at a constant temperature, and as a thermally induced transition at a constant field. In this state, JV system is not in an ordered state in 3D, along the *c*-axis or along the *a*- or *b*-axis also. This state may be in a 2D quasi-long-range ordered state suggested in Ref. (Hu &Tachiki 2000). Further experiments should be made to make

at 5.8 kOe with decreasing temperature.

clear the phase diagram of JVs.

**4.2 JV flow resistance at higher magnetic fields** 

nature of JV-system in a strongly anisotropic superconductors, and hence can be applied to determine the magnetic phases of JV-system and the magnetic field precisely, which will be disccussed in section 4 and 5, respectively.

## **4. Magnetic phases of Josephson vortices**

Magnetic phases of HTSCs have been well studied in the weakly anisotropic superconductors such as YBCO (Blatter et al. 1994). The first-order melting transition is one of the pronounced characters in HTSCs. In strongly anisoropic superconductors such as Bi-2212, theoretical approaches have been made extensively, but few experimental studies have been made due to the little change in free energy to confirm the phase boundary, as mentioned in the Introduction. One of the useful methods is to measure a flow resistance of JVs. This method is useful even in the vortex solid phase. In the solid phase, it is difficult to study the state because of that resistivity usually goes to zero due to the pinning effects from quenched disorder. However, a collective JV flow appears in the *c*-axis resistance even at low temperatures (Lee et al. 1995). The periodic oscillations mentionend above is an indirect experimental evidence of forming a triangular lattice in JV system of strongly anisotropic superconductors, but is considered to be a unique experimental probe to confirm the 3Dordered state in JVs. Therefore we applied this method to determine magnetic phases in Bi-2212.

## **4.1 JV flow resistance at lower magnetic fields**

To determine magnetic phases in Bi-2212, we apply the JV flow resistance method. The magnetic field region, in which we can observe the periodic oscillations, is assigned to the 3D-ordered state of JV system. In the measurements, the effect of the PVs is crucial, which are easily excited thermally in HTSCs with a strong anisotropy. As PVs are also incorporated with the misalignment of the parallel fields to the superconducting layers. However, the effect is much enhanced at higher magnetic fields, and there is no need to consider about the existence of PVs at lower fields. Figure 9 shows a JV flow resistance of the sample (width= 17 m, length = 14 m and thickness= 1 m) as a function of the lower parallel field at the temperature close to *T*c (85K) with an ac current of 1.0 A. The periodic oscillations start at around 80.5 K, and are clearly seen from 80.0 K at 5.8 kOe, which is denoted as an oscillation-starting magnetic field of *H*s. This magnetic field *H*<sup>s</sup> corresponds to the boundary between a liquid state and a triangular lattice state. *H*s has little temperature dependence seen from Fig.9. The boundary is almost independent of temperature. It is noted here that, at the temperatures without the oscillations in the flow resistance, the flow resistance increases proportional to the field and we can see a development of a hump shown as *H*h in Fig.9. Just before the appearance of the oscillations, the hump *H*h seemes to jump to the starting field *H*s of the oscillations with decreasing temperature. *H*h also appears continuously as a kink in the flow resistance at about 3 kOe independent of temperature at lower temperatures (not shown here). This value of *H*h is almost half of the starting field *H*s, and is considered to relate to the formation of the JV lattice with a half filling to the layers (Ikeda 2002). As the half filling state is one of ordered state (Ikeda 2002), the increment of the flow resistance will be stopped as indicated in the experiments, considering from the matching effect of the JV half filling lattice to the sample width.

nature of JV-system in a strongly anisotropic superconductors, and hence can be applied to determine the magnetic phases of JV-system and the magnetic field precisely, which will be

Magnetic phases of HTSCs have been well studied in the weakly anisotropic superconductors such as YBCO (Blatter et al. 1994). The first-order melting transition is one of the pronounced characters in HTSCs. In strongly anisoropic superconductors such as Bi-2212, theoretical approaches have been made extensively, but few experimental studies have been made due to the little change in free energy to confirm the phase boundary, as mentioned in the Introduction. One of the useful methods is to measure a flow resistance of JVs. This method is useful even in the vortex solid phase. In the solid phase, it is difficult to study the state because of that resistivity usually goes to zero due to the pinning effects from quenched disorder. However, a collective JV flow appears in the *c*-axis resistance even at low temperatures (Lee et al. 1995). The periodic oscillations mentionend above is an indirect experimental evidence of forming a triangular lattice in JV system of strongly anisotropic superconductors, but is considered to be a unique experimental probe to confirm the 3Dordered state in JVs. Therefore we applied this method to determine magnetic phases in Bi-

To determine magnetic phases in Bi-2212, we apply the JV flow resistance method. The magnetic field region, in which we can observe the periodic oscillations, is assigned to the 3D-ordered state of JV system. In the measurements, the effect of the PVs is crucial, which are easily excited thermally in HTSCs with a strong anisotropy. As PVs are also incorporated with the misalignment of the parallel fields to the superconducting layers. However, the effect is much enhanced at higher magnetic fields, and there is no need to consider about the existence of PVs at lower fields. Figure 9 shows a JV flow resistance of the sample (width= 17 m, length = 14 m and thickness= 1 m) as a function of the lower parallel field at the temperature close to *T*c (85K) with an ac current of 1.0 A. The periodic oscillations start at around 80.5 K, and are clearly seen from 80.0 K at 5.8 kOe, which is denoted as an oscillation-starting magnetic field of *H*s. This magnetic field *H*<sup>s</sup> corresponds to the boundary between a liquid state and a triangular lattice state. *H*s has little temperature dependence seen from Fig.9. The boundary is almost independent of temperature. It is noted here that, at the temperatures without the oscillations in the flow resistance, the flow resistance increases proportional to the field and we can see a development of a hump shown as *H*h in Fig.9. Just before the appearance of the oscillations, the hump *H*h seemes to jump to the starting field *H*s of the oscillations with decreasing temperature. *H*h also appears continuously as a kink in the flow resistance at about 3 kOe independent of temperature at lower temperatures (not shown here). This value of *H*h is almost half of the starting field *H*s, and is considered to relate to the formation of the JV lattice with a half filling to the layers (Ikeda 2002). As the half filling state is one of ordered state (Ikeda 2002), the increment of the flow resistance will be stopped as indicated in the experiments, considering from the matching effect of the JV

disccussed in section 4 and 5, respectively.

2212.

**4. Magnetic phases of Josephson vortices** 

**4.1 JV flow resistance at lower magnetic fields** 

half filling lattice to the sample width.

Fig. 9. Josephson-vortex flow resistance as a function of the parallel fields at the temperature close to *T*c. We can find a hump, continuously connected to the beginning of the oscillations at 5.8 kOe with decreasing temperature.

#### **4.2 JV flow resistance at higher magnetic fields**

Figures 10 (a) and (b) show the flow resistance as a function of the parallel field at temperatures from 4.2 K to 80K on the same sample in Fig.9. Below 20 K, we can obseve a constant period *H*<sup>p</sup> (*H*<sup>p</sup> = 0/(2*ws*)) of the oscillations till 70 kOe. The flow resistance still increases in this temperature range. At intermediate temperatures from 30 K to 50K, it shows a maximum and decreases and becomes constant at higher magnetic fields. But at higher temperatures it increases again. The decrease in flow resistance at the intermediate temperatures is not caused by the misalignment of the sample as following. In the case of the misalignment about 0.03°, on which the results are shown in Fig.3 (c), the resistance shows a maximum around 20 kOe and a sharp drop to zero at about 23 kOe. Incorporation of the PVs with the *c*-axis component of the field, JVs are easily pinned to the pinning sites, because of the strong pinning of the PVs at lower temperatures in Bi-2212 and of considering the free energy of the pinned JVs to the PVs (Koshelev 1999). Decrease of the flow resistance after the maximum may be caused by a disordered state of JVs magnetically and thermally from the 3D-ordered state. Above 60 K, the pinning strength of the PVs in Bi-2212 is decreased, which can explain the increment of the flow resistance. Then, it is considered that, when the sample is close to the aligned configuration like this experiment, the finite flow resistance in the intermediate temperatures at higher magnetic fields is an intrinsic one in JV system. With increasing temperature and magnetic field, it is suggested that JV system changes from a triangular-lattice state to another one. The phase boundaries are drawn in Fig. 11 as a magnetic phase diagram of JV system. This may be considered as a magnetically induced transition at a constant temperature, and as a thermally induced transition at a constant field. In this state, JV system is not in an ordered state in 3D, along the *c*-axis or along the *a*- or *b*-axis also. This state may be in a 2D quasi-long-range ordered state suggested in Ref. (Hu &Tachiki 2000). Further experiments should be made to make clear the phase diagram of JVs.

Josephson Vortices in High Tc Superconductors 149

get rid of the effect of the PVs caused by the misalignment in the higher magnetic fields. Then, the decrease of the resistance in this region may be caused by the excitation of abundant amount of pancake-vortex/anti-vortex pairs, thermally and magnetically, because the pinning force of the PVs is very weak in this magnetic field and temperature range far

Fig. 11. Magnetic phase diagram of Josephson vortices in Bi-2212, obtained from the flowresistance measurements. Open and solid symbols correspond to the boundaries defined on

To study the JV state above/below the upper boundary *H*e, shown in Fig.11 as the region A/B, respectively, *I-V* measurements have been made. In Fig.12, the power-law dependence

the *I-V* characteristics. In the figure, the data are plotted every 10 kOe, and are shifted in both sides at every magnetic field to see clearly. The measurements were made in the current range well below the critical current density of Bi-2212, in which the periodic

Hu and Tachiki (Hu & Tachiki 2000) have studied on JV system, the first-order melting transition, phase transition between 3D ordered state and 2D QLRO (quasi-long-range ordered) state, and also the power-law dependence in K-T phase with Monte Carlo simulations. The region B in Fig.11 obtained from the present measurements is considered to correspond to the K-T phase in Fig.10 of Ref. (Hu & Tachiki 2000). Temperature dependence

which decreases with increasing temperature. Assuming that the behavior of *Y* in tempera-

, respectively.

(T,H) of *V=*A*I*

=1+*Y*/(kB*T*) of Ref. (Hu & Tachiki 2004).

(T,H) is inferred from the temperature dependence

= 8 and 20 in Fig.10 of Ref. (Hu & Tachiki 2000),

, obtained from

(T,H) to 1.6 and

is well

(*T*,*H*) decreases with

above the irreversibility region of the PVs.

the sample in Fig. 10 and another sample with smaller

to the magnetic field in sample A is shown as the exponent

coincided with the magnetic phase diagram shown in Fig.11.

of (*T*,*H*) is not explicitly shown in Eq.18;

However, the behavior of the exponent

of the calculated helicity modulus *Y* with

oscillations can be observed in the region A. It shows a distinct change in

2.0 above 50 kOe, compared with those at lower magnetic fields. The change in

ture is the same as that in a larger anisotropic value of Bi-2212,

Fig. 10. Josephson-vortex flow resistance as a function of the parallel fields at various temperatures (a) and the enlarged part of (b) at higher magnetic fields (b). *H*e denotes the end of the periodic oscillations.

The flow resistance increases monotonously with the periodic oscillations at lower temperatures with increasing magnetic field. With increasing temperature, the flow resistance always shows a maximum and decreases to a finite value in the flow resistance, in which the oscillations stop just before reaching to the finite flow resistance. To define the characteristic magnetic field *H*e, a criterion was taken at some magnitude of the resistivity in the amplitude of the oscillations. Reproducibility was confirmed several times with changing the angle between the superconducting layers and the magnetic field, which showed a few kOe as in experimental error. Therefore, this characteristic feature of the flow resistance at higher temperature is considered as an intrinsic behaviour in JV flow.

In Fig.11, the phase boundaries obtained from the flow resistance measurements are shown. The magnetic field *H*e in Fig.10 (b), indicated by the arrow, corresponds to the upper boundary of the 3D-ordered phase, which is defined as the magnetic field for the oscillations to be stopped. The boundary *H*e decreases steeply at the temperatures lower than 40 K, is saturated at about 50kOe, and suddenly decreases at about 75 K towards 0 kOe at *T*c. This means that the oscillations continue even in higher magnetic fields than 70kOe below 40 K, and JV system is in the 3D-ordered state below *H*e and above *H*s boundaries. It is noted here that *H*s is almost constant with temperature because of depending only on the anisotropy and the samples width *w*.

Considering about the JV state in this magnetic field and temperature region, it is suggested that the JV state may really change from the 3D-ordered state to the other. In the flow resistance above *H*e, the resistance always decreases to a finite value after the periodic oscillations in the region B of Fig.11, which may suggest that the JV state is in disordered along the *c*-axis, as the *in-plane* disorder causes an enhancement of the flow resistance with less effective potential barrier to the dynamical movement of JVs. With changing the angle between the layers and the magnetic field, little effect and the less systematic changes to the flow resistance were observed concerning on the decrement of the resistance. Then, we can

Fig. 10. Josephson-vortex flow resistance as a function of the parallel fields at various temperatures (a) and the enlarged part of (b) at higher magnetic fields (b). *H*e denotes the

higher temperature is considered as an intrinsic behaviour in JV flow.

The flow resistance increases monotonously with the periodic oscillations at lower temperatures with increasing magnetic field. With increasing temperature, the flow resistance always shows a maximum and decreases to a finite value in the flow resistance, in which the oscillations stop just before reaching to the finite flow resistance. To define the characteristic magnetic field *H*e, a criterion was taken at some magnitude of the resistivity in the amplitude of the oscillations. Reproducibility was confirmed several times with changing the angle between the superconducting layers and the magnetic field, which showed a few kOe as in experimental error. Therefore, this characteristic feature of the flow resistance at

In Fig.11, the phase boundaries obtained from the flow resistance measurements are shown. The magnetic field *H*e in Fig.10 (b), indicated by the arrow, corresponds to the upper boundary of the 3D-ordered phase, which is defined as the magnetic field for the oscillations to be stopped. The boundary *H*e decreases steeply at the temperatures lower than 40 K, is saturated at about 50kOe, and suddenly decreases at about 75 K towards 0 kOe at *T*c. This means that the oscillations continue even in higher magnetic fields than 70kOe below 40 K, and JV system is in the 3D-ordered state below *H*e and above *H*s boundaries. It is noted here that *H*s is almost constant with temperature because of depending only on the anisotropy

Considering about the JV state in this magnetic field and temperature region, it is suggested that the JV state may really change from the 3D-ordered state to the other. In the flow resistance above *H*e, the resistance always decreases to a finite value after the periodic oscillations in the region B of Fig.11, which may suggest that the JV state is in disordered along the *c*-axis, as the *in-plane* disorder causes an enhancement of the flow resistance with less effective potential barrier to the dynamical movement of JVs. With changing the angle between the layers and the magnetic field, little effect and the less systematic changes to the flow resistance were observed concerning on the decrement of the resistance. Then, we can

end of the periodic oscillations.

and the samples width *w*.

get rid of the effect of the PVs caused by the misalignment in the higher magnetic fields. Then, the decrease of the resistance in this region may be caused by the excitation of abundant amount of pancake-vortex/anti-vortex pairs, thermally and magnetically, because the pinning force of the PVs is very weak in this magnetic field and temperature range far above the irreversibility region of the PVs.

Fig. 11. Magnetic phase diagram of Josephson vortices in Bi-2212, obtained from the flowresistance measurements. Open and solid symbols correspond to the boundaries defined on the sample in Fig. 10 and another sample with smaller , respectively.

To study the JV state above/below the upper boundary *H*e, shown in Fig.11 as the region A/B, respectively, *I-V* measurements have been made. In Fig.12, the power-law dependence to the magnetic field in sample A is shown as the exponent (T,H) of *V=*A*I* , obtained from the *I-V* characteristics. In the figure, the data are plotted every 10 kOe, and are shifted in both sides at every magnetic field to see clearly. The measurements were made in the current range well below the critical current density of Bi-2212, in which the periodic oscillations can be observed in the region A. It shows a distinct change in (T,H) to 1.6 and 2.0 above 50 kOe, compared with those at lower magnetic fields. The change in is well coincided with the magnetic phase diagram shown in Fig.11.

Hu and Tachiki (Hu & Tachiki 2000) have studied on JV system, the first-order melting transition, phase transition between 3D ordered state and 2D QLRO (quasi-long-range ordered) state, and also the power-law dependence in K-T phase with Monte Carlo simulations. The region B in Fig.11 obtained from the present measurements is considered to correspond to the K-T phase in Fig.10 of Ref. (Hu & Tachiki 2000). Temperature dependence of (*T*,*H*) is not explicitly shown in Eq.18; =1+*Y*/(kB*T*) of Ref. (Hu & Tachiki 2004). However, the behavior of the exponent (T,H) is inferred from the temperature dependence of the calculated helicity modulus *Y* with = 8 and 20 in Fig.10 of Ref. (Hu & Tachiki 2000), which decreases with increasing temperature. Assuming that the behavior of *Y* in temperature is the same as that in a larger anisotropic value of Bi-2212, (*T*,*H*) decreases with

Josephson Vortices in High Tc Superconductors 151

Fig. 13. Temperature dependence of the flow resistance at a constant magnetic field measured in a zero-field cooling&warming-up process. Inset shows the flow resistance at

However, the magnetic field, at which the flow resistance suddenly drops, cannot be seen in the temperature dependence of magnetic-field cycles at every temperature. There exists only a boundary between the regions on the periodic oscillations and of the finite flow resistance without the oscillations at high temperatures and magnetic fields (Hirata et al. 2004). This boundary may correspond to the continuously melting transition (Hu & Tachiki 2004). In the field-cooling process, it is considered that these PV pairs are excited thermally at higher temperatures than the irreversibility temperature, they are frozen to the pinning centers with decreasing temperature, and the flow resistance does not show any sharp increment at low temperatures. Although there are still incomprehensive problems in the JV system, for the application, we have to avoid a temperature increase of the sensor higher than 50 K at 20 kOe,

Using a pair of small coils, we have measured the JV flow resistance. Figure 14 shows the results of the JV flow resistance measured at the main magnetic field *H*m with an interval of 20 Oe, an ac current of 1.0 A, and the size of the junction *t* = 1.5 m, *w* = 11.2m, *l* = 12.4 m at 70.0 K as an example. The period *H*p is obtained as about 900 Oe independent of temperature and magnetic field, which is well coincided with the calculated value *H*p as 924 Oe, considering about Josephson penetration length and the damaged width by the FIBfabrication of the sample (Hirata et al. 2004). Fixing the field *H*m at 15.2 kOe, a modulation field *H*i was applied to measure a resistance change by the JV flow. The signal was directly obtained from LR-700 as a raw output voltage without any filters, which is shown in Fig.14. To compare the magnitude of the flow resistance at 15.2 kOe of 70.0 K in Fig.13 with that in Fig.14, it is just a bit smaller than that about 30 ohm. Carrying out the measurements of the flow resistance, we have recognized that the *c*-axis resistance becomes very sensitive in superconducting state. By the extra set-up of the coils to the normal measurements, it may

In Fig.14, it is apparently seen that the signal is well scattered. This means that the JV flow velocity changes within less than 0.1 sec, because the measurements were made with the ac current of 13.6 Hz. This has been discussed by Machida (Machida 2003). The JV flow is a

lower magnetic field region.

for example, as shown in Fig.13.

cause some noise in the circuit of LR-700.

increasing temperature. This shows a little different behavior from the experimental results. Furthermore, there have not been observed the oscillatory behavior of the melting transition and the first-order phase transition in the experiments on Bi-2212. It may need much more sensitive measurements than the present experimental set-up. Further experiments will be needed to make this JV state clear.

Fig. 12. Magnetic field dependence of the exponent (*T*,*H*) in the *I-V* characteristics at 5, 30, 35, 60 K with *V=*A*I .*

We have tried to confirm the effect of the anisotropy in the boundary with changing the anisotropy (Yu et al. 2005 & 2007). The tendency of the change of *H*e is a good indication to increase the doping level of carriers into Bi-2212, namely, decreasing the anisotropic parameter , which can be related to the weaker anisotropic superconductors such as YBCO continuously.

### **5. Application of periodic oscillations for magnetic sensor**

With magnetic-field cycles, the flow resistance reproduces quite well in a wide range of temperature, in which the oscillations can be observed. However, with heating/cooling cycles at a constant magnetic field, the flow resistance shows an irreversible feature as shown in Fig.13. After applying the magnetic field up to 20 kOe at 5 K with a zero-field cooling and confirming the periodic oscillations, for example, with increasing temperature, the flow resistance suddenly drops close to zero at about 50 K, even if a finite flow resistance can be observed at the same temperature and magnetic field with the zero-field cooling and magnetic-field cycles. This can be seen at every magnetic field where the oscillations can be observed. This may be closely related to the JV states, which have never been made clear before on the strongly anisotropic HTSCs experimentally. It has been theoretically discussed just after the discovery of HTSCs (Chakravaty et al. 1990; Blatter et al. 1994), and recently with Monte-Carlo simulations (Hu & Tachiki 2004). In the JV system, it is predicted that pairs of pancake vortex-anti-vortex are easily excited thermally. If these pancake pairs are excited in the JV system, they are easily pinned at the intrinsic pinning centers of Bi-2212, because this region belongs to the irreversible one in the magnetic phase diagram of Bi-2212 with the perpendicular field.

increasing temperature. This shows a little different behavior from the experimental results. Furthermore, there have not been observed the oscillatory behavior of the melting transition and the first-order phase transition in the experiments on Bi-2212. It may need much more sensitive measurements than the present experimental set-up. Further experiments will be

Fig. 12. Magnetic field dependence of the exponent (*T*,*H*) in the *I-V* characteristics at 5, 30,

anisotropy (Yu et al. 2005 & 2007). The tendency of the change of *H*e is a good indication to increase the doping level of carriers into Bi-2212, namely, decreasing the anisotropic

With magnetic-field cycles, the flow resistance reproduces quite well in a wide range of temperature, in which the oscillations can be observed. However, with heating/cooling cycles at a constant magnetic field, the flow resistance shows an irreversible feature as shown in Fig.13. After applying the magnetic field up to 20 kOe at 5 K with a zero-field cooling and confirming the periodic oscillations, for example, with increasing temperature, the flow resistance suddenly drops close to zero at about 50 K, even if a finite flow resistance can be observed at the same temperature and magnetic field with the zero-field cooling and magnetic-field cycles. This can be seen at every magnetic field where the oscillations can be observed. This may be closely related to the JV states, which have never been made clear before on the strongly anisotropic HTSCs experimentally. It has been theoretically discussed just after the discovery of HTSCs (Chakravaty et al. 1990; Blatter et al. 1994), and recently with Monte-Carlo simulations (Hu & Tachiki 2004). In the JV system, it is predicted that pairs of pancake vortex-anti-vortex are easily excited thermally. If these pancake pairs are excited in the JV system, they are easily pinned at the intrinsic pinning centers of Bi-2212, because this region belongs to the irreversible one in the magnetic phase diagram of Bi-2212

, which can be related to the weaker anisotropic superconductors such as YBCO

in the boundary with changing the

needed to make this JV state clear.

35, 60 K with *V=*A*I*

with the perpendicular field.

parameter

continuously.

*.*

We have tried to confirm the effect of the anisotropy

**5. Application of periodic oscillations for magnetic sensor** 

Fig. 13. Temperature dependence of the flow resistance at a constant magnetic field measured in a zero-field cooling&warming-up process. Inset shows the flow resistance at lower magnetic field region.

However, the magnetic field, at which the flow resistance suddenly drops, cannot be seen in the temperature dependence of magnetic-field cycles at every temperature. There exists only a boundary between the regions on the periodic oscillations and of the finite flow resistance without the oscillations at high temperatures and magnetic fields (Hirata et al. 2004). This boundary may correspond to the continuously melting transition (Hu & Tachiki 2004). In the field-cooling process, it is considered that these PV pairs are excited thermally at higher temperatures than the irreversibility temperature, they are frozen to the pinning centers with decreasing temperature, and the flow resistance does not show any sharp increment at low temperatures. Although there are still incomprehensive problems in the JV system, for the application, we have to avoid a temperature increase of the sensor higher than 50 K at 20 kOe, for example, as shown in Fig.13.

Using a pair of small coils, we have measured the JV flow resistance. Figure 14 shows the results of the JV flow resistance measured at the main magnetic field *H*m with an interval of 20 Oe, an ac current of 1.0 A, and the size of the junction *t* = 1.5 m, *w* = 11.2m, *l* = 12.4 m at 70.0 K as an example. The period *H*p is obtained as about 900 Oe independent of temperature and magnetic field, which is well coincided with the calculated value *H*p as 924 Oe, considering about Josephson penetration length and the damaged width by the FIBfabrication of the sample (Hirata et al. 2004). Fixing the field *H*m at 15.2 kOe, a modulation field *H*i was applied to measure a resistance change by the JV flow. The signal was directly obtained from LR-700 as a raw output voltage without any filters, which is shown in Fig.14. To compare the magnitude of the flow resistance at 15.2 kOe of 70.0 K in Fig.13 with that in Fig.14, it is just a bit smaller than that about 30 ohm. Carrying out the measurements of the flow resistance, we have recognized that the *c*-axis resistance becomes very sensitive in superconducting state. By the extra set-up of the coils to the normal measurements, it may cause some noise in the circuit of LR-700.

In Fig.14, it is apparently seen that the signal is well scattered. This means that the JV flow velocity changes within less than 0.1 sec, because the measurements were made with the ac current of 13.6 Hz. This has been discussed by Machida (Machida 2003). The JV flow is a

Josephson Vortices in High Tc Superconductors 153

the sensors at 4.2 K and a fixed magnetic field, we can get a precise control of the magnetic

In conclusion, we have measured JV flow-resistance on the in-line symmetric IJJs of Bi-2212 single crystals to study on the flow mechanism of JVs and on the dynamics of moving JVs. A pronounced feature has been found in the flow resistance. Periodic oscillations have been observed with sweeping the magnetic field parallel to the layers. The oscillations have been found in the wide range of the parallel fields larger than 6kOe and only in the smaller current region than about 10 A/cm2. The period of the oscillations is inversely proportional to the width of the samples, and it corresponds to adding "one" flux quantum per "two" IJJs. These results suggest that the JVs form a triangular lattice as a ground state, which is related to the commensurability between the sample width and the lattice spacing of JVs along the layers in the present experiments

We have studied the JV states with measuring the flow resistance. From the measurements, we can determine the 3D-ordered phase of JVs, in which JVs are well ordered. This has been never determined with other methods in strongly anisotropic superconductors. Furthermore, by the measurements of temperature dependence of the flow resistance, we find a sudden drop in the resistance in the zero-field cooling&warming-up process. The sudden drop is explained by the thermal and magnetical excitation of pancake vortex/antivortex pairs, which is an intrinsic property of Bi-2212. The boundary of the excitation exists between 20-30K above 15kOe and above 30K at lower magnetic fields. This boundary limits

We also have characterized the JV flow device as a magnetic sensor to measure the flowresistance in Bi-2212 IJJs, using a remarkable feature of the periodic oscillations. The device has an irreversible character in temperature-cycles at high magnetic fields, which is caused by the thermal excitation of pancake vortex-anti-vortex pairs. Using a pair of small coils under high magnetic fields, good linearity of the flow-resistance was obtained to the internal field *H*i. There is also a problem with noise, which may be caused by the intrinsic behavior of JV flow. However, it can be improved easily by the measurement method. The linearity of the flow-resistance to the small field *H*i indicate that, once the device is calibrated, it has a high potential to measure and control high magnetic fields in Tesla range with a T

I acknowledged to Dr. S. Ooi for the fabrications of junctions and for useful discussions, and

Bean C.P. & Livingstone J.D. (1964). Surface Barrier in Type-II Superconductors, *Phys. Rev.* 

the application of HTSC materials in the magnetic fields.

Dr. T. Mochiku for providing us single crystals of Bi-2212.

*Letters* Vol.12, pp.14-16

field within a T range.

**6. Conclusion** 

using small IJJs.

accuracy.

**7. Acknowledgment** 

**8. References** 

dynamical effect, interacting to each other. And JVs go over the potential barriers at the both edges of the sample, which is one of the principal reasons for the periodic oscillations, while inside the sample edges JVs move almost freely accompanied with some dissipation by the quasi-particles (Koshelev 2000). When we take a snapshot to see the behavior of the JV flow, there might be some distribution in velocity of JV-lattice flow. The velocity is determined mainly by the matching effect of JV lattice to the width of the sample (Machida 2003). Making these noise characteristics clear, it may be effective to take a noise spectrum of the flow voltage, which will be developed and obtained near future.

Fig. 14. Flow-resistance (a), (b) and (c) at the main fields *H*m=15.2 kOe as a function of a modulation field *H*i of small coils at 5, 50 and 70 K, respectively. The solid lines show a fitting line to the data, and show a good linearlity with a slope of 0.1776, 0.1757 and 0.2586 ohm/Oe, respectively.

It is worth for being noted that the linearity of the flow resistance to *H*i is relatively good, which is shown as solid lines in Fig.14. The solid lines were obtained with a square-root minimum fitting method. The slope of the line is calculated as 0.2568 ohm/Oe at 70 K shown in Fig.14 (c), for example. The fitting line is considered to indicate average values in time at each field. And the measurement is easily improved to make averaging the voltage in the interval of a few second or longer. When we need to measure a high magnetic field precisely, we are usually using the magnetic field in a steady state; apparatus of nuclear magnetic resonance and magnetic resonance imaging, for example. It is also benefit for this magnetic-field sensor to put into the low *T*c superconducting magnets, as they are using at a liquid He temperature, and the temperature is very stable. Once if we make a calibration of the sensors at 4.2 K and a fixed magnetic field, we can get a precise control of the magnetic field within a T range.

## **6. Conclusion**

152 Superconductors – Properties, Technology, and Applications

dynamical effect, interacting to each other. And JVs go over the potential barriers at the both edges of the sample, which is one of the principal reasons for the periodic oscillations, while inside the sample edges JVs move almost freely accompanied with some dissipation by the quasi-particles (Koshelev 2000). When we take a snapshot to see the behavior of the JV flow, there might be some distribution in velocity of JV-lattice flow. The velocity is determined mainly by the matching effect of JV lattice to the width of the sample (Machida 2003). Making these noise characteristics clear, it may be effective to take a noise spectrum of the

**Magnetic Field** *H* **i [Oe]** 

It is worth for being noted that the linearity of the flow resistance to *H*i is relatively good, which is shown as solid lines in Fig.14. The solid lines were obtained with a square-root minimum fitting method. The slope of the line is calculated as 0.2568 ohm/Oe at 70 K shown in Fig.14 (c), for example. The fitting line is considered to indicate average values in time at each field. And the measurement is easily improved to make averaging the voltage in the interval of a few second or longer. When we need to measure a high magnetic field precisely, we are usually using the magnetic field in a steady state; apparatus of nuclear magnetic resonance and magnetic resonance imaging, for example. It is also benefit for this magnetic-field sensor to put into the low *T*c superconducting magnets, as they are using at a liquid He temperature, and the temperature is very stable. Once if we make a calibration of

Fig. 14. Flow-resistance (a), (b) and (c) at the main fields *H*m=15.2 kOe as a function of a modulation field *H*i of small coils at 5, 50 and 70 K, respectively. The solid lines show a fitting line to the data, and show a good linearlity with a slope of 0.1776, 0.1757 and 0.2586


**15.2 kOe**

**0.1757 ohm/Oe**

**0.1776 ohm/Oe**

**0.2586 ohm/Oe**

flow voltage, which will be developed and obtained near future.

**(a)** 

**(b)** 

**(c)** 

170

ohm/Oe, respectively.

170

171

**Flow-resistance [ohm]** 

**50.0 K** 

**5.0 K** 

**70.0 K** 

In conclusion, we have measured JV flow-resistance on the in-line symmetric IJJs of Bi-2212 single crystals to study on the flow mechanism of JVs and on the dynamics of moving JVs. A pronounced feature has been found in the flow resistance. Periodic oscillations have been observed with sweeping the magnetic field parallel to the layers. The oscillations have been found in the wide range of the parallel fields larger than 6kOe and only in the smaller current region than about 10 A/cm2. The period of the oscillations is inversely proportional to the width of the samples, and it corresponds to adding "one" flux quantum per "two" IJJs. These results suggest that the JVs form a triangular lattice as a ground state, which is related to the commensurability between the sample width and the lattice spacing of JVs along the layers in the present experiments using small IJJs.

We have studied the JV states with measuring the flow resistance. From the measurements, we can determine the 3D-ordered phase of JVs, in which JVs are well ordered. This has been never determined with other methods in strongly anisotropic superconductors. Furthermore, by the measurements of temperature dependence of the flow resistance, we find a sudden drop in the resistance in the zero-field cooling&warming-up process. The sudden drop is explained by the thermal and magnetical excitation of pancake vortex/antivortex pairs, which is an intrinsic property of Bi-2212. The boundary of the excitation exists between 20-30K above 15kOe and above 30K at lower magnetic fields. This boundary limits the application of HTSC materials in the magnetic fields.

We also have characterized the JV flow device as a magnetic sensor to measure the flowresistance in Bi-2212 IJJs, using a remarkable feature of the periodic oscillations. The device has an irreversible character in temperature-cycles at high magnetic fields, which is caused by the thermal excitation of pancake vortex-anti-vortex pairs. Using a pair of small coils under high magnetic fields, good linearity of the flow-resistance was obtained to the internal field *H*i. There is also a problem with noise, which may be caused by the intrinsic behavior of JV flow. However, it can be improved easily by the measurement method. The linearity of the flow-resistance to the small field *H*i indicate that, once the device is calibrated, it has a high potential to measure and control high magnetic fields in Tesla range with a T accuracy.

## **7. Acknowledgment**

I acknowledged to Dr. S. Ooi for the fabrications of junctions and for useful discussions, and Dr. T. Mochiku for providing us single crystals of Bi-2212.

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**1. Introduction**

Josephson junction.

In 1963, Shapiro did an experiment for the microwave irradiation effect on the *I*-*V* characteristics of a Josephson junction Al/Al2O3/Sn and showed that when the microwave is impressed to the Josephson junction, clear current steps are observed on the quasiparticle (QP) branches of the *I*-*V* characteristics and those steps, which are now called "Shapiro steps", are observed on constant voltages given by *n*Φ<sup>0</sup> *fr* (*n* = ±1, ±2, ...) [1]. Here the Φ<sup>0</sup> is a flux quantum (<sup>=</sup> 2.06785 <sup>×</sup> <sup>10</sup>−<sup>15</sup> Wb) defined by *<sup>h</sup>*/2*<sup>e</sup>* using Planck constant *<sup>h</sup>* and the *fr* is the frequency of the impressed microwave. The interactions between the Josephson junction and the electromagnetic field are important and intersting problems on not only the fundamental but also the applicational points of view. The Shapiro steps are one of the important phenomena observed on a superconductor-insulator-superconductor (SIS)-type

*Department of Electrical and Electronic System Engineering, Utsunomiya University,* 

**Shapiro Steps in BSCCO Intrinsic** 

**Josephson Junctions** 

Michihide Kitamura

*Japan* 

**8**

*Yoto, Utsunomiya, Tochigi* 

It is well known that the high-*Tc* superconductors such as Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (BSCCO) and (Pb*y*Bi1−*<sup>y</sup>*)2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (PBSCCO) consist of a stack of atomic-scale Josephson junctions called intrinsic Josephson junctions (IJJs) and these IJJs are SIS-type Josephson junctions. [2– 8] The stacking structure of IJJs along the crystallographic *c*-axis can be made *naturally*, so that these natural super-lattices can be regarded as a stack of good quality Josephson devices made under the atomic-scale. This is just a reason why the studies of the current flow along the *c*-axis have been done extensively for the high-*Tc* superconductors. It has been already established that the high-*Tc* superconductors such as BSCCO and PBSCCO are characterized by the *dx*<sup>2</sup>−*y*<sup>2</sup> -symmetry superconducting gap rather than the *<sup>s</sup>*-one. [9–11] There are some

When a *large* SIS Josephson junction is embedded in the magnetic field applied parallel to the junction surface, the fluxon dynamics in the SIS Josephson junction can be understood by

where *c*¯ is Swihart velocity, *λ<sup>J</sup>* is Josephson penetration depth and *ϕ* is the phase difference of a SIS Josephson junction as a function of spacial variable *x* and real time *t*. As already stated,

sin *ϕ* , (1)

*∂*2*ϕ <sup>∂</sup>t*<sup>2</sup> <sup>−</sup> *<sup>β</sup> c*¯2 *∂ϕ <sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>1</sup> *λ*2 *J*

excellent works for Shapiro steps on BSCCO single crystals. [12–18]

solving a following nonlinear partial differential equation. [19]

*∂*2*ϕ <sup>∂</sup>x*<sup>2</sup> <sup>−</sup> <sup>1</sup> *c*¯2


## **Shapiro Steps in BSCCO Intrinsic Josephson Junctions**

Michihide Kitamura

*Department of Electrical and Electronic System Engineering, Utsunomiya University, Yoto, Utsunomiya, Tochigi Japan* 

#### **1. Introduction**

156 Superconductors – Properties, Technology, and Applications

Zeldov E.; Majer D., Konczykowski M., Geshkenbein V.B., Vinokur V.M. & Shtrikman H.

Yethiraj M.; Mook H.A., Wignall G.D., Cubitt R., Forgan E.M., Paul D.M. & Armstrong T.

Yu S.; Ooi S., Mochiku T. & Hirata K. (2005). Anisotropy Dependence of Josephson-Vortex

Yu S.; Ooi S., Mochiku T. & Hirata K. (2007). Anisotropy Dependence of Traiangular lattice

Transition, *Nature* (London) Vol.375, pp.373-376

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092505-1-4

(1995). Thermodynamic Observation of First-Order Vortex-Lattice Melting

(1993). Small-Angle Neutron Scattering Study of Flux Line Lattices in Twinned

Formation of Josephson Vortices in Bi2Sr2CaCu2O8+y, *Phys. Rev.* B Vol. 76 pp.

In 1963, Shapiro did an experiment for the microwave irradiation effect on the *I*-*V* characteristics of a Josephson junction Al/Al2O3/Sn and showed that when the microwave is impressed to the Josephson junction, clear current steps are observed on the quasiparticle (QP) branches of the *I*-*V* characteristics and those steps, which are now called "Shapiro steps", are observed on constant voltages given by *n*Φ<sup>0</sup> *fr* (*n* = ±1, ±2, ...) [1]. Here the Φ<sup>0</sup> is a flux quantum (<sup>=</sup> 2.06785 <sup>×</sup> <sup>10</sup>−<sup>15</sup> Wb) defined by *<sup>h</sup>*/2*<sup>e</sup>* using Planck constant *<sup>h</sup>* and the *fr* is the frequency of the impressed microwave. The interactions between the Josephson junction and the electromagnetic field are important and intersting problems on not only the fundamental but also the applicational points of view. The Shapiro steps are one of the important phenomena observed on a superconductor-insulator-superconductor (SIS)-type Josephson junction.

It is well known that the high-*Tc* superconductors such as Bi2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (BSCCO) and (Pb*y*Bi1−*<sup>y</sup>*)2Sr2CaCu2O8<sup>+</sup>*<sup>δ</sup>* (PBSCCO) consist of a stack of atomic-scale Josephson junctions called intrinsic Josephson junctions (IJJs) and these IJJs are SIS-type Josephson junctions. [2– 8] The stacking structure of IJJs along the crystallographic *c*-axis can be made *naturally*, so that these natural super-lattices can be regarded as a stack of good quality Josephson devices made under the atomic-scale. This is just a reason why the studies of the current flow along the *c*-axis have been done extensively for the high-*Tc* superconductors. It has been already established that the high-*Tc* superconductors such as BSCCO and PBSCCO are characterized by the *dx*<sup>2</sup>−*y*<sup>2</sup> -symmetry superconducting gap rather than the *<sup>s</sup>*-one. [9–11] There are some excellent works for Shapiro steps on BSCCO single crystals. [12–18]

When a *large* SIS Josephson junction is embedded in the magnetic field applied parallel to the junction surface, the fluxon dynamics in the SIS Josephson junction can be understood by solving a following nonlinear partial differential equation. [19]

$$\frac{\partial^2 \varrho}{\partial \mathbf{x}^2} - \frac{1}{\mathcal{E}^2} \frac{\partial^2 \varrho}{\partial t^2} - \frac{\beta}{\mathcal{E}^2} \frac{\partial \varrho}{\partial t} = \frac{1}{\lambda\_f^2} \sin \varrho \,\,\,\,\tag{1}$$

where *c*¯ is Swihart velocity, *λ<sup>J</sup>* is Josephson penetration depth and *ϕ* is the phase difference of a SIS Josephson junction as a function of spacial variable *x* and real time *t*. As already stated,

normalized to the critical current *Ic*(= *JcS*), therefore, the current J�(*x*, *t*) can be written in SI

Shapiro Steps in BSCCO Intrinsic Josephson Junctions 159

current of the �-th SIS Josephson junction whose general form has been already presented in

current *jext*(*t*) is the same as the normalized external current *iext*(*t*) presented in my previous

where *i*<sup>0</sup> is the normalized external dc, *ir* sin *ωrt* is the normalized external ac modulation current with the frequency *ω<sup>r</sup>* and *inoise*(*t*, *T*) is the normalized current due to thermal noise at a sample temperature *T*. As a noise, we consider here a "white noise" so that the *Inoise*(*t*, *T*) =

�

where < *A*(*t*) >*<sup>t</sup>* means the time average of a time dependent function *A*(*t*). The *RJ* is the

+

The white noises are made numerically by using random numbers. We write here the random

the library program. The normalized white noise *inoise*(*t*, *T*) defined by *Inoise*(*t*, *T*)/*Ic*(*T*) must

Here we define an absolute time resolution Δ*t*. By doing some numerical calculations for the kernel defined by Eqs.(27) and (28), I found that the time resolution Δ*t* with the value of 1 × 10−14s(=10fs) is small enough to resolve the structure of the kernel. [35] The autocorrelation <

*inoise*(*t*, *T*) = *η*(*T*)*i*

) >*t*. Therefore, by using relation (8) and doing an integral over the *t*

 ∞ −∞ < *i* (*Cal*) *Random*(*t*)*i*

= *η*(*T*)<sup>2</sup>

�

1 *Rshunt*

*Random*(*t*), which have been made by the normal random number generator in

(*Cal*)

, *T*) >*<sup>t</sup>* is equal to *η*(*T*)<sup>2</sup> < *i*

(*Cal*) *Random*(*t* + *t*

, *T*) >*t*= <sup>2</sup>*kBT*

*RJ δ*(*t* � ) 

<sup>≡</sup> <sup>1</sup>

*∂ϕ*�(*x*, *t*) *<sup>∂</sup><sup>t</sup>* <sup>+</sup>

*N* ∑ ��=1

(�)

*jext*(*t*) = *i*<sup>0</sup> + *ir* sin *ωrt* + *inoise*(*t*, *T*) ≡ *jext*(*t*, *T*) , (7)

{**Σc**}�,�� *i*

*CP*(*x*, *t*) is the normalized Cooper-pair (CP) tunneling

(�� )

*<sup>c</sup>* and conductivity *σ*/*d*�

*CP*(*x*, *t*) is sin *ϕ*�(*x*, *t*). The external

*RJ*(*I*0) , (9)

*Random*(*t*) . (10)

� , we get

�

(*Cal*) *Random*(*t*)*i*

*CP* (*x*, *t*) − *jext*(*t*) , (6)

, (8)

*QP* (*I*0) is an effective

(*Cal*) *Random*(*t* +

) >*<sup>t</sup> dt*� . (11)

*<sup>c</sup>* per

 *G*<sup>∗</sup> *u Jc*

*<sup>u</sup>* are the effective unit area capacitance *εε*0/*d*�

(�)

 Φ<sup>0</sup> 2*π*

units as

where *C*∗

paper, that is,

numbers as *i*

*t* �

be proportional to *i*

<sup>J</sup>�(*x*, *<sup>t</sup>*) = <sup>Φ</sup><sup>0</sup>

2*π*

*<sup>u</sup>* and *G*<sup>∗</sup>

 *C*<sup>∗</sup> *u Jc*

a junction, respectively. The current *i*

*inoise*(*t*, *T*)*Ic*(*T*) satisfies following relations:

resistance of a junction due to QP tunneling.

(*Cal*)

The autocorrelation < *inoise*(*t*, *T*)*inoise*(*t* + *t*

2*kBT Ic*(*T*)<sup>2</sup>*RJ*

*Random*(*t*), that is,

(*Cal*)

*∂*2*ϕ*�(*x*, *t*) *<sup>∂</sup>t*<sup>2</sup> <sup>+</sup>

my previous paper. [35] Note that the simplest form of *i*

< *Inoise*(*t*, *T*) >*t*= 0 < *Inoise*(*t*, *T*)*Inoise*(*t* + *t*

resistance of a junction and is given as a function of *I*<sup>0</sup> = *i*<sup>0</sup> *Ic*(*T*);

<sup>=</sup> <sup>1</sup> *R*(*eff*) *QP* (*I*0)

where *Rshunt* is a shunt resistance added externally per a junction, and *<sup>R</sup>*(*eff*)

1 *RJ*

BSCCO consists of a stack of SIS-type IJJs, so it is expected that the BSCCO should be regarded as a good sample to study the fluxon dynamics in the SIS Josephson junctions. Actually, there are many excellent works for the effect of magnetic field on the BSCCO IJJs. [20–34] It is sure that a pioneering work done by Sakai, Bodin and Pedersen(SBP) [20] is the starting point to understand the fluxon dynamics in BSCCO IJJs with a stacking structure.

We adopt here a unified theory proposed by Machida and Sakai (MS) [29], which includes both the electric and magnetic field couplings between neighboring IJJs in multistacked structure. The MS-unified theory is a further extension of the model proposed by Sakai, Bodin and Pedersen (SBP). [20] Present paper is mainly based on my previous jobs. [35, 36]

#### **2. Theory**

Let us consider a stacked system consisting of *N*-identical SIS-Josephson junctions such as IJJs in BSCCO high-*Tc* superconductors, and set spacial coodinates *x*, *y* and *z* parallel to *a*, *b* and *c* axes of a BSCCO single crystal. If the external magnetic induction **B***ext* is selected parallel to *y*-axis, i.e., **B***ext* = (0, *Bext*, 0), and the spacial variation of the gauge-invariant phase difference *ϕ* for the -th SIS Josephson junction is assumed as a function of only *x*, then the *ϕ*(*x*, *t*) as a function of *x* and real time *t* satisfies a following coupled sine-Gordon(CSG) equation with a matrix form. [29]

$$
\left(\Sigma\_{\mathbf{c}}\lambda\_{\mathbf{f}}^{2}\frac{\partial^{2}}{\partial\mathbf{x}^{2}}\begin{pmatrix}\varphi\_{1}(\mathbf{x},t)\\ \cdot\\ \varphi\_{\ell}(\mathbf{x},t)\\ \cdot\\ \varphi\_{N}(\mathbf{x},t)\end{pmatrix}\right) = \Sigma\_{\mathbf{L}}\begin{pmatrix}\mathcal{J}\_{1}(\mathbf{x},t)\\ \cdot\\ \mathcal{J}\_{\ell}(\mathbf{x},t)\\ \cdot\\ \mathcal{J}\_{N}(\mathbf{x},t)\end{pmatrix},\tag{2}$$

where *<sup>λ</sup><sup>J</sup>* is the Josephson penetration depth given by � Φ0/2*πμ*0*d*� *<sup>L</sup> Jc* using a flux quantum Φ0, the vacuum permeability *μ*<sup>0</sup> and the critical current density *Jc*. The **Σ<sup>c</sup>** and **Σ<sup>L</sup>** are matrices describing the electric and magnetic interactions between neighboring Josephson junctions and are written as

$$
\Sigma\_{\mathbf{c}(L)} = \begin{pmatrix}
1 & \Sigma\_{\mathbf{c}(L)} & 0 & \cdot & \cdot \\
\Sigma\_{\mathbf{c}(L)} & 1 & \Sigma\_{\mathbf{c}(L)} & 0 & \cdot \\
\cdot & \Sigma\_{\mathbf{c}(L)} & 1 & \Sigma\_{\mathbf{c}(L)} & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & 0 & \Sigma\_{\mathbf{c}(L)} & 1
\end{pmatrix} \tag{3}
$$

where the coupling constants Σ*<sup>c</sup>* and Σ*<sup>L</sup>* are given by [29]

$$\Sigma\_{\mathfrak{c}(L)} = -\frac{\lambda\_{\mathfrak{c}(L)}}{d\_{\mathfrak{c}(L)}' \sinh(w/\lambda\_{\mathfrak{c}(L)})} \,. \tag{4}$$

The *d*� *<sup>c</sup>* and *d*� *<sup>L</sup>* are the effective electric and magnetic thickness given by [29]

$$d'\_{\varepsilon(L)} = d + 2\lambda\_{\varepsilon(L)} \coth(w/\lambda\_{\varepsilon(L)}) \, , \tag{5}$$

using electrode thickness *w* and barrier thickness *d*. The *λ<sup>e</sup>* and *λ<sup>L</sup>* are the Debye screening length and the London penetration depth, respectively. The J(*x*, *t*) in Eq.(2) is the current defined by *j* () *<sup>J</sup>* (*x*, *t*) − *jext*(*t*). The *j* () *<sup>J</sup>* (*x*, *t*) and *jext*(*t*) are junction and external currents 2 Will-be-set-by-IN-TECH

BSCCO consists of a stack of SIS-type IJJs, so it is expected that the BSCCO should be regarded as a good sample to study the fluxon dynamics in the SIS Josephson junctions. Actually, there are many excellent works for the effect of magnetic field on the BSCCO IJJs. [20–34] It is sure that a pioneering work done by Sakai, Bodin and Pedersen(SBP) [20] is the starting point to

We adopt here a unified theory proposed by Machida and Sakai (MS) [29], which includes both the electric and magnetic field couplings between neighboring IJJs in multistacked structure. The MS-unified theory is a further extension of the model proposed by Sakai, Bodin and

Let us consider a stacked system consisting of *N*-identical SIS-Josephson junctions such as IJJs in BSCCO high-*Tc* superconductors, and set spacial coodinates *x*, *y* and *z* parallel to *a*, *b* and *c* axes of a BSCCO single crystal. If the external magnetic induction **B***ext* is selected parallel to *y*-axis, i.e., **B***ext* = (0, *Bext*, 0), and the spacial variation of the gauge-invariant phase difference *ϕ* for the -th SIS Josephson junction is assumed as a function of only *x*, then the *ϕ*(*x*, *t*) as a function of *x* and real time *t* satisfies a following coupled sine-Gordon(CSG) equation with a

⎞

Φ0, the vacuum permeability *μ*<sup>0</sup> and the critical current density *Jc*. The **Σ<sup>c</sup>** and **Σ<sup>L</sup>** are matrices describing the electric and magnetic interactions between neighboring Josephson junctions

> 1 Σ*c*(*L*) 0 · · Σ*c*(*L*) 1 Σ*c*(*L*) 0 · · Σ*c*(*L*) 1 Σ*c*(*L*) · · · · ·· · · 0 Σ*c*(*L*) 1

⎟⎟⎟⎟⎠ = **ΣL**

⎛

J1(*x*, *t*) · J(*x*, *t*) · J*N*(*x*, *t*)

Φ0/2*πμ*0*d*�

⎞

⎟⎟⎟⎟⎟⎠

*<sup>c</sup>*(*L*) = *d* + 2*λe*(*L*) coth(*w*/*λe*(*L*)) , (5)

*<sup>J</sup>* (*x*, *t*) and *jext*(*t*) are junction and external currents

*<sup>c</sup>*(*L*) sinh(*w*/*λe*(*L*)) . (4)

⎞

⎟⎟⎟⎟⎠ , (2)

*<sup>L</sup> Jc* using a flux quantum

, (3)

⎜⎜⎜⎜⎝

�

understand the fluxon dynamics in BSCCO IJJs with a stacking structure.

**Σc***λ*<sup>2</sup> *J ∂*2 *∂x*<sup>2</sup>

where *λ<sup>J</sup>* is the Josephson penetration depth given by

**Σc**(**L**) =

where the coupling constants Σ*<sup>c</sup>* and Σ*<sup>L</sup>* are given by [29]

*d*�

*<sup>J</sup>* (*x*, *t*) − *jext*(*t*). The *j*

⎛

*ϕ*1(*x*, *t*) · *ϕ*(*x*, *t*) · *ϕN*(*x*, *t*)

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎜⎝

<sup>Σ</sup>*c*(*L*) <sup>=</sup> <sup>−</sup> *<sup>λ</sup>e*(*L*) *d*�

()

*<sup>L</sup>* are the effective electric and magnetic thickness given by [29]

using electrode thickness *w* and barrier thickness *d*. The *λ<sup>e</sup>* and *λ<sup>L</sup>* are the Debye screening length and the London penetration depth, respectively. The J(*x*, *t*) in Eq.(2) is the current

**2. Theory**

matrix form. [29]

and are written as

The *d*�

*<sup>c</sup>* and *d*�

defined by *j*

()

Pedersen (SBP). [20] Present paper is mainly based on my previous jobs. [35, 36]

normalized to the critical current *Ic*(= *JcS*), therefore, the current J�(*x*, *t*) can be written in SI units as

$$\mathcal{J}\_{\ell}(\mathbf{x},t) = \left(\frac{\Phi\_{0}}{2\pi}\right)\frac{\mathbb{C}\_{\mathsf{u}}^{\*}}{f\_{\mathsf{c}}}\frac{\partial^{2}\varrho\_{\ell}(\mathbf{x},t)}{\partial t^{2}} + \left(\frac{\Phi\_{0}}{2\pi}\right)\frac{\mathbb{G}\_{\mathsf{u}}^{\*}}{f\_{\mathsf{c}}}\frac{\partial\varrho\_{\ell}(\mathbf{x},t)}{\partial t} + \sum\_{\ell'=1}^{N}\{\mathsf{L}\_{\mathsf{c}}\}\_{\ell,\ell'}i\_{\mathsf{c}P}^{(\ell')}(\mathbf{x},t) - j\_{\mathsf{ext}}(t) \,,\tag{6}$$

where *C*∗ *<sup>u</sup>* and *G*<sup>∗</sup> *<sup>u</sup>* are the effective unit area capacitance *εε*0/*d*� *<sup>c</sup>* and conductivity *σ*/*d*� *<sup>c</sup>* per a junction, respectively. The current *i* (�) *CP*(*x*, *t*) is the normalized Cooper-pair (CP) tunneling current of the �-th SIS Josephson junction whose general form has been already presented in my previous paper. [35] Note that the simplest form of *i* (�) *CP*(*x*, *t*) is sin *ϕ*�(*x*, *t*). The external current *jext*(*t*) is the same as the normalized external current *iext*(*t*) presented in my previous paper, that is,

$$j\_{\rm ext}(t) = i\_0 + i\_r \sin \omega\_r t + i\_{\rm noise}(t, T) \equiv j\_{\rm ext}(t, T) \,, \tag{7}$$

where *i*<sup>0</sup> is the normalized external dc, *ir* sin *ωrt* is the normalized external ac modulation current with the frequency *ω<sup>r</sup>* and *inoise*(*t*, *T*) is the normalized current due to thermal noise at a sample temperature *T*. As a noise, we consider here a "white noise" so that the *Inoise*(*t*, *T*) = *inoise*(*t*, *T*)*Ic*(*T*) satisfies following relations:

$$\begin{cases} < I\_{\text{noise}}(t, T) >\_{t} = 0\\ < I\_{\text{noise}}(t, T) I\_{\text{noise}}(t + t', T) >\_{t} = \frac{2k\_{\mathbb{R}}T}{\mathbb{R}\_{\text{I}}} \delta(t') \end{cases} \tag{8}$$

where < *A*(*t*) >*<sup>t</sup>* means the time average of a time dependent function *A*(*t*). The *RJ* is the resistance of a junction and is given as a function of *I*<sup>0</sup> = *i*<sup>0</sup> *Ic*(*T*);

$$\frac{1}{R\_I} = \frac{1}{R\_{QP}^{(eff)}(I\_0)} + \frac{1}{R\_{shunt}} \equiv \frac{1}{R\_I(I\_0)}\,'\,\tag{9}$$

where *Rshunt* is a shunt resistance added externally per a junction, and *<sup>R</sup>*(*eff*) *QP* (*I*0) is an effective resistance of a junction due to QP tunneling.

The white noises are made numerically by using random numbers. We write here the random numbers as *i* (*Cal*) *Random*(*t*), which have been made by the normal random number generator in the library program. The normalized white noise *inoise*(*t*, *T*) defined by *Inoise*(*t*, *T*)/*Ic*(*T*) must be proportional to *i* (*Cal*) *Random*(*t*), that is,

$$i\_{\rm noise}(t, T) = \eta(T) i\_{\rm Random}^{(\rm CaI)}(t) \,. \tag{10}$$

The autocorrelation < *inoise*(*t*, *T*)*inoise*(*t* + *t* � , *T*) >*<sup>t</sup>* is equal to *η*(*T*)<sup>2</sup> < *i* (*Cal*) *Random*(*t*)*i* (*Cal*) *Random*(*t* + *t* � ) >*t*. Therefore, by using relation (8) and doing an integral over the *t* � , we get

$$\frac{2k\_BT}{I\_{\mathbb{C}}(T)^2R\_I} = \eta(T)^2 \int\_{-\infty}^{\infty} < i\_{\text{Random}}^{\text{(Cal)}}(t)i\_{\text{Random}}^{\text{(Cal)}}(t+t') >\_I dt'.\tag{11}$$

Here we define an absolute time resolution Δ*t*. By doing some numerical calculations for the kernel defined by Eqs.(27) and (28), I found that the time resolution Δ*t* with the value of 1 × 10−14s(=10fs) is small enough to resolve the structure of the kernel. [35] The autocorrelation <

transition". The first crossover thickness *d*

+ 2*π f* ∗ *p fr*

The *Q*(*ν*, *τ*) in Eq.(13) is

by *I*

(*F*) *<sup>c</sup>* (*dF*)/*<sup>I</sup>*

be written as

where

review paper by Buzdin. [40]

of the CP tunneling current is considered and other is not.

*∂τ*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>π</sup>*

<sup>2</sup> *<sup>N</sup>* ∑ �=1

(*NM*) *<sup>c</sup>* using the critical currents *<sup>I</sup>*

*χ*(*y*) = 4*y*

By using the McCumber parameter *βc*(*i*0, *Jc*, *SRshunt*), [35] the *β*<sup>∗</sup>

1

*<sup>c</sup>* (*i*0, *Jc*, *SRshunt*) <sup>=</sup> <sup>Φ</sup><sup>0</sup> *<sup>f</sup>* <sup>∗</sup>

*β*<sup>∗</sup>

 

no F-layer. For example, *χ*(*dF*) can be written as [40]

*<sup>Q</sup>*(*ν*, *<sup>τ</sup>*) = *<sup>∂</sup>*2*<sup>ϕ</sup>*(*ν*, *<sup>τ</sup>*)

(1)

0 to the *π* state, is about 50Å for the SFS-junction using an NiCu-system ferromagnetic metal. [40, 43] As can be seen later, the barrier thickness *d* of BSCCO IJJs is 12Å. In the present paper, therefore, two cases are considered; one is the case in which the correction for the amplitude

Shapiro Steps in BSCCO Intrinsic Josephson Junctions 161

*f* ∗ *p fr*

*β*<sup>∗</sup>

{**Σc**},� *χ*�(*dF*)*i*

Here note that we generally consider the case including the F-layer, so that the correction term *χ*(*dF*) which has not been written in Eq.(6) has been added into Eq.(16). The *χ*(*dF*) is defined

junctions, and the value of *χ*(*dF*) oscillatory varies as a function of *dF* bewteen 0 to 1 when the -th junction includes F-layer. Here it is clear that *χ*(*dF*) = 1 when the -th junction includes

where *y* ≡ *dF*/*ξ<sup>F</sup>* and *ξ<sup>F</sup>* is the characteristic length of the superconducting correlation decay with oscillations in the F layer. [40] The *χ*(*y*) as a function of 2*y* is drawn in Fig.11 in the

= *γ<sup>p</sup>*

It should be noted here that the *μ*(*i*0) in Eq.(18) is a universal curve as a function of only the normalized external dc *i*<sup>0</sup> so that this curve is always valid for all the IJJs in BSCCO characterized by the *dx*<sup>2</sup>−*y*<sup>2</sup> symmetry superconducting gap. [35] Namely, if the no shunt case, i.e., the case of *Rshunt* → ∞, is considered, then the McCumber parameter of BSCCO is given as a function of only the *i*<sup>0</sup> such as *μ*−2(*i*0). [35] Here I wish to say that when the effect of electric field coupling is not considered, the plasma frequencies ratio *γ<sup>p</sup>* is equal to 1 and

*<sup>γ</sup><sup>p</sup>* <sup>≡</sup> *<sup>f</sup>* <sup>∗</sup> *p fp* = *d*� *c* 1

(� )

(*F*) *<sup>c</sup>* (*dF*) and *<sup>I</sup>*

cos 2*y* sinh 2*y* + sin 2*y* cosh 2*y* cosh 4*y* − cos 4*y*

> *p JcSRshunt*

> > Φ<sup>0</sup> *fp JcSRshunt*

<sup>=</sup> *<sup>γ</sup><sup>p</sup>*

+

Φ<sup>0</sup> *f* ∗ *p*

+ *μ*(*i*0)

*<sup>β</sup>c*(*i*0, *Jc*, *SRshunt*) , (18)

*<sup>d</sup>* . (19)

*IcR*(*eff*) *QP* (*I*0)

*<sup>c</sup>* (*i*0, *Jc*, *SRshunt*)

*CP* (*ν*, *τ*) − *iext*(*τ*)

 

*<sup>F</sup>* , in which the nature of junction changes from the

*∂ϕ*(*ν*, *τ*) *∂τ*

(*NM*) *<sup>c</sup>* of the SFS and SIS Josephson

, (17)

*<sup>c</sup>* (*i*0, *Jc*, *SRshunt*) in Eq.(16) can

. (16)

*i* (*Cal*) *Random*(*t*)*i* (*Cal*) *Random*(*t* + *t* � ) >*<sup>t</sup>* is also proportional to the delta function *δ*(*t* � ), so that the integral in Eq.(11) can be well evaluated as < *i* (*Cal*) *Random*(*t*)<sup>2</sup> <sup>&</sup>gt;*<sup>t</sup>* <sup>Δ</sup>*t*. Therefore, the *<sup>η</sup>*(*T*) is given by

$$\eta(T) = \sqrt{\frac{2k\_B T}{t} \, \Delta t \, I\_c(T)^2 R\_I}}\,\,\,\,\tag{12}$$

so that from Eq.(10) we can get the normalized white noise *inoise*(*t*, *T*) at a finite temperature *T* as full numerical data with a function of time *t* defined by *n*Δ*t*. All the experiments always include the effect of thermal noise. Therefore, all the calculations presented in the present paper also include the effect of the thermal noise, except for the special case that I notice. In the present study, the sample temperature *T* has been set to 4.2K.

By using the dimensionless time *τ* defined by *ωrt*/2*π*(= *frt*) and the normalized distance *ν* defined by *x*/*λJ*, equation (2) can be rewritten as

$$
\begin{pmatrix} Q\_1(\nu,\tau) \\ \cdot \\ Q\_\ell(\nu,\tau) \\ \cdot \\ Q\_N(\nu,\tau) \end{pmatrix} = \left( 2\pi \frac{f\_p^\*}{f\_r} \right)^2 \mathbf{\hat{\Sigma}\_\mathbf{L}^{-1} \mathbf{\hat{\Sigma}\_\mathbf{c}} \frac{\partial^2}{\partial \nu^2} \begin{pmatrix} \varrho\_1(\nu,\tau) \\ \cdot \\ \varrho\_\ell(\nu,\tau) \\ \cdot \\ \varrho\_N(\nu,\tau) \end{pmatrix} \tag{13}
$$

where *f* ∗ *<sup>p</sup>* is the effective plasma frequency of a junction calculated as �*Jcd*� *<sup>c</sup>*/2*π*Φ0*εε*0. In the MS-unified theory, the effect of the magnetic field is taken into account via the boundary condition at the junction edge. Therefore, we solve Eq.(13) under the boundary condition such that

$$\frac{\partial \rho\_{\ell}(\nu, \tau)}{\partial \nu}\Big|\_{\mathrm{edge}} = \frac{B\_{\mathrm{eff}}^{(\ell)}}{\mu\_{0} l\_{\ell} \lambda\_{I}} (1 + 2\Sigma\_{L}) \; , \tag{14}$$

where *B*(�) *eff* is an effective magnetic induction at the edge of the �-th IJJ. By solving Eq.(13) full numerically, we can get the gauge-invariant phase differences *ϕ*�(*ν*, *τ*) of all the junctions � = 1, 2, ··, *N* as a function of *ν* and *τ*. In the present paper, not only the case such as SIS-IJJs but also the case in which some insulating(I) layers in IJJs are replaced by ferromagnetic(F) layers is considered. The F-layer makes a magnetization *M*, therefore, if the �-th I-layer is replaced by the F-layer with *M* = *M*(�), then the *B*(�) *eff* in Eq.(14) is given in SI-units as

$$B\_{eff}^{(\ell)} = B\_{\text{ext}} + \mathbf{M}^{(\ell)} \,. \tag{15}$$

For the SFS-junctions, there are many excellent works. [37–44] Especially, two review papers by Golubov et al. [39] and Buzdin [40] include many valuable information. Recently, the phase dynamics induced by spin waves in *a* SFS Josephson junction consisting of *s*-wave superconductors have been numerically studied within the framework of the resistively shunted junction (RSJ) model. [44] The calculations by Hikino et al. [44] have told us that the magnetization of F-layer plays an essential role to understand the nature of phase dynamics. In the SFS-junction, it is well known that the amplitude of the CP-tunneling critical current *oscillatory* varies as a function of the thickness *dF* of the F-layer [40], which is called as "0-*π*

transition". The first crossover thickness *d* (1) *<sup>F</sup>* , in which the nature of junction changes from the 0 to the *π* state, is about 50Å for the SFS-junction using an NiCu-system ferromagnetic metal. [40, 43] As can be seen later, the barrier thickness *d* of BSCCO IJJs is 12Å. In the present paper, therefore, two cases are considered; one is the case in which the correction for the amplitude of the CP tunneling current is considered and other is not.

The *Q*(*ν*, *τ*) in Eq.(13) is

4 Will-be-set-by-IN-TECH

) >*<sup>t</sup>* is also proportional to the delta function *δ*(*t*

so that from Eq.(10) we can get the normalized white noise *inoise*(*t*, *T*) at a finite temperature *T* as full numerical data with a function of time *t* defined by *n*Δ*t*. All the experiments always include the effect of thermal noise. Therefore, all the calculations presented in the present paper also include the effect of the thermal noise, except for the special case that I notice. In

By using the dimensionless time *τ* defined by *ωrt*/2*π*(= *frt*) and the normalized distance *ν*

�<sup>2</sup> **ΣL** <sup>−</sup>1**Σ<sup>c</sup>** *∂*2 *∂ν*<sup>2</sup>

*<sup>p</sup>* is the effective plasma frequency of a junction calculated as �*Jcd*�

the MS-unified theory, the effect of the magnetic field is taken into account via the boundary condition at the junction edge. Therefore, we solve Eq.(13) under the boundary condition such

> <sup>=</sup> *<sup>B</sup>*(�) *eff μ*<sup>0</sup> *Jcλ<sup>J</sup>*

full numerically, we can get the gauge-invariant phase differences *ϕ*�(*ν*, *τ*) of all the junctions � = 1, 2, ··, *N* as a function of *ν* and *τ*. In the present paper, not only the case such as SIS-IJJs but also the case in which some insulating(I) layers in IJJs are replaced by ferromagnetic(F) layers is considered. The F-layer makes a magnetization *M*, therefore, if the �-th I-layer is

For the SFS-junctions, there are many excellent works. [37–44] Especially, two review papers by Golubov et al. [39] and Buzdin [40] include many valuable information. Recently, the phase dynamics induced by spin waves in *a* SFS Josephson junction consisting of *s*-wave superconductors have been numerically studied within the framework of the resistively shunted junction (RSJ) model. [44] The calculations by Hikino et al. [44] have told us that the magnetization of F-layer plays an essential role to understand the nature of phase dynamics. In the SFS-junction, it is well known that the amplitude of the CP-tunneling critical current *oscillatory* varies as a function of the thickness *dF* of the F-layer [40], which is called as "0-*π*

*eff* is an effective magnetic induction at the edge of the �-th IJJ. By solving Eq.(13)

2*kBT*

*Random*(*t*)<sup>2</sup> >*<sup>t</sup>* <sup>Δ</sup>*t Ic*(*T*)<sup>2</sup>*RJ*

(*Cal*)

�

*Random*(*t*)<sup>2</sup> <sup>&</sup>gt;*<sup>t</sup>* <sup>Δ</sup>*t*. Therefore, the *<sup>η</sup>*(*T*) is given by

⎛

*ϕ*1(*ν*, *τ*) · *ϕ*�(*ν*, *τ*) · *ϕN*(*ν*, *τ*) ⎞

(1 + 2Σ*L*) , (14)

*eff* in Eq.(14) is given in SI-units as

*eff* <sup>=</sup> *Bext* <sup>+</sup> *<sup>M</sup>*(�) . (15)

⎜⎜⎜⎜⎝

), so that the integral

, (12)

⎟⎟⎟⎟⎠ , (13)

*<sup>c</sup>*/2*π*Φ0*εε*0. In

*i* (*Cal*) *Random*(*t*)*i*

where *f* ∗

where *B*(�)

that

(*Cal*) *Random*(*t* + *t*

�

*η*(*T*) =

����

the present study, the sample temperature *T* has been set to 4.2K.

⎞

⎟⎟⎟⎟⎠ =

*∂ϕ*�(*ν*, *τ*) *∂ν*

� � � � *edge*

*B*(�)

� 2*π f* ∗ *p fr*

defined by *x*/*λJ*, equation (2) can be rewritten as

*Q*1(*ν*, *τ*) · *Q*�(*ν*, *τ*) · *QN*(*ν*, *τ*)

replaced by the F-layer with *M* = *M*(�), then the *B*(�)

⎛

⎜⎜⎜⎜⎝

< *i* (*Cal*)

in Eq.(11) can be well evaluated as < *i*

$$\begin{split} Q\_{\ell}(\boldsymbol{\nu},\boldsymbol{\tau}) &= \frac{\partial^{2}\varrho\_{\ell}(\boldsymbol{\nu},\boldsymbol{\tau})}{\partial\tau^{2}} + 2\pi \frac{f\_{p}^{\*}}{f\_{r}} \frac{1}{\sqrt{\mathfrak{F}\_{c}^{\*}(\boldsymbol{i}\_{0\prime}\boldsymbol{f}\_{\ell}\boldsymbol{\cdot}\boldsymbol{S}\boldsymbol{R}\_{\mathrm{shunt}})}} \frac{\partial\varrho\_{\ell}(\boldsymbol{\nu},\boldsymbol{\tau})}{\partial\boldsymbol{\tau}} \\ &+ \left(2\pi\frac{f\_{p}^{\*}}{f\_{r}}\right)^{2} \left\{\sum\_{\ell'=1}^{N} \{\mathbf{E}\_{\mathbf{c}}\}\_{\ell,\ell'} \chi\_{\ell'}(d\_{F}) i\_{\mathrm{CP}}^{(\ell')}(\boldsymbol{\nu},\boldsymbol{\tau}) - i\_{\mathrm{ext}}(\boldsymbol{\tau})\right\} \,. \end{split} \tag{16}$$

Here note that we generally consider the case including the F-layer, so that the correction term *χ*(*dF*) which has not been written in Eq.(6) has been added into Eq.(16). The *χ*(*dF*) is defined by *I* (*F*) *<sup>c</sup>* (*dF*)/*<sup>I</sup>* (*NM*) *<sup>c</sup>* using the critical currents *<sup>I</sup>* (*F*) *<sup>c</sup>* (*dF*) and *<sup>I</sup>* (*NM*) *<sup>c</sup>* of the SFS and SIS Josephson junctions, and the value of *χ*(*dF*) oscillatory varies as a function of *dF* bewteen 0 to 1 when the -th junction includes F-layer. Here it is clear that *χ*(*dF*) = 1 when the -th junction includes no F-layer. For example, *χ*(*dF*) can be written as [40]

$$\chi(y) = 4y \left| \frac{\cos 2y \sinh 2y + \sin 2y \cosh 2y}{\cosh 4y - \cos 4y} \right| \,, \tag{17}$$

where *y* ≡ *dF*/*ξ<sup>F</sup>* and *ξ<sup>F</sup>* is the characteristic length of the superconducting correlation decay with oscillations in the F layer. [40] The *χ*(*y*) as a function of 2*y* is drawn in Fig.11 in the review paper by Buzdin. [40]

By using the McCumber parameter *βc*(*i*0, *Jc*, *SRshunt*), [35] the *β*<sup>∗</sup> *<sup>c</sup>* (*i*0, *Jc*, *SRshunt*) in Eq.(16) can be written as

$$\begin{split} \frac{1}{\sqrt{\mathcal{P}\_c^\*(i\_0, f\_{\rm c}, SR\_{\rm shunt})}} &= \frac{\Phi\_0 f\_p^\*}{I\_c SR\_{\rm shunt}} + \frac{\Phi\_0 f\_p^\*}{I\_c R\_{QP}^{(eff)}(I\_0)} \\ &= \gamma\_p \left\{ \frac{\Phi\_0 f\_p}{I\_c SR\_{\rm shunt}} + \mu(i\_0) \right\} \\ &= \frac{\gamma\_p}{\sqrt{\mathcal{P}\_c(i\_0, f\_{\rm c}, SR\_{\rm shunt})}} . \end{split} \tag{18}$$

where

$$
\gamma\_p \equiv \frac{f\_p^\*}{f\_p} = \sqrt{\frac{d\_c'}{d}} \,. \tag{19}
$$

It should be noted here that the *μ*(*i*0) in Eq.(18) is a universal curve as a function of only the normalized external dc *i*<sup>0</sup> so that this curve is always valid for all the IJJs in BSCCO characterized by the *dx*<sup>2</sup>−*y*<sup>2</sup> symmetry superconducting gap. [35] Namely, if the no shunt case, i.e., the case of *Rshunt* → ∞, is considered, then the McCumber parameter of BSCCO is given as a function of only the *i*<sup>0</sup> such as *μ*−2(*i*0). [35] Here I wish to say that when the effect of electric field coupling is not considered, the plasma frequencies ratio *γ<sup>p</sup>* is equal to 1 and

is,

The general form of *i*

where the kernel *K*(�)

*κ* (�) *CP*(*τ* − *τ*�

*H*�(�)

*τ*�

that the *H*�(�)

evaluate *i*

**3. Calculation**

(�)

<sup>∑</sup>*BZ*<sup>1</sup> **<sup>k</sup>** <sup>∑</sup>*BZ*<sup>1</sup> **k**� � � � *H*�(�) **k**,**k**� � � � <sup>2</sup> <sup>|</sup><sup>Δ</sup>**k**Δ**k**� <sup>|</sup> *<sup>E</sup>***k***E***k**�

<sup>&</sup>lt; *<sup>V</sup>*�(*x*, *<sup>t</sup>*) <sup>&</sup>gt;*t*<sup>=</sup> <sup>Φ</sup><sup>0</sup> *fr*

*CP*(*ν*, *τ*) is given by [35]

−∞

*K*(�) *CP*(*τ* − *τ*�

⎡ ⎢ ⎣

<sup>∑</sup>*BZ*<sup>1</sup> **<sup>k</sup>** <sup>∑</sup>*BZ*<sup>1</sup> **k**� � � � *H*�(�) **k**,**k**� � � � <sup>2</sup> <sup>|</sup><sup>Δ</sup>**k**Δ**k**� <sup>|</sup> *<sup>E</sup>***k***E***k**�

one is given by a constant irrespective of **k** and **k**�

) defined by Eq.(28), therefore, *H*�(�)

*K*(�) *CP*(*τ* − *τ*�

) can be written as

the incoherent one. The quasiparticle excitation energy *<sup>E</sup>***<sup>k</sup>** is given by �

band structure calculation, I got the values of *ξ***<sup>k</sup>** as full numerical data. [46]

value for an averaging between *τ*<sup>1</sup> and *τ*2. [35]

*CP*(*ν*, *<sup>τ</sup>*) = � *<sup>τ</sup>*

*CP*(*τ* − *τ*�

) =

(�)

*i* (�) 2*π*

Here, we must remember that *τ*<sup>1</sup> and *τ*<sup>2</sup> must be intergers in order to get a very well converged

Shapiro Steps in BSCCO Intrinsic Josephson Junctions 163

) = <sup>2</sup>*<sup>π</sup> h*¯ *ω<sup>r</sup> κ* (�) *CP*(*τ* − *τ*�

{ *<sup>f</sup>*(*E***k**) <sup>−</sup> *<sup>f</sup>*(*E***k**�)} sin � *<sup>E</sup>***k**−*E***k**�

where BZ1 means the 1st Brillouin zone, *f*(*E*) is the Fermi-Dirac distribution function and

is the superconducting energy gap and it is well known that the Δ**<sup>k</sup>** of *d*-wave symmetry superconductor is represented as Δ(*T*) cos 2*φk*, where *φ<sup>k</sup>* is the angle of **k**-vector measured from *x*-axis. [10] The *ξ***<sup>k</sup>** is the one-electron energy relative to the Fermi level. By doing the

In my previous paper, [35] I have carried out the sophisticated numerical calculations in which two types of CP tunneling currents such as coherent and incoherent ones have been correctly calculated within the framework of the *dx*<sup>2</sup>−*y*<sup>2</sup> symmetry superconducting gap, and found that the resultant *I*-*V* characteristics very well coincide with the result calculated in a usual way such as *iCP*(*τ*) = sin *ϕ*(*τ*). [35] This finding is come from the fact that Shapiro steps are observed on the quasiparticle (QP) branch of the *I*-*V* characteristics so that the Shapiro steps are not so sensitive to the CP tunneling *mechanism*. [35] In the present paper, therefore, we

We consider here the BSCCO high-*Tc* superconductors. The typical values of electrode thickness *w*, barrier thickness *d* and London penetration depth *λ<sup>L</sup>* are 3, 12 and 1700Å,

*CP*(*ν*, *τ*) as sin *ϕ*�(*ν*, *τ*), in order to save the CPU times on the numerical calculations.

**<sup>k</sup>**,**k**� is the matrix element due to the **k** → **k**� tunneling in the �-th IJJ. Here it should be noted

**<sup>k</sup>**,**k**� for the coherent tunneling is proportional to *δ***k**,**k**� and that for the incoherent

<sup>+</sup> {<sup>1</sup> <sup>−</sup> *<sup>f</sup>*(*E***k**) <sup>−</sup> *<sup>f</sup>*(*E***k**�)} sin � *<sup>E</sup>***k**+*E***k**�

� *<sup>f</sup>*(*E***k**)−*f*(*E***k**� )

*<sup>E</sup>***k**−*E***k**� <sup>+</sup> <sup>1</sup>−*f*(*E***k**)−*f*(*E***k**� ) *<sup>E</sup>***k**+*E***k**�

) sin *<sup>ϕ</sup>*�(*ν*, *<sup>τ</sup>*�

*ϕ*�(*x*, *τ*2) − *ϕ*�(*x*, *τ*1) *τ*<sup>2</sup> − *τ*<sup>1</sup>

) + *ϕ*�(*ν*, *τ*)

*<sup>h</sup>*¯ *<sup>ω</sup><sup>r</sup>* 2*π*(*τ* − *τ*�

. (25)

<sup>2</sup> *<sup>d</sup>τ*� , (26)

) , (27)

) � ⎤ ⎥ ⎦

*ξ*2 **<sup>k</sup>** <sup>+</sup> <sup>Δ</sup><sup>2</sup>

, (28)

(�) *CP*(*τ* −

**<sup>k</sup>**. The Δ**<sup>k</sup>**

) �

*<sup>h</sup>*¯ *<sup>ω</sup><sup>r</sup>* 2*π*(*τ* − *τ*�

�

. In the practical calculations of *κ*

**<sup>k</sup>**,**k**� is set to *δ***k**,**k**� for the coherent tunneling and to 1 for

the matrix **Σ<sup>c</sup>** is equal to the unit matrix **1**, therefore, the *Q*�(*ν*, *τ*) for this case is just equal to the left-hand side of Eq.(31) in my recent paper. [35] We will see later that *γ<sup>p</sup>* � 1 and **Σ<sup>c</sup>** � **1** even if the effect of the electric field coupling is considered for BSCCO IJJs.

Kautz and Monaco extensively studied the nature of Shapiro steps on the SIS Josephson junction [45]. As a result, they found that there is a chaotic region in which Shapiro steps cannot be observed. The chaotic region is found between *fRC* and *fp*, i.e., *fRC* ≤ *fr* ≤ *fp*. In my previous paper, [35] I have already pointed out that the lower frequency *fRC* of the chaotic region is given by

$$\begin{array}{l} f\_{\rm R\overline{C}} = f\_{\rm R\overline{C}}^{\rm (NS)}(i\_0) + f\_{\rm R\overline{C}}^{\rm (Shunt)}(SR\_{\rm slunt})\\ f\_{\rm R\overline{C}}^{\rm (Shunt)}(SR\_{\rm slunt}) = \frac{1}{2\pi R\_{\rm shunt}\overline{C}}\end{array} \tag{20}$$

Moreover, I have mentioned that the frequency *f* (*NS*) *RC* (*i*0) for the case of no-shunt (NS) resistance is given by using a universal curve *μ*(*i*0) as

$$f\_{RC}^{(NS)}(i\_0) = f\_p \mu(i\_0) \,. \tag{21}$$

Above two equations (20) and (21) tell us that the *SRshunt*-product is an important parameter to understand the nature of Shapiro step observed on the BSCCO IJJ.

If the case such that *Bext* = 0 and there are no F-layers in IJJs is considered, then the *∂ϕ*�(*ν*, *τ*)/*∂ν* is zero for all junctions. Therefore, the equation (13) for the present case is equivalent to the equation *Q*�(*ν*, *τ*) = 0. Equation (16) tells us that the equation *Q*�(*ν*, *τ*) = 0 reflects *only* the electric interaction between neighboring Josephson junctions. It is clear that if **Σ<sup>c</sup>** = **1** and *χ*�(*dF*) = 1 for all �, then the equation *Q*�(*ν*, *τ*) = 0 is just equal to equation(31) in my recent paper. [35] In order to see the effects originated from both the electric and magnetic interactions, we must solve Eq.(13) with a finite value of *Bext*, so that the Eq.(13) becomes a matrix form including not only **Σ<sup>c</sup>** but also **Σ<sup>L</sup>** describing the magnetic interaction between neighboring Josephson junctions.

The dc voltage < *V*� > observed on the �-th IJJ satisfies

$$
\varepsilon \varepsilon\_0 \frac{L\_x L\_y}{d} < V\_\ell > = \int d\mathbf{Q} = \int\_0^{L\_x} \varepsilon \varepsilon\_0 \frac{L\_y}{d} < V\_\ell(\mathbf{x}, t) >\_t d\mathbf{x} \,, \tag{22}
$$

where *Lx* and *Ly* are junction lengths along the *x* and *y* axes, i.e., *S* = *LxLy*, and < *V*�(*x*, *t*) >*<sup>t</sup>* means the time average of *V*�(*x*, *t*) which is a voltage of the �-th IJJ as a function of *x* and *t*. Therefore, the dc voltage < *V*� > for the �-th IJJ is simply given by

$$ = \frac{1}{L\_{\chi}} \int\_{0}^{L\_{\chi}} \_{t} d\mathbf{x} \,\,\,\,\tag{23}$$

so that the total dc voltage < *V* >*total* observed experimentally is given by

$$\_{total} = \sum\_{\ell=1}^{N}  \equiv N < V>\_{reduce} \tag{24}$$

where < *V* >*reduce* is the reduced voltage when < *V* >*total* is assumed as a simple sum of *N*-junctions system. The < *V*�(*x*, *t*) >*<sup>t</sup>* is the same as Eq.(33) in my previous paper [35], that is,

6 Will-be-set-by-IN-TECH

the matrix **Σ<sup>c</sup>** is equal to the unit matrix **1**, therefore, the *Q*�(*ν*, *τ*) for this case is just equal to the left-hand side of Eq.(31) in my recent paper. [35] We will see later that *γ<sup>p</sup>* � 1 and **Σ<sup>c</sup>** � **1**

Kautz and Monaco extensively studied the nature of Shapiro steps on the SIS Josephson junction [45]. As a result, they found that there is a chaotic region in which Shapiro steps cannot be observed. The chaotic region is found between *fRC* and *fp*, i.e., *fRC* ≤ *fr* ≤ *fp*. In my previous paper, [35] I have already pointed out that the lower frequency *fRC* of the chaotic

Above two equations (20) and (21) tell us that the *SRshunt*-product is an important parameter

If the case such that *Bext* = 0 and there are no F-layers in IJJs is considered, then the *∂ϕ*�(*ν*, *τ*)/*∂ν* is zero for all junctions. Therefore, the equation (13) for the present case is equivalent to the equation *Q*�(*ν*, *τ*) = 0. Equation (16) tells us that the equation *Q*�(*ν*, *τ*) = 0 reflects *only* the electric interaction between neighboring Josephson junctions. It is clear that if **Σ<sup>c</sup>** = **1** and *χ*�(*dF*) = 1 for all �, then the equation *Q*�(*ν*, *τ*) = 0 is just equal to equation(31) in my recent paper. [35] In order to see the effects originated from both the electric and magnetic interactions, we must solve Eq.(13) with a finite value of *Bext*, so that the Eq.(13) becomes a matrix form including not only **Σ<sup>c</sup>** but also **Σ<sup>L</sup>** describing the magnetic interaction between

(*Shunt*)

2*πRshuntC*

*RC* (*SRshunt*)

(*NS*)

*RC* (*i*0) = *fpμ*(*i*0) . (21)

. (20)

*RC* (*i*0) for the case of no-shunt (NS)

*<sup>d</sup>* <sup>&</sup>lt; *<sup>V</sup>*�(*x*, *<sup>t</sup>*) <sup>&</sup>gt;*<sup>t</sup> dx* , (22)

< *V*�(*x*, *t*) >*<sup>t</sup> dx* , (23)

< *V*� >≡ *N* < *V* >*reduce* , (24)

even if the effect of the electric field coupling is considered for BSCCO IJJs.

(*NS*) *RC* (*i*0) + *f*

> *f* (*NS*)

to understand the nature of Shapiro step observed on the BSCCO IJJ.

*RC* (*SRshunt*) = <sup>1</sup>

*fRC* = *f*

*f* (*Shunt*)

Moreover, I have mentioned that the frequency *f*

resistance is given by using a universal curve *μ*(*i*0) as

The dc voltage < *V*� > observed on the �-th IJJ satisfies

*<sup>d</sup>* <sup>&</sup>lt; *<sup>V</sup>*� <sup>&</sup>gt;<sup>=</sup>

Therefore, the dc voltage < *V*� > for the �-th IJJ is simply given by

< *V* >*total*=

<sup>&</sup>lt; *<sup>V</sup>*� <sup>&</sup>gt;<sup>=</sup> <sup>1</sup>

so that the total dc voltage < *V* >*total* observed experimentally is given by

*Lx*

*N* ∑ �=1

*dQ* =

 *Lx* 0 *εε*0 *Ly*

where *Lx* and *Ly* are junction lengths along the *x* and *y* axes, i.e., *S* = *LxLy*, and < *V*�(*x*, *t*) >*<sup>t</sup>* means the time average of *V*�(*x*, *t*) which is a voltage of the �-th IJJ as a function of *x* and *t*.

> *Lx* 0

where < *V* >*reduce* is the reduced voltage when < *V* >*total* is assumed as a simple sum of *N*-junctions system. The < *V*�(*x*, *t*) >*<sup>t</sup>* is the same as Eq.(33) in my previous paper [35], that

neighboring Josephson junctions.

*εε*0 *LxLy*

region is given by

$$\_{l} = \frac{\Phi\_{0}f\_{r}}{2\pi} \frac{\varphi\_{\ell}(\mathbf{x},\mathbf{\tau}\_{2}) - \varphi\_{\ell}(\mathbf{x},\mathbf{\tau}\_{1})}{\tau\_{2} - \tau\_{1}} \,. \tag{25}$$

Here, we must remember that *τ*<sup>1</sup> and *τ*<sup>2</sup> must be intergers in order to get a very well converged value for an averaging between *τ*<sup>1</sup> and *τ*2. [35]

The general form of *i* (�) *CP*(*ν*, *τ*) is given by [35]

$$\mathrm{id}\_{\mathrm{CP}}^{(\ell)}(\nu,\tau) = \int\_{-\infty}^{\tau} K\_{\mathrm{CP}}^{(\ell)}(\tau-\tau') \sin \frac{\varphi\_{\ell}(\nu,\tau') + \varphi\_{\ell}(\nu,\tau)}{2} d\tau' \,\,\,\,\,\tag{26}$$

where the kernel *K*(�) *CP*(*τ* − *τ*� ) can be written as

$$K\_{\rm CP}^{(\ell)}(\tau-\tau') = \frac{2\pi}{\hbar\omega\_{\rm P}}\kappa\_{\rm CP}^{(\ell)}(\tau-\tau')\,. \tag{27}$$

$$\begin{split} \kappa\_{\rm CP}^{(\ell)}(\tau-\tau') &= \\ \frac{\sum\_{\mathbf{k}}^{\rm BZ}\sum\_{\mathbf{k}'}^{\rm BZ}\left|H\_{\mathbf{k},\mathbf{k}'}^{(\ell)}\right|^{2}\frac{|\mathbf{k}\_{\mathbf{k}}\mathbf{k}\_{\mathbf{k}'}|}{\hat{\varepsilon}\_{\mathbf{k}}\hat{\varepsilon}\_{\mathbf{k}'}}}{\left|\frac{\sum\_{\mathbf{k}}^{\rm BZ}\sum\_{\mathbf{k}'}^{\rm BZ}\left|H\_{\mathbf{k},\mathbf{k}'}^{(\ell)}\right|^{2}}\right|^{2}+\left\{1-f(\mathbf{E}\_{\mathbf{k}})-f(\mathbf{E}\_{\mathbf{k}'})\right\}\sin\left\{\frac{\mathbb{E}\_{\mathbf{k}}-\mathbb{E}\_{\mathbf{k}'}}{\hbar\omega\_{r}}2\pi(\tau-\tau')\right\}}\end{split} \tag{28}$$

where BZ1 means the 1st Brillouin zone, *f*(*E*) is the Fermi-Dirac distribution function and *H*�(�) **<sup>k</sup>**,**k**� is the matrix element due to the **k** → **k**� tunneling in the �-th IJJ. Here it should be noted that the *H*�(�) **<sup>k</sup>**,**k**� for the coherent tunneling is proportional to *δ***k**,**k**� and that for the incoherent one is given by a constant irrespective of **k** and **k**� . In the practical calculations of *κ* (�) *CP*(*τ* − *τ*� ) defined by Eq.(28), therefore, *H*�(�) **<sup>k</sup>**,**k**� is set to *δ***k**,**k**� for the coherent tunneling and to 1 for the incoherent one. The quasiparticle excitation energy *<sup>E</sup>***<sup>k</sup>** is given by � *ξ*2 **<sup>k</sup>** <sup>+</sup> <sup>Δ</sup><sup>2</sup> **<sup>k</sup>**. The Δ**<sup>k</sup>** is the superconducting energy gap and it is well known that the Δ**<sup>k</sup>** of *d*-wave symmetry superconductor is represented as Δ(*T*) cos 2*φk*, where *φ<sup>k</sup>* is the angle of **k**-vector measured from *x*-axis. [10] The *ξ***<sup>k</sup>** is the one-electron energy relative to the Fermi level. By doing the band structure calculation, I got the values of *ξ***<sup>k</sup>** as full numerical data. [46]

In my previous paper, [35] I have carried out the sophisticated numerical calculations in which two types of CP tunneling currents such as coherent and incoherent ones have been correctly calculated within the framework of the *dx*<sup>2</sup>−*y*<sup>2</sup> symmetry superconducting gap, and found that the resultant *I*-*V* characteristics very well coincide with the result calculated in a usual way such as *iCP*(*τ*) = sin *ϕ*(*τ*). [35] This finding is come from the fact that Shapiro steps are observed on the quasiparticle (QP) branch of the *I*-*V* characteristics so that the Shapiro steps are not so sensitive to the CP tunneling *mechanism*. [35] In the present paper, therefore, we evaluate *i* (�) *CP*(*ν*, *τ*) as sin *ϕ*�(*ν*, *τ*), in order to save the CPU times on the numerical calculations.

#### **3. Calculation**

We consider here the BSCCO high-*Tc* superconductors. The typical values of electrode thickness *w*, barrier thickness *d* and London penetration depth *λ<sup>L</sup>* are 3, 12 and 1700Å,

the 1@5 and 5@5 junctions, which are surface junctions, show the same characteristics and the

Shapiro Steps in BSCCO Intrinsic Josephson Junctions 165

In the following, previous results calculated for no F-layers, i.e., *M*(�) = 0 for all the IJJs, are first presented, and next the results for *<sup>M</sup>*(�) �<sup>=</sup> 0 are given. Here note that in the present calculations, *an* �-th insulating (I) layer in BSCCO IJJs is replaced by a F-layer which makes a

It is important to check the effect of junction length *Lx* along the *x*-axis into the *I*-*V* characteristics. In my previous paper, [36] I have calculated the *I*-*V* characteristics for three shapes (*Lx*, *Ly*)=(2.5, 10), (5, 5) and (10, 2.5) in *μ*m with the same junction cross section *S*(= *LxLy*) = 25*μ*m2. The calculated results have told us that the overall profiles of the *I*-*V* characteristics calculated for the same external magnetic induction are fairly similar each other. This finding is reasonable since all the junction lengths *Lx* considered here satisfy the condition of *Lx* > *λJ*(= 0.3687*μ*m), that is, all the junctions adopted in the calculations belong to the category of so-called "large junction". In the following, therefore, only the square

The *SRshunt*-product is an important parameter to understand the nature of Shapiro step. In my previous papers, [35, 36] I *extensively* studied the effect of the *SRshunt*-product to the *I*-*V* characteristics. In the present paper, therefore, only the conclusions obtained are briefly

First, from the calculations for a single junction, I have found that (I) the value of shunt resistance *Rshunt* added externally should be small such as 1Ω/junction, i.e., *SRshunt*/junction= 25*μ*m2Ω, if we wish to make a Shapiro step device which is *hard* for the external magnetic disturbance, and (II) in the case of the large *Rshunt*-value such as 5Ω/junction, not only the 1st-order Shapiro steps but also the Fiske steps [47, 48] are found, especially, in the case of the low magnetic field. Next, from the calculations for multi junctions such that *N* = 5, 6 and *Rshunt*/junction= 1, 2, 3, 4 and 5Ω, I have found that (III) the 1st-order Shapiro steps are found when *Rshunt*/junction≤ 3Ω, but no detectable Shapiro steps are found when *Rshunt*/junction= 4 and 5Ω, and (IV) this result directly reflects the nature of

From the above results, I have concluded that it is essential to add the shunt resistance with a low-value into the BSCCO IJJs, if we wish to make a good quality Shapiro step device. In the

We consider here the effect of the normalized amplitude *ir* of the external ac modulation current *ir* sin *ωrt*. In order to do so, first, I have calculated the *I*-*V* characteristics with no external magnetic field, i.e., *α* = 0, under the condition such that *Jc* = 1000A/cm2,

the magnetic interactions between the neighboring Josephson junctions.

present study, therefore, the value of *SRshunt*/junction is fixed to 50*μ*m2Ω.

2@5 and 4@5 ones (intermediate junctions) also show the same nature.

magnetization with the value of *M*(�).

junction (*Lx* = *Ly* = 5*μ*m) is considered for simplicity.

**4. Results and discussion**

**4.2 Effect of** *SRshunt***-product**

**4.1 Effect of** *Lx*

presented.

**4.3 Effect of** *ir*

respectively [21], and that of the Debye screening length *λ<sup>e</sup>* is 1Å. [29] Therefore, the effective electric and magnetic thickness *d*� *<sup>c</sup>* and *d*� *<sup>L</sup>* defined by Eq.(5) are calculated as 14.0 and 1.93 <sup>×</sup> <sup>10</sup>6Å , and the coupling constants <sup>Σ</sup>*<sup>c</sup>* and <sup>Σ</sup>*<sup>L</sup>* defined by Eq.(4) are calculated as <sup>−</sup>7.13 <sup>×</sup> <sup>10</sup>−<sup>3</sup> and <sup>−</sup>0.499996, respectively. Therefore, we can see that the plasma frequencies ratio *<sup>γ</sup><sup>p</sup>* calculated as *<sup>d</sup>*� *<sup>c</sup>*/*d* is 1.08 and the matrix **Σ<sup>c</sup>** describing the electric interaction between the neighboring Josephson junctions is very well approximated by the unit matrix **1**. This fact means that the effect of electric field coupling, the charge coupling in other words, could not be so crucial in BSCCO IJJs. In my previous paper, [36] I have checked this point *quantitatively*, and found that the effect of charge coupling is not crucial in the BSCCO IJJs. In the present calculations, therefore, *γ<sup>p</sup>* is set to 1 and **Σ<sup>c</sup>** is set to **1**. This reduction largely saves the CPU times, since the term ∑*<sup>N</sup>* �=<sup>1</sup> {**Σc**},� *χ*�(*dF*)*i* (� ) *CP* (*ν*, *τ*) in Eq.(16) is simply given by *χ*(*dF*)*i* () *CP*(*ν*, *τ*) when **Σ<sup>c</sup>** = **1**.

We adopt here 1000(A/cm2) again as the value of critical current density *Jc*, therefore, the Josephson penetration depth *λ<sup>J</sup>* and the plasma frequency *fp* are calculated as 3687Å and 122GHz, respectively, using *ε* = 7. The *fp* and *Jc* are given so that the *SRshunt*-product becomes an essential value to decide the value of McCumber parameter *βc*(*i*0, *Jc*, *SRshunt*). Recently, I have numerically studied the conditions for observing Shapiro steps in BSCCO IJJs and found that clear and stable Shapiro steps with good responses are obtained when the IJJ is operated under the condition such that the shunt resistance *Rshunt* is added and the external ac modulation frequency *fr* is higher than the plasma frequency *fp*. [35] In the present paper, therefore, the value of *fr* is set to 200GHz again because of *fp* = 122GHz.

The values of *λJ*, *d*� *c*(*L*) , Σ*c*(*L*), *fr*, *fp* and *Jc* are given, the effect of the normalized current *inoise*(*τ*) due to the thermal noise is considered by using normal random numbers, and the normalized CP tunneling current *i* () *CP*(*ν*, *τ*) is evaluated as *χ*(*dF*) sin *ϕ*(*ν*, *τ*), therefore, *SRshunt*-product, *i*0, *ir*, *Bext*, *M*(), *χ*(*dF*) and the number of junctions *N* become the important variables to be considered. Here, note that the *i*<sup>0</sup> is the basic variable in the *I*-*V* characteristics that I present here. The junction cross section *S*(= *LxLy*) is set to 25*μ*m2 again and 1, 2, 3, 4 and 5Ω are selected as the values of shunt resistance *Rshunt* per a junction, i.e., the *SRshunt*-products are set to 25, 50 75, 100 and 125*μ*m2Ω/junction. We write here the external magnetic induction *Bext* as *αB*0(*Lx*, *d*). Here the *B*0(*Lx*, *d*) is the magnetic induction needed to insert a flux quantum Φ<sup>0</sup> into a SIS-Josephson junction and is given by Φ0/*Lxd* as a function of *Lx* and *d*. For example, the *B*0(*Lx*, *d*) is calculated as 0.345T for *Lx* = 5*μ*m and *d* = 12Å. In the present paper, the value of barrier thickness *d* is fixed to 12Å, so that the *B*0(*Lx*, *d*) is simply written as *B*0(*Lx*).

In my previous paper, [36] I have studied the effect of *N* on the *I*-*V* characteristics very carefully. The calculated results have told us that (I) a stacking system consisting of 5 or 6-identical SIS Josephson junctions is enough to see the *I*-*V* characteristics of BSCCO IJJs with odd or even number of junctions, and (II) if the stacked system consists of 5 junctions, then *a* flux quantum enters an insulating layer of the 3@5 central junction, but if the system consists of 6 junctions, then *two* flux quanta must enter the insulating layers in such a way that one flux quantum enters the insulating layer of 3@6 junction and the other the 4@6 one, because of its symmetry. In the present paper, the number *N* of junction is fixed to 5. Therefore, if the case in which there is no ferromagnetic (F)-layer is considered, then the symmetry says that the 1@5 and 5@5 junctions, which are surface junctions, show the same characteristics and the 2@5 and 4@5 ones (intermediate junctions) also show the same nature.

In the following, previous results calculated for no F-layers, i.e., *M*(�) = 0 for all the IJJs, are first presented, and next the results for *<sup>M</sup>*(�) �<sup>=</sup> 0 are given. Here note that in the present calculations, *an* �-th insulating (I) layer in BSCCO IJJs is replaced by a F-layer which makes a magnetization with the value of *M*(�).

## **4. Results and discussion**

## **4.1 Effect of** *Lx*

8 Will-be-set-by-IN-TECH

respectively [21], and that of the Debye screening length *λ<sup>e</sup>* is 1Å. [29] Therefore, the

and 1.93 <sup>×</sup> <sup>10</sup>6Å , and the coupling constants <sup>Σ</sup>*<sup>c</sup>* and <sup>Σ</sup>*<sup>L</sup>* defined by Eq.(4) are calculated as <sup>−</sup>7.13 <sup>×</sup> <sup>10</sup>−<sup>3</sup> and <sup>−</sup>0.499996, respectively. Therefore, we can see that the plasma frequencies

between the neighboring Josephson junctions is very well approximated by the unit matrix **1**. This fact means that the effect of electric field coupling, the charge coupling in other words, could not be so crucial in BSCCO IJJs. In my previous paper, [36] I have checked this point *quantitatively*, and found that the effect of charge coupling is not crucial in the BSCCO IJJs. In the present calculations, therefore, *γ<sup>p</sup>* is set to 1 and **Σ<sup>c</sup>** is set to **1**. This reduction largely

�=<sup>1</sup> {**Σc**},� *χ*�(*dF*)*i*

We adopt here 1000(A/cm2) again as the value of critical current density *Jc*, therefore, the Josephson penetration depth *λ<sup>J</sup>* and the plasma frequency *fp* are calculated as 3687Å and 122GHz, respectively, using *ε* = 7. The *fp* and *Jc* are given so that the *SRshunt*-product becomes an essential value to decide the value of McCumber parameter *βc*(*i*0, *Jc*, *SRshunt*). Recently, I have numerically studied the conditions for observing Shapiro steps in BSCCO IJJs and found that clear and stable Shapiro steps with good responses are obtained when the IJJ is operated under the condition such that the shunt resistance *Rshunt* is added and the external ac modulation frequency *fr* is higher than the plasma frequency *fp*. [35] In the present paper,

*inoise*(*τ*) due to the thermal noise is considered by using normal random numbers, and

*SRshunt*-product, *i*0, *ir*, *Bext*, *M*(), *χ*(*dF*) and the number of junctions *N* become the important variables to be considered. Here, note that the *i*<sup>0</sup> is the basic variable in the *I*-*V* characteristics that I present here. The junction cross section *S*(= *LxLy*) is set to 25*μ*m2 again and 1, 2, 3, 4 and 5Ω are selected as the values of shunt resistance *Rshunt* per a junction, i.e., the *SRshunt*-products are set to 25, 50 75, 100 and 125*μ*m2Ω/junction. We write here the external magnetic induction *Bext* as *αB*0(*Lx*, *d*). Here the *B*0(*Lx*, *d*) is the magnetic induction needed to insert a flux quantum Φ<sup>0</sup> into a SIS-Josephson junction and is given by Φ0/*Lxd* as a function of *Lx* and *d*. For example, the *B*0(*Lx*, *d*) is calculated as 0.345T for *Lx* = 5*μ*m and *d* = 12Å. In the present paper, the value of barrier thickness *d* is fixed to 12Å, so that the *B*0(*Lx*, *d*) is

In my previous paper, [36] I have studied the effect of *N* on the *I*-*V* characteristics very carefully. The calculated results have told us that (I) a stacking system consisting of 5 or 6-identical SIS Josephson junctions is enough to see the *I*-*V* characteristics of BSCCO IJJs with odd or even number of junctions, and (II) if the stacked system consists of 5 junctions, then *a* flux quantum enters an insulating layer of the 3@5 central junction, but if the system consists of 6 junctions, then *two* flux quanta must enter the insulating layers in such a way that one flux quantum enters the insulating layer of 3@6 junction and the other the 4@6 one, because of its symmetry. In the present paper, the number *N* of junction is fixed to 5. Therefore, if the case in which there is no ferromagnetic (F)-layer is considered, then the symmetry says that

therefore, the value of *fr* is set to 200GHz again because of *fp* = 122GHz.

()

*<sup>c</sup>* and *d*�

*<sup>L</sup>* defined by Eq.(5) are calculated as 14.0

*CP* (*ν*, *τ*) in Eq.(16) is simply given

*<sup>c</sup>*/*d* is 1.08 and the matrix **Σ<sup>c</sup>** describing the electric interaction

(� )

, Σ*c*(*L*), *fr*, *fp* and *Jc* are given, the effect of the normalized current

*CP*(*ν*, *τ*) is evaluated as *χ*(*dF*) sin *ϕ*(*ν*, *τ*), therefore,

effective electric and magnetic thickness *d*�

saves the CPU times, since the term ∑*<sup>N</sup>*

*CP*(*ν*, *τ*) when **Σ<sup>c</sup>** = **1**.

*c*(*L*)

the normalized CP tunneling current *i*

ratio *<sup>γ</sup><sup>p</sup>* calculated as *<sup>d</sup>*�

()

The values of *λJ*, *d*�

simply written as *B*0(*Lx*).

by *χ*(*dF*)*i*

It is important to check the effect of junction length *Lx* along the *x*-axis into the *I*-*V* characteristics. In my previous paper, [36] I have calculated the *I*-*V* characteristics for three shapes (*Lx*, *Ly*)=(2.5, 10), (5, 5) and (10, 2.5) in *μ*m with the same junction cross section *S*(= *LxLy*) = 25*μ*m2. The calculated results have told us that the overall profiles of the *I*-*V* characteristics calculated for the same external magnetic induction are fairly similar each other. This finding is reasonable since all the junction lengths *Lx* considered here satisfy the condition of *Lx* > *λJ*(= 0.3687*μ*m), that is, all the junctions adopted in the calculations belong to the category of so-called "large junction". In the following, therefore, only the square junction (*Lx* = *Ly* = 5*μ*m) is considered for simplicity.

## **4.2 Effect of** *SRshunt***-product**

The *SRshunt*-product is an important parameter to understand the nature of Shapiro step. In my previous papers, [35, 36] I *extensively* studied the effect of the *SRshunt*-product to the *I*-*V* characteristics. In the present paper, therefore, only the conclusions obtained are briefly presented.

First, from the calculations for a single junction, I have found that (I) the value of shunt resistance *Rshunt* added externally should be small such as 1Ω/junction, i.e., *SRshunt*/junction= 25*μ*m2Ω, if we wish to make a Shapiro step device which is *hard* for the external magnetic disturbance, and (II) in the case of the large *Rshunt*-value such as 5Ω/junction, not only the 1st-order Shapiro steps but also the Fiske steps [47, 48] are found, especially, in the case of the low magnetic field. Next, from the calculations for multi junctions such that *N* = 5, 6 and *Rshunt*/junction= 1, 2, 3, 4 and 5Ω, I have found that (III) the 1st-order Shapiro steps are found when *Rshunt*/junction≤ 3Ω, but no detectable Shapiro steps are found when *Rshunt*/junction= 4 and 5Ω, and (IV) this result directly reflects the nature of the magnetic interactions between the neighboring Josephson junctions.

From the above results, I have concluded that it is essential to add the shunt resistance with a low-value into the BSCCO IJJs, if we wish to make a good quality Shapiro step device. In the present study, therefore, the value of *SRshunt*/junction is fixed to 50*μ*m2Ω.

### **4.3 Effect of** *ir*

We consider here the effect of the normalized amplitude *ir* of the external ac modulation current *ir* sin *ωrt*. In order to do so, first, I have calculated the *I*-*V* characteristics with no external magnetic field, i.e., *α* = 0, under the condition such that *Jc* = 1000A/cm2,

but not so clear when *α* = 2.

*i*<sup>0</sup> is about 0.6.

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Shapiro Steps in BSCCO Intrinsic Josephson Junctions 167

Fig. 2. The *I*-*V* characteristics of BSCCO IJJs calculated for the same condition as in Fig.1, except for the external magnetic induction *Bext*. The *Bext* is defined by *αB*0(*Lx*) and the *B*0(*Lx*) is 0.345T because of *Lx* = 5*μ*m. The (*ir*, *α*) of (a), (b), (c), (d), (e), (f), (g), (h) and (i) are (0, 0.5), (0, 1), (0, 2), (1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2), respectively. The reduced voltage < *V* >*reduce* defined by Eq.(24) is also shown. Shapiro steps are found when *ir* = 0,

of the odd number of junctions, then it is sure that *its* flux quantum must enter the insulating layer of central SIS Josephson junction such as 3@5 junction because of the symmetry of the odd number of junctions. This is reasonable to the finding (III). Actually, in the case of Fig.2(a), we can see that only the 3@5 central junction just shows the voltage due to the flux flow when

 

 

 

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*fp* = 122GHz, *fr* = 200GHz, *SRshunt*/junction= 50*μ*m2Ω and *N* = 5. Those are drawn as (i), (ii) and (iii) in Fig.1 for *ir* = 0, 1 and 2 which are denoted as (*ir*, *α*)=(0, 0), (1, 0) and (2, 0), respectively. Next, I have calculated the *I*-*V* characteristics with finite external magnetic fields, i.e., *α* � 0, under the same condition as in Fig.1. Here note again that the external magnetic induction *Bext* is defined by *αB*0(*Lx*) and the *B*0(*Lx*) is 0.345T for *Lx* = 5*μ*m. The *I*-*V* characteristics calculated for (*ir*, *α*)=(0, 0.5), (0, 1), (0, 2), (1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2) are shown in Fig.2(a), (b), (c), (d), (e), (f), (g), (h) and (i), respectively.

Fig. 1. The *I*-*V* characteristics with no external magnetic field, i.e., *α* = 0, calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Lx* = 5*μ*m, *S*(= *LxLy*) = 25*μ*m2, and *SRshunt*/junction= 50*μ*m2Ω. The effect of charge coupling is not taken into account. (i), (ii) and (iii) are those for *ir* = 0, 1 and 2, and are denoted as (*ir*, *α*)=(0, 0), (1, 0) and (2, 0), respectively. Note that the *I*-*V* characteristics (ii) and (iii) have been shifted by 1 and 2 along the vertical axis, and that the (ii) is basically the same as the *I*-*V* characteristics (b) in Fig.10 in my previous paper. [35] It is also noted that no external magnetic field is applied, so that all the SIS-junctions show exactly the same characteristics because the no charge coupling case is considered. Namely, a relation "1@*N* = 2@*N* = ·· = *N*@*N*" is satisfied for the *I*-*V* characteristics of respective junctions. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found when the external ac modulation current is applied, i.e., *ir* � 0.

First, let us focus our attention to the results calculated for *ir* = 0, i.e., (*ir*, *α*)=(0, 0) in (i) in Fig.1 and (0, 0.5), (0, 1) and (0, 2) in Fig.2(a), (b) and (c), respectively. Since *ir* = 0, these results directly show the change of the *I*-*V* characteristics due to the change of the external magnetic field. We can see that (I) the critical current decreases with the increasing the external magnetic induction *Bext*, (II) this trend is remarkable in inner junctions 2@5, 3@5 and 4@5 rather than surface junctions 1@5 and 5@5, and (III) the central junction 3@5 shows a large change due to the change of the *Bext*. Flux quanta enter the insulating layers when *Bext* � 0. If we consider the case when *a* flux quantum just enters an insulating layer in the stacked junctions consisting 10 Will-be-set-by-IN-TECH

*fp* = 122GHz, *fr* = 200GHz, *SRshunt*/junction= 50*μ*m2Ω and *N* = 5. Those are drawn as (i), (ii) and (iii) in Fig.1 for *ir* = 0, 1 and 2 which are denoted as (*ir*, *α*)=(0, 0), (1, 0) and (2, 0), respectively. Next, I have calculated the *I*-*V* characteristics with finite external magnetic fields, i.e., *α* � 0, under the same condition as in Fig.1. Here note again that the external magnetic induction *Bext* is defined by *αB*0(*Lx*) and the *B*0(*Lx*) is 0.345T for *Lx* = 5*μ*m. The *I*-*V* characteristics calculated for (*ir*, *α*)=(0, 0.5), (0, 1), (0, 2), (1, 0.5), (1, 1), (1, 2), (2, 0.5),

(2, 1) and (2, 2) are shown in Fig.2(a), (b), (c), (d), (e), (f), (g), (h) and (i), respectively.

Fig. 1. The *I*-*V* characteristics with no external magnetic field, i.e., *α* = 0, calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Lx* = 5*μ*m, *S*(= *LxLy*) = 25*μ*m2, and *SRshunt*/junction= 50*μ*m2Ω. The effect of charge coupling is not taken into account. (i), (ii) and (iii) are those for *ir* = 0, 1 and 2, and are denoted as (*ir*, *α*)=(0, 0), (1, 0) and (2, 0), respectively. Note that the *I*-*V* characteristics (ii) and (iii) have been shifted by 1 and 2 along the vertical axis, and that the (ii) is basically the same as the *I*-*V* characteristics (b) in Fig.10 in my previous paper. [35] It is also noted that no external magnetic field is applied, so that all the SIS-junctions show exactly the same characteristics because the no charge coupling case

is considered. Namely, a relation "1@*N* = 2@*N* = ·· = *N*@*N*" is satisfied for the *I*-*V* characteristics of respective junctions. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are

First, let us focus our attention to the results calculated for *ir* = 0, i.e., (*ir*, *α*)=(0, 0) in (i) in Fig.1 and (0, 0.5), (0, 1) and (0, 2) in Fig.2(a), (b) and (c), respectively. Since *ir* = 0, these results directly show the change of the *I*-*V* characteristics due to the change of the external magnetic field. We can see that (I) the critical current decreases with the increasing the external magnetic induction *Bext*, (II) this trend is remarkable in inner junctions 2@5, 3@5 and 4@5 rather than surface junctions 1@5 and 5@5, and (III) the central junction 3@5 shows a large change due to the change of the *Bext*. Flux quanta enter the insulating layers when *Bext* � 0. If we consider the case when *a* flux quantum just enters an insulating layer in the stacked junctions consisting

clearly found when the external ac modulation current is applied, i.e., *ir* � 0.

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Fig. 2. The *I*-*V* characteristics of BSCCO IJJs calculated for the same condition as in Fig.1, except for the external magnetic induction *Bext*. The *Bext* is defined by *αB*0(*Lx*) and the *B*0(*Lx*) is 0.345T because of *Lx* = 5*μ*m. The (*ir*, *α*) of (a), (b), (c), (d), (e), (f), (g), (h) and (i) are (0, 0.5), (0, 1), (0, 2), (1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2), respectively. The reduced voltage < *V* >*reduce* defined by Eq.(24) is also shown. Shapiro steps are found when *ir* = 0, but not so clear when *α* = 2.

of the odd number of junctions, then it is sure that *its* flux quantum must enter the insulating layer of central SIS Josephson junction such as 3@5 junction because of the symmetry of the odd number of junctions. This is reasonable to the finding (III). Actually, in the case of Fig.2(a), we can see that only the 3@5 central junction just shows the voltage due to the flux flow when *i*<sup>0</sup> is about 0.6.

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Shapiro Steps in BSCCO Intrinsic Josephson Junctions 169

Fig. 3. Normalized height Δ*i*1(*α*, *N*) of the 1st-order Shapiro step and the normalized critical current *ic*(*α*, *N*) as a function of *α* and *N*. In this figure, the values of *α* have been selected from 0 to 2 with 0.1-step, and two values 1 and 5 have been adopted as the number *N* of junctions. The calculation condition is that *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Lx* = 5*μ*m, *Rshunt*/junction= 2Ω, and *ir* = 1. As already stated, the external magnetic induction *Bext* is defined by *αB*0(*Lx*) and the *B*0(*Lx*) is 0.345T because of *Lx* = 5*μ*m. The Δ*i*1(*α*, 1) and Δ*i*1(*α*, 5) are indicated by open and solid circles, respectively, and the *ic*(*α*, 1) and *ic*(*α*, 5) are drawn by open and solid squares, respectively. In the present paper, the effect of thermal noise has been always taken into account in all the calculations, so it must be stated that both the values of *ic*(0, 1) and *ic*(0, 5) are somewhat smaller than 1. The effect of

*Msat* = *M*(*Bext* ≥ *BL*), but the strength of the *BL* strongly depends on the magnetic property of the F-layer. On the contrary to the case of *Msat*, the *Mres* is obtained when *Bext* = 0, i.e., *Mres* = *M*(0), so it is expected that it could be much easier to control the sample condition as

The sizable effect for the magnetic field is mainly found in the 3@5 central junction as can be seen from Fig.2 for *N* = 5. In the following, therefore, first we focus our attention to the case such that the I-layer in the 3@5 central SIS-Josephson junction is replaced by the F-layer, hereafter we call it as "central case", and next the cases for 1@5 and 2@5 junctions are considered, called as "surface and intermediate cases". Equation (15) tells us that the effective

The external magnetic induction *Bext* is written as *αB*0(*Lx*) using a constant magnetic induction *B*0(*Lx*) so that we write the magnetization *M*(*Bext*) as *m*(*α*)*B*0(*Lx*) using a dimensionless parameter *m*(*α*) as a function of *α*. The in Eq.(29) is 3, 1 and 2 for the

*eff* at the edge of the *j*-th IJJ is given by

*B*(*j*)

charge coupling is not taken into account.

compared with the case of *Msat*.

magnetic induction *B*(*j*)

*eff* = *Bext* + *M*(*Bext*)*δj*, . (29)

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As can be seen in Fig.1, the height of Shapiro steps increases with the increasing the *ir*. By comparing the calculated results for *ir* 0 shown in Fig.2(d), (e), (f), (g), (h) and (i) with those for *ir* = 0 shown in Fig.2(a), (b) and (c), we can say that the *I*-*V* characteristics calculated for (*ir*, *α*)=(1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2) could be regarded as those for (*ir*, *α*) = (0, 0.5), (0, 1) and (0, 2) *added* Shapiro steps which have been calculated by taking into account the effect of the external magnetic field. The *I*-*V* characteristics for *ir* = 0 shown in Fig.2(a), (b) and (c) *directly* reflect the nature of flux flow, and the Shapiro steps shown in Fig.2(d), (e), (f), (g), (h) and (i) are originated from the coupling between the ac Josephson effect and the external ac modulation current with *ir* 0. Therefore, the calculated results shown in Fig.2 seem to show that the Shapiro step and flux flow could be treated separately under the presence of the external magnetic field.

## **4.4 Effect of** *Bext*

We shall consider here the effect of the external magnetic field *Bext* on the reduced voltage < *V* >*reduce* defined by Eq.(24) which is directly connected with the experimental *I*-*V* characteristics. By setting the calculation condition to such that *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1 and *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T, I have calculated the *I*-*V* characteristics as a function of *α*, which defines the external magnetic induction *Bext* as *αB*0(*Lx*), and the number *N* of stacked junctions. The values from 0 to 2 with 0.1-step have been selected as the *α* and two values 1 and 5 have been adopted as the *N*. The normalized height Δ*i*1(*α*, *N*) of the 1st-order Shapiro step and the normalized critical current *ic*(*α*, *N*) evaluated from the reduced voltage < *V* >*reduce* are plotted in Fig.3 as a function of *α*.

Figure 3 shows that (I) for almost all the *α*-values, the values of Δ*i*1(*α*, 1) drawn by open circles are somewhat larger than those of Δ*i*1(*α*, 5) by solid circles, but its difference is not so large, i.e., the *α*-dependence of Δ*i*1(*α*, *N*) seems to be very similar even if the value of *N* differs each other such as *N* = 1 and 5. For the *ic*(*α*, *N*), we can see that (II) the *α*-dependence of the *ic*(*α*, 1) drawn by open squares largely differs from that of the *ic*(*α*, 5) by solid squares. In the BSCCO IJJs with a stacking structure, there are considerable magnetic interactions between neighboring Josephson junctions when *α* 0. Therefore, the finding (II) clearly tells us that (III) the normalized critical current is strongly affected by the magnetic interaction between neighboring Josephson junctions.

#### **4.5 Effect of ferromagnetic layer**

As the case in which some insulating(I) layers in BSCCO-IJJs are replaced by ferromagnetic(F) ones, there are many cases. In this section, for the simplicity we consider the case such that only the �-th I-layer in the BSCCO-IJJs is replaced by a F-layer whose value of the magnetization is *M*(*Bext*) as a function of the external magnetic field *Bext*. The magnetization curve *M*(*Bext*) is characterized by three points; one is the saturation magnetization *Msat*( 0) in which *dM*/*dBext* = 0, two is the residual magnetization *Mres*( 0) in which *Bext* = 0, and three is the coercive force in which the magnetization is zero for the non-zero external field, i.e., *M*(*Bext* = *Bcoe* 0) = 0. We consider here the non-zero magnetization case so that the case of coercive force is not considered here. The *Msat* is obtained when a magnetic field *BL* which is large enough to satisfy the condition of *dM*/*dBext* = 0 is applied to the F-layer, i.e., 12 Will-be-set-by-IN-TECH

As can be seen in Fig.1, the height of Shapiro steps increases with the increasing the *ir*. By comparing the calculated results for *ir* 0 shown in Fig.2(d), (e), (f), (g), (h) and (i) with those for *ir* = 0 shown in Fig.2(a), (b) and (c), we can say that the *I*-*V* characteristics calculated for (*ir*, *α*)=(1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2) could be regarded as those for (*ir*, *α*) = (0, 0.5), (0, 1) and (0, 2) *added* Shapiro steps which have been calculated by taking into account the effect of the external magnetic field. The *I*-*V* characteristics for *ir* = 0 shown in Fig.2(a), (b) and (c) *directly* reflect the nature of flux flow, and the Shapiro steps shown in Fig.2(d), (e), (f), (g), (h) and (i) are originated from the coupling between the ac Josephson effect and the external ac modulation current with *ir* 0. Therefore, the calculated results shown in Fig.2 seem to show that the Shapiro step and flux flow could be treated separately under the

We shall consider here the effect of the external magnetic field *Bext* on the reduced voltage < *V* >*reduce* defined by Eq.(24) which is directly connected with the experimental *I*-*V* characteristics. By setting the calculation condition to such that *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1 and *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T, I have calculated the *I*-*V* characteristics as a function of *α*, which defines the external magnetic induction *Bext* as *αB*0(*Lx*), and the number *N* of stacked junctions. The values from 0 to 2 with 0.1-step have been selected as the *α* and two values 1 and 5 have been adopted as the *N*. The normalized height Δ*i*1(*α*, *N*) of the 1st-order Shapiro step and the normalized critical current *ic*(*α*, *N*) evaluated from the reduced voltage < *V* >*reduce*

Figure 3 shows that (I) for almost all the *α*-values, the values of Δ*i*1(*α*, 1) drawn by open circles are somewhat larger than those of Δ*i*1(*α*, 5) by solid circles, but its difference is not so large, i.e., the *α*-dependence of Δ*i*1(*α*, *N*) seems to be very similar even if the value of *N* differs each other such as *N* = 1 and 5. For the *ic*(*α*, *N*), we can see that (II) the *α*-dependence of the *ic*(*α*, 1) drawn by open squares largely differs from that of the *ic*(*α*, 5) by solid squares. In the BSCCO IJJs with a stacking structure, there are considerable magnetic interactions between neighboring Josephson junctions when *α* 0. Therefore, the finding (II) clearly tells us that (III) the normalized critical current is strongly affected by the magnetic interaction between

As the case in which some insulating(I) layers in BSCCO-IJJs are replaced by ferromagnetic(F) ones, there are many cases. In this section, for the simplicity we consider the case such that only the �-th I-layer in the BSCCO-IJJs is replaced by a F-layer whose value of the magnetization is *M*(*Bext*) as a function of the external magnetic field *Bext*. The magnetization curve *M*(*Bext*) is characterized by three points; one is the saturation magnetization *Msat*( 0) in which *dM*/*dBext* = 0, two is the residual magnetization *Mres*( 0) in which *Bext* = 0, and three is the coercive force in which the magnetization is zero for the non-zero external field, i.e., *M*(*Bext* = *Bcoe* 0) = 0. We consider here the non-zero magnetization case so that the case of coercive force is not considered here. The *Msat* is obtained when a magnetic field *BL* which is large enough to satisfy the condition of *dM*/*dBext* = 0 is applied to the F-layer, i.e.,

presence of the external magnetic field.

are plotted in Fig.3 as a function of *α*.

neighboring Josephson junctions.

**4.5 Effect of ferromagnetic layer**

**4.4 Effect of** *Bext*

Fig. 3. Normalized height Δ*i*1(*α*, *N*) of the 1st-order Shapiro step and the normalized critical current *ic*(*α*, *N*) as a function of *α* and *N*. In this figure, the values of *α* have been selected from 0 to 2 with 0.1-step, and two values 1 and 5 have been adopted as the number *N* of junctions. The calculation condition is that *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Lx* = 5*μ*m, *Rshunt*/junction= 2Ω, and *ir* = 1. As already stated, the external magnetic induction *Bext* is defined by *αB*0(*Lx*) and the *B*0(*Lx*) is 0.345T because of *Lx* = 5*μ*m. The Δ*i*1(*α*, 1) and Δ*i*1(*α*, 5) are indicated by open and solid circles, respectively, and the *ic*(*α*, 1) and *ic*(*α*, 5) are drawn by open and solid squares, respectively. In the present paper, the effect of thermal noise has been always taken into account in all the calculations, so it must be stated that both the values of *ic*(0, 1) and *ic*(0, 5) are somewhat smaller than 1. The effect of charge coupling is not taken into account.

*Msat* = *M*(*Bext* ≥ *BL*), but the strength of the *BL* strongly depends on the magnetic property of the F-layer. On the contrary to the case of *Msat*, the *Mres* is obtained when *Bext* = 0, i.e., *Mres* = *M*(0), so it is expected that it could be much easier to control the sample condition as compared with the case of *Msat*.

The sizable effect for the magnetic field is mainly found in the 3@5 central junction as can be seen from Fig.2 for *N* = 5. In the following, therefore, first we focus our attention to the case such that the I-layer in the 3@5 central SIS-Josephson junction is replaced by the F-layer, hereafter we call it as "central case", and next the cases for 1@5 and 2@5 junctions are considered, called as "surface and intermediate cases". Equation (15) tells us that the effective magnetic induction *B*(*j*) *eff* at the edge of the *j*-th IJJ is given by

$$B\_{eff}^{(j)} = B\_{\rm ext} + M(B\_{\rm ext}) \delta\_{j, \ell} \,. \tag{29}$$

The external magnetic induction *Bext* is written as *αB*0(*Lx*) using a constant magnetic induction *B*0(*Lx*) so that we write the magnetization *M*(*Bext*) as *m*(*α*)*B*0(*Lx*) using a dimensionless parameter *m*(*α*) as a function of *α*. The in Eq.(29) is 3, 1 and 2 for the

(c), (d), (e) and (f) are for *mres* = 0, 0.1, 0.2, 0.5, 1 and 2, respectively. From the symmetry, only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found except for the 3@5 central junction

in not only the junctions without F-layer but also the junction including F-layer, (II) the *I*-*V* characteristics of the 3@5 central junction including F-layer are surely changed due to the increasing the value of *m*(1), but (III) those of other junctions without F-layer almost remain the same for the change of *m*(1). Findings (II) and (III) mentioned now are fairly similar to those for the case of no external magnetic field, however, we wish to say that Shapiro step is found even in the 3@5 junction shown in (e) whose the **Γ** is (1, 1, 2, 1, 1), in spite of the fact that

The *I*-*V* characteristics calculated for the surface case (1 + *m*(1), 1, 1, 1, 1) are shown in Fig.8 and those for the intermediate one (1, 1 + *m*(1), 1, 1, 1) are in Fig.9, where (a), (b) and (c) are for *m*(1) = 0.2, 0.5 and 1, respectively. Figure 8 shows that (I) Shapiro steps are clearly found in all the junctions even in the case of *m*(1) = 1, and (II) the *I*-*V* characteristics of 1@5 surface

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*eff* /*B*0(*Lx*) is (0, 0, *mres*, 0, 0). (a), (b),

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when *mres* = 2 shown in (f).

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> 

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dimensionless effective magnetic induction defined by *B*(*j*)

clear Shapiro step cannot be found in (f) whose the **Γ** is (2, 2, 2, 2, 2).

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Shapiro Steps in BSCCO Intrinsic Josephson Junctions 171

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Fig. 4. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T and *χ*(*dF*) = 1 for all junctions. The case such that there is no external magnetic field, i.e., *Bext* = 0, and the I-layer in the 3@5 central SIS-Josephson junction is replaced by a F-layer is considered, so that the configuration **Γ** for the

 

central, surface and intermediate cases, respectively. Therefore, the configuration **Γ** for the dimensionless effective magnetic induction defined by *B*(*j*) *eff* /*B*0(*Lx*) can be written as (*α*, *α*, *α* + *m*(*α*), *α*, *α*), (*α* + *m*(*α*), *α*, *α*, *α*, *α*) and (*α*, *α* + *m*(*α*), *α*, *α*, *α*) for the central, surface and intermediate cases, respectively.

## **4.5.1 Case of** *χ*(*dF*) = 1 **for the -th SFS junction**

First, we consider the case of the residual magnetization *Mres*, i.e., *α* = 0. The configurations **Γ** for the central, surface and intermediate cases are therefore (0, 0, *m*(0), 0, 0), (*m*(0), 0, 0, 0, 0) and (0, *m*(0), 0, 0, 0), respectively. By adopting the condition such that *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1 and *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T, I have calculated the *I*-*V* characteristics for *m*(0) ≡ *mres* = 0.1, 0.2, 0.5, 1 and 2. The *I*-*V* characteristics of the central case calculated for *mres* = 0, 0.1, 0.2, 0.5, 1 and 2 are shown in Fig.4(a), (b), (c), (d), (e) and (f), respectively. Here note that only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown from a symmetry consideration. Figure 4 shows that (I) Shapiro steps are clearly found except for the 3@5 central junction when *mres* = 2, and due to the increasing the *mres*, i.e., the residual magnetization *Mres* (≡ *mresB*0(*Lx*)), (II) the *I*-*V* characteristics of 3@5 central junction are gradually changed, but (III) those of other junctions including no F-layer almost remain the same, that is, no remarkable change is found for the junctions without the F-layer. These findings are interesting. We are now solving a matrix equation (13) with **Σ<sup>c</sup>** = **1**. The matrix **Σ<sup>L</sup>** describing the magnetic interactions between neighboring junctions is given by Eq.(3) so that the inverse matrix **Σ<sup>L</sup>** <sup>−</sup><sup>1</sup> is not a diagonal but a general form in which all the matrix elements are not equal to zero. Therefore, except for the case of **Γ** = (0, 0, 0, 0, 0), it seems to be reasonable to suppose that the effect of ferromagnetic layer should be observed on not only the 3@5 central junction but also the others. The calculated results are far from such a conjecture. The results calculated for the surface and intermediate cases, namely, **Γ** = (*mres*, 0, 0, 0, 0) and (0, *mres*, 0, 0, 0), are shown in Figs.5 and 6, respectively. From these figures, we can see a similar result as in Fig.4 such that (I) Shapiro steps are clearly found except for the junction including F-layer when *mres* = 2, (II) the *I*-*V* characteristics of the junction including F-layer are gradually changed due to the increasing the *mres*, but (III) no remarkable change is found for others.

From the above, we can conclude that if the case such that an I-layer in the BSCCO-IJJs is replaced by a F-layer is considered, then the effect of the F-layer to the *I*-*V* characteristics is restricted when *Bext* = 0, namely, no remarkable change is found for the junctions without the F-layer.

Finally, let us consider the case of *Bext* �= 0, i.e., *α* �= 0. The configurations **Γ** to be considered are (*α*, *α*, *α* + *m*(*α*), *α*, *α*), (*α* + *m*(*α*), *α*, *α*, *α*, *α*) and (*α*, *α* + *m*(*α*), *α*, *α*, *α*) for the central, surface and intermediate cases. As an example, in the following the *α* is set to 1 tentatively, and 0.1, 0.2, 0.5 and 1 are selected as the value of *m*(1).

The *I*-*V* characteristics calculated for the central cases (1, 1, 1 + *m*(1), 1, 1) with *m*(1) = 0, 0.1, 0.2, 0.5 and 1 are shown in Fig.7(a), (b), (c), (d) and (e), respectively, together with those for **Γ** = (2, 2, 2, 2, 2) shown in (f). Figure 7 shows that (I) Shapiro steps are clearly found 14 Will-be-set-by-IN-TECH

central, surface and intermediate cases, respectively. Therefore, the configuration **Γ** for

(*α*, *α*, *α* + *m*(*α*), *α*, *α*), (*α* + *m*(*α*), *α*, *α*, *α*, *α*) and (*α*, *α* + *m*(*α*), *α*, *α*, *α*) for the central, surface and

First, we consider the case of the residual magnetization *Mres*, i.e., *α* = 0. The configurations **Γ** for the central, surface and intermediate cases are therefore (0, 0, *m*(0), 0, 0), (*m*(0), 0, 0, 0, 0) and (0, *m*(0), 0, 0, 0), respectively. By adopting the condition such that *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1 and *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T, I have calculated the *I*-*V* characteristics for *m*(0) ≡ *mres* = 0.1, 0.2, 0.5, 1 and 2. The *I*-*V* characteristics of the central case calculated for *mres* = 0, 0.1, 0.2, 0.5, 1 and 2 are shown in Fig.4(a), (b), (c), (d), (e) and (f), respectively. Here note that only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown from a symmetry consideration. Figure 4 shows that (I) Shapiro steps are clearly found except for the 3@5 central junction when *mres* = 2, and due to the increasing the *mres*, i.e., the residual magnetization *Mres* (≡ *mresB*0(*Lx*)), (II) the *I*-*V* characteristics of 3@5 central junction are gradually changed, but (III) those of other junctions including no F-layer almost remain the same, that is, no remarkable change is found for the junctions without the F-layer. These findings are interesting. We are now solving a matrix equation (13) with **Σ<sup>c</sup>** = **1**. The matrix **Σ<sup>L</sup>** describing the magnetic interactions between neighboring

form in which all the matrix elements are not equal to zero. Therefore, except for the case of **Γ** = (0, 0, 0, 0, 0), it seems to be reasonable to suppose that the effect of ferromagnetic layer should be observed on not only the 3@5 central junction but also the others. The calculated results are far from such a conjecture. The results calculated for the surface and intermediate cases, namely, **Γ** = (*mres*, 0, 0, 0, 0) and (0, *mres*, 0, 0, 0), are shown in Figs.5 and 6, respectively. From these figures, we can see a similar result as in Fig.4 such that (I) Shapiro steps are clearly found except for the junction including F-layer when *mres* = 2, (II) the *I*-*V* characteristics of the junction including F-layer are gradually changed due to the increasing the *mres*, but (III)

From the above, we can conclude that if the case such that an I-layer in the BSCCO-IJJs is replaced by a F-layer is considered, then the effect of the F-layer to the *I*-*V* characteristics is restricted when *Bext* = 0, namely, no remarkable change is found for the junctions without the

Finally, let us consider the case of *Bext* �= 0, i.e., *α* �= 0. The configurations **Γ** to be considered are (*α*, *α*, *α* + *m*(*α*), *α*, *α*), (*α* + *m*(*α*), *α*, *α*, *α*, *α*) and (*α*, *α* + *m*(*α*), *α*, *α*, *α*) for the central, surface and intermediate cases. As an example, in the following the *α* is set to 1 tentatively, and 0.1,

The *I*-*V* characteristics calculated for the central cases (1, 1, 1 + *m*(1), 1, 1) with *m*(1) = 0, 0.1, 0.2, 0.5 and 1 are shown in Fig.7(a), (b), (c), (d) and (e), respectively, together with those for **Γ** = (2, 2, 2, 2, 2) shown in (f). Figure 7 shows that (I) Shapiro steps are clearly found

*eff* /*B*0(*Lx*) can be written as

<sup>−</sup><sup>1</sup> is not a diagonal but a general

the dimensionless effective magnetic induction defined by *B*(*j*)

**4.5.1 Case of** *χ*(*dF*) = 1 **for the -th SFS junction**

junctions is given by Eq.(3) so that the inverse matrix **Σ<sup>L</sup>**

no remarkable change is found for others.

0.2, 0.5 and 1 are selected as the value of *m*(1).

F-layer.

intermediate cases, respectively.

Fig. 4. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T and *χ*(*dF*) = 1 for all junctions. The case such that there is no external magnetic field, i.e., *Bext* = 0, and the I-layer in the 3@5 central SIS-Josephson junction is replaced by a F-layer is considered, so that the configuration **Γ** for the dimensionless effective magnetic induction defined by *B*(*j*) *eff* /*B*0(*Lx*) is (0, 0, *mres*, 0, 0). (a), (b), (c), (d), (e) and (f) are for *mres* = 0, 0.1, 0.2, 0.5, 1 and 2, respectively. From the symmetry, only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found except for the 3@5 central junction when *mres* = 2 shown in (f).

in not only the junctions without F-layer but also the junction including F-layer, (II) the *I*-*V* characteristics of the 3@5 central junction including F-layer are surely changed due to the increasing the value of *m*(1), but (III) those of other junctions without F-layer almost remain the same for the change of *m*(1). Findings (II) and (III) mentioned now are fairly similar to those for the case of no external magnetic field, however, we wish to say that Shapiro step is found even in the 3@5 junction shown in (e) whose the **Γ** is (1, 1, 2, 1, 1), in spite of the fact that clear Shapiro step cannot be found in (f) whose the **Γ** is (2, 2, 2, 2, 2).

The *I*-*V* characteristics calculated for the surface case (1 + *m*(1), 1, 1, 1, 1) are shown in Fig.8 and those for the intermediate one (1, 1 + *m*(1), 1, 1, 1) are in Fig.9, where (a), (b) and (c) are for *m*(1) = 0.2, 0.5 and 1, respectively. Figure 8 shows that (I) Shapiro steps are clearly found in all the junctions even in the case of *m*(1) = 1, and (II) the *I*-*V* characteristics of 1@5 surface

Fig. 6. The same as in Fig.5 but for **Γ** = (0, *mres*, 0, 0, 0), that is, the I-layer in 2@5 intermediate SIS junction is replaced by a F-layer. Shapiro steps are clearly found except for the 2@5

we consider only the central case with *m*(0) = 0.5. Namely, only an I-layer in the central SIS Josephson junction is replaced by a F-layer, therefore, the corresponding configuration **Γ** is (0, 0, 0.5, 0, 0). The aim of the present section is to see the effect of *χ*(*dF*) defined in Eq.(16) to

The central case for *N* = 5 is considered so that the value of is 3. By using 0, 0.2, 0.4, 0.6 and 0.8 as a value of *χ*3(*dF*), I have calculated the *I*-*V* characteristics. The *I*-*V* characteristics calculated for *χ*3(*dF*) = 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 are shown in Fig.10(a), (b), (c), (d), (e) and (f), respectively. Here note that from a symmetry consideration, only the *I*-*V* characteristics of 1@5-surface, 2@5-intermediate and 3@5-central junctions are shown in the respective figure. Figure 10 shows that (I) as is expected, the *I*-*V* characteristic of 3@5-central junction including F-layer surely changes due to the change of the value of *χ*3(*dF*), especially, the height of the Shapiro step decreases with the decreasing the *χ*3(*dF*) and reaches to zero when *χ*3(*dF*) = 0, however, (II) those of 1@5-surface and 2@5-intermediate junctions without the F-layer fairly well remain the same for the changing the *χ*3(*dF*), and (III) clear Shapiro steps are found in 1@5-surface and 2@5-intermediate junctions irrespective of the value of *χ*3(*dF*). Equation (16) tells us that when *χ*3(*dF*) = 0, the 3@5-central junction can be regarded as a simple non-superconducting RC-junction. This fact is clearly found in the *I*-*V* characteristic of

3@5-junction shown in (f), that is, in which a purely ohmic characterictic is found.

Next, let us consider the case of *α* 0 for the same calculation condition mentioned above. We consider again the central case with *m*(*α*) = 0.5. As an example, we set the *α* to 1 so that the corresponding configuration **Γ** is (1, 1, 1.5, 1, 1). The *I*-*V* characteristics calculated for *χ*3(*dF*) = 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 are shown in Fig.11(a), (b), (c), (d), (e) and (f),

 **-**

**-**

Shapiro Steps in BSCCO Intrinsic Josephson Junctions 173

**-**

**-**

 **-**

the *I*-*V* characteristics.

**-**  **-**

 **-**

**-**


 **-**

intermediate junction when *mres* = 2 shown in (c).

**-** 

**-**

 **-**

**-** 

**-**

 **-**

Fig. 5. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T, *Bext* = 0 and *χ*�(*dF*) = 1 for all junctions. The I-layer in 1@5 surface SIS junction is replaced by a F-layer so that the corresponding configuration **Γ** is (*mres*, 0, 0, 0, 0). Therefore, the *I*-*V* characteristics of all junctions are shown together with the reduced voltage < *V* >*reduce*. (a), (b) and (c) are for *mres* = 0.5, 1 and 2, respectively. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found except for the 1@5 surface junction when *mres* = 2 shown in (c).

junction including F-layer are gradually changed due to the increasing the value of *m*(1). These findings are basically the same as the finding obtained in the central case. For the 2@5, 3@5, 4@5 and 5@5 junctions without F-layer, we can see that (III) the *I*-*V* characteristics of 2@5 and 3@5 junctions, which are close to the 1@5 junction including the F-layer, are fairly changed due to the increasing the *m*(1), but (IV) those of 4@5 and 5@5 junctions, which are far from the 1@5 junction, almost remain the same as well as the central case. Finding (III) clearly tells us that the magnetic interaction between neighboring junctions is not negligible in BSCCO IJJs when *Bext* 0. From Fig.9, we can see a similar result as in the surface case, excepting 1@5 and 5@5 surface junctions in which the *I*-*V* characteristics are almost the same for the change of the value of *m*(1).

## **4.5.2 Case of** *χ*�(*dF*) < 1 **for the** �**-th SFS junction**

First, we consider the case of no external magnetic field, i.e., *α* = 0. The calculation condition is the same as above, that is, *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1 and *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T. There are three cases as the configurations that we should study, but in the following, 16 Will-be-set-by-IN-TECH

**-**

**-**

**-**

 **-**

*mres* = 2 shown in (c).

of the value of *m*(1).

**-**  **-**

 **-**

**-**


**-** 

**-**

 **-**

Fig. 5. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T, *Bext* = 0 and *χ*�(*dF*) = 1 for all junctions. The I-layer in 1@5 surface SIS junction is replaced by a F-layer so that the corresponding configuration **Γ** is (*mres*, 0, 0, 0, 0). Therefore, the *I*-*V* characteristics of all junctions are shown together with the reduced voltage < *V* >*reduce*. (a), (b) and (c) are for *mres* = 0.5, 1 and 2, respectively. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found except for the 1@5 surface junction when

 **-**

**4.5.2 Case of** *χ*�(*dF*) < 1 **for the** �**-th SFS junction**

junction including F-layer are gradually changed due to the increasing the value of *m*(1). These findings are basically the same as the finding obtained in the central case. For the 2@5, 3@5, 4@5 and 5@5 junctions without F-layer, we can see that (III) the *I*-*V* characteristics of 2@5 and 3@5 junctions, which are close to the 1@5 junction including the F-layer, are fairly changed due to the increasing the *m*(1), but (IV) those of 4@5 and 5@5 junctions, which are far from the 1@5 junction, almost remain the same as well as the central case. Finding (III) clearly tells us that the magnetic interaction between neighboring junctions is not negligible in BSCCO IJJs when *Bext* 0. From Fig.9, we can see a similar result as in the surface case, excepting 1@5 and 5@5 surface junctions in which the *I*-*V* characteristics are almost the same for the change

First, we consider the case of no external magnetic field, i.e., *α* = 0. The calculation condition is the same as above, that is, *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1 and *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T. There are three cases as the configurations that we should study, but in the following,

 **-**

**-** 

**-**

 **-**

Fig. 6. The same as in Fig.5 but for **Γ** = (0, *mres*, 0, 0, 0), that is, the I-layer in 2@5 intermediate SIS junction is replaced by a F-layer. Shapiro steps are clearly found except for the 2@5 intermediate junction when *mres* = 2 shown in (c).

we consider only the central case with *m*(0) = 0.5. Namely, only an I-layer in the central SIS Josephson junction is replaced by a F-layer, therefore, the corresponding configuration **Γ** is (0, 0, 0.5, 0, 0). The aim of the present section is to see the effect of *χ*(*dF*) defined in Eq.(16) to the *I*-*V* characteristics.

The central case for *N* = 5 is considered so that the value of is 3. By using 0, 0.2, 0.4, 0.6 and 0.8 as a value of *χ*3(*dF*), I have calculated the *I*-*V* characteristics. The *I*-*V* characteristics calculated for *χ*3(*dF*) = 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 are shown in Fig.10(a), (b), (c), (d), (e) and (f), respectively. Here note that from a symmetry consideration, only the *I*-*V* characteristics of 1@5-surface, 2@5-intermediate and 3@5-central junctions are shown in the respective figure. Figure 10 shows that (I) as is expected, the *I*-*V* characteristic of 3@5-central junction including F-layer surely changes due to the change of the value of *χ*3(*dF*), especially, the height of the Shapiro step decreases with the decreasing the *χ*3(*dF*) and reaches to zero when *χ*3(*dF*) = 0, however, (II) those of 1@5-surface and 2@5-intermediate junctions without the F-layer fairly well remain the same for the changing the *χ*3(*dF*), and (III) clear Shapiro steps are found in 1@5-surface and 2@5-intermediate junctions irrespective of the value of *χ*3(*dF*). Equation (16) tells us that when *χ*3(*dF*) = 0, the 3@5-central junction can be regarded as a simple non-superconducting RC-junction. This fact is clearly found in the *I*-*V* characteristic of 3@5-junction shown in (f), that is, in which a purely ohmic characterictic is found.

Next, let us consider the case of *α* 0 for the same calculation condition mentioned above. We consider again the central case with *m*(*α*) = 0.5. As an example, we set the *α* to 1 so that the corresponding configuration **Γ** is (1, 1, 1.5, 1, 1). The *I*-*V* characteristics calculated for *χ*3(*dF*) = 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 are shown in Fig.11(a), (b), (c), (d), (e) and (f),

Fig. 8. The same as in Fig.7 but for the surface case, i.e., **Γ** = (1 + *m*(1), 1, 1, 1, 1). The I-layer in 1@5 surface SIS junction is replaced by a F-layer so that the *I*-*V* characteristics of all junctions are shown together with the reduced voltage < *V* >*reduce*. (a), (b) and (c) are for *m*(1) = 0.2, 0.5 and 1, respectively. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found in all

Shapiro steps in BSCCO IJJs have been studied full numerically. In the study, the *I*-*V* characteristics of not only the case of no external magnetic field but also the case in which an external magnetic field is applied parallel to the junction surface have been calculated by using the MS-unified theory including both the electric and magnetic field couplings between

For the effect of the junction length *Lx* along the *x*-axis, we have found that the overall profiles of the *I*-*V* characteristics including Shapiro steps calculated for the same external magnetic induction are fairly similar each other even if the junction length *Lx* differs each other.

For the effect of *SRshunt*-product, we have found that it is essential to add the shunt resistance with a low-value into the BSCCO IJJs, if we wish to make a good quality Shapiro step device which is *hard* for the external magnetic disturbance. In the present study, therefore, the value

In order to see the effect of *ir* to the *I*-*V* characteristics including Shapiro steps, I have calculated the *I*-*V* characteristics of nine cases such that (*ir*, *α*)=(0, 0.5), (0, 1), (0, 2), (1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2). Here note that the *ir* is the normalized amplitude of the external ac modulation current with the frequency *ω<sup>r</sup>* and the *α* defines the external magnetic induction *Bext* as *αB*0(*Lx*) using a constant magnetic induction *B*0(*Lx*). We have found that the

**-**

 **-**

Shapiro Steps in BSCCO Intrinsic Josephson Junctions 175

  

the *I*-*V* characteristics.

neighboring Josephson junctions.

of *SRshunt*-product has been fixed to 50*μ*m2Ω/junction.

**5. Summary**

**-**

**-** 

**-**

**-**


 

 

Fig. 7. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T and *χ*(*dF*) = 1 for all junctions. The external magnetic induction *Bext* is equal to *B*0(*Lx*) because of *α* = 1. The I-layer in the 3@5 central SIS-Josephson junction is replaced by a F-layer so that the corresponding configuration **Γ** is (1, 1, 1 + *m*(1), 1, 1). (a), (b), (c), (d) and (e) are for *m*(1) = 0, 0.1, 0.2, 0.5 and 1, respectively. For the comparison, the *I*-*V* characteristics calculated for **Γ** = (2, 2, 2, 2, 2) are also shown in (f). From the symmetry, only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. (a) and (f) are equal to Fig.2(e) and (f), respectively. Shapiro steps are clearly found in all the *I*-*V* characteristics drawn in (a) to (e).

respectively. Figure 11 leads us a same conclusion mentioned above such that due to the decreasing the value of *χ*3(*dF*), (I) the *I*-*V* characteristic of 3@5 junction including the F-layer gradually changes, but those of 1@5 and 2@5 junctions without the F-layer well remain the same, and (II) Shapiro steps are clearly found in 1@5 and 2@5 junctions, but the height of the Shapiro step found in 3@5 junction decreases and reaches to zero when *χ*3(*dF*) = 0.

From the above, we can say that the effect of *χ*3(*dF*) defined in the 3@5-central junction to the other junctions is negligible small irrespective of the value of the external magnetic field *Hext*. This finding tells us that even if the BSCCO junction array with a multi-stacking structure is considered, the influence of the F-layer is restricted within a SFS-junction *itself*, i.e., that is negligible for others.

Fig. 8. The same as in Fig.7 but for the surface case, i.e., **Γ** = (1 + *m*(1), 1, 1, 1, 1). The I-layer in 1@5 surface SIS junction is replaced by a F-layer so that the *I*-*V* characteristics of all junctions are shown together with the reduced voltage < *V* >*reduce*. (a), (b) and (c) are for *m*(1) = 0.2, 0.5 and 1, respectively. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found in all the *I*-*V* characteristics.

## **5. Summary**

18 Will-be-set-by-IN-TECH

**-**

 **-**

**-**

 

 

**-**

**-** 

**-**

  

  

negligible for others.

**-** 

**-**

**-**

 

 

 **-**

 

 

 

**-**

Fig. 7. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T and *χ*(*dF*) = 1 for all junctions. The external magnetic induction *Bext* is equal to *B*0(*Lx*) because of *α* = 1. The I-layer in the 3@5 central

SIS-Josephson junction is replaced by a F-layer so that the corresponding configuration **Γ** is (1, 1, 1 + *m*(1), 1, 1). (a), (b), (c), (d) and (e) are for *m*(1) = 0, 0.1, 0.2, 0.5 and 1, respectively. For the comparison, the *I*-*V* characteristics calculated for **Γ** = (2, 2, 2, 2, 2) are also shown in (f). From the symmetry, only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. (a) and (f) are equal to Fig.2(e) and (f), respectively. Shapiro steps are clearly found in all the *I*-*V* characteristics drawn in (a) to (e).

respectively. Figure 11 leads us a same conclusion mentioned above such that due to the decreasing the value of *χ*3(*dF*), (I) the *I*-*V* characteristic of 3@5 junction including the F-layer gradually changes, but those of 1@5 and 2@5 junctions without the F-layer well remain the same, and (II) Shapiro steps are clearly found in 1@5 and 2@5 junctions, but the height of the

From the above, we can say that the effect of *χ*3(*dF*) defined in the 3@5-central junction to the other junctions is negligible small irrespective of the value of the external magnetic field *Hext*. This finding tells us that even if the BSCCO junction array with a multi-stacking structure is considered, the influence of the F-layer is restricted within a SFS-junction *itself*, i.e., that is

Shapiro step found in 3@5 junction decreases and reaches to zero when *χ*3(*dF*) = 0.

Shapiro steps in BSCCO IJJs have been studied full numerically. In the study, the *I*-*V* characteristics of not only the case of no external magnetic field but also the case in which an external magnetic field is applied parallel to the junction surface have been calculated by using the MS-unified theory including both the electric and magnetic field couplings between neighboring Josephson junctions.

For the effect of the junction length *Lx* along the *x*-axis, we have found that the overall profiles of the *I*-*V* characteristics including Shapiro steps calculated for the same external magnetic induction are fairly similar each other even if the junction length *Lx* differs each other.

For the effect of *SRshunt*-product, we have found that it is essential to add the shunt resistance with a low-value into the BSCCO IJJs, if we wish to make a good quality Shapiro step device which is *hard* for the external magnetic disturbance. In the present study, therefore, the value of *SRshunt*-product has been fixed to 50*μ*m2Ω/junction.

In order to see the effect of *ir* to the *I*-*V* characteristics including Shapiro steps, I have calculated the *I*-*V* characteristics of nine cases such that (*ir*, *α*)=(0, 0.5), (0, 1), (0, 2), (1, 0.5), (1, 1), (1, 2), (2, 0.5), (2, 1) and (2, 2). Here note that the *ir* is the normalized amplitude of the external ac modulation current with the frequency *ω<sup>r</sup>* and the *α* defines the external magnetic induction *Bext* as *αB*0(*Lx*) using a constant magnetic induction *B*0(*Lx*). We have found that the

case of *α* 0, the findings (II) and (III) mentioned above are basically held again, but for the finging (I) we have found that (V) clear Shapiro steps are found in not only the junction without F-layer but also the junction including F-layer. In addition to the above calculations, I have further calculated the *I*-*V* characteristics for the surface and intermediate cases when *α* = 1, whose the configurations are (1 + *m*(1), 1, 1, 1, 1) and (1, 1 + *m*(1), 1, 1, 1), respectively. We have concluded that even in the case in which a junction in BSCCO IJJs includes a F-layer with a finite magnetization, (I) the effect of the magnetic interaction between neighboring junctions is not so clearly found when *Bext* = 0, but (II) when *Bext* 0, that is found clearly. Conclusion (II) obtained here is the same as a previous statement such that in the BSCCO IJJs with a stacking structure, there are considerable magnetic interactions between neighboring

In order to see the effect of the F-layer in the SFS-junction to the *I*-*V* characteristics of BSCCO IJJs with a stacking structure, I have calculated the *I*-*V* characteristics as a function of *χ*(*dF*) defined in Eq.(16). From the results calculated for the no external magnetic field and the

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Shapiro Steps in BSCCO Intrinsic Josephson Junctions 177


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Fig. 10. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T. In this figure, the central case has been considered and the configuration **Γ** has been tentatively selected as (0, 0, 0.5, 0, 0). For the *χ*(*dF*) defined in Eq.(16), only the value for = 3 has been corrected, i.e., that of others has been held to 1. The value of *χ*3(*dF*) is 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 for (a), (b), (c), (d), (e) and (f), respectively. From the symmetry, only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the

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Josephson junctions when *α* 0. [36]

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vertical one shows the normalized dc *i*0. Note that (a) is equal to Fig.4(d).

 

Fig. 9. The same as in Fig.7 but for the intermediate case, i.e., **Γ** = (1, 1 + *m*(1), 1, 1, 1). The I-layer in 2@5 intermediate SIS junction is replaced by a F-layer so that the *I*-*V* characteristics of all junctions are shown together with the reduced voltage < *V* >*reduce*. (a), (b) and (c) are for *m*(1) = 0.2, 0.5 and 1, respectively. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are clearly found in all the *I*-*V* characteristics.

Shapiro step and the flux flow could be treated separately under the presence of the external magnetic field.

From the *I*-*V* characteristics as a function of *α*, we have found that (I) the *α*-dependence of the normalized height Δ*i*1(*α*, *N*) of the 1st-order Shapiro step is very similar even if the *N* differs each other, however, (II) the *α*-dependence of the normalized critical current *ic*(*α*, *N*) largely differs for *N* = 1 and 5, that is, the *α*-dependence of the *ic*(*α*, *N*) is strongly affected by the magnetic interaction between neighboring Josephson junctions.

Within the framework of the MS-unified theory, I have studied the effect of ferromagnetic(F) layers into the *I*-*V* characteristics of BSCCO IJJs. In the present paper, we have studied the simplest case such that only the �-th insulating(I)-layer in the BSCCO-IJJs is replaced by a F-layer whose the value of the magnetization is *M*(*Bext*) as a function of the external magnetic induction *Bext*. The effect of the magnetic field is taken into account via the boundary condition at the junction edge, so that the effective magnetic induction *B*(*j*) *eff* at the edge of the *j*-th IJJ is given by *Bext* + *M*(*Bext*)*δj*,�. We have found that when *Bext* = 0, (I) Shapiro steps are clearly found except for the junction including F-layer with a large residual magnetization *Mres* ≡ *mresB*0(*Lx*) such as *mres* = 2, (II) the *I*-*V* characteristics of the junction including F-layer are gradually changed due to the increasing the value of *mres*, but (III) no remarkable change is found for the other junctions without F-layer. For *Bext* �= 0, i.e., *α* �= 0, in which the magnetization *M*(*Bext*) is written as *m*(*α*)*B*0(*Lx*), we have found that (IV) even in the 20 Will-be-set-by-IN-TECH

**-**

 **-**

*eff* at the edge of the

  

magnetic field.

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**-** 

**-**

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clearly found in all the *I*-*V* characteristics.

 

magnetic interaction between neighboring Josephson junctions.

condition at the junction edge, so that the effective magnetic induction *B*(*j*)

Fig. 9. The same as in Fig.7 but for the intermediate case, i.e., **Γ** = (1, 1 + *m*(1), 1, 1, 1). The I-layer in 2@5 intermediate SIS junction is replaced by a F-layer so that the *I*-*V* characteristics of all junctions are shown together with the reduced voltage < *V* >*reduce*. (a), (b) and (c) are

Shapiro step and the flux flow could be treated separately under the presence of the external

From the *I*-*V* characteristics as a function of *α*, we have found that (I) the *α*-dependence of the normalized height Δ*i*1(*α*, *N*) of the 1st-order Shapiro step is very similar even if the *N* differs each other, however, (II) the *α*-dependence of the normalized critical current *ic*(*α*, *N*) largely differs for *N* = 1 and 5, that is, the *α*-dependence of the *ic*(*α*, *N*) is strongly affected by the

Within the framework of the MS-unified theory, I have studied the effect of ferromagnetic(F) layers into the *I*-*V* characteristics of BSCCO IJJs. In the present paper, we have studied the simplest case such that only the �-th insulating(I)-layer in the BSCCO-IJJs is replaced by a F-layer whose the value of the magnetization is *M*(*Bext*) as a function of the external magnetic induction *Bext*. The effect of the magnetic field is taken into account via the boundary

*j*-th IJJ is given by *Bext* + *M*(*Bext*)*δj*,�. We have found that when *Bext* = 0, (I) Shapiro steps are clearly found except for the junction including F-layer with a large residual magnetization *Mres* ≡ *mresB*0(*Lx*) such as *mres* = 2, (II) the *I*-*V* characteristics of the junction including F-layer are gradually changed due to the increasing the value of *mres*, but (III) no remarkable change is found for the other junctions without F-layer. For *Bext* �= 0, i.e., *α* �= 0, in which the magnetization *M*(*Bext*) is written as *m*(*α*)*B*0(*Lx*), we have found that (IV) even in the

for *m*(1) = 0.2, 0.5 and 1, respectively. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Shapiro steps are

 

Fig. 10. The *I*-*V* characteristics of BSCCO IJJs calculated for *N* = 5, *Jc* = 1000A/cm2, *fp* = 122GHz, *fr* = 200GHz, *Rshunt*/junction= 2Ω, *SRshunt*/junction= 50*μ*m2Ω, *ir* = 1, *Lx* = 5*μ*m, i.e., *B*0(*Lx*) = 0.345T. In this figure, the central case has been considered and the configuration **Γ** has been tentatively selected as (0, 0, 0.5, 0, 0). For the *χ*(*dF*) defined in Eq.(16), only the value for = 3 has been corrected, i.e., that of others has been held to 1. The value of *χ*3(*dF*) is 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 for (a), (b), (c), (d), (e) and (f), respectively. From the symmetry, only the *I*-*V* characteristics of 1@5 surface, 2@5 intermediate and 3@5 central junctions are shown. The horizontal axis indicates the dc voltage normalized to Φ<sup>0</sup> *fr*, and the vertical one shows the normalized dc *i*0. Note that (a) is equal to Fig.4(d).

case of *α* 0, the findings (II) and (III) mentioned above are basically held again, but for the finging (I) we have found that (V) clear Shapiro steps are found in not only the junction without F-layer but also the junction including F-layer. In addition to the above calculations, I have further calculated the *I*-*V* characteristics for the surface and intermediate cases when *α* = 1, whose the configurations are (1 + *m*(1), 1, 1, 1, 1) and (1, 1 + *m*(1), 1, 1, 1), respectively. We have concluded that even in the case in which a junction in BSCCO IJJs includes a F-layer with a finite magnetization, (I) the effect of the magnetic interaction between neighboring junctions is not so clearly found when *Bext* = 0, but (II) when *Bext* 0, that is found clearly. Conclusion (II) obtained here is the same as a previous statement such that in the BSCCO IJJs with a stacking structure, there are considerable magnetic interactions between neighboring Josephson junctions when *α* 0. [36]

In order to see the effect of the F-layer in the SFS-junction to the *I*-*V* characteristics of BSCCO IJJs with a stacking structure, I have calculated the *I*-*V* characteristics as a function of *χ*(*dF*) defined in Eq.(16). From the results calculated for the no external magnetic field and the

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89, 137007 (2002).

(1994).

Fig. 11. The same as in Fig.10 but for the case of **Γ** = (1, 1, 1.5, 1, 1). Note that (a) is equal to Fig.7(d).

nonzero field such as *α* = 0 and 1, we have found that the effect of *χ*3(*dF*) defined in the 3@5-central junction to the other junctions is negligible small irrespective of the value of the external magnetic field *Hext*, namely, the influence of the F-layer is restricted within a SFS-junction *itself*, i.e., that is negligible for others.

#### **6. Acknowledgments**

The author would like to thank Professors G. Oya and A. Irie, Utsunomiya University, for their valuable comments and discussion.

#### **7. References**


22 Will-be-set-by-IN-TECH




 **-**

Fig. 11. The same as in Fig.10 but for the case of **Γ** = (1, 1, 1.5, 1, 1). Note that (a) is equal to

nonzero field such as *α* = 0 and 1, we have found that the effect of *χ*3(*dF*) defined in the 3@5-central junction to the other junctions is negligible small irrespective of the value of the external magnetic field *Hext*, namely, the influence of the F-layer is restricted within a

The author would like to thank Professors G. Oya and A. Irie, Utsunomiya University, for

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[5] A. Yurgens, D. Winkler, N. V. Zavaritsky, and T. Claeson, Phys. Rev. B 53, R8887 (1996). [6] Yu. I. Latyshev, J. E. Nevelskaya and P. Monceau, Phys. Rev. Lett. 77, 932 (1996). [7] K. Schlenga, R. Kleiner, G. Hechtfischer, M. Mößle, S. Schmitt, P. Müller, Ch. Helm, Ch. Preis, F. Forsthofer, J. Keller, H. L. Johnson, M. Veith, and E. Steinbeiß, Phys. Rev. B 57,

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**-**

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**-**

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**6. Acknowledgments**

**7. References**

(1992).

14518 (1998).

**-** 

Fig.7(d).

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> -

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SFS-junction *itself*, i.e., that is negligible for others.

their valuable comments and discussion.

[1] S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).

[4] R. Kleiner and P. Müller, Phys. Rev. B 49, 1327 (1994).

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**9** 

*1France 2Brazil 3Spain* 

**Magnetical Response and Mechanical** 

**Superconductors, YBaCu3O7-X Materials** 

*Université de Poitiers – ENSMA, SP2MI - Téléport 2 - BP 30179,* 

*3DIOPMA Centre. Universitat de Barcelona, Facultat de Quimica,* 

*Departament de Ciència dels Materials i Enginyeria Metal·lúrgica, Barcelona* 

*1Institut P' (UPR 3346), Département de Physique et Mécanique des Matériaux, CNRS -* 

*2Federal University of Pelotas, Physics and Mathematics Institute, Department of Physics,* 

Since the discovery of the second generation High Superconductor Materials (2G-HTSC) in 1986, their magnetic properties have been widely studied by different research groups around the world. During the last years, the mechanical properties at micro-/ and nanometric scale are starting to be studied in order to know and improve the durability of

In this book chapter, we would like to focus our attention on the magnetical response and

In relation to the magnetic response, the Meissner effect is one important signature of the superconductivity. In this case, a diamagnetic response is observed due to exclusion of the magnetic flux of the interior of the superconducting material when the temperature is below the critical temperature. This important property allows to distinguish a superconducting material from a conducting one. But in several cases the superconducting materials exhibit a paramagnetic response instead of the conventional diamagnetism. This effect is frequently called *paramagnetic Meissner effect (PME).* In this case, the magnetic flux is not expelled, and a paramagnetic state can be originated. This effect is observed in several magnetic field regimes, and in some cases the paramagnetic response increases with the applied magnetic field, but in others the paramagnetic response decreases when the magnetic field increases. The paramagnetic Meissner effect (PME) was first observed in polycrystalline samples of high temperature superconductors. The discovery of this effect has originated an enormous effort of investigation by several groups around the world and several theories and models have been proposed to explain this interesting and controversial behavior, but without a

**1. Introduction** 

conventional devices.

also the mechanical properties of 2G-HTSC.

**Properties of High Temperature** 

J.J. Roa1,\*, F.T. Dias2 and M. Segarra3

*F-86962 Futuroscope, Chasseneuil Cedex* 


## **Magnetical Response and Mechanical Properties of High Temperature Superconductors, YBaCu3O7-X Materials**

J.J. Roa1,\*, F.T. Dias2 and M. Segarra3

*1Institut P' (UPR 3346), Département de Physique et Mécanique des Matériaux, CNRS - Université de Poitiers – ENSMA, SP2MI - Téléport 2 - BP 30179, F-86962 Futuroscope, Chasseneuil Cedex 2Federal University of Pelotas, Physics and Mathematics Institute, Department of Physics, 3DIOPMA Centre. Universitat de Barcelona, Facultat de Quimica, Departament de Ciència dels Materials i Enginyeria Metal·lúrgica, Barcelona 1France 2Brazil 3Spain* 

## **1. Introduction**

24 Will-be-set-by-IN-TECH

180 Superconductors – Properties, Technology, and Applications

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[43] M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D. Koelle, R. Kleiner, and E. Goldobin,

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177003 (2006).

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[45] R. L. Kautz and R. Monaco, J. Appl. Phys. 57, 875 (1985). [46] M. Kitamura, A. Irie, and G. Oya, Phys. Rev. B 66, 054519 (2002).

> Since the discovery of the second generation High Superconductor Materials (2G-HTSC) in 1986, their magnetic properties have been widely studied by different research groups around the world. During the last years, the mechanical properties at micro-/ and nanometric scale are starting to be studied in order to know and improve the durability of conventional devices.

> In this book chapter, we would like to focus our attention on the magnetical response and also the mechanical properties of 2G-HTSC.

> In relation to the magnetic response, the Meissner effect is one important signature of the superconductivity. In this case, a diamagnetic response is observed due to exclusion of the magnetic flux of the interior of the superconducting material when the temperature is below the critical temperature. This important property allows to distinguish a superconducting material from a conducting one. But in several cases the superconducting materials exhibit a paramagnetic response instead of the conventional diamagnetism. This effect is frequently called *paramagnetic Meissner effect (PME).* In this case, the magnetic flux is not expelled, and a paramagnetic state can be originated. This effect is observed in several magnetic field regimes, and in some cases the paramagnetic response increases with the applied magnetic field, but in others the paramagnetic response decreases when the magnetic field increases.

> The paramagnetic Meissner effect (PME) was first observed in polycrystalline samples of high temperature superconductors. The discovery of this effect has originated an enormous effort of investigation by several groups around the world and several theories and models have been proposed to explain this interesting and controversial behavior, but without a

Magnetical Response and Mechanical Properties of

**1.1.1 The Meissner effect** 

conducting material.

the superconductor.

**1.1 Magnetic response: The paramagnetic Meissner effect** 

Fig. 1. Magnetic behaviour of a type-II superconductor.

High Temperature Superconductors, YBaCu3O7-X Materials 183

up to now of a ceramic superconductor material (YBa2Cu3O7-x or Y-123) textured by Bridgman and Top-Seeding Melt Growth techniques. This material presents a phase transition from tetragonal to orthorhombic that promotes a change in its electrical properties, from insulating to superconductor, and that can be achieved by partially oxygenating the material. On the other hand, the structure of the texture material presents a heterogeneous distribution of two different phases: Y-123) as a matrix and Y-211 as quasi spherical inclusions. So, a review and a discussion about the different equations and method employed to isolate the mechanical response for each phase and for the whole material is presented. Finally, the different fracture mechanisms activated during the indentation process such as chipping, or radial cracks at the corners of the imprints will be presented.

This section is dedicated to the description of the magnetic response exhibited by some

The Meissner effect is one important signature of the superconductivity. In this case, a diamagnetic response is observed due to exclusion of the magnetic flux from inside the superconducting material when the temperature is below its critical temperature. This important property allows us to distinguish a superconducting material from a perfectly

Type-II superconductors present a perfect diamagnetism (Meissner effect) just when the magnetic applied field is lower than *HC1(T)*, known as the lower critical field, as showed by **Fig. 1**. Consequently this region is known as Meissner state. For magnetic applied fields higher than *HC1(T)* but lower than *HC2(T)*, or upper critical field, the magnetic field can penetrate inside the superconductor in the form of vortices. This state is known as mixed state, and the *HC1(T)* value determines the penetration of the first magnetic flux-line inside

superconducting materials, which is known as the paramagnetic Meissner effect.

conclusive understanding yet. One of the first interpretations is based on the occurrence of Josephson junctions between superconducting grains into the sample, and in this case the superconducting samples showing the PME can be modeled as a Josephson medium where the junctions are randomly distributed.

Some experimental results have motivated the proposal of mechanisms based on flux trapping and flux compression into the samples, and a non-equilibrium compressed flux states can be stabilized by inhomogeneously cooling the sample, thus yielding to a paramagnetic response. The formation of a giant vortex state on the sample, or the flux capture by samples with a particular shape, may produce a strong compression of the trapped flux in their interior. Several groups have used models based on these ideas to explain their results mainly when high magnetic fields are applied.

Other models based on arrays of multiply-connected superconductors have been proposed to explain the PME. Josephson junctions are taken into account and a paramagnetic response appears when inner currents may flow through these junctions under special conditions. Many groups have pointed that the PME observed in granular superconductors can be described as an intrinsic effect of multiply-connected superconductors, since the granularity emulates a network of Josephson junctions.

An important result observed in superconducting samples that exhibit the PME is the anomalous time-dependence on the magnetization. In some cases, the paramagnetic moments obtained after field cooling process relax monotonically to increasingly positive values. Some results show that this tendency remains unaltered for large time intervals. In other cases, the relaxation may show a change of sign in the magnetization from negative to positive. This behavior is opposite to expectations based on a flux-creep scenario, and some models have been proposed in order to explain this anomalous effect.

One of the sections of this chapter is dedicated to a review of the experimental results and main models and ideas about the paramagnetic Meissner effect in high-Tc materials. A special attention to results observed in YBa2Cu3O7-x superconductor prepared by melttextured techniques will be paid. These materials are very important to technological applications, and in some cases the observed PME is very strong. In relation to melttextured samples, the importance of the Y2BaCuO5 (Y-211) phase added during the growth is also discussed in order to connect with the PME causes.

Concerning the mechanical properties, attention will be focused on the following: hardness, Elastic modulus, toughness fracture and the stress-strain curve, taking into account the different works published since the discovery of this material. However, their determination at nanometric scale is highlighted, as this technique is very functional and fast, and can give us a lot of information about the mechanical behavior of each phase present in this kind of ceramic materials. The principle of the technique is the evaluation of the response of a material to an applied load, it allows to extract the mechanical properties of an heterogeneous material at very low applied loads. Depending on the tip employed, different equations should be applied to study the response of the material (Oliver and Pharr or Hertz equations for elastic-to-plastic and elastic regimen respectively), and thus calculate the stress-strain curves in order to obtain the yield strength of the material, and conventional parameters like hardness, Elastic modulus and toughness. In this chapter we would like to make a state of the art of the different mechanical properties obtained using hardness tests up to now of a ceramic superconductor material (YBa2Cu3O7-x or Y-123) textured by Bridgman and Top-Seeding Melt Growth techniques. This material presents a phase transition from tetragonal to orthorhombic that promotes a change in its electrical properties, from insulating to superconductor, and that can be achieved by partially oxygenating the material. On the other hand, the structure of the texture material presents a heterogeneous distribution of two different phases: Y-123) as a matrix and Y-211 as quasi spherical inclusions. So, a review and a discussion about the different equations and method employed to isolate the mechanical response for each phase and for the whole material is presented. Finally, the different fracture mechanisms activated during the indentation process such as chipping, or radial cracks at the corners of the imprints will be presented.

## **1.1 Magnetic response: The paramagnetic Meissner effect**

This section is dedicated to the description of the magnetic response exhibited by some superconducting materials, which is known as the paramagnetic Meissner effect.

## **1.1.1 The Meissner effect**

182 Superconductors – Properties, Technology, and Applications

conclusive understanding yet. One of the first interpretations is based on the occurrence of Josephson junctions between superconducting grains into the sample, and in this case the superconducting samples showing the PME can be modeled as a Josephson medium where

Some experimental results have motivated the proposal of mechanisms based on flux trapping and flux compression into the samples, and a non-equilibrium compressed flux states can be stabilized by inhomogeneously cooling the sample, thus yielding to a paramagnetic response. The formation of a giant vortex state on the sample, or the flux capture by samples with a particular shape, may produce a strong compression of the trapped flux in their interior. Several groups have used models based on these ideas to

Other models based on arrays of multiply-connected superconductors have been proposed to explain the PME. Josephson junctions are taken into account and a paramagnetic response appears when inner currents may flow through these junctions under special conditions. Many groups have pointed that the PME observed in granular superconductors can be described as an intrinsic effect of multiply-connected superconductors, since the granularity

An important result observed in superconducting samples that exhibit the PME is the anomalous time-dependence on the magnetization. In some cases, the paramagnetic moments obtained after field cooling process relax monotonically to increasingly positive values. Some results show that this tendency remains unaltered for large time intervals. In other cases, the relaxation may show a change of sign in the magnetization from negative to positive. This behavior is opposite to expectations based on a flux-creep scenario, and some

One of the sections of this chapter is dedicated to a review of the experimental results and main models and ideas about the paramagnetic Meissner effect in high-Tc materials. A special attention to results observed in YBa2Cu3O7-x superconductor prepared by melttextured techniques will be paid. These materials are very important to technological applications, and in some cases the observed PME is very strong. In relation to melttextured samples, the importance of the Y2BaCuO5 (Y-211) phase added during the growth is

Concerning the mechanical properties, attention will be focused on the following: hardness, Elastic modulus, toughness fracture and the stress-strain curve, taking into account the different works published since the discovery of this material. However, their determination at nanometric scale is highlighted, as this technique is very functional and fast, and can give us a lot of information about the mechanical behavior of each phase present in this kind of ceramic materials. The principle of the technique is the evaluation of the response of a material to an applied load, it allows to extract the mechanical properties of an heterogeneous material at very low applied loads. Depending on the tip employed, different equations should be applied to study the response of the material (Oliver and Pharr or Hertz equations for elastic-to-plastic and elastic regimen respectively), and thus calculate the stress-strain curves in order to obtain the yield strength of the material, and conventional parameters like hardness, Elastic modulus and toughness. In this chapter we would like to make a state of the art of the different mechanical properties obtained using hardness tests

explain their results mainly when high magnetic fields are applied.

models have been proposed in order to explain this anomalous effect.

also discussed in order to connect with the PME causes.

the junctions are randomly distributed.

emulates a network of Josephson junctions.

The Meissner effect is one important signature of the superconductivity. In this case, a diamagnetic response is observed due to exclusion of the magnetic flux from inside the superconducting material when the temperature is below its critical temperature. This important property allows us to distinguish a superconducting material from a perfectly conducting material.

Type-II superconductors present a perfect diamagnetism (Meissner effect) just when the magnetic applied field is lower than *HC1(T)*, known as the lower critical field, as showed by **Fig. 1**. Consequently this region is known as Meissner state. For magnetic applied fields higher than *HC1(T)* but lower than *HC2(T)*, or upper critical field, the magnetic field can penetrate inside the superconductor in the form of vortices. This state is known as mixed state, and the *HC1(T)* value determines the penetration of the first magnetic flux-line inside the superconductor.

Fig. 1. Magnetic behaviour of a type-II superconductor.

Magnetical Response and Mechanical Properties of

al., 1999) prepared by several techniques.

another YBa2Cu3O7-x single-crystal.

High Temperature Superconductors, YBaCu3O7-X Materials 185

The PME has been observed in several superconducting systems, since low-TC materials such as Nb (Thompson et al., 1996; Minhaj et al., 1994), high-Tc materials such as Bi2Sr2CaCu2O8 and YBa2Cu3O7-x (Braunisch et al., 1992; Dias et al., 2000), exotic systems such as Ba1-xKxBiO3 (Kim et al., 1996), multiphase In-Sn alloys (Schwartz et al., 2006), YBa2Cu3O7-x /La0.7Ca0.3MnO3 superlattices (Arias et al., 2006), organic compounds (Lebed, 2008) and many others. Recently, the PME has been also observed in Pb films by Brandt et al. (Brandt et al., 2011). The effect can be observed in polycrystalline sintered samples (Svendlindh et al., 1989), melt-textured samples (Pureur et al., 2001), single-crystals (Kusmartsev et al., 1997; Dias et al., 2010), alloys (Schwartz et al., 2006) and thin films (Pan et al., 2006; Terentiev et

The PME is qualitatively distinct in relation to the applied magnetic field, and can be observed from few Oe up to several kOe. In some cases the effect decreases when the magnetic field increases, as can be seen in **Fig. 3** for an YBa2Cu3O7-x single-crystal. However, in other cases the effect increases when the magnetic field increases, as showed by **Fig. 4** for

Fig. 3. Magnetic susceptibility in an YBa2Cu3O7-x single crystal. The magnetic susceptibility

decreases when the magnetic field increases. From Bräuchle et al., 1994.

The Meissner effect can be observed by the field-cooled warming (FCW) or field-cooled cooling (FCC) prescriptions. The FCW prescription consists of a first cooling down of the superconducting material to temperatures well below the critical temperature (TC) in magnetic applied field; then the magnetization can be measured under constant magnetic field while the sample warms up to temperatures above the critical temperature. The FCC prescription is similar, but in this case the magnetization is measured while cooling the sample back to temperatures well below the critical temperature in magnetic applied field. In these conditions the magnetic flux uniformly distributes into the sample, and when the sample is cooled below TC a diamagnetic moment appears as a consequence of the expulsion of magnetic flux from within it. Due to vortex pinning effects, in general the field-cooled (FC) response is lower than the zero-field cooling (ZFC - diamagnetic shielding) response, as showed by **Fig. 2**.

Fig. 2. ZFC and FC curves for a melt-textured YBa2Cu3O7-x sample.

#### **1.1.2 Paramagnetic Meissner effect**

The Meissner effect is characterized by a diamagnetic response of the superconducting material when a magnetic field is applied, but in several cases this magnetic response in FC experiments can be paramagnetic, challenging the conventional Meissner effect. This effect is known as paramagnetic Meissner effect (PME), but sometimes is called Wohlleben effect in honor of some of its discoverers (Bräuchle et al., 1994; Khomskii, 1994).

The first observations of the PME were reported in a pioneer work by P. Svedlindh et al. (Svedlindh et al., 1989) in the BiSrCaCuO superconductor. The discovery of the PME has prompted an enormous effort of investigation due to the striking contrast of this effect with the expected diamagnetic response. Initially, the effect was credited as an artifact of the measurement procedure (Blunt et al., 1991), but several further results showed that the PME is a genuine response of the superconducting material.

The Meissner effect can be observed by the field-cooled warming (FCW) or field-cooled cooling (FCC) prescriptions. The FCW prescription consists of a first cooling down of the superconducting material to temperatures well below the critical temperature (TC) in magnetic applied field; then the magnetization can be measured under constant magnetic field while the sample warms up to temperatures above the critical temperature. The FCC prescription is similar, but in this case the magnetization is measured while cooling the sample back to temperatures well below the critical temperature in magnetic applied field. In these conditions the magnetic flux uniformly distributes into the sample, and when the sample is cooled below TC a diamagnetic moment appears as a consequence of the expulsion of magnetic flux from within it. Due to vortex pinning effects, in general the field-cooled (FC) response is lower than the zero-field cooling (ZFC - diamagnetic shielding) response, as

Fig. 2. ZFC and FC curves for a melt-textured YBa2Cu3O7-x sample.

in honor of some of its discoverers (Bräuchle et al., 1994; Khomskii, 1994).

is a genuine response of the superconducting material.

The Meissner effect is characterized by a diamagnetic response of the superconducting material when a magnetic field is applied, but in several cases this magnetic response in FC experiments can be paramagnetic, challenging the conventional Meissner effect. This effect is known as paramagnetic Meissner effect (PME), but sometimes is called Wohlleben effect

The first observations of the PME were reported in a pioneer work by P. Svedlindh et al. (Svedlindh et al., 1989) in the BiSrCaCuO superconductor. The discovery of the PME has prompted an enormous effort of investigation due to the striking contrast of this effect with the expected diamagnetic response. Initially, the effect was credited as an artifact of the measurement procedure (Blunt et al., 1991), but several further results showed that the PME

**1.1.2 Paramagnetic Meissner effect** 

showed by **Fig. 2**.

The PME has been observed in several superconducting systems, since low-TC materials such as Nb (Thompson et al., 1996; Minhaj et al., 1994), high-Tc materials such as Bi2Sr2CaCu2O8 and YBa2Cu3O7-x (Braunisch et al., 1992; Dias et al., 2000), exotic systems such as Ba1-xKxBiO3 (Kim et al., 1996), multiphase In-Sn alloys (Schwartz et al., 2006), YBa2Cu3O7-x /La0.7Ca0.3MnO3 superlattices (Arias et al., 2006), organic compounds (Lebed, 2008) and many others. Recently, the PME has been also observed in Pb films by Brandt et al. (Brandt et al., 2011). The effect can be observed in polycrystalline sintered samples (Svendlindh et al., 1989), melt-textured samples (Pureur et al., 2001), single-crystals (Kusmartsev et al., 1997; Dias et al., 2010), alloys (Schwartz et al., 2006) and thin films (Pan et al., 2006; Terentiev et al., 1999) prepared by several techniques.

The PME is qualitatively distinct in relation to the applied magnetic field, and can be observed from few Oe up to several kOe. In some cases the effect decreases when the magnetic field increases, as can be seen in **Fig. 3** for an YBa2Cu3O7-x single-crystal. However, in other cases the effect increases when the magnetic field increases, as showed by **Fig. 4** for another YBa2Cu3O7-x single-crystal.

Fig. 3. Magnetic susceptibility in an YBa2Cu3O7-x single crystal. The magnetic susceptibility decreases when the magnetic field increases. From Bräuchle et al., 1994.

Magnetical Response and Mechanical Properties of

High Temperature Superconductors, YBaCu3O7-X Materials 187

Fig. 5. Time dependence of normalized FC magnetization in a melt-textured YBa2Cu3O7-x sample exhibiting the PME. The inset shows the sign reversal obtained in a relaxation

The paramagnetic Meissner effect is a controversial and challenging effect that can be observed in a variety of superconducting systems. However, in several cases there is a strong dependence on the applied magnetic field (magnitude and configuration), cooling rate during measurements, time effects, pinning effects, and others circumstances, and hence there is no a definitive model that can explain the paramagnetic moments observed

After the first observations of the PME, an enormous effort of investigation has emerged in order to explain this anomalous effect. The initial motivation was given by its interpretation based on the occurrence of junctions between the superconducting grains in the material

According to this, the sample showing PME may be modeled as a Josephson medium where the junctions are randomly distributed. Assuming that closed superconducting loops containing an odd number of these anomalous junctions are likely to occur in these networks, spontaneous and polarizable orbital currents may be generated, giving rise to the PME when very low magnetic fields are applied. The required -phase shifts in the Josephson links were proposed to arise from *d*-wave pairing symmetry in the high-*Tc* 

The **Fig. 6** shows PME results obtained in a polycrystalline Bi2Sr2CaCu2O8 sample by

Braunisch et al. (Braunisch et al., 1992) and explained on the basis of this model.

experiment in the same sample. From Pureur et al., 2001.

**1.1.3 Models and experimental results** 

superconducting materials (Sigrist & Rice, 1995).

during the FC experiments.

(Kusmartsev et al., 1992).

Fig. 4. FC magnetizations in an overdoped YBa2Cu3O7-x single crystal. In this case the magnetization increases when the magnetic field increases. From Kusmartsev et al., 1992.

Sometimes the PME presents a curious time dependence of the FC magnetization, as showed by **Fig. 5** for a melt-textured YBa2Cu3O7-x sample. In **Fig. 5** a paramagnetic signal was observed when the FC moment relaxes at constant magnetic field and temperature. The inset of this figure shows the sign reversal obtained in a relaxation experiment in the same sample, but cooled under another rate and magnetic field configuration.

It is possible to observe that an originally negative FC moment relaxes towards positive values and reverses sign after a wait time of about 3000 s. The time dependence of the PME has been related by other authors in different superconducting materials (Pan et al., 2006; Terentiev et al., 1999; Dias et al., 2004).

Fig. 4. FC magnetizations in an overdoped YBa2Cu3O7-x single crystal. In this case the magnetization increases when the magnetic field increases. From Kusmartsev et al., 1992.

sample, but cooled under another rate and magnetic field configuration.

Terentiev et al., 1999; Dias et al., 2004).

Sometimes the PME presents a curious time dependence of the FC magnetization, as showed by **Fig. 5** for a melt-textured YBa2Cu3O7-x sample. In **Fig. 5** a paramagnetic signal was observed when the FC moment relaxes at constant magnetic field and temperature. The inset of this figure shows the sign reversal obtained in a relaxation experiment in the same

It is possible to observe that an originally negative FC moment relaxes towards positive values and reverses sign after a wait time of about 3000 s. The time dependence of the PME has been related by other authors in different superconducting materials (Pan et al., 2006;

Fig. 5. Time dependence of normalized FC magnetization in a melt-textured YBa2Cu3O7-x sample exhibiting the PME. The inset shows the sign reversal obtained in a relaxation experiment in the same sample. From Pureur et al., 2001.

## **1.1.3 Models and experimental results**

The paramagnetic Meissner effect is a controversial and challenging effect that can be observed in a variety of superconducting systems. However, in several cases there is a strong dependence on the applied magnetic field (magnitude and configuration), cooling rate during measurements, time effects, pinning effects, and others circumstances, and hence there is no a definitive model that can explain the paramagnetic moments observed during the FC experiments.

After the first observations of the PME, an enormous effort of investigation has emerged in order to explain this anomalous effect. The initial motivation was given by its interpretation based on the occurrence of junctions between the superconducting grains in the material (Kusmartsev et al., 1992).

According to this, the sample showing PME may be modeled as a Josephson medium where the junctions are randomly distributed. Assuming that closed superconducting loops containing an odd number of these anomalous junctions are likely to occur in these networks, spontaneous and polarizable orbital currents may be generated, giving rise to the PME when very low magnetic fields are applied. The required -phase shifts in the Josephson links were proposed to arise from *d*-wave pairing symmetry in the high-*Tc*  superconducting materials (Sigrist & Rice, 1995).

The **Fig. 6** shows PME results obtained in a polycrystalline Bi2Sr2CaCu2O8 sample by Braunisch et al. (Braunisch et al., 1992) and explained on the basis of this model.

Magnetical Response and Mechanical Properties of

et al., 1996.

High Temperature Superconductors, YBaCu3O7-X Materials 189

Fig. 7. FC (a) and ZFC (b) experiments in a Nb disk. The circles were taken before polishing the sample surface, and the triangles were taken after polishing the sample surface. In this case, the polishing of the surface affects the PME, leading to its disappearance. From Kostic´

Similarly, the formation of a giant vortex state on the sample surface proposed by Moshchalkov et al. (Moshchalkov et al., 1997) may produce a compression of the trapped flux. According to the authors, the PME observed in high-TC and Nb superconducting samples can be explained by the persistence of a giant vortex state with a fixed orbital quantum number *L,*  formed during the FC regime at the third critical field. However, the amplitude of the PME is suppressed by applying a magnetic field. The giant vortex state was also shown to be responsible for PME in small disks of Al samples (Geim et al., 1998), where transitions between metastable giant vortex states having different angular momenta (*L*) have been observed. The transitions compress the giant vortex into smaller volumes, allowing extra flux to penetrate in

the sample surface and producing the paramagnetic moments.

Fig. 6. ZFC and FC signals in a polycrystalline Bi2Sr2CaCu2O8 sample exhibiting the PME. From Braunisch et al., 1992.

Several results can be explained based on this model, specially when low magnetic fields are applied. The concept of polarized orbital currents to explain the low field PME was further developed by studies of the characteristic microstructure of samples showing this effect (Freitag et al., 1999), and by investigations on the time dependence of the paramagnetic moment (Magnusson et al., 1995), ac susceptibility (Nordblad et al., 1998) and magnetic aging (Svedlindh et al., 1999).

A controversial point to this model arises from similar results obtained in Nb samples, because some authors (Sigrist & Rice, 1995) attribute that junctions are a consequence of the *d*-wave pairing symmetry, which is nonexistent in this system. The PME observed in Nb (Thompson et al., 1996; Minhaj et al., 1994; Kostic et al., 1996) and Nb thin films (Ortiz et al., 2001; Terentiev et al., 1999) was demonstrated to be strongly influenced by the sample geometry and surface. In some cases, the polishing of the surfaces affects the PME significantly, leading to its disappearance. **Fig. 7** shows the magnetization results of a Nb disk before and after polishing the sample surface, where it is possible to see that the PME disappears after the sample is polished. A surface dependent PME in YBa2Cu3O7-x single crystals was also found (Lucht et al., 1995).

These results motivated the proposal of mechanisms to explain the PME without taking into account effects from intrinsically nonconventional superconductivity, and models based on flux trapping and flux compression effects were proposed. According to these models, a nonequilibrium compressed flux state may be generated and stabilized by inhomogeneous cooling of the sample, so that its surface becomes superconducting prior to the bulk (Koshelev & Larkin, 1995). Then, the magnetic flux may be pushed into the sample, creating a flux compressed state upon further cooling.

Fig. 6. ZFC and FC signals in a polycrystalline Bi2Sr2CaCu2O8 sample exhibiting the PME.

Several results can be explained based on this model, specially when low magnetic fields are applied. The concept of polarized orbital currents to explain the low field PME was further developed by studies of the characteristic microstructure of samples showing this effect (Freitag et al., 1999), and by investigations on the time dependence of the paramagnetic moment (Magnusson et al., 1995), ac susceptibility (Nordblad et al., 1998) and magnetic

A controversial point to this model arises from similar results obtained in Nb samples, because some authors (Sigrist & Rice, 1995) attribute that junctions are a consequence of the *d*-wave pairing symmetry, which is nonexistent in this system. The PME observed in Nb (Thompson et al., 1996; Minhaj et al., 1994; Kostic et al., 1996) and Nb thin films (Ortiz et al., 2001; Terentiev et al., 1999) was demonstrated to be strongly influenced by the sample geometry and surface. In some cases, the polishing of the surfaces affects the PME significantly, leading to its disappearance. **Fig. 7** shows the magnetization results of a Nb disk before and after polishing the sample surface, where it is possible to see that the PME disappears after the sample is polished. A surface dependent PME in YBa2Cu3O7-x single

These results motivated the proposal of mechanisms to explain the PME without taking into account effects from intrinsically nonconventional superconductivity, and models based on flux trapping and flux compression effects were proposed. According to these models, a nonequilibrium compressed flux state may be generated and stabilized by inhomogeneous cooling of the sample, so that its surface becomes superconducting prior to the bulk (Koshelev & Larkin, 1995). Then, the magnetic flux may be pushed into the sample, creating

From Braunisch et al., 1992.

aging (Svedlindh et al., 1999).

crystals was also found (Lucht et al., 1995).

a flux compressed state upon further cooling.

Fig. 7. FC (a) and ZFC (b) experiments in a Nb disk. The circles were taken before polishing the sample surface, and the triangles were taken after polishing the sample surface. In this case, the polishing of the surface affects the PME, leading to its disappearance. From Kostic´ et al., 1996.

Similarly, the formation of a giant vortex state on the sample surface proposed by Moshchalkov et al. (Moshchalkov et al., 1997) may produce a compression of the trapped flux. According to the authors, the PME observed in high-TC and Nb superconducting samples can be explained by the persistence of a giant vortex state with a fixed orbital quantum number *L,*  formed during the FC regime at the third critical field. However, the amplitude of the PME is suppressed by applying a magnetic field. The giant vortex state was also shown to be responsible for PME in small disks of Al samples (Geim et al., 1998), where transitions between metastable giant vortex states having different angular momenta (*L*) have been observed. The transitions compress the giant vortex into smaller volumes, allowing extra flux to penetrate in the sample surface and producing the paramagnetic moments.

Magnetical Response and Mechanical Properties of

Kusmartsev et al.

High Temperature Superconductors, YBaCu3O7-X Materials 191

from the low-field PME. The HFPME was found to depend on the cooling rate and the sample size, which suggests that the paramagnetic moment in this case might be induced by a flux compression mechanism. The **Fig. 4** shows representative results obtained by

Fig. 9. FCC and FCW magnetizations in a melt-textured YBa2Cu3O7-x sample exhibiting a

The PME at high magnetic fields was also observed by Dias et al. (Dias et al., 2004) in different melt-textured YBa2Cu3O7-x samples containing large amounts of the Y2BaCuO5 (Y-211) phase. Magnetic fields up to 50 kOe were applied either parallel or perpendicular to the *ab* planes and the magnitude of the high-field paramagnetic moment increases when the

strong PME. From Dias et al., 2004.

The flux capture by samples with a special shape proposed by Obukhov (Obukhov et al., 1998), may also produce compression of the trapped flux. According to the author, an unusual behavior of the long-time relaxation of the magnetization after a FC procedure may show not only an increase in the magnetization over the time but a change of sign from negative to positive. This interesting result was observed in melt-textured YBa2Cu3O7-x samples (Pureur et al., 2001), as shown by the inset in the **Fig. 5**.

A model to explain the PME observed in Josephson-junctions arrays, such as Nb-Al2O3-Nb (Nielsen et al., 2000), was proposed to account for the paramagnetism without invoking the role of junctions or *d*-wave superconductivity (Araujo-Moreira, 1997; Barbara, 1999; Nielsen et al., 2000). According to this model, the screening of the array of these multiplyconnected superconductors is provided by the loops at its boundaries, and the diamagnetic currents running at the external perimeter induce currents in the opposite sense inside the array. These inner currents generate a paramagnetic scenario under certain conditions. This mechanism is shown in **Fig. 8**.

Fig. 8. Mechanism for generation of a paramagnetic current inside a multiply-connected superconducting sample. From Nielsen et al., 2000.

According to this model, only the exterior plaquettes create a diamagnetic screening current on the outside of the sample, and as a consequence a paramagnetic current of the same magnitude is created inside the sample. Simulations taking into account the mutualinductance interactions between loops in the array confirm that PME might be the dominant response, depending on the field strength and parameters as the self-inductance and capacitance of the Josephson junctions (Rotoli et al., 2001).

The granularity in superconductors emulates a disordered network of Josephson junctions, and the paramagnetic moments observed in granular thin films of Nb and YBa2Cu3O7-x has also been described as an intrinsic effect of multiply-connected superconductors.

Important results were obtained by Kusmartsev et al. (Kusmartsev et al., 1997) after cooling large YBa2Cu3O7-x single crystals in strong applied fields. The authors called this phenomenon the high-field paramagnetic effect (HFPME) so that it may be distinguished

The flux capture by samples with a special shape proposed by Obukhov (Obukhov et al., 1998), may also produce compression of the trapped flux. According to the author, an unusual behavior of the long-time relaxation of the magnetization after a FC procedure may show not only an increase in the magnetization over the time but a change of sign from negative to positive. This interesting result was observed in melt-textured YBa2Cu3O7-x

A model to explain the PME observed in Josephson-junctions arrays, such as Nb-Al2O3-Nb (Nielsen et al., 2000), was proposed to account for the paramagnetism without invoking the role of junctions or *d*-wave superconductivity (Araujo-Moreira, 1997; Barbara, 1999; Nielsen et al., 2000). According to this model, the screening of the array of these multiplyconnected superconductors is provided by the loops at its boundaries, and the diamagnetic currents running at the external perimeter induce currents in the opposite sense inside the array. These inner currents generate a paramagnetic scenario under certain conditions. This

Fig. 8. Mechanism for generation of a paramagnetic current inside a multiply-connected

According to this model, only the exterior plaquettes create a diamagnetic screening current on the outside of the sample, and as a consequence a paramagnetic current of the same magnitude is created inside the sample. Simulations taking into account the mutualinductance interactions between loops in the array confirm that PME might be the dominant response, depending on the field strength and parameters as the self-inductance and

The granularity in superconductors emulates a disordered network of Josephson junctions, and the paramagnetic moments observed in granular thin films of Nb and YBa2Cu3O7-x has

Important results were obtained by Kusmartsev et al. (Kusmartsev et al., 1997) after cooling large YBa2Cu3O7-x single crystals in strong applied fields. The authors called this phenomenon the high-field paramagnetic effect (HFPME) so that it may be distinguished

also been described as an intrinsic effect of multiply-connected superconductors.

superconducting sample. From Nielsen et al., 2000.

capacitance of the Josephson junctions (Rotoli et al., 2001).

samples (Pureur et al., 2001), as shown by the inset in the **Fig. 5**.

mechanism is shown in **Fig. 8**.

from the low-field PME. The HFPME was found to depend on the cooling rate and the sample size, which suggests that the paramagnetic moment in this case might be induced by a flux compression mechanism. The **Fig. 4** shows representative results obtained by Kusmartsev et al.

Fig. 9. FCC and FCW magnetizations in a melt-textured YBa2Cu3O7-x sample exhibiting a strong PME. From Dias et al., 2004.

The PME at high magnetic fields was also observed by Dias et al. (Dias et al., 2004) in different melt-textured YBa2Cu3O7-x samples containing large amounts of the Y2BaCuO5 (Y-211) phase. Magnetic fields up to 50 kOe were applied either parallel or perpendicular to the *ab* planes and the magnitude of the high-field paramagnetic moment increases when the

Magnetical Response and Mechanical Properties of

the indirect measurement of the contact area.

testing.

applied load employed. **1.2.1.1 Microindentation** 

more detail in the present section.

**1.2.1 Theoretical fundamentals** 

**1.2 Mechanical properties of YBCO-bulk materials** 

High Temperature Superconductors, YBaCu3O7-X Materials 193

Indentation testing is a simple method that consists essentially of touching the material of interest, whose mechanical properties such as elastic modulus and hardness are unknown, with another material whose properties are known (Fischer-Cripps et al., 2004). The technique has its origins in Mohs' hardness scale of 1822, in which materials that are able to leave a permanent scratch in another were ranked as harder materials, with diamond assigned the maximum value of 10 on the scale. The establishment of the Brinell, Knoop, Vickers and Rockwell tests all follow from a refinement of the method of indentation test, in which the length scale of the penetration is measured in nanometers rather than microns or millimeters, the latter being common in conventional hardness tests. Apart from the displacement scale involved, the distinguishing feature of most nanoindentation testing is

In conventional hardness tests, the area of contact is calculated from direct measurements of the dimensions of the residual impression in the specimen surface upon the removal of load. In nanoindentation tests, the size of the residual impression is of the order of microns and too small to be conveniently directly measured. Thus, it is usual to determine the area of contact by measuring the depth of indentation of the indenter into the specimen of interest. This, together with the known geometry of the indenter, provides an indirect measurement of the contact area at full load. For this reason, nanoindentation can be considered a special case of depth-sensing indentation or instrumented indentation

In this section, we would like to do a brief explanation of the different hardness tests employed to characterize the YBa2Cu3O7-x samples textured by Bridgman and TSMG technique. Concretely, we will explain in detail the micro-, nano- and even picoindentation, according to the characteristic indentation sizes that fall into a particular size range and the

The purpose of microindentation hardness testing is to study fine scale changes in hardness. This technique is also commonly called microhardness testing, although this term is misleading because it implies that the hardness is extremely low. The Vickers and the Knoop tests, which are the two most common microindentation tests, will be described in

The term microhardness test usually refers to static indentations made with loads not exceeding 1kgf. The indenter is either the Vickers diamond pyramid or the Knoop elongated diamond pyramid. The surface being tested generally requires a metallographic finishing;

The Knoop indenter is a diamond ground to pyramidal form that produces a diamond shaped indentation, having an approximate ratio between long and short diagonals of 7:1. The depth of indentation is about 1/30 of its length. The Knoop hardness number (*KHN*) is

the ratio of the load applied to the indenter to the unrecovered projected area as:

the smaller the load used, the higher the surface finish required.

field is augmented. The **Fig. 9** shows FCC and FCW magnetizations as a function of the temperature in a melt-textured YBa2Cu3O7-x sample measured in the indicated fields applied parallel and perpendicular to the *ab* planes. In the **Fig. 9a** the data are obtained when the field is applied parallel to the *ab* plane. These results present the general trends of the PME shown by the melt-textured YBaCuO samples. In the **Fig. 9b** are depicted the peculiar results obtained when the field is applied along the *c*-axis. In this case the positive contribution to the moment rises abruptly at an almost field-independent temperature around 55 K, then shows a tendency to saturate at low temperatures at values varying roughly proportional to the applied field. When the magnetic field increases the magnetization also increases and for magnetic fields below 5 kOe the magnetization is negative, as shown in the **Fig 9b**.

The effect shows a strong and anomalous relaxation, such that the paramagnetic moment increases as a function of the time, as shown in **Fig. 10**.

According to the authors, the pinning by the Y-211 particles is relevant and the formation of a flux-compressed scenario modulated by pinning and a strong flux-creep is a possible origin of the observed PME. Recently, the high-field PME was also observed in an Y0.98Ca0.02Ba2Cu3O7-x single crystal (Dias et al., 2010), and the results were attributed to strong flux compression state into the sample modulated by pinning. An important review of the paramagnetic Meissner effect and related dynamical phenomena was presented by M.S. Li (Li, 2003), with emphasis on the theoretical description of the junctions.

Fig. 10. Strong time dependence of the FCC magnetization in a melt-textured YBa2Cu3O7-x sample exhibiting the PME.

## **1.2 Mechanical properties of YBCO-bulk materials**

## **1.2.1 Theoretical fundamentals**

192 Superconductors – Properties, Technology, and Applications

field is augmented. The **Fig. 9** shows FCC and FCW magnetizations as a function of the temperature in a melt-textured YBa2Cu3O7-x sample measured in the indicated fields applied parallel and perpendicular to the *ab* planes. In the **Fig. 9a** the data are obtained when the field is applied parallel to the *ab* plane. These results present the general trends of the PME shown by the melt-textured YBaCuO samples. In the **Fig. 9b** are depicted the peculiar results obtained when the field is applied along the *c*-axis. In this case the positive contribution to the moment rises abruptly at an almost field-independent temperature around 55 K, then shows a tendency to saturate at low temperatures at values varying roughly proportional to the applied field. When the magnetic field increases the magnetization also increases and for magnetic fields below 5 kOe the magnetization is

The effect shows a strong and anomalous relaxation, such that the paramagnetic moment

According to the authors, the pinning by the Y-211 particles is relevant and the formation of a flux-compressed scenario modulated by pinning and a strong flux-creep is a possible origin of the observed PME. Recently, the high-field PME was also observed in an Y0.98Ca0.02Ba2Cu3O7-x single crystal (Dias et al., 2010), and the results were attributed to strong flux compression state into the sample modulated by pinning. An important review of the paramagnetic Meissner effect and related dynamical phenomena was presented by

M.S. Li (Li, 2003), with emphasis on the theoretical description of the junctions.

Fig. 10. Strong time dependence of the FCC magnetization in a melt-textured YBa2Cu3O7-x

negative, as shown in the **Fig 9b**.

sample exhibiting the PME.

increases as a function of the time, as shown in **Fig. 10**.

Indentation testing is a simple method that consists essentially of touching the material of interest, whose mechanical properties such as elastic modulus and hardness are unknown, with another material whose properties are known (Fischer-Cripps et al., 2004). The technique has its origins in Mohs' hardness scale of 1822, in which materials that are able to leave a permanent scratch in another were ranked as harder materials, with diamond assigned the maximum value of 10 on the scale. The establishment of the Brinell, Knoop, Vickers and Rockwell tests all follow from a refinement of the method of indentation test, in which the length scale of the penetration is measured in nanometers rather than microns or millimeters, the latter being common in conventional hardness tests. Apart from the displacement scale involved, the distinguishing feature of most nanoindentation testing is the indirect measurement of the contact area.

In conventional hardness tests, the area of contact is calculated from direct measurements of the dimensions of the residual impression in the specimen surface upon the removal of load. In nanoindentation tests, the size of the residual impression is of the order of microns and too small to be conveniently directly measured. Thus, it is usual to determine the area of contact by measuring the depth of indentation of the indenter into the specimen of interest. This, together with the known geometry of the indenter, provides an indirect measurement of the contact area at full load. For this reason, nanoindentation can be considered a special case of depth-sensing indentation or instrumented indentation testing.

In this section, we would like to do a brief explanation of the different hardness tests employed to characterize the YBa2Cu3O7-x samples textured by Bridgman and TSMG technique. Concretely, we will explain in detail the micro-, nano- and even picoindentation, according to the characteristic indentation sizes that fall into a particular size range and the applied load employed.

## **1.2.1.1 Microindentation**

The purpose of microindentation hardness testing is to study fine scale changes in hardness. This technique is also commonly called microhardness testing, although this term is misleading because it implies that the hardness is extremely low. The Vickers and the Knoop tests, which are the two most common microindentation tests, will be described in more detail in the present section.

The term microhardness test usually refers to static indentations made with loads not exceeding 1kgf. The indenter is either the Vickers diamond pyramid or the Knoop elongated diamond pyramid. The surface being tested generally requires a metallographic finishing; the smaller the load used, the higher the surface finish required.

The Knoop indenter is a diamond ground to pyramidal form that produces a diamond shaped indentation, having an approximate ratio between long and short diagonals of 7:1. The depth of indentation is about 1/30 of its length. The Knoop hardness number (*KHN*) is the ratio of the load applied to the indenter to the unrecovered projected area as:

Magnetical Response and Mechanical Properties of

yield strength (

examine the

et al., 1997).

during the indentation tests:

characterizing the mechanical properties of a given component.

*ys*), or shear stress (

*ys* and obtain the

High Temperature Superconductors, YBaCu3O7-X Materials 195

Finally, this technique eliminates the need to visualize the imprints produced during the test for homogeneous materials, which makes the extraction of mechanical properties easier. Due to all these advantages, nanoindentation could be a good and versatile technique for

The nanoindenter is an equipment which measures load and maximum indentation depth as a function of time during loading and unloading, and displays load-displacement data. Depending on the details of the specific testing systems, loads as small as 1 nN can be

mechanical properties such as hardness (*H*), elastic modulus (*E*), fracture toughness (*KIC*),

corresponding tip indenter. The different indenters can be classified in four different groups: i. *Pyramidal:* the most used sharp indenter is the Berkovich. It is a three-sided pyramid with the same depth-to-area relation as the four sided Vickers pyramid commonly used in microhardness. This indenter allows obtaining the *H* and *E*. The *KIC* of brittle

ii. *Spherical:* for this kind of indenters, the contact stress is initially small and produces only elastic deformation. As the spherical indenter is driven into the surface, a transition from elastic-to-plastic deformation occurs, which can theoretically be used to

iii. *Cube-corner:* a three-sided pyramid with mutually perpendicular faces arranged in geometry like the corner of a cube. The center-line-to-face angle for this indenter is 34.3°. The sharper cube-corner indenter produces much higher stress and strain in the vicinity of the contact, which is useful, for example, in producing very small and well

iv. *Conical indenters:* this type of indenter is interesting due to the existence of stress concentrations at the sharp edges of the indenter. It is difficult to manufacture conical diamonds with sharp tips, making them of little use in nanoindentation technique (Tsui

Moreover, in order to obtain reproducible results, several factors should be controlled

 *Choosing an appropriate indenter:* this is one of the most important steps and requires the consideration of a number of factors. One of them is the strain the tip imposes on the test materials. Although the indentation process produces a complex strain field beneath the indenter, it has been proven to be useful to quantify this field with a single quantity, often termed as the characteristic strain. There are problems, however, in obtaining accurate measurements of *H* and *E* with cube-corner indenters (Hay et al., 1999). Although not entirely understood, the problems appear to have two separate origins: first, as the angle of the indenter decreases, friction in the specimen-indenter interphase and its influence on the contact mechanics becomes increasingly important; on the other hand, to obtain the relation between the contact stiffness (*S*), contact area (*A(hc)*) and effective elastic modulus (*Eeff*), corrections are required, and the magnitude

 *Environmental control:* to take full advantage of the fine displacement resolution available in most ITT systems, several precautions must be taken when choosing and

curve (Field et al., 1993).

), can be obtained from the P-h data and with the

) and

applied, and displacements of 0.1 nm can be measured. Stress-strain curves (

materials can also be determined with this indenter without high accuracy.

of the correction factor depends on the angle of the indenter.

defined cracks at the corners of the residual imprints in brittle materials.

$$\text{KHN} = \frac{F}{A} = \frac{P}{\text{CL}^2} \tag{1}$$

Where *F* or *P* is the applied load in *kgf*, *A* is the unrecovered projected area of the indentation in *mm2*, *L* is the measured length of the long diagonal of indentation in *mm*, and *C* is a constant of the indenter relating the projected area of the indentation to the square of the length of the long diagonal, and equal to 0.07028.

The Vickers diamond pyramidal indenter is ground in the form of a square pyramid with an angle of 136° between faces. The depth of indentation is about 1/7 of the diagonal length. The Vickers diamond pyramid hardness number (*HV*) can be obtained by using the following equation:

$$HV = \frac{2\text{Fsin}\frac{136^\circ}{2}}{d^2} \quad or \quad HV \approx 1.854\frac{F}{d^2} \tag{2}$$

Where *F* is the load in *kgf*, *d* is the arithmetic mean of the two diagonals, *d1* and *d2* in *mm* and *HV* is the Vickers hardness.

Comparing the indentations made with Knoop and Vickers diamond pyramid indenters for a given load and test material we can conclude:


#### **1.2.1.2 Nanoindentation**

During the last years, Instrumented Indentation Testing has been widely employed. Also known as depth-sensing indentation, continuous-recording indentation, ultra-low load indentation or nanoindentation, ITT is a relatively new form of mechanical testing, which significantly expands the capabilities of traditional hardness tests at macro- and microscale. It employs high-resolution instrumentation to continuously control and monitor the loads and displacements of an indenter (Golovin et al., 2008; Pharr et al., 1992). This technique presents several advantages compared to the microindentation technique:


Moreover, this technique is especially well suited for the characterization of small volumes of material, such as single grains or phases in a composite (Roa et al., 2007), dislocation dynamics (Gaillard et al., 2006), small structures (Choi et al., 2003) or thin films and coatings (Gaillard et al., 2008; Beegan et al., 2005; Roa et al., 2009a, 2011a; Rayon et al., 2011).

*F P KHN = =*

Where *F* or *P* is the applied load in *kgf*, *A* is the unrecovered projected area of the indentation in *mm2*, *L* is the measured length of the long diagonal of indentation in *mm*, and *C* is a constant of the indenter relating the projected area of the indentation to the square of

The Vickers diamond pyramidal indenter is ground in the form of a square pyramid with an angle of 136° between faces. The depth of indentation is about 1/7 of the diagonal length. The Vickers diamond pyramid hardness number (*HV*) can be obtained by using the

Where *F* is the load in *kgf*, *d* is the arithmetic mean of the two diagonals, *d1* and *d2* in *mm* and

Comparing the indentations made with Knoop and Vickers diamond pyramid indenters for

During the last years, Instrumented Indentation Testing has been widely employed. Also known as depth-sensing indentation, continuous-recording indentation, ultra-low load indentation or nanoindentation, ITT is a relatively new form of mechanical testing, which significantly expands the capabilities of traditional hardness tests at macro- and microscale. It employs high-resolution instrumentation to continuously control and monitor the loads and displacements of an indenter (Golovin et al., 2008; Pharr et al., 1992). This technique

<sup>136</sup> 2Fsin




presents several advantages compared to the microindentation technique:



(Gaillard et al., 2008; Beegan et al., 2005; Roa et al., 2009a, 2011a; Rayon et al., 2011).


Moreover, this technique is especially well suited for the characterization of small volumes of material, such as single grains or phases in a composite (Roa et al., 2007), dislocation dynamics (Gaillard et al., 2006), small structures (Choi et al., 2003) or thin films and coatings


*° <sup>F</sup> HV = or HV*

2 2

2 1.854

the length of the long diagonal, and equal to 0.07028.

a given load and test material we can conclude:


following equation:

*HV* is the Vickers hardness.

**1.2.1.2 Nanoindentation** 

*2*

*<sup>A</sup> CL* (1)

*d d* (2)

Finally, this technique eliminates the need to visualize the imprints produced during the test for homogeneous materials, which makes the extraction of mechanical properties easier. Due to all these advantages, nanoindentation could be a good and versatile technique for characterizing the mechanical properties of a given component.

The nanoindenter is an equipment which measures load and maximum indentation depth as a function of time during loading and unloading, and displays load-displacement data. Depending on the details of the specific testing systems, loads as small as 1 nN can be applied, and displacements of 0.1 nm can be measured. Stress-strain curves () and mechanical properties such as hardness (*H*), elastic modulus (*E*), fracture toughness (*KIC*), yield strength (*ys*), or shear stress (), can be obtained from the P-h data and with the corresponding tip indenter. The different indenters can be classified in four different groups:


Moreover, in order to obtain reproducible results, several factors should be controlled during the indentation tests:


Magnetical Response and Mechanical Properties of

where the contact depth (*hc*) is determined as:

being *hmax* the maximum indentation depth,

applied load, *Pmax*:

2011b).

where

where *E* and

properties (*Ei =* 1141GPa and

*b. Fracture toughness* 

High Temperature Superconductors, YBaCu3O7-X Materials 197

The *H* is calculated by dividing the load by the projected contact area (*A(hc)*) at maximum

max *<sup>c</sup>* max *<sup>P</sup> h =h <sup>ε</sup>*

(*dP/dh*, see **Fig. 11**). In the case of a perfect Berkovich tip, the *A(hc)* is determined as:

The effective elastic modulus (*Eeff*) has been expressed in terms of *S* and *A(hc)* as:

(Oliver et al., 1992), the Elastic modulus can be obtained as:

1 2 *eff*

*<sup>π</sup> <sup>S</sup> E =*

symmetry of the Berkovich indenter. Then, according to the Oliver and Pharr approach

<sup>2</sup> <sup>2</sup> 1 1 1 *<sup>i</sup> eff i*

2007, 2009d, 2010a, 2010c, 2011b, 2011d), respectively. The subindex *i* denotes the indenter

Moreover, nanoindentation provides information about contact mechanisms taking place in the material. In brittle materials indented with a Berkovich tip, surface examination of residual imprints has revealed the appearance of several typical features: surface deformation effects such as *sink-in* (Johnson et al., 1985; Alcala et al., 2000), microcracks or damage inside the

*= + E EE*

*c*

are the Elastic modulus and the Poisson's ratio (νYBa2Cu3O7-x *=* 0.3 (Roa et al,

) (Oliver et al., 1992; Pharr et al., 1992).

is a geometrical factor (1.034). This parameter takes into account the lack of

*ν ν*

Berkovich tip indenter, and *S* the unloading contact stiffness at maximum indentation depth

However, in practice the projected contact area differs from this value when the tip is worn out, because the different angles of the faces may be somehow different from the nominal. Hence, the projected contact area must be regularly calibrated. When catastrophic fracture mechanisms occur, the *A(hc)* obtained by this theoretical method is overestimated, and results in an *H* underestimation. To avoid this problem, the real contact area should be determined using a technique of microscopy (as Scanning Electron Microscopy or Atomic Force Microscopy) and the contact area must be recalculated in order to avoid the sink-in or pile-up effects for brittle and/or ductile materials, respectively (Roa et al., 2009d, 2010c,

*<sup>P</sup> H =*

max *c*

*A(h )* (3)

*<sup>S</sup>* (4)

a parameter approximately equal to 0.75 for a

<sup>2</sup> 24.56 *A c c (h ) = h* (5)

*<sup>β</sup> A(h )* (6)

(7)

preparing the testing environment. Uncertainties and mistakes in measured displacement arise from two separate environmental sources: vibration and temperature. To reduce vibration, the nanoindenter should be located on quite, solid foundation and mounted on vibration-isolation system. Thermal stability can be provided by closing the apparatus in a thermally buffered cabin and controlling the room temperature to within ± 0.5°C.


In this section we will explain the method developed by Oliver and Pharr (Pharr et al., 1992; Oliver et al., 1992) to determine the hardness (*H*) and the elastic modulus (*E*) from the unloading part of recorded indentation load/unload (or *P-h* curves, such as **Fig. 11**).

Fig. 11. Schematic representation of indentation P-h curve showing important measured parameters (From Roa et al., 2007).

 *Surface preparation:* one of the most important factors is the surface roughness of the samples, which is extremely important during the nanoindentation test because the mechanical properties are obtained from contact areas, which are calculated from the contact depth and area function. The main problems are found when the characteristic

 *Testing procedure:* to avoid interference from previous assays, and ensure the independence of the measurement, successive indentations should be separated up to 20-30 times the maximum indentation depth (*hmax*) when Berkovich or Vickers indenters are employed. However, for other geometries, the rule is to perform indentations

 *Detecting the surface:* the most important step of any good nanoindentation test procedure is to accurate identificate the location of the specimen's surface. For hard and stiff materials, such as hardened metals and ceramics, the load and/or contact stiffness, both of which increase upon contact, are often employed. However, for soft compliant materials (like polymers and biological tissues), the rate of increase in load and contact stiffness is often too small to allow for accurate surface identification. In these situations, a better method is sometimes offered by dynamic stiffness measurement

In this section we will explain the method developed by Oliver and Pharr (Pharr et al., 1992; Oliver et al., 1992) to determine the hardness (*H*) and the elastic modulus (*E*) from the

*S* 

unloading part of recorded indentation load/unload (or *P-h* curves, such as **Fig. 11**).

Fig. 11. Schematic representation of indentation P-h curve showing important measured

wavelength of the roughness is comparable to the contact diameter.

separated from 7 to 10 times the maximum contact radius.

room temperature to within ± 0.5°C.

(Pethica et al., 1989; Lucas et al., 1998). *a. Hardness and Elastic modulus determination* 

parameters (From Roa et al., 2007).

preparing the testing environment. Uncertainties and mistakes in measured displacement arise from two separate environmental sources: vibration and temperature. To reduce vibration, the nanoindenter should be located on quite, solid foundation and mounted on vibration-isolation system. Thermal stability can be provided by closing the apparatus in a thermally buffered cabin and controlling the The *H* is calculated by dividing the load by the projected contact area (*A(hc)*) at maximum applied load, *Pmax*:

$$H = \frac{P\_{\text{max}}}{A(h\_c)}\tag{3}$$

where the contact depth (*hc*) is determined as:

$$\eta\_{\rm li} = \eta\_{\rm max} - \varepsilon \frac{P\_{\rm max}}{S} \tag{4}$$

being *hmax* the maximum indentation depth, a parameter approximately equal to 0.75 for a Berkovich tip indenter, and *S* the unloading contact stiffness at maximum indentation depth (*dP/dh*, see **Fig. 11**). In the case of a perfect Berkovich tip, the *A(hc)* is determined as:

$$A(\text{l}\_c) = 24.56 \text{l}\_c^{2} \tag{5}$$

However, in practice the projected contact area differs from this value when the tip is worn out, because the different angles of the faces may be somehow different from the nominal. Hence, the projected contact area must be regularly calibrated. When catastrophic fracture mechanisms occur, the *A(hc)* obtained by this theoretical method is overestimated, and results in an *H* underestimation. To avoid this problem, the real contact area should be determined using a technique of microscopy (as Scanning Electron Microscopy or Atomic Force Microscopy) and the contact area must be recalculated in order to avoid the sink-in or pile-up effects for brittle and/or ductile materials, respectively (Roa et al., 2009d, 2010c, 2011b).

The effective elastic modulus (*Eeff*) has been expressed in terms of *S* and *A(hc)* as:

$$E\_{eff} = \frac{1}{\beta} \frac{\sqrt{\pi}}{2} \frac{S}{\sqrt{A(\text{h}\_c)}} \tag{6}$$

where is a geometrical factor (1.034). This parameter takes into account the lack of symmetry of the Berkovich indenter. Then, according to the Oliver and Pharr approach (Oliver et al., 1992), the Elastic modulus can be obtained as:

$$\frac{1}{E\_{\rm eff}} = \frac{1-\nu^2}{E} + \frac{1-\nu\_i^2}{E\_i} \tag{7}$$

where *E* and are the Elastic modulus and the Poisson's ratio (νYBa2Cu3O7-x *=* 0.3 (Roa et al, 2007, 2009d, 2010a, 2010c, 2011b, 2011d), respectively. The subindex *i* denotes the indenter properties (*Ei =* 1141GPa and ) (Oliver et al., 1992; Pharr et al., 1992).

#### *b. Fracture toughness*

Moreover, nanoindentation provides information about contact mechanisms taking place in the material. In brittle materials indented with a Berkovich tip, surface examination of residual imprints has revealed the appearance of several typical features: surface deformation effects such as *sink-in* (Johnson et al., 1985; Alcala et al., 2000), microcracks or damage inside the

Magnetical Response and Mechanical Properties of

**a** 

Hertzian equations as:

expressed as (Fischer-Cripps, 1999):

2011).

High Temperature Superconductors, YBaCu3O7-X Materials 199

**D**

**2a**

where *he* is the elastic indentation depth. Recently, Barsoum *et al.* (Moseson et al., 2008) proposed the fitting of the stiffness vs. indentation depth data, and extrapolating to zero, as predicated by a relation between *Eeff* and *a*, as follows (Roa et al., 2010b, 2010c, 2011b):

Hertz found that the radius of the circle of contact, *a*, is related to the indentation load, *P*, the indenter radius, *R*, and the elastic properties for small indentation depths of the material by:

3

2

*p= =*

*m*

The mean contact pressure, *pm*, between the indenter and the material is the applied load divided by the contact area. For small indentation depths, it can be obtained from the

> 4 3π *eff*

*P a E*

*πa R*

The mean contact pressure is referred to as the "*indentation stress*" and the quantity *a/R* as the "*indentation strain*". This equation allows us to plot the stress-strain curves when we use a spherical tip. The left side of this equation represents the indentation stress or mean contact pressure, also referred to as the Meyer hardness (Basu et al., 2006). The expression in parentheses on the right side of this equation represents the indentation strain (Rayon et al.,

The maximum tensile stress occurs at the edge of the contact circle in the surface and can be

*PR PR <sup>j</sup> <sup>E</sup> <sup>π</sup> <sup>S</sup>*

2

*β A h*

3 3 4 4 1

*eff*

Fig. 12. Schematic representation of spherical indentation (From Roa et al., 2010b).

3

**R**

**he/2** 

**htot hmax**

**hc**

2 *S= E a eff* (10)

(12)

*c*

(11)

imprints (Burghard et al., 2004); and fracture mechanisms such as *radial cracks* emanating from the imprint corners (Roa et al., 2008a, 2008b), and *chipping* (Lawn et al., 1975).

The resistance of the material against crack propagation, characterised by the fracture toughness *K*IC (Anderson et al., 2005), is a critical mechanical parameter in brittle materials. Fracture toughness estimation by microfracture Vickers indentation is a well-known and broadly employed technique in ceramic materials. Usually, a Vickers diamond pyramid indenter at loads greater than 1 N is used to nucleate radial cracks at the imprint corners. The cracks emanating from the corners of a Vickers or Berkovich indenter are arrested when the residual stress driving force at the crack tip is in equilibrium with the *KIC*. Several expressions are widely employed to extract the fracture toughness, depending on the indenter geometry and crack morphology. In this experimental work, the method we are pursuing is based on the radial cracking which occurs when brittle materials are indented by sharp indenter such as Vickers or Berkovich diamond (Casellas et al., 2007; Jang et al., 2008). *KIC* is then evaluated using the equation developed by Lawn, Evans, and Marshall (Lawn et al., 1980):

$$K\_{IC} = a \left(\frac{E}{H}\right)^{1/2} \left(\frac{P}{c^{32}}\right) \tag{8}$$

where is an empirical constant depending on the indenter geometry (for pyramidal tips), *P* is the peak indentation load, *c* is the length of the radial cracks, *E* is the elastic modulus, and *H* is the hardness value. The fracture toughness of YBa2Cu3O7-x melttextured samples has been determined by Vickers indentation (Leblond-Harnois et al., 2000; Li et al., 1997; Yoshino et al., 2001; Leenders et al., 1999; Cook et al., 1987; Fujimoto et al., 1992), and also by the single-edge notch beam (Joo et al., 1998) and bending (Giese et al., 1990) techniques. Values in the range of *KIC=*0.4-2.8 MPa·m1/2 have been found, with a scattering attributed to differences between the techniques employed to obtain this parameter. In addition, *KIC* is influenced by exact matrix composition. For instance, Joo *et al.* (Joo et al., 1998), reported that the fracture toughness of YBa2Cu3O7-x increased from *KIC*=1.60 MPa·m1/2 to 2.80 MPa·m1/2 by adding a 15% in volume of silver to the matrix.

#### *c. Yield strength and stress-strain curve*

The stress and deflection arising from the contact between two elastic solids are of particular interest to understand the first steps of indentation testing. The contact depth between a rigid sphere and a flat surface for small indentation depths was described by Hertz (Hertz et al., 1881). This phenomenon is shown in **Fig. 12**. The radius of the circle of contact is *a*, the total indentation depth is *hmax*, the depth of the circle of contact from the specimen free surface is *ha*, and *hp* is the distance from the bottom of the contact to the contact circle.

When using spherical tips, the contact point is more difficult to be determined, due to the moderate increase in stiffness during the initial contact, and the interaction between the tip and samples before contacting. The indentation load-displacement data were analyzed based on the Hertz equation in the elastic region as follows (Oliver et al., 2004; Field et al., 1993; Field et al., 1995):

$$P = \frac{3}{4} E\_{eff} \sqrt{R} \sqrt{R} \sqrt{h\_e} \tag{9}$$

imprints (Burghard et al., 2004); and fracture mechanisms such as *radial cracks* emanating from

The resistance of the material against crack propagation, characterised by the fracture toughness *K*IC (Anderson et al., 2005), is a critical mechanical parameter in brittle materials. Fracture toughness estimation by microfracture Vickers indentation is a well-known and broadly employed technique in ceramic materials. Usually, a Vickers diamond pyramid indenter at loads greater than 1 N is used to nucleate radial cracks at the imprint corners. The cracks emanating from the corners of a Vickers or Berkovich indenter are arrested when the residual stress driving force at the crack tip is in equilibrium with the *KIC*. Several expressions are widely employed to extract the fracture toughness, depending on the indenter geometry and crack morphology. In this experimental work, the method we are pursuing is based on the radial cracking which occurs when brittle materials are indented by sharp indenter such as Vickers or Berkovich diamond (Casellas et al., 2007; Jang et al., 2008). *KIC* is then evaluated

> 1 2 *IC* 32 *E P K = <sup>α</sup> H c*

is an empirical constant depending on the indenter geometry (

pyramidal tips), *P* is the peak indentation load, *c* is the length of the radial cracks, *E* is the elastic modulus, and *H* is the hardness value. The fracture toughness of YBa2Cu3O7-x melttextured samples has been determined by Vickers indentation (Leblond-Harnois et al., 2000; Li et al., 1997; Yoshino et al., 2001; Leenders et al., 1999; Cook et al., 1987; Fujimoto et al., 1992), and also by the single-edge notch beam (Joo et al., 1998) and bending (Giese et al., 1990) techniques. Values in the range of *KIC=*0.4-2.8 MPa·m1/2 have been found, with a scattering attributed to differences between the techniques employed to obtain this parameter. In addition, *KIC* is influenced by exact matrix composition. For instance, Joo *et al.* (Joo et al., 1998), reported that the fracture toughness of YBa2Cu3O7-x increased from *KIC*=1.60 MPa·m1/2 to 2.80 MPa·m1/2 by adding a 15% in volume of silver to the matrix.

The stress and deflection arising from the contact between two elastic solids are of particular interest to understand the first steps of indentation testing. The contact depth between a rigid sphere and a flat surface for small indentation depths was described by Hertz (Hertz et al., 1881). This phenomenon is shown in **Fig. 12**. The radius of the circle of contact is *a*, the total indentation depth is *hmax*, the depth of the circle of contact from the specimen free surface is *ha*, and *hp* is the distance from the bottom of the contact to the contact circle.

When using spherical tips, the contact point is more difficult to be determined, due to the moderate increase in stiffness during the initial contact, and the interaction between the tip and samples before contacting. The indentation load-displacement data were analyzed based on the Hertz equation in the elastic region as follows (Oliver et al., 2004; Field et al.,

<sup>3</sup> <sup>3</sup> <sup>2</sup>

*P= E R h eff <sup>e</sup>* (9)

4

(8)

for

the imprint corners (Roa et al., 2008a, 2008b), and *chipping* (Lawn et al., 1975).

using the equation developed by Lawn, Evans, and Marshall (Lawn et al., 1980):

where

*c. Yield strength and stress-strain curve* 

1993; Field et al., 1995):

Fig. 12. Schematic representation of spherical indentation (From Roa et al., 2010b).

where *he* is the elastic indentation depth. Recently, Barsoum *et al.* (Moseson et al., 2008) proposed the fitting of the stiffness vs. indentation depth data, and extrapolating to zero, as predicated by a relation between *Eeff* and *a*, as follows (Roa et al., 2010b, 2010c, 2011b):

$$S = \mathcal{D}E\_{\rm eff}a\tag{10}$$

Hertz found that the radius of the circle of contact, *a*, is related to the indentation load, *P*, the indenter radius, *R*, and the elastic properties for small indentation depths of the material by:

$$j = \sqrt[3]{\frac{3}{4} \frac{PR}{E\_{eff}}} = \sqrt[3]{\frac{3}{4} \frac{PR}{1 \frac{1}{\beta} \frac{\sqrt{\pi}}{2} \frac{S}{\sqrt{A(h\_c)}}}} \tag{11}$$

The mean contact pressure, *pm*, between the indenter and the material is the applied load divided by the contact area. For small indentation depths, it can be obtained from the Hertzian equations as:

$$p\_m = \frac{P}{\pi a^2} = \frac{4E\_{eff}}{\Im \mathbf{n}} \left(\frac{a}{R}\right) \tag{12}$$

The mean contact pressure is referred to as the "*indentation stress*" and the quantity *a/R* as the "*indentation strain*". This equation allows us to plot the stress-strain curves when we use a spherical tip. The left side of this equation represents the indentation stress or mean contact pressure, also referred to as the Meyer hardness (Basu et al., 2006). The expression in parentheses on the right side of this equation represents the indentation strain (Rayon et al., 2011).

The maximum tensile stress occurs at the edge of the contact circle in the surface and can be expressed as (Fischer-Cripps, 1999):

Magnetical Response and Mechanical Properties of

Processor, see **Fig. 13** (Roa et al., 2011e).

light.

High Temperature Superconductors, YBaCu3O7-X Materials 201

1. **Cleaning the tip:** it is important to remove contaminants covering the tip. First, rinse the tip with miliQ water, acetone and ethanol and dry it under a nitrogen flow. After that, put the tip in an ozone cleaning chamber during 30-35 minutes, and ultraviolet

2. **Measuring the tip radius by means of a NiO Pattern:** A NiO sample that features extremely sharp crystallographic ridges is used for measuring the tip radius. Then, as the sample ridges are nominally sharper than the tip apex, it is possible to obtain a "tip image" using blond reconstruction software. AFM images of the pattern should be collected in tapping mode with scan of 0.5µm by 0.5µm, scan rates of 1 Hz and 512 by 512 pixels. The blind reconstruction software employed is Scanning Probe Image

Fig. 13. Software reconstruction images of the AFM tip, (a) before the picoindentation test,

and (b) after the picoindentation test. From Roa et al., 2011e.

$$
\sigma\_{\text{max}} = \frac{1}{2} (1 - 2\text{v}) p\_m \tag{13}
$$

The yield stress of the material, *ys*, can be obtained using the following expression:

$$p\_m = 1.1 \sigma\_{ys} \tag{14}$$

This parameter will give information about the transition from elastic to elastic-to-plastic response (Jiménez-Piqué et al., 2007). The maximum shear stress (*max*) is produced beneath the indentation axis at a depth close to *0.5a*, and equal to:

$$
\pi\_{\text{max}} = 0.46 p\_m \tag{15}
$$

The maximum shear stress is also known as the Tresca criterion. This assumes that yield occurs when the shear stress exceeds the shear yield strength, *ys* (Tsui et al., 1997; Roa et al., 2008a):

$$
\sigma = \frac{\sigma\_1 - \sigma\_3}{2} \le \mathfrak{r}\_{ys} \tag{16}
$$

Where *1* and *3* are the principal stresses (also, can be re-written as *x* and *z*, respectively).

#### **1.2.1.3 Picoindentation**

The applied load in this case is much lower than in the previous one. Normally, this technique produces some indentations of few nanometers (generally fully elastic) obtaining the Elastic modulus. Picoindentation by means of atomic force microscopy-force spectroscopy (AFM-FS) presents several advantages over standard methods with regard to the quantification of *E* values:


These advantages mean that picoindentation is a suitable and reliable technique for measuring E values while applying loads ranging from a few nN down to the pN level. The study of force versus displacement (F-z) curves can shed new light on the elastic properties of hard materials, as these curves are similar to those obtained with classic microscopic or nanoscopic indentation tests.

The different steps of the method to extract the elastic modulus value using the AFM-FS and Hertz equations are described below (Roa et al., 2011e, 2011f):

max <sup>1</sup> 1 2<sup>ν</sup> 2

This parameter will give information about the transition from elastic to elastic-to-plastic

The maximum shear stress is also known as the Tresca criterion. This assumes that yield

1 3 2 *ys*

The applied load in this case is much lower than in the previous one. Normally, this technique produces some indentations of few nanometers (generally fully elastic) obtaining the Elastic modulus. Picoindentation by means of atomic force microscopy-force spectroscopy (AFM-FS) presents several advantages over standard methods with regard to




These advantages mean that picoindentation is a suitable and reliable technique for measuring E values while applying loads ranging from a few nN down to the pN level. The study of force versus displacement (F-z) curves can shed new light on the elastic properties of hard materials, as these curves are similar to those obtained with classic microscopic or

The different steps of the method to extract the elastic modulus value using the AFM-FS and

*σ σ*

exceeds the shear yield strength,

*3* are the principal stresses (also, can be re-written as

response (Jiménez-Piqué et al., 2007). The maximum shear stress (

the indentation axis at a depth close to *0.5a*, and equal to:

The yield stress of the material,

occurs when the shear stress

al., 2008a):

Where

*1* and

regimen).

**1.2.1.3 Picoindentation** 

the quantification of *E* values:

nanoscopic indentation tests.

mechanical properties of the samples.

Hertz equations are described below (Roa et al., 2011e, 2011f):

respectively).

*σ = pm* (13)

*<sup>m</sup>* 1.1σ*ys p =* (14)

max *τ = p* 0.46 *<sup>m</sup>* (15)

*<sup>τ</sup> <sup>=</sup> <sup>τ</sup>* (16)

*max*) is produced beneath

*ys* (Tsui et al., 1997; Roa et

*x* and *z*,

*ys*, can be obtained using the following expression:


Fig. 13. Software reconstruction images of the AFM tip, (a) before the picoindentation test, and (b) after the picoindentation test. From Roa et al., 2011e.

Magnetical Response and Mechanical Properties of

equation 20 (see F**ig. 14d**):

**1.2.2 State of the art** 

the yield stress value,

*and 4,* respectively.

*Güçlü et al.* 

*Roa et al.*

<sup>1</sup> *Texture process:* Bridgman technique

sample.

and 

High Temperature Superconductors, YBaCu3O7-X Materials 203

order to ensure that the sample had not undergone any plastic deformation. Only *F-z* curves that presented negligible lateral deflection should be considered to extract the mechanical response. *F-z* curves obtained at a certain *F* value were analyzed using the Hertz model in the elastic region (Oliver et al., 2004; Field et al., 1993, Field et al., 1995) by means of

3

*ys*. Some authors studied the mechanical properties at room or at

indentation 2005 Polycrystalline, 160K 29.63

Texturized 182 Nanoindentation 1991

Vickers

Nanoindentation 2007

*F= E R eff <sup>δ</sup>* (20)

<sup>2</sup> 3

where *Eeff* can be obtained using the equation 7. In this case the subindex *i* corresponds to the mechanical properties of the AFM probe (SiO2 with *E =* 76 GPa, and *i =* 0.17 (Oliver et al., 1992)). The quality of *S* value relies on the absence of elastic deformation of the reference

The mechanical properties of YBa2Cu3O7-x (Y-123 or YBCO) samples have been studied during the last years. The most important properties studied have been the elastic modulus, *E*, the hardness, *H* (*at micro-/ nanometric scale*), the toughness fracture, *KIC*, and

cryogenic temperatures, also known as work temperature (*Tw*). Lots of different techniques could be used to determine these parameters. The reported values of *E*, *H, KIC*

**Author Material E (GPa) Method Year**  *Lucas et al.* YBa2Cu3O7-x 154.30 ± 16.34 Indentation 1996

Polycrystalline, 50K 47.20

Polycrystalline, 180K 28.47 Polycrystalline, 293K 9.39 *Goyal et al.* Y2BaCuO5 (Y-211) 213 Nanoindentation 1991

Y-1231 at 5mN 197 ± 7

Y-2111 at 5mN 194 ± 9 Y-123/Y-2111 at 5mN 201 ± 7 Y-1231 at 10mN 189 ± 4 Y-2111 at 10 mN 206 ± 4 Y-123/Y-2111 at 10mN 204 ± 6 Y-123/Y-2111 at 30mN 180 ± 5 Y-123/Y-2111 at 100mN 173 ± 3 Table 1. Literature values of *E* for YBa2Cu3O7-x obtained by indentation techniques.

*ys* of YBa2Cu3O7-x samples (bulk materials) obtained using indentation techniques (macro-, micro-, nano- and/or picoindentation techniques) are summarized in *table 1, 2, 3* 

4

3. **Indentation process:** after that, an image of the sample should be taken, and several spots (at least 40 spots) should be chosen in order to perform one indentation per spot, see **Fig. 14a** (from Roa et al., 2011e).

Fig. 14. Images of the different steps taking place during the picoindentation process. (a) AFM image with 40 different spots. (b) *F* vs. *z* cantilever deflection. (c) *F* vs , and (d) *F* vs. *3/2* from the slope of Elastic modulus, which is calculated using Hertzian equations. From Roa et al., 2011e.

*F* is calculated as (see **Fig. 14b**):

$$F = k\_v \Delta z \tag{17}$$

where *z* is the cantilever deflection, defined as:

$$
\Delta z = \frac{\Delta V}{S} \tag{18}
$$

where *V* is the increment in photodetector vertical signal, as the tip comes into contact with the sample, and *S* is the sensitivity, which is the slope of the contact region of a force curve performed on a rigid sample.

The indentation depth () due to F is calculated as (see **Fig. 14c**):

$$
\delta = z - \Delta z \tag{19}
$$

where *z* represents the piezo-scanner displacement along the axis perpendicular to the sample plane. After puncturing the sample, a topographic image should be captured in order to ensure that the sample had not undergone any plastic deformation. Only *F-z* curves that presented negligible lateral deflection should be considered to extract the mechanical response. *F-z* curves obtained at a certain *F* value were analyzed using the Hertz model in the elastic region (Oliver et al., 2004; Field et al., 1993, Field et al., 1995) by means of equation 20 (see F**ig. 14d**):

$$F = \left(\frac{3}{4}E\_{eff}\sqrt{R}\right)\delta^{\frac{3}{2}}\tag{20}$$

where *Eeff* can be obtained using the equation 7. In this case the subindex *i* corresponds to the mechanical properties of the AFM probe (SiO2 with *E =* 76 GPa, and *i =* 0.17 (Oliver et al., 1992)). The quality of *S* value relies on the absence of elastic deformation of the reference sample.

## **1.2.2 State of the art**

202 Superconductors – Properties, Technology, and Applications

3. **Indentation process:** after that, an image of the sample should be taken, and several spots (at least 40 spots) should be chosen in order to perform one indentation per spot,

Fig. 14. Images of the different steps taking place during the picoindentation process. (a)

*3/2* from the slope of Elastic modulus, which is calculated using Hertzian equations. From

*<sup>Δ</sup><sup>V</sup> <sup>Δ</sup>z =*

the sample, and *S* is the sensitivity, which is the slope of the contact region of a force curve

where *z* represents the piezo-scanner displacement along the axis perpendicular to the sample plane. After puncturing the sample, a topographic image should be captured in

) due to F is calculated as (see **Fig. 14c**):

*V* is the increment in photodetector vertical signal, as the tip comes into contact with

*F=kvΔz* (17)

*z z* (19)

*S* (18)

, and (d) *F* vs.

AFM image with 40 different spots. (b) *F* vs. *z* cantilever deflection. (c) *F* vs

*z* is the cantilever deflection, defined as:

see **Fig. 14a** (from Roa et al., 2011e).

where 

where 

Roa et al., 2011e.

*F* is calculated as (see **Fig. 14b**):

performed on a rigid sample.

The indentation depth (

The mechanical properties of YBa2Cu3O7-x (Y-123 or YBCO) samples have been studied during the last years. The most important properties studied have been the elastic modulus, *E*, the hardness, *H* (*at micro-/ nanometric scale*), the toughness fracture, *KIC*, and the yield stress value, *ys*. Some authors studied the mechanical properties at room or at cryogenic temperatures, also known as work temperature (*Tw*). Lots of different techniques could be used to determine these parameters. The reported values of *E*, *H, KIC* and *ys* of YBa2Cu3O7-x samples (bulk materials) obtained using indentation techniques (macro-, micro-, nano- and/or picoindentation techniques) are summarized in *table 1, 2, 3 and 4,* respectively.


Table 1. Literature values of *E* for YBa2Cu3O7-x obtained by indentation techniques.

<sup>1</sup> *Texture process:* Bridgman technique

Magnetical Response and Mechanical Properties of

*Li et al.* 

*Güçlü et al.* 

*Roa et al.*

*Roa et al.* 

<sup>6</sup> *Texture process:* Bridgman technique

High Temperature Superconductors, YBaCu3O7-X Materials 205

*Goyal et al.* Textures 6.7 1992 10.8 Nanoindentation *Goyal et al.* Y-211 14.0 Nanoindentation 1992 *Lucas et al.* Y-123 10.28 ± 1.67 Indentation 1996

*Lim et al.* Single crystal 7.81 ± 0.23 Nanoindentation 2001

*Cook et al.* YBCO 8.7 Vickers

MTG-1100ºC, 5min 5.4

MTG-1100ºC, 15min 10.3 Solid State reaction 10.3

*Yoshimo et al.* YBCO 40K 18 ± 2.5 Vickers

Polycrystalline, 50K 3.58

Polycrystalline, 180K 0.95 Polycrystalline, 293K 0.53

*Roa et al.* Y-123/Y-2116 at 100mN (O) 8.9 ± 0.2

*Foerster et al.* YBCO-TSMG (*ab*-plane) 8

YBCO-TSMG + 5% Ag2O (*ab*-

YBCO-TSMG + 5% Ag2O (*ca(b)*-

YBCO-TSMG + 10% Ag2O (*ab*-

YBCO-TSMG + 10% Ag2O

Y-1236 at 5mN (O) 10.4 ± 0.5

Y-2116 at 5mN (O) 19.1 ± 0.8 Y-123/Y-2116 at 5mN (O) 14.8 ± 0.5 Y-1236 at 10mN (O) 11.2 ± 0.4 Y-2116 at 10 mN (O) 16.9 ± 0.6 Y-123/Y-2116 at 10mN (O) 15.1 ± 0.3 Y-123/Y-2116 at 30mN (O) 11.1 ± 0.3

Y-1236 at 5mN (T) 11.9 ± 0.7

Y-2116 at 5mN (T) 17.9 ± 1.1 Y-123/Y-2116 at 5mN (T) 15.4 ± 0.9 Y-1236 at 10mN (T) 10.2 ± 0.7 Y-2116 at 10 mN (T) 17.0 ± 0.9 Y-123/Y-2116 at 10 mN (T) 14.8 ± 0.4 Y-123/Y-2116 at 30 mN (T) 8.3 ± 0.2 Y-123/Y-2116 at 100 mN (T) 8.1 ± 0.5

YBCO-TSMG (*ca(b)*-plane) 7

plane) <sup>9</sup>

plane) <sup>8</sup>

plane) 6.5

(*ca(b)*-plane) <sup>7</sup>

Table 2. Literature values of *H* for YBa2Cu3O7-x obtained by indentation techniques.

**Author Material H (GPa) Method Year** 

indentation 1997 MTG-1100ºC, 10min 11.0

Indentation 2001 YBCO 293K 5.2 ± 0.5

Indentation 2005 Polycrystalline, 160K 1.03

Indentation

Vickers

Vickers

Nanoindentation

Nanoindentation (Berkovich)

Nanoindentation 2008

(Berkovich) <sup>2007</sup>

2007

2008a, b

1987


Table 1. Literature values of E for YBa2Cu3O7-x obtained by indentation techniques (continuation).

 2At maximum indentation depth (around *1300nm*)


Nanoindentation 2008

Nanoindentation 2009d

FS**\***-AFM 2011c

**Author Material E (GPa) Method Year** 

(*ab*-plane) <sup>220</sup>

(*ab*-plane) 149 ± 4

(*ab*-plane) <sup>150</sup>

(*ca(b)*-plane) <sup>175</sup>

(*ca(b)*-plane) <sup>175</sup>

(*ca(b)*-plane) <sup>150</sup>

Y-1233 at 5mN 176 ± 15

Y-2113 at 5mN 224 ± 18 Y-123/Y-2113 at 5mN 208 ± 21 Y-1233 at 10mN 174 ± 17 Y-2113 at 10 mN 207 ± 11 Y-123/Y-2113 at 10mN 190 ± 16 Y-123/Y-2113 at 30mN 140 ± 14 Y-123/Y-2113 at 100mN 129 ± 6

*Roa et al.* YBa1.75Sr0.25Cu3O7-� 106 ± 4 FS4-AFM 2009b *Roa et al* Y-123/Y-211-Bridgman 123.5 ± 3.4 Nanoindentation 2010b,c *Bartolomé et al.* YBCO-TSMG 120 ± 5 Nanoindentation 2010a *Roa et al.* YBCO-Bridgman 128 ± 5 Nanoindentation 2011b,d

Table 1. Literature values of E for YBa2Cu3O7-x obtained by indentation techniques (continuation).

2At maximum indentation depth (around *1300nm*)

<sup>3</sup> *Texture process:* TSMG technique

<sup>5</sup> *Texture process:* Self-flux technique

<sup>4</sup> *FS:* Force Spectroscopy

Y-123 at 300K (tetragonal, T)**<sup>3</sup>** 185 ± 7

Y-211 at 300K (T) 3 202 ± 5 Y-123 at 300K (orthorhombic, O)**<sup>3</sup>** 189 ± 4 Y-211 at 300K (O) 3 201 ± 6 Y-123 at 300K (T) 3 192 ± 3 Y-211 at 300K (T) 3 208 ± 7 Y-123 at 300K (O) 3 193 ± 4 Y-211 at 300K (O) <sup>3</sup> 205 ± 5 Y-123 at 300K (T)5 179 ± 4 Y-123 at 300K (O)5 182 ± 5 YBa1.75Sr0.25Cu3O7-x5 106 ± 4 Y0.98Ca0.02Ba2Cu3O7-x5 100 ± 4

YBCO-TSMG (*ca(b)*-plane) 177 ± 102

YBCO-TSMG (*ab*-plane) 177 ± 102

YBCO-TSMG + 5% Ag2O

YBCO-TSMG + 10% Ag2O

YBCO-TSMG + 15% Ag2O

YBCO-TSMG + 5% Ag2O

YBCO-TSMG + 10% Ag2O

YBCO-TSMG + 15% Ag2O

*Foerster et al.* 

*Roa et al.* 

*Roa et al.* 


Table 2. Literature values of *H* for YBa2Cu3O7-x obtained by indentation techniques.

<sup>6</sup> *Texture process:* Bridgman technique

Magnetical Response and Mechanical Properties of

Table 4. Literature values of yield stress,

**1.2.3 Mechanical properties** 

techniques.

Finally, the

transition (from Roa et al., 2011b).

High Temperature Superconductors, YBaCu3O7-X Materials 207

**Author Material ys (GPa) Method Year** 

*Roa et al.* YBCO-TSMG 3.58 ± 0.10 2009c

The elastic modulus values reported by Roa et al. (Roa et al., 2007, 2009d, 2011b) presented in **Table 1** for Bridgman and TSMG texture processes present a dependency with the applied load. This phenomenon is far from what are expected, as elastic modulus is an intrinsic property of each material and only depends on its composition. Then, this

 Firstly, the contact area value, which is strongly affected by the presence of macro- /microcracks, porosity, roughness and a wide variety of defects. In order to avoid this

After considering all the different parameters that can modify the elastic modulus, we believe that the main contribution to change it is the contact area, and then the scattered values presented in the **Table 1** and reported by Roa et al. (Roa et al., 2007, 2008b) are attributed to this effect. Moreover, the Bridgman samples present elastic modulus values higher than for TSMG, as this last presents a higher density of macro- and microcracks,

Fig. 15. FE-SEM image of the *ab*-plane of TSMG samples, which present a high microcracks density performed during the oxygenation process due to the tetragonal to orthorhombic

factor could not modify the elastic modulus because it remains equal to

**66,7 m**

66.7 m

(Spherical)

*ys*, for YBa2Cu3O7-x obtained by indentation

2010b

*Roa et al.* YBCO-Bridgman 3.2 Nanoindentation

difference of elastic modulus is able to be attributed to one of the following factors:

effect, the polishing process before the indentation should be carefully done. Secondly, the elastic modulus is obtained using the unloading slope of the loadingunloading curve, also denoted as *S*. If the contact point (indenter-sample) is not well

estimated, the *S* value will be underestimated, yielding to a lower *Eeff*.

which can produce a relaxation of the elastic deformation range, see **Fig. 15.** 

1.034 for each test performed (Oliver et al., 1992).


Table 2. Literature values of *H* for YBCO obtained by indentation techniques. (continuation)


Table 3. Literature values of toughness fracture, *KIC*, for YBa2Cu3O7-x obtained by indentation techniques.

 <sup>7</sup> *Texture process:* TSMG technique

<sup>8</sup> Hardness value in the welding region (YBCO/Ag/YBCO)


Table 4. Literature values of yield stress, *ys*, for YBa2Cu3O7-x obtained by indentation techniques.

## **1.2.3 Mechanical properties**

206 Superconductors – Properties, Technology, and Applications

**Author Material H (GPa) Method Year** 

YBCO-TSMG + 5% Ag2O 5.2 ± 0.2 Microindentation YBCO-TSMG + 10% Ag2O 4.1 ± 0.4 (Vickers)

Nanoindentation

Vickers Indentation

Indentation

(Berkovich) 2010a

plane) 6.5

YBCO7 (T) 10.6 ± 0.4

YBCO7 (O) 7.6 ± 0.4

misorientation) (T) 8 6.5 ± 0.4

misorientation) (O) 8 4.4 ± 0.5

misorientation) (O) 8 5.2 ± 0.3

*Roa et al.* YBCO-TSMG (T) 10.5 ± 0.1 Nanoindentation

MTG-1100ºC, 5min 1.9

MTG-1100ºC, 15min 1.7 Solid State Reaction 1.3

*Lenders et al.* Y-123 + 30 mol% Y-211 1.01 Vickers

*Foerster et al.* YBCO-TSMG pure 1.43 ± 0.02 Vickers

*Fujimoto et al.* Textured 0.99-1.20 Vickers

*Yoshino et al.* YBCO 40K 0.4 2001 YBCO 293K 1.3

*Cook et al.* YBCO 1.10 1998

Table 3. Literature values of toughness fracture, *KIC*, for YBa2Cu3O7-x obtained by indentation

(T)8 9.5 ± 0.3

(O) <sup>8</sup> 6.8 ± 0.5

*Roa et al.* YBCO7 (O) 8.8 ± 0.3 Nanoindentation 2011b

Table 2. Literature values of *H* for YBCO obtained by indentation techniques. (continuation)

**Author Material KIC (MPa·m1/2) Method Year** 

(Berkovich) 2011c YBCO-TSMG (O) 7.8 ± 0.1

2000 Y-123 + 5% wt Ag 1.88

1997 MTG-1100ºC, 10min 1.7

1999 Y-123 + 60 mol % Y-211 1.44

Indentation 2008 YBCO-TSMG + 10% Ag2O 1.6 ± 0.1

Indentation 1992 Y-123 with Ag 1.60-2.10

(*ca(b)*-plane) <sup>6</sup>

YBCO-TSMG + 15% Ag2O (*ab*-

YBCO-TSMG + 15% Ag2O

YBCO weld (0° misorientation)

YBCO weld (14°

YBCO weld (0° misorientation)

YBCO weld (14°

YBCO weld (15°

*Lenblond-Harnois et al* YBCO 1.53

*Bartolomé et al.* 

*Li et al.* 

techniques.

<sup>7</sup> *Texture process:* TSMG technique

8 Hardness value in the welding region (YBCO/Ag/YBCO)

The elastic modulus values reported by Roa et al. (Roa et al., 2007, 2009d, 2011b) presented in **Table 1** for Bridgman and TSMG texture processes present a dependency with the applied load. This phenomenon is far from what are expected, as elastic modulus is an intrinsic property of each material and only depends on its composition. Then, this difference of elastic modulus is able to be attributed to one of the following factors:


After considering all the different parameters that can modify the elastic modulus, we believe that the main contribution to change it is the contact area, and then the scattered values presented in the **Table 1** and reported by Roa et al. (Roa et al., 2007, 2008b) are attributed to this effect. Moreover, the Bridgman samples present elastic modulus values higher than for TSMG, as this last presents a higher density of macro- and microcracks, which can produce a relaxation of the elastic deformation range, see **Fig. 15.** 

Fig. 15. FE-SEM image of the *ab*-plane of TSMG samples, which present a high microcracks density performed during the oxygenation process due to the tetragonal to orthorhombic transition (from Roa et al., 2011b).

Magnetical Response and Mechanical Properties of

value than Bridgman samples.

**1.2.4 Fracture mechanisms** 

been previously described (Gaillard et al., 2008).

within the imprint when observed from the surface.

2011b).

cryogenic temperature due to the effect explained above.

High Temperature Superconductors, YBaCu3O7-X Materials 209

2008). Moreover, several studies performed at cryogenic temperature (Güçlü et al., 2005) exhibit that the mechanical properties tend to be higher due to the nitrogen going inside the pre-existing cracks, yielding a homogeneous material. As we can observe in **Table 2,** the different YBa2Cu3O7-x samples studied at room temperature are much brittle than at

**Table 3** shows the different toughness fracture values reported in the literature and obtained using Vickers and Nanoindentation tests. The different results reported in this table present a high scattered due to the YBa2Cu3O7-x samples are strongly heterogeneous due to the presence of internal cracks, secondary phases and grain boundaries. Due to all this defects the different toughness values do not present any tendency. Lenblond-Harnois et al. (Lenblond-Harnois et al., 2000) and Foester et al. (Foester et al., 2008) obtained a toughness fracture of 1.88 and 1.6 MPam1/2 for YBa2Cu3O7-x with 5 and 10% of Ag and Ag2O, respectively. In that case, this value is higher than the others reported in this table due to the silver introduced inside of the YBa2Cu3O7-x samples reduce the fragility of these materials. Finally, another mechanical parameter to take into account is the yield stress, see **Table 4**. As can be observed in this table, the intrinsic properties obtained using the stress-strain plots present similar values for both samples. The TSMG samples, present a higher density of macro-/microcracks and pore density (see **Fig. 15**) along the *c-*axis than Bridgman samples. Then, during the first contact between the indenter and the samples, the energy applied by the tip onto the surface is employed to close the different cracks produced during the oxygenation process. For this reason, TSMG samples present a higher contact radius

In this section we have analysed the different brittle effects appearing in as-grown samples textured by Bridgman and TSMG techniques. **Fig. 16** and **Fig. 17**, show Field Emission Scanning Electron Microscopy images of some residual imprints performed at 10 mN of applied load for Bridgman and TSMG samples, respectively (from Roa et al.,

All the different imprints in **Fig. 16** present the *sink-in* effect, typically found in elastic, brittle materials (Oliver et al., 1992). Sink-in is an elastic displacement of the surface at the contact perimeter. This effect can lead to an overestimation of the contact area, and thus an underestimation of the hardness value obtained by nanoindentation. As we can observe in this **Figure**, we noted a high density of micrometric pores around the residual imprint. Presumably, this micrometric porosity is created during the recrystalization process. The nanoindentation process would render the pores exposed to the surface, an effect that has

In addition, all these images show the presence of microcracks inside the imprints, generated during the indentation process. This kind of damage is found in brittle materials containing inclusions harder than the matrix, in this case Y-211 (see **table 2**). Upon indentation, microcracks tend to be created at inclusions, and are recognised as microflaws

Moreover, the **Table 1** presents several measurements of the elastic modulus for Y-123, which resulted in values scattered within the range between 40 – 200 GPa, caused by residual porosity and bad contact between the grains, as reported Johansen et al. (Johansen et al., 2001). If we compare all the elastic modulus reported in **Table 1**, we can conclude that the broad distribution observed can be attributed to the different measuring techniques (nanoindentation, AFM-FS) and to the different quality of the studied YBa2Cu3O7-x samples (grain structure, texture, and others).

**Table 2** exhibits the hardness values for YBa2Cu3O7-x samples using different techniques. It is well known that the YBa2Cu3O7-x samples textured by Bridgman and TSMG techniques present a heterogeneous microstructure with Y-123 matrix and a minor phase (Y-211 or inclusions). Using the conventional micro hardness technique it is not possible to isolate the hardness value for each phase due to that the size of the residual imprint is higher than the size of the different inclusions. However, in 2007, Roa et al. (Roa et al., 2007) using the Nanoindentation technique obtained the hardness value for each phase when the applied load was lower or equal to 10mN. With this applied load, the residual contact area is lower than the size of the inclusions. Then, the mechanical property can be obtained, see **Table 2**. We can observe that the hardness value for Y-211 is around twice times that for Y-123. This fact could be due to different reasons (Roa et al., 2007):


If the different residual imprints have been performed near the grain boundary, the plastic deformation mechanisms under the imprint could be affected by the Y-123 matrix, and the hardness value would be modified, thus giving us an average of the studied property.

At very small applied loads (lower than 10 mN), the hardness value can be strongly affected by the presence of defects and impurities that can cause almost no change in dislocation movement. In other words, hardness does not remain constant at different applied loads as corroborate Roa *et al.* in 2007 and 2008 (Roa et al., 2007, 2008b). Using Nanoindentation technique this effect can be related to one physical phenomenon known as *Indentation Size Effect*. Normally, hardness presents a maximum value for low loads. However, when the applied load increases this property is widely modified by the superficial defects, such as roughness, macro-cracks, and others. When this phenomenon takes place, the contact area could be over- or underestimated yielding to a false hardness value.

When the applied load using Nanoindentation technique is high (100 mN), overall nanohardness is very similar for both texturing techniques (8.9 GPa and 7.9 GPa for Bridgman and TSMG techniques, respectively). However, these values are lower than the nanohardness value of the Y-123 phase. The different hardness values reported in **Table 2** by Micro or Nanohardness tests at room temperature present the same value at high applied loads. However, when the YBa2Cu3O7-x samples are doped with a ductile material as silver oxide, the mechanical response tends to decrease, as reported by Foester et al. (Foester et al.,

Moreover, the **Table 1** presents several measurements of the elastic modulus for Y-123, which resulted in values scattered within the range between 40 – 200 GPa, caused by residual porosity and bad contact between the grains, as reported Johansen et al. (Johansen et al., 2001). If we compare all the elastic modulus reported in **Table 1**, we can conclude that the broad distribution observed can be attributed to the different measuring techniques (nanoindentation, AFM-FS) and to the different quality of the studied YBa2Cu3O7-x samples

**Table 2** exhibits the hardness values for YBa2Cu3O7-x samples using different techniques. It is well known that the YBa2Cu3O7-x samples textured by Bridgman and TSMG techniques present a heterogeneous microstructure with Y-123 matrix and a minor phase (Y-211 or inclusions). Using the conventional micro hardness technique it is not possible to isolate the hardness value for each phase due to that the size of the residual imprint is higher than the size of the different inclusions. However, in 2007, Roa et al. (Roa et al., 2007) using the Nanoindentation technique obtained the hardness value for each phase when the applied load was lower or equal to 10mN. With this applied load, the residual contact area is lower than the size of the inclusions. Then, the mechanical property can be obtained, see **Table 2**. We can observe that the hardness value for Y-211 is around twice times that for Y-123. This

ionic bond of Y-211 is stronger than Y-123 (related to the different melting point of the

high anisotropy of dislocations confined onto a (001) plane (Sandiumenge et al., 2000),

 the melt-processed ceramic composites contain a dense population of fine peritectic particles, which drastically affects the microstructure acting as nucleation sites for

If the different residual imprints have been performed near the grain boundary, the plastic deformation mechanisms under the imprint could be affected by the Y-123 matrix, and the hardness value would be modified, thus giving us an average of the studied property.

At very small applied loads (lower than 10 mN), the hardness value can be strongly affected by the presence of defects and impurities that can cause almost no change in dislocation movement. In other words, hardness does not remain constant at different applied loads as corroborate Roa *et al.* in 2007 and 2008 (Roa et al., 2007, 2008b). Using Nanoindentation technique this effect can be related to one physical phenomenon known as *Indentation Size Effect*. Normally, hardness presents a maximum value for low loads. However, when the applied load increases this property is widely modified by the superficial defects, such as roughness, macro-cracks, and others. When this phenomenon takes place, the contact area

When the applied load using Nanoindentation technique is high (100 mN), overall nanohardness is very similar for both texturing techniques (8.9 GPa and 7.9 GPa for Bridgman and TSMG techniques, respectively). However, these values are lower than the nanohardness value of the Y-123 phase. The different hardness values reported in **Table 2** by Micro or Nanohardness tests at room temperature present the same value at high applied loads. However, when the YBa2Cu3O7-x samples are doped with a ductile material as silver oxide, the mechanical response tends to decrease, as reported by Foester et al. (Foester et al.,

two phases, *TY-123 =* 1010°C, and *TY-211 =* 1200°C (Aselage et al., 1988)),

could be over- or underestimated yielding to a false hardness value.

(grain structure, texture, and others).

and/or

fact could be due to different reasons (Roa et al., 2007):

dislocations (Sandiumenge et al., 2000).

2008). Moreover, several studies performed at cryogenic temperature (Güçlü et al., 2005) exhibit that the mechanical properties tend to be higher due to the nitrogen going inside the pre-existing cracks, yielding a homogeneous material. As we can observe in **Table 2,** the different YBa2Cu3O7-x samples studied at room temperature are much brittle than at cryogenic temperature due to the effect explained above.

**Table 3** shows the different toughness fracture values reported in the literature and obtained using Vickers and Nanoindentation tests. The different results reported in this table present a high scattered due to the YBa2Cu3O7-x samples are strongly heterogeneous due to the presence of internal cracks, secondary phases and grain boundaries. Due to all this defects the different toughness values do not present any tendency. Lenblond-Harnois et al. (Lenblond-Harnois et al., 2000) and Foester et al. (Foester et al., 2008) obtained a toughness fracture of 1.88 and 1.6 MPam1/2 for YBa2Cu3O7-x with 5 and 10% of Ag and Ag2O, respectively. In that case, this value is higher than the others reported in this table due to the silver introduced inside of the YBa2Cu3O7-x samples reduce the fragility of these materials.

Finally, another mechanical parameter to take into account is the yield stress, see **Table 4**. As can be observed in this table, the intrinsic properties obtained using the stress-strain plots present similar values for both samples. The TSMG samples, present a higher density of macro-/microcracks and pore density (see **Fig. 15**) along the *c-*axis than Bridgman samples. Then, during the first contact between the indenter and the samples, the energy applied by the tip onto the surface is employed to close the different cracks produced during the oxygenation process. For this reason, TSMG samples present a higher contact radius value than Bridgman samples.

## **1.2.4 Fracture mechanisms**

In this section we have analysed the different brittle effects appearing in as-grown samples textured by Bridgman and TSMG techniques. **Fig. 16** and **Fig. 17**, show Field Emission Scanning Electron Microscopy images of some residual imprints performed at 10 mN of applied load for Bridgman and TSMG samples, respectively (from Roa et al., 2011b).

All the different imprints in **Fig. 16** present the *sink-in* effect, typically found in elastic, brittle materials (Oliver et al., 1992). Sink-in is an elastic displacement of the surface at the contact perimeter. This effect can lead to an overestimation of the contact area, and thus an underestimation of the hardness value obtained by nanoindentation. As we can observe in this **Figure**, we noted a high density of micrometric pores around the residual imprint. Presumably, this micrometric porosity is created during the recrystalization process. The nanoindentation process would render the pores exposed to the surface, an effect that has been previously described (Gaillard et al., 2008).

In addition, all these images show the presence of microcracks inside the imprints, generated during the indentation process. This kind of damage is found in brittle materials containing inclusions harder than the matrix, in this case Y-211 (see **table 2**). Upon indentation, microcracks tend to be created at inclusions, and are recognised as microflaws within the imprint when observed from the surface.

Magnetical Response and Mechanical Properties of

et al., 2011b, 2011d).

of the imprint (from Roa et al., 2011d).

summarized in the **table 3**, are not reproducible.

shown in this work.

**2. Conclusions** 

high magnetic fields.

High Temperature Superconductors, YBaCu3O7-X Materials 211

Another common feature in all imprints is the presence of radial cracks emanating from the corners of the imprints. This fracture mechanism, typical for ceramic materials, is nucleated

Another common fracture mechanism is the *chipping*, i.e. a partial removal of surface material around the imprint. This phenomenon occurs as a result of Palqmvist cracks

In order to understand the fracture mechanisms that take place under the residual imprint, one nanoindentation performed at 10mN has been visualized by FIB, see **Fig. 18** (from Roa

Fig. 18. Cross-sectioning and imaging of damage under an indentation imprint performed at 10mN of applied load in Bridgman samples using FIB-SEM. Cross-sectioning in the middle

**Fig. 18** shows a heterogeneous distribution of Y-211 particles around the matrix, with different sizes from 1 to 5 m. Moreover, no cracks or failure events can be appreciated under the nanoindentation imprint, thus implying that the deformation can be attributed to the closing of the oxygenation micro-cracks. The TSMG samples present the same effect –not

Due to all of these fracture mechanisms, the different values reported in the literature and

The paramagnetic Meissner effect has been observed in several superconducting materials grown by different techniques since their discovery, and different models and explanations have been proposed in order to explain their origin. Apparently, there is no a definitive model neither a tendency to saturation and the paramagnetic moments persist up to very

**1 µm** 

due to the tensile strength originated during the indentation process.

nucleating at the lateral sides of the imprint, beneath the surface.

Fig. 16. Micrographs of nanoindentation imprints obtained by FE-SEM when the applied load was 10 mN for Bridgman samples, a) Y-123, b) Y-211 and c) Y-123/Y-211 (from Roa et al., 2011b).

Fig. 17. Micrographs of nanoindentation imprints obtained by FE-SEM when the applied load was 10 mN for TSMG samples, a) Y-123, b) Y-211 and c) Y-123/Y-211 (from Roa et al., 2011b).

**1.2 µm 1.2 µm** 

**1.2 µm** 

1.50 m

1.50 m

1.50 m

1.50 m

Fig. 16. Micrographs of nanoindentation imprints obtained by FE-SEM when the applied load was 10 mN for Bridgman samples, a) Y-123, b) Y-211 and c) Y-123/Y-211 (from Roa et

**b**

**b)**

1.50 m

Fig. 17. Micrographs of nanoindentation imprints obtained by FE-SEM when the applied load was 10 mN for TSMG samples, a) Y-123, b) Y-211 and c) Y-123/Y-211 (from Roa et al., 2011b).

**c)**

**a) b)**

**c)**

al., 2011b).

**a)** 

Another common feature in all imprints is the presence of radial cracks emanating from the corners of the imprints. This fracture mechanism, typical for ceramic materials, is nucleated due to the tensile strength originated during the indentation process.

Another common fracture mechanism is the *chipping*, i.e. a partial removal of surface material around the imprint. This phenomenon occurs as a result of Palqmvist cracks nucleating at the lateral sides of the imprint, beneath the surface.

In order to understand the fracture mechanisms that take place under the residual imprint, one nanoindentation performed at 10mN has been visualized by FIB, see **Fig. 18** (from Roa et al., 2011b, 2011d).

Fig. 18. Cross-sectioning and imaging of damage under an indentation imprint performed at 10mN of applied load in Bridgman samples using FIB-SEM. Cross-sectioning in the middle of the imprint (from Roa et al., 2011d).

**Fig. 18** shows a heterogeneous distribution of Y-211 particles around the matrix, with different sizes from 1 to 5 m. Moreover, no cracks or failure events can be appreciated under the nanoindentation imprint, thus implying that the deformation can be attributed to the closing of the oxygenation micro-cracks. The TSMG samples present the same effect –not shown in this work.

Due to all of these fracture mechanisms, the different values reported in the literature and summarized in the **table 3**, are not reproducible.

## **2. Conclusions**

The paramagnetic Meissner effect has been observed in several superconducting materials grown by different techniques since their discovery, and different models and explanations have been proposed in order to explain their origin. Apparently, there is no a definitive model neither a tendency to saturation and the paramagnetic moments persist up to very high magnetic fields.

Magnetical Response and Mechanical Properties of

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With a cross-section, and using FIB technique, we can observe that the samples present a heterogenous distribution of Y-211 particles imbedded in the Y-123 matrix. No generation of porosity or microcracking have been observed, so the deformation mechanisms can be attributed to dislocations movement and twin generation in the *ab*-plane and *c-*axis, respectively.

## **3. Acknowledgments**

The author thanks the Brazilian Ministry of Science and Technology and the State of Rio Grande do Sul, under the Grant PRONEX FAPERGS/CNPq 10/0009-2, and Ministry of Science and Technology, under the Grant CAPES/Procad 059/2007.

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The mechanical properties (hardness, elastic modulus, toughness fracture and yield stress) for YBCO samples can be determined using the nanoindentation technique. This technique allows us to isolate the mechanical behavior of each phase, independently of the growing technique used. The mechanical properties for Bridgman samples are higher than that for TSMG samples. This phenomenon can be attributed to the texturing process, as TSMG process creates a high density of macro- and microcracks during the texture and

The values for yield stress for both texturing methods (Bridgman and TSMG), cannot be

With a cross-section, and using FIB technique, we can observe that the samples present a heterogenous distribution of Y-211 particles imbedded in the Y-123 matrix. No generation of porosity or microcracking have been observed, so the deformation mechanisms can be attributed to dislocations movement and twin generation in the *ab*-plane and *c-*axis,

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**10** 

*Brazil* 

Fábio Teixeira Dias

*Universidade Federal de Pelotas* 

**Magnetic Irreversibility and Resistive** 

**Transition in YBaCuO Superconductors:** 

**Interpretations and Possible Correlations** 

The second generation of high-temperature superconductors (HTSC) have been extensively studied since the discovery. Because of the potential for technological applications several research groups around the world have studied the electric and magnetic properties of HTSC that can provide important information about the performance of these materials when, e.g., magnetic fields are applied in many different

This chapter will be dedicated to describe the main results obtained with YBaCuO superconductors and the interpretations on the resistive transition and magnetic irreversibility limit with special attention to their correlations. It will be presented experimental results obtained by magnetization and magnetotransport measurements in several magnetic field regimes with the purpose of disclosing the connection between the magnetic irreversibility limit and the zero resistance temperature as a function of applied

The discussion will be centered in experimental evidences obtained not only in pure YBaCuO materials but also in doped materials. It will be presented results obtained with single crystals, poly-crystals (sintered and melt-textured) and thin film materials grown by different techniques aiming to discuss the influence of the microstructure in the electric and

Initially it will be done a rapid review about the YBaCuO superconductor and its basic properties, with special attention to magnetic and electric properties, such as magnetic

The next topic to be approached is the central theme proposed in this chapter i.e. the correlation between the magnetic irreversibility and the zero resistance in YBaCuO superconductors and the subsequent fundamental interpretations. Several experimental results will be presented and a special attention will be done in relation to results obtained in grain-oriented samples, which allow us to investigate anisotropic effects of the zero

**1. Introduction** 

applications.

magnetic field.

magnetic properties.

irreversibility and resistive transition.

resistance line and the magnetic irreversibility line.


## **Magnetic Irreversibility and Resistive Transition in YBaCuO Superconductors: Interpretations and Possible Correlations**

Fábio Teixeira Dias *Universidade Federal de Pelotas Brazil* 

## **1. Introduction**

218 Superconductors – Properties, Technology, and Applications

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The second generation of high-temperature superconductors (HTSC) have been extensively studied since the discovery. Because of the potential for technological applications several research groups around the world have studied the electric and magnetic properties of HTSC that can provide important information about the performance of these materials when, e.g., magnetic fields are applied in many different applications.

This chapter will be dedicated to describe the main results obtained with YBaCuO superconductors and the interpretations on the resistive transition and magnetic irreversibility limit with special attention to their correlations. It will be presented experimental results obtained by magnetization and magnetotransport measurements in several magnetic field regimes with the purpose of disclosing the connection between the magnetic irreversibility limit and the zero resistance temperature as a function of applied magnetic field.

The discussion will be centered in experimental evidences obtained not only in pure YBaCuO materials but also in doped materials. It will be presented results obtained with single crystals, poly-crystals (sintered and melt-textured) and thin film materials grown by different techniques aiming to discuss the influence of the microstructure in the electric and magnetic properties.

Initially it will be done a rapid review about the YBaCuO superconductor and its basic properties, with special attention to magnetic and electric properties, such as magnetic irreversibility and resistive transition.

The next topic to be approached is the central theme proposed in this chapter i.e. the correlation between the magnetic irreversibility and the zero resistance in YBaCuO superconductors and the subsequent fundamental interpretations. Several experimental results will be presented and a special attention will be done in relation to results obtained in grain-oriented samples, which allow us to investigate anisotropic effects of the zero resistance line and the magnetic irreversibility line.

Magnetic Irreversibility and Resistive Transition in

Fig. 1. The crystal structure of YBa2Cu3O7-superconductor.

temperature at around 93 K.

**2.2 Growth techniques** 

crystalline samples.

1988b).

YBaCuO Superconductors: Interpretations and Possible Correlations 221

The typical lattice parameters are *a* = 3.82 Å, *b* = 3.89 Å and *c* = 11.68 Å. As shown in Figure 1, the YBa2Cu3O7- superconductor is a perovskite-type cuprate with orthorhombic symmetry and the main structural characteristic is the presence of Cu-O2 double planes separated by an yttrium atomic layer. The Cooper pairs, responsible to superconducting properties, are placed in the CuO2 double planes. The atomic structures between the conducting planes are denominated charge reservoir. The oxygen content can vary between 6 and 7 atoms by unit cell. Deoxygenated systems ( = 1 or simply 6 oxygen atoms per unit cell) has tetragonal structure and insulating anti-ferromagnet behaviour. For the optimum oxygen content of the YBa2Cu3O7- system has the biggest superconducting

Several techniques have been employed to grow superconducting samples of YBa2Cu3O7-. It's possible to prepare samples as bulk (polycrystalline sintered or melt-textured), thin films and single crystals, among others. Initially the sintering procedure was the most promising technique because of its simplicity and possibility to obtain samples in a specific desired shape. The samples obtained by sintering techniques in general present a microstructure characterized by reduced grain size (few m), porosity, cracks or microcracks and an aleatory grain alignment. This microstructure is responsible for the low critical current density values observed in sintered samples as compared to single-

The melt-textured techniques have arisen as an alternative way to produce samples with high critical current density values needed for some technological applications. The pioneer work by Jin et al. (Jin et al., 1988a) set the basis to the melt-textured growth of YBa2Cu3O7- samples. Melt-textured samples exhibit a microstructure characterized by large grains, low-angle grain boundaries, directional alignment (c-axis oriented), reduced porosity and dense structure. This microstructure plays an important role at the high critical density current values observed in melt-textured samples. Figure 2 shows a comparative image of (a) sintered and (b) melt-textured YBa2Cu3O7- samples (Jin et al.,

The idea that the magnetic irreversibility line is a boundary which an electric current may flow without electric resistance comes from the conventional homogeneous metallic superconductors. On the another hand, the high-temperature superconductors such as YBaCuO and other cuprates are usually granular superconductors and in this case the onset of magnetic irreversibility takes place when the first loop of coupled grains is formed. An array of coupled grains is responsible to trap the Josephson flux and establish an irreversible behavior. However in this case the electric transport remains resistive because the well-coupled grain clusters (which are responsible for the magnetic irreversibility) still remain disconnected from each other. It's important to observe that the resistivity vanishes only when the grain coupling strength overcomes the phase entropy and leads to a long-range coherence of the order parameter. As a consequence the electrical resistance in granular superconductors is expected to persist down to temperatures below the irreversibility line and to vanish only after the phase coherence percolates through the whole sample. In materials who exhibit a weak superconducting granularity the zero resistance data are expected to fall closely below the irreversibility line. Although in some cases the zero resistance line falls below the magnetic irreversibility line, some authors admitted that in high magnetic fields, where the flux dynamics is dominated by the intragrain Abrikosov flux, the zero resistance line again follows the irreversibility line. But apparently some results obtained in grain-oriented materials exclude this possibility when high magnetic fields are applied and in this case the expected behavior is not observed.

These important aspects of the correlation between magnetic irreversibility and zero resistance line will be discussed based on the main models and ideas currently accepted.

## **2. YBaCuO superconductor**

It will be described in this section the structural properties, the main growth techniques and some potential technological applications of the YBaCuO superconductor.

## **2.1 Structural properties**

The discovery of superconducting oxides of high critical temperature occurred in 1986 with the pioneer work by Müller and Bednorz (Müller & Bednorz, 1986), who reported the observation of the superconductivity around 30 K in a compound containing La, Ba, Cu and O. The intense interest in these systems occurred a year later, after the work of Wu et al. (Wu et al., 1987) who substituted lanthanum by yttrium in the original formula of Müller and Bednorz. The critical temperature increased to 93 K, well above the boiling point of liquid nitrogen. This material, YBa2Cu3O7-, or simply YBaCuO, was the responsible to inaugurate a new era in the materials research. The relative simplicity in growth of ceramic samples was crucial to mobilize several research groups around the world in a deep study for its physical properties, as well as possible technological applications.

Unlike conventional metallic superconductors, the crystal structure of the HTSC is complex. One of the most important characteristic in high-temperature superconductors is the planar anisotropy. Figure 1 shows the crystal structure of YBa2Cu3O7-superconductor.

Fig. 1. The crystal structure of YBa2Cu3O7-superconductor.

The typical lattice parameters are *a* = 3.82 Å, *b* = 3.89 Å and *c* = 11.68 Å. As shown in Figure 1, the YBa2Cu3O7- superconductor is a perovskite-type cuprate with orthorhombic symmetry and the main structural characteristic is the presence of Cu-O2 double planes separated by an yttrium atomic layer. The Cooper pairs, responsible to superconducting properties, are placed in the CuO2 double planes. The atomic structures between the conducting planes are denominated charge reservoir. The oxygen content can vary between 6 and 7 atoms by unit cell. Deoxygenated systems ( = 1 or simply 6 oxygen atoms per unit cell) has tetragonal structure and insulating anti-ferromagnet behaviour. For the optimum oxygen content of the YBa2Cu3O7- system has the biggest superconducting temperature at around 93 K.

## **2.2 Growth techniques**

220 Superconductors – Properties, Technology, and Applications

The idea that the magnetic irreversibility line is a boundary which an electric current may flow without electric resistance comes from the conventional homogeneous metallic superconductors. On the another hand, the high-temperature superconductors such as YBaCuO and other cuprates are usually granular superconductors and in this case the onset of magnetic irreversibility takes place when the first loop of coupled grains is formed. An array of coupled grains is responsible to trap the Josephson flux and establish an irreversible behavior. However in this case the electric transport remains resistive because the well-coupled grain clusters (which are responsible for the magnetic irreversibility) still remain disconnected from each other. It's important to observe that the resistivity vanishes only when the grain coupling strength overcomes the phase entropy and leads to a long-range coherence of the order parameter. As a consequence the electrical resistance in granular superconductors is expected to persist down to temperatures below the irreversibility line and to vanish only after the phase coherence percolates through the whole sample. In materials who exhibit a weak superconducting granularity the zero resistance data are expected to fall closely below the irreversibility line. Although in some cases the zero resistance line falls below the magnetic irreversibility line, some authors admitted that in high magnetic fields, where the flux dynamics is dominated by the intragrain Abrikosov flux, the zero resistance line again follows the irreversibility line. But apparently some results obtained in grain-oriented materials exclude this possibility when high magnetic fields are applied and in this case

These important aspects of the correlation between magnetic irreversibility and zero resistance line will be discussed based on the main models and ideas currently accepted.

It will be described in this section the structural properties, the main growth techniques and

The discovery of superconducting oxides of high critical temperature occurred in 1986 with the pioneer work by Müller and Bednorz (Müller & Bednorz, 1986), who reported the observation of the superconductivity around 30 K in a compound containing La, Ba, Cu and O. The intense interest in these systems occurred a year later, after the work of Wu et al. (Wu et al., 1987) who substituted lanthanum by yttrium in the original formula of Müller and Bednorz. The critical temperature increased to 93 K, well above the boiling point of liquid nitrogen. This material, YBa2Cu3O7-, or simply YBaCuO, was the responsible to inaugurate a new era in the materials research. The relative simplicity in growth of ceramic samples was crucial to mobilize several research groups around the world in a deep study for its physical properties, as well as possible technological

Unlike conventional metallic superconductors, the crystal structure of the HTSC is complex. One of the most important characteristic in high-temperature superconductors is the planar

anisotropy. Figure 1 shows the crystal structure of YBa2Cu3O7-superconductor.

some potential technological applications of the YBaCuO superconductor.

the expected behavior is not observed.

**2. YBaCuO superconductor** 

**2.1 Structural properties** 

applications.

Several techniques have been employed to grow superconducting samples of YBa2Cu3O7-. It's possible to prepare samples as bulk (polycrystalline sintered or melt-textured), thin films and single crystals, among others. Initially the sintering procedure was the most promising technique because of its simplicity and possibility to obtain samples in a specific desired shape. The samples obtained by sintering techniques in general present a microstructure characterized by reduced grain size (few m), porosity, cracks or microcracks and an aleatory grain alignment. This microstructure is responsible for the low critical current density values observed in sintered samples as compared to singlecrystalline samples.

The melt-textured techniques have arisen as an alternative way to produce samples with high critical current density values needed for some technological applications. The pioneer work by Jin et al. (Jin et al., 1988a) set the basis to the melt-textured growth of YBa2Cu3O7- samples. Melt-textured samples exhibit a microstructure characterized by large grains, low-angle grain boundaries, directional alignment (c-axis oriented), reduced porosity and dense structure. This microstructure plays an important role at the high critical density current values observed in melt-textured samples. Figure 2 shows a comparative image of (a) sintered and (b) melt-textured YBa2Cu3O7- samples (Jin et al., 1988b).

Magnetic Irreversibility and Resistive Transition in

**3. Fundamental properties** 

**3.1 Electrical transport properties** 

and in the granularity and disorder effects.

YBa2Cu3O7- single crystal. From Friedmann et al.

**3.1.2 Granularity and disorder** 

**3.1.1 Anisotropy of the normal state** 

2011).

YBaCuO Superconductors: Interpretations and Possible Correlations 223

large-scale superconducting electric devices for power industry, such as power cables, magnetic energy-storage devices, transformers, fault current limiters, etc. YBa2Cu3O7-

The importance of high-performance in current transport associated with a low-cost process to the fabrication of single grain bulk superconductors is also pointed out as one of main factors to effective applications in engineering devices by Hari Babu et al. (Hari Babu et al.,

This section is dedicated to a review about electrical and magnetic properties of the high-

In this sub-section it will be discussed some fundamental aspects of the electric properties of the high-temperature superconductors, with emphasis in the anisotropy of the normal state

The transport properties of the high-temperature superconductors are highly anisotropic due to their crystalline structure. The CuO2 atomic planes (ab planes) are good conductors but separated by highly resistive sheets, resulting in a strong planar anisotropy, with a low conductivity along the c-axis. The electrical current transport in YBa2Cu3O7- superconductor is metallic-type along the ab plane. The same transport behaviour along the c*-*axis is also achieved in good quality samples. Figure 3 shows the electrical resistivity behaviour along the three crystallographic axes for an untwinned YBa2Cu3O7- single crystal. The strong anisotropic character can be observed with a ratio at room temperatures of a/b 2.5 and c/a 30.

Fig. 3. Electrical resistivity behaviour along the three crystallographic axes for an untwinned

The coherence length (T = 0 K) of an YBa2Cu3O7- superconductor is approximately 12-20 Å along the ab plane and 1-3 Å along the c-axis, while the penetration length is about 1000 Å.

superconductor is pointed as one of the best candidates for these applications.

temperature superconductors, with special attention to the YBa2Cu3O7- system.

Fig. 2. Comparative image of (a) sintered and (b) melt-textured YBaCuO samples. From Jin et al.

Due to the large size of the oriented single domains obtained by melt-textured processes is possible to access directly the physics properties along the c-axis, such as electrical transport. Some works have been done using this important characteristic to study, for example, conductivity fluctuations in melt-textured YBa2Cu3O7- samples (Jurelo et al., 2008; Dias et al., 2009).

YBa2Cu3O7- samples can be grown as thin films and several techniques have been employed. Recently the chemical solution deposition (CSD technique) has been the more attractive technique due to its low cost and high performance of the samples obtained by this method. The growth of YBa2Cu3O7- samples by metal-organic decomposition (MOD) using trifluoroacetate precursor solutions is used to produce high-performance superconductor thin films, as reviewed by some authors (Sandiumenge et al., 2006; Araki & I. Hirabayashi, 2003; Obradors et al., 2004).

### **2.3 Technological applications**

The YBa2Cu3O7- superconductor is the more used high-TC system for technological applications due to its high critical current density and high-performance in applications involving high-magnetic fields. The relative facility in the growth of samples with specific shape permits their use in large scale. Due to the high performance in magnetic and electrical properties the materials prepared by melt-textured and thin films techniques have contributed decisively in this direction.

Several applications involving YBa2Cu3O7- superconductor have been made since its discovery in 1987. Masato Murakami (Murakami, 2000) proposed two definitions for classify the applications of bulk high-temperature superconductors, named passive applications and magnet applications. In passive applications the high-temperature superconductors can be used in superconducting bearing systems, flywheels, cryogenic pumps, hysteresis motors (as rotor), etc. Examples for magnet applications include quasi-permanent magnets, maglev trains, magnetic clamps, motors (as stator), magnetic bumpers, in high-field physics, magnetization devices.

An important review by David Larbalestier et al. (Larbalestier et al., 2001) on power applications of high-TC superconductors points out the need to low-cost fabrication associated to high-performance in electric current transport. This would allow producing large-scale superconducting electric devices for power industry, such as power cables, magnetic energy-storage devices, transformers, fault current limiters, etc. YBa2Cu3O7 superconductor is pointed as one of the best candidates for these applications.

The importance of high-performance in current transport associated with a low-cost process to the fabrication of single grain bulk superconductors is also pointed out as one of main factors to effective applications in engineering devices by Hari Babu et al. (Hari Babu et al., 2011).

## **3. Fundamental properties**

222 Superconductors – Properties, Technology, and Applications

Fig. 2. Comparative image of (a) sintered and (b) melt-textured YBaCuO samples. From Jin

Due to the large size of the oriented single domains obtained by melt-textured processes is possible to access directly the physics properties along the c-axis, such as electrical transport. Some works have been done using this important characteristic to study, for example, conductivity fluctuations in melt-textured YBa2Cu3O7- samples (Jurelo et al., 2008;

YBa2Cu3O7- samples can be grown as thin films and several techniques have been employed. Recently the chemical solution deposition (CSD technique) has been the more attractive technique due to its low cost and high performance of the samples obtained by this method. The growth of YBa2Cu3O7- samples by metal-organic decomposition (MOD) using trifluoroacetate precursor solutions is used to produce high-performance superconductor thin films, as reviewed by some authors (Sandiumenge et al., 2006; Araki &

The YBa2Cu3O7- superconductor is the more used high-TC system for technological applications due to its high critical current density and high-performance in applications involving high-magnetic fields. The relative facility in the growth of samples with specific shape permits their use in large scale. Due to the high performance in magnetic and electrical properties the materials prepared by melt-textured and thin films techniques have

Several applications involving YBa2Cu3O7- superconductor have been made since its discovery in 1987. Masato Murakami (Murakami, 2000) proposed two definitions for classify the applications of bulk high-temperature superconductors, named passive applications and magnet applications. In passive applications the high-temperature superconductors can be used in superconducting bearing systems, flywheels, cryogenic pumps, hysteresis motors (as rotor), etc. Examples for magnet applications include quasi-permanent magnets, maglev trains, magnetic clamps, motors (as stator), magnetic bumpers, in high-field physics,

An important review by David Larbalestier et al. (Larbalestier et al., 2001) on power applications of high-TC superconductors points out the need to low-cost fabrication associated to high-performance in electric current transport. This would allow producing

et al.

Dias et al., 2009).

I. Hirabayashi, 2003; Obradors et al., 2004).

contributed decisively in this direction.

magnetization devices.

**2.3 Technological applications** 

This section is dedicated to a review about electrical and magnetic properties of the hightemperature superconductors, with special attention to the YBa2Cu3O7- system.

## **3.1 Electrical transport properties**

In this sub-section it will be discussed some fundamental aspects of the electric properties of the high-temperature superconductors, with emphasis in the anisotropy of the normal state and in the granularity and disorder effects.

## **3.1.1 Anisotropy of the normal state**

The transport properties of the high-temperature superconductors are highly anisotropic due to their crystalline structure. The CuO2 atomic planes (ab planes) are good conductors but separated by highly resistive sheets, resulting in a strong planar anisotropy, with a low conductivity along the c-axis. The electrical current transport in YBa2Cu3O7- superconductor is metallic-type along the ab plane. The same transport behaviour along the c*-*axis is also achieved in good quality samples. Figure 3 shows the electrical resistivity behaviour along the three crystallographic axes for an untwinned YBa2Cu3O7- single crystal. The strong anisotropic character can be observed with a ratio at room temperatures of a/b 2.5 and c/a 30.

Fig. 3. Electrical resistivity behaviour along the three crystallographic axes for an untwinned YBa2Cu3O7- single crystal. From Friedmann et al.

## **3.1.2 Granularity and disorder**

The coherence length (T = 0 K) of an YBa2Cu3O7- superconductor is approximately 12-20 Å along the ab plane and 1-3 Å along the c-axis, while the penetration length is about 1000 Å.

Magnetic Irreversibility and Resistive Transition in

**3.2 Magnetic properties** 

magnetization.

**3.2.1 Phase diagram** 

YBaCuO Superconductors: Interpretations and Possible Correlations 225

In this section it will be discussed some fundamental aspects of the magnetic properties of the high-temperature superconductors, focusing in the irreversibility behaviour of the

The zero resistivity and the perfect diamagnetism (Meissner effect) are the main characteristics to any material be considered a superconductor. The Figure 5 shows a meanfield type phase diagram for a type-II superconductor, such as YBa2Cu3O7-. Type-II superconductors present a perfect diamagnetism (Meissner effect) just when the applied magnetic field is lower than HC1(T), called as lower critical field. Consequently this region is known as Meissner state. For applied magnetic fields in the region between HC1(T) and HC2(T), the magnetic flux can penetrate inside the superconductor in form of filamentary structures known as vortices. This state is titled as mixed state and the HC1(T) value marks the penetration of the first magnetic flux-line inside the superconductor. When the external magnetic field is higher than HC2(T) (also known as upper critical field), the

superconductivity is suppressed and the material passes to the normal state.

Fig. 5. Mean-field type phase diagram for a type-II superconductor.

superparticles density (superelectrons density or Cooper pairs density).

The Figure 6 shows the mixed state structure in a superconducting sample where Ha is the applied magnetic field (Ha > HC1). The Figure 6(a) shows a lattice of normal cores and the respective associated vortices, while the Figure 6(b) shows the variation with position of

The Figure 6(c) shows the variation of flux density with position. The superparticles density, designed by nS, falls to zero at the centre of each isolated vortex, and the dip in the nS is about two coherence lengths wide. The flux density in not cancelled inside the normal cores, but decreases into a small value in a distance about (London penetration length) away from the normal cores, as shown in Figure 6(c). Each individual vortex is surrounding by the shielding

currents and the total flux generated is just one fluxon 0, given by 2.067x10-15 Weber.

**3.2.2 Mixed state and vortex structure** 

As a consequence the YBa2Cu3O7- system is classified as an extreme type-II superconductor. Therefore defects of the same (or higher) order of magnitude are very important in these materials. The high-temperature superconductors present several intrinsic defects in different size scales: macroscopic (grain-boundaries, porosity, non-superconducting phases, such as the Y2Ba1Cu1O5 phase in some melt-textured YBa2Cu3O7- samples), mesoscopic (twin-planes, dislocations, stacking-faults, columnar defects) and microscopic (oxygen vacancy, substituting atoms). The existence of these defect levels is responsible for the complex topology of the order parameter. Samples with higher disorder degree present an intrinsic and complex granularity, and this inhomogeneous character reflects in the magnetic and transport properties. Disorder is relevant since discovery of the HTSC's, when its presence was associated to the superconducting-glass state (Ebner & Stroud, 1985).

The resistive transition to the superconducting state in granular systems occurs in two stepslike transition, defining two important temperatures, namely TC and TC0, as presented in Figure 4.

Fig. 4. Resistive transition in a granular superconductor.

The region above TC is called "paraconducting" region and the excess of conductivity is attributed to thermal fluctuations of the amplitude of the order parameter. TC is practically coincident with the bulk critical temperature, and the superconductivity stabilizes in homogeneous and mesoscopic regions of the sample (grains). Between TC and TC0 the superconducting grains are weakly coupled to each other by Josephson effect and thermal fluctuations occur in the phase of the order parameter of the grains. This region is called "paracoherent" region and the electrical resistivity is not zero. The long-range superconducting state is achieved at TC0 through a percolation process at the grains. This process controls the activation of weak links between the superconducting grains and is called coherence transition. A detailed investigation of this process is given by Rosenblatt et al. (Rosenblatt et al., 1988). At the critical temperature (TC0) where the coherence transition sets in, the fluctuating phases of the order parameter in each grain couple to each other into a long-range ordered state and a zero resistivity state is established.

## **3.2 Magnetic properties**

224 Superconductors – Properties, Technology, and Applications

As a consequence the YBa2Cu3O7- system is classified as an extreme type-II superconductor. Therefore defects of the same (or higher) order of magnitude are very important in these materials. The high-temperature superconductors present several intrinsic defects in different size scales: macroscopic (grain-boundaries, porosity, non-superconducting phases, such as the Y2Ba1Cu1O5 phase in some melt-textured YBa2Cu3O7- samples), mesoscopic (twin-planes, dislocations, stacking-faults, columnar defects) and microscopic (oxygen vacancy, substituting atoms). The existence of these defect levels is responsible for the complex topology of the order parameter. Samples with higher disorder degree present an intrinsic and complex granularity, and this inhomogeneous character reflects in the magnetic and transport properties. Disorder is relevant since discovery of the HTSC's, when its presence was associated to the superconducting-glass state (Ebner & Stroud, 1985).

The resistive transition to the superconducting state in granular systems occurs in two stepslike transition, defining two important temperatures, namely TC and TC0, as presented in

The region above TC is called "paraconducting" region and the excess of conductivity is attributed to thermal fluctuations of the amplitude of the order parameter. TC is practically coincident with the bulk critical temperature, and the superconductivity stabilizes in homogeneous and mesoscopic regions of the sample (grains). Between TC and TC0 the superconducting grains are weakly coupled to each other by Josephson effect and thermal fluctuations occur in the phase of the order parameter of the grains. This region is called "paracoherent" region and the electrical resistivity is not zero. The long-range superconducting state is achieved at TC0 through a percolation process at the grains. This process controls the activation of weak links between the superconducting grains and is called coherence transition. A detailed investigation of this process is given by Rosenblatt et al. (Rosenblatt et al., 1988). At the critical temperature (TC0) where the coherence transition sets in, the fluctuating phases of the order parameter in each grain couple to each other into

Fig. 4. Resistive transition in a granular superconductor.

a long-range ordered state and a zero resistivity state is established.

Figure 4.

In this section it will be discussed some fundamental aspects of the magnetic properties of the high-temperature superconductors, focusing in the irreversibility behaviour of the magnetization.

## **3.2.1 Phase diagram**

The zero resistivity and the perfect diamagnetism (Meissner effect) are the main characteristics to any material be considered a superconductor. The Figure 5 shows a meanfield type phase diagram for a type-II superconductor, such as YBa2Cu3O7-. Type-II superconductors present a perfect diamagnetism (Meissner effect) just when the applied magnetic field is lower than HC1(T), called as lower critical field. Consequently this region is known as Meissner state. For applied magnetic fields in the region between HC1(T) and HC2(T), the magnetic flux can penetrate inside the superconductor in form of filamentary structures known as vortices. This state is titled as mixed state and the HC1(T) value marks the penetration of the first magnetic flux-line inside the superconductor. When the external magnetic field is higher than HC2(T) (also known as upper critical field), the superconductivity is suppressed and the material passes to the normal state.

Fig. 5. Mean-field type phase diagram for a type-II superconductor.

### **3.2.2 Mixed state and vortex structure**

The Figure 6 shows the mixed state structure in a superconducting sample where Ha is the applied magnetic field (Ha > HC1). The Figure 6(a) shows a lattice of normal cores and the respective associated vortices, while the Figure 6(b) shows the variation with position of superparticles density (superelectrons density or Cooper pairs density).

The Figure 6(c) shows the variation of flux density with position. The superparticles density, designed by nS, falls to zero at the centre of each isolated vortex, and the dip in the nS is about two coherence lengths wide. The flux density in not cancelled inside the normal cores, but decreases into a small value in a distance about (London penetration length) away from the normal cores, as shown in Figure 6(c). Each individual vortex is surrounding by the shielding currents and the total flux generated is just one fluxon 0, given by 2.067x10-15 Weber.

Magnetic Irreversibility and Resistive Transition in

superconductor.

behaviour.

the sample.

(Müller et al., 1987).

**3.2.3 Magnetic irreversibilities** 

**3.2.3.1 Magnetic irreversibility line** 

YBaCuO Superconductors: Interpretations and Possible Correlations 227

Fig. 7. Representative scheme showing intragranular and intergranular vortices in a type-II

The magnetization of high-TC superconductors shows an irreversible behaviour due to flux pinning combined with thermal activation of the vortices. In the next sections will be discussed the origin, the characteristics and some theoretical models used to explain this

Through the realization of experiments based on ZFC (zero-field cooling) and FC (field-cooling) procedures in a superconducting sample it's possible to observe strong magnetic irreversibility effects, such as presented by Figure 8, due to flux pinning into

Below a temperature called irreversibility temperature (Tirr), the ZFC and FC curves present distinct behaviours. Below Tirr the vortices have its mobility strongly decreased due to pinning centres. As Tirr is dependent on the magnetic applied field, it´s possible to obtain a H versus T phase diagram similar to the one presented in Figure 9, where the line separate the diagram in two distinct regions: a region of high temperatures (T > Tirr) where the magnetization is reversible, and a region of low temperatures (T < Tirr) where the magnetization is irreversible. This line is known as magnetic irreversibility line or simply irreversibility line (IL). The study of the irreversibility line was originated with the pioneer work by Müller et al. in granular samples of La2-xBaxCuO4-superconductor

Fig. 6. The mixed state structure in a type-II superconductor. The Figure (a) shows a lattice of normal cores and the respective vortices, while the Figure (b) shows the variation with position of superparticles density and the Figure (c) shows the variation of flux density with position. From Rose-Innes & Rhoderick.

The interaction between vortices is repulsive and mediated by the Lorenz force. In a homogeneous superconductor the cores are arranged in a hexagonal (triangular) periodic array due to lower energy state. This array is known as Abrikosov lattice and can be observed in the Figure 6(a). In granular superconductors the vortices placed inside the superconducting grains are denominated intragranular vortices and are arranged in this lattice type.

The vortices can be pinned by defects of the crystal lattice. This effect is known as flux pinning and the mechanism can be improved by introducing microstructural or compositional changes like irradiation, chemical doping or precipitates.

In an isotropic and perfect superconductor (without the presence of defects) the vortices can be arranged in an Abrikosov lattice (hexagonal lattice). In a granular material, the superconducting grains can be coupled by Josephson effect, and based on this idea John R. Clem proposed a theoretical model to describe a granular superconductor using the Josephson effect as starting point (Clem, 1988). In this model a granular superconductor can be represented as anisotropic grains coupled by Josephson junctions (junctions based on Josephson effect).

The magnetic field can penetrate into a granular superconductor in the region between the grains, in the form of intergranular vortices, known as Josephson vortices. An intergranular Josephson vortex transports just one fluxon, similar to intragranular Abrikosov vortex and the magnetic structures are similar. The main difference is the region where both vortices are located. The Figure 7 shows a scheme of both vortices in a superconducting granular sample.

Fig. 6. The mixed state structure in a type-II superconductor. The Figure (a) shows a lattice of normal cores and the respective vortices, while the Figure (b) shows the variation with position of superparticles density and the Figure (c) shows the variation of flux density with

The interaction between vortices is repulsive and mediated by the Lorenz force. In a homogeneous superconductor the cores are arranged in a hexagonal (triangular) periodic array due to lower energy state. This array is known as Abrikosov lattice and can be observed in the Figure 6(a). In granular superconductors the vortices placed inside the superconducting grains are denominated intragranular vortices and are arranged in this

The vortices can be pinned by defects of the crystal lattice. This effect is known as flux pinning and the mechanism can be improved by introducing microstructural or

In an isotropic and perfect superconductor (without the presence of defects) the vortices can be arranged in an Abrikosov lattice (hexagonal lattice). In a granular material, the superconducting grains can be coupled by Josephson effect, and based on this idea John R. Clem proposed a theoretical model to describe a granular superconductor using the Josephson effect as starting point (Clem, 1988). In this model a granular superconductor can be represented as anisotropic grains coupled by Josephson junctions (junctions based on

The magnetic field can penetrate into a granular superconductor in the region between the grains, in the form of intergranular vortices, known as Josephson vortices. An intergranular Josephson vortex transports just one fluxon, similar to intragranular Abrikosov vortex and the magnetic structures are similar. The main difference is the region where both vortices are located. The Figure 7 shows a scheme of both vortices in a superconducting granular

compositional changes like irradiation, chemical doping or precipitates.

position. From Rose-Innes & Rhoderick.

lattice type.

Josephson effect).

sample.

Fig. 7. Representative scheme showing intragranular and intergranular vortices in a type-II superconductor.

## **3.2.3 Magnetic irreversibilities**

The magnetization of high-TC superconductors shows an irreversible behaviour due to flux pinning combined with thermal activation of the vortices. In the next sections will be discussed the origin, the characteristics and some theoretical models used to explain this behaviour.

## **3.2.3.1 Magnetic irreversibility line**

Through the realization of experiments based on ZFC (zero-field cooling) and FC (field-cooling) procedures in a superconducting sample it's possible to observe strong magnetic irreversibility effects, such as presented by Figure 8, due to flux pinning into the sample.

Below a temperature called irreversibility temperature (Tirr), the ZFC and FC curves present distinct behaviours. Below Tirr the vortices have its mobility strongly decreased due to pinning centres. As Tirr is dependent on the magnetic applied field, it´s possible to obtain a H versus T phase diagram similar to the one presented in Figure 9, where the line separate the diagram in two distinct regions: a region of high temperatures (T > Tirr) where the magnetization is reversible, and a region of low temperatures (T < Tirr) where the magnetization is irreversible. This line is known as magnetic irreversibility line or simply irreversibility line (IL). The study of the irreversibility line was originated with the pioneer work by Müller et al. in granular samples of La2-xBaxCuO4-superconductor (Müller et al., 1987).

Magnetic Irreversibility and Resistive Transition in

**3.2.3.2 Theoretical models** 

irreversibility line (IL). *3.2.3.2.1 Giant flux creep* 

superconductors.

pinned.

*3.2.3.2.2 Vortex melting* 

follow the power law

Lindemann criterion (Lindemann, 1910).

where *t = T/TC* is the reduced temperature.

diagram obeys the power-law

YBaCuO Superconductors: Interpretations and Possible Correlations 229

This section is dedicated to describe some of the main models used to explain the magnetic

The model was proposed by Yeshurun and Malozemoff (Yeshurun & Malozemoff, 1988) based on the flux creep model initially developed by Anderson and Kim (Anderson & Kim, 1964). This model is based on the thermal activation effects of the vortex lattice. The model assumes that a vortex can be thermally activated over a pinning barrier even if the Lorentz force exceeds the pinning force. The main differences between both models are the high temperature and the low energy of the vortex pinning presented by the high-temperature

According to the authors the experimental behaviour presented by the IL in a H versus T

2/3 *H ( t)* 1 , (1)

According to the giant flux creep model the IL is essentially a depinning line in a H versus T phase diagram. Above the IL the vortices can move freely resulting in a zero critical current density while below the IL a finite value occurs once the vortices are

According to this model the thermal fluctuations are responsible by the melting of the Abrikosov lattice, and the IL behaves as a boundary between two regions, namely vortex solid and vortex liquid. The instability produced by thermal fluctuations results in a second order phase transition of the vortex lattice at a temperature called vortex lattice melting temperature, or TM. The aleatory dislocations produced by instabilities can promote collisions and loss of correlations between vortices. According to Nelson and Seung (Nelson & Seung, 1989), when the lattice melts two new regimes can arise, namely disentangled flux liquid and entangled flux liquid. The criterion used to define the vortex melting is known as

The Figure 10 shows a representation of the vortex lattice melting phases. The disentangled flux liquid is characterized by high temperatures and low flux-line densities, as represented in Figure 10(b). The entangled flux liquid is reached when the flux-line density is high, as presented in Fig. 10(c). According to Houghton et al. (Houghton et al., 1989) the melting temperature (TM) is slightly lower than the critical temperature (TC) and the behaviour

<sup>2</sup> *( t) H* 1 , (2)

where *t = Tirr/TC* is the reduced temperature and *Tirr* is the irreversibility temperature.

Fig. 8. Magnetic irreversibility effects in a superconducting sample obtained through the realization of ZFC (zero-field cooling) and FC (field-cooling) procedures.

Fig. 9. The anisotropic character of the magnetic irreversibility line in a melt-textured sample.

The Figure 9 shows the anisotropic character of the magnetic irreversibility, attributed to intrinsic planar structure of the high-temperature superconductors. When H // ab the intrinsic pinning mechanism is evidenced and the vortices have their mobility restricted to the ab plane, reinforcing the effective pinning. By another hand, when H // c the lattice take the form of a stacked two-dimensional pancake-type vortices confined to the ab planes weakly coupled. At low temperatures the vortex lattice can be stabilized in a threedimensional arrangement, but at high temperatures the thermal depinning effects passes to be relevant and a crossover from 3D to 2D occurs, resulting in a two-dimensional lattice of pancake vortices (Vinokur et al., 1990).

## **3.2.3.2 Theoretical models**

228 Superconductors – Properties, Technology, and Applications

Fig. 8. Magnetic irreversibility effects in a superconducting sample obtained through the

Fig. 9. The anisotropic character of the magnetic irreversibility line in a melt-textured

The Figure 9 shows the anisotropic character of the magnetic irreversibility, attributed to intrinsic planar structure of the high-temperature superconductors. When H // ab the intrinsic pinning mechanism is evidenced and the vortices have their mobility restricted to the ab plane, reinforcing the effective pinning. By another hand, when H // c the lattice take the form of a stacked two-dimensional pancake-type vortices confined to the ab planes weakly coupled. At low temperatures the vortex lattice can be stabilized in a threedimensional arrangement, but at high temperatures the thermal depinning effects passes to be relevant and a crossover from 3D to 2D occurs, resulting in a two-dimensional lattice of

sample.

pancake vortices (Vinokur et al., 1990).

realization of ZFC (zero-field cooling) and FC (field-cooling) procedures.

This section is dedicated to describe some of the main models used to explain the magnetic irreversibility line (IL).

## *3.2.3.2.1 Giant flux creep*

The model was proposed by Yeshurun and Malozemoff (Yeshurun & Malozemoff, 1988) based on the flux creep model initially developed by Anderson and Kim (Anderson & Kim, 1964). This model is based on the thermal activation effects of the vortex lattice. The model assumes that a vortex can be thermally activated over a pinning barrier even if the Lorentz force exceeds the pinning force. The main differences between both models are the high temperature and the low energy of the vortex pinning presented by the high-temperature superconductors.

According to the authors the experimental behaviour presented by the IL in a H versus T diagram obeys the power-law

$$H^{2/3} \propto (1-t) \tag{1}$$

where *t = Tirr/TC* is the reduced temperature and *Tirr* is the irreversibility temperature.

According to the giant flux creep model the IL is essentially a depinning line in a H versus T phase diagram. Above the IL the vortices can move freely resulting in a zero critical current density while below the IL a finite value occurs once the vortices are pinned.

## *3.2.3.2.2 Vortex melting*

According to this model the thermal fluctuations are responsible by the melting of the Abrikosov lattice, and the IL behaves as a boundary between two regions, namely vortex solid and vortex liquid. The instability produced by thermal fluctuations results in a second order phase transition of the vortex lattice at a temperature called vortex lattice melting temperature, or TM. The aleatory dislocations produced by instabilities can promote collisions and loss of correlations between vortices. According to Nelson and Seung (Nelson & Seung, 1989), when the lattice melts two new regimes can arise, namely disentangled flux liquid and entangled flux liquid. The criterion used to define the vortex melting is known as Lindemann criterion (Lindemann, 1910).

The Figure 10 shows a representation of the vortex lattice melting phases. The disentangled flux liquid is characterized by high temperatures and low flux-line densities, as represented in Figure 10(b). The entangled flux liquid is reached when the flux-line density is high, as presented in Fig. 10(c). According to Houghton et al. (Houghton et al., 1989) the melting temperature (TM) is slightly lower than the critical temperature (TC) and the behaviour follow the power law

$$\left(\mathbf{1} - \mathbf{t}\right)^{2} \approx \mathbf{H}\_{\text{\textquotedblleft}}\tag{2}$$

where *t = T/TC* is the reduced temperature.

Magnetic Irreversibility and Resistive Transition in

Almeida & Thouless, 1978), and it's given by

where 

where 

law

where 

when H 0 and H = 0. *3.2.3.2.4 Vortex glass* 

the centres of grains *i* and *j*.

cos *ij <sup>i</sup> <sup>j</sup> ij*

YBaCuO Superconductors: Interpretations and Possible Correlations 231

phase difference of the Ginzburg-Landau parameter on the grains *i* and *j*. The phase

0 <sup>2</sup><sup>π</sup> . *j*

This model predicts the existence of a low temperature phase called superconducting glass, where disorder and frustration are dominants. The irreversibility line (IL) can be interpreted as a separation between two phases: namely superconducting glass and a system of decoupled grains. According to Müller et al. (Müller et al., 1987) it's possible to correlate the functionality of the IL with the same functionality used in spin glass systems. Using this analogy Müller et al. proposed that the behaviour of the IL at low applied magnetic fields in high-TC superconductors can be described by de Almeida-Thouless type power law (de

1

12 shows experimental results obtained in a La2BaCuO4- sample. The solid curve represents a numerical fit using equation (5). Another similarity occurs when higher magnetic fields are applied. In this case it's possible to observe a crossover from de Almeida-Thouless to Gabay-Toulouse (Gabay & Toulouse, 1981), where the behaviour can be described by the power

*T (H) H=H*

*0*

1

0

*g*

*g*

*T( )* 

 *= 3/2* and *Tg(H)* can be interpret as *Tirr(H)* in superconducting materials. The Figure

*γ*

1 2

0 *irr irr T (H) H = <sup>β</sup> T()* 

is a constant and *Tirr(H)* and *Tirr(0)* are the irreversibility temperatures obtained

The vortex glass model was proposed by Fisher (Fisher, 1989) to describe the crossover from reversible to irreversible magnetization in high-TC superconducting oxides using disorder effects. This model predicts the existence of a vortex glass phase between the Meissner state and the line defined by the temperature TG(H), as showed in Figure 13. The Abrikosov lattice in the vortex glass state has no long-range symmetry and the vortices freeze in a 2D lattice exhibiting a low-range order. The disorder and frustration are related to aleatory pinning centres distribution. As the temperature increases the vortex glass can be

*i A = A dl φ*

*<sup>0</sup>* is the elementary flux quantum (fluxon), and the line integral is evaluated between

*H= J ( θ θ A )* , (3)

*i - <sup>j</sup>* is the

, (4)

, (5)

, (6)

*i, j*

where *Jij* is the phase coupling energies between neighboring grains *i* and *j* and

*ij*

displacements *Aij*, caused by the applied magnetic field, are given by

Fig. 10. Representation of the vortex lattice melting phases, showing (a) the Abrikosov lattice, (b) the disentangled and (c) the entangled flux liquid. From Nelson & Seung.

The Figure 11 shows experimental results obtained in an untwinned single crystal of YBa2Cu3O7-. The solid curve represents a fitting using equation (2).

Fig. 11. H versus T phase diagram for the flux melting regime obtained in an untwinned single crystal of YBa2Cu3O7-. The solid curve represents a fitting using equation (2). From Farrell et al.

#### *3.2.3.2.3 Superconducting glass*

This model was based on the propositions of Ebner and Stroud (Ebner & Stroud, 1985) to study the diamagnetic susceptibility of superconducting clusters. The superconducting glass model proposes the existence of a disordered array of superconducting grains weakly coupled by Josephson effect or proximity effect. The Hamiltonian used to describe a system of weakly coupled grains is given by

Fig. 10. Representation of the vortex lattice melting phases, showing (a) the Abrikosov lattice, (b) the disentangled and (c) the entangled flux liquid. From Nelson & Seung.

YBa2Cu3O7-. The solid curve represents a fitting using equation (2).

Farrell et al.

*3.2.3.2.3 Superconducting glass* 

of weakly coupled grains is given by

The Figure 11 shows experimental results obtained in an untwinned single crystal of

Fig. 11. H versus T phase diagram for the flux melting regime obtained in an untwinned single crystal of YBa2Cu3O7-. The solid curve represents a fitting using equation (2). From

This model was based on the propositions of Ebner and Stroud (Ebner & Stroud, 1985) to study the diamagnetic susceptibility of superconducting clusters. The superconducting glass model proposes the existence of a disordered array of superconducting grains weakly coupled by Josephson effect or proximity effect. The Hamiltonian used to describe a system

$$H = -\sum\_{i,j} I\_{ij} \cos(\theta\_i - \theta\_j - A\_{ij}) \, \, \, \, \, \tag{3}$$

where *Jij* is the phase coupling energies between neighboring grains *i* and *j* and *i - <sup>j</sup>* is the phase difference of the Ginzburg-Landau parameter on the grains *i* and *j*. The phase displacements *Aij*, caused by the applied magnetic field, are given by

$$A\_{i\circ} = \frac{2\text{m}}{\text{op}\_0} \Big| \vec{A}.d\vec{l} \; \tag{4}$$

where *<sup>0</sup>* is the elementary flux quantum (fluxon), and the line integral is evaluated between the centres of grains *i* and *j*.

This model predicts the existence of a low temperature phase called superconducting glass, where disorder and frustration are dominants. The irreversibility line (IL) can be interpreted as a separation between two phases: namely superconducting glass and a system of decoupled grains. According to Müller et al. (Müller et al., 1987) it's possible to correlate the functionality of the IL with the same functionality used in spin glass systems. Using this analogy Müller et al. proposed that the behaviour of the IL at low applied magnetic fields in high-TC superconductors can be described by de Almeida-Thouless type power law (de Almeida & Thouless, 1978), and it's given by

$$H = H\_0 \left( 1 - \frac{T\_{\text{g}}(H)}{T\_{\text{g}}(0)} \right)^{\text{y}},\tag{5}$$

where  *= 3/2* and *Tg(H)* can be interpret as *Tirr(H)* in superconducting materials. The Figure 12 shows experimental results obtained in a La2BaCuO4- sample. The solid curve represents a numerical fit using equation (5). Another similarity occurs when higher magnetic fields are applied. In this case it's possible to observe a crossover from de Almeida-Thouless to Gabay-Toulouse (Gabay & Toulouse, 1981), where the behaviour can be described by the power law

$$H = \beta \left( 1 - \frac{T\_{irr}(H)}{T\_{irr}(0)} \right)^{1/2},\tag{6}$$

where is a constant and *Tirr(H)* and *Tirr(0)* are the irreversibility temperatures obtained when H 0 and H = 0.

#### *3.2.3.2.4 Vortex glass*

The vortex glass model was proposed by Fisher (Fisher, 1989) to describe the crossover from reversible to irreversible magnetization in high-TC superconducting oxides using disorder effects. This model predicts the existence of a vortex glass phase between the Meissner state and the line defined by the temperature TG(H), as showed in Figure 13. The Abrikosov lattice in the vortex glass state has no long-range symmetry and the vortices freeze in a 2D lattice exhibiting a low-range order. The disorder and frustration are related to aleatory pinning centres distribution. As the temperature increases the vortex glass can be

Magnetic Irreversibility and Resistive Transition in

**transition** 

**4.1 Brief considerations** 

YBaCuO Superconductors: Interpretations and Possible Correlations 233

This section will be dedicated to present and to discuss experimental results on the magnetic irreversibility line, on the zero resistance line and on the respective correlations in hightemperature superconductors. The results will be focused in the YBa2Cu3O7-superconductor.

The magnetic irreversibility line in homogeneous superconductors defines a boundary in the H versus T plane below which the magnetization is irreversible and up to which a nonzero electric current can flow without any electrical resistance. Above this line the magnetization is reversible and all electric transport is resistive due to dissipation by flux dynamic effects. Very clean and well-oxygenated YBa2Cu3O7- single crystals can be considered homogeneous superconductors and in this case the electrical resistivity is expected to vanish at the irreversibility line and even to comply with the planar anisotropy of the magnetic irreversibility. However high-temperature superconducting cuprates are generally inhomogeneous

superconductors (Vieira & Schaf, 2002) or intrinsically inhomogeneous (Dagotto, 2005).

after the achievement of a long-range coherence state.

The magnetic irreversibility and the electrical resistivity in granular superconductors do not depend on the same parts of the superconducting sample. While the irreversibility depends on well-coupled grain clusters, the electrical resistance depends on grain arrays traversing the whole sample. Along such long-range paths, zero electrical resistance can be attained only at some temperature below the irreversibility temperature (Rosenblatt et al., 1988). On the other hand, at magnetic fields above several kOe the magnetic field penetrates the grains and the magnetic irreversibility is dominated by the Abrikosov flux dynamics (intragrain flux) while the electrical resistivity is still ruled by the grain junctions and can vanish only

Magnetic irreversibility studies on pure and doped YBa2Cu3O7- single crystals (Schaf et al., 2001, 2002) show the signature of superconducting granularity in the low-field region, dominated by Josephson flux (intergrain flux). Although in some cases the data for zero resistance falls below the magnetic irreversibility line, the authors admitted that in high magnetic fields, where the flux dynamics is dominated by the Abrikosov flux, the zero

High-quality melt-textured YBa2Cu3O7- materials normally do not exhibit the signature of superconducting granularity, in spite of the polycrystalline structure. The linking between the crystallites is so strong that in some cases the coupling between superconducting grains

The results in this section are presented for several YBa2Cu3O7- samples grown by different techniques, and some of experimental results and discussions were previously published by

The Figure 14 shows results in an YBa2Cu2.97Zn0.03O7- single crystal grown by self-flux method. The continuous lines represent the irreversibility lines, Tirr(H), for H // ab and

resistance line again follows the irreversibility line (Pureur et al., 2000, 2001).

occurs almost simultaneously with the superconducting transition.

**4.2 Experimental results in YBaCuO superconductors** 

Schaf et al. (Schaf et al., 2008) and Dias et al. (Dias et al., 2008).

**4. Correlations between the magnetic irreversibility and the resistive** 

destabilized by thermal fluctuations and it melts above the TG(H) line, defining a state known as vortex liquid, as showed in Figure 13. The TG(H) line can be identified as the irreversibility line (IL).

Fig. 12. Irreversibility line obtained in a La2BaCuO4- sample. The solid curve represents a numerical fit using equation (5). From Müller et al.

Fig. 13. Phase diagram for the vortex glass model. The TG(H) line can be identified as the irreversibility line (IL). From Fisher.

#### *3.2.3.2.5 Bose glass*

The Bose glass theory was developed by Nelson and Vinokur (Nelson & Vinokur, 1992) to treat the effects due to flux pinning by correlated defects (columnar defects, twin planes, grain boundaries, etc.). For a high-temperature superconductor with strongly correlated disorder the model predicts a H versus T diagram divided into three phases: superfluid, Bose glass and Mott insulator.

The superfluid phase is characterized by high temperature and viscous motion of vortices, which can jump across the correlated defects trough the sample. The Bose glass phase is essentially characterized by the strongly anisotropic pinning due to extended and correlated defects in the sample. This behaviour is strongly dependent on the alignment between the applied magnetic field and the direction of the defects dispersed in the superconducting sample. Misalignments greater than 5° may remove this phase in accordance with experimental results obtained by Safar et al. (Safar et al., 1996) in twinned YBa2Cu3O7- single crystals. The Mott insulator phase occurs at low temperatures and can be characterized by a highly non-linear vortex dynamics governed by similar properties to the Meissner state.

## **4. Correlations between the magnetic irreversibility and the resistive transition**

This section will be dedicated to present and to discuss experimental results on the magnetic irreversibility line, on the zero resistance line and on the respective correlations in hightemperature superconductors. The results will be focused in the YBa2Cu3O7-superconductor.

## **4.1 Brief considerations**

232 Superconductors – Properties, Technology, and Applications

destabilized by thermal fluctuations and it melts above the TG(H) line, defining a state known as vortex liquid, as showed in Figure 13. The TG(H) line can be identified as the

Fig. 12. Irreversibility line obtained in a La2BaCuO4- sample. The solid curve represents a

Fig. 13. Phase diagram for the vortex glass model. The TG(H) line can be identified as the

The Bose glass theory was developed by Nelson and Vinokur (Nelson & Vinokur, 1992) to treat the effects due to flux pinning by correlated defects (columnar defects, twin planes, grain boundaries, etc.). For a high-temperature superconductor with strongly correlated disorder the model predicts a H versus T diagram divided into three phases: superfluid,

The superfluid phase is characterized by high temperature and viscous motion of vortices, which can jump across the correlated defects trough the sample. The Bose glass phase is essentially characterized by the strongly anisotropic pinning due to extended and correlated defects in the sample. This behaviour is strongly dependent on the alignment between the applied magnetic field and the direction of the defects dispersed in the superconducting sample. Misalignments greater than 5° may remove this phase in accordance with experimental results obtained by Safar et al. (Safar et al., 1996) in twinned YBa2Cu3O7- single crystals. The Mott insulator phase occurs at low temperatures and can be characterized by a highly non-linear vortex dynamics governed by similar properties to the Meissner state.

numerical fit using equation (5). From Müller et al.

irreversibility line (IL). From Fisher.

Bose glass and Mott insulator.

*3.2.3.2.5 Bose glass* 

irreversibility line (IL).

The magnetic irreversibility line in homogeneous superconductors defines a boundary in the H versus T plane below which the magnetization is irreversible and up to which a nonzero electric current can flow without any electrical resistance. Above this line the magnetization is reversible and all electric transport is resistive due to dissipation by flux dynamic effects. Very clean and well-oxygenated YBa2Cu3O7- single crystals can be considered homogeneous superconductors and in this case the electrical resistivity is expected to vanish at the irreversibility line and even to comply with the planar anisotropy of the magnetic irreversibility. However high-temperature superconducting cuprates are generally inhomogeneous superconductors (Vieira & Schaf, 2002) or intrinsically inhomogeneous (Dagotto, 2005).

The magnetic irreversibility and the electrical resistivity in granular superconductors do not depend on the same parts of the superconducting sample. While the irreversibility depends on well-coupled grain clusters, the electrical resistance depends on grain arrays traversing the whole sample. Along such long-range paths, zero electrical resistance can be attained only at some temperature below the irreversibility temperature (Rosenblatt et al., 1988). On the other hand, at magnetic fields above several kOe the magnetic field penetrates the grains and the magnetic irreversibility is dominated by the Abrikosov flux dynamics (intragrain flux) while the electrical resistivity is still ruled by the grain junctions and can vanish only after the achievement of a long-range coherence state.

Magnetic irreversibility studies on pure and doped YBa2Cu3O7- single crystals (Schaf et al., 2001, 2002) show the signature of superconducting granularity in the low-field region, dominated by Josephson flux (intergrain flux). Although in some cases the data for zero resistance falls below the magnetic irreversibility line, the authors admitted that in high magnetic fields, where the flux dynamics is dominated by the Abrikosov flux, the zero resistance line again follows the irreversibility line (Pureur et al., 2000, 2001).

High-quality melt-textured YBa2Cu3O7- materials normally do not exhibit the signature of superconducting granularity, in spite of the polycrystalline structure. The linking between the crystallites is so strong that in some cases the coupling between superconducting grains occurs almost simultaneously with the superconducting transition.

## **4.2 Experimental results in YBaCuO superconductors**

The results in this section are presented for several YBa2Cu3O7- samples grown by different techniques, and some of experimental results and discussions were previously published by Schaf et al. (Schaf et al., 2008) and Dias et al. (Dias et al., 2008).

The Figure 14 shows results in an YBa2Cu2.97Zn0.03O7- single crystal grown by self-flux method. The continuous lines represent the irreversibility lines, Tirr(H), for H // ab and

Magnetic Irreversibility and Resistive Transition in

YBaCuO Superconductors: Interpretations and Possible Correlations 235

Fig. 15. Irreversibility lines obtained in a melt-textured YBa2Cu3O7-, together with the zero resistance data TC0. The closed circles are for fields applied parallel to the ab plane and the

The twinning domains in the investigated melt-textured sample are nearly random in the ab plane and their direction also must be scattered somewhat about the average c-axis, due to the misalignment of the c-axis. Therefore, while flux pinning by the twinning planes certainly contributes somewhat for fields applied along the c-axis, its contribution for fields along the ab plane must be irrelevant. In the Figure 15 the irreversibility data show a much reduced anisotropy, due to the spreading out of the c-axes of the different crystallites and

Similar results (not published) are showed in the Figure 16, for a YBa2Cu3O7- thin film grown by chemical solution deposition. It can be observed that the zero resistance data for H // c and H J split away from the irreversibility line (H // ab) towards lower

The data in the Figure 17 are representative for a polycrystalline YBa1.75Sr0.25Cu3O7 sample grown by conventional sintered technique using the standard solid state reaction method (Schaf et al., 2008). This polycrystalline sample exhibits a strong superconducting granularity, as shown by the presence of the de Almeida-Thouless and Gabay-Toulouse regimes in the low-field irreversibility data. The asterisks in the Figure 17 denote the approximate superconducting temperature transition (TC); the open circles are the irreversibility data; the closed circles are the zero resistance data (TC0) and the continuous lines are fits to de Almeida-Thouless, Gabay-Toulouse and giant-flux creep

In spite of the magnetic field being applied parallel to the measuring current, H // J, the zero resistance data of this polycrystalline sample fall considerably below the irreversibility line already in low magnetic fields and split away from this line steeply when the applied magnetic field increases. In polycrystalline samples the current flow through the grain aggregate is highly dispersive and therefore the H // J configuration is only a microscopic approximation. The data in the Figure 17 also show no indication the zero resistance data might come back and meet the irreversibility line somewhere in

open circles are for fields along the c-axis. From Schaf et al.

the consequent noncoplanarity of the ab planes.

temperatures.

power laws.

higher magnetic fields.

H // c and are fits to the power law predicted by the giant-flux-creep theory, while the circles represent the zero resistance data, TC0(H), for H // ab and H // c. The inset in the Figure 14 highlights the data at low magnetic fields where the continuous lines are fits to the de Almeida-Thouless and Gabay-Toulouse power laws. The Tirr(H) data show a weak superconducting granularity and consequently the zero resistance data are expected to fall closely below the irreversibility line, as showed in the Figure 14. While at high magnetic fields the TC0(H) data fall about the irreversibility lines, in the low-field region they fall systematically underneath the irreversibility line for both field configurations (see inset).

Fig. 14. Irreversibility lines in an YBa2Cu2.97Zn0.03O7- single crystal, together with the zero resistance data TC0. The closed circles are for fields applied parallel to the ab plane and the open circles are for fields along the c-axis. The inset highlights the data in low fields. From Schaf et al.

The data in the Figure 15 illustrates the case of a melt-textured YBa2Cu3O7- sample grown by top-seeding technique and containing 30 wt% of Y2BaCuO5 (Y211 phase) and 1 wt% of CeO2. The purpose of the Y211 phase is to introduce pinning centers and the CeO2 physically stabilizes the melt at high temperatures during the growth, to limit the size of the Y211 particles and to improve the distribution of the Y211 phase. This procedure strongly enhances the flux pinning potential in melt-textured YBa2Cu3O7- samples. On the other hand, high-quality melt-textured samples in general do not display superconducting granularity. However, the high concentration of Y211 phase causes considerable misalignment of the c-axis, thereby encumbers the grain coupling and reduces the planar anisotropy (Carrillo et al., 2000). The weak superconducting granularity presented by the melt-textured sample, whose results are presented in the Figure 15, arises mainly from misalignment of the c*-*axis, resulting in a reduced planar anisotropy.

In the Figure 15 the irreversibility lines (H // ab and H // c) are represented by the experimental data and the continuous lines are fits to the power law predicted by the giantflux-creep theory. The measuring current in the zero resistance data flows along the same ab planes for both field orientations (H // ab and H // c). While for H // ab plane and H // J the zero resistance data (closed circles) fall closely underneath the corresponding irreversibility line, they split strongly away from the irreversibility line toward lower temperatures for H // c and H J (open circles).

H // c and are fits to the power law predicted by the giant-flux-creep theory, while the circles represent the zero resistance data, TC0(H), for H // ab and H // c. The inset in the Figure 14 highlights the data at low magnetic fields where the continuous lines are fits to the de Almeida-Thouless and Gabay-Toulouse power laws. The Tirr(H) data show a weak superconducting granularity and consequently the zero resistance data are expected to fall closely below the irreversibility line, as showed in the Figure 14. While at high magnetic fields the TC0(H) data fall about the irreversibility lines, in the low-field region they fall systematically underneath the irreversibility line for both field configurations

Fig. 14. Irreversibility lines in an YBa2Cu2.97Zn0.03O7- single crystal, together with the zero resistance data TC0. The closed circles are for fields applied parallel to the ab plane and the open circles are for fields along the c-axis. The inset highlights the data in low fields. From

The data in the Figure 15 illustrates the case of a melt-textured YBa2Cu3O7- sample grown by top-seeding technique and containing 30 wt% of Y2BaCuO5 (Y211 phase) and 1 wt% of CeO2. The purpose of the Y211 phase is to introduce pinning centers and the CeO2 physically stabilizes the melt at high temperatures during the growth, to limit the size of the Y211 particles and to improve the distribution of the Y211 phase. This procedure strongly enhances the flux pinning potential in melt-textured YBa2Cu3O7- samples. On the other hand, high-quality melt-textured samples in general do not display superconducting granularity. However, the high concentration of Y211 phase causes considerable misalignment of the c-axis, thereby encumbers the grain coupling and reduces the planar anisotropy (Carrillo et al., 2000). The weak superconducting granularity presented by the melt-textured sample, whose results are presented in the Figure 15, arises mainly from

In the Figure 15 the irreversibility lines (H // ab and H // c) are represented by the experimental data and the continuous lines are fits to the power law predicted by the giantflux-creep theory. The measuring current in the zero resistance data flows along the same ab planes for both field orientations (H // ab and H // c). While for H // ab plane and H // J the zero resistance data (closed circles) fall closely underneath the corresponding irreversibility line, they split strongly away from the irreversibility line toward lower

misalignment of the c*-*axis, resulting in a reduced planar anisotropy.

temperatures for H // c and H J (open circles).

(see inset).

Schaf et al.

Fig. 15. Irreversibility lines obtained in a melt-textured YBa2Cu3O7-, together with the zero resistance data TC0. The closed circles are for fields applied parallel to the ab plane and the open circles are for fields along the c-axis. From Schaf et al.

The twinning domains in the investigated melt-textured sample are nearly random in the ab plane and their direction also must be scattered somewhat about the average c-axis, due to the misalignment of the c-axis. Therefore, while flux pinning by the twinning planes certainly contributes somewhat for fields applied along the c-axis, its contribution for fields along the ab plane must be irrelevant. In the Figure 15 the irreversibility data show a much reduced anisotropy, due to the spreading out of the c-axes of the different crystallites and the consequent noncoplanarity of the ab planes.

Similar results (not published) are showed in the Figure 16, for a YBa2Cu3O7- thin film grown by chemical solution deposition. It can be observed that the zero resistance data for H // c and H J split away from the irreversibility line (H // ab) towards lower temperatures.

The data in the Figure 17 are representative for a polycrystalline YBa1.75Sr0.25Cu3O7 sample grown by conventional sintered technique using the standard solid state reaction method (Schaf et al., 2008). This polycrystalline sample exhibits a strong superconducting granularity, as shown by the presence of the de Almeida-Thouless and Gabay-Toulouse regimes in the low-field irreversibility data. The asterisks in the Figure 17 denote the approximate superconducting temperature transition (TC); the open circles are the irreversibility data; the closed circles are the zero resistance data (TC0) and the continuous lines are fits to de Almeida-Thouless, Gabay-Toulouse and giant-flux creep power laws.

In spite of the magnetic field being applied parallel to the measuring current, H // J, the zero resistance data of this polycrystalline sample fall considerably below the irreversibility line already in low magnetic fields and split away from this line steeply when the applied magnetic field increases. In polycrystalline samples the current flow through the grain aggregate is highly dispersive and therefore the H // J configuration is only a microscopic approximation. The data in the Figure 17 also show no indication the zero resistance data might come back and meet the irreversibility line somewhere in higher magnetic fields.

Magnetic Irreversibility and Resistive Transition in

Figure 15.

**5. Conclusion** 

**6. Acknowledgment** 

**7. References** 

1751-8113

YBaCuO Superconductors: Interpretations and Possible Correlations 237

Due to the dependence of the phase displacements Aij and the associated grain decoupling effect on the direction of the applied field with respect to the weak links, the magnetic irreversibility line and the zero resistance of granular superconductors depend strongly on the field-current configuration. For instance, if the magnetic field is applied along the c-axis and the current flows in the ab plane, the degradation of the conducting junctions is expected to be strong and leads to a considerable increase of the resistivity and a large displacement of the zero resistance temperature towards lower temperatures. A representative result can be observed in the Figure 15. On the other hand, if the magnetic field orientation is parallel to the current in the ab plane, the effect of the field on the weak links carrying the current is expected to be much smaller, and this can be seen also in the

In the case of the polycrystalline sintered sample, in which the links between the grains are much weaker, the effects of the applied field on the grain coupling is drastic even for fields applied parallel to the intended measuring current, as can be seen in the Figure 17. In these granular superconductors the direction of the current is disperse due to the coupled grain

The connection between the magnetic irreversibility limit and the zero resistance in high-TC superconductors is very important in order to understand the mechanisms of the electric current transport in these systems. In granular superconductors the zero resistance state is ruled by grain couplings and the effect of an applied magnetic field on the electrical resistivity depends strongly on the field-current configuration. In a superconducting sample which has a weak superconducting granularity, the electrical resistance vanishes very close to and underneath the magnetic irreversibility limit. On the other hand, for samples with a strong superconducting granularity, the magnetic field severely degrades the grain couplings. The electrical resistance, for current flowing along the weak links affected by the

Because of the potential for technological applications, the correlation between electric and magnetic properties in HTSC systems can provide important information on the performance of these materials when magnetic fields are applied in different configurations.

I'm very grateful to Dr. Frederik Wolff-Fabris for a careful reading of the text and to Dr. Jacob Schaf, Dr. Paulo Pureur and Dr. Valdemar Vieira for important discussions. This work was partially supported by the Brazilian Ministry of Science and Technology and the State of Rio Grande do Sul, under the Grant PRONEX FAPERGS/CNPq 10/0009-2, and Ministry

Almeida, J., & Thouless, D. (1978). Stability of the Scherrington-Kirkpatrick solution of a

spin glass model. *Journal of Physics A,* Vol.11, No.5, (May 1978), pp. 983-990, ISSN

of Science and Technology, under the Grant CAPES/Procad 059/2007.

arrays that percolate through the whole sample in randomic loops.

magnetic field, vanishes far below the irreversibility line.

Fig. 16. Irreversibility line (Tirr) and zero resistance data (TC0) obtained in a YBa2Cu3O7- thin film. The closed circles are for fields applied parallel to the ab plane and the closed squares are for fields along the c-axis. Not published.

An applied magnetic field to a granular superconductor increases the phase entropy (Stroud et al., 1984). This magnetic field, while randomly produces a distortion of the phase of the order parameter, weakens and frustrates the grain coupling and favors the phase fluctuations that can result in the increase of the magnetoresistance. The random phase distortions also weakens the ability of the grain aggregate to pin the intergrain Josephson flux that dominates in the low-field regime. Therefore, in the low-field regime the irreversibility line exhibits the de Almeida-Thouless and Gabay-Toulouse power laws that are the signature of frustrated systems.

The response of a granular superconductor to a magnetic field is usually described in terms of the Josephson coupling Hamiltonian given by the equation (3) together with the equation (4). The equation (4) shows that a magnetic field causes phase displacements of the Ginzburg-Landau order parameter along those weak links which extend along the vector potential A , i.e. the Aij are large along weak links that lie transversely to the magnetic field but may vanish along the weak links oriented parallel to the magnetic field.

Fig. 17. Irreversibility line (Tirr) and zero resistance data (TC0) in a polycrystalline YBa1.75Sr0.25Cu3O7- sample. The asterisks indicate the approximate superconducting transition temperatures. From Schaf et al.

Due to the dependence of the phase displacements Aij and the associated grain decoupling effect on the direction of the applied field with respect to the weak links, the magnetic irreversibility line and the zero resistance of granular superconductors depend strongly on the field-current configuration. For instance, if the magnetic field is applied along the c-axis and the current flows in the ab plane, the degradation of the conducting junctions is expected to be strong and leads to a considerable increase of the resistivity and a large displacement of the zero resistance temperature towards lower temperatures. A representative result can be observed in the Figure 15. On the other hand, if the magnetic field orientation is parallel to the current in the ab plane, the effect of the field on the weak links carrying the current is expected to be much smaller, and this can be seen also in the Figure 15.

In the case of the polycrystalline sintered sample, in which the links between the grains are much weaker, the effects of the applied field on the grain coupling is drastic even for fields applied parallel to the intended measuring current, as can be seen in the Figure 17. In these granular superconductors the direction of the current is disperse due to the coupled grain arrays that percolate through the whole sample in randomic loops.

## **5. Conclusion**

236 Superconductors – Properties, Technology, and Applications

Fig. 16. Irreversibility line (Tirr) and zero resistance data (TC0) obtained in a YBa2Cu3O7- thin film. The closed circles are for fields applied parallel to the ab plane and the closed squares

An applied magnetic field to a granular superconductor increases the phase entropy (Stroud et al., 1984). This magnetic field, while randomly produces a distortion of the phase of the order parameter, weakens and frustrates the grain coupling and favors the phase fluctuations that can result in the increase of the magnetoresistance. The random phase distortions also weakens the ability of the grain aggregate to pin the intergrain Josephson flux that dominates in the low-field regime. Therefore, in the low-field regime the irreversibility line exhibits the de Almeida-Thouless and Gabay-Toulouse power laws that

The response of a granular superconductor to a magnetic field is usually described in terms of the Josephson coupling Hamiltonian given by the equation (3) together with the equation (4). The equation (4) shows that a magnetic field causes phase displacements of the Ginzburg-Landau order parameter along those weak links which extend along the vector potential A , i.e. the Aij are large along weak links that lie transversely to the magnetic field but may vanish along the weak links oriented parallel to the magnetic

Fig. 17. Irreversibility line (Tirr) and zero resistance data (TC0) in a polycrystalline YBa1.75Sr0.25Cu3O7- sample. The asterisks indicate the approximate superconducting

are for fields along the c-axis. Not published.

are the signature of frustrated systems.

transition temperatures. From Schaf et al.

field.

The connection between the magnetic irreversibility limit and the zero resistance in high-TC superconductors is very important in order to understand the mechanisms of the electric current transport in these systems. In granular superconductors the zero resistance state is ruled by grain couplings and the effect of an applied magnetic field on the electrical resistivity depends strongly on the field-current configuration. In a superconducting sample which has a weak superconducting granularity, the electrical resistance vanishes very close to and underneath the magnetic irreversibility limit. On the other hand, for samples with a strong superconducting granularity, the magnetic field severely degrades the grain couplings. The electrical resistance, for current flowing along the weak links affected by the magnetic field, vanishes far below the irreversibility line.

Because of the potential for technological applications, the correlation between electric and magnetic properties in HTSC systems can provide important information on the performance of these materials when magnetic fields are applied in different configurations.

## **6. Acknowledgment**

I'm very grateful to Dr. Frederik Wolff-Fabris for a careful reading of the text and to Dr. Jacob Schaf, Dr. Paulo Pureur and Dr. Valdemar Vieira for important discussions. This work was partially supported by the Brazilian Ministry of Science and Technology and the State of Rio Grande do Sul, under the Grant PRONEX FAPERGS/CNPq 10/0009-2, and Ministry of Science and Technology, under the Grant CAPES/Procad 059/2007.

## **7. References**

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**11** 

*Russia* 

**Manufacturing Technology** 

**of the (Bi,Pb)2Sr2Ca2Cu3O10** 

*National Research University of Electronic Technology* 

Yury Evgenevich Grigorashvili

**High-Temperature Superconductor** 

Practical use of high-temperature superconductors is considerably limited by the available technology to fabricate superconductors with characteristics, suitable for use in electrical products and electronics applications. All known high-temperature superconductors, which have critical temperature above the boiling point of nitrogen (77 K in normal conditions), consist of complex elemental composition and crystal lattice. All types of high-temperature

Nowadays the most advanced technology in manufacturing the high-temperature superconductors and their products is done for the Y1Ba2Cu3O7 (Y-123) system. These are conductors of the second generation, multi-layer electronic structures, and devices with SQUIDs base. However, the temperature change of this system into the superconductive state is about 90 K. If affordable nitrogen is used to cool down the system, the working temperature difference would be just 14 percent shy of the critical one. Also, operational

The high-temperature superconductor of the following composition - Bi2Sr2Ca2Cu3O10 (Bi-2223) is the only one having the temperature in superconductive state above 100 K. Experimental data and research allow us to consider the creation of commercially-viable technologies based on the use of this superconductor. This superconductor has 30 percent operational reliability in temperature. It is also degradation-resistant under normal weather conditions. Yet, very

This chapter contains research results that focus on commercial implementation of technology that is based on manufacturing structures with the Bi-2223 superconductor, which are used in electronic industry. In this chapter we report the study results of the volume formation of the

We show the process of obtaining thick films on inert substrate. Major attention is given to a manufacturing technology of epitaxial, thin, single-phased and defect-free layers situated on a monocrystal lattice. Also, additional information is included on its electrical and magnetic properties, as well as the analysis of possible implementation into electronic technology.

limited research records exist in forming this high-temperature superconductor.

110 K superconducting phase, including influences of lead, silver, and fluorine.

**1. Introduction** 

superconductors are fragile.

reliability is insufficient.


## **Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor**

Yury Evgenevich Grigorashvili *National Research University of Electronic Technology Russia* 

## **1. Introduction**

240 Superconductors – Properties, Technology, and Applications

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flux dynamics in homogeneous and granular high-*TC* superconductors. *Physical* 

dynamics in YBa2-xSrxCu3O7- single crystals: Bose-glass and vortex-glass features. *Physical Review B,* Vol.65, No.14, (April 2002), pp. 144531-1-9, ISSN 1098-0121 Schaf, J., Pureur, P., & Vieira, V. (2002). Effects of Zn and Mg in Cu sites of YBa2Cu3O7-

single crystals on the resistive transition, fluctuation conductivity, and magnetic irreversibilities. *Physical Review B,* Vol.66, No.22, (December 2002), pp. 224506-1-11,

between the magnetic irreversibility limit and the zero resistance point in different granular YBa2Cu3O7- superconductors. *Physical Review B,* Vol.77, No.13, (April

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Stroud, D., Ebner, C., & Shih, W. (1984). Frustration and disorder in granular

Vinokur, V., Kes, P., & Koshelev, A. (1990). Flux pinning and creep in the very anisotropic

Yeshurun, Y., & Malozemoff, A. (1988). Giant flux creep and irreversibility in an Y-Ba-Cu-O

Vol.60, No.21, (May 1988), pp. 2202-2205, ISSN 0031-9007

Practical use of high-temperature superconductors is considerably limited by the available technology to fabricate superconductors with characteristics, suitable for use in electrical products and electronics applications. All known high-temperature superconductors, which have critical temperature above the boiling point of nitrogen (77 K in normal conditions), consist of complex elemental composition and crystal lattice. All types of high-temperature superconductors are fragile.

Nowadays the most advanced technology in manufacturing the high-temperature superconductors and their products is done for the Y1Ba2Cu3O7 (Y-123) system. These are conductors of the second generation, multi-layer electronic structures, and devices with SQUIDs base. However, the temperature change of this system into the superconductive state is about 90 K. If affordable nitrogen is used to cool down the system, the working temperature difference would be just 14 percent shy of the critical one. Also, operational reliability is insufficient.

The high-temperature superconductor of the following composition - Bi2Sr2Ca2Cu3O10 (Bi-2223) is the only one having the temperature in superconductive state above 100 K. Experimental data and research allow us to consider the creation of commercially-viable technologies based on the use of this superconductor. This superconductor has 30 percent operational reliability in temperature. It is also degradation-resistant under normal weather conditions. Yet, very limited research records exist in forming this high-temperature superconductor.

This chapter contains research results that focus on commercial implementation of technology that is based on manufacturing structures with the Bi-2223 superconductor, which are used in electronic industry. In this chapter we report the study results of the volume formation of the 110 K superconducting phase, including influences of lead, silver, and fluorine.

We show the process of obtaining thick films on inert substrate. Major attention is given to a manufacturing technology of epitaxial, thin, single-phased and defect-free layers situated on a monocrystal lattice. Also, additional information is included on its electrical and magnetic properties, as well as the analysis of possible implementation into electronic technology.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 243

Right after the discovery of Bismuth-based superconductor it was determined that the lead presence considerably speeded up the process of superconductive phase formation (Sunshine S., et al., 1988). In our experiments lead was injected either as an additional element or as partial substitute for Bismuth. The amount of lead and Bismuth varied

Just like in previous case the pellets were pressed from powder, which were annealed in argon, oxygen, and air atmospheres. It was shown that the argon atmosphere annealing produced slight melting of pellets. Yet, under the oxygen atmosphere annealing the pellets' sides remained sharp and even. However, the amount of superconductive phases was too small in both cases. The best results were obtained during the annealing of argon and oxygen mixture with a partial pressure of 0.8 and 0.2 respectively. Similar results were

(a) (b)

It's important to notice that the formation of Bi- 2212 phase happens when the temperature range of annealing is between 820 - 840 0С. The surface morphology of such samples has needle-like crystallite appearance (Fig. 1а). Needles' diameter is 2 to 4 μm and their length is 10 to 15 μm. The resistance dependence on temperature is shown in a spike that starts at 90 K temperature and ends at around 80 K (Fig. 2). The increased time of annealing does not lead to a temperature increase in the beginning of transition into a superconductive state. The process of annealing at 840-860 0С temperature range forms a two-phase system of Bi-2212 and Bi- 2223. It is shown with two little steps situated on curves, explaining resistance dependence on temperature (Fig. 2). There is a formation of the through channel at a superconducting temperature reaching near 110 K during the 30-hour annealing at 860 0 С. Critical current density is above 104 А/сm2. It's typical for these samples to have crystallite formation in a shape of flat disks (platelet crystallites) that are 15-20 μm in diameter and 1-

The results of experimental research show that it's difficult to make the Bi-2223 mono-phased material. Diamagnetic response measurements prove existence of two phases with critical temperatures measured at 110 K and 85 К. The synthesis of superconductive phase is very

long and demands thermal treatment maintenance locked in a very narrow range.

a – Bi2Sr2Ca1Cu2O8 Phase. Crystals are in a shape of needle-like crystallites. b – Bi2Sr2Ca2Cu3O10 Phase. Crystals are in a shape of platelet crystallites. Fig. 1. The superconductor's surface morphology of Bismuth system.

between 2/0.4 and 1.4/0.7.

2μm thickness (Fig. 1b).

received after annealing in air atmosphere.

## **2. The manufacturing technology of superconductive structures with (Bi,Pb)2Sr2Ca2Cu3O10 composition**

In order to make high-quality superconductive structures on a monocrystal substrate we need to collect the information on processes happening during the crystal structure formation, the influences of material composition and dopants, manufacturing methods and technological conditions, as well as obtaining of electro-physical and magnetic properties in samples. In this chapter we describe the manufacturing technology of Bi – 2223 hightemperature superconductor and its structures that can be used in electronic industry (Grigorashvili, Y.E., Volkov, S.I., 1999; Grigorashvili, Y.E., 2005; Grigorashvily, Yu.E., Ichkitidze, L.P. & Volik, N.N., 2006).

## **2.1 The formation of superconductive phases in bulk material**

Bulk material synthesis allows us gather information easily on the formation of hightemperature superconductors of various compositions and on optimal technological modes of thermal treatment. During the manufacturing process bulk material presence neutralizes changes in material surface structure. Prepared samples have dominant bulk properties as opposed to boundary properties.

This section describes the technology of making bulk superconductors of Bi-2223 composition, using methods of solid-phase synthesis. The original research results show the impact of developed technology on the formation of different superconducting phases in the BSCCO system. The conducted research determined the effect of lead and silver doping onto electrical characteristics of samples, including critical temperatures in the beginning and end of the superconductor's transition and critical currents in magnetic fields.

The first stage in bulk superconductors manufacturing is the production of micro dispersive charge of stoichiometric composition. In our research the charge was obtained from nitrates in metals. First, the manufacturing of nitrates mixture was done in proportions stated in 2223 formula for metals. In order to get rid of water and partial nitrates' decomposition the mixture has been annealed at 400 0С for two hours.

Formed spec grinded and the mixture stirred. The average size of grains in powdered mixture measured 4.5 μm. The second thermal treatment has been completed in oxygen atmosphere in 16 hours at 810-850 0С temperature range. The manufactured charge had superconductive properties. It was registered with diamagnetic response. The method of such diagnostics is presented in (Afanasev & Chaplygin, 1992).

Pellets were pressed from the charge into shapes sized 12-20 mm in diameter and 3-5 mm in thickness. The pellets were annealed in a gas mixture of nitrogen and oxygen in 10 to 50 hours under 800-850 0С. It was experimentally proven that the superconductive phases appear after a 20-hour annealing in temperatures registered above 8200С. It is proven by temperature dependence on diamagnetic response and direct current resistance. The content of superconductive phases rises in relation to time of thermal treatment. Yet, their amount is small in comparison to the volume of charge. Based on measured results of diamagnetic response, their bulk makes about 15 percent at 849 degree C thermal treatment in 50 hours.

In order to make high-quality superconductive structures on a monocrystal substrate we need to collect the information on processes happening during the crystal structure formation, the influences of material composition and dopants, manufacturing methods and technological conditions, as well as obtaining of electro-physical and magnetic properties in samples. In this chapter we describe the manufacturing technology of Bi – 2223 hightemperature superconductor and its structures that can be used in electronic industry (Grigorashvili, Y.E., Volkov, S.I., 1999; Grigorashvili, Y.E., 2005; Grigorashvily, Yu.E.,

Bulk material synthesis allows us gather information easily on the formation of hightemperature superconductors of various compositions and on optimal technological modes of thermal treatment. During the manufacturing process bulk material presence neutralizes changes in material surface structure. Prepared samples have dominant bulk properties as

This section describes the technology of making bulk superconductors of Bi-2223 composition, using methods of solid-phase synthesis. The original research results show the impact of developed technology on the formation of different superconducting phases in the BSCCO system. The conducted research determined the effect of lead and silver doping onto electrical characteristics of samples, including critical temperatures in the beginning

The first stage in bulk superconductors manufacturing is the production of micro dispersive charge of stoichiometric composition. In our research the charge was obtained from nitrates in metals. First, the manufacturing of nitrates mixture was done in proportions stated in 2223 formula for metals. In order to get rid of water and partial nitrates' decomposition the

Formed spec grinded and the mixture stirred. The average size of grains in powdered mixture measured 4.5 μm. The second thermal treatment has been completed in oxygen atmosphere in 16 hours at 810-850 0С temperature range. The manufactured charge had superconductive properties. It was registered with diamagnetic response. The method of

Pellets were pressed from the charge into shapes sized 12-20 mm in diameter and 3-5 mm in thickness. The pellets were annealed in a gas mixture of nitrogen and oxygen in 10 to 50 hours under 800-850 0С. It was experimentally proven that the superconductive phases appear after a 20-hour annealing in temperatures registered above 8200С. It is proven by temperature dependence on diamagnetic response and direct current resistance. The content of superconductive phases rises in relation to time of thermal treatment. Yet, their amount is small in comparison to the volume of charge. Based on measured results of diamagnetic response, their bulk makes about 15 percent at 849 degree C thermal

and end of the superconductor's transition and critical currents in magnetic fields.

**2. The manufacturing technology of superconductive structures with** 

**2.1 The formation of superconductive phases in bulk material** 

**(Bi,Pb)2Sr2Ca2Cu3O10 composition** 

Ichkitidze, L.P. & Volik, N.N., 2006).

opposed to boundary properties.

treatment in 50 hours.

mixture has been annealed at 400 0С for two hours.

such diagnostics is presented in (Afanasev & Chaplygin, 1992).

Right after the discovery of Bismuth-based superconductor it was determined that the lead presence considerably speeded up the process of superconductive phase formation (Sunshine S., et al., 1988). In our experiments lead was injected either as an additional element or as partial substitute for Bismuth. The amount of lead and Bismuth varied between 2/0.4 and 1.4/0.7.

Just like in previous case the pellets were pressed from powder, which were annealed in argon, oxygen, and air atmospheres. It was shown that the argon atmosphere annealing produced slight melting of pellets. Yet, under the oxygen atmosphere annealing the pellets' sides remained sharp and even. However, the amount of superconductive phases was too small in both cases. The best results were obtained during the annealing of argon and oxygen mixture with a partial pressure of 0.8 and 0.2 respectively. Similar results were received after annealing in air atmosphere.

a – Bi2Sr2Ca1Cu2O8 Phase. Crystals are in a shape of needle-like crystallites. b – Bi2Sr2Ca2Cu3O10 Phase. Crystals are in a shape of platelet crystallites.

Fig. 1. The superconductor's surface morphology of Bismuth system.

It's important to notice that the formation of Bi- 2212 phase happens when the temperature range of annealing is between 820 - 840 0С. The surface morphology of such samples has needle-like crystallite appearance (Fig. 1а). Needles' diameter is 2 to 4 μm and their length is 10 to 15 μm. The resistance dependence on temperature is shown in a spike that starts at 90 K temperature and ends at around 80 K (Fig. 2). The increased time of annealing does not lead to a temperature increase in the beginning of transition into a superconductive state.

The process of annealing at 840-860 0С temperature range forms a two-phase system of Bi-2212 and Bi- 2223. It is shown with two little steps situated on curves, explaining resistance dependence on temperature (Fig. 2). There is a formation of the through channel at a superconducting temperature reaching near 110 K during the 30-hour annealing at 860 0 С. Critical current density is above 104 А/сm2. It's typical for these samples to have crystallite formation in a shape of flat disks (platelet crystallites) that are 15-20 μm in diameter and 1- 2μm thickness (Fig. 1b).

The results of experimental research show that it's difficult to make the Bi-2223 mono-phased material. Diamagnetic response measurements prove existence of two phases with critical temperatures measured at 110 K and 85 К. The synthesis of superconductive phase is very long and demands thermal treatment maintenance locked in a very narrow range.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 245

Widely accepted substrates of monocrystal silicon, oxidized silicon, and sapphire get mixed with superconductor's metals and its oxidizers at temperatures above 500 0С. Chemically

Aerosol method is used to spray the oxides of metals onto the substrate. It is achieved by sputtering the microdrops of the solution with metal salts and their oxidation onto a hot substrate. The composition of the solution was chosen in such a way that when the microdrops hit the hot substrate, evaporation of a solvent, decomposition of metals' salts

The diagram of a unit, illustrating the manufacturing of films using aerosol method is shown at Fig. 3. The source is 0.05M water solution of lead nitrate, strontium, calcium, bismuth, and copper with added nitric acid and acetic acid. Ultrasonic disperser was used to make the aerosol. The aerosol mixture was delivered to the hot substrate via inert gas. In experimental research the substrate was heat-treated at 820-830 0С temperature. It required use of a silicon carbide heater that was able to work for a long time in aggressive atmosphere. The evaporation of water occurred at high temperature when water-drops reached the substrate. After that the nitrates decomposed and oxides developed. To increase the size of area of homogeneous deposition and maintain high temperature of a substrate,

In experimental research the power of ultrasonic transmitter, speed of gas current, temperature of a substrate, frequency and range of scanning, and composition of a solution were varied. The composition of obtained films was controlled by chemical methods and the

1- Control unit of the mechanical scanning system. 2- The temperature controlling unit. 3- Highfrequency generator 12.6 MHz, 4- piezoceramic transmitter, 5- membrane, 6- the salt solution of basic metals. 7- water. 8- water refrigerator. 9- nozzle. 10- substrate. 11- Lamellar heater. 12- step motor. Fig. 3. The diagram of a device showing the deposition of a bulk-layered superconductor on

inert substrates are MgO and SrTiO3 monocrystal substrates.

and the formation of oxide mixture took place.

the aerosol nozzle scans the surface in two coordinates.

X-ray specter microanalysis method.

a substrate.

With a temperature range at +/-0.5 at 860 0С and the process of annealing around 100 hours the samples can be made with the superconducting state transition beginning at 115 K and completing at 108 K respectively. General findings in literature mention the development of bismuth-system phase with a critical temperature transition at only 110 К.

a – Bi2Sr2Ca1Cu2O8 phase; b– mixture of two phases c - Bi2Sr2Ca2Cu3O10 phase.

Fig. 2. Ceramic pellets resistivity dependence on temperature at various modes of annealing.

The author researched the opportunity of getting new characteristics of Bismuth system where the oxygen was replaced by fluorine. The samples were manufactured using the same technology mentioned above. Considerable superconductor volumes were not present. Alloying of silver powder with an atomic fraction of up to 0.5 increased the critical current density over 10 times.

## **2.2 Formation of bulk superconductors on a solid substrate**

Next step in technological development of Bi-2223 formation is the creation of a bulk superconductor on a solid substrate. In this case, the processes described above should remain the same but the new sample would have a double-layer structure. To meet the described conditions, it's necessary to make an inert substrate relative to metals and their oxidations, entering the superconductor.

Bulk-layered Bi-2223 superconductor with a solid substrate was manufactured using a twostep system. First, the mixture was placed on a substrate, and then, a high-temperature annealing was carried out. This approach allowed us to divide the problem into two parts and research the conditions of each part separately.

The thickness of a deposited layer was several hundred microns. Such layer may be considered as a bulk one with the condition that there is neither interaction nor interfusion of the mixture at the superconductor-substrate border.

With a temperature range at +/-0.5 at 860 0С and the process of annealing around 100 hours the samples can be made with the superconducting state transition beginning at 115 K and completing at 108 K respectively. General findings in literature mention the development of

bismuth-system phase with a critical temperature transition at only 110 К.

a – Bi2Sr2Ca1Cu2O8 phase; b– mixture of two phases c - Bi2Sr2Ca2Cu3O10 phase.

**2.2 Formation of bulk superconductors on a solid substrate** 

density over 10 times.

oxidations, entering the superconductor.

and research the conditions of each part separately.

of the mixture at the superconductor-substrate border.

Fig. 2. Ceramic pellets resistivity dependence on temperature at various modes of annealing.

The author researched the opportunity of getting new characteristics of Bismuth system where the oxygen was replaced by fluorine. The samples were manufactured using the same technology mentioned above. Considerable superconductor volumes were not present. Alloying of silver powder with an atomic fraction of up to 0.5 increased the critical current

Next step in technological development of Bi-2223 formation is the creation of a bulk superconductor on a solid substrate. In this case, the processes described above should remain the same but the new sample would have a double-layer structure. To meet the described conditions, it's necessary to make an inert substrate relative to metals and their

Bulk-layered Bi-2223 superconductor with a solid substrate was manufactured using a twostep system. First, the mixture was placed on a substrate, and then, a high-temperature annealing was carried out. This approach allowed us to divide the problem into two parts

The thickness of a deposited layer was several hundred microns. Such layer may be considered as a bulk one with the condition that there is neither interaction nor interfusion Widely accepted substrates of monocrystal silicon, oxidized silicon, and sapphire get mixed with superconductor's metals and its oxidizers at temperatures above 500 0С. Chemically inert substrates are MgO and SrTiO3 monocrystal substrates.

Aerosol method is used to spray the oxides of metals onto the substrate. It is achieved by sputtering the microdrops of the solution with metal salts and their oxidation onto a hot substrate. The composition of the solution was chosen in such a way that when the microdrops hit the hot substrate, evaporation of a solvent, decomposition of metals' salts and the formation of oxide mixture took place.

The diagram of a unit, illustrating the manufacturing of films using aerosol method is shown at Fig. 3. The source is 0.05M water solution of lead nitrate, strontium, calcium, bismuth, and copper with added nitric acid and acetic acid. Ultrasonic disperser was used to make the aerosol. The aerosol mixture was delivered to the hot substrate via inert gas. In experimental research the substrate was heat-treated at 820-830 0С temperature. It required use of a silicon carbide heater that was able to work for a long time in aggressive atmosphere. The evaporation of water occurred at high temperature when water-drops reached the substrate. After that the nitrates decomposed and oxides developed. To increase the size of area of homogeneous deposition and maintain high temperature of a substrate, the aerosol nozzle scans the surface in two coordinates.

In experimental research the power of ultrasonic transmitter, speed of gas current, temperature of a substrate, frequency and range of scanning, and composition of a solution were varied. The composition of obtained films was controlled by chemical methods and the X-ray specter microanalysis method.

1- Control unit of the mechanical scanning system. 2- The temperature controlling unit. 3- Highfrequency generator 12.6 MHz, 4- piezoceramic transmitter, 5- membrane, 6- the salt solution of basic metals. 7- water. 8- water refrigerator. 9- nozzle. 10- substrate. 11- Lamellar heater. 12- step motor.

Fig. 3. The diagram of a device showing the deposition of a bulk-layered superconductor on a substrate.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 247

During our preliminary research for this chapter we studied the available publications on manufacturing technology of Bi-2223 thin films. Publications were studied for 1990-2011 period in the following journals: Supercond. Sci. Technol., J. Appl. Phys., Appl. Phys. Lett., IEEE Transactions on Magnetics, IEEE Transactions on Applied Superconductivity, Physica C: Superconductivity and its Applications, Journal of Alloys and Compounds, Journal of Crystal Growth, Thin Solid Films, Journal of Vacuum Science & Technology A, Semiconductors, Technical Physics. Besides journal publications, we also studied the U.S.,

It's worth mentioning that almost all publications on thin film manufacturing are for the Bi-2212 system. There are just a few publications available that have printed the results on thin film manufacturing with thickness less than 100 nm and critical temperature above 100 K. The problem of the formation of a superconductor Bi-2223 with a superconducting

Method of molecular-beam epitaxy (MBE) was developed in the beginning of the 70-s for manufacturing of high-quality, very thin films (Cho, A. Y., Arthur, J. R., 1975). This technology was used by a number of authors for BiSrCaCuO superconductive film manufacturing (Steinbeck, J., 1989; Yoon, D.H., Agung, I., Saito, M., Yoshizawa, M., 1997; Zakharov, N.D., Hoffschulz, H. et al., 1997). In this technology, the evaporation of individual components of the superconductor material is made from separate the Effusion (Knudsen cells). The composition and structure get formed by continuous opening of these cells for a required amount of time. (Varilci, A., Altunbas, M. et al., 2002) Working in MBE method BiPbSrCaCuO films were manufactured on MgO substrates (001). There were received critical temperatures of 105 К and current of 6х104 А/сm2. However, MBE method uses very expensive and complex equipment. Thus it's practically important to develop the

The technology of laser deposition (PLD) of BiSrCaCuO system films was used in this research (Ivanov, Z. et al., 1989). The process of film deposition consists of target material evaporation with a laser-radiation at energy density above 1 J/сm2. There are excimer lasers - ArF ( = 193 nm), KrF ( = 248 nm), and XeCl ( = 308nm) used for film deposition as well as infrared СО<sup>2</sup> lasers ( = 193 nm). HTSC-film growth occurs at molecular oxygen pressures around 10 Pa.

This method has a number of important advantages. Thanks to having a laser radiation source outside the working chamber, the deposition process occurs in "clean" conditions. Laser beam with optical system is focused onto a small area using small-diameter targets that increase the process of efficiency in superconducting film manufacturing. Laser evaporation lets us manufacture multi-layer structures easily that consist of multicomponent material (Jannah, A.N., 2009). Pulse-controlled laser radiation lets us determine

In a study (Pilosyan, S.H., 1990) the manufacturing of (BiPb)2Sr2Ca2Cu3O10 films is on MgO substrates (100) with Тсо=110 К and Тсе= 105 К parameters. Working pressure in a chamber during the deposition is 10-1 Torr. The impulse energy is 0.05-0.11 J at the frequency of 15 Hz

transition temperature above 100 K on single-crystal substrate remains topical today.

alternative technologies of manufacturing superconductive thin film structures.

**2.3 The technology of manufacturing thin film superconductors Bi-2223** 

European and Russian patents.

**2.3.1 Basic methods of thin film manufacturing** 

precisely the thickness of the growing film.

Main advantage of the described method is the simplicity of making changes in film composition. It is achieved by a varied concentration of basic metals' nitrate salts in a solution. The second important advantage of this method is the ability to make compositionally homogeneous layers in a big area.

Manufactured layers had amorphous structure of oxide mixture of basic metals. To form the crystal structure of Bi-2223 superconductive phase we used annealing in a mixture of either oxygen and argon, or 0.2/0.4 nitrogen. During the process of annealing there is a partial loss of bismuth, lead, and copper. That is why the aerosol mixture had enrichment with salts of listed metals at the depositional stage.

Film resistivity dependence on temperature with annealing modes that result in Bi-2223 phase formation is shown on Fig. 4. As seen on this picture the formation of superconductive phase is noticed after 10 hours of annealing. It is accompanied by appearance of graph zones with sharp decline of resistance at 110 K. There is a proportional increase of a superconductive phase during further annealing. Fair superconductive through channel forms after 50 hours of annealing.

Major disadvantage of this method is the difficulty of forming films with even relief. Superconductive 1-2 μm-thick layer had a bumpy surface of about the same height. The reduction in thickness by up to 0.5 μm decreased the height of unevenness. However, some zones appeared free of any superconductive phase on a substrate's surface.

Period of annealing is 3, 5, 10, 20, 50, 100 hours. Curves are 1….6 respectively.

Fig. 4. Sample resistivity dependence on temperature at various annealing periods. Annealing temperature is 860 0 С. R0 – is the resistance at room temperature.

## **2.3 The technology of manufacturing thin film superconductors Bi-2223**

During our preliminary research for this chapter we studied the available publications on manufacturing technology of Bi-2223 thin films. Publications were studied for 1990-2011 period in the following journals: Supercond. Sci. Technol., J. Appl. Phys., Appl. Phys. Lett., IEEE Transactions on Magnetics, IEEE Transactions on Applied Superconductivity, Physica C: Superconductivity and its Applications, Journal of Alloys and Compounds, Journal of Crystal Growth, Thin Solid Films, Journal of Vacuum Science & Technology A, Semiconductors, Technical Physics. Besides journal publications, we also studied the U.S., European and Russian patents.

It's worth mentioning that almost all publications on thin film manufacturing are for the Bi-2212 system. There are just a few publications available that have printed the results on thin film manufacturing with thickness less than 100 nm and critical temperature above 100 K. The problem of the formation of a superconductor Bi-2223 with a superconducting transition temperature above 100 K on single-crystal substrate remains topical today.

## **2.3.1 Basic methods of thin film manufacturing**

246 Superconductors – Properties, Technology, and Applications

Main advantage of the described method is the simplicity of making changes in film composition. It is achieved by a varied concentration of basic metals' nitrate salts in a solution. The second important advantage of this method is the ability to make

Manufactured layers had amorphous structure of oxide mixture of basic metals. To form the crystal structure of Bi-2223 superconductive phase we used annealing in a mixture of either oxygen and argon, or 0.2/0.4 nitrogen. During the process of annealing there is a partial loss of bismuth, lead, and copper. That is why the aerosol mixture had enrichment with salts of

Film resistivity dependence on temperature with annealing modes that result in Bi-2223 phase formation is shown on Fig. 4. As seen on this picture the formation of superconductive phase is noticed after 10 hours of annealing. It is accompanied by appearance of graph zones with sharp decline of resistance at 110 K. There is a proportional increase of a superconductive phase during further annealing. Fair superconductive through

Major disadvantage of this method is the difficulty of forming films with even relief. Superconductive 1-2 μm-thick layer had a bumpy surface of about the same height. The reduction in thickness by up to 0.5 μm decreased the height of unevenness. However, some

zones appeared free of any superconductive phase on a substrate's surface.

Period of annealing is 3, 5, 10, 20, 50, 100 hours. Curves are 1….6 respectively.

Fig. 4. Sample resistivity dependence on temperature at various annealing periods. Annealing temperature is 860 0 С. R0 – is the resistance at room temperature.

compositionally homogeneous layers in a big area.

listed metals at the depositional stage.

channel forms after 50 hours of annealing.

Method of molecular-beam epitaxy (MBE) was developed in the beginning of the 70-s for manufacturing of high-quality, very thin films (Cho, A. Y., Arthur, J. R., 1975). This technology was used by a number of authors for BiSrCaCuO superconductive film manufacturing (Steinbeck, J., 1989; Yoon, D.H., Agung, I., Saito, M., Yoshizawa, M., 1997; Zakharov, N.D., Hoffschulz, H. et al., 1997). In this technology, the evaporation of individual components of the superconductor material is made from separate the Effusion (Knudsen cells). The composition and structure get formed by continuous opening of these cells for a required amount of time. (Varilci, A., Altunbas, M. et al., 2002) Working in MBE method BiPbSrCaCuO films were manufactured on MgO substrates (001). There were received critical temperatures of 105 К and current of 6х104 А/сm2. However, MBE method uses very expensive and complex equipment. Thus it's practically important to develop the alternative technologies of manufacturing superconductive thin film structures.

The technology of laser deposition (PLD) of BiSrCaCuO system films was used in this research (Ivanov, Z. et al., 1989). The process of film deposition consists of target material evaporation with a laser-radiation at energy density above 1 J/сm2. There are excimer lasers - ArF ( = 193 nm), KrF ( = 248 nm), and XeCl ( = 308nm) used for film deposition as well as infrared СО<sup>2</sup> lasers ( = 193 nm). HTSC-film growth occurs at molecular oxygen pressures around 10 Pa.

This method has a number of important advantages. Thanks to having a laser radiation source outside the working chamber, the deposition process occurs in "clean" conditions. Laser beam with optical system is focused onto a small area using small-diameter targets that increase the process of efficiency in superconducting film manufacturing. Laser evaporation lets us manufacture multi-layer structures easily that consist of multicomponent material (Jannah, A.N., 2009). Pulse-controlled laser radiation lets us determine precisely the thickness of the growing film.

In a study (Pilosyan, S.H., 1990) the manufacturing of (BiPb)2Sr2Ca2Cu3O10 films is on MgO substrates (100) with Тсо=110 К and Тсе= 105 К parameters. Working pressure in a chamber during the deposition is 10-1 Torr. The impulse energy is 0.05-0.11 J at the frequency of 15 Hz

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 249

Magnetron sputtering for superconductive film manufacturing is used in modes of constant current (Schultz et al., 2001) and alternating current (Grigorashvily, Y.E., Volkov, S.I., Sotnikov, I.L. & Mingazin, V.T., 1999; Grigorashvili Y.E., Bukhlin, A.V. & Veryuzhskii, I.V., 2010) with the use of one or several sources (Kuroda, K., Kojima, K. et al., 1991). In this research the following strategy of forming Bi-2223 thin films with superconductive

1. Film manufacturing was complete in two stages. First stage included the deposition onto a substrate's surface of a mechanical mixture of metal atoms and their oxides of stechometric composition. The second stage consisted of the crystal lattice formation of

The construction and technology of magnetron sputtering unit should take into account features of a problem, such as the formation of a five-component mixture of oxide metals on a substrate with various physical and chemical characteristics. To justify the construction of the sputtering system via experimental research is extremely expensive. Thus this research process consisted of experiments with the use of a mathematical model of transporting

The most advanced mathematical model of the process of solution transportation with magnetron sputtering belongs to Volpyas (Volpyas, V.A. & Kozirev, A.B., 1997). It was

The computer model of the sputtering process used in this research had the following




The model allows to determine the thickness of sputtered layer precisely and the stoichiometry of composition depending on coordinates on a substrate. Partial pressures of argon and oxygen can vary in range widely, depending on the conditions of collision-free

2. Method of magnetron sputtering is used for deposition of metals onto a substrate. 3. Target composition corresponds to Bi2Sr2Ca2Cu3O10 superconductor composition. 4. Annealing is used to form structure. During this process the stimulation of epitaxial

expansion of superconductor's thin layer takes place on a substrate.

**2.3.2 The magnetron sputtering method – The computer model for the sputtering** 

temperature around 110 K on a 40mm-radius substrate was used:

supplemented in accordance with this research problem-solving.

depending on the coordinates with axisymmetric system.

**process** 

algorithm:

superconductive phase.

metal atoms from a target onto a substrate.

configuration of a magnetic conductor.

quasi-hard sphere potential (Volpyas, 2000).

sputtered atoms until their complete thermalisation.

pulse repetition rate. Target-substrate distance is 20-25mm. Thermal treatment of films occurs at 850 0С temperature. It is 3 hours long. It's necessary to point out that in this process the target of Bi 2Pb0.5Sr1.4Ca2Cu3.6Ox composition was used.

Main disadvantage of the PLD method is the existence of clusters (drops) found in evaporated material. It happens because of high density of laser- radiation energy. Attempts to reduce the amount of microdrops via the reduction of energetic density current, lead to an increase in a non-stoichiometric component in a depositional condensate on a substrate. It limits dimensions of substrates that can be used for high-quality film manufacturing. Usually they don't exceed 10 mm in diameter.

Among chemical methods used for Bi-2223 film manufacturing we can highlight Liquid epitaxy (Pandey, et al., 1994), Spray Pyrolysis and the deposition from gas-vapor mixture (Kimura, T., Nakao, H. et al., 1991), including the MOCVD metalorganic compounds (Stejskal, J., Leitner, J.J. et al., 2000). In the MOCVD method, the components of superconductive film are transported into a reactor in a shape of gases of metalorganic volatile compounds. They are mixed with gaseous oxidizer resulting in gases' decay either inside the reactor with hot walls or on a heated substrate and the HTSC-film formation.

The advantages of this method include: the ability to make deposition of material onto substrates of large scale, production of homogenous layers in thickness and composition, and the ease of equipment use. At the same time, data found in publications shows that manufactured films are of low-quality. They contain non-superconductive inclusions, the surface has considerable unevenness, and there is no data available on the Bi-2223 phase formation.

Method of thermal deposition of material is the most simple and inexpensive resource for BiPbSrCaCuO film manufacturing. There are several types of used sources, including resistive evaporators, flash evaporation, ion beam (electron beam) evaporators and crucibles with radioactive and high-frequency inductive heating.

In a study (Azoulay, J., 1989) the researchers made BiSrCaCuO films using resistive heating of tungsten boat, with a source crucible of Bi2.3Sr1.5Ca1Cu3 material composition. After the deposition process, films were thermally processed in an oven at 725 0С for 15 minutes and 840 0С for 5 minutes. Typical superconductive temperature reaching the superconductive state Тсе was around 78-К for ZrO2 substrates and 88-К for SrTiO3 substrates.

Similar film parameters (Тcе = 78К) are listed in (Patil, J.M., Bhangale, A.R. et al., 1993) that were manufactured with thermal deposition onto MgO substrate (100). In a study (Silver, R.M., Ogawa, E.T., Pan. S. de Lozanne, A.L., 1991) the BiSrCaCuO manufacturing in situ was completed using thermal evaporation method with high-frequency plasma generator. The device was constructed to make plasma at a distance from a substrate in order to stop recombination of oxygen atoms. After the deposition at 600-660 0С temperatures, films were annealed at 850 0С. The dominant phase was 2212, while the 2223 phase was present in small quantities. Similar critical temperatures were produced in studies (Basturk, N., 2005).

The analysis of conducted studies shows that high-quality HTSC-film with Bi-2223 phase and Тс around 110 К remains unresolved. To find a compromise solving a problem of Bi-2223 superconductor manufacturing might be the use of magnetron sputtering method.

pulse repetition rate. Target-substrate distance is 20-25mm. Thermal treatment of films occurs at 850 0С temperature. It is 3 hours long. It's necessary to point out that in this

Main disadvantage of the PLD method is the existence of clusters (drops) found in evaporated material. It happens because of high density of laser- radiation energy. Attempts to reduce the amount of microdrops via the reduction of energetic density current, lead to an increase in a non-stoichiometric component in a depositional condensate on a substrate. It limits dimensions of substrates that can be used for high-quality film manufacturing.

Among chemical methods used for Bi-2223 film manufacturing we can highlight Liquid epitaxy (Pandey, et al., 1994), Spray Pyrolysis and the deposition from gas-vapor mixture (Kimura, T., Nakao, H. et al., 1991), including the MOCVD metalorganic compounds (Stejskal, J., Leitner, J.J. et al., 2000). In the MOCVD method, the components of superconductive film are transported into a reactor in a shape of gases of metalorganic volatile compounds. They are mixed with gaseous oxidizer resulting in gases' decay either inside the reactor with hot walls or on a heated substrate and the HTSC-film formation.

The advantages of this method include: the ability to make deposition of material onto substrates of large scale, production of homogenous layers in thickness and composition, and the ease of equipment use. At the same time, data found in publications shows that manufactured films are of low-quality. They contain non-superconductive inclusions, the surface has considerable unevenness, and there is no data available on the Bi-2223 phase

Method of thermal deposition of material is the most simple and inexpensive resource for BiPbSrCaCuO film manufacturing. There are several types of used sources, including resistive evaporators, flash evaporation, ion beam (electron beam) evaporators and crucibles

In a study (Azoulay, J., 1989) the researchers made BiSrCaCuO films using resistive heating of tungsten boat, with a source crucible of Bi2.3Sr1.5Ca1Cu3 material composition. After the deposition process, films were thermally processed in an oven at 725 0С for 15 minutes and 840 0С for 5 minutes. Typical superconductive temperature reaching the superconductive

Similar film parameters (Тcе = 78К) are listed in (Patil, J.M., Bhangale, A.R. et al., 1993) that were manufactured with thermal deposition onto MgO substrate (100). In a study (Silver, R.M., Ogawa, E.T., Pan. S. de Lozanne, A.L., 1991) the BiSrCaCuO manufacturing in situ was completed using thermal evaporation method with high-frequency plasma generator. The device was constructed to make plasma at a distance from a substrate in order to stop recombination of oxygen atoms. After the deposition at 600-660 0С temperatures, films were annealed at 850 0С. The dominant phase was 2212, while the 2223 phase was present in small quantities. Similar critical temperatures were produced in studies (Basturk, N., 2005). The analysis of conducted studies shows that high-quality HTSC-film with Bi-2223 phase and Тс around 110 К remains unresolved. To find a compromise solving a problem of Bi-2223 superconductor manufacturing might be the use of magnetron sputtering method.

state Тсе was around 78-К for ZrO2 substrates and 88-К for SrTiO3 substrates.

process the target of Bi 2Pb0.5Sr1.4Ca2Cu3.6Ox composition was used.

Usually they don't exceed 10 mm in diameter.

with radioactive and high-frequency inductive heating.

formation.

## **2.3.2 The magnetron sputtering method – The computer model for the sputtering process**

Magnetron sputtering for superconductive film manufacturing is used in modes of constant current (Schultz et al., 2001) and alternating current (Grigorashvily, Y.E., Volkov, S.I., Sotnikov, I.L. & Mingazin, V.T., 1999; Grigorashvili Y.E., Bukhlin, A.V. & Veryuzhskii, I.V., 2010) with the use of one or several sources (Kuroda, K., Kojima, K. et al., 1991). In this research the following strategy of forming Bi-2223 thin films with superconductive temperature around 110 K on a 40mm-radius substrate was used:


The construction and technology of magnetron sputtering unit should take into account features of a problem, such as the formation of a five-component mixture of oxide metals on a substrate with various physical and chemical characteristics. To justify the construction of the sputtering system via experimental research is extremely expensive. Thus this research process consisted of experiments with the use of a mathematical model of transporting metal atoms from a target onto a substrate.

The most advanced mathematical model of the process of solution transportation with magnetron sputtering belongs to Volpyas (Volpyas, V.A. & Kozirev, A.B., 1997). It was supplemented in accordance with this research problem-solving.

The computer model of the sputtering process used in this research had the following algorithm:


The model allows to determine the thickness of sputtered layer precisely and the stoichiometry of composition depending on coordinates on a substrate. Partial pressures of argon and oxygen can vary in range widely, depending on the conditions of collision-free sputtered atoms until their complete thermalisation.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 251

In order to shorten time and reduce material expenses while searching for optimal gas pressures, we used method of planned experiment (Adler, Yu.P., Markova, E.V. & Granovskiy, Yu.V., 1976). The following arguments were chosen: the relationship of partial pressures of argon and oxygen - P(O2/Ar) and the overall pressure of gas mixture (Psum). The response function consisted of atomic concentration of each component of

Method of planned experiment assumes to find the analytical dependence between function of response and varied arguments (Mongomery, D.C., 1980). The input information for obtaining such correlations comes from experimental results, received in limited quantities of points of argument values. In our research we needed to find concentration correlations of each component (5 in total) in relationship to partial pressures of oxygen/argon and the summarized pressure of gas mixture. Next, we used the optimal plan D that works well is conditions under considerable influence of the noise

During the development of matrix of planed experiment it was important to consider that the expected effect of one factor depends on the level at which the other factor is located. Thus we have the effect of reciprocity of two factors. To quantitatively determine the effects of reciprocity is to use a full-factor experiment. The planning matrix is shown in a table 1. It

number X0 Psum P(O2/Ar) Psum \*P(O2/Ar) Ci

1 +1 +1 +1 +1 Сi (1)

2 +1 -1 +1 -1 Сi (2)

3 +1 -1 -1 +1 Сi (3)

4 +1 +1 -1 -1 Сi (4)

Here Ci (N) is the atomic concentration of i-component in experiment number N. When i equals 1,2,3,4 (i = 1,2,3,4) the concentration corresponds to (Bi,Pb), Sr, Ca, Cu elements. Arguments' values are both on the upper level (+1) and lower level (-1). Quantitative values of these levels are determined by expert estimation. The regression equation for this

Coefficients of variables b0, b1, b2, b12 are determined by standard procedures method of

Сi=b0X0 + b1Psum + b2P(O2/Ar) + b12Psum· P(O2/Ar) (1)

**2.3.3 The influence of technological regimes onto composition of the multi-**

**component mixture** 

sputtered layer.

factor (Mongomery, D.C., 1980).

consists of four real experiments.

Table 1. The matrix of planned experiment

experiment looks like this:

planned experiment.

Experiment's

Computer model was used to calculate the construction of magnetron source, substrate's position, and the pressure range of gas mixture. The magnetron source type, magnetic fields' configuration, and calculated profiles of sputtered film in a multi-component system are shown at Fig. 5.

Fig. 5. The construction of magnetron sputtering system. The magnetic field induction is shown in magnetic conductor and inter-electrode space. The graphs illustrate the thickness of sputtered layer dependence on the target-substrate distance.

Computer model was used to calculate the construction of magnetron source, substrate's position, and the pressure range of gas mixture. The magnetron source type, magnetic fields' configuration, and calculated profiles of sputtered film in a multi-component system

Fig. 5. The construction of magnetron sputtering system. The magnetic field induction is shown in magnetic conductor and inter-electrode space. The graphs illustrate the thickness

of sputtered layer dependence on the target-substrate distance.

are shown at Fig. 5.

## **2.3.3 The influence of technological regimes onto composition of the multicomponent mixture**

In order to shorten time and reduce material expenses while searching for optimal gas pressures, we used method of planned experiment (Adler, Yu.P., Markova, E.V. & Granovskiy, Yu.V., 1976). The following arguments were chosen: the relationship of partial pressures of argon and oxygen - P(O2/Ar) and the overall pressure of gas mixture (Psum). The response function consisted of atomic concentration of each component of sputtered layer.

Method of planned experiment assumes to find the analytical dependence between function of response and varied arguments (Mongomery, D.C., 1980). The input information for obtaining such correlations comes from experimental results, received in limited quantities of points of argument values. In our research we needed to find concentration correlations of each component (5 in total) in relationship to partial pressures of oxygen/argon and the summarized pressure of gas mixture. Next, we used the optimal plan D that works well is conditions under considerable influence of the noise factor (Mongomery, D.C., 1980).

During the development of matrix of planed experiment it was important to consider that the expected effect of one factor depends on the level at which the other factor is located. Thus we have the effect of reciprocity of two factors. To quantitatively determine the effects of reciprocity is to use a full-factor experiment. The planning matrix is shown in a table 1. It consists of four real experiments.


Table 1. The matrix of planned experiment

Here Ci (N) is the atomic concentration of i-component in experiment number N. When i equals 1,2,3,4 (i = 1,2,3,4) the concentration corresponds to (Bi,Pb), Sr, Ca, Cu elements. Arguments' values are both on the upper level (+1) and lower level (-1). Quantitative values of these levels are determined by expert estimation. The regression equation for this experiment looks like this:

$$\mathbf{C}\_{i} = \mathbf{b}\_{0}\mathbf{X}\_{0} + \mathbf{b}\_{1}\mathbf{P}\_{\text{sum}} + \mathbf{b}\_{2}\mathbf{P}(\mathbf{O}\_{2}/\text{Ar}) + \mathbf{b}\_{12}\mathbf{P}\_{\text{sum}} \cdot \mathbf{P}(\mathbf{O}\_{2}/\text{Ar})\tag{1}$$

Coefficients of variables b0, b1, b2, b12 are determined by standard procedures method of planned experiment.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 253

concentrations were measured with the x-ray spectral microanalysis method in ten points.

Thus, the goal was accomplished by obtaining a thin layer of oxide mixture on a substrate, in which metals content corresponded with the formula of a high-temperature 2223

We should point out that during our calculation of regression equations we used a variable summing up of bismuth and lead concentrations, which equals in formula to (2). In experimentally produced films we found the right correlation between bismuth and lead concentrations. In particular, the average value of bismuth is 1.7 atomic fracture and of lead

Average values corresponded with this formula - Bi1.71Pb0.42Sr2.02Ca1.99Cu3.05Ox .

Fig. 6. Dependence of concentration of elements of a film from Psum and P(O2/Ar).

superconductor.

is 0.4.

To lay out a plan of the experiment it's necessary to determine a varied range of arguments. The criteria for range determination include the following: stability of magnetron discharge, absence of electrical gaps between the electrodes, reasonable timeframe of the process, and the formation of the depositional layer in a shape of metal oxides. Last criterion is determined by stage characteristics of a crystal structure formation. If the depositional film has oxygen deficit, then the intensive evaporation of Bismuth and lead takes place during the annealing stage. It leads to stoichiometry disturbance.

For the first approximation the following values of arguments were chosen: lower level - P(O2/Ar) = 0.2; Psum = 5 Pa and the upper level P(O2/Ar) = 0.8; Psum = 8 Pa.

Four experimental tests were performed according to a plan shown in a table 1. Before each experiment the target was sputtered onto a shutter for 50 hours to level out the surface concentration of components. In all four experiments, atomic concentrations of metals did not match the 2223 stoichiometric composition. Thus, it's necessary to search for possible manufacturing regimes (modes) to make the essential mixture. To make it happen we used the regression equation analysis. Standard programs of data analysis let us receive the following equations for atomic percentages of the components:

For Bismuth and lead:

$$\mathbf{C\_{Hi,Pb}} = 2.26 + 2.28 \cdot \mathbf{P\_{sum}} - 0.068 \cdot \mathbf{P(O\_2/Ar)} - 0.443 \cdot \mathbf{P\_{sum}} \cdot \mathbf{P(O\_2/Ar)};\tag{2}$$

For strontium :

$$\mathbf{C\_{Sr}} = 5.32 - 2.38 \cdot \mathbf{P\_{sum}} \cdot 0.56 \cdot \mathbf{P(O\_2/Ar)} + 0.239 \cdot \mathbf{P\_{sum}} \cdot \mathbf{P(O\_2/Ar)};\tag{3}$$

For calcium:

$$\mathbf{C\_{Ca}} = 0.73 - 2.6 \cdot \mathbf{P\_{sum}} + 0.31 \cdot \mathbf{P(O\_2/Ar)} + 0.62 \cdot \mathbf{P\_{sum}} \cdot \mathbf{P(O\_2/Ar)};\tag{4}$$

For copper:

$$\mathbf{C\_{Cu}} = 1.72 + 2.37 \cdot \mathbf{P\_{sum}} + 0.31 \cdot \mathbf{P(O\_2/Ar)} - 0.34 \cdot \mathbf{P\_{sum}} \cdot \mathbf{P(O\_2/Ar)};\tag{5}$$

Regression equations can be presented in graphs that let us find visually a compromised version in a choice of pressures. Color "green" represents values of stoichiometric composition components (Fig.6).

To determine the pressure area (Psum and P (O2/Ar)) simultaneously with all film variables corresponding to a required mixture - (Bi,Pb)2Sr2Ca2Cu3O10, the data array analysis was completed, taken from the regression equation. Area configuration where the mixture conforms to required accuracy is shown at Fig. 7.

Our study of the results has shown that in order to maintain film stoichiometry - (BiPb)2Sr2Ca2Cu3O10, the optimal pressure variables are Psum = 4.7 Pa, P(O2/Ar) = 0.35. We must mention that this point is situated on the outside of the square, rather than its inside. The square shows conditions of the initial experiment.

To experimentally test our conclusions, the target was adjusted and the sputtering was completed at the following pressures - Psum – 4.7 Pa, P (O2/Ar) – 0.35. Component

To lay out a plan of the experiment it's necessary to determine a varied range of arguments. The criteria for range determination include the following: stability of magnetron discharge, absence of electrical gaps between the electrodes, reasonable timeframe of the process, and the formation of the depositional layer in a shape of metal oxides. Last criterion is determined by stage characteristics of a crystal structure formation. If the depositional film has oxygen deficit, then the intensive evaporation of Bismuth and lead takes place during

For the first approximation the following values of arguments were chosen: lower level -

Four experimental tests were performed according to a plan shown in a table 1. Before each experiment the target was sputtered onto a shutter for 50 hours to level out the surface concentration of components. In all four experiments, atomic concentrations of metals did not match the 2223 stoichiometric composition. Thus, it's necessary to search for possible manufacturing regimes (modes) to make the essential mixture. To make it happen we used the regression equation analysis. Standard programs of data analysis let us receive the

CBi,Pb = 2.26 + 2.28· Psum – 0.068· P(O2/Ar) – 0.443· Psum· P(O2/Ar); (2)

CSr = 5.32 – 2.38· Psum -0.56· P(O2/Ar)+0.239· Psum· P(O2/Ar); (3)

CCa = 0.73 – 2.6· Psum + 0.31· P(O2/Ar) + 0.62· Psum · P(O2/Ar); (4)

 CCu = 1.72 + 2.37· Psum + 0.31· P(O2/Ar) – 0.34· Psum · P(O2/Ar); (5) Regression equations can be presented in graphs that let us find visually a compromised version in a choice of pressures. Color "green" represents values of stoichiometric

To determine the pressure area (Psum and P (O2/Ar)) simultaneously with all film variables corresponding to a required mixture - (Bi,Pb)2Sr2Ca2Cu3O10, the data array analysis was completed, taken from the regression equation. Area configuration where the mixture

Our study of the results has shown that in order to maintain film stoichiometry - (BiPb)2Sr2Ca2Cu3O10, the optimal pressure variables are Psum = 4.7 Pa, P(O2/Ar) = 0.35. We must mention that this point is situated on the outside of the square, rather than its inside.

To experimentally test our conclusions, the target was adjusted and the sputtering was completed at the following pressures - Psum – 4.7 Pa, P (O2/Ar) – 0.35. Component

P(O2/Ar) = 0.2; Psum = 5 Pa and the upper level P(O2/Ar) = 0.8; Psum = 8 Pa.

the annealing stage. It leads to stoichiometry disturbance.

following equations for atomic percentages of the components:

For Bismuth and lead:

For strontium :

For calcium:

For copper:

composition components (Fig.6).

conforms to required accuracy is shown at Fig. 7.

The square shows conditions of the initial experiment.

concentrations were measured with the x-ray spectral microanalysis method in ten points. Average values corresponded with this formula - Bi1.71Pb0.42Sr2.02Ca1.99Cu3.05Ox .

Thus, the goal was accomplished by obtaining a thin layer of oxide mixture on a substrate, in which metals content corresponded with the formula of a high-temperature 2223 superconductor.

We should point out that during our calculation of regression equations we used a variable summing up of bismuth and lead concentrations, which equals in formula to (2). In experimentally produced films we found the right correlation between bismuth and lead concentrations. In particular, the average value of bismuth is 1.7 atomic fracture and of lead is 0.4.

Fig. 6. Dependence of concentration of elements of a film from Psum and P(O2/Ar).

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 255

a

b

c Fig. 8. Morphology of a surface and temperature dependence of resistance of films after

Finally, the optimal annealing modes stimulate 2223 phase growth along the substrate's surface. To obtain such results we used the difference in growth speed of 2223 phase along the axis a, b and c. Grains looked disc-shaped of minor height yet big dimension,

various modes of annealing

Fig. 8 c.

Fig. 7. The area of pressure in which deposited composition corresponds stoichiometric with accuracy: а - 4 % and b - 5 %.

## **2.3.4 The formation of crystal structure**

It was shown in a 2.1 section that in the oxide mixtures of stoichiometric composition and temperatures above 820 С, the formation of superconductive phases takes place. The phase of Bi-2223 composition is formed at higher temperature rate of 860 0С and a narrow temperature interval. These results are taken to validate the manufacturing technology of thin film superconductive structures. Yet, there are a few considerable differences that must be taken into account.

High-temperature annealing leads to changes in composition not only at its surface, but also at its entire thickness. If the film is annealed at 850…..860 0С temperatures in an air atmosphere, then it will evaporate from a substrate's surface in a few hours of annealing. Thinly-layered silver sputtering over it doesn't impede this process.

The conducted experiments have shown that the evaporation process depends on both bismuth and lead losses. If the annealing is done in clean oxygen, then it's difficult to form 2223 phase. If the annealing is done in inert atmosphere, then bismuth, copper, and lead evaporate quickly. To stop the evaporation process by regulating the oxygen levels is not possible.

In our research to hold the mixture at substrate's surface, we used oxidized atmosphere with partial pressure of bismuth and lead equal to saturated vapors of these components at annealing temperature. If the process managed to form 2223 phase, then the structure remained stable even at high temperatures.

The Fig 8 shows the surface morphology and film resistivity dependence on temperature after various annealing stages. Low-temperature annealing leads to phase formations that become conductors in 77….300-К temperature range, Fig. 8 a. The annealing at close to optimal conditions forms 2223 phase. Yet, its grains have chaotic orientation. Moreover, there are parts with 2212 phase and other conductive phases, Fig. 8 b.

Fig. 7. The area of pressure in which deposited composition corresponds stoichiometric with

It was shown in a 2.1 section that in the oxide mixtures of stoichiometric composition and temperatures above 820 С, the formation of superconductive phases takes place. The phase of Bi-2223 composition is formed at higher temperature rate of 860 0С and a narrow temperature interval. These results are taken to validate the manufacturing technology of thin film superconductive structures. Yet, there are a few considerable differences that must

High-temperature annealing leads to changes in composition not only at its surface, but also at its entire thickness. If the film is annealed at 850…..860 0С temperatures in an air atmosphere, then it will evaporate from a substrate's surface in a few hours of annealing.

The conducted experiments have shown that the evaporation process depends on both bismuth and lead losses. If the annealing is done in clean oxygen, then it's difficult to form 2223 phase. If the annealing is done in inert atmosphere, then bismuth, copper, and lead evaporate quickly. To stop the evaporation process by regulating the oxygen levels is not

In our research to hold the mixture at substrate's surface, we used oxidized atmosphere with partial pressure of bismuth and lead equal to saturated vapors of these components at annealing temperature. If the process managed to form 2223 phase, then the structure

The Fig 8 shows the surface morphology and film resistivity dependence on temperature after various annealing stages. Low-temperature annealing leads to phase formations that become conductors in 77….300-К temperature range, Fig. 8 a. The annealing at close to optimal conditions forms 2223 phase. Yet, its grains have chaotic orientation. Moreover,

Thinly-layered silver sputtering over it doesn't impede this process.

there are parts with 2212 phase and other conductive phases, Fig. 8 b.

accuracy: а - 4 % and b - 5 %.

be taken into account.

possible.

**2.3.4 The formation of crystal structure** 

remained stable even at high temperatures.

Fig. 8. Morphology of a surface and temperature dependence of resistance of films after various modes of annealing

Finally, the optimal annealing modes stimulate 2223 phase growth along the substrate's surface. To obtain such results we used the difference in growth speed of 2223 phase along the axis a, b and c. Grains looked disc-shaped of minor height yet big dimension, Fig. 8 c.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 257

All researched films had critical temperature around 110 К. Yet, technical parameter variations could considerably change the size of critical current and magnetic characteristics. The track consists of continuously connected grains of monocrystal superconductor and intergranular boundaries. These two elements behave differently. It was experimentally proven that grains remain to be superconductive at current density 105 А/сm2 and constant magnetic filed 1 T. Intergranular boundaries can be either dialectical, conductive or Josephson elements. If the boundaries are resistive fields, then during the increase in current density, there is a residual current registration at temperatures below critical ones. Every boundary behaves as the Josephson junction in the intermediate condition. Overall, the entire structure can be considered as the Josephson environment. Although such boundaries reduce critical current density, they simultaneously increase the responsiveness to magnetic

It was experimentally shown that magnetic responsiveness - S depends on sample's temperature, value of a measuring current, and the average value of magnetic field at measurements. All values had extremum in function of third variable at two other fixed variables. Fig. 10 shows magnetic responsiveness dependence on a measuring current in

Manufactured films have anisotropic properties in relation to the magnetic field. The Fig. 11 illustrates resistance of the microbridge change depending on the angle between magnetic field direction and substrate's surface. The resistance difference increases with the increase

Value S is managed by technological regimes in broad limits. It's possible to pick up such options when the microbridge in resistance measurement mode can be used as a magnetic field device with high sensitivity. Samples were manufactured with ~ 4 103 T-1 magnetic sensitivity at 77 К temperature in magnetic fields measuring less than 100 μT. Magnetic flux

Fig. 10. The dependence of the magnetic sensitivity on the value of the measuring current in

S= [R(B)/R(0) – 1]/dB (6)

field. Usually magnetic response value is determined as:

various ranges of measured magnetic fields.

of magnetic field's absolute value.

sensitivity of this device is about 0.1 Ф0.

static magnetic field. The temperature is 77K.

Here R(0) – is the sample resistance in external magnetic field B0.

R(B) – is the sample resistance in magnetic field that equals to B0+dB.

Fig. 9. Morphology of a surface of a film with the oriented grains

Using structures with sputtered films less then 100nm in thickness, the superconductive layer becomes shaped in the polycrystal units. Its orientation repeats the structure of MgO monocrystal. The surface morphology of such samples is very smooth in both electron and tunnel microscopes. During the increase in sputtered layer thickness structures arise having several storey grain formations. The example of such formation is shown on the Fig 9. The sample has continuous superconductive layer with 110 K critical temperature. Small grains located on the surface have orientation of a superconductive crystal.

## **2.4 Electrical and magnetic characteristics of Bi-2223 thin films**

To research both electrical and magnetic characteristics of Bi-2223 films (Grigorashvily, Y.E., Volkov et. al., 2000; Grigorashvili, Y.E., Ichkitidze, L.P., Mingazin, V.T., 2004) using photolithography methods, the bridge-shaped structures were made 0.5….2.0 mm in length and 5….50 μm in width. Wide areas (200х200 μm2.) were formed on the edges of bridges. There were two contact areas made on each field with sputtering and the following silver annealing. Measurements were taken by four probe method at a computer device. In all measurements magnetic field vector B was directed perpendicular to a transport current. Current density varied through sample in a range between 101…..106 А/сm2.

Fig. 9. Morphology of a surface of a film with the oriented grains

located on the surface have orientation of a superconductive crystal.

**2.4 Electrical and magnetic characteristics of Bi-2223 thin films** 

Current density varied through sample in a range between 101…..106 А/сm2.

Using structures with sputtered films less then 100nm in thickness, the superconductive layer becomes shaped in the polycrystal units. Its orientation repeats the structure of MgO monocrystal. The surface morphology of such samples is very smooth in both electron and tunnel microscopes. During the increase in sputtered layer thickness structures arise having several storey grain formations. The example of such formation is shown on the Fig 9. The sample has continuous superconductive layer with 110 K critical temperature. Small grains

To research both electrical and magnetic characteristics of Bi-2223 films (Grigorashvily, Y.E., Volkov et. al., 2000; Grigorashvili, Y.E., Ichkitidze, L.P., Mingazin, V.T., 2004) using photolithography methods, the bridge-shaped structures were made 0.5….2.0 mm in length and 5….50 μm in width. Wide areas (200х200 μm2.) were formed on the edges of bridges. There were two contact areas made on each field with sputtering and the following silver annealing. Measurements were taken by four probe method at a computer device. In all measurements magnetic field vector B was directed perpendicular to a transport current. All researched films had critical temperature around 110 К. Yet, technical parameter variations could considerably change the size of critical current and magnetic characteristics. The track consists of continuously connected grains of monocrystal superconductor and intergranular boundaries. These two elements behave differently. It was experimentally proven that grains remain to be superconductive at current density 105 А/сm2 and constant magnetic filed 1 T. Intergranular boundaries can be either dialectical, conductive or Josephson elements. If the boundaries are resistive fields, then during the increase in current density, there is a residual current registration at temperatures below critical ones. Every boundary behaves as the Josephson junction in the intermediate condition. Overall, the entire structure can be considered as the Josephson environment. Although such boundaries reduce critical current density, they simultaneously increase the responsiveness to magnetic field. Usually magnetic response value is determined as:

$$\mathbf{S} \triangleq \left[ \mathbf{R}(\mathbf{B}) / \mathbf{R}(\mathbf{0}) - 1 \right] / \mathbf{dB} \tag{6}$$

Here R(0) – is the sample resistance in external magnetic field B0.

R(B) – is the sample resistance in magnetic field that equals to B0+dB.

It was experimentally shown that magnetic responsiveness - S depends on sample's temperature, value of a measuring current, and the average value of magnetic field at measurements. All values had extremum in function of third variable at two other fixed variables. Fig. 10 shows magnetic responsiveness dependence on a measuring current in various ranges of measured magnetic fields.

Manufactured films have anisotropic properties in relation to the magnetic field. The Fig. 11 illustrates resistance of the microbridge change depending on the angle between magnetic field direction and substrate's surface. The resistance difference increases with the increase of magnetic field's absolute value.

Value S is managed by technological regimes in broad limits. It's possible to pick up such options when the microbridge in resistance measurement mode can be used as a magnetic field device with high sensitivity. Samples were manufactured with ~ 4 103 T-1 magnetic sensitivity at 77 К temperature in magnetic fields measuring less than 100 μT. Magnetic flux sensitivity of this device is about 0.1 Ф0.

Fig. 10. The dependence of the magnetic sensitivity on the value of the measuring current in static magnetic field. The temperature is 77K.

Manufacturing Technology of the (Bi,Pb)2Sr2Ca2Cu3O10 High-Temperature Superconductor 259

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Fig. 11. Structure resistance dependence on the angle between substrate's surface and magnetic field direction. Magnetic field is perpendicular to transport current. The microbridge temperature is 77-К.

## **3. Conclusion**

High-temperature superconductor of (Bi,PB)2Sr2Ca2Cu3O10 (Bi-2223) composition can become a useful material for creation of commercial technologies. This chapter contains research results aiming at creation of methods to form superconductor thin films for electronic structures. Thin films were fabricated on the MgO monocrystal substrate that had the most considerable content of 2223 phase. The result is achieved by precise lead doping, stoichiometry maintenance at all stages of formation, as well as creation of special conditions when growing superconductor film epitaxialy repeats the MgO monocrystal substrate and has the structure of strongly textural polycrystal. Changing technological modes, it is possible to control effectively parameters of boundaries between grains, saving thus critical temperature 110К and changing a critical current with magnetic sensitivity that is used for creation of sensors of physical values.

#### **4. Acknowledgments**

This chapter's research results were commissioned by the administration of the Moscow National Research University of Electronic Technology and financed by the Ministry of education and science of the Russian Federation. The author is grateful to the "Superconductive electronics" laboratory employees for their creative input in hightemperature superconductors' problem-solving for many years.

## **5. References**

Adler, Yu.P., Markova, E.V. & Granovskiy, Yu.V. (1976). The design of experiments to find optimal conditions, Publishing house "Science", Moscow

Afanasev, N.V., Chaplygin, Yu. A. (1992). Hysteresis losses in high-temperature superconductors, In: *High-temperature superconductors. Obtaining. Properties. Application,* Grigorashvily Yu. E., pp. 17-27, MIEE, Moscow

Fig. 11. Structure resistance dependence on the angle between substrate's surface and magnetic field direction. Magnetic field is perpendicular to transport current. The

High-temperature superconductor of (Bi,PB)2Sr2Ca2Cu3O10 (Bi-2223) composition can become a useful material for creation of commercial technologies. This chapter contains research results aiming at creation of methods to form superconductor thin films for electronic structures. Thin films were fabricated on the MgO monocrystal substrate that had the most considerable content of 2223 phase. The result is achieved by precise lead doping, stoichiometry maintenance at all stages of formation, as well as creation of special conditions when growing superconductor film epitaxialy repeats the MgO monocrystal substrate and has the structure of strongly textural polycrystal. Changing technological modes, it is possible to control effectively parameters of boundaries between grains, saving thus critical temperature 110К and changing a critical current with magnetic sensitivity that

This chapter's research results were commissioned by the administration of the Moscow National Research University of Electronic Technology and financed by the Ministry of education and science of the Russian Federation. The author is grateful to the "Superconductive electronics" laboratory employees for their creative input in high-

Adler, Yu.P., Markova, E.V. & Granovskiy, Yu.V. (1976). The design of experiments to find

Afanasev, N.V., Chaplygin, Yu. A. (1992). Hysteresis losses in high-temperature

superconductors, In: *High-temperature superconductors. Obtaining. Properties.* 

microbridge temperature is 77-К.

is used for creation of sensors of physical values.

temperature superconductors' problem-solving for many years.

optimal conditions, Publishing house "Science", Moscow

*Application,* Grigorashvily Yu. E., pp. 17-27, MIEE, Moscow

**3. Conclusion** 

**4. Acknowledgments** 

**5. References** 


**0**

**12**

*Colombia*

**Properties of YBa**2**Cu**3**O**7−<sup>δ</sup> **Superconducting**

Omar Ortiz-Diaz1, David A. Landinez Tellez2 and Jairo Roa-Rojas<sup>2</sup> <sup>1</sup>*Grupo de Fisica de Materiales, Universidad Pedagogica y Tecnologica de Colombia*

In a previously published work we have shown that the Sr2YSbO6 material can be successfully used as a buffer layer for the epitaxial growth of YBa2Cu3O7−*<sup>δ</sup>* films with high density current value in self-field at 77 K (Ortiz-Diaz et al., 2010). The layer of Sr2YSbO6 material plays the role of a buffer layer because the negative effects of MgO over the superconducting properties of YBa2Cu3O7−*<sup>δ</sup>* films were eliminated, and because it evidences an excellent structural matching with YBa2Cu3O7−*<sup>δ</sup>* and with MgO. These results show that Sr2YSbO6 can be an excellent substrate material for the YBa2Cu3O7−*<sup>δ</sup>* layers in coated conductors, using the IBAD–MgO templates. Besides which, with the Sr2YSbO6 material the architecture of the coated conductors can be simplified, because only a single Sr2YSbO6 buffer layer is used. The importance of this work is the fact that during the last decade, significant advances in the performance levels of high–temperature superconducting (HTS) wire have made it suitable for commercially viable applications such as electric power cables, fault current limiters, motors, and generators (Maguire&Yuan, 2009; Shiohara et al., 2009). For instance, both the United Sates Department of Energy and private industry have been developing a key superconductor cable and fault current limiter projects (Maguire&Yuan, 2009), and there is a five year Japanese national project for materials and power applications of coated conductors, which was started in 2008 (Shiohara et al., 2009). These power applications share a common requirement: that the superconducting material is formed into a long, strong, and flexible conductor so that it can be used like the copper wire it is intended to replace. And this is where the problems began, because the HTS materials are ceramics that are more like a piece of chalk than the ductile metal copper (Foltyn et al., 2007). The first solution to this problem, the so–called first generation wire, was a tape that was made packing Bi–Sr–Ca–Cu–O (BSCCO) superconducting powder into a silver tube, following a series of rolling and heating steps (Heine et al., 1989). In spite of successful applications, this type of

conductor is expensive for most commercial applications due to the use of silver.

Further, BSCCO is not suitable for applications such as motors and magnets at liquid nitrogen temperature; it loses its ability to carry super current in a magnetic field (Foltyn et al., 2007; Heine et al., 1989). High critical current density in more successfully multifilamentary BSCCO wires, with magnetic field, happen at helium liquid temperatures (Goyal et al., 1996; Shen et al., 2010). The alternative approach, known as the second generation wire, uses the epitaxial growth of an YBa2Cu3O7−*<sup>δ</sup>* superconducting coating on a thin metal tape. The

**1. Introduction**

<sup>2</sup>*Grupo de Fisica de Nuevos Materiales, Universidad Nacional de Colombia*

**Films on Sr**2**YSbO**<sup>6</sup> **Buffer Layers**


## **Properties of YBa**2**Cu**3**O**7−<sup>δ</sup> **Superconducting Films on Sr**2**YSbO**<sup>6</sup> **Buffer Layers**

Omar Ortiz-Diaz1, David A. Landinez Tellez2 and Jairo Roa-Rojas<sup>2</sup> <sup>1</sup>*Grupo de Fisica de Materiales, Universidad Pedagogica y Tecnologica de Colombia* <sup>2</sup>*Grupo de Fisica de Nuevos Materiales, Universidad Nacional de Colombia Colombia*

#### **1. Introduction**

260 Superconductors – Properties, Technology, and Applications

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Substrates by Electron Beam Deposition. *Physica Status Solidi (a)*, Vol. 194, No. 1, pp.

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layer-by-layer molecular-beam-epitaxy and high-resolution transmission electron microscopy analysis of the film/substrate interface and of growth defects. *Journal of* 

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*Alloys and Compounds,* Vol. 251, No. 1-2, pp. 44-47

27, No 2, pp. 1215-1218

5-89028-005-8, St.-Petersburg

206-215

51

In a previously published work we have shown that the Sr2YSbO6 material can be successfully used as a buffer layer for the epitaxial growth of YBa2Cu3O7−*<sup>δ</sup>* films with high density current value in self-field at 77 K (Ortiz-Diaz et al., 2010). The layer of Sr2YSbO6 material plays the role of a buffer layer because the negative effects of MgO over the superconducting properties of YBa2Cu3O7−*<sup>δ</sup>* films were eliminated, and because it evidences an excellent structural matching with YBa2Cu3O7−*<sup>δ</sup>* and with MgO. These results show that Sr2YSbO6 can be an excellent substrate material for the YBa2Cu3O7−*<sup>δ</sup>* layers in coated conductors, using the IBAD–MgO templates. Besides which, with the Sr2YSbO6 material the architecture of the coated conductors can be simplified, because only a single Sr2YSbO6 buffer layer is used.

The importance of this work is the fact that during the last decade, significant advances in the performance levels of high–temperature superconducting (HTS) wire have made it suitable for commercially viable applications such as electric power cables, fault current limiters, motors, and generators (Maguire&Yuan, 2009; Shiohara et al., 2009). For instance, both the United Sates Department of Energy and private industry have been developing a key superconductor cable and fault current limiter projects (Maguire&Yuan, 2009), and there is a five year Japanese national project for materials and power applications of coated conductors, which was started in 2008 (Shiohara et al., 2009). These power applications share a common requirement: that the superconducting material is formed into a long, strong, and flexible conductor so that it can be used like the copper wire it is intended to replace. And this is where the problems began, because the HTS materials are ceramics that are more like a piece of chalk than the ductile metal copper (Foltyn et al., 2007). The first solution to this problem, the so–called first generation wire, was a tape that was made packing Bi–Sr–Ca–Cu–O (BSCCO) superconducting powder into a silver tube, following a series of rolling and heating steps (Heine et al., 1989). In spite of successful applications, this type of conductor is expensive for most commercial applications due to the use of silver.

Further, BSCCO is not suitable for applications such as motors and magnets at liquid nitrogen temperature; it loses its ability to carry super current in a magnetic field (Foltyn et al., 2007; Heine et al., 1989). High critical current density in more successfully multifilamentary BSCCO wires, with magnetic field, happen at helium liquid temperatures (Goyal et al., 1996; Shen et al., 2010). The alternative approach, known as the second generation wire, uses the epitaxial growth of an YBa2Cu3O7−*<sup>δ</sup>* superconducting coating on a thin metal tape. The

effect on *Jc*, based on unpublished experimental results of curves current-voltage (*I*-*V*), which permit to calculate the critical current. Also is shown the relevancy of the critical parameters for applications of superconducting films and layers. Section 3 present a short overview of state of the art about the coated conductors (CC), relevance of the buffer layers for the increase of *Jc* in the second generation of superconducting wires. Special emphasis is made on importance the single buffer layers in order to simplify the fabrication process of CC. We are show the relevance of our Sr2YSbO6 perovskite for this goal. In section 4 We will describe details of the evaluation of the properties of Sr2YSbO6 polycristalline material, such as crystallographic matching and chemical stability between Sr2YSbO6 and YBa2Cu3O7−*δ*; also We show details of the fabrication of Sr2YSbO6 films on MgO single crystal by magnetron sputtering, besides the structural matching of Sr2YSbO6with MgO. Section 5 describes the fabrication of YBa2Cu3O7−*<sup>δ</sup>* target and the growth proccess of films on MgO and SrTiO3 (STO) single crystals and on the Sr2YSbO6 buffer layer. Also is shown the preparation of the conductor path, for electrical measurements, by means of photolitography, and show the results of critical temperature for all the films, and the measurements *I*–*V* based on which

Superconducting Films on Sr2YSbO6 Buffer Layers 263

**2. Superconductivity elementary notions and relevant properties for applications** A superconductor is the material that show two important properties: zero DC resistivity and magnetic induction zero inside the material when it is cooled below temperature *Tc*, known as critical temperature. For temperatures *T* > *Tc*, the superconductor it is in the normal state, and is like normal metal with poor conductivity (Cyrot & Pavuna, 1992; Pool et al., 1995). For a superconductor with a high *Tc* value, in the normal state, the resistance depends linearly on temperature, such as a typical conductor, as shown in figure 1(a). On the other hand, zero magnetic induction in the superconducting state means that the magnetization takes negative values below *Tc* and it is usually zero for temperatures above *Tc*, as shown in figure 1(b).

> 0 10 20 30 40 50 60 70 80 90 100 Temperature (K)

> > (b) Magnetization.


Fig. 1. Resistance and magnetization curves for a YBa2Cu3O7−*<sup>δ</sup>* samples, in function of

The second property of the superconductors is called diamagnetism. When it is perfect a superconducting material does not permit an externally applied magnetic field to penetrate into its interior. However, superconductivity dissapears and the material returns to the normal state if one applies an external magnetic field of the strength greater than some critical value, *Bc*, called the critical thermodynamic field. It worth to say that we are calling magnetic field to the magnetic induction field B. The superconducting state can also destroyed by passing an




Magnetization (emu/g)


0

were determined the critical current density.

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Temperature (K)

(a) Resistance.

0

temperature.

50

100

Resistance (Ω)

150

Properties of YBa2Cu3O7–

advantages of this wire are that very little silver is needed, making it inexpensive, and that the compound YBa2Cu3O7−*<sup>δ</sup>* retains much higher current carrying ability in a magnetic field.

Despite these advantages of superconductors, the ability to carry current without loss is limited to current densities lower than a critical value, *Jc*. In order to carry a higher current in a wire, the objective of research efforts is to increase *Jc*. In this context, the preparation of bi–axially textured substrates and subsequent epitaxial buffer layers is very important for the realization of long–length YBa2Cu3O7−*<sup>δ</sup>* coated conductors. The buffer layer should not only satisfy chemical stability, but structural matching with YBa2Cu3O7−*<sup>δ</sup>* as well because the alignment of the superconductor is required for high *Jc* values (Foltyn et al., 2007; Ying et al., 2009).

Different oxide materials have been successfully used as a buffer layer to fulfill these requirements (Foltyn et al., 2007; Huhne et al., 2006; Jia et al., 2002; 2003; Nishikawa et al., 2003; Parans et al., 2003; Sathyamurthy et al., 2003; Wang et al., 2005; Wee et al., 2005). However, most of them are really multilayer architectures, which significantly increase the complexity as well as the cost of production (Ying et al., 2009). Therefore, the development of a single buffer layer is of great interest, as this might simplify the preparation process and lead to a more cost–effective fabrication of coated conductors. To fabricate templates of great length, the most promising approach is, generally, with ion beam assisted deposition, IBAD yttria–stabilized zirconia (YSZ), Gd2Zr2O7 or MgO (Arendt et al., 2004; Groves et al., 2002; Hanyu et al., 2007; Koa et al., 2007). Of these, the best is IBAD–MgO because very good biaxial texture can be obtained with films only 10 nm thick, which reduces the production costs (Arendt et al., 2004; Foltyn et al., 2007; Koa et al., 2007). Section 3 is extended a bit the issue of buffer layers and materials used in them.

Sr2YSbO6 appears to be a promising material for fulfill these requirements as a buffer layer, as it has a lattice parameter exhibiting a good lattice match with YBa2Cu3O7−*<sup>δ</sup>* (mismatch between *<sup>a</sup>* and *<sup>b</sup>* YBa2Cu3O7−*<sup>δ</sup>* parameters and *<sup>a</sup>* of Sr2YSbO6 is 5%). Previously we have showed that Sr2YSbO6 has been applied effectively as a buffer layer for Sr2YSbO6 film growth by DC sputtering (Ortiz-Diaz et al., 2010). This superconducting film has a *Jc* value 103 times higher than one grown on MgO. The Sr2YSbO6 films were deposited over MgO single–crystal substrate, because Sr2YSbO6 has a good match with MgO, which is the material of the IBAD–MgO tapes. Other applications for the Sr2YSbO6 material are in a Josephson junction because they are an insulating material for the deposition of superconductor films in microwave applications, and for the elaboration of crucibles for the preparation of superconductors due to their chemical non reactivity with YBa2Cu3O7−*δ*.

The Sr2YSbO6 material was chosen because we had been working to find new substrates for YBa2Cu3O7−*<sup>δ</sup>* within the perovskite family A2BB� O6 since by means of substitutions they permit adjusting the lattice parameters (Ortiz-Diaz et al., 2004a). Preliminary studies of the material in polycrystalline form showed that it was viable as a substrate for the growth of YBa2Cu3O7−*<sup>δ</sup>* films (Ortiz-Diaz et al., 2004b; 2010).

The goal of this chapter is to show a summary of these works, which were taken from a period of 2004–2010, showing the technology of manufacturing, since synthesis and evaluation of a material as a potential substrate for superconducting films, until the effective utilization as a substrate for films with excellent superconducting properties.

Section 2 begin with definition of elementary concepts of critical temperature, critical current density *Jc* and critical magnetic field. One of these topics concerns about the temperature 2 Will-be-set-by-IN-TECH

advantages of this wire are that very little silver is needed, making it inexpensive, and that the compound YBa2Cu3O7−*<sup>δ</sup>* retains much higher current carrying ability in a magnetic field. Despite these advantages of superconductors, the ability to carry current without loss is limited to current densities lower than a critical value, *Jc*. In order to carry a higher current in a wire, the objective of research efforts is to increase *Jc*. In this context, the preparation of bi–axially textured substrates and subsequent epitaxial buffer layers is very important for the realization of long–length YBa2Cu3O7−*<sup>δ</sup>* coated conductors. The buffer layer should not only satisfy chemical stability, but structural matching with YBa2Cu3O7−*<sup>δ</sup>* as well because the alignment of the superconductor is required for high *Jc* values (Foltyn et al., 2007; Ying et al.,

Different oxide materials have been successfully used as a buffer layer to fulfill these requirements (Foltyn et al., 2007; Huhne et al., 2006; Jia et al., 2002; 2003; Nishikawa et al., 2003; Parans et al., 2003; Sathyamurthy et al., 2003; Wang et al., 2005; Wee et al., 2005). However, most of them are really multilayer architectures, which significantly increase the complexity as well as the cost of production (Ying et al., 2009). Therefore, the development of a single buffer layer is of great interest, as this might simplify the preparation process and lead to a more cost–effective fabrication of coated conductors. To fabricate templates of great length, the most promising approach is, generally, with ion beam assisted deposition, IBAD yttria–stabilized zirconia (YSZ), Gd2Zr2O7 or MgO (Arendt et al., 2004; Groves et al., 2002; Hanyu et al., 2007; Koa et al., 2007). Of these, the best is IBAD–MgO because very good biaxial texture can be obtained with films only 10 nm thick, which reduces the production costs (Arendt et al., 2004; Foltyn et al., 2007; Koa et al., 2007). Section 3 is extended a bit the

Sr2YSbO6 appears to be a promising material for fulfill these requirements as a buffer layer, as it has a lattice parameter exhibiting a good lattice match with YBa2Cu3O7−*<sup>δ</sup>* (mismatch between *<sup>a</sup>* and *<sup>b</sup>* YBa2Cu3O7−*<sup>δ</sup>* parameters and *<sup>a</sup>* of Sr2YSbO6 is 5%). Previously we have showed that Sr2YSbO6 has been applied effectively as a buffer layer for Sr2YSbO6 film growth by DC sputtering (Ortiz-Diaz et al., 2010). This superconducting film has a *Jc* value 103 times higher than one grown on MgO. The Sr2YSbO6 films were deposited over MgO single–crystal substrate, because Sr2YSbO6 has a good match with MgO, which is the material of the IBAD–MgO tapes. Other applications for the Sr2YSbO6 material are in a Josephson junction because they are an insulating material for the deposition of superconductor films in microwave applications, and for the elaboration of crucibles for the preparation of superconductors due to their chemical non reactivity with YBa2Cu3O7−*δ*. The Sr2YSbO6 material was chosen because we had been working to find new substrates for

permit adjusting the lattice parameters (Ortiz-Diaz et al., 2004a). Preliminary studies of the material in polycrystalline form showed that it was viable as a substrate for the growth of

The goal of this chapter is to show a summary of these works, which were taken from a period of 2004–2010, showing the technology of manufacturing, since synthesis and evaluation of a material as a potential substrate for superconducting films, until the effective utilization as a

Section 2 begin with definition of elementary concepts of critical temperature, critical current density *Jc* and critical magnetic field. One of these topics concerns about the temperature

O6 since by means of substitutions they

issue of buffer layers and materials used in them.

YBa2Cu3O7−*<sup>δ</sup>* within the perovskite family A2BB�

YBa2Cu3O7−*<sup>δ</sup>* films (Ortiz-Diaz et al., 2004b; 2010).

substrate for films with excellent superconducting properties.

2009).

effect on *Jc*, based on unpublished experimental results of curves current-voltage (*I*-*V*), which permit to calculate the critical current. Also is shown the relevancy of the critical parameters for applications of superconducting films and layers. Section 3 present a short overview of state of the art about the coated conductors (CC), relevance of the buffer layers for the increase of *Jc* in the second generation of superconducting wires. Special emphasis is made on importance the single buffer layers in order to simplify the fabrication process of CC. We are show the relevance of our Sr2YSbO6 perovskite for this goal. In section 4 We will describe details of the evaluation of the properties of Sr2YSbO6 polycristalline material, such as crystallographic matching and chemical stability between Sr2YSbO6 and YBa2Cu3O7−*δ*; also We show details of the fabrication of Sr2YSbO6 films on MgO single crystal by magnetron sputtering, besides the structural matching of Sr2YSbO6with MgO. Section 5 describes the fabrication of YBa2Cu3O7−*<sup>δ</sup>* target and the growth proccess of films on MgO and SrTiO3 (STO) single crystals and on the Sr2YSbO6 buffer layer. Also is shown the preparation of the conductor path, for electrical measurements, by means of photolitography, and show the results of critical temperature for all the films, and the measurements *I*–*V* based on which were determined the critical current density.

### **2. Superconductivity elementary notions and relevant properties for applications**

A superconductor is the material that show two important properties: zero DC resistivity and magnetic induction zero inside the material when it is cooled below temperature *Tc*, known as critical temperature. For temperatures *T* > *Tc*, the superconductor it is in the normal state, and is like normal metal with poor conductivity (Cyrot & Pavuna, 1992; Pool et al., 1995). For a superconductor with a high *Tc* value, in the normal state, the resistance depends linearly on temperature, such as a typical conductor, as shown in figure 1(a). On the other hand, zero magnetic induction in the superconducting state means that the magnetization takes negative values below *Tc* and it is usually zero for temperatures above *Tc*, as shown in figure 1(b).

Fig. 1. Resistance and magnetization curves for a YBa2Cu3O7−*<sup>δ</sup>* samples, in function of temperature.

The second property of the superconductors is called diamagnetism. When it is perfect a superconducting material does not permit an externally applied magnetic field to penetrate into its interior. However, superconductivity dissapears and the material returns to the normal state if one applies an external magnetic field of the strength greater than some critical value, *Bc*, called the critical thermodynamic field. It worth to say that we are calling magnetic field to the magnetic induction field B. The superconducting state can also destroyed by passing an

AAA <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup>

0

0

10000

20000

30000

Critical current density (A/cm²)

YBa2Cu3O7−*<sup>δ</sup>* on MgO single crystal.

40000

50000

0,005 Voltage (V)

Properties of YBa2Cu3O7–

<sup>A</sup> <sup>A</sup> A A A A A A A A A A A A <sup>A</sup> <sup>A</sup> A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A A AA

A A AA <sup>A</sup> AA<sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup> <sup>A</sup>

AA <sup>A</sup> <sup>A</sup> AA <sup>A</sup> <sup>A</sup> A

A

A

A A

A

A

A

AA

AA

Superconducting Films on Sr2YSbO6 Buffer Layers 265

A <sup>A</sup>

AA <sup>A</sup> A

Current (mA)

 <sup>A</sup> A

<sup>A</sup> <sup>A</sup> A A A A A <sup>A</sup> <sup>A</sup> <sup>A</sup> A <sup>A</sup> <sup>A</sup> A <sup>A</sup> <sup>A</sup> A A A A A A A A A A A A A A A A <sup>A</sup> <sup>A</sup> A A A

77K 82K 70K 65K 60K 50K <sup>A</sup> 45K 40K 35K 30K 25K 20K 15K 10K <sup>A</sup> 5K

<sup>A</sup> AAA AA

024

(a) *I*–*V* curves.

0 20 40 60 80 Temperature (K)

(b) Critical current.

Fig. 3. *I*–*V* curves for several temperatures and critical current in function of temperature for

current as a function of temperature, calculated for each temperature in the same way as was done for the data in Figure 2. As the temperature increases, the current value for which the potential drop is 1*μ*V decreases dramatically, so that at 82 K the curve approaches the linear behavior as a sign that is disappearing superconducting state. This is equivalent to the reduction of critical current density with increasing temperature as shown in Figure 3(b). In contrast, as shown in Figures 2, for this temperature of 82 K the superconductor film grown on Sr2YSbO6 buffer layer maintains a relatively high critical current value. From applications viewpoint, is important to determine the critical current value for at determined temperature. For HTS superconductors often the critical current is measured at the reference value of 77 K.

A

excessive current greater than some value *Ic*, called critical current, which creates a magnetic field at the surface. This limits the maximun current value that the material can sustain and it is a crucial problem for applications of superconducting materials. Those materials that totally exclude an applied magnetic flux, for *T* < *Tc*, are known as type I superconductors.

There are other superconductors, known as type II superconductors, that totally exclude magnetic field when the applied magnetic field is low, but only partially exclude it when the applied field is higher. For this superconductors there are tow critical fields: the lower *Bc*<sup>1</sup> and the upper *Bc*2. The flux is totally expelled only up to the lower critical field *Bc*1. Then, in applied fields smaller than *Bc*1, the type II superconductor behaves like a type I superconductor below *Bc*. Above *Bc*<sup>1</sup> the flux penetrate into the material until the upper critical field *Bc*<sup>2</sup> is reached. Above *Bc*<sup>2</sup> the material returns to the normal state.

Between *Bc*<sup>1</sup> and *Bc*2, the type II superconductor is said to be in the mixed state. The Meissner effect is only partial. For this range of fields magnetic flux partially penetrates the superconducting sample in the form tiny filaments known as vortices. A vortex consists of a normal core in which the magnetic field is large, surrounded by a superconducting region in which flows supercurrent.

Critical magnetic field and, obviously, critical current depends on temperature. Because of that it is important to report the temperature value at which the measures of critical current are carried out. Figure 2(a) shows a typical curve of voltage *V* in function on applied current *<sup>I</sup>* at temperature of 82 K for a YBa2Cu3O7−*<sup>δ</sup>* film which was growth on Sr2YSbO6 buffer layer. Note that this temperature is higher than 77 K, usually reported for critical current measurements in high temperature superconductors. In these measurements one applies current *I* and takes a voltage value between two electrodes. In the so–called criterion of a microvolt/cm (*μ*V/cm), the current value for which the voltage drop in a 1 cm length is 1*μ* V is defined as a critical current value. Figure 2(b) shows an expanded region of the figure 2(a), in which the red dashed line is a visual guide to determine the current value for which the drop voltage is 1*μ*V. For this example, the critical current value is 4.14 mA, from which by dividing by the cross–sectional area of the sample, yields a relatively high value of *Jc* <sup>∼</sup> 0.2 <sup>×</sup> 106 A/cm2.

Fig. 2. Voltage in function of current, at 82 K for a YBa2Cu3O7−*<sup>δ</sup>* film growth on Sr2YSbO6/MgO substrate.

Figure 3(a) shows the voltage curves as a function of current for YBa2Cu3O7−*<sup>δ</sup>* film on MgO at different temperatures from 5 K to 82 K and, Figure 3(b), the behavior of the density critical 4 Will-be-set-by-IN-TECH

excessive current greater than some value *Ic*, called critical current, which creates a magnetic field at the surface. This limits the maximun current value that the material can sustain and it is a crucial problem for applications of superconducting materials. Those materials that totally

There are other superconductors, known as type II superconductors, that totally exclude magnetic field when the applied magnetic field is low, but only partially exclude it when the applied field is higher. For this superconductors there are tow critical fields: the lower *Bc*<sup>1</sup> and the upper *Bc*2. The flux is totally expelled only up to the lower critical field *Bc*1. Then, in applied fields smaller than *Bc*1, the type II superconductor behaves like a type I superconductor below *Bc*. Above *Bc*<sup>1</sup> the flux penetrate into the material until the upper

Between *Bc*<sup>1</sup> and *Bc*2, the type II superconductor is said to be in the mixed state. The Meissner effect is only partial. For this range of fields magnetic flux partially penetrates the superconducting sample in the form tiny filaments known as vortices. A vortex consists of a normal core in which the magnetic field is large, surrounded by a superconducting region in

Critical magnetic field and, obviously, critical current depends on temperature. Because of that it is important to report the temperature value at which the measures of critical current are carried out. Figure 2(a) shows a typical curve of voltage *V* in function on applied current *<sup>I</sup>* at temperature of 82 K for a YBa2Cu3O7−*<sup>δ</sup>* film which was growth on Sr2YSbO6 buffer layer. Note that this temperature is higher than 77 K, usually reported for critical current measurements in high temperature superconductors. In these measurements one applies current *I* and takes a voltage value between two electrodes. In the so–called criterion of a microvolt/cm (*μ*V/cm), the current value for which the voltage drop in a 1 cm length is 1*μ* V is defined as a critical current value. Figure 2(b) shows an expanded region of the figure 2(a), in which the red dashed line is a visual guide to determine the current value for which the drop voltage is 1*μ*V. For this example, the critical current value is 4.14 mA, from which by dividing by the cross–sectional area of the sample, yields a relatively high value of

> 01234 5 6 Current (mA)

(b) Detailed section of curve.

0

1e-06

Voltage (V)

2e-06

3e-06

exclude an applied magnetic flux, for *T* < *Tc*, are known as type I superconductors.

critical field *Bc*<sup>2</sup> is reached. Above *Bc*<sup>2</sup> the material returns to the normal state.

which flows supercurrent.

*Jc* <sup>∼</sup> 0.2 <sup>×</sup> 106 A/cm2.


Sr2YSbO6/MgO substrate.


0

Voltage (V)

0,0001

0,0002


Fig. 2. Voltage in function of current, at 82 K for a YBa2Cu3O7−*<sup>δ</sup>* film growth on

Figure 3(a) shows the voltage curves as a function of current for YBa2Cu3O7−*<sup>δ</sup>* film on MgO at different temperatures from 5 K to 82 K and, Figure 3(b), the behavior of the density critical

(a) Complete curve.

Fig. 3. *I*–*V* curves for several temperatures and critical current in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* on MgO single crystal.

current as a function of temperature, calculated for each temperature in the same way as was done for the data in Figure 2. As the temperature increases, the current value for which the potential drop is 1*μ*V decreases dramatically, so that at 82 K the curve approaches the linear behavior as a sign that is disappearing superconducting state. This is equivalent to the reduction of critical current density with increasing temperature as shown in Figure 3(b). In contrast, as shown in Figures 2, for this temperature of 82 K the superconductor film grown on Sr2YSbO6 buffer layer maintains a relatively high critical current value. From applications viewpoint, is important to determine the critical current value for at determined temperature. For HTS superconductors often the critical current is measured at the reference value of 77 K.

of its structure. The unit crystallographic structure of YBa2Cu3O7−*δ*, shown in figure 5, can be seen as a stacking of three perovskites. So, crystallographers classify the structure of YBa2Cu3O7−*<sup>δ</sup>* as the perovskite type with vacancies of oxygen; the oxygen contents it is in the range 6 to 7, as indicates the subscript 7 − *δ*. The lattice parameter *c* is around three times

a

Ba

Thus, its structure consists of a sequence of oxide layers perpendicular to the *c*-axis separated by Y and Ba atoms, which can be seen in figure 6. Figure 6(a) corresponds to *δ* = 1, that is, to YBa2Cu3O6 and, figure 6(b) is the representation of YBa2Cu3O7. The sequence of oxide layers

• a Cu–O layer with two vacancies of oxygen for figure 6(b); each Cu is surrounded by 4 O. Thus the typical octahedral coordination has been replaced for a square planar coordination. So along the *b*-axis are formed the called chain copper oxygen connect by oxygen atoms. In the YBa2Cu3O6 there is not chains because each copper ion has

• a Cu2O layer where each copper exhibit fivefold pyramidal coordination. The basis of pyramids linked by oxygen atoms, forms the called copper oxygen planes through the

The tetragonal YBa2Cu3O6 compound is an insulator. By increasing of oxygen concentration the *ab*-planes are gradually doped with holes and eventually it reaches the YBa2Cu3O7 composition in which there are not vacancies of oxygen. The maximun *Tc* = 94 K is obtained for YBa2Cu3O6.93 and for YBa2Cu3O7 the critical temperature is lower (Cyrot & Pavuna, 1992;

• a layer of Y with for oxygen vacancies sandwiched between two Cu–O planes.

These sequence is duplicated in the upper half of the structure.

Y

Cu

Superconducting Films on Sr2YSbO6 Buffer Layers 267

O

b

*c*

Fig. 5. Schematic representation of YBa2Cu3O7−*<sup>δ</sup>* structure.

*b a*

larger than *a* and *b* parameters.

Properties of YBa2Cu3O7–

are as follows:

• a barium layer,

*ab*-plane,

surrounded by two oxygen,

It worth noting that the issue of the temperature is crucial factor for the cost of practical uses of superconductors. For the called conventional superconductors is required the liquid helium coolant which have very higher cost. Nitrogen becomes liquid at 77 K; it is far less costly to liquefy this gas than to liquefy helium. For any application in which liquid nitrogen can replaced liquid helium, the refrigeration cost will be about 1000 times less (Sheahen, 2002). High-*Tc* oxides have critical temperatures above liquid nitrogen. For instance, the YBa2Cu3O7−*<sup>δ</sup>* has been found to be superconducting up to ∼ 90 K; then liquid nitrogen is sufficient to cool YBa2Cu3O7−*<sup>δ</sup>* in the superconductor range.

The maximum current density, *Jc*, that a superconductor can carry without exceeding a voltage drop criterion (1*μ* V/cm) depends both on the superconductor temperature and the applied magnetic field. From the practical point of view the critical temperature, *Tc*, is the temperature at which a very small *Jc* may flow at zero applied field. At a fixed temperature less than *Tc*, *Jc* will usually drop very rapidly above some threshold value of applied field. Thus, it is common to see a critical *B* plotted as a function of *T*, defined at some value of *Jc*. For an HTS, this critical magnetic field is called the irreversibility field, *Birr*, which is the field at which flux lines or vortices to flow and causes dissipation. This value is often considerably lower than the *Bc*<sup>2</sup> of the material (Hull, 2003). Figure 4(a) shows *Bc*<sup>2</sup> in function of temperature for some common superconductor materials. For YBa2Cu3O7−*<sup>δ</sup>* and BSCCO *Bc*<sup>2</sup> has high values at liquid nitrogen temperature. Figure 4(b) shows *Birr* in function of temperature for some materials. The *Birr* curve for BSCCO is noteworthy because it is relatively low at 77 K but starts to rise rapidly as temperatures drop below about 30 K. Thus, at temperatures near 77 K, applications for BSSCO will be limited to low magnetic field applications, such as transmission lines. Higher field applications, such as motors and generators, will require the expensive helium cooling of the BSSCO. This mean that the YBa2Cu3O7−*<sup>δ</sup>* is an ideal candidate for these applications at liquid nitrogen temperature, which appears as a dashed line in figure 4(b). This is the main reason why the YBa2Cu3O7−*<sup>δ</sup>* is the preferred material for applications involving high magnetic fields, and therefore, the reason why we decided to use YBa2Cu3O7−*<sup>δ</sup>* and no other superconductor material for our study.

Fig. 4. Magnetic field in function of the temperature for some superconductors. Data adapted from (Hull, 2003).

Structural and transport properties of the HTS oxide materials is highly anisotropic. For this reason it is helpful, for the understanding of the properties of the YBa2Cu3O7−*δ*, a description of its structure. The unit crystallographic structure of YBa2Cu3O7−*δ*, shown in figure 5, can be seen as a stacking of three perovskites. So, crystallographers classify the structure of YBa2Cu3O7−*<sup>δ</sup>* as the perovskite type with vacancies of oxygen; the oxygen contents it is in the range 6 to 7, as indicates the subscript 7 − *δ*. The lattice parameter *c* is around three times larger than *a* and *b* parameters.

Fig. 5. Schematic representation of YBa2Cu3O7−*<sup>δ</sup>* structure.

Thus, its structure consists of a sequence of oxide layers perpendicular to the *c*-axis separated by Y and Ba atoms, which can be seen in figure 6. Figure 6(a) corresponds to *δ* = 1, that is, to YBa2Cu3O6 and, figure 6(b) is the representation of YBa2Cu3O7. The sequence of oxide layers are as follows:


6 Will-be-set-by-IN-TECH

It worth noting that the issue of the temperature is crucial factor for the cost of practical uses of superconductors. For the called conventional superconductors is required the liquid helium coolant which have very higher cost. Nitrogen becomes liquid at 77 K; it is far less costly to liquefy this gas than to liquefy helium. For any application in which liquid nitrogen can replaced liquid helium, the refrigeration cost will be about 1000 times less (Sheahen, 2002). High-*Tc* oxides have critical temperatures above liquid nitrogen. For instance, the YBa2Cu3O7−*<sup>δ</sup>* has been found to be superconducting up to ∼ 90 K; then liquid nitrogen is

The maximum current density, *Jc*, that a superconductor can carry without exceeding a voltage drop criterion (1*μ* V/cm) depends both on the superconductor temperature and the applied magnetic field. From the practical point of view the critical temperature, *Tc*, is the temperature at which a very small *Jc* may flow at zero applied field. At a fixed temperature less than *Tc*, *Jc* will usually drop very rapidly above some threshold value of applied field. Thus, it is common to see a critical *B* plotted as a function of *T*, defined at some value of *Jc*. For an HTS, this critical magnetic field is called the irreversibility field, *Birr*, which is the field at which flux lines or vortices to flow and causes dissipation. This value is often considerably lower than the *Bc*<sup>2</sup> of the material (Hull, 2003). Figure 4(a) shows *Bc*<sup>2</sup> in function of temperature for some common superconductor materials. For YBa2Cu3O7−*<sup>δ</sup>* and BSCCO *Bc*<sup>2</sup> has high values at liquid nitrogen temperature. Figure 4(b) shows *Birr* in function of temperature for some materials. The *Birr* curve for BSCCO is noteworthy because it is relatively low at 77 K but starts to rise rapidly as temperatures drop below about 30 K. Thus, at temperatures near 77 K, applications for BSSCO will be limited to low magnetic field applications, such as transmission lines. Higher field applications, such as motors and generators, will require the expensive helium cooling of the BSSCO. This mean that the YBa2Cu3O7−*<sup>δ</sup>* is an ideal candidate for these applications at liquid nitrogen temperature, which appears as a dashed line in figure 4(b). This is the main reason why the YBa2Cu3O7−*<sup>δ</sup>* is the preferred material for applications involving high magnetic fields, and therefore, the reason why we decided to use YBa2Cu3O7−*<sup>δ</sup>* and no other superconductor material for our

sufficient to cool YBa2Cu3O7−*<sup>δ</sup>* in the superconductor range.

0 20 40 60 80 100 120 Temperature (K)

(a) Magnetic field.

YBCO Bi-2223 MgB2 Nb3Sn NbTi

> 20 30 40 50 60 70 80 90 100 Temperature (K)

> > (b) Irreversivility field.

YBCO MgB2 Bi-2223

0

Fig. 4. Magnetic field in function of the temperature for some superconductors. Data adapted

Structural and transport properties of the HTS oxide materials is highly anisotropic. For this reason it is helpful, for the understanding of the properties of the YBa2Cu3O7−*δ*, a description

5

10

B irrev. (Tesla)

15

20

study.

B field (Tesla)

from (Hull, 2003).


These sequence is duplicated in the upper half of the structure.

The tetragonal YBa2Cu3O6 compound is an insulator. By increasing of oxygen concentration the *ab*-planes are gradually doped with holes and eventually it reaches the YBa2Cu3O7 composition in which there are not vacancies of oxygen. The maximun *Tc* = 94 K is obtained for YBa2Cu3O6.93 and for YBa2Cu3O7 the critical temperature is lower (Cyrot & Pavuna, 1992;

To produce films is necessary to use a substrate, which must satisfies basic requirements to promote epitaxial growth and micro structural properties to obtain high current density values: must have good matching of lattice constants with superconductor; must be chemically non reactant with YBa2Cu3O7−*<sup>δ</sup>* even under heat treatment at temperature of film deposition and; should not affect the values of critical temperature and critical current density of the superconductor. It worth to say that the recent analyses, about current limiting defects in coated conductors, have shown that the major local current–limiting defects are *a*–axis grains, which exert a big effect on current flow and local dissipation in the coated conductor samples studied (Li et al., 2012). By reducing the amount of grains grown along the *a* axis, the quality of the superconducting layers substantially improves. To achieve this purpose,

Superconducting Films on Sr2YSbO6 Buffer Layers 269

besides optimizing the deposition conditions, also, the substrate plays a crucial role.

as low as possible (Koren & Polturak, 2002).

Properties of YBa2Cu3O7–

rotated 45◦ with respect to MgO (1 0 0) surface.

films (Hollmannt et al., 1994).

The crystallography and preparation of the substrate is of primary importance in determining the quality of the film deposition. Some of the highest critical current densities at 77 K and highest film uniformities have been achieved in YBa2Cu3O7−*<sup>δ</sup>* films grown on commercially available (1 0 0) SrTiO3 substrates (Huhne et al., 2006; Jia et al., 2002; 2003; Ortiz-Diaz et al., 2004a; Sathyamurthy et al., 2003; Varanasi et al., 2008; Wang et al., 2005; Wee et al., 2005; Wu et al., 2005; Ying et al., 2009). However, SrTiO3 is fairly expensive, has high dielectric constant, and undergoes phase transition at ∼ 110 K. LaGaO3 with ∼ 0.5% mismatch it is neither cheap nor twin free (Cyrot & Pavuna, 1992). Other commercial substrate, LaAlO3 was reported as a problematic substrate for the deposition of thin YBa2Cu3O7−*<sup>δ</sup>* films, especially when a thick substrate is required, because detrimental extended defects are developed. The use of LaAlO3 layers of 0.5 mm is advantageous, provided the deposition temperature is kept

MgO single crystal substrates have been widely used for its economy and acceptable matching of lattice constants. However, it is widely accepted that MgO affects the superconducting properties. In particular, *Tc* and *Jc* values for YBa2Cu3O7−*<sup>δ</sup>* and REBa2Cu3O7−*<sup>δ</sup>* films are generally lower when are deposited on MgO than the values for films grown on other commercial substrates (Hollmannt et al., 1994; Wee et al., 2005). This has been attributed to chemical reaction between the superconductor and the MgO, which causes the formation of third phases, which act as "non-superconducting dead layers" and; grains (0 0 1) growth

Sapphire is another cheap substrate, free twining. However, has relatively high mismatch with YBa2Cu3O7−*<sup>δ</sup>* and reacts chemically with superconductor, thus, the interfacial BaAl2O4 layer is formed, which affects the epitaxial growth of the film and result in poor superconducting

The list of materials tested as substrates for superconducting films is broad and includes a variety of compounds and crystal structures, ranging from oxides and simple perovskites to complex perovskites and pyrochlore structures. For instance, Yttrium Stabilized Zirconium Oxide (YSZ) and cerium oxide, CeO2 have been widely used, sometimes as a combination of layers. Among simple perovskites that have been proposed, perhaps the most successful are SrTiO3, LaAlO3 and SrRuO3, which are commercially availables as single crystals. Examples of other simple perovskites proposed as substrate materials are YAlO3, GdAlO3, EuAlO3, SmAlO3, LaGaO3, NdGaO3, PrGaO3. Also, were proposed to make substitutions with the idea that variation in the composition allows the lattice parameters vary, and then the matching can be improved, but question arises about the chemical compatibility with YBa2Cu3O7−*<sup>δ</sup>* (Hollmannt et al., 1994); thus, are tested as potential substrates, for

Fig. 6. Schematic representation of YBa2Cu3O7−*<sup>δ</sup>* structures, shown the planes and chains.

Pool et al., 1995). Those compounds YBa2Cu3O7−*<sup>δ</sup>* with optimally oxygen contents have orthorombic structure and are superconductors. They are called YBa2Cu3O7−*<sup>δ</sup>* optimally doped.

Conductivity in *ab*-planes are around 100 times than *c*-axis conductivity. For this fact is accepted that superconductivity essentially takes place within quasi dimensional planes, and YBa2Cu3O7−*<sup>δ</sup>* is considered as Cu–O planes separated by a charge reservoir (the chains).

As the conductivity, the *Jc* of the ceramic oxide materials is too highly anisotropic. For this reason a good alignment of the crystalline axes of all the grains is essential to pass high current density from one superconducting grain to another. For polycristalline specimens the *Jc* is very low, of order hundred A/cm2. In contrast, higher *Jc* values, around <sup>∼</sup> 107 A/cm2, are obtained in single crystals and a good films.

To explain the low value of *Jc* in polycrystalline samples and understand how to improve these values by using the superconductor in the form of films, it is worth noting, as mentioned in section 4 that for the production of a polycrystalline sample in the form of pellet, one mix a precursor components, which are subjected to heat treatment and appropriate oxygenation for optimal superconducting properties. However, even the best polycrystalline samples consist of many randomly oriented polycrystalline grains, each of them with their anisotropic layered structure. It also contains grain boundaries, structural defects such as twins, voids, which act as non conducting grains. All this conspires against applications that require high critical current densities. In contrast, with single crystals and epitaxially grown films are obtained large values of *Jc* at 77 K and self field, because superconductivity occurs through the *ab* planes.

8 Will-be-set-by-IN-TECH

(a) Tetragonal YBa2Cu3O6. (b) Orthorombic YBa2Cu3O7.

Fig. 6. Schematic representation of YBa2Cu3O7−*<sup>δ</sup>* structures, shown the planes and chains.

doped.

planes.

obtained in single crystals and a good films.

Pool et al., 1995). Those compounds YBa2Cu3O7−*<sup>δ</sup>* with optimally oxygen contents have orthorombic structure and are superconductors. They are called YBa2Cu3O7−*<sup>δ</sup>* optimally

Conductivity in *ab*-planes are around 100 times than *c*-axis conductivity. For this fact is accepted that superconductivity essentially takes place within quasi dimensional planes, and YBa2Cu3O7−*<sup>δ</sup>* is considered as Cu–O planes separated by a charge reservoir (the chains). As the conductivity, the *Jc* of the ceramic oxide materials is too highly anisotropic. For this reason a good alignment of the crystalline axes of all the grains is essential to pass high current density from one superconducting grain to another. For polycristalline specimens the *Jc* is very low, of order hundred A/cm2. In contrast, higher *Jc* values, around <sup>∼</sup> 107 A/cm2, are

To explain the low value of *Jc* in polycrystalline samples and understand how to improve these values by using the superconductor in the form of films, it is worth noting, as mentioned in section 4 that for the production of a polycrystalline sample in the form of pellet, one mix a precursor components, which are subjected to heat treatment and appropriate oxygenation for optimal superconducting properties. However, even the best polycrystalline samples consist of many randomly oriented polycrystalline grains, each of them with their anisotropic layered structure. It also contains grain boundaries, structural defects such as twins, voids, which act as non conducting grains. All this conspires against applications that require high critical current densities. In contrast, with single crystals and epitaxially grown films are obtained large values of *Jc* at 77 K and self field, because superconductivity occurs through the *ab*

To produce films is necessary to use a substrate, which must satisfies basic requirements to promote epitaxial growth and micro structural properties to obtain high current density values: must have good matching of lattice constants with superconductor; must be chemically non reactant with YBa2Cu3O7−*<sup>δ</sup>* even under heat treatment at temperature of film deposition and; should not affect the values of critical temperature and critical current density of the superconductor. It worth to say that the recent analyses, about current limiting defects in coated conductors, have shown that the major local current–limiting defects are *a*–axis grains, which exert a big effect on current flow and local dissipation in the coated conductor samples studied (Li et al., 2012). By reducing the amount of grains grown along the *a* axis, the quality of the superconducting layers substantially improves. To achieve this purpose, besides optimizing the deposition conditions, also, the substrate plays a crucial role.

The crystallography and preparation of the substrate is of primary importance in determining the quality of the film deposition. Some of the highest critical current densities at 77 K and highest film uniformities have been achieved in YBa2Cu3O7−*<sup>δ</sup>* films grown on commercially available (1 0 0) SrTiO3 substrates (Huhne et al., 2006; Jia et al., 2002; 2003; Ortiz-Diaz et al., 2004a; Sathyamurthy et al., 2003; Varanasi et al., 2008; Wang et al., 2005; Wee et al., 2005; Wu et al., 2005; Ying et al., 2009). However, SrTiO3 is fairly expensive, has high dielectric constant, and undergoes phase transition at ∼ 110 K. LaGaO3 with ∼ 0.5% mismatch it is neither cheap nor twin free (Cyrot & Pavuna, 1992). Other commercial substrate, LaAlO3 was reported as a problematic substrate for the deposition of thin YBa2Cu3O7−*<sup>δ</sup>* films, especially when a thick substrate is required, because detrimental extended defects are developed. The use of LaAlO3 layers of 0.5 mm is advantageous, provided the deposition temperature is kept as low as possible (Koren & Polturak, 2002).

MgO single crystal substrates have been widely used for its economy and acceptable matching of lattice constants. However, it is widely accepted that MgO affects the superconducting properties. In particular, *Tc* and *Jc* values for YBa2Cu3O7−*<sup>δ</sup>* and REBa2Cu3O7−*<sup>δ</sup>* films are generally lower when are deposited on MgO than the values for films grown on other commercial substrates (Hollmannt et al., 1994; Wee et al., 2005). This has been attributed to chemical reaction between the superconductor and the MgO, which causes the formation of third phases, which act as "non-superconducting dead layers" and; grains (0 0 1) growth rotated 45◦ with respect to MgO (1 0 0) surface.

Sapphire is another cheap substrate, free twining. However, has relatively high mismatch with YBa2Cu3O7−*<sup>δ</sup>* and reacts chemically with superconductor, thus, the interfacial BaAl2O4 layer is formed, which affects the epitaxial growth of the film and result in poor superconducting films (Hollmannt et al., 1994).

The list of materials tested as substrates for superconducting films is broad and includes a variety of compounds and crystal structures, ranging from oxides and simple perovskites to complex perovskites and pyrochlore structures. For instance, Yttrium Stabilized Zirconium Oxide (YSZ) and cerium oxide, CeO2 have been widely used, sometimes as a combination of layers. Among simple perovskites that have been proposed, perhaps the most successful are SrTiO3, LaAlO3 and SrRuO3, which are commercially availables as single crystals. Examples of other simple perovskites proposed as substrate materials are YAlO3, GdAlO3, EuAlO3, SmAlO3, LaGaO3, NdGaO3, PrGaO3. Also, were proposed to make substitutions with the idea that variation in the composition allows the lattice parameters vary, and then the matching can be improved, but question arises about the chemical compatibility with YBa2Cu3O7−*<sup>δ</sup>* (Hollmannt et al., 1994); thus, are tested as potential substrates, for

with (0 0 1)NdBa2Cu3O7−*<sup>δ</sup>*(1 1 0)MgO (Wee et al., 2005). NdBa2Cu3O7−*<sup>δ</sup>* film properties were improved too using a buffer layer of Ba2NdTaO6 on MgO (Kurian & Morishita, 2003).

Superconducting Films on Sr2YSbO6 Buffer Layers 271

The above example is a sample of the use of compounds different to the commercial substrates as a material for buffer layers. Also, with the emergence of the second generation of superconducting wires took great importance to study and search for new buffer layers. For instance, Biaxially textured La2Zr2O7 with CeO2 and Ce0.7Gd0.3O3 buffer layers were grown on Ni–RABiTS (rolling-assisted biaxially textured substrates) to obtain suitable buffer layer architectures for YBa2Cu3O7−*<sup>δ</sup>* coated conductors (Knoth et al., 2005). More recently, it has been used as textured substrate for coated conductors a NiW alloy with a Gd2Zr2O7 buffered Ce0.9La0.1O2−*<sup>y</sup>* for the growth of YBa2Cu3O7−*<sup>δ</sup>* films with *Jc* <sup>∼</sup> <sup>1</sup> <sup>×</sup> 106 A/cm<sup>2</sup> (Zhao et al., 2012). This work is an example of current interest in improving the manufacturing process of superconducting tapes with different combinations of deposition technique and combinations

YBCO film

Fig. 7. Schematic representation of the YBa2Cu3O7−*<sup>δ</sup>* film on Sr2YSbO6 buffer layer.

Film deposition Buffer layer *Jc* (×106 A/cm2) Reference Sputtering RF SrTiO3 0.4 (Lucia et al., 1993) PLD Sr2AlTaO6/LaAlO3 1.3 (Findikoglu et al., 1992) PLD CeO2/IBAD–YSZ 2.2 (Li et al., 2012) PLD SrO/Sr2AlTaO6 ∼ 0.5 (Takahashi et al., 2003) PLD Ba2LaNbO6 <sup>5</sup> <sup>×</sup> 106 (Pai et al., 1997) PLD SrTiO3 4 (Proteau et al., 1995) PLD Ce0.9La0.1O2−*<sup>y</sup>*/Gd2Zr2O7 ∼ 1 (Zhao et al., 2012) Sputtering RF Sr2YSbO6/MgO 0.86 (Ortiz-Diaz et al., 2010)

Table 1. Values of critical current density for YBa2Cu3O7−*<sup>δ</sup>* films on some buffer layers at

We have mentioned a few examples of materials used for buffer layers; a more extensive list can be consulted by the interested reader in an excellent review of coated conductors (Foltyn et al., 2007). With all this it can see two things: first, the problem of the epitaxial growth appears overcomed and; second, the optimization and reduction of cost of deposition can be obtained with the uses of buffer layer sandwiched between the single substrate and superconducting film, as is shown in figure 7 (Foltyn et al., 2007; Hanyu et al.,

Most promising applications of type II superconductors are in the power area, where the most advantageous superconductor material is YBa2Cu3O7−*δ*, because high critical current densities needed for this applications can be achieved with it, with acceptably low energy dissipation, even with magnetic fields, as was shown in section 2. These applications require that the superconductor be formed as a long coated conductor tape so that it can be used

MgO substrate

buffer layer Sr2YSbO<sup>6</sup>

77 K and self–field. PLD means Pulsed Laser Deposition.

2007; Nishikawa et al., 2003; Ortiz-Diaz et al., 2010; Parans et al., 2003).

of buffer layers.

Properties of YBa2Cu3O7–

YBa2Cu3O7−*<sup>δ</sup>* applications in microwaves and electronics, many complex perovskites with formula A2BB'O6, besides, some pyrochlore structures. Some examples of complex perovskites proposed are Sr2AlTaO6 for superconducting devices (Findikoglu et al., 1992; Ying & Hilbert, 1994); Ca2GaNbO6; Sr2GaTaO6 (Brandle, 1996); Ba2LaNbO6 (Pai et al., 1997); DyBa2SnO5.5 (Koshy et al., 1995); DyBa2ZrO5.5 (Yadava et al, 1998); HoBa2SbO6 (Aguiar et al., 1998) and Ba2NdTaO6 (Kurian et al., 2002; Kurian & Morishita, 2003).

As can to see, the search for new substrates for the deposition of superconducting films is continuous, since it is hard to find the substrate to produce the perfect film.

## **3. Buffer layers for superconducting films**

As was say in previous section, there are many potential substrates for YBa2Cu3O7−*<sup>δ</sup>* films deposition, but only a few commercially available in single crystal form. Also, each substrate has a problem or disadvantage to the film that has to grow on. To solve this problem without having to wait for the manufacture of new mono–crystalline substrates, some groups began to use these materials in the form of film on any of the commercial MgO and SrTiO3 substrates for growth of superconducting films. The idea was to eliminate the negative effect of single–crystal substrate by using a buffer layer that acts as a chemical barrier that eliminates the reaction between the crystal and the superconductor. In addition, the buffer layer plays the role of template to facilitate the epitaxial growth of superconducting film with improved superconducting and microstructural properties. RF sputtered and laser ablated films of YBa2Cu3O7−*<sup>δ</sup>* on a MgO single crystal substrate using a buffer layer of SrTiO3 began to be manufactured in the 1990's.

In 1993 SrTiO3 buffer layers were grown on MgO (1 0 0) substrates to provide a better match to RF sputtered YBa2Cu3O7−*<sup>δ</sup>* films. This heterostructure allows a highly textured growth to be achieved over thickness as high as 1 *μ*m. The improvement of lattice matching makes the critical current density increase from 10<sup>3</sup> A/cm<sup>2</sup> for 1 *<sup>μ</sup>*m films grown to 4 <sup>×</sup> 105 A/cm<sup>2</sup> for films grown with a SrTiO3 buffer layer (Lucia et al., 1993).

Laser ablated thin films of YBa2Cu3O7−*<sup>δ</sup>* on a MgO substrate using a SrTiO3 buffer layer was grown, which is perfectly oriented with respect to the MgO substrate. Superconductivity is improved in a spectacular manner with respect to YBa2Cu3O7−*<sup>δ</sup>* directly deposited on MgO. A critical temperature *Tc* <sup>=</sup> 92 K and a critical current density of 4 <sup>×</sup> 106 A/cm<sup>2</sup> at 82 K are reached for the first time for films deposited on MgO substrate (Proteau et al., 1995).

In 1996 were grown epitaxial YBa2Cu3O7−*<sup>δ</sup>* films by pulsed laser ablation on SrTiO3 buffered (1 0 0) MgO and was found that the SrTiO3 buffer layer provide a better lattice match to the YBa2Cu3O7−*<sup>δ</sup>* film and play a crucial role to prevent the interaction between YBa2Cu3O7−*<sup>δ</sup>* and MgO. Thus, were obtained YBa2Cu3O7−*<sup>δ</sup>* films with *Jc* <sup>∼</sup> 106 A/cm<sup>2</sup> at 77 K and zero magnetic field (Boffa et al., 1996).

A comparative study of NdBa2Cu3O7−*<sup>δ</sup>* films deposited, by laser ablation, on different substrates showed that the transport *Jc* value of 3.5 <sup>×</sup> 106 A/cm<sup>2</sup> at 77 K in self–field was obtained from the NdBa2Cu3O7−*<sup>δ</sup>* on SrTiO3 with *Tc* value of 91.2 K, while the NdBa2Cu3O7−*<sup>δ</sup>* on MgO with *Tc* of 86.8 K exhibited the low *Jc* value of 0.25 <sup>×</sup> 106 A/cm2. On the other hand, *Jc* value of 1.55 <sup>×</sup> 106 A/cm<sup>2</sup> was obtained from NdBa2Cu3O7−*<sup>δ</sup>* film on SrTiO3 buffered MgO(1 0 0) substrate with *Tc* of 91.5 K. The low *Jc* value of the NdBa2Cu3O7−*<sup>δ</sup>* on MgO can be attributed to both depressed *Tc* value and the existence of 45◦ rotated grains 10 Will-be-set-by-IN-TECH

YBa2Cu3O7−*<sup>δ</sup>* applications in microwaves and electronics, many complex perovskites with formula A2BB'O6, besides, some pyrochlore structures. Some examples of complex perovskites proposed are Sr2AlTaO6 for superconducting devices (Findikoglu et al., 1992; Ying & Hilbert, 1994); Ca2GaNbO6; Sr2GaTaO6 (Brandle, 1996); Ba2LaNbO6 (Pai et al., 1997); DyBa2SnO5.5 (Koshy et al., 1995); DyBa2ZrO5.5 (Yadava et al, 1998); HoBa2SbO6 (Aguiar et al.,

As can to see, the search for new substrates for the deposition of superconducting films is

As was say in previous section, there are many potential substrates for YBa2Cu3O7−*<sup>δ</sup>* films deposition, but only a few commercially available in single crystal form. Also, each substrate has a problem or disadvantage to the film that has to grow on. To solve this problem without having to wait for the manufacture of new mono–crystalline substrates, some groups began to use these materials in the form of film on any of the commercial MgO and SrTiO3 substrates for growth of superconducting films. The idea was to eliminate the negative effect of single–crystal substrate by using a buffer layer that acts as a chemical barrier that eliminates the reaction between the crystal and the superconductor. In addition, the buffer layer plays the role of template to facilitate the epitaxial growth of superconducting film with improved superconducting and microstructural properties. RF sputtered and laser ablated films of YBa2Cu3O7−*<sup>δ</sup>* on a MgO single crystal substrate using a buffer layer of SrTiO3 began to be

In 1993 SrTiO3 buffer layers were grown on MgO (1 0 0) substrates to provide a better match to RF sputtered YBa2Cu3O7−*<sup>δ</sup>* films. This heterostructure allows a highly textured growth to be achieved over thickness as high as 1 *μ*m. The improvement of lattice matching makes the critical current density increase from 10<sup>3</sup> A/cm<sup>2</sup> for 1 *<sup>μ</sup>*m films grown to 4 <sup>×</sup> 105 A/cm<sup>2</sup> for

Laser ablated thin films of YBa2Cu3O7−*<sup>δ</sup>* on a MgO substrate using a SrTiO3 buffer layer was grown, which is perfectly oriented with respect to the MgO substrate. Superconductivity is improved in a spectacular manner with respect to YBa2Cu3O7−*<sup>δ</sup>* directly deposited on MgO. A critical temperature *Tc* <sup>=</sup> 92 K and a critical current density of 4 <sup>×</sup> 106 A/cm<sup>2</sup> at 82 K are

In 1996 were grown epitaxial YBa2Cu3O7−*<sup>δ</sup>* films by pulsed laser ablation on SrTiO3 buffered (1 0 0) MgO and was found that the SrTiO3 buffer layer provide a better lattice match to the YBa2Cu3O7−*<sup>δ</sup>* film and play a crucial role to prevent the interaction between YBa2Cu3O7−*<sup>δ</sup>* and MgO. Thus, were obtained YBa2Cu3O7−*<sup>δ</sup>* films with *Jc* <sup>∼</sup> 106 A/cm<sup>2</sup> at 77 K and zero

A comparative study of NdBa2Cu3O7−*<sup>δ</sup>* films deposited, by laser ablation, on different substrates showed that the transport *Jc* value of 3.5 <sup>×</sup> 106 A/cm<sup>2</sup> at 77 K in self–field was obtained from the NdBa2Cu3O7−*<sup>δ</sup>* on SrTiO3 with *Tc* value of 91.2 K, while the NdBa2Cu3O7−*<sup>δ</sup>* on MgO with *Tc* of 86.8 K exhibited the low *Jc* value of 0.25 <sup>×</sup> 106 A/cm2. On the other hand, *Jc* value of 1.55 <sup>×</sup> 106 A/cm<sup>2</sup> was obtained from NdBa2Cu3O7−*<sup>δ</sup>* film on SrTiO3 buffered MgO(1 0 0) substrate with *Tc* of 91.5 K. The low *Jc* value of the NdBa2Cu3O7−*<sup>δ</sup>* on MgO can be attributed to both depressed *Tc* value and the existence of 45◦ rotated grains

reached for the first time for films deposited on MgO substrate (Proteau et al., 1995).

1998) and Ba2NdTaO6 (Kurian et al., 2002; Kurian & Morishita, 2003).

**3. Buffer layers for superconducting films**

films grown with a SrTiO3 buffer layer (Lucia et al., 1993).

manufactured in the 1990's.

magnetic field (Boffa et al., 1996).

continuous, since it is hard to find the substrate to produce the perfect film.

with (0 0 1)NdBa2Cu3O7−*<sup>δ</sup>*(1 1 0)MgO (Wee et al., 2005). NdBa2Cu3O7−*<sup>δ</sup>* film properties were improved too using a buffer layer of Ba2NdTaO6 on MgO (Kurian & Morishita, 2003).

The above example is a sample of the use of compounds different to the commercial substrates as a material for buffer layers. Also, with the emergence of the second generation of superconducting wires took great importance to study and search for new buffer layers. For instance, Biaxially textured La2Zr2O7 with CeO2 and Ce0.7Gd0.3O3 buffer layers were grown on Ni–RABiTS (rolling-assisted biaxially textured substrates) to obtain suitable buffer layer architectures for YBa2Cu3O7−*<sup>δ</sup>* coated conductors (Knoth et al., 2005). More recently, it has been used as textured substrate for coated conductors a NiW alloy with a Gd2Zr2O7 buffered Ce0.9La0.1O2−*<sup>y</sup>* for the growth of YBa2Cu3O7−*<sup>δ</sup>* films with *Jc* <sup>∼</sup> <sup>1</sup> <sup>×</sup> 106 A/cm<sup>2</sup> (Zhao et al., 2012). This work is an example of current interest in improving the manufacturing process of superconducting tapes with different combinations of deposition technique and combinations of buffer layers.

Fig. 7. Schematic representation of the YBa2Cu3O7−*<sup>δ</sup>* film on Sr2YSbO6 buffer layer.


Table 1. Values of critical current density for YBa2Cu3O7−*<sup>δ</sup>* films on some buffer layers at 77 K and self–field. PLD means Pulsed Laser Deposition.

We have mentioned a few examples of materials used for buffer layers; a more extensive list can be consulted by the interested reader in an excellent review of coated conductors (Foltyn et al., 2007). With all this it can see two things: first, the problem of the epitaxial growth appears overcomed and; second, the optimization and reduction of cost of deposition can be obtained with the uses of buffer layer sandwiched between the single substrate and superconducting film, as is shown in figure 7 (Foltyn et al., 2007; Hanyu et al., 2007; Nishikawa et al., 2003; Ortiz-Diaz et al., 2010; Parans et al., 2003).

Most promising applications of type II superconductors are in the power area, where the most advantageous superconductor material is YBa2Cu3O7−*δ*, because high critical current densities needed for this applications can be achieved with it, with acceptably low energy dissipation, even with magnetic fields, as was shown in section 2. These applications require that the superconductor be formed as a long coated conductor tape so that it can be used

20 30 40 50 60 70 80 2θ (degrees)

For the study of chemical and physical compatibility, we have synthesized several composites of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* with 15 to 90 vol% of Sr2YSbO6 component in the composite. For synthesis of each composite, the component materials were mixed in desired vol% ratios and mixture was pelletized as circular discs at a pressure of 1.6 ton/cm2. These pellets were heat treated at 900◦C for 10 hour in oxygen and cooled down slowly at a rate of 0.5◦C/min for proper oxygenation. Chemical stability of Sr2YSbO6 with YBa2Cu3O7−*<sup>δ</sup>* was examined by X–ray diffractometry of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites. XRD patterns of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites are shown in figure 9. As seen from these XRD patterns, all the XRD peaks could be indexed for either Sr2YSbO6 or YBa2Cu3O7−*<sup>δ</sup>* and there is no extra peak corresponding to impurity phase. Within the accuracy of the XRD technique, these results show that there is no chemical interaction between these materials and Sr2YSbO6 is

The effect of Sr2YSbO6 addition on the superconductivity of YBa2Cu3O7−*<sup>δ</sup>* superconductors was investigated by measuring dc magnetization of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites in the temperature range 5–100 K using a Quantum Design SQUID magnetometer. Figure 10 shows the magnetization for Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites for 0, 45 and 90 vol% of Sr2YSbO6. As shown from these figures all the Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites gave a superconducting transition temperature *Tc* ∼ 90 K as that of the pure YBa2Cu3O7−*<sup>δ</sup>* superconductor. A saturated diamagnetic transition is clearly observed in every sample at temperatures well bellow *Tc*. However, with decreasing YBa2Cu3O7−*<sup>δ</sup>* superconductor volume fraction the magnitude of magnetization decreases in all Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>*

These remarkable results guarantee that indeed the material does not react chemically with YBa2Cu3O7−*δ*, in spite of the severe heat treatment made to the composites at temperatures above the deposition of YBa2Cu3O7−*<sup>δ</sup>* films. For this reason the Sr2YSbO6 material can be

It worth noting that the Sr2YSbO6 material has good structural matching with MgO. Thus, next step was the epitaxial growth of Sr2YSbO6 films on (1 0 0) MgO commercial single crystal substrate, which were performed by magnetron sputtering (13.56 MHz, 70 watt) using a polycrystalline target, which was fabricated, like the firsts samples, by the solid state reaction

proposed as a potential substrate for the growth of superconducting films.

(400)

experimental calculated

Superconducting Films on Sr2YSbO6 Buffer Layers 273

0

Fig. 8. XRD pattern of Sr2YSbO6 sample after the Rietveld refinement.

chemically compatible with YBa2Cu3O7−*<sup>δ</sup>* superconductors.

composite samples.

5000

Intensity (counts)

10000

Properties of YBa2Cu3O7–

like the copper wires. The architecture of the superconducting tapes is like the buffered YBa2Cu3O7−*<sup>δ</sup>* films showing in figure 7, but with additional protective layer of gold or silver and with the single substrate replaced for a flexible tape of commercially alloy textured with an oxide layer. The most promising technology for the manufacture of coated conductor tape is based on Ion Beam Assisted Deposition (IBAD) (Arendt et al., 2004; Foltyn et al., 2007), where MgO and Gd2Zr2O7 are successfully used for texturing. Intermediate oxide layers are epitaxially grown on this textured substrate. The architecture of the tapes can be simplified and, then, the cost of manufacture can be reduced if these intermediate layers are replaced with a single buffer layer. It is the principal motivation for the evaluation of new materials, such as Sr2YSbO6. Also, the evaluation is made using the MgO single substrate, because the MgO is one of the oxides successfully used in IBAD technology. Generally, the route followed to evaluate a material as a possible constituent of the buffer layer in a superconducting tape begins with tests on a small sample of single crystalline substrate. That is why in our work the films are grown on single crystal, instead of on a textured tape directly.

## **4. Evaluation of Sr**2**YSbO**6**, step by step**

The evaluation of Sr2YSbO6 stars with the preparation, which was made by the solid state reaction method. Stoichiometric mixtures of high purity (99.99%) commercial precursor oxides Y2O3, SrO and Sb2O3 in adequate amounts are mixed thoroughly, pelletized and calcined at 1100◦C for 18 h. The calcined material was reground, pressed as circular discs and sintered at 1090◦C for 135 h. All the above processing was carried out in ambient atmosphere.

For the compatibility studies, single phase YBa2Cu3O7−*<sup>δ</sup>* superconducting material was prepared by the solid state reaction method. High purity (99.99%) constituent commercial chemicals Y2O3, BaCO3, and CuO were mixed in stoichiometric ratio. The mixed powder was finely ground and calcined at 900◦C for 24 h at ambient atmosphere. The calcined material was reground, pressed as circular discs at a pressure of 1.6 ton/cm2. The pellets were sintered at 930◦C for 24 h, followed by slow cooling up to 500◦C for 13 h and annealing at this temperature for 24 h at O2 atmosphere. Samples were finally furnace cooled to room temperature over a span of 12 h.

For the study of the structural characteristics of the materials, X Ray Difraction (XRD) patterns of the samples were recorded by a Siemens D5000 X Ray diffractometer, using Cu-K*α* radiation (*λ* = 1.5406 Å) and studied by Rietveld method with the programs EXPGUI and GSAS (Larson & Dreele, 2000; Toby, 2001).

Powder XRD pattern of Sr2YSbO6 is shown in figure 8. It consists of strong peaks which are characteristics of a primitive cubic perovskite plus a few weak line reflections arising from the superlattice. Thus, the whole XRD pattern of Sr2YSbO6 can be indexed in a A2BB� O6 cubic cell with the cell parameter *a* = 8.2561 Å.

Taking into account the doubling of the basic perovskite lattice parameter, the lattice constant *a* = 8.2561 Å (*a*/2 = 4.128 Å) of Sr2YSbO6 is comparable to lattice constants *a* and *b* of YBa2Cu3O7−*<sup>δ</sup>*. Thus, Sr2YSbO6 has a lattice parameter *<sup>a</sup>* which presents a good match with the lattices parameters *<sup>a</sup>* and *<sup>b</sup>* of YBa2Cu3O7−*<sup>δ</sup>* superconductors. This first result was crucial for work purposes. First, the Sr2YSbO6 show acceptable fit with YBa2Cu3O7−*δ*, one of the basic requirements of a substrate. On the other hand, the matching is also suitable with MgO, which would ensure the epitaxial growth of Sr2YSbO6 films.

12 Will-be-set-by-IN-TECH

like the copper wires. The architecture of the superconducting tapes is like the buffered YBa2Cu3O7−*<sup>δ</sup>* films showing in figure 7, but with additional protective layer of gold or silver and with the single substrate replaced for a flexible tape of commercially alloy textured with an oxide layer. The most promising technology for the manufacture of coated conductor tape is based on Ion Beam Assisted Deposition (IBAD) (Arendt et al., 2004; Foltyn et al., 2007), where MgO and Gd2Zr2O7 are successfully used for texturing. Intermediate oxide layers are epitaxially grown on this textured substrate. The architecture of the tapes can be simplified and, then, the cost of manufacture can be reduced if these intermediate layers are replaced with a single buffer layer. It is the principal motivation for the evaluation of new materials, such as Sr2YSbO6. Also, the evaluation is made using the MgO single substrate, because the MgO is one of the oxides successfully used in IBAD technology. Generally, the route followed to evaluate a material as a possible constituent of the buffer layer in a superconducting tape begins with tests on a small sample of single crystalline substrate. That is why in our work

The evaluation of Sr2YSbO6 stars with the preparation, which was made by the solid state reaction method. Stoichiometric mixtures of high purity (99.99%) commercial precursor oxides Y2O3, SrO and Sb2O3 in adequate amounts are mixed thoroughly, pelletized and calcined at 1100◦C for 18 h. The calcined material was reground, pressed as circular discs and sintered at 1090◦C for 135 h. All the above processing was carried out in ambient atmosphere.

For the compatibility studies, single phase YBa2Cu3O7−*<sup>δ</sup>* superconducting material was prepared by the solid state reaction method. High purity (99.99%) constituent commercial chemicals Y2O3, BaCO3, and CuO were mixed in stoichiometric ratio. The mixed powder was finely ground and calcined at 900◦C for 24 h at ambient atmosphere. The calcined material was reground, pressed as circular discs at a pressure of 1.6 ton/cm2. The pellets were sintered at 930◦C for 24 h, followed by slow cooling up to 500◦C for 13 h and annealing at this temperature for 24 h at O2 atmosphere. Samples were finally furnace cooled to room

For the study of the structural characteristics of the materials, X Ray Difraction (XRD) patterns of the samples were recorded by a Siemens D5000 X Ray diffractometer, using Cu-K*α* radiation (*λ* = 1.5406 Å) and studied by Rietveld method with the programs EXPGUI and

Powder XRD pattern of Sr2YSbO6 is shown in figure 8. It consists of strong peaks which are characteristics of a primitive cubic perovskite plus a few weak line reflections arising from the

Taking into account the doubling of the basic perovskite lattice parameter, the lattice constant *a* = 8.2561 Å (*a*/2 = 4.128 Å) of Sr2YSbO6 is comparable to lattice constants *a* and *b* of YBa2Cu3O7−*<sup>δ</sup>*. Thus, Sr2YSbO6 has a lattice parameter *<sup>a</sup>* which presents a good match with the lattices parameters *<sup>a</sup>* and *<sup>b</sup>* of YBa2Cu3O7−*<sup>δ</sup>* superconductors. This first result was crucial for work purposes. First, the Sr2YSbO6 show acceptable fit with YBa2Cu3O7−*δ*, one of the basic requirements of a substrate. On the other hand, the matching is also suitable with MgO,

O6 cubic

superlattice. Thus, the whole XRD pattern of Sr2YSbO6 can be indexed in a A2BB�

the films are grown on single crystal, instead of on a textured tape directly.

**4. Evaluation of Sr**2**YSbO**6**, step by step**

temperature over a span of 12 h.

GSAS (Larson & Dreele, 2000; Toby, 2001).

cell with the cell parameter *a* = 8.2561 Å.

which would ensure the epitaxial growth of Sr2YSbO6 films.

Fig. 8. XRD pattern of Sr2YSbO6 sample after the Rietveld refinement.

For the study of chemical and physical compatibility, we have synthesized several composites of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* with 15 to 90 vol% of Sr2YSbO6 component in the composite. For synthesis of each composite, the component materials were mixed in desired vol% ratios and mixture was pelletized as circular discs at a pressure of 1.6 ton/cm2. These pellets were heat treated at 900◦C for 10 hour in oxygen and cooled down slowly at a rate of 0.5◦C/min for proper oxygenation. Chemical stability of Sr2YSbO6 with YBa2Cu3O7−*<sup>δ</sup>* was examined by X–ray diffractometry of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites. XRD patterns of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites are shown in figure 9. As seen from these XRD patterns, all the XRD peaks could be indexed for either Sr2YSbO6 or YBa2Cu3O7−*<sup>δ</sup>* and there is no extra peak corresponding to impurity phase. Within the accuracy of the XRD technique, these results show that there is no chemical interaction between these materials and Sr2YSbO6 is chemically compatible with YBa2Cu3O7−*<sup>δ</sup>* superconductors.

The effect of Sr2YSbO6 addition on the superconductivity of YBa2Cu3O7−*<sup>δ</sup>* superconductors was investigated by measuring dc magnetization of Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites in the temperature range 5–100 K using a Quantum Design SQUID magnetometer. Figure 10 shows the magnetization for Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites for 0, 45 and 90 vol% of Sr2YSbO6. As shown from these figures all the Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites gave a superconducting transition temperature *Tc* ∼ 90 K as that of the pure YBa2Cu3O7−*<sup>δ</sup>* superconductor. A saturated diamagnetic transition is clearly observed in every sample at temperatures well bellow *Tc*. However, with decreasing YBa2Cu3O7−*<sup>δ</sup>* superconductor volume fraction the magnitude of magnetization decreases in all Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composite samples.

These remarkable results guarantee that indeed the material does not react chemically with YBa2Cu3O7−*δ*, in spite of the severe heat treatment made to the composites at temperatures above the deposition of YBa2Cu3O7−*<sup>δ</sup>* films. For this reason the Sr2YSbO6 material can be proposed as a potential substrate for the growth of superconducting films.

It worth noting that the Sr2YSbO6 material has good structural matching with MgO. Thus, next step was the epitaxial growth of Sr2YSbO6 films on (1 0 0) MgO commercial single crystal substrate, which were performed by magnetron sputtering (13.56 MHz, 70 watt) using a polycrystalline target, which was fabricated, like the firsts samples, by the solid state reaction

0 10 20 30 40 50 60 70 80 90 100 Temperature (K)

> 41 42 43 44 45 46 2θ (degrees)

> > (b) Short XRD.

(400) Sr2YSbO6

(200) MgO

0

5e+05

1e+06

Intensity (counts)

Fig. 11. XRD patterns for Sr2YSbO6 film growth by magnetron sputtering on MgO.

In conclusion, in this section we have showed the preliminary study of structural characteristics of a complex ordered perovskite Sr2YSbO6 for its use as substrate material for the fabrication of YBa2Cu3O7−*<sup>δ</sup>*. Sr2YSbO6 has a fairly good lattice match (lattice mismatch ∼ 8%) with this superconductor. X-ray diffractometry and magnetic measurements made on Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites show that Sr2YSbO6 is chemically and physically compatible with YBa2Cu3O7−*<sup>δ</sup>* material, even after a severe heat treatment at 900◦C, processing temperature of YBa2Cu3O7−*<sup>δ</sup>*. These favorable characteristics of Sr2YSbO6 show that it can be used as a buffer layer for deposition of superconductor YBa2Cu3O7−*<sup>δ</sup>* films using

1,5e+06

2e+06

Fig. 10. Magnetization in function of temperature for composites Sr2YSbO6–YBa2Cu3O7−*δ*. The X–ray diffraction pattern for films, were recorded by a PHILLIPS PW1710 diffractometer using Cu–K*α* radiation (*λ* = 1.5406 Å). Figures 11 show XRD pattern for 2*θ* between 10 and 80 degrees, and a short detailed scan for 2*θ* between 41 and 46 degrees. Pattern of figure 11(a) consists of strong peaks (2 0 0) of MgO and (4 0 0) of Sr2YSbO6. Figure 11(b) shows the MgO peak in 2*θ* = 43◦ and the Sr2YSbO6 peak in 2*θ* = 43.1◦. This result reveals the successfully


epitaxial growth of Sr2YSbO6 films on MgO (1 0 0) substrate.

20 30 40 50 60 70 80 2θ (degrees)

(a) XRD for 10◦ < 2*θ* < 90◦.

0 1e+05 2e+05 3e+05 4e+05 5e+05 6e+05

MgO single substrate.

Intensity (counts)




YBCO 100%

YBCO 55% Sr2YSbO6 45% YBCO 10% Sr2YSbO6 90%

Superconducting Films on Sr2YSbO6 Buffer Layers 275

Magnetization (emu/g)


0

Properties of YBa2Cu3O7–

Fig. 9. XRD patterns for several composites Sr2YSbO6–YBa2Cu3O7−*δ*. First plot show, as a reference, XRD patterns of single phases of YBa2Cu3O7−*<sup>δ</sup>* and of Sr2YSbO6.

method, based on SrO, Sb2O3 and Y2O3, powder oxides. The substrate temperature and oxygen pressure for the Sr2YSbO6 growth were kept at 800◦C and 7 <sup>×</sup> <sup>10</sup>−<sup>3</sup> mbar, respectively. 14 Will-be-set-by-IN-TECH

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80 Angle (2θ)

Fig. 9. XRD patterns for several composites Sr2YSbO6–YBa2Cu3O7−*δ*. First plot show, as a

method, based on SrO, Sb2O3 and Y2O3, powder oxides. The substrate temperature and oxygen pressure for the Sr2YSbO6 growth were kept at 800◦C and 7 <sup>×</sup> <sup>10</sup>−<sup>3</sup> mbar, respectively.

reference, XRD patterns of single phases of YBa2Cu3O7−*<sup>δ</sup>* and of Sr2YSbO6.

Sr2YSbO6 90%

Sr2YSbO6 80%

Sr2YSbO6 70%

Sr2YSbO6 60%

Sr2YSbO6 45%

YSrSbO 30%

YSrSbO 15%

YBaCuO 100% YSrSbO 100%

Intensity (counts)

Intensity (counts)

Intensity (counts)

Intensity (counts)

Intensity (counts)

Intensity (counts)

Intensity (counts)

Intensity (counts)

Fig. 10. Magnetization in function of temperature for composites Sr2YSbO6–YBa2Cu3O7−*δ*.

The X–ray diffraction pattern for films, were recorded by a PHILLIPS PW1710 diffractometer using Cu–K*α* radiation (*λ* = 1.5406 Å). Figures 11 show XRD pattern for 2*θ* between 10 and 80 degrees, and a short detailed scan for 2*θ* between 41 and 46 degrees. Pattern of figure 11(a) consists of strong peaks (2 0 0) of MgO and (4 0 0) of Sr2YSbO6. Figure 11(b) shows the MgO peak in 2*θ* = 43◦ and the Sr2YSbO6 peak in 2*θ* = 43.1◦. This result reveals the successfully epitaxial growth of Sr2YSbO6 films on MgO (1 0 0) substrate.

Fig. 11. XRD patterns for Sr2YSbO6 film growth by magnetron sputtering on MgO.

In conclusion, in this section we have showed the preliminary study of structural characteristics of a complex ordered perovskite Sr2YSbO6 for its use as substrate material for the fabrication of YBa2Cu3O7−*<sup>δ</sup>*. Sr2YSbO6 has a fairly good lattice match (lattice mismatch ∼ 8%) with this superconductor. X-ray diffractometry and magnetic measurements made on Sr2YSbO6-YBa2Cu3O7−*<sup>δ</sup>* composites show that Sr2YSbO6 is chemically and physically compatible with YBa2Cu3O7−*<sup>δ</sup>* material, even after a severe heat treatment at 900◦C, processing temperature of YBa2Cu3O7−*<sup>δ</sup>*. These favorable characteristics of Sr2YSbO6 show that it can be used as a buffer layer for deposition of superconductor YBa2Cu3O7−*<sup>δ</sup>* films using MgO single substrate.

(a) YBa2Cu3O7−*<sup>δ</sup>* film after

Fig. 13. YBa2Cu3O7−*<sup>δ</sup>* film after the photholitographic process for preparation of micro

Fig. 14. YBa2Cu3O7−*<sup>δ</sup>* with contacts for superconducting measurements.

shown for comparison of the resistance behavior.

films growth over MgO and on SrTiO3, with the same conditions, are shown. Figure 17 is

It worth to say about the ways of defining the sharpness and superconducting transition temperature. There is two criteria widely used. Some authors talk in terms on the onset, 5%, 10%, midpoint, 90%, 95%, and zero resistance points (Pool et al., 1995). The onset, or 0% point is where the experimental curve begins to drop below the extrapolated linear behavior. With this criterion, the *Tc* value is midpoint at which the resistivity has decreases by 50% below onset. The point at which the first derivative of the resistive transition curve reaches its maximum value could be selected as defining *Tc*, since is the point of most rapid change from the normal to superconducting phase. Also, the width Δ*T* between the half–amplitude points of the first derivative curve is good quantitative measure of the width of the transition.

(b) YBa2Cu3O7−*<sup>δ</sup>* micro bridge.

Superconducting Films on Sr2YSbO6 Buffer Layers 277

photolithography.

Properties of YBa2Cu3O7–

bridge.

## **5. Sr**2**YSbO**<sup>6</sup> **as a successfully buffer layer for YBa**2**Cu**3**O**7−*<sup>δ</sup>* **films**

In the previous section We show that Sr2YSbO6 satisfies the requirements for YBa2Cu3O7−*<sup>δ</sup>* films deposition. For this reason, was made the growth and characterization of YBa2Cu3O7−*<sup>δ</sup>* film on buffered layer of Sr2YSbO6. This film, and additional films on commercial substrates of MgO and STO, for comparative studies, were carried by sputtering DC (∼ 30 watt) at an optimized substrate temperature of 850◦C an O2 pressure of 3.5 mbar for 1 hour, followed by cooling up to 550◦C in 30 min at O2 pressure of ∼ 850 mbar and therefore were annealed at 550◦C for 30 min at the same O2 pressure.

Figures 12 show the XRD pattern of YBa2Cu3O7−*<sup>δ</sup>* film over Sr2YSbO6 buffer layer, for 2*<sup>θ</sup>* between 10 and 90 degrees. It consists of peaks (0 0 *l*) of YBa2Cu3O7−*δ*, besides the MgO and Sr2YSbO6 peaks, such as is detailed in figure 12(b). These result reveals the epitaxial growth of YBa2Cu3O7−*<sup>δ</sup>* over Sr2YSbO6/MgO buffered substrate.

Fig. 12. XRD patterns for YBa2Cu3O7−*<sup>δ</sup>* film growth on Sr2YSbO6 buffer layer.

The superconducting properties of YBa2Cu3O7−*<sup>δ</sup>* films were characterized by measurements of the transition temperature (*Tc*) and critical current density (*Jc*) at 77 K in self field, by means of ACT measurements (bias AC current of 30 Hz) with four probe method, using the PPMS system of Quantum Design. These measurements were performed on YBa2Cu3O7−*<sup>δ</sup>* micro bridges, with 20 *μ*m of width and 100 nm of thickness, which were prepared by UV photo lithography.

For the photolithography, the films were coated with a layer of photolack. Then, the coated surface was put over a mask and was irradiated for 12 min with UV radiation. Irradiated film was submerged and moved into mix of H2O and NaOH, until we can saw a marks of mask. Then, the film was pasted to another recipient with 60 drops of H2O and five drops of phosphoric acid, and move until the YBa2Cu3O7−*<sup>δ</sup>* superconductor paths that were used for electrical measurements were clears. Finally, film was retired and cleaned. Figures 13 show a YBa2Cu3O7−*<sup>δ</sup>* film after the photolithography and detailed microscopic image of the micro brigde. Films of YBa2Cu3O7−*<sup>δ</sup>* were contacted by means of indium leads such as is show in figure 14, by measurements both resistance in function of temperature and voltage *V* in function of current *I*.

Figures 15 to 17 show the behavior of YBa2Cu3O7−*<sup>δ</sup>* films resistance as a function of temperature. For films growth over Sr2YSbO6 the curve exhibits linear behavior up to a transition temperature *Tc*. In figure 16 the measurements corresponding to a YBa2Cu3O7−*<sup>δ</sup>*

16 Will-be-set-by-IN-TECH

In the previous section We show that Sr2YSbO6 satisfies the requirements for YBa2Cu3O7−*<sup>δ</sup>* films deposition. For this reason, was made the growth and characterization of YBa2Cu3O7−*<sup>δ</sup>* film on buffered layer of Sr2YSbO6. This film, and additional films on commercial substrates of MgO and STO, for comparative studies, were carried by sputtering DC (∼ 30 watt) at an optimized substrate temperature of 850◦C an O2 pressure of 3.5 mbar for 1 hour, followed by cooling up to 550◦C in 30 min at O2 pressure of ∼ 850 mbar and therefore were annealed at

Figures 12 show the XRD pattern of YBa2Cu3O7−*<sup>δ</sup>* film over Sr2YSbO6 buffer layer, for 2*<sup>θ</sup>* between 10 and 90 degrees. It consists of peaks (0 0 *l*) of YBa2Cu3O7−*δ*, besides the MgO and Sr2YSbO6 peaks, such as is detailed in figure 12(b). These result reveals the epitaxial growth

0

5e+05

Intensity (counts)

1e+06

41 42 43 44 45 2θ (degrees)

(b) Short XRD.

**5. Sr**2**YSbO**<sup>6</sup> **as a successfully buffer layer for YBa**2**Cu**3**O**7−*<sup>δ</sup>* **films**

550◦C for 30 min at the same O2 pressure.

(200) MgO(400) Sr2YSbO6

(006) YBCO

of YBa2Cu3O7−*<sup>δ</sup>* over Sr2YSbO6/MgO buffered substrate.

20 30 40 50 60 70 80 90 2θ (degrees)

Fig. 12. XRD patterns for YBa2Cu3O7−*<sup>δ</sup>* film growth on Sr2YSbO6 buffer layer.

The superconducting properties of YBa2Cu3O7−*<sup>δ</sup>* films were characterized by measurements of the transition temperature (*Tc*) and critical current density (*Jc*) at 77 K in self field, by means of ACT measurements (bias AC current of 30 Hz) with four probe method, using the PPMS system of Quantum Design. These measurements were performed on YBa2Cu3O7−*<sup>δ</sup>* micro bridges, with 20 *μ*m of width and 100 nm of thickness, which were prepared by UV photo

For the photolithography, the films were coated with a layer of photolack. Then, the coated surface was put over a mask and was irradiated for 12 min with UV radiation. Irradiated film was submerged and moved into mix of H2O and NaOH, until we can saw a marks of mask. Then, the film was pasted to another recipient with 60 drops of H2O and five drops of phosphoric acid, and move until the YBa2Cu3O7−*<sup>δ</sup>* superconductor paths that were used for electrical measurements were clears. Finally, film was retired and cleaned. Figures 13 show a YBa2Cu3O7−*<sup>δ</sup>* film after the photolithography and detailed microscopic image of the micro brigde. Films of YBa2Cu3O7−*<sup>δ</sup>* were contacted by means of indium leads such as is show in figure 14, by measurements both resistance in function of temperature and voltage *V* in

Figures 15 to 17 show the behavior of YBa2Cu3O7−*<sup>δ</sup>* films resistance as a function of temperature. For films growth over Sr2YSbO6 the curve exhibits linear behavior up to a transition temperature *Tc*. In figure 16 the measurements corresponding to a YBa2Cu3O7−*<sup>δ</sup>*

(a) XRD for 10◦ < 2*θ* < 90◦.

YBCO (005)

0

lithography.

function of current *I*.

5e+05

Intensity (counts)

1e+06

Fig. 13. YBa2Cu3O7−*<sup>δ</sup>* film after the photholitographic process for preparation of micro bridge.

Fig. 14. YBa2Cu3O7−*<sup>δ</sup>* with contacts for superconducting measurements.

films growth over MgO and on SrTiO3, with the same conditions, are shown. Figure 17 is shown for comparison of the resistance behavior.

It worth to say about the ways of defining the sharpness and superconducting transition temperature. There is two criteria widely used. Some authors talk in terms on the onset, 5%, 10%, midpoint, 90%, 95%, and zero resistance points (Pool et al., 1995). The onset, or 0% point is where the experimental curve begins to drop below the extrapolated linear behavior. With this criterion, the *Tc* value is midpoint at which the resistivity has decreases by 50% below onset. The point at which the first derivative of the resistive transition curve reaches its maximum value could be selected as defining *Tc*, since is the point of most rapid change from the normal to superconducting phase. Also, the width Δ*T* between the half–amplitude points of the first derivative curve is good quantitative measure of the width of the transition.

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Temperature (K)

Although the *Tc* values are similar for films over STO and over Sr2YSbO6 buffer layer, the resistance in the normal zone is less for the YBa2Cu3O7−*<sup>δ</sup>* growth on buffer layer; also, the extrapolated residual resistance for this film is less than the residual resistance of

Results of measurements for voltage *V* in function of current *I* (*I*–*V* curves) are shown in figure 18 for the films of YBa2Cu3O7−*<sup>δ</sup>* over buffered substrate Sr2YSbO6/MgO. Based on this *<sup>I</sup>*–*<sup>V</sup>* data and on the *<sup>I</sup>*–*<sup>V</sup>* curves for YBa2Cu3O7−*δ*/MgO, of the figure 3, with the 1 *μ*V/cm criterion, the critical current values, at 77 K, were determined in ∼ 0.013 mA for YBa2Cu3O7−*<sup>δ</sup>* on MgO and ∼ 17 mA for YBa2Cu3O7−*<sup>δ</sup>* on Sr2YSbO6 buffer layer. So, the critical current density value for YBa2Cu3O7−*<sup>δ</sup>* films growth over Sr2YSbO6 buffer layer is *Jc* <sup>∼</sup> 0.86 <sup>×</sup> 106 A/cm<sup>2</sup> which is three order of magnitud times the *Jc* of YBa2Cu3O7−*δ*/MgO films. The *Jc* value for YBa2Cu3O7−*<sup>δ</sup>*/Sr2YSbO6/MgO film growth over buffer layer appears to be less than those reported in the literature (*Jc* <sup>∼</sup> 107 A/cm2). However, it is worth saying that the value for YBa2Cu3O7−*δ*/MgO is less too in comparison with references (*Jc* ∼ 106 A/cm2). Thus, the sputtering deposition conditions perhaps are not yet optimized, and we believe that with other methods of deposition, such as laser ablation, we could improve the *Jc* results. The results reported in the literature are for films deposited in wealthy laboratories

The figure 18 also shows the voltage curves as a function of the applied current for temperatures of 82, 85 and 90 K. We observe a decrease in the value of critical current with increasing temperature and how at 90 K the linear behavior is similar to of conductors at this

temperature, which is a sign that the YBa2Cu3O7−*<sup>δ</sup>* is in the normal state.

On STO On MgO On Sr2YSbO6

Superconducting Films on Sr2YSbO6 Buffer Layers 279

*R* = 5.1 + 0.50*T*, (3)

0

for YBa2Cu3O7−*δ*/MgO film, and

YBa2Cu3O7−*<sup>δ</sup>* film growth on SrTiO3.

that have optimized deposition conditions.

for YBa2Cu3O7−*δ*/STO film.

Fig. 17. Resistance in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* films.

50

100

Resistance (

Ω)

Properties of YBa2Cu3O7–

150

200

Fig. 15. Resistance in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* film on Sr2YSbO6 buffer layer.

Fig. 16. Resistance in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* films on STO and MgO single crystals.

Based on the first derivative criterion, we determined the *Tc* values as *Tc* = 86.6 ± 6.6 K for YBa2Cu3O7−*<sup>δ</sup>* on MgO; *Tc* = <sup>88</sup> ± 3 K for YBa2Cu3O7−*<sup>δ</sup>* on STO and; *Tc* = <sup>88</sup> ± 2 K for YBa2Cu3O7−*<sup>δ</sup>* on Sr2YSbO6 buffer layer.

Linear fit for 140 K< *T* <280 K; 99 K< *T* <147 K and 150 K< *T* <250 K were made for *R*–*T* data for YBa2Cu3O7−*<sup>δ</sup>* on Sr2YSbO6, MgO and STO, respectively. The linear fit results are

$$R = -0.11 + 0.41T\_{\prime} \tag{1}$$

for YBa2Cu3O7−*<sup>δ</sup>*/Sr2YSbO6 film,

$$R = 45.4 + 0.85T\_{\prime} \tag{2}$$

Fig. 17. Resistance in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* films.

for YBa2Cu3O7−*δ*/MgO film, and

$$R = 5.1 + 0.50T\_{\prime} \tag{3}$$

for YBa2Cu3O7−*δ*/STO film.

18 Will-be-set-by-IN-TECH

Experimental First derivative Linear fit

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Temperature (K)

0

50

100

Resistance (

Ω)

150

Experimental First derivative Linear fit

200

0 50 100 150 200 Temperature (K)

(b) YBa2Cu3O7−*<sup>δ</sup>* on STO.

*R* = −0.11 + 0.41*T*, (1)

*R* = 45.4 + 0.85*T*, (2)

Fig. 15. Resistance in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* film on Sr2YSbO6 buffer

Fig. 16. Resistance in function of temperature for YBa2Cu3O7−*<sup>δ</sup>* films on STO and MgO

Based on the first derivative criterion, we determined the *Tc* values as *Tc* = 86.6 ± 6.6 K for YBa2Cu3O7−*<sup>δ</sup>* on MgO; *Tc* = <sup>88</sup> ± 3 K for YBa2Cu3O7−*<sup>δ</sup>* on STO and; *Tc* = <sup>88</sup> ± 2 K for

Linear fit for 140 K< *T* <280 K; 99 K< *T* <147 K and 150 K< *T* <250 K were made for *R*–*T* data for YBa2Cu3O7−*<sup>δ</sup>* on Sr2YSbO6, MgO and STO, respectively. The linear fit results are

0

Experimental First derivative Linear fit

0 100 200 300 Temperature (K)

(a) YBa2Cu3O7−*<sup>δ</sup>* on MgO.

YBa2Cu3O7−*<sup>δ</sup>* on Sr2YSbO6 buffer layer.

for YBa2Cu3O7−*<sup>δ</sup>*/Sr2YSbO6 film,

layer.

0

single crystals.

50

100

Resistance (

Ω)

150

200

50

Resistance (

Ω)

100

150

Although the *Tc* values are similar for films over STO and over Sr2YSbO6 buffer layer, the resistance in the normal zone is less for the YBa2Cu3O7−*<sup>δ</sup>* growth on buffer layer; also, the extrapolated residual resistance for this film is less than the residual resistance of YBa2Cu3O7−*<sup>δ</sup>* film growth on SrTiO3.

Results of measurements for voltage *V* in function of current *I* (*I*–*V* curves) are shown in figure 18 for the films of YBa2Cu3O7−*<sup>δ</sup>* over buffered substrate Sr2YSbO6/MgO. Based on this *<sup>I</sup>*–*<sup>V</sup>* data and on the *<sup>I</sup>*–*<sup>V</sup>* curves for YBa2Cu3O7−*δ*/MgO, of the figure 3, with the 1 *μ*V/cm criterion, the critical current values, at 77 K, were determined in ∼ 0.013 mA for YBa2Cu3O7−*<sup>δ</sup>* on MgO and ∼ 17 mA for YBa2Cu3O7−*<sup>δ</sup>* on Sr2YSbO6 buffer layer. So, the critical current density value for YBa2Cu3O7−*<sup>δ</sup>* films growth over Sr2YSbO6 buffer layer is *Jc* <sup>∼</sup> 0.86 <sup>×</sup> 106 A/cm<sup>2</sup> which is three order of magnitud times the *Jc* of YBa2Cu3O7−*δ*/MgO films. The *Jc* value for YBa2Cu3O7−*<sup>δ</sup>*/Sr2YSbO6/MgO film growth over buffer layer appears to be less than those reported in the literature (*Jc* <sup>∼</sup> 107 A/cm2). However, it is worth saying that the value for YBa2Cu3O7−*δ*/MgO is less too in comparison with references (*Jc* ∼ 106 A/cm2). Thus, the sputtering deposition conditions perhaps are not yet optimized, and we believe that with other methods of deposition, such as laser ablation, we could improve the *Jc* results. The results reported in the literature are for films deposited in wealthy laboratories that have optimized deposition conditions.

The figure 18 also shows the voltage curves as a function of the applied current for temperatures of 82, 85 and 90 K. We observe a decrease in the value of critical current with increasing temperature and how at 90 K the linear behavior is similar to of conductors at this temperature, which is a sign that the YBa2Cu3O7−*<sup>δ</sup>* is in the normal state.

with YBa2Cu3O7−*<sup>δ</sup>* despite the severe heat treatment applied to the two materials in contact with each other, at temperature above the deposition of the films, were crucial to ensure that the Sr2YSbO6 buffer layer to fulfill its role of chemical barrier to eliminate the negative effect of MgO on the superconducting properties of YBCO film. In fact, the critical current density at 77 K of the film on the Sr2YSbO6 buffer layer was three orders of magnitude larger than *Jc* of the films deposited on MgO directly under the same conditions, as has been widely

Superconducting Films on Sr2YSbO6 Buffer Layers 281

The value of critical current density of YBa2Cu3O7−*<sup>δ</sup>* film on the Sr2YSbO6 buffer layer is smaller than other values reported in the literature, as is shown in table 1, which can be explained by the deposition conditions not yet optimized and by use of magnetron as a technique of deposition. To support this assumption is worth mentioning, for example, that the laser ablated YBa2Cu3O7−*<sup>δ</sup>* films on LaAlO3 have 2 <sup>×</sup> 106 A/cm2 and the quality of this films is better than of the films grown by sputtering DC (Koren & Polturak, 2002). In this reference Koren mentions that substrate thickness influences the formation of structural defects in the superconducting film, besides the size of superconducting grains is one order of magnitude lower for films deposited by laser ablation than for deposited with DC magnetron, so that their quality improvement. The optimum temperature of deposition is another factor to review, as there is evidence that it may affect the *Jc* value. For example, buffered substrate SrRuO3/MgO was used for growth of YBa2Cu3O7−*<sup>δ</sup>* films with *Jc* ∼ 2.5 <sup>×</sup> 106 A/cm<sup>2</sup> at temperature of 770◦C, while for deposition at 790◦<sup>C</sup> *Jc* decreased to <sup>6</sup> <sup>×</sup> 105 A/cm<sup>2</sup> (Uprety et al., 2004). Also, in a study by Takahashi is reported the effect of buffer layer thickness on the superconducting properties of YBa2Cu3O7−*δ*. There is a minimum thickness that ensures chemical isolation between YBa2Cu3O7−*<sup>δ</sup>* and other layers, but an increase of buffer layer thickness results in decreased critical current density. There is an optimum thickness of this layer that produces the films with the highest value of *Jc* (Takahashi et al., 2003). In our study, Sr2YSbO6 film was deposited with any thickness,

As the emphasis was made on the application in coated conductors, it is worth noting that, once found the optimal conditions of manufacture, the Sr2YSbO6 could be used as a single buffer layer, thereby simplifying the manufacturing process of the superconductor tapes. Although it appears that the workhorse to discuss possible applications of our material is the field of superconducting tapes, it is interesting to note that the material could be used in

Aguiar, J. A.; de Souza Silva, C. C.; Yadava, Y. P.; Landinez Tellez. D. A.; Ferreira, J. M.;

Arendt, P. N.; Foltyn, S. R. Civale, L.; DePaula, R. F.; Dowden, P. C.; Groves, J. R.; Holesinger,

Boffa, V.; Petrisor, T.; Ciontea, L.; Gambardilla, U. & Barbanera, S. (1996). Properties of in–situ

Brandle, C. D. (1996). Preparation of perovskite oxides for high *Tc* temperature. *J. Mater. Res.*,

Guzman, J. & Chavira, E. (1998). Structure, microstructure, magnetic properties and chemical stability of Ho2Ba2SbO6 with YBa2Cu3O7−*<sup>δ</sup>* superconductor. *Physica C*,

T. G.; Jia, Q. X.; Kreiskott, S.; Stan, L.; Usov, I.; Wang H. & Coulter, J. Y. (2004). High critical current YBCO coated conductors based on IBAD MgO. *Physica C*,

laser–pulsed deposited YBCO thin films on MgO with SrTiO3 buffer layer. *Physica C*,

other application fields such as electronic devices and Josephson junctions.

mentioned in the literature.

Properties of YBa2Cu3O7–

which is not necessarily optimal.

Vol(307) 189–196

Vol(260) 111–116

Vol(5)No 5, 2160

Vol(412–414), 795–800

**7. References**

Fig. 18. *<sup>I</sup>*–*<sup>V</sup>* curves for YBa2Cu3O7−*<sup>δ</sup>* films on Sr2YSbO6 buffer layer at different temperatures.

### **6. Conclusion**

In this chapter We have showed some relevant properties of type II superconductors for applications that requires high critical current densities even with applied magnetic fields. Special attention was dedicated to structural properties of the substrates for YBa2Cu3O7−*<sup>δ</sup>* films in order to improve the *Jc* values. In this context, in this chapter was showed a case of study: the evaluation of the Sr2YSbO6 as a potential material for buffer layer in growth of superconducting films, since preliminary studies of polycrystalline samples until the effective application of this material for deposition of high quality superconducting films. It worth noting that this chapter concerns only on structural issues that limits the *Jc* values, which is a problem practically solved. Nothing were said about the flux pinning, another property that can be improved in order to obtain *Jc* values higher that reported at the present.

There has been show a review of the role of substrate in the successful deposition of superconducting films for applications. Furthermore, it has illustrated the convenience of using buffer layers for the growth of superconducting layers. For these purposes also We showed the different steps in the manufacture of superconducting film on the Sr2YSbO6 buffer layer used, with a focus on techniques to evaluate the material as a potential substrate for the successful growth of YBa2Cu3O7−*<sup>δ</sup>* films.

Previous studies on a polycrystalline Sr2YSbO6 material showed an acceptable structural matching with the YBa2Cu3O7−*<sup>δ</sup>* and the MgO, allowing the epitaxial growth of Sr2YSbO6 film on MgO and subsequently, the superconducting film. Studies on chemical non-reactivity with YBa2Cu3O7−*<sup>δ</sup>* despite the severe heat treatment applied to the two materials in contact with each other, at temperature above the deposition of the films, were crucial to ensure that the Sr2YSbO6 buffer layer to fulfill its role of chemical barrier to eliminate the negative effect of MgO on the superconducting properties of YBCO film. In fact, the critical current density at 77 K of the film on the Sr2YSbO6 buffer layer was three orders of magnitude larger than *Jc* of the films deposited on MgO directly under the same conditions, as has been widely mentioned in the literature.

The value of critical current density of YBa2Cu3O7−*<sup>δ</sup>* film on the Sr2YSbO6 buffer layer is smaller than other values reported in the literature, as is shown in table 1, which can be explained by the deposition conditions not yet optimized and by use of magnetron as a technique of deposition. To support this assumption is worth mentioning, for example, that the laser ablated YBa2Cu3O7−*<sup>δ</sup>* films on LaAlO3 have 2 <sup>×</sup> 106 A/cm2 and the quality of this films is better than of the films grown by sputtering DC (Koren & Polturak, 2002). In this reference Koren mentions that substrate thickness influences the formation of structural defects in the superconducting film, besides the size of superconducting grains is one order of magnitude lower for films deposited by laser ablation than for deposited with DC magnetron, so that their quality improvement. The optimum temperature of deposition is another factor to review, as there is evidence that it may affect the *Jc* value. For example, buffered substrate SrRuO3/MgO was used for growth of YBa2Cu3O7−*<sup>δ</sup>* films with *Jc* ∼ 2.5 <sup>×</sup> 106 A/cm<sup>2</sup> at temperature of 770◦C, while for deposition at 790◦<sup>C</sup> *Jc* decreased to <sup>6</sup> <sup>×</sup> 105 A/cm<sup>2</sup> (Uprety et al., 2004). Also, in a study by Takahashi is reported the effect of buffer layer thickness on the superconducting properties of YBa2Cu3O7−*δ*. There is a minimum thickness that ensures chemical isolation between YBa2Cu3O7−*<sup>δ</sup>* and other layers, but an increase of buffer layer thickness results in decreased critical current density. There is an optimum thickness of this layer that produces the films with the highest value of *Jc* (Takahashi et al., 2003). In our study, Sr2YSbO6 film was deposited with any thickness, which is not necessarily optimal.

As the emphasis was made on the application in coated conductors, it is worth noting that, once found the optimal conditions of manufacture, the Sr2YSbO6 could be used as a single buffer layer, thereby simplifying the manufacturing process of the superconductor tapes. Although it appears that the workhorse to discuss possible applications of our material is the field of superconducting tapes, it is interesting to note that the material could be used in other application fields such as electronic devices and Josephson junctions.

### **7. References**

20 Will-be-set-by-IN-TECH

0


Voltage (V)

0,0001 Voltage (V)

> 0 5 10 Current (mA)


(d) *I*–*V* curve at 90 K..

(b) *I*–*V* curve at 82 K.

024 6 8 10 12 14 16 18 20 Current (mA)

0 0,2 0,4 0,6 0,8 1 Current (mA)

Fig. 18. *<sup>I</sup>*–*<sup>V</sup>* curves for YBa2Cu3O7−*<sup>δ</sup>* films on Sr2YSbO6 buffer layer at different

can be improved in order to obtain *Jc* values higher that reported at the present.

In this chapter We have showed some relevant properties of type II superconductors for applications that requires high critical current densities even with applied magnetic fields. Special attention was dedicated to structural properties of the substrates for YBa2Cu3O7−*<sup>δ</sup>* films in order to improve the *Jc* values. In this context, in this chapter was showed a case of study: the evaluation of the Sr2YSbO6 as a potential material for buffer layer in growth of superconducting films, since preliminary studies of polycrystalline samples until the effective application of this material for deposition of high quality superconducting films. It worth noting that this chapter concerns only on structural issues that limits the *Jc* values, which is a problem practically solved. Nothing were said about the flux pinning, another property that

There has been show a review of the role of substrate in the successful deposition of superconducting films for applications. Furthermore, it has illustrated the convenience of using buffer layers for the growth of superconducting layers. For these purposes also We showed the different steps in the manufacture of superconducting film on the Sr2YSbO6 buffer layer used, with a focus on techniques to evaluate the material as a potential substrate for the

Previous studies on a polycrystalline Sr2YSbO6 material showed an acceptable structural matching with the YBa2Cu3O7−*<sup>δ</sup>* and the MgO, allowing the epitaxial growth of Sr2YSbO6 film on MgO and subsequently, the superconducting film. Studies on chemical non-reactivity

(c) *I*–*V* curve at 85 K..

successful growth of YBa2Cu3O7−*<sup>δ</sup>* films.

(a) *I*–*V* curve at 77 K.


0

1e-05

Voltage (V)

temperatures.

**6. Conclusion**

2e-05

Voltage (V)


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Ce–Gd–O buffer layers for YBCO coated conductors using chemical solution

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**13** 

*1Japan 2Australia* 

**Structural Characteristic and Superconducting** 

The phenomenon of superconductivity, which arises from an unusual quantum mechanism below a superconducting critical transition temperature (*T*c), shows unique physical properties, namely, exactly zero direct current (dc) electrical resistance, the Meissner effect associated with nearly perfect diamagnetism, and the Josephson effect in two weakly coupled superconductors which may have applicability in high-speed, high-sensitivity, lowvoltage sensors and switches. Thanks to these valuable characteristics, today, superconducting materials are having a considerable influence on the development of the next generation of medical services, electrical power systems, transportation, communication technology, and integrated circuits. Thus, our energy-hungry world increasingly requires those technologies and superconductors are becoming indispensable

Superconductivity was discovered by Heike Kamerlingh Onnes in 1911. It is one of the greatest discoveries in the history of science. After that, he tried to fabricate a superconducting magnet using lead (Pb), but it failed. From the failure of his experiment, he learned of the existence of the critical field of superconductors, which completely suppresses superconductivity. His dream was that a high magnetic field could be excited by magnets using superconductors, but it was left unfulfilled. In 1947, a compact helium liquefier was invented by Collins and cowokers, and it made possible some advance in theoretical and experimental studies related to superconducting magnets. From 1960 to 1990, niobium titanium (NbTi) and niobium tin (Nb3Sn) superconductors were employed in many kinds of approaches towards technical development for real applications. These efforts formed the basis of technical application for magnets using metallic superconducting materials, and even now, many researchers are still making steady progress on it. As a result, superconducting materials have become widely applied in not only research magnets for laboratory use, but also various other instruments, which is more than what was expected. Specifically, magnetic resonance imaging (MRI)

**1. Introduction** 

materials.

**Performance of MgB2 Fabricated by** 

*1Department of Physics, College of Science and Technology, Nihon University* 

 *Australian Institute of Innovative Materials, University of Wollongong* 

Minoru Maeda1, Jung Ho Kim2 and Shi Xue Dou2

*2Institute for Superconducting and Electronic Materials,* 

**Mg Diffusion Process** 

as a Single Buffer Layers for High *Ic* YBCO Coated Conductors. *IEEE Trans. Appl. Supercond.*, Vol(13) No 2, 2658–2660


## **Structural Characteristic and Superconducting Performance of MgB2 Fabricated by Mg Diffusion Process**

Minoru Maeda1, Jung Ho Kim2 and Shi Xue Dou2

*1Department of Physics, College of Science and Technology, Nihon University 2Institute for Superconducting and Electronic Materials, Australian Institute of Innovative Materials, University of Wollongong 1Japan 2Australia* 

## **1. Introduction**

24 Will-be-set-by-IN-TECH

284 Superconductors – Properties, Technology, and Applications

Sheahen, T. P. (2002). *Selected topics in superconductivity, Introduction to High–temperature*

Shen, T.; Jiang, J.; Kametani, F.; Trociewitz, U. P.; Larbaliester, D. C.; Schwartz, J.& Hellstrom,

Shiohara, Y.; Fujiwara, N.; Hayashi, H.; Nagaya, S.; Izumi, T. & Yoshizumi (2009). Japanese

Takahashi, Y.; Wakana, H.; Ogawa, A.; Morishita, T. & Tanabe, K. (2003). Effects

Toby, B. H. (2001). EXPGUI, a graphical user interface for GSAS. *J. Appl. Cryst.*, Vol(34), 210–213 Uprety, K. K.; Ma, B.; Koritala, R. E.; Baurceanu, R. M.; Weber, T. P.; Fisher, B. L.; Dorris,

Wang, H.; Foltyn, S. R.; Arendt, P. N.; Jia, Q. X.; Li, Y. & Zhang, X. (2005) Thickness effects

Wee, S. H.; Moon, S. H.; Park, C. & Yoo, S. I. (2005) Effects of substrates on the NdBa2Cu3O7−*<sup>δ</sup>* films fabricated by the PLD process. *Physica C*, Vol(426-431), 996–1000 Wu, C. H.; Chen, M. J.; Hsu, M. H.; Chen, J. C.; Chen, K. L.; Chen, J. H.; Jeng, J. T.; Lai, T. S.;

Yadava, Y. P; Landinez Tellez, D. A.; Melo, M. T.; Ferreira, J. M. & Aguiar, J. A. (1998). Structural

Ying, Q. Y. & Hilbert, C. (1994). All–YBa2Cu3O7−*<sup>δ</sup>* trilayer tunnel junctions with Sr2AlTaO6

Ying, L. L.; Lu, Y.M.; Liu, Z. Y.; Fan, F.; Gao, B.; Cai, C. B.; Thersleff, T.; Reich, E.; Huhne

Zhao, Y,; Li, X-F.; Khoryushin, A.; He, D.; Andersen, N. H.; Hansen, J. B. & Grivel. J-C.

SrRuO3–buffered MgO substrate. *Supercond. Sci. Tecnol.*, Vol(17) 671–675 Varanasi, C. V.; Burke, J.; Lu, R.; Wu, J.; Brunke, L.; Chuck, L.; Smith, H. E.; Maartense,

*Supercond.*, Vol(13) No 2, 2658–2660

*Sci. Technol.*, Vol(23), 025009–025018

*C*, Vol(469), 863–867

Vol(468), 1070–1077

conductors. *Physica C*, Vol(433), 43–49

barrier. *Appl. Phys. Lett.*, Vol(65) 3005–3007

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doi:10.1088/0953-2048/25/1/015008

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YBa2Cu3O7−*<sup>δ</sup>* superconductors. *Appl. Phys. A*, Vol(66) 455–458

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as a Single Buffer Layers for High *Ic* YBCO Coated Conductors. *IEEE Trans. Appl.*

E. E. (2010). Filament to filament bridging and its influence on developing high critical current density in multifilamentary Bi2Sr2CaCu2O*<sup>x</sup>* round wires. *Supercond.*

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of thin SrO buffer layer on dielectric and superconducting properties of YBa2Cu3O7−*<sup>δ</sup>*/Sr2AlTaO6/YBa2Cu3O7−*<sup>δ</sup>* multilayers. *Jpn. J. Appl. Phys.*, Vol(42)

S. E.; Erck, R. A.; Maroni, V. A. & Balachandran, U. (2004). Growth of YBCO film on

I. & Barnes, P. N. (2008). Biaxially textured YBa2Cu3O7−*<sup>δ</sup>* films deposited on polycrystalline flexible yttria–stabilized zirconia ceramic substrates. *Physica C*,

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Horng, H. E. & Yang, H. C. (2005). Optimization of step–edge substrates for high–TC

ordering and chemical stability of a complex perovskite oxide DyBa2ZrO5.5 with

R. & Holzapfel, B. (2009). Thickness effect of La2Zr2O7 single buffers on metallic substrates using pulsed laser deposition for YBa2Cu3O7−*δ*–coated conductors.

(2012). Development of all chemical solution derived Ce0.9La0.1O2*y*/Gd2Zr2O7 buffer layer stack for coated conductors: influence of the post–annealing process on surface crystallinity. *Supercond. Sci. Technol.*, Vol(25) 015008 The phenomenon of superconductivity, which arises from an unusual quantum mechanism below a superconducting critical transition temperature (*T*c), shows unique physical properties, namely, exactly zero direct current (dc) electrical resistance, the Meissner effect associated with nearly perfect diamagnetism, and the Josephson effect in two weakly coupled superconductors which may have applicability in high-speed, high-sensitivity, lowvoltage sensors and switches. Thanks to these valuable characteristics, today, superconducting materials are having a considerable influence on the development of the next generation of medical services, electrical power systems, transportation, communication technology, and integrated circuits. Thus, our energy-hungry world increasingly requires those technologies and superconductors are becoming indispensable materials.

Superconductivity was discovered by Heike Kamerlingh Onnes in 1911. It is one of the greatest discoveries in the history of science. After that, he tried to fabricate a superconducting magnet using lead (Pb), but it failed. From the failure of his experiment, he learned of the existence of the critical field of superconductors, which completely suppresses superconductivity. His dream was that a high magnetic field could be excited by magnets using superconductors, but it was left unfulfilled. In 1947, a compact helium liquefier was invented by Collins and cowokers, and it made possible some advance in theoretical and experimental studies related to superconducting magnets. From 1960 to 1990, niobium titanium (NbTi) and niobium tin (Nb3Sn) superconductors were employed in many kinds of approaches towards technical development for real applications. These efforts formed the basis of technical application for magnets using metallic superconducting materials, and even now, many researchers are still making steady progress on it. As a result, superconducting materials have become widely applied in not only research magnets for laboratory use, but also various other instruments, which is more than what was expected. Specifically, magnetic resonance imaging (MRI)

Structural Characteristic and Superconducting

**2.1 Crystal and electrical structure** 

**2. MgB2 material** 

2005).

at the Fermi energy (*E*F).

Performance of MgB2 Fabricated by Mg Diffusion Process 287

In 1954, the crystal structure of MgB2 was first identified by an X-ray diffraction investigation using nickel-filtered Cu Kα radiation (Jones & Marsh, 1954). MgB2 possesses the simple AlB2 type structure, and its corresponding space group, crystal system, and Lauesymmetry class are P6/mmm, hexagonal, and 6/mmm, respectively. The structure consists of alternating layers of close-packed ionized Mg2+ atoms, which are separated by honeycomb-type boron sheets. In the case of pure MgB2 samples without external forcing such as hydrostatic pressure, the *a*-lattice parameter is estimated to range from ~ 3.083 to ~ 3.086 Å (Jones & Marsh, 1954; Jorgensen et al., 2001), with the variation most likely to be due to disorder in the honeycomb lattice. By contrast, the *c*-lattice parameter shows nearly the same value of ~ 3.521 Å. This may reflect the fact that in the case of close-packed hexagonal structure, the slip directions are essentially �1�1�0�, �21�0�, and �1�20� in the *ab* plane, while expansion and contraction along the *c*-axis direction hardly occurs. The fractional coordinates (*x*, *y*, *z*) for Mg and B atoms are allocated to the 1*a* site of (0, 0, 0) and the 2*d* site of (1/3, 2/3, 1/2). The site occupancies of Mg and B are observed to be ~ 0.95-1.00 and 1.00, respectively (Mori et al., 2002; Tsirelson et al., 2003; Kazakov et al., 2005). The atomic displacement parameters *U* of Mg and B atoms are also observed to be ~ 0.0049-0.0060 Å2 and ~ 0.0042-0.0080 Å2, respectively (Mori et al., 2002; Tsirelson et al., 2003; Kazakov et al.,

Band structure calculations on MgB2 (An & Pickett 2001; Kortus et al., 2001) suggest that the bands at the Fermi level (*E*F) mainly originate from the orbitals of boron due to substantially ionized Mg, and there are four conduction bands, namely, two *σ* bands delivered from the *σ* bonding 2*s*/2*px*,*y* orbitals and two *π* bands delivered from the *π*-bonding and the antibonding 2*pz* orbitals. The former bands are hole types and occur in two-dimensional (2D) cylindrical forms. The latter bands are both hole type and electron type, and form threedimensional (3D) tubular networks. The *σ* states are localized and confined in the boron layers, while the *π* states are delocalized in all directions. The *σ* bands within boron sheets are strongly covalent, whereas the *π* bands over the whole crystal show metallic behaviour, reflecting the lack of covalent bonding (Mazin & Antropov, 2003). The 2D strongly covalent states and the 3D metallic-type states mainly contribute to the total density of states (DOS)

Calculations of both the band structure and the lattice dynamics for MgB2 indicate that there are four distinct phonon modes at the Brillouin zone center (Γ), namely the *A*2u, *B*1g, *E*1u, and *E*2g modes (Bohnen et al., 2001; Yildirim et al., 2001). The *A*2u singly degenerate mode involves the vibration of Mg and B layers in opposite directions along the *c* axis. The *B*1g singly degenerate mode is responsible for the vibrations of B atoms in opposite directions along the *c* axis, with the Mg atoms stationary. On the other hand, the others are doubly degenerate modes and involve only in-plane motions along the *x* or *y* axis. For the *E*1u mode, the Mg and B layers move in opposite directions. The *E*2g mode involves the vibration of the B ions in opposite directions, with the Mg ions stationary. The *E*2g in-plane boron phonons show giant anharmonicity, are strongly coupled to the *σ* bands, and thus change the 2*s*/2*px*,*y* orbital overlap. The strong coupling between the holes of the *σ* bands and the *E*2g mode is responsible for the high *T*c in MgB2. The *π* band charge carriers also become superconducting and

equipment provides image contrast between different kinds of soft issues in the human body and it is essential to confirm the diagnosis of spinal disc herniation, anterior cruciate ligament injury, tumour, and stroke. Nuclear magnetic resonance (NMR) equipment is utilized on the cutting edge of genome-related research. In addition, superconducting magnetic energy storage (SMES), magnetic levitation systems for transportation, and large-scale magnets for confinement of plasma in fusion reactors are expected as the next generation of devices for energy production, its storage, and its distribution, even though they are still at the experimental stage of development. On the other hand, cooling technology, which is crucial for superconducting applications, has also maintained steady progress for the past 15 years. The most recent helium liquefier has smaller scale, longer operation life, and higher energy-saving compared with those a decade ago. Small, convenient cryocoolers without liquid helium have been put to practical use and developed into lower power consumption versions. A unique small pulse tube refrigerator without moving pistons in the low-temperature part of the device is also coming into actual utilization and can easily achieve a low temperature of 10 ~ 20 K. Thus, the development of cooling technology has given an impulse to superconducting applications.

Despite the recent progress of cooling technology, today, the majority of current practical superconducting devices are still made of NbTi and Nb3Sn superconductors, which have the low *T*c values below 20 K. MRI is one of the conventional superconducting devices and has been critical to our high-quality healthcare. However, only 10 % of human beings utilize the superconducting technology in the world. The most serious cause of disturbing its largely widespread use is to need liquid Helium for activating MRI magnets at 4.2 K. The cryogen is expensive and unmanageable, and its resource is exhaustible. In order to promote its widespread use and reduce operating cost, the next generation of superconducting technology must eliminate the use of liquid cryogen bath cooling and its real application has been strongly required. Currently, the most promising materials are thought to be bismuth strontium calcium copper oxide (BSCCO), yttrium barium cooper oxide (YBCO), and magnesium diboride (MgB2). MgB2, a metallic compound, shows *T*c of ~ 39 K, the highest value that has been reported for conventional metallic superconductors, and its superconductivity was discovered by Jun Akimitsu's group in 2001 (Nagamatsu et al., 2001). Owing to its large thermal margin than Nb based superconductors, MgB2 superconducting materials have been judged potentially capable of meeting the needs of the next-generation superconducting applications operated at 15~30 K. MgB2 materials also show superior characteristics for industrial applications including that (1) the manufacture method is much easier to that of NbTi wires, (2) the raw materials are much cheaper than that of YBCO thin films and BSCCO wires, and the manufacturing cost would potentially be lower than that of NbTi wires, (3) the density is one third of NbTi and it can yield triple length wire using the same mass of raw materials, (4) the charging rate of the magnet is faster than that of NbTi magnet, (5) the lower anisotropy and the lager coherence length do not make the weak link problem severe compared with YBCO thin films and BSCCO wires (Larbalestier et al., 2001). Although the *T*c values of YBCO and BSCCO show 93 K and 110 K, respectively, and those are much higher than that of MgB2, the prices of those conductors seem not to be reduced in spite of enormous efforts for more than 20 years. In these regards, therefore, MgB2 materials have been energetically studied for industrial applications in the world since the discovery of the superconductivity.

## **2. MgB2 material**

286 Superconductors – Properties, Technology, and Applications

equipment provides image contrast between different kinds of soft issues in the human body and it is essential to confirm the diagnosis of spinal disc herniation, anterior cruciate ligament injury, tumour, and stroke. Nuclear magnetic resonance (NMR) equipment is utilized on the cutting edge of genome-related research. In addition, superconducting magnetic energy storage (SMES), magnetic levitation systems for transportation, and large-scale magnets for confinement of plasma in fusion reactors are expected as the next generation of devices for energy production, its storage, and its distribution, even though they are still at the experimental stage of development. On the other hand, cooling technology, which is crucial for superconducting applications, has also maintained steady progress for the past 15 years. The most recent helium liquefier has smaller scale, longer operation life, and higher energy-saving compared with those a decade ago. Small, convenient cryocoolers without liquid helium have been put to practical use and developed into lower power consumption versions. A unique small pulse tube refrigerator without moving pistons in the low-temperature part of the device is also coming into actual utilization and can easily achieve a low temperature of 10 ~ 20 K. Thus, the development of cooling technology has given an impulse to

Despite the recent progress of cooling technology, today, the majority of current practical superconducting devices are still made of NbTi and Nb3Sn superconductors, which have the low *T*c values below 20 K. MRI is one of the conventional superconducting devices and has been critical to our high-quality healthcare. However, only 10 % of human beings utilize the superconducting technology in the world. The most serious cause of disturbing its largely widespread use is to need liquid Helium for activating MRI magnets at 4.2 K. The cryogen is expensive and unmanageable, and its resource is exhaustible. In order to promote its widespread use and reduce operating cost, the next generation of superconducting technology must eliminate the use of liquid cryogen bath cooling and its real application has been strongly required. Currently, the most promising materials are thought to be bismuth strontium calcium copper oxide (BSCCO), yttrium barium cooper oxide (YBCO), and magnesium diboride (MgB2). MgB2, a metallic compound, shows *T*c of ~ 39 K, the highest value that has been reported for conventional metallic superconductors, and its superconductivity was discovered by Jun Akimitsu's group in 2001 (Nagamatsu et al., 2001). Owing to its large thermal margin than Nb based superconductors, MgB2 superconducting materials have been judged potentially capable of meeting the needs of the next-generation superconducting applications operated at 15~30 K. MgB2 materials also show superior characteristics for industrial applications including that (1) the manufacture method is much easier to that of NbTi wires, (2) the raw materials are much cheaper than that of YBCO thin films and BSCCO wires, and the manufacturing cost would potentially be lower than that of NbTi wires, (3) the density is one third of NbTi and it can yield triple length wire using the same mass of raw materials, (4) the charging rate of the magnet is faster than that of NbTi magnet, (5) the lower anisotropy and the lager coherence length do not make the weak link problem severe compared with YBCO thin films and BSCCO wires (Larbalestier et al., 2001). Although the *T*c values of YBCO and BSCCO show 93 K and 110 K, respectively, and those are much higher than that of MgB2, the prices of those conductors seem not to be reduced in spite of enormous efforts for more than 20 years. In these regards, therefore, MgB2 materials have been energetically studied for industrial applications in the world since the discovery

superconducting applications.

of the superconductivity.

## **2.1 Crystal and electrical structure**

In 1954, the crystal structure of MgB2 was first identified by an X-ray diffraction investigation using nickel-filtered Cu Kα radiation (Jones & Marsh, 1954). MgB2 possesses the simple AlB2 type structure, and its corresponding space group, crystal system, and Lauesymmetry class are P6/mmm, hexagonal, and 6/mmm, respectively. The structure consists of alternating layers of close-packed ionized Mg2+ atoms, which are separated by honeycomb-type boron sheets. In the case of pure MgB2 samples without external forcing such as hydrostatic pressure, the *a*-lattice parameter is estimated to range from ~ 3.083 to ~ 3.086 Å (Jones & Marsh, 1954; Jorgensen et al., 2001), with the variation most likely to be due to disorder in the honeycomb lattice. By contrast, the *c*-lattice parameter shows nearly the same value of ~ 3.521 Å. This may reflect the fact that in the case of close-packed hexagonal structure, the slip directions are essentially �1�1�0�, �21�0�, and �1�20� in the *ab* plane, while expansion and contraction along the *c*-axis direction hardly occurs. The fractional coordinates (*x*, *y*, *z*) for Mg and B atoms are allocated to the 1*a* site of (0, 0, 0) and the 2*d* site of (1/3, 2/3, 1/2). The site occupancies of Mg and B are observed to be ~ 0.95-1.00 and 1.00, respectively (Mori et al., 2002; Tsirelson et al., 2003; Kazakov et al., 2005). The atomic displacement parameters *U* of Mg and B atoms are also observed to be ~ 0.0049-0.0060 Å2 and ~ 0.0042-0.0080 Å2, respectively (Mori et al., 2002; Tsirelson et al., 2003; Kazakov et al., 2005).

Band structure calculations on MgB2 (An & Pickett 2001; Kortus et al., 2001) suggest that the bands at the Fermi level (*E*F) mainly originate from the orbitals of boron due to substantially ionized Mg, and there are four conduction bands, namely, two *σ* bands delivered from the *σ* bonding 2*s*/2*px*,*y* orbitals and two *π* bands delivered from the *π*-bonding and the antibonding 2*pz* orbitals. The former bands are hole types and occur in two-dimensional (2D) cylindrical forms. The latter bands are both hole type and electron type, and form threedimensional (3D) tubular networks. The *σ* states are localized and confined in the boron layers, while the *π* states are delocalized in all directions. The *σ* bands within boron sheets are strongly covalent, whereas the *π* bands over the whole crystal show metallic behaviour, reflecting the lack of covalent bonding (Mazin & Antropov, 2003). The 2D strongly covalent states and the 3D metallic-type states mainly contribute to the total density of states (DOS) at the Fermi energy (*E*F).

Calculations of both the band structure and the lattice dynamics for MgB2 indicate that there are four distinct phonon modes at the Brillouin zone center (Γ), namely the *A*2u, *B*1g, *E*1u, and *E*2g modes (Bohnen et al., 2001; Yildirim et al., 2001). The *A*2u singly degenerate mode involves the vibration of Mg and B layers in opposite directions along the *c* axis. The *B*1g singly degenerate mode is responsible for the vibrations of B atoms in opposite directions along the *c* axis, with the Mg atoms stationary. On the other hand, the others are doubly degenerate modes and involve only in-plane motions along the *x* or *y* axis. For the *E*1u mode, the Mg and B layers move in opposite directions. The *E*2g mode involves the vibration of the B ions in opposite directions, with the Mg ions stationary. The *E*2g in-plane boron phonons show giant anharmonicity, are strongly coupled to the *σ* bands, and thus change the 2*s*/2*px*,*y* orbital overlap. The strong coupling between the holes of the *σ* bands and the *E*2g mode is responsible for the high *T*c in MgB2. The *π* band charge carriers also become superconducting and

Structural Characteristic and Superconducting

use of Nb-based superconducting materials.

**2.3 Current status in development for superconducting application** 

Two major critical issues facing MgB2 polycrystalline materials such as wires and bulks for the next generation of superconducting applications are weak grain connectivity and lack of pinning centers (Rowell, J. M. 2003; Eisterer, 2007). In order to improve rapid *J*c degradation caused by its weak pinning behaviour in pure MgB2 wires and bulks, so far, many doping materials have been tested since the discovery of its superconductivity. The first successful dopant for this was silicon carbide (SiC) and its additive improved the irreversibility field (*B*irr) and *J*c (Dou et al., 2002). Numerous researchers focused on the impressive results and then started to do many studies related to carbon (C) containing dopants for real superconducting applications. Even now, SiC is still known to be one of the most effective dopants for MgB2 materials. Specifically, *B*c2 values near 0 K for the SiC-doped MgB2 wires fabricated by the *in situ* method (Sumption et al., 2005) and coated-conductors on SiC fibers prepared from hybrid physical-chemical vapor deposition (HPCVD) (Ferrando et al., 2005) were 33 T and 55 T, respectively. These values are much higher than those of NbTi and Nb3Sn. Similar doping materials, nano-carbon (Soltanian et al., 2003), carbon nanotubes

Performance of MgB2 Fabricated by Mg Diffusion Process 289

behaviour from each other. In particular, the contribution of the *π* band to the total condensation energy at 0 T in temperatures above 30 K rapidly increases, while that below 30 K is negligible. This is the reason why the *B*c2(T) of MgB2 shows a positive curvature near *T*c. The unique slope can make the zero-temperature *B*c2(0) significantly higher than that estimated from a single-gap dirty superconductor model (Gurevich, 2003). In addition, an interesting scenario relating to *B*c2(T) for dirty two-gap superconductors was pointed out by Gurevich (Gurevich, 2007). It is assumed that MgB2 can be identified as a simple bilayer sample where two thin films corresponding to the *σ* and *π* bands are separated by a Josephson contact, which reflects the weak interband coupling in MgB2. From the two-gap dirty-limit theory for the BCS matrix constants (Golubov et al., 2002), the temperature dependence of *B*c2 is mostly determined by whether the *σ* film or the *π* film is dirtier. Provided that the *π* film is much dirtier than the *σ* film, the global *B*c2(T) of the bilayer at higher temperature is mainly controlled by the *σ* film with the higher *B*c2, while the *π* film takes over at lower temperatures. This leads to a considerable upturn in the global *B*c2 curve at low temperatures, which can not be estimated from a single-gap dirty-limit superconductor. This interesting scenario for *B*c2 at low temperatures delivered from the simple bilayer model is indirectly demonstrated by experimentally observed *B*c2 curves for alloyed MgB2 thin films (Braccini et al., 2005). Specifically, an anomalous upward curvature of *B*c2(T) at low temperatures observed in a carbon-alloyed thin film provides a record high value of *B*c2(4.2K) = 51 T parallel to the *ab* plane. The extrapolation of the *B*c2(T) curve to *T* = 0 K may reach over ~ 60 T, approaching the paramagnetic limit of 65 T which is calculated from the conventional BCS theory (Sarma, 1963). This value is significantly higher than those for NbTi and Nb3Sn superconductors, which are widely used in superconducting applications now. Moreover, Gurevich pointed out that given the case of electron-phonon coupling, the paramagnetic limit of MgB2 is estimated to be ~ 130 T (Gurevich, 2007). This means that there is still large room for a further increase in *B*c2 from optimum dirty conditions of the intraband and the interband behaviour. Therefore, we can extract a conclusion that the two-gap superconductor MgB2 is suitable for superconducting applications in higher operating temperatures and fields, which can not be reached via the

contribute to unusual superconducting behaviour in MgB2 which is not observed in conventional superconductors. Since the two properties of the *σ* bands are very similar and those of the *π* bands are very similar, even through one *π* band is hole type and the other is electron type, the electronic structure is usually simplified to one *σ* band and one *π* band. Therefore, MgB2 is known and described as a two-band superconductor (Liu et al., 2001).

#### **2.2 Unique characteristic caused by two-band and two-gap structure**

MgB2 is known to be theoretically and experimentally identified as the first superconductor where two superconducting gaps clearly exist (Xi, 2008; Tajima, 2005). The unique two-band and two-gap structure has an influence on the resistivity behaviour in the normal state. In the year of discovery, it became well known that the residual resistivity ratio (RRR) was independent of *T*c in polycrystalline MgB2 samples. Specifically, many MgB2 samples showed widely different RRR values despite having similar *T*c values around ~ 39 K (Buzea & Yamashita, 2001). In the case of the high *T*c, the disorder effects on it (Eisterer, 2007) due to poor crystallinity and lattice strain can be negligible, and thus, the RRR value calculated from the residual resistivity was thought to depend on impurity contamination in a sample. This indicates that lower RRR is caused by a larger amount of impurity phase, even though it should be noted that the residual resistivity is partially attributable to extrinsic effects such as lack of connectivity associated with cracks, voids, and low mass density (Rowell, J. M. 2003). In the case of the two-gap superconductivity model (Golubov & Mazin, 1997), nonmagnetic impurities may contribute to pair breaking, which is similar to the effects of magnetic impurities in a conventional *s*-wave superconductor. Thus, it was difficult to theoretically understand the observed unclear correlation between the RRR and the *T*c. Regarding this issue, Mazin et al. clearly explained the mechanism by pointing out the very weak impurity scattering between the *σ* and the *π* bands compared with the intraband impurity scattering between the *σσ* and *ππ* sub-bands, due to the particular electronic structure in MgB2 (Mazin et al., 2002). It was also found that the *ππ* scattering rate is larger than the *σσ* scattering rate. In addition, they indicated that the *T*c is given by the maximum eigenvalue of the superconducting coupling constant *λ* for the *σπ* and the *πσ* interband coupling, and the *σσ* and *ππ* intraband coupling, while the conductivity is the sum of all the conducting channels responsible for the interband and intraband couplings. The estimated *λ* for the *σσ* intraband coupling showed the highest value, suggesting that the *T*c is affected by the *σ* band. These results provide the conclusion that because impurities have a stronger effect on the scattering rate of the *π* band in preference to the *σ* band, and thus the residual resistivity, they may not contribute to the interband scattering and *T*c.

The upper critical field behaviour of MgB2 was also found to be very different from that of a conventional phonon-mediated Bardeen, Cooper, and Schrieffer (BCS) superconductor with a single-band structure. A unique upward curvature of *B*c2(T) parallel to the *ab* plane near *T*<sup>c</sup> was observed in single crystal MgB2 samples (Zehetmayer et al., 2002; Lyard et al., 2002). This upward curvature can be well explained by using the Ginzburg-Landau theory, suggesting that the *B*c2 is determined by the Ginzburg-Landau parameter and the condensation energy (Eisterer, 2007). In the case of MgB2, which has the two-band structure, the total condensation energy depends on the sum of all the energies arising from the *σ* band, the *π* band, and the interaction between the two bands (Eisterer et al., 2005). These condensation energies in the temperature range from absolute zero to *T*c show different

contribute to unusual superconducting behaviour in MgB2 which is not observed in conventional superconductors. Since the two properties of the *σ* bands are very similar and those of the *π* bands are very similar, even through one *π* band is hole type and the other is electron type, the electronic structure is usually simplified to one *σ* band and one *π* band. Therefore, MgB2 is known and described as a two-band superconductor (Liu et al., 2001).

MgB2 is known to be theoretically and experimentally identified as the first superconductor where two superconducting gaps clearly exist (Xi, 2008; Tajima, 2005). The unique two-band and two-gap structure has an influence on the resistivity behaviour in the normal state. In the year of discovery, it became well known that the residual resistivity ratio (RRR) was independent of *T*c in polycrystalline MgB2 samples. Specifically, many MgB2 samples showed widely different RRR values despite having similar *T*c values around ~ 39 K (Buzea & Yamashita, 2001). In the case of the high *T*c, the disorder effects on it (Eisterer, 2007) due to poor crystallinity and lattice strain can be negligible, and thus, the RRR value calculated from the residual resistivity was thought to depend on impurity contamination in a sample. This indicates that lower RRR is caused by a larger amount of impurity phase, even though it should be noted that the residual resistivity is partially attributable to extrinsic effects such as lack of connectivity associated with cracks, voids, and low mass density (Rowell, J. M. 2003). In the case of the two-gap superconductivity model (Golubov & Mazin, 1997), nonmagnetic impurities may contribute to pair breaking, which is similar to the effects of magnetic impurities in a conventional *s*-wave superconductor. Thus, it was difficult to theoretically understand the observed unclear correlation between the RRR and the *T*c. Regarding this issue, Mazin et al. clearly explained the mechanism by pointing out the very weak impurity scattering between the *σ* and the *π* bands compared with the intraband impurity scattering between the *σσ* and *ππ* sub-bands, due to the particular electronic structure in MgB2 (Mazin et al., 2002). It was also found that the *ππ* scattering rate is larger than the *σσ* scattering rate. In addition, they indicated that the *T*c is given by the maximum eigenvalue of the superconducting coupling constant *λ* for the *σπ* and the *πσ* interband coupling, and the *σσ* and *ππ* intraband coupling, while the conductivity is the sum of all the conducting channels responsible for the interband and intraband couplings. The estimated *λ* for the *σσ* intraband coupling showed the highest value, suggesting that the *T*c is affected by the *σ* band. These results provide the conclusion that because impurities have a stronger effect on the scattering rate of the *π* band in preference to the *σ* band, and thus the residual

**2.2 Unique characteristic caused by two-band and two-gap structure** 

resistivity, they may not contribute to the interband scattering and *T*c.

The upper critical field behaviour of MgB2 was also found to be very different from that of a conventional phonon-mediated Bardeen, Cooper, and Schrieffer (BCS) superconductor with a single-band structure. A unique upward curvature of *B*c2(T) parallel to the *ab* plane near *T*<sup>c</sup> was observed in single crystal MgB2 samples (Zehetmayer et al., 2002; Lyard et al., 2002). This upward curvature can be well explained by using the Ginzburg-Landau theory, suggesting that the *B*c2 is determined by the Ginzburg-Landau parameter and the condensation energy (Eisterer, 2007). In the case of MgB2, which has the two-band structure, the total condensation energy depends on the sum of all the energies arising from the *σ* band, the *π* band, and the interaction between the two bands (Eisterer et al., 2005). These condensation energies in the temperature range from absolute zero to *T*c show different behaviour from each other. In particular, the contribution of the *π* band to the total condensation energy at 0 T in temperatures above 30 K rapidly increases, while that below 30 K is negligible. This is the reason why the *B*c2(T) of MgB2 shows a positive curvature near *T*c. The unique slope can make the zero-temperature *B*c2(0) significantly higher than that estimated from a single-gap dirty superconductor model (Gurevich, 2003). In addition, an interesting scenario relating to *B*c2(T) for dirty two-gap superconductors was pointed out by Gurevich (Gurevich, 2007). It is assumed that MgB2 can be identified as a simple bilayer sample where two thin films corresponding to the *σ* and *π* bands are separated by a Josephson contact, which reflects the weak interband coupling in MgB2. From the two-gap dirty-limit theory for the BCS matrix constants (Golubov et al., 2002), the temperature dependence of *B*c2 is mostly determined by whether the *σ* film or the *π* film is dirtier. Provided that the *π* film is much dirtier than the *σ* film, the global *B*c2(T) of the bilayer at higher temperature is mainly controlled by the *σ* film with the higher *B*c2, while the *π* film takes over at lower temperatures. This leads to a considerable upturn in the global *B*c2 curve at low temperatures, which can not be estimated from a single-gap dirty-limit superconductor. This interesting scenario for *B*c2 at low temperatures delivered from the simple bilayer model is indirectly demonstrated by experimentally observed *B*c2 curves for alloyed MgB2 thin films (Braccini et al., 2005). Specifically, an anomalous upward curvature of *B*c2(T) at low temperatures observed in a carbon-alloyed thin film provides a record high value of *B*c2(4.2K) = 51 T parallel to the *ab* plane. The extrapolation of the *B*c2(T) curve to *T* = 0 K may reach over ~ 60 T, approaching the paramagnetic limit of 65 T which is calculated from the conventional BCS theory (Sarma, 1963). This value is significantly higher than those for NbTi and Nb3Sn superconductors, which are widely used in superconducting applications now. Moreover, Gurevich pointed out that given the case of electron-phonon coupling, the paramagnetic limit of MgB2 is estimated to be ~ 130 T (Gurevich, 2007). This means that there is still large room for a further increase in *B*c2 from optimum dirty conditions of the intraband and the interband behaviour. Therefore, we can extract a conclusion that the two-gap superconductor MgB2 is suitable for superconducting applications in higher operating temperatures and fields, which can not be reached via the use of Nb-based superconducting materials.

## **2.3 Current status in development for superconducting application**

Two major critical issues facing MgB2 polycrystalline materials such as wires and bulks for the next generation of superconducting applications are weak grain connectivity and lack of pinning centers (Rowell, J. M. 2003; Eisterer, 2007). In order to improve rapid *J*c degradation caused by its weak pinning behaviour in pure MgB2 wires and bulks, so far, many doping materials have been tested since the discovery of its superconductivity. The first successful dopant for this was silicon carbide (SiC) and its additive improved the irreversibility field (*B*irr) and *J*c (Dou et al., 2002). Numerous researchers focused on the impressive results and then started to do many studies related to carbon (C) containing dopants for real superconducting applications. Even now, SiC is still known to be one of the most effective dopants for MgB2 materials. Specifically, *B*c2 values near 0 K for the SiC-doped MgB2 wires fabricated by the *in situ* method (Sumption et al., 2005) and coated-conductors on SiC fibers prepared from hybrid physical-chemical vapor deposition (HPCVD) (Ferrando et al., 2005) were 33 T and 55 T, respectively. These values are much higher than those of NbTi and Nb3Sn. Similar doping materials, nano-carbon (Soltanian et al., 2003), carbon nanotubes

Structural Characteristic and Superconducting

high-field *J*c of MgB2.

application.

**2.4 Mg diffusion method** 

tube (PIT) processed MgB2 wires (Hossain et al., 2009).

Performance of MgB2 Fabricated by Mg Diffusion Process 291

4.2 K and 6 T for *in situ* processed C4H6O5-doped MgB2 wires exceeds 1.0 × 105 A/cm2, which is comparable to that for commercial NbTi wires (Kim et al., 2010). The application of cold high pressure densification to those MgB2 wires yielded a high *J*c value of 4.0 × 104 A/cm2 at 4.2 K and 10 T, which is the record value among conventional *in situ* powder-in-

According to recent advanced microscopic analysis by using scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS), un-reacted carbon agglomerates are found to be formed at grain boundaries in C-doped MgB2 (Kim et al., 2012). The residual impurity arises from inhomogeneous mixing of starting materials including carbon compound as dopant and impedes supercurrent flow. In order to address the issue, recently, a carbon containing gas diffusion method by using oxygen-free pyrene (C16H10) was proposed (Maeda et al., 2011b). Pyrene is an aromatic hydrocarbon and its boiling temperature is 404 ºC lower than the temperature of MgB2 formation. Owing to the characteristic, molecular carbon delinked from pyrene gas homogenously distributes into dense MgB2 materials and it causes intensive structural disorder without any severe deterioration of grain connectivity. In addition, more recently, an alternative carbon doping method by using carbon encapsulated boron nanopowder made from a radio frequency plasma process was also proposed and the resultant C-doped MgB2 wires showed high *J*<sup>c</sup> performance due to very thin carbon layers on boron surface for its carbon homogenous distribution (Kim et al., 2011). Thus, the weak flux pinning problem has been well addressed by these recent advanced carbon doping methods, resulting in considerable improvement of

On the other hand, the other major issue, the weak grain connectivity, has also known to be critical for the superconducting performance of MgB2 since the discovery of its superconductivity (Rowell, J. M. 2003; Eisterer, 2007). This is more serious compared with the former major issue and it occurs in MgB2 polycrystalline materials prepared from not only *in situ* reaction of Mg and B mixtures but also *ex situ* fabrication through the use of MgB2 powder. The *in situ* PIT method is the most popular methods for fabricating MgB2 conductors and the *J*c performance is currently higher than that of *ex situ* processed conductors. However, the *in situ* processed conductors show numerous voids caused by MgB2 formation (Kim et al., 2007). This is because the theoretical volume of the mixture (VMg + V2B) is larger than the volume of the reacted MgB2. The resultant mass density is limited to ~ 50 % of the theoretical density, and the porous structure significantly reduces the supercurrent-carrying area and the *J*c for MgB2 materials. In order to counter the severe problem, *in situ* Mg diffusion methods have been proposed (Canfield et al., 2001; Kang et al., 2001) and it seems to be the most effective method even though there is still room for real

Mg diffusion into a B matrix for fabricating dense MgB2 materials was suggested by Canfield et al. and Kang et al. around the same time (Canfield et al., 2001; Kang et al., 2001), soon after the discovery of superconductivity in MgB2. Canfield et al. successfully fabricated the first MgB2 wire using the Mg diffusion method (Canfield et al., 2001). The specific technique was to expose boron filaments to Mg vapour or liquid at 950 ºC for 2 hours. The obtained conductor showed a very low residual resistivity of ~ 0.38 cm and a high

(Yeoh et al., 2004), and boron carbide (Yamamoto et al., 2005a) were also found to effectively improve the in-field *J*c in MgB2 materials. Thus, it has been found that the carbon (C) element is indispensable for superconducting applications of MgB2.

From structural analysis of C-doped MgB2 single crystals (Kazakov et al., 2005), the *a*-lattice parameter was observed to decrease drastically due to the C doping effect. The lattice shrinkage is most likely due to C substitution at B sites and to the difference in atomic radius between these two atoms, which may yield substitutional defects in the crystal lattice. This is because the C replacement effect accumulates in the B layer, and dislocations and its associated stacking faults may form when the accumulated strain exceeds the critical value. The existence of lattice defects in C-doped MgB2 samples has been proven by experimental results (Kazakov et al., 2005; Avdeev et al., 2003; Zhu et al., 2007; Kim et al., 2012). Specifically, the considerable increase in the atomic displacement parameter at the Mg and B sites in C-doped single crystals reflects the enhancement of lattice disorder (Kazakov et al., 2005). The reason why reliability factors for Rietveld refinement in C-doped samples show high values compared with those for pure samples may be explained by the presence of local disorder in the lattice, which can not be described using conventional refined parameters (Kazakov et al., 2005; Avdeev et al., 2003). The C-doped thin films deposited by HPCVD, which shows a high *B*c2 at 0 K value of 70 T, are found to have intensive structural disorder caused by dislocations (Zhu et al., 2007a). In the case of SiC-doped tapes showing relatively low mass density compared with MgB2 thin films, the MgB2 grains are also found to show a large amount of interior contrast generated by intragrain defects that are likely due to lattice disorder (Zhu et al., 2007b). In addition, the relationship between lattice change and not only substitutional, but also interstitial defects in C-doped MgB2 materials can be theoretically explained by Vienna *ab* initio calculations (Bengtson et al., 2010). Recently, boron vacancies were clearly observed in C-doped MgB2 wires and it has been found that those vacancies generate intrinsic stacking faults within the MgB2 grains, together with associated lattice distortion (Kim et al., 2012). Thus, in C-doped MgB2 materials, the reduced unit cell volume caused by the lattice shrinkage leads to introduction of lattice disorder effects, which result in increasing scattering of charge carriers, reducing the mean free path, and thereby enhancing *B*c2 and *J*c (Kazakov et al., 2005; Eisterer, 2007).

However, in the case of inorganic carbon compounds as C doping materials, due to its high melting points or its high decomposition temperature, high-temperature reaction is usually required to effectively cause the lattice shrinkage of the *a*-lattice parameter. These sintering conditions generally offer the disadvantage of grain growth associated with weak grain boundary pinning, because this pinning mechanism is also known to be effective in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). To address the problem, organic compounds as C doping materials with relatively low decomposition temperatures have been intensively studied from around 2006 (Kim et al., 2006; Hossain et al., 2009; Kim et al., 2010; Maeda et al., 2011a). One of the best known organic dopants that has been suggested to date is malic acid (C4H6O5) (Kim et al., 2006). The advantage is that the carbohydrate melts and decomposes at a low temperature below the temperature of MgB2 formation. The decomposition produces highly reactive and fresh C on the atomic scale for C substitution on B site in the lattice. Due to these excellent characteristics, the *a*-lattice parameter decreases even under low-temperature sintering conditions, and grain growth is suppressed, resulting in considerable enhancement of *J*c and *B*c2. Specifically, the *J*c value at

(Yeoh et al., 2004), and boron carbide (Yamamoto et al., 2005a) were also found to effectively improve the in-field *J*c in MgB2 materials. Thus, it has been found that the carbon (C)

From structural analysis of C-doped MgB2 single crystals (Kazakov et al., 2005), the *a*-lattice parameter was observed to decrease drastically due to the C doping effect. The lattice shrinkage is most likely due to C substitution at B sites and to the difference in atomic radius between these two atoms, which may yield substitutional defects in the crystal lattice. This is because the C replacement effect accumulates in the B layer, and dislocations and its associated stacking faults may form when the accumulated strain exceeds the critical value. The existence of lattice defects in C-doped MgB2 samples has been proven by experimental results (Kazakov et al., 2005; Avdeev et al., 2003; Zhu et al., 2007; Kim et al., 2012). Specifically, the considerable increase in the atomic displacement parameter at the Mg and B sites in C-doped single crystals reflects the enhancement of lattice disorder (Kazakov et al., 2005). The reason why reliability factors for Rietveld refinement in C-doped samples show high values compared with those for pure samples may be explained by the presence of local disorder in the lattice, which can not be described using conventional refined parameters (Kazakov et al., 2005; Avdeev et al., 2003). The C-doped thin films deposited by HPCVD, which shows a high *B*c2 at 0 K value of 70 T, are found to have intensive structural disorder caused by dislocations (Zhu et al., 2007a). In the case of SiC-doped tapes showing relatively low mass density compared with MgB2 thin films, the MgB2 grains are also found to show a large amount of interior contrast generated by intragrain defects that are likely due to lattice disorder (Zhu et al., 2007b). In addition, the relationship between lattice change and not only substitutional, but also interstitial defects in C-doped MgB2 materials can be theoretically explained by Vienna *ab* initio calculations (Bengtson et al., 2010). Recently, boron vacancies were clearly observed in C-doped MgB2 wires and it has been found that those vacancies generate intrinsic stacking faults within the MgB2 grains, together with associated lattice distortion (Kim et al., 2012). Thus, in C-doped MgB2 materials, the reduced unit cell volume caused by the lattice shrinkage leads to introduction of lattice disorder effects, which result in increasing scattering of charge carriers, reducing the mean free path, and thereby enhancing *B*c2 and *J*c (Kazakov et al., 2005; Eisterer, 2007). However, in the case of inorganic carbon compounds as C doping materials, due to its high melting points or its high decomposition temperature, high-temperature reaction is usually required to effectively cause the lattice shrinkage of the *a*-lattice parameter. These sintering conditions generally offer the disadvantage of grain growth associated with weak grain boundary pinning, because this pinning mechanism is also known to be effective in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). To address the problem, organic compounds as C doping materials with relatively low decomposition temperatures have been intensively studied from around 2006 (Kim et al., 2006; Hossain et al., 2009; Kim et al., 2010; Maeda et al., 2011a). One of the best known organic dopants that has been suggested to date is malic acid (C4H6O5) (Kim et al., 2006). The advantage is that the carbohydrate melts and decomposes at a low temperature below the temperature of MgB2 formation. The decomposition produces highly reactive and fresh C on the atomic scale for C substitution on B site in the lattice. Due to these excellent characteristics, the *a*-lattice parameter decreases even under low-temperature sintering conditions, and grain growth is suppressed, resulting in considerable enhancement of *J*c and *B*c2. Specifically, the *J*c value at

element is indispensable for superconducting applications of MgB2.

According to recent advanced microscopic analysis by using scanning transmission electron microscopy (STEM) and electron energy loss spectroscopy (EELS), un-reacted carbon agglomerates are found to be formed at grain boundaries in C-doped MgB2 (Kim et al., 2012). The residual impurity arises from inhomogeneous mixing of starting materials including carbon compound as dopant and impedes supercurrent flow. In order to address the issue, recently, a carbon containing gas diffusion method by using oxygen-free pyrene (C16H10) was proposed (Maeda et al., 2011b). Pyrene is an aromatic hydrocarbon and its boiling temperature is 404 ºC lower than the temperature of MgB2 formation. Owing to the characteristic, molecular carbon delinked from pyrene gas homogenously distributes into dense MgB2 materials and it causes intensive structural disorder without any severe deterioration of grain connectivity. In addition, more recently, an alternative carbon doping method by using carbon encapsulated boron nanopowder made from a radio frequency plasma process was also proposed and the resultant C-doped MgB2 wires showed high *J*<sup>c</sup> performance due to very thin carbon layers on boron surface for its carbon homogenous distribution (Kim et al., 2011). Thus, the weak flux pinning problem has been well addressed by these recent advanced carbon doping methods, resulting in considerable improvement of high-field *J*c of MgB2.

On the other hand, the other major issue, the weak grain connectivity, has also known to be critical for the superconducting performance of MgB2 since the discovery of its superconductivity (Rowell, J. M. 2003; Eisterer, 2007). This is more serious compared with the former major issue and it occurs in MgB2 polycrystalline materials prepared from not only *in situ* reaction of Mg and B mixtures but also *ex situ* fabrication through the use of MgB2 powder. The *in situ* PIT method is the most popular methods for fabricating MgB2 conductors and the *J*c performance is currently higher than that of *ex situ* processed conductors. However, the *in situ* processed conductors show numerous voids caused by MgB2 formation (Kim et al., 2007). This is because the theoretical volume of the mixture (VMg + V2B) is larger than the volume of the reacted MgB2. The resultant mass density is limited to ~ 50 % of the theoretical density, and the porous structure significantly reduces the supercurrent-carrying area and the *J*c for MgB2 materials. In order to counter the severe problem, *in situ* Mg diffusion methods have been proposed (Canfield et al., 2001; Kang et al., 2001) and it seems to be the most effective method even though there is still room for real application.

## **2.4 Mg diffusion method**

Mg diffusion into a B matrix for fabricating dense MgB2 materials was suggested by Canfield et al. and Kang et al. around the same time (Canfield et al., 2001; Kang et al., 2001), soon after the discovery of superconductivity in MgB2. Canfield et al. successfully fabricated the first MgB2 wire using the Mg diffusion method (Canfield et al., 2001). The specific technique was to expose boron filaments to Mg vapour or liquid at 950 ºC for 2 hours. The obtained conductor showed a very low residual resistivity of ~ 0.38 cm and a high

Structural Characteristic and Superconducting

**3. Experimental** 

evaporation processes.

Performance of MgB2 Fabricated by Mg Diffusion Process 293

and *J*c in MgB2 prepared from Mg diffusion process. In this study, therefore, we have systematically evaluated structural and superconducting properties in Mg diffusion

All samples were fabricated via Mg diffusion and evaporation processes, as shown in Fig, 1.

Fig. 1. Schematic diagram showing a fabrication method of MgB2 via Mg diffusion and

Mg ingots (purity 99.99 %, size 2-5 mm) and amorphous B powder (purity 99.995 %, size 0.25 m) were used as starting materials and the molar ratio was chosen to be 1 : 1. The reason for using the large amount of Mg ingots is to eliminate Mg loss caused by MgO formation and to more homogeneously diffuse Mg into B matrix in comparison with what occurs in a Mg diffusion process with the stoichiometric molar ratio. Amorphous B powder was pressed into slabs with dimensions of 4.5 x 21.5 x 0.5 mm under a pressure of 100 MPa. The compacted B powder and Mg ingots were placed in a Ta tube. Both ends of the Ta tube were sealed by a metal inert gas (MIG) welding method in a glove box filled with high purity Ar gas. During the MIG welding process, the Ta tube was heat-sunk with ice-cooled copper block to avoid heating the slab and Mg ingots. The sealed Ta tube was vacuumsealed in a quartz tube and then heat-treated under four major types of sintering conditions that are (1) an one-step sintering process at 1100 ºC which is higher than the boiling point of

processed MgB2 bulks sintered under various heat-treated conditions.

**3.1 Sample preparation via Mg diffusion and evaporation processes** 

density above 80 % of theoretical. Even now, MgB2 conductors prepared by conventional *in situ* or *ex situ* PIT methods do not reach either value. On the other hand, Kang et al. first made high-quality MgB2 thin films via a similar method (Kang et al., 2001). The successful technique was to deposit amorphous B thin films and sinter them in Mg vapor at 950 ºC for 10-30 minutes. The self-field *J*c at 5 K for the obtained film showed a high value of ~ 6.0 × 106 A/cm2 and a low residual resistivity of ~ 2.0 cm, which was likely due to its highly dense structure. Thus, the Mg diffusion method was found to yield higher mass density of MgB2 matrix in materials compared with conventional *in situ* PIT method, and it has been modified by many researchers for further enhancement of *J*c and real applications of various materials such as wires and bulks since it was first discovered (Giunchi, 2003; Ueda et al., 2005; Togano et al., 2009).

For the synthesis of dense MgB2 bulks, the reactive liquid Mg infiltration (RLI) technique was proposed by Giunchi (Giunchi, 2003). The method has the advantage that desired forms can be fabricated on a large scale and in an inexpensive way without hot pressing methods. The following is a brief procedure of the fabrication method. First, compacted B powders in contact with an appropriate amount of Mg bulk pieces are inserted into steel containers with desirable shapes. The containers are arc-welded in Ar atmosphere using a high-temperature resistant technique and then sintered at 750-950 ºC. After sintering, the MgB2 bulks are obtained by removal of the containers. The mass density of the obtained samples is ~ 2.40 g/cm3, and the transport *J*c values at 4.2 K and 10 T reach 3.0 × 103 A/cm2. Moreover, the RLI technique in combination with *in situ* PIT methods can be applied to the fabrication of monofilament and multifilament MgB2 hollow wires (Giunchi et al., 2003). It is found that hollow cores are formed in the sintered wire due to Mg diffusion into the B matrix. The formed MgB2 core was observed to have highly dense structure from SEM observation.

In order to address the problem of the low *J*c at self field in MgB2 bulks, the powder-inclosed-tube (PICT)-diffusion method was suggested by Ueda et al. (Ueda et al., 2005). The bulks were prepared by the following procedure. The starting Mg and B powders in appropriate amounts were separately packed into a stainless steel tube. The tube was deformed into a tape-shaped sample and both ends were sealed by bending and pressing. The samples were vacuum-sealed in a silicon dioxide (SiO2) tube and then sintered at 800- 900 ºC. The obtained sample sintered at 800 ºC for 60 hours exhibits well-connected small grains with sizes of 30-100 nm. The mass density is above 95 % of the theoretical density, and the self-field *J*c value at 20 K reaches ~ 8.6 × 105 A/cm2 due to its good connectivity.

To achieve much less MgO secondary phase, as well as highly dense structure in MgB2 conductors, an interface diffusion process was proposed by Togano et al. (Togano et al., 2006). The conductors were fabricated from an interface diffusive reaction between an Fe-Mg alloy substrate and B at 950 ºC for 12 hours. The formed MgB2 layer was observed to be composed of densely packed equi-axial grains of submicron size and without oxygen contamination. This group also suggested an internal Mg diffusion (IMD) process for further enhancement of *J*c in MgB2 wires (Hur et al., 2008). Specifically, 19-filament dense MgB2 wires are fabricated by the internal Mg diffusion (IMD) process and the transport *J*c values at 4.2 K and 10 T reach ~ 1.0 × 105 A/cm2 (Togano et al., 2009). This performance is the best *J*c reported so far in MgB2 wires. However, despite these attractive performances of Mg diffusion processed materials, very little work has been done on the effects of sintering temperature on the phase composition, lattice parameters, mass density, grain size, *T*c, *B*c2, and *J*c in MgB2 prepared from Mg diffusion process. In this study, therefore, we have systematically evaluated structural and superconducting properties in Mg diffusion processed MgB2 bulks sintered under various heat-treated conditions.

## **3. Experimental**

292 Superconductors – Properties, Technology, and Applications

density above 80 % of theoretical. Even now, MgB2 conductors prepared by conventional *in situ* or *ex situ* PIT methods do not reach either value. On the other hand, Kang et al. first made high-quality MgB2 thin films via a similar method (Kang et al., 2001). The successful technique was to deposit amorphous B thin films and sinter them in Mg vapor at 950 ºC for 10-30 minutes. The self-field *J*c at 5 K for the obtained film showed a high value of ~ 6.0 × 106 A/cm2 and a low residual resistivity of ~ 2.0 cm, which was likely due to its highly dense structure. Thus, the Mg diffusion method was found to yield higher mass density of MgB2 matrix in materials compared with conventional *in situ* PIT method, and it has been modified by many researchers for further enhancement of *J*c and real applications of various materials such as wires and bulks since it was first discovered (Giunchi, 2003; Ueda et al.,

For the synthesis of dense MgB2 bulks, the reactive liquid Mg infiltration (RLI) technique was proposed by Giunchi (Giunchi, 2003). The method has the advantage that desired forms can be fabricated on a large scale and in an inexpensive way without hot pressing methods. The following is a brief procedure of the fabrication method. First, compacted B powders in contact with an appropriate amount of Mg bulk pieces are inserted into steel containers with desirable shapes. The containers are arc-welded in Ar atmosphere using a high-temperature resistant technique and then sintered at 750-950 ºC. After sintering, the MgB2 bulks are obtained by removal of the containers. The mass density of the obtained samples is ~ 2.40 g/cm3, and the transport *J*c values at 4.2 K and 10 T reach 3.0 × 103 A/cm2. Moreover, the RLI technique in combination with *in situ* PIT methods can be applied to the fabrication of monofilament and multifilament MgB2 hollow wires (Giunchi et al., 2003). It is found that hollow cores are formed in the sintered wire due to Mg diffusion into the B matrix. The formed MgB2 core was observed to have highly dense structure from SEM observation.

In order to address the problem of the low *J*c at self field in MgB2 bulks, the powder-inclosed-tube (PICT)-diffusion method was suggested by Ueda et al. (Ueda et al., 2005). The bulks were prepared by the following procedure. The starting Mg and B powders in appropriate amounts were separately packed into a stainless steel tube. The tube was deformed into a tape-shaped sample and both ends were sealed by bending and pressing. The samples were vacuum-sealed in a silicon dioxide (SiO2) tube and then sintered at 800- 900 ºC. The obtained sample sintered at 800 ºC for 60 hours exhibits well-connected small grains with sizes of 30-100 nm. The mass density is above 95 % of the theoretical density, and the self-field *J*c value at 20 K reaches ~ 8.6 × 105 A/cm2 due to its good connectivity.

To achieve much less MgO secondary phase, as well as highly dense structure in MgB2 conductors, an interface diffusion process was proposed by Togano et al. (Togano et al., 2006). The conductors were fabricated from an interface diffusive reaction between an Fe-Mg alloy substrate and B at 950 ºC for 12 hours. The formed MgB2 layer was observed to be composed of densely packed equi-axial grains of submicron size and without oxygen contamination. This group also suggested an internal Mg diffusion (IMD) process for further enhancement of *J*c in MgB2 wires (Hur et al., 2008). Specifically, 19-filament dense MgB2 wires are fabricated by the internal Mg diffusion (IMD) process and the transport *J*c values at 4.2 K and 10 T reach ~ 1.0 × 105 A/cm2 (Togano et al., 2009). This performance is the best *J*c reported so far in MgB2 wires. However, despite these attractive performances of Mg diffusion processed materials, very little work has been done on the effects of sintering temperature on the phase composition, lattice parameters, mass density, grain size, *T*c, *B*c2,

2005; Togano et al., 2009).

## **3.1 Sample preparation via Mg diffusion and evaporation processes**

All samples were fabricated via Mg diffusion and evaporation processes, as shown in Fig, 1.

Fig. 1. Schematic diagram showing a fabrication method of MgB2 via Mg diffusion and evaporation processes.

Mg ingots (purity 99.99 %, size 2-5 mm) and amorphous B powder (purity 99.995 %, size 0.25 m) were used as starting materials and the molar ratio was chosen to be 1 : 1. The reason for using the large amount of Mg ingots is to eliminate Mg loss caused by MgO formation and to more homogeneously diffuse Mg into B matrix in comparison with what occurs in a Mg diffusion process with the stoichiometric molar ratio. Amorphous B powder was pressed into slabs with dimensions of 4.5 x 21.5 x 0.5 mm under a pressure of 100 MPa. The compacted B powder and Mg ingots were placed in a Ta tube. Both ends of the Ta tube were sealed by a metal inert gas (MIG) welding method in a glove box filled with high purity Ar gas. During the MIG welding process, the Ta tube was heat-sunk with ice-cooled copper block to avoid heating the slab and Mg ingots. The sealed Ta tube was vacuumsealed in a quartz tube and then heat-treated under four major types of sintering conditions that are (1) an one-step sintering process at 1100 ºC which is higher than the boiling point of

Structural Characteristic and Superconducting

Performance of MgB2 Fabricated by Mg Diffusion Process 295

Fig. 2. Microstructure of Mg diffusion processed samples via various sintering conditions including (a) 1100 ºC 0.5 h, (b) 1100 ºC 1 h, (c) 1100 ºC 3 h, (d) 660 ºC 24 h, (e) 660 ºC 48 h, (f) 660 ºC 96 h, (g) 1100 ºC 0.1h - 660 ºC 3 h, (h) 1100 ºC 0.1 h - 660 ºC 6 h, (i) 1100 ºC 0.1h - 660 ºC 12 h, (k) 660 ºC 12 h - 1100 ºC 0.25 h, and (l) 660 ºC 12 h - 1100 ºC 0.5 h. (j) Microstructure

of compacted B powder before sintering.

Mg, (2) an one-step sintering process at 660 ºC which is higher than the melting point of Mg, (3) a two-step process sintered at 1100 ºC for a short period of 6 minutes followed by second sintering at 660 ºC, and (4) a two-step process sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC. After each sintering process for MgB2 formation, a large amount of unreacted Mg remains in the sealed Ta tubes. In order to remove the residual Mg from MgB2 sample, the Ta tube was pierced through and again vacuum-sealed in a quartz tube to undergo an Mg evaporation process at 680 °C for 0.5 hour.

## **3.2 Sample characterization**

The phase composition was investigated from X-ray powder diffraction (XRD) patterns which were collected with a Rigaku RINT 2000 diffractmeter using CuK radiation. Rietveld refinement of the crystal structure was carried out with the RIETAN-2000 program (Izumi & Ikeda, 2000) and the structural parameters were refined by fitting the observed XRD patterns. The morphology and grain size were examined using a HITACHI S–4500 Scanning Electron Microscopy (SEM). The real and the imaginary component curves of ac susceptibility and the dc magnetization loops were evaluated by using a SQUID magnetometer (MPMS-5T, Quantum Design). The *T*c and *B*c2 were determined by ac susceptibility measurements at *f* = 76.97 Hz with 0*H*ac = 10 T and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T. The *J*c values were derived from the magnetization hysteresis loops using the Bean critical state model (Chen & Goldfarb, 1989).

## **4. Results and discussion**

## **4.1 Morphology and density**

Figure 2 shows the SEM images of the surface of Mg diffusion processed samples via various sintering conditions. For reference, compacted amorphous B powder was also observed, as can be seen in Fig. 2(j). It was found that the one-step sintering process at 1100 ºC yields welldeveloped crystalline grains, as shown in Fig. 2(a), (b), and (c). The crystal grains were also found to grow rapidly with an increase in sintering time. Specifically, the average grain sizes of samples sintered at 1100 ºC for 0.5, 1, and 3 hours were observed to be approximately 0.8, 1.5, and 3.0 m, respectively. The mass densities of the corresponding samples are 2.2, 2.4, and 1.9 g/cm3, respectively, and close to the theoretical density of MgB2. However, the drastic decrease in mass density from 2.4 g/cm3 to 1.9 g/cm3 is most likely due to the observed rapid growth caused by the increase in sintering time. It is well known that grain boundaries can act as pinning sites in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). The considerably large grains offer the disadvantage of lowering in-field *J*c, even though high mass density is easily achieved by the one-step process at 1100 ºC for the short sintering time. Contrary to the grain growth, the one-step sintering process at 660 ºC for long sintering time ranging from 24 hours to 96 hours was found to leave grain size more or less the same, as can be seen in Fig. 2(d), (e), and (f). It is important to note that the average grain size is much smaller than that of Mg diffusion processed samples at 1100 ºC and the grain shape shows poorly crystallized form. It is also well known that crystallinity associated with structural disorder in the lattice as well as grain boundaries can enhance the flux pinning effect (Yamamoto et al., 2005b; Eisterer, 2007). The low-temperature sintering process is advantageous in terms of both poor crystallinity and small grains.

Mg, (2) an one-step sintering process at 660 ºC which is higher than the melting point of Mg, (3) a two-step process sintered at 1100 ºC for a short period of 6 minutes followed by second sintering at 660 ºC, and (4) a two-step process sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC. After each sintering process for MgB2 formation, a large amount of unreacted Mg remains in the sealed Ta tubes. In order to remove the residual Mg from MgB2 sample, the Ta tube was pierced through and again vacuum-sealed in a quartz tube to

The phase composition was investigated from X-ray powder diffraction (XRD) patterns which were collected with a Rigaku RINT 2000 diffractmeter using CuK radiation. Rietveld refinement of the crystal structure was carried out with the RIETAN-2000 program (Izumi & Ikeda, 2000) and the structural parameters were refined by fitting the observed XRD patterns. The morphology and grain size were examined using a HITACHI S–4500 Scanning Electron Microscopy (SEM). The real and the imaginary component curves of ac susceptibility and the dc magnetization loops were evaluated by using a SQUID magnetometer (MPMS-5T, Quantum Design). The *T*c and *B*c2 were determined by ac susceptibility measurements at *f* = 76.97 Hz with 0*H*ac = 10 T and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T. The *J*c values were derived from the magnetization hysteresis loops using the

Figure 2 shows the SEM images of the surface of Mg diffusion processed samples via various sintering conditions. For reference, compacted amorphous B powder was also observed, as can be seen in Fig. 2(j). It was found that the one-step sintering process at 1100 ºC yields welldeveloped crystalline grains, as shown in Fig. 2(a), (b), and (c). The crystal grains were also found to grow rapidly with an increase in sintering time. Specifically, the average grain sizes of samples sintered at 1100 ºC for 0.5, 1, and 3 hours were observed to be approximately 0.8, 1.5, and 3.0 m, respectively. The mass densities of the corresponding samples are 2.2, 2.4, and 1.9 g/cm3, respectively, and close to the theoretical density of MgB2. However, the drastic decrease in mass density from 2.4 g/cm3 to 1.9 g/cm3 is most likely due to the observed rapid growth caused by the increase in sintering time. It is well known that grain boundaries can act as pinning sites in MgB2 materials (Larbalestier et al., 2001; Eisterer, 2007). The considerably large grains offer the disadvantage of lowering in-field *J*c, even though high mass density is easily achieved by the one-step process at 1100 ºC for the short sintering time. Contrary to the grain growth, the one-step sintering process at 660 ºC for long sintering time ranging from 24 hours to 96 hours was found to leave grain size more or less the same, as can be seen in Fig. 2(d), (e), and (f). It is important to note that the average grain size is much smaller than that of Mg diffusion processed samples at 1100 ºC and the grain shape shows poorly crystallized form. It is also well known that crystallinity associated with structural disorder in the lattice as well as grain boundaries can enhance the flux pinning effect (Yamamoto et al., 2005b; Eisterer, 2007). The low-temperature sintering process is advantageous in terms of both poor

undergo an Mg evaporation process at 680 °C for 0.5 hour.

Bean critical state model (Chen & Goldfarb, 1989).

**3.2 Sample characterization** 

**4. Results and discussion 4.1 Morphology and density** 

crystallinity and small grains.

Fig. 2. Microstructure of Mg diffusion processed samples via various sintering conditions including (a) 1100 ºC 0.5 h, (b) 1100 ºC 1 h, (c) 1100 ºC 3 h, (d) 660 ºC 24 h, (e) 660 ºC 48 h, (f) 660 ºC 96 h, (g) 1100 ºC 0.1h - 660 ºC 3 h, (h) 1100 ºC 0.1 h - 660 ºC 6 h, (i) 1100 ºC 0.1h - 660 ºC 12 h, (k) 660 ºC 12 h - 1100 ºC 0.25 h, and (l) 660 ºC 12 h - 1100 ºC 0.5 h. (j) Microstructure of compacted B powder before sintering.

Structural Characteristic and Superconducting

diffusion process could not avoid MgO formation.

Radiation source CuK

Temperature (K) 298

and the calculated one was evaluated by goodness of fit *s*.

(º) 0.01

Sintering

2

**4.3 Influence on** *T***<sup>c</sup>**

Performance of MgB2 Fabricated by Mg Diffusion Process 297

level of disorder and small grains in the high-density MgB2 samples, which are essential for enhancing the flux pinning effect. The MgO nanoparticles are also well known to act as pinning sites in MgB2 thin films (Eom et al., 2001; Singh et al., 2008), whereas it is difficult to control the size in polycrystalline MgB2 samples and large MgO grains could impede supercurrent flow (Maeda et al., 2011a). In this respect, it is necessary to eliminate the MgO impurity from polycrystalline MgB2 materials. However, the MgO secondary phase of the four MgB2 samples is not sensitive to sintering conditions. This indicates that even Mg

condition 1100ºC1h 660ºC48h 1100ºC0.1h - 660ºC6h 660ºC12h - 1100ºC0.5h X-ray powder diffraction data

Goodness of fit *s* 1.32 1.27 1.23 1.23 Lattice parameters *a* (Å) 3.0864 (2) 3.0850 (2) 3.0859 (2) 3.0868 (2) *c* (Å) 3.5210 (1) 3.5210 (1) 3.5210 (1) 3.5211 (2) Unit cell volume *V* (Å3) 29.047 (2) 29.020 (3) 29.038 (3) 29.054 (3) Reflection of MgB2 (101) FWHM(º) 0.15 0.31 0.29 0.10 Weight fraction MgB2 (wt%) 95.9 95.8 96.3 96.1 MgO (wt%) 4.1 4.2 3.7 3.9

Table 1. Results of Rietveld refinement on XRD data for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.5 hour. The fitting quality between the observed pattern

Figure 3 shows the real and imaginary components of the ac susceptibility for four MgB2 samples sintered via four major conditions. It was found that the onset of diamagnetism at the dc field of 0 T for the MgB2 sample sintered at 660 ºC for 48 hours is ~ 38 K and 1 K lower than those for the other samples. The decrease in *T*c is most likely due to lattice disorder caused by the reduction of unit cell volume, as indicated by the refinement result listed in Table 1. It is noteworthy that although the other three samples show ~ 39 K of the onset diamagnetism at the dc field of 0 T, the superconducting transition curves are different. Specifically, the transition width of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is ~ 1.5 K and approximately two times broader than that of the MgB2 sample sintered by the one-step process at either 1100 ºC or 660 ºC. It was

(Å) 1.54060 (CuKα1), 1.54443 (CuKα2)

However, the Mg diffusion processed samples at 660 ºC were brittle and the mass densities are ~ 1.5 g/cm3. The value is almost half of the theoretical density, even though the SEM images of the corresponding samples seem to be dense. The porous structure caused by the low density may intensively limit grain connectivity and hence supercurrent flow. On the other hand, the two-step process first sintered at 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 3 -12 hours was found to form well-developed crystalline grains, as can be seen in Fig. 2(g), (h), and (i). The average grain size is below 1 m and smaller than that of Mg diffusion processed samples via the one-step sintering at 1100 ºC. However, unlike Mg diffusion processed samples via the one-step sintering at 660 ºC which have small grains, the mass densities are ~ 2.4 -2.5 g/cm3 and those are above 90 % of the theoretical density. The two-step sintering process achieves both high density and small grains, which arise from the natures of high-temperature sintering and low-temperature sintering, respectively. In contrast to the highly dense structure related to the well-connected small grains, the two-step process first sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.25 - 0.5 hour was found to result in loosely connected grains, as can be seen in Fig. 2(k) and (l). The mass densities are ~ 1.7 g/cm3 and this result also confirm its porous structure. The corresponding samples were observed to have well-developed crystalline and relatively large grains. The two-step sintering process yields low-density structure and large crystal grains due to low-temperature sintering and high-temperature sintering, respectively.

#### **4.2 Phase composition and lattice structure**

All Mg diffusion processed samples were observed to have the same phase composition, which consists of MgB2 with a trace amount of MgO from XRD -2 scans. Hereafter the samples will be referred as MgB2 samples. In order to examine deeper insight of microstructure, structural parameters of four MgB2 samples sintered via four major conditions were refined and its result is listed in Table 1. It was found that the *a*-lattice parameter is slightly changed owing to various sintering effects, while leaving the *c*-lattice parameter the same. The *a*-lattice parameter of the MgB2 sample sintered at 660 ºC for 48 hours shows 3. 0850 Å, and it is the lowest value of the four samples. The decrease in the *a*lattice parameter reduces the unit cell volume of MgB2. The reduction may introduce structural disorder in the lattice and decrease the DOS at the *E*F responsible for a considerable change in *T*c and thus *B*c2. The XRD full width at half maximum (FWHM) is well known to be closely related to crystallite size and structural disorder and it is also listed in Table 1. The one-step sintering process at 1100 ºC and the two-step process first sintered at 660 ºC followed by second sintering at 1100 ºC were found to obtain relatively narrow FWHM reflecting well-crystallized and large grain. This is consistent with the SEM result, as can be seen in Fig. 2. On the other hand, the FWHM values of the rest of MgB2 samples were observed to be much broader than those of the other MgB2 samples. The broad FWHM of the MgB2 sample sintered at the one-step process at 660 ºC is caused by poorly crystallized and small grains, as also indicated by the SEM observation. However, it should be noted that the FWHM of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is comparable to that of the sample sintered at 660 ºC for 48 hours, although the corresponding sample seems to have well crystallized grains which are distinctly lager than that of MgB2 samples sintered at the one-step process at 660 ºC. This indicates that thanks to the second sintering at the low temperature, the two-step process could achieve both a high

However, the Mg diffusion processed samples at 660 ºC were brittle and the mass densities are ~ 1.5 g/cm3. The value is almost half of the theoretical density, even though the SEM images of the corresponding samples seem to be dense. The porous structure caused by the low density may intensively limit grain connectivity and hence supercurrent flow. On the other hand, the two-step process first sintered at 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 3 -12 hours was found to form well-developed crystalline grains, as can be seen in Fig. 2(g), (h), and (i). The average grain size is below 1 m and smaller than that of Mg diffusion processed samples via the one-step sintering at 1100 ºC. However, unlike Mg diffusion processed samples via the one-step sintering at 660 ºC which have small grains, the mass densities are ~ 2.4 -2.5 g/cm3 and those are above 90 % of the theoretical density. The two-step sintering process achieves both high density and small grains, which arise from the natures of high-temperature sintering and low-temperature sintering, respectively. In contrast to the highly dense structure related to the well-connected small grains, the two-step process first sintered at 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.25 - 0.5 hour was found to result in loosely connected grains, as can be seen in Fig. 2(k) and (l). The mass densities are ~ 1.7 g/cm3 and this result also confirm its porous structure. The corresponding samples were observed to have well-developed crystalline and relatively large grains. The two-step sintering process yields low-density structure and large crystal grains due to low-temperature sintering and high-temperature sintering,

All Mg diffusion processed samples were observed to have the same phase composition,

samples will be referred as MgB2 samples. In order to examine deeper insight of microstructure, structural parameters of four MgB2 samples sintered via four major conditions were refined and its result is listed in Table 1. It was found that the *a*-lattice parameter is slightly changed owing to various sintering effects, while leaving the *c*-lattice parameter the same. The *a*-lattice parameter of the MgB2 sample sintered at 660 ºC for 48 hours shows 3. 0850 Å, and it is the lowest value of the four samples. The decrease in the *a*lattice parameter reduces the unit cell volume of MgB2. The reduction may introduce structural disorder in the lattice and decrease the DOS at the *E*F responsible for a considerable change in *T*c and thus *B*c2. The XRD full width at half maximum (FWHM) is well known to be closely related to crystallite size and structural disorder and it is also listed in Table 1. The one-step sintering process at 1100 ºC and the two-step process first sintered at 660 ºC followed by second sintering at 1100 ºC were found to obtain relatively narrow FWHM reflecting well-crystallized and large grain. This is consistent with the SEM result, as can be seen in Fig. 2. On the other hand, the FWHM values of the rest of MgB2 samples were observed to be much broader than those of the other MgB2 samples. The broad FWHM of the MgB2 sample sintered at the one-step process at 660 ºC is caused by poorly crystallized and small grains, as also indicated by the SEM observation. However, it should be noted that the FWHM of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is comparable to that of the sample sintered at 660 ºC for 48 hours, although the corresponding sample seems to have well crystallized grains which are distinctly lager than that of MgB2 samples sintered at the one-step process at 660 ºC. This indicates that thanks to the second sintering at the low temperature, the two-step process could achieve both a high


scans. Hereafter the

respectively.

**4.2 Phase composition and lattice structure** 

which consists of MgB2 with a trace amount of MgO from XRD

level of disorder and small grains in the high-density MgB2 samples, which are essential for enhancing the flux pinning effect. The MgO nanoparticles are also well known to act as pinning sites in MgB2 thin films (Eom et al., 2001; Singh et al., 2008), whereas it is difficult to control the size in polycrystalline MgB2 samples and large MgO grains could impede supercurrent flow (Maeda et al., 2011a). In this respect, it is necessary to eliminate the MgO impurity from polycrystalline MgB2 materials. However, the MgO secondary phase of the four MgB2 samples is not sensitive to sintering conditions. This indicates that even Mg diffusion process could not avoid MgO formation.


Table 1. Results of Rietveld refinement on XRD data for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.5 hour. The fitting quality between the observed pattern and the calculated one was evaluated by goodness of fit *s*.

## **4.3 Influence on** *T***<sup>c</sup>**

Figure 3 shows the real and imaginary components of the ac susceptibility for four MgB2 samples sintered via four major conditions. It was found that the onset of diamagnetism at the dc field of 0 T for the MgB2 sample sintered at 660 ºC for 48 hours is ~ 38 K and 1 K lower than those for the other samples. The decrease in *T*c is most likely due to lattice disorder caused by the reduction of unit cell volume, as indicated by the refinement result listed in Table 1. It is noteworthy that although the other three samples show ~ 39 K of the onset diamagnetism at the dc field of 0 T, the superconducting transition curves are different. Specifically, the transition width of the MgB2 sample first sintered at 1100 ºC followed by second sintering at 660 ºC is ~ 1.5 K and approximately two times broader than that of the MgB2 sample sintered by the one-step process at either 1100 ºC or 660 ºC. It was

Structural Characteristic and Superconducting

**4.4 Influence on** *B***c2**

conditions.

conditions.

Performance of MgB2 Fabricated by Mg Diffusion Process 299

Figure 4 shows the temperature dependence of *B*c2 for MgB2 samples sintered via various

Fig. 4. The temperature dependence of *B*c2 derived from ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T for MgB2 samples sintered via various

It was found that although all MgB2 samples has positive curvatures of *B*c2 at *T* ~ *T*<sup>c</sup> reflecting suppression of the diffusivity in the *σ* bands compared to that in the *π* bands (Gurevich, 2003), the behaviours of curves at lower temperature are considerably different. The *B*c2 slopes of the MgB2 samples first sintered at 1100 ºC followed by second sintering at 660 ºC or sintered only at 660 ºC are steeper in comparison with those of the MgB2 samples sintered at 1100 ºC. The structural disorder of both samples may be responsible for this. By contrast, the *B*c2 slopes of the MgB2 samples sintered at 660 ºC followed by second sintering

reported that the broad transition width arises from the nature of two kinds of superconducting MgB2 phases, which are good ordering phase and a relatively lower level of ordering phase owing to the two-step sintering process (Maeda et al., 2008). The lower level of ordering phase could be a cause for the broad FWHM responsible for structural disorder, as observed in the XRD pattern (Table 1). Contrary to this, the transition width of the MgB2 sample first sintered at 660 ºC followed by second sintering at 1100 ºC is ~ 0.3 K, suggesting that the sample consists of nearly perfect crystallite MgB2 grains without any lattice disorder.

Fig. 3. The real and imaginary components of ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0 T for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at 1100 ºC for 0.5 hour.

## **4.4 Influence on** *B***c2**

298 Superconductors – Properties, Technology, and Applications

reported that the broad transition width arises from the nature of two kinds of superconducting MgB2 phases, which are good ordering phase and a relatively lower level of ordering phase owing to the two-step sintering process (Maeda et al., 2008). The lower level of ordering phase could be a cause for the broad FWHM responsible for structural disorder, as observed in the XRD pattern (Table 1). Contrary to this, the transition width of the MgB2 sample first sintered at 660 ºC followed by second sintering at 1100 ºC is ~ 0.3 K, suggesting that the sample consists of nearly perfect crystallite MgB2 grains without any

Fig. 3. The real and imaginary components of ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0 T for Mg diffusion processed samples via four major sintering conditions that are 1100 ºC for 1 hour, 660 ºC for 48 hours, 1100 ºC for 0.1 hour followed by second sintering at 660 ºC for 6 hours, and 660 ºC for 12 hours followed by second sintering at

lattice disorder.

1100 ºC for 0.5 hour.

Figure 4 shows the temperature dependence of *B*c2 for MgB2 samples sintered via various conditions.

Fig. 4. The temperature dependence of *B*c2 derived from ac susceptibility with *f* = 76.97 Hz, 0*H*ac = 10 T, and 0*H*dc = 0, 0.2, 0.5, 1, 2, 3, 4, and 5 T for MgB2 samples sintered via various conditions.

It was found that although all MgB2 samples has positive curvatures of *B*c2 at *T* ~ *T*<sup>c</sup> reflecting suppression of the diffusivity in the *σ* bands compared to that in the *π* bands (Gurevich, 2003), the behaviours of curves at lower temperature are considerably different. The *B*c2 slopes of the MgB2 samples first sintered at 1100 ºC followed by second sintering at 660 ºC or sintered only at 660 ºC are steeper in comparison with those of the MgB2 samples sintered at 1100 ºC. The structural disorder of both samples may be responsible for this. By contrast, the *B*c2 slopes of the MgB2 samples sintered at 660 ºC followed by second sintering

Structural Characteristic and Superconducting

samples.

conditions.

**6. References** 

**5. Conclusion** 

Performance of MgB2 Fabricated by Mg Diffusion Process 301

second sintering at 660 ºC is significantly higher than those of the other MgB2 samples. The enhancement of *J*c at not only low field but also high field is clearly due to the twostep sintering process which yields well-connected small grains with a high level of disorder in highly dense MgB2 samples. It is also interesting to note that even though the *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is nearly the same as that of the MgB2 sample sintered at 660 ºC for 48 hours, the behaviours of the curves are somewhat different. Specifically, the low-field *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is slightly higher than that of the MgB2 sample sintered at 660 ºC for 48 hours, whereas the corresponding behaviours of high-field *J*c are distinctly opposite. This results from high density and lack of pinning sites caused by good crystallinity, which are responsible for the increase in self-field *J*c and the decrease in high-field *J*c, respectively. Owing to the low density and nearly perfect crystallization, the MgB2 samples sintered at 660 ºC followed by second sintering at 1100 ºC show the lowest values of in-field *J*c in all MgB2

In summary, this chapter has introduced structural characteristic and superconducting performance of pure MgB2 polycrystalline samples fabricated by Mg diffusion processes. The Mg diffusion processes easily yield MgB2 phase and a trace amount of MgO phase in polycrystalline samples. The microstructure and mass density of the MgB2 samples can be controlled by Mg diffusion processes via various sintering conditions. The two-step sintering process, namely, the sintering for a short period of 0.1 hour at 1100 ºC first, followed by the second sintering at 660 ºC, produces highly dense MgB2 samples characterized by well-connected small grains with a high level of disorder. The unique feature results in considerable improvement of *B*c2 and in-field *J*c. Therefore, Mg diffusion process is critical for controlling the structure of MgB2 polycrystalline samples, and there is great potential for further enhancement of *J*c by careful optimisation of sintering

An, J. M. & Pickett, W. E. (2001). Superconductivity of MgB2: Covalent bonds driven

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at 1100 ºC are intensively shelving compared to those of the other MgB2 samples. This indicates that the MgB2 samples do not have any lattice defects, as suggested by the result of the narrow superconducting transition (Figure 3).

#### **4.5 Influence on** *J***<sup>c</sup>**

Figure 5 shows the magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various conditions. It was found that the *J*c curves are highly sensitive to sintering conditions which lead to considerably differences in MgB2 structure, as shown in the results of SEM observation, XRD analysis, and ac susceptibility. The most noticeable feature shown in this figure is that the in-field *J*c of the MgB2 samples sintered at 1100 ºC followed by

Fig. 5. Magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various conditions.

second sintering at 660 ºC is significantly higher than those of the other MgB2 samples. The enhancement of *J*c at not only low field but also high field is clearly due to the twostep sintering process which yields well-connected small grains with a high level of disorder in highly dense MgB2 samples. It is also interesting to note that even though the *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is nearly the same as that of the MgB2 sample sintered at 660 ºC for 48 hours, the behaviours of the curves are somewhat different. Specifically, the low-field *J*c of the MgB2 sample sintered at 1100 ºC for 1 hour is slightly higher than that of the MgB2 sample sintered at 660 ºC for 48 hours, whereas the corresponding behaviours of high-field *J*c are distinctly opposite. This results from high density and lack of pinning sites caused by good crystallinity, which are responsible for the increase in self-field *J*c and the decrease in high-field *J*c, respectively. Owing to the low density and nearly perfect crystallization, the MgB2 samples sintered at 660 ºC followed by second sintering at 1100 ºC show the lowest values of in-field *J*c in all MgB2 samples.

## **5. Conclusion**

300 Superconductors – Properties, Technology, and Applications

at 1100 ºC are intensively shelving compared to those of the other MgB2 samples. This indicates that the MgB2 samples do not have any lattice defects, as suggested by the result of

Figure 5 shows the magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various conditions. It was found that the *J*c curves are highly sensitive to sintering conditions which lead to considerably differences in MgB2 structure, as shown in the results of SEM observation, XRD analysis, and ac susceptibility. The most noticeable feature shown in this figure is that the in-field *J*c of the MgB2 samples sintered at 1100 ºC followed by

Fig. 5. Magnetic field dependence of *J*c at 20 K for MgB2 samples sintered via various

the narrow superconducting transition (Figure 3).

**4.5 Influence on** *J***<sup>c</sup>**

conditions.

In summary, this chapter has introduced structural characteristic and superconducting performance of pure MgB2 polycrystalline samples fabricated by Mg diffusion processes. The Mg diffusion processes easily yield MgB2 phase and a trace amount of MgO phase in polycrystalline samples. The microstructure and mass density of the MgB2 samples can be controlled by Mg diffusion processes via various sintering conditions. The two-step sintering process, namely, the sintering for a short period of 0.1 hour at 1100 ºC first, followed by the second sintering at 660 ºC, produces highly dense MgB2 samples characterized by well-connected small grains with a high level of disorder. The unique feature results in considerable improvement of *B*c2 and in-field *J*c. Therefore, Mg diffusion process is critical for controlling the structure of MgB2 polycrystalline samples, and there is great potential for further enhancement of *J*c by careful optimisation of sintering conditions.

## **6. References**


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**14** 

İbrahim Karaca

 *Turkey* 

**Measurement of Levitation Forces in** 

**High Temperature Superconductors** 

Magnetic Levitation (MagLev) has become a popular of new technological applications since the discovery of the new superconductor materials, in 1987. MagLev transportation systems have been ignited the human's imagination about levitated motion. MagLev systems have become very important in the transportation and other applications in the future life in the world is generally based on the superconductor technology. These applications are ranging from in an energy efficient prototype of a cryogen transfer line to in space energy storage

The superconducting behaviors of materials originated from two basic physical properties. One of them is disappearance of the electrical resistance below the transition temperature ( ) *TC* in the superconducting materials. The another of them is special magnetic properties in an applied magnetic field ( ) *H* including perfect diamagnetism at low fields (low temperature superconductors (LTS) and high temperature superconductors (HTSs)) and the penetration of quantized magnetic flux (vortices or flux lines) at higher fields regions (HTSs). The electrons in the superconducting state can be described by a macroscopic wave function which is undisturbed by scattering and, in the absence of applied fields and currents, uniform in space over macroscopic distances. This coherence of the superconducting state is preserved even in the presence of weak currents and magnetic fields (below the upper critical field <sup>2</sup> ( ) *HC* , which is a necessary condition for the practical use of the high temperatures superconductors (HTSs) in strong magnetic fields. The additional condition for new applications in these superconductors materials have been used to avoid any motion of flux lines. This can be achieved through the introduction of pinning centers in the superconductor, interacting, in most cases, with the normal conducting core of the flux lines. A flux line is composed of a normal conducting core regions and a surrounding circulating supercurrents area where the magnetic field and supercurrents fall off within penetration depth. The critical current density ( ) *CJ* increases with the applied magnetic field ( ) *H* increases and penetration depth ( )

decreases. The force-distance (permanent magnets and superconductor) hysteresis loops during the descending and ascending process expanded with critical current density

**1. Introduction** 

systems.

increases.

*Department of Physics, Niğde University, Niğde* 

Zhu, Y.; Matsumoto, A.; Senkowicz, B. J.; Kumakura, H.; Kitaguchi, H.; Jewell, M. C.; Hellstrom, E. E.; Larbalestier, D. C. & Voyles, P. M. (2007b). Microstructures of SiC nanoparticle-doped MgB2/Fe tapes. *Journal of Applied Physics*, 102(1): 013913, ISSN 0021-8979.

## **Measurement of Levitation Forces in High Temperature Superconductors**

## İbrahim Karaca

*Department of Physics, Niğde University, Niğde Turkey* 

## **1. Introduction**

306 Superconductors – Properties, Technology, and Applications

Zhu, Y.; Matsumoto, A.; Senkowicz, B. J.; Kumakura, H.; Kitaguchi, H.; Jewell, M. C.;

0021-8979.

Hellstrom, E. E.; Larbalestier, D. C. & Voyles, P. M. (2007b). Microstructures of SiC nanoparticle-doped MgB2/Fe tapes. *Journal of Applied Physics*, 102(1): 013913, ISSN

> Magnetic Levitation (MagLev) has become a popular of new technological applications since the discovery of the new superconductor materials, in 1987. MagLev transportation systems have been ignited the human's imagination about levitated motion. MagLev systems have become very important in the transportation and other applications in the future life in the world is generally based on the superconductor technology. These applications are ranging from in an energy efficient prototype of a cryogen transfer line to in space energy storage systems.

> The superconducting behaviors of materials originated from two basic physical properties. One of them is disappearance of the electrical resistance below the transition temperature ( ) *TC* in the superconducting materials. The another of them is special magnetic properties in an applied magnetic field ( ) *H* including perfect diamagnetism at low fields (low temperature superconductors (LTS) and high temperature superconductors (HTSs)) and the penetration of quantized magnetic flux (vortices or flux lines) at higher fields regions (HTSs). The electrons in the superconducting state can be described by a macroscopic wave function which is undisturbed by scattering and, in the absence of applied fields and currents, uniform in space over macroscopic distances. This coherence of the superconducting state is preserved even in the presence of weak currents and magnetic fields (below the upper critical field <sup>2</sup> ( ) *HC* , which is a necessary condition for the practical use of the high temperatures superconductors (HTSs) in strong magnetic fields. The additional condition for new applications in these superconductors materials have been used to avoid any motion of flux lines. This can be achieved through the introduction of pinning centers in the superconductor, interacting, in most cases, with the normal conducting core of the flux lines. A flux line is composed of a normal conducting core regions and a surrounding circulating supercurrents area where the magnetic field and supercurrents fall off within penetration depth. The critical current density ( ) *CJ* increases with the applied magnetic field ( ) *H* increases and penetration depth ( ) decreases. The force-distance (permanent magnets and superconductor) hysteresis loops during the descending and ascending process expanded with critical current density increases.

Measurement of Levitation Forces in High Temperature Superconductors

critical current ( ) *CJ* as constant.

behavior of the HTSs.

materials;

ii. History of applied field; iii. The initial state of the system; iv. The boundary conditions.

penetrated (Johansen, *et al.*, 1994; Riise, *et al.,* 1994).

order to study the influence of orientation, material, or shape of superconducting samples, comparing, sometimes, thin films with bulk samples (Navau & Sanchez, 1998). On the other hand, a few models have been developed, most of them studying the interaction between a superconductor and a permanent magnet. Hellman *et al.* and Yang presented models based on total flux exclusion of the samples that thus describe the behavior of LTSs or HTSs materials with higher critical current densities (Navau & Sanchez, 1998). Schönhuber and Moon, Chan *et al.*, and Torng and Chen have taken into account the penetration of supercurrents, the latter with an applied field provided by a pair of oppositely wound coils (Schonhuber & Moon, 1994; Chan, *et al.,* 1994; Torng & Chen, 1992). Some papers have considered granular structure for the superconductors when grains are completely

All over the models mentioned above assume that a superconducting sample is small enough to consider the magnetic field gradient constant along it. Demagnetization field of the sample due to the finite dimensions of the HTSs has been neglected by the same time. Demagnetization effects have been take into only account in theoretical calculation so far Tsuchimoto *et al.* have calculated the dependence of levitation force upon different geometrical parameters of a PM-HTSs system by using the an axisymmetric boundary element analysis (Tsuchimoto, *et al.,*1994). Portabella *et al.* have been used finite element calculations for the levitation force-distance ( ) *F x* hysteresis loop in order to estimate the value of critical current of a HTSs. Navau and Sanchez calculated the dependence of maximum vertical force upon the length of superconductor, considering both the demagnetization effect and the nonuniformity of the field gradient. In spite of these efforts, a systematic treatment of vertical levitation force for which the applied magnetic field can not be assumed to have a constant gradient has not been developed yet. In the scope of this chapter, we have been observed vertical levitation force as a function of distance between PM and HTSs. Moreover, most of the developments consider the superconductor to be in the critical state and describe it by means of Bean's critical-state model considering the

The most HTSs present a dependence of their critical current density ( ) *CJ* upon the internal magnetic field. Some analytical expressions have been proposed and successfully applied to describe the magnetic properties of superconductors (Navau & Sanchez, 1998). Therefore it is necessary to include such dependence in the study of the levitation

Kim's model dependence on the critical current density was developed by Tsuchimoto *et al.*  with numerical (Navau & Sanchez, 1998). But, physical properties dependence on critical

i. Character of response to applied field, equations of dependence on properties of

current density ( ) *CJ* in levitation behavior has not been studied in general so far.

To obtain critical current density ( ) *CJ* , the consideration includes:

309

The mechanism of superconductivity in high temperature superconductor (HTS) is still controversially discussed. The Bardeen-Cooper-Schrieffer (BCS) concept of a macroscopic quantum state of Cooper pairs remains the common basis in almost all theories proposed for HTS. However, there is no consensus about the origin of the pairing so far. The lack of an isotope effect in HTS of highest *TC* would favor a nonphonon pairing mechanism as, for instance, d-wave pairing, whereas its presence in several HTS with lower*TC* suggests a phonon-mediated s-wave pairing. The most experimental features favor d-wave pairing. A direct test of the phonon mechanism for pairing in HTSs is to check the sign change in the energy gap ( ( )) *E k <sup>g</sup>* as function of momentum ( ) *k* . A d-state energy gap is composed of two pairs of positive and negative leaves forming a four-leaf clover. In contrast, the magnitude of an s-state may be anisotropic but always remains positive.

## **2. Levitation with superconductors**

Superconducting levitation by using the permanent magnet (PM) is a fascinating property of the magnetic behavior of these materials. It can be mainly defined in two regimes for HTSs. In the first regime, the magnetic induction applied an external magnetic field to samples are expelled from the superconductor and do not penetrate inside of the superconductor materials, which is defined as Meissner effect. This effect is possible only if the applied magnetic field does not exceed the first critical field *(*so called *HC*<sup>1</sup> *)* of the HTS. To obtain experimentally levitation in the Meissner state, it is convenient to close the magnet very carefully, after cooling the superconductor. In the second regime of the levitation, i.e., the mixed state, a vortex penetration through for HTSs, occurs only for fields between the first ( *HC*<sup>1</sup> ) and the second critical magnetic field ( *HC*<sup>2</sup> ) of the HTSs. This is so because magnetic lines could be pinned if they penetrate in the sample and the superconductor would be switched to the mixed state. It is clearly that the levitation measurement is finding in the mixed state acceptable. Magnetic levitation effects are seen Meissner states as the same in HTSs and LTSs. But, magnetic levitation effects originated from flux lines for only HTSs.

When a superconductor is in the presence of a changing applied magnetic field, supercurrents ( *CI* ) are induced inside the sample producing a diamagnetic response of the material. The interaction of the supercurrents with an inhomogeneous magnetic field can produce stable levitation. The flux lines in the small currents densities ( *<sup>C</sup> J J* for a certain critical current *CJ* ) for the real HTSs can be pinned by inhomogenities. Under this condition, when an external applied field increases or decreases, the magnetic flux moves within the superconductor towards its interior or exterior until a critical slope is reached in the flux profile. The current density in this critical state attains its maximum value *CJ* . The critical slope behavior has been modeled by Bean's critical-state model and its extensions, phenomenological. Brandt showed qualitatively how from the pinning of flux lines some properties of levitation of HTSs can be explained in detail (Brandt, 1998).

Moon *et al.* measured the vertical forces on a levitating superconductor showing some of the main properties (Moon, 1988). This measurement has been extended by several groups in

The mechanism of superconductivity in high temperature superconductor (HTS) is still controversially discussed. The Bardeen-Cooper-Schrieffer (BCS) concept of a macroscopic quantum state of Cooper pairs remains the common basis in almost all theories proposed for HTS. However, there is no consensus about the origin of the pairing so far. The lack of an isotope effect in HTS of highest *TC* would favor a nonphonon pairing mechanism as, for instance, d-wave pairing, whereas its presence in several HTS with lower*TC* suggests a phonon-mediated s-wave pairing. The most experimental features favor d-wave pairing. A direct test of the phonon mechanism for pairing in HTSs is to check the sign change in the energy gap ( ( )) *E k <sup>g</sup>* as function of momentum ( ) *k* . A d-state energy gap is composed of two pairs of positive and negative leaves forming a four-leaf clover. In contrast, the magnitude of an s-state may be

Superconducting levitation by using the permanent magnet (PM) is a fascinating property of the magnetic behavior of these materials. It can be mainly defined in two regimes for HTSs. In the first regime, the magnetic induction applied an external magnetic field to samples are expelled from the superconductor and do not penetrate inside of the superconductor materials, which is defined as Meissner effect. This effect is possible only if the applied magnetic field does not exceed the first critical field *(*so called *HC*<sup>1</sup> *)* of the HTS. To obtain experimentally levitation in the Meissner state, it is convenient to close the magnet very carefully, after cooling the superconductor. In the second regime of the levitation, i.e., the mixed state, a vortex penetration through for HTSs, occurs only for fields between the first ( *HC*<sup>1</sup> ) and the second critical magnetic field ( *HC*<sup>2</sup> ) of the HTSs. This is so because magnetic lines could be pinned if they penetrate in the sample and the superconductor would be switched to the mixed state. It is clearly that the levitation measurement is finding in the mixed state acceptable. Magnetic levitation effects are seen Meissner states as the same in HTSs and LTSs. But, magnetic levitation effects originated

When a superconductor is in the presence of a changing applied magnetic field, supercurrents ( *CI* ) are induced inside the sample producing a diamagnetic response of the material. The interaction of the supercurrents with an inhomogeneous magnetic field can produce stable levitation. The flux lines in the small currents densities ( *<sup>C</sup> J J* for a certain critical current *CJ* ) for the real HTSs can be pinned by inhomogenities. Under this condition, when an external applied field increases or decreases, the magnetic flux moves within the superconductor towards its interior or exterior until a critical slope is reached in the flux profile. The current density in this critical state attains its maximum value *CJ* . The critical slope behavior has been modeled by Bean's critical-state model and its extensions, phenomenological. Brandt showed qualitatively how from the pinning of flux lines some

Moon *et al.* measured the vertical forces on a levitating superconductor showing some of the main properties (Moon, 1988). This measurement has been extended by several groups in

properties of levitation of HTSs can be explained in detail (Brandt, 1998).

anisotropic but always remains positive.

**2. Levitation with superconductors** 

from flux lines for only HTSs.

309

order to study the influence of orientation, material, or shape of superconducting samples, comparing, sometimes, thin films with bulk samples (Navau & Sanchez, 1998). On the other hand, a few models have been developed, most of them studying the interaction between a superconductor and a permanent magnet. Hellman *et al.* and Yang presented models based on total flux exclusion of the samples that thus describe the behavior of LTSs or HTSs materials with higher critical current densities (Navau & Sanchez, 1998). Schönhuber and Moon, Chan *et al.*, and Torng and Chen have taken into account the penetration of supercurrents, the latter with an applied field provided by a pair of oppositely wound coils (Schonhuber & Moon, 1994; Chan, *et al.,* 1994; Torng & Chen, 1992). Some papers have considered granular structure for the superconductors when grains are completely penetrated (Johansen, *et al.*, 1994; Riise, *et al.,* 1994).

All over the models mentioned above assume that a superconducting sample is small enough to consider the magnetic field gradient constant along it. Demagnetization field of the sample due to the finite dimensions of the HTSs has been neglected by the same time. Demagnetization effects have been take into only account in theoretical calculation so far Tsuchimoto *et al.* have calculated the dependence of levitation force upon different geometrical parameters of a PM-HTSs system by using the an axisymmetric boundary element analysis (Tsuchimoto, *et al.,*1994). Portabella *et al.* have been used finite element calculations for the levitation force-distance ( ) *F x* hysteresis loop in order to estimate the value of critical current of a HTSs. Navau and Sanchez calculated the dependence of maximum vertical force upon the length of superconductor, considering both the demagnetization effect and the nonuniformity of the field gradient. In spite of these efforts, a systematic treatment of vertical levitation force for which the applied magnetic field can not be assumed to have a constant gradient has not been developed yet. In the scope of this chapter, we have been observed vertical levitation force as a function of distance between PM and HTSs. Moreover, most of the developments consider the superconductor to be in the critical state and describe it by means of Bean's critical-state model considering the critical current ( ) *CJ* as constant.

The most HTSs present a dependence of their critical current density ( ) *CJ* upon the internal magnetic field. Some analytical expressions have been proposed and successfully applied to describe the magnetic properties of superconductors (Navau & Sanchez, 1998). Therefore it is necessary to include such dependence in the study of the levitation behavior of the HTSs.

Kim's model dependence on the critical current density was developed by Tsuchimoto *et al.*  with numerical (Navau & Sanchez, 1998). But, physical properties dependence on critical current density ( ) *CJ* in levitation behavior has not been studied in general so far.

To obtain critical current density ( ) *CJ* , the consideration includes:


Measurement of Levitation Forces in High Temperature Superconductors

from Eq. (1-4) as the next form,

using the Eqs. (1) and (2), respectively if the

where

*H*<sup>0</sup> /

identical.

1994),

Where

the 

*et al*., 1994).

**d. Yin model,** 

A general critical-state model was developed for the critical current density. It is obtained

( ) ( ,)

condition *H H <sup>i</sup>* / 1 0 is satisfied in eq. (5) and the *K* and *n* are set to <sup>0</sup> *J T <sup>C</sup>* () H

, respectively (Xu, et al., 1990). In the conditions of *H H <sup>i</sup>* / 1 <sup>0</sup> and

*J T J HT*

*C i*

with the preceding conditions and letting *HH x <sup>i</sup>* / <sup>0</sup> , it has

0 0

 

> 

(Moon, 1990; Sanchez & Navau, 2001; Brandt, 1990; De Medeiros, *et al.* 1998),

i. Biot–Savart's law: Calculating *BPM* with Biot–Savart's law.

<sup>0</sup> ,*U*<sup>0</sup> , ( ) *W P E J HP V f* 

term in the *WV* corresponds to <sup>0</sup>

 *HH H C n* <sup>2</sup> 

coefficient (with <sup>2</sup> 

*x x*

lim 1 lim 1 

 *C C*

0

1 / () *C*

is a dimensionless constant. Eq. (5) can be converted to Bean and Kim models by

*H HT*

*i*

**H**0, where **H**0 is a finite value with dimension of field, and taking the limit of ( ) *C i J H*

( ) ( ) lim ( , ) ( )exp( / )

It can be seen that Eq. (6) has the same form as Eq. (4) and that their physical meanings are

Material's equation, known as Yin model, has been developed in the form as (Yin *et al*.,

0 <sup>0</sup> ( ) 2 exp sinh *U WV WL EJ H kT kT*

velocity, the pinning potential, the viscous dissipation term of flux motion with viscosity

of the volume of the moving flux bundle and the range of the force action, respectively. In

Yin model includes pinning barrier term, viscous dissipation term and direction dependent. It provides a unified description which covers all regimes of flux motion: thermally activated flux flow, flux creep, critical state and flux flow. Yin model contain Anderson–Kim model, Bardeen–Steven model and critical state model ( ) *FJH P C* as its specific forms (Yin

There are different ways of calculation of magnetic field ( ) *HPM* of permanent magnets

*J T J T J HT JT H H*

*C i xH H C i*

 

311

and

1 , with

(5)

0 and 1. Eq. (3) is obtained when the

<sup>0</sup> (1/ )( / ) 0

*<sup>x</sup> <sup>x</sup>* <sup>0</sup> (6)

(7)

*<sup>f</sup>* ), the energy due to Lorentz driving force, a product

exp ( ) *U J kT* . *U* represents the activation barrier.

, *W JHP <sup>L</sup>* and *P* are a prefactor with dimension of

Models on magnetization of the HTSs as can be seen in below.

#### **a. Bean model,**

Bean proposed a so-called critical-state model in which the hysteresis loop is plugged in a macroscopic parameter critical current density *CJ* (Bean, 1964). According to the Bean model, critical current density *CJ* presumes a constant and given as

$$\mathbf{J}\_{\mathbb{C}}(\mathcal{H}\_{i'}T) = \mathbf{J}\_{\mathbb{C}}(T) \tag{1}$$

where *Hi* and *T* are the local magnetic field and the temperature with respectively. In the model also pointed out that *CJ* was directly determined by the microstructure of the superconductors. The relationship of *CJ* versus temperature can be expressed as 0 0 ( ) ( )/( ) *C CC C JT J T T T T* (Berger, *et al.* 2007), where *TC* is the critical temperature and *<sup>C</sup>*<sup>0</sup> *J* the critical current density according to the reference temperature <sup>0</sup> ( ) *T* .

#### **b. Kim model,**

Anderson and Kim modified the Bean model (Anderson & Kim, 1964; Kim, et al. 1962) and *CJ* was suggested that it should vary with the local magnetic field and should have the form

$$f\_{\mathbb{C}}(H\_i, T) = \frac{f\_{\mathbb{C}}(T)}{1 + H\_i \, / \, H\_0} \tag{2}$$

where *H*0 is a macroscopic materials parameter with the dimension of field. The model was found to fit the experimental results for some conventional superconductors for a solid cylinder shaped superconductors.

According to Kim model the *CJ* depend on the term *H H <sup>i</sup>* / <sup>0</sup> , which may vary considerably among different materials. Watson showed that, for some systems, the condition *H H <sup>i</sup>* <sup>0</sup> is fulfilled. Eq. (2) shows a simple linear field dependence of *CJ* , ( ) *Ci i J H A CH* , where *A* and *C* are constants (Hindley & Watson, 1969 ).

#### **c. Power law model,**

Irie and Yamafuji are focus on the specific pinning mechanisms, a power-law field dependence of *CJ* was suggested (Irie & Yamafuji, 1967),

$$f\_{\mathcal{C}}(H\_i, T) = \frac{K(T)}{H\_i^n} \tag{3}$$

where *K* is a materials parameter and *n* directly reflects pinning strength. An empirical formula based on experimental results developed by Fietz et al. (Fietz, *et al.,* 1964),

$$\mathcal{J}\_{\mathbb{C}}(H\_{i'}T) = \mathcal{J}\_{\mathbb{C}}(T) \exp\left(-H\_i \,/\, H\_0\right). \tag{4}$$

Bean proposed a so-called critical-state model in which the hysteresis loop is plugged in a macroscopic parameter critical current density *CJ* (Bean, 1964). According to the Bean

where *Hi* and *T* are the local magnetic field and the temperature with respectively. In the model also pointed out that *CJ* was directly determined by the microstructure of the superconductors. The relationship of *CJ* versus temperature can be expressed as 0 0 ( ) ( )/( ) *C CC C JT J T T T T* (Berger, *et al.* 2007), where *TC* is the critical temperature and

Anderson and Kim modified the Bean model (Anderson & Kim, 1964; Kim, et al. 1962) and *CJ* was suggested that it should vary with the local magnetic field and should have the

( ) ( ,) 1 /

where *H*0 is a macroscopic materials parameter with the dimension of field. The model was found to fit the experimental results for some conventional superconductors for a solid

According to Kim model the *CJ* depend on the term *H H <sup>i</sup>* / <sup>0</sup> , which may vary considerably among different materials. Watson showed that, for some systems, the condition *H H <sup>i</sup>* <sup>0</sup> is fulfilled. Eq. (2) shows a simple linear field dependence of *CJ* , ( ) *Ci i J H A CH* , where *A*

Irie and Yamafuji are focus on the specific pinning mechanisms, a power-law field

( ) ( ,) *C i <sup>n</sup>*

where *K* is a materials parameter and *n* directly reflects pinning strength. An empirical

formula based on experimental results developed by Fietz et al. (Fietz, *et al.,* 1964),

*i*

*K T J HT <sup>H</sup>* (3)

<sup>0</sup> ( , ) ( )exp ( / ) *Ci C <sup>i</sup> J HT J T H H* . (4)

*J T J HT H H*

*C*

*i*

0

( ,) () *Ci C J HT J T* (1)

(2)

Models on magnetization of the HTSs as can be seen in below.

model, critical current density *CJ* presumes a constant and given as

*<sup>C</sup>*<sup>0</sup> *J* the critical current density according to the reference temperature <sup>0</sup> ( ) *T* .

*C i*

**a. Bean model,** 

**b. Kim model,** 

cylinder shaped superconductors.

**c. Power law model,** 

and *C* are constants (Hindley & Watson, 1969 ).

dependence of *CJ* was suggested (Irie & Yamafuji, 1967),

form

A general critical-state model was developed for the critical current density. It is obtained from Eq. (1-4) as the next form,

$$J\_{\mathbb{C}}(H\_{i'}T) = \frac{J\_{\mathbb{C}}(T)}{\left[1 + H\_i \,/\, H\_0(T)\right]^\beta} \tag{5}$$

where is a dimensionless constant. Eq. (5) can be converted to Bean and Kim models by using the Eqs. (1) and (2), respectively if the 0 and 1. Eq. (3) is obtained when the condition *H H <sup>i</sup>* / 1 0 is satisfied in eq. (5) and the *K* and *n* are set to <sup>0</sup> *J T <sup>C</sup>* () H and , respectively (Xu, et al., 1990). In the conditions of *H H <sup>i</sup>* / 1 <sup>0</sup> and 1 , with *H*<sup>0</sup> / **H**0, where **H**0 is a finite value with dimension of field, and taking the limit of ( ) *C i J H* with the preceding conditions and letting *HH x <sup>i</sup>* / <sup>0</sup> , it has

$$\lim\_{i} I\_{\mathbb{C}}(H\_{i},T) = \frac{I\_{\mathbb{C}}(T)}{\lim\_{\mathbf{x} \to \mathbf{0}} \mathbb{E} \mathbf{1} + \mathbf{x} \mathbf{J}^{\mathcal{S}}} = \frac{I\_{\mathbb{C}}(T)}{\lim\_{\mathbf{x} \to \mathbf{0}} \mathbb{E} \mathbf{1} + \mathbf{x} \mathbf{J}^{(1/\mathbf{x})(H\_{0}/H\_{0})}} = I\_{\mathbb{C}}(T) \exp(-H\_{i} \mid H\_{0}) \tag{6}$$

It can be seen that Eq. (6) has the same form as Eq. (4) and that their physical meanings are identical.

#### **d. Yin model,**

Material's equation, known as Yin model, has been developed in the form as (Yin *et al*., 1994),

$$E(f) = 2\nu\_0 H \exp\left[-\left(\frac{\mathcal{U}\_0 + \mathcal{W}\_V}{kT}\right)\right] \sinh\left[\frac{\mathcal{W}\_L}{kT}\right] \tag{7}$$

Where<sup>0</sup> ,*U*<sup>0</sup> , ( ) *W P E J HP V f* , *W JHP <sup>L</sup>* and *P* are a prefactor with dimension of velocity, the pinning potential, the viscous dissipation term of flux motion with viscosity coefficient (with <sup>2</sup> *HH H C n* <sup>2</sup> *<sup>f</sup>* ), the energy due to Lorentz driving force, a product of the volume of the moving flux bundle and the range of the force action, respectively. In the term in the *WV* corresponds to <sup>0</sup> exp ( ) *U J kT* . *U* represents the activation barrier. Yin model includes pinning barrier term, viscous dissipation term and direction dependent. It provides a unified description which covers all regimes of flux motion: thermally activated flux flow, flux creep, critical state and flux flow. Yin model contain Anderson–Kim model, Bardeen–Steven model and critical state model ( ) *FJH P C* as its specific forms (Yin *et al*., 1994).

There are different ways of calculation of magnetic field ( ) *HPM* of permanent magnets (Moon, 1990; Sanchez & Navau, 2001; Brandt, 1990; De Medeiros, *et al.* 1998),

i. Biot–Savart's law: Calculating *BPM* with Biot–Savart's law.

Measurement of Levitation Forces in High Temperature Superconductors

Levitation force (N)

0,0

0,5

1,0

1,5

2,0

Airgap(mm) 0 5 10 15 20 25

1. The HTS's are of HTS with a strongly irreversible magnetic behavior, i.e., their

2. The levitation phenomenon between HTS and PM materials has been studied for magnetic granularity or polycrystalline bulk materials in general. The magnetic

During the past years, many research groups have done extensive studies of levitation forces using HTS bulk materials of various kinds (Takao *et al.* 2011; Lu *et al.,* 2011; Krabbes *et al.*, 2000). The higher levitation force and stable levitation or suspension between a PM and a bulk superconductor, makes superconducting bulks possible for various applications such as superconducting magnetic bearings, flywheel, magnetic levitation transportation

The levitation forces measure between permanent magnet and superconducting high temperature, bulk or film in general. The critical current has the crucial role in the levitation force measuring system. The levitation force between HTS and PM originated from by the interaction between the induced electric current in the HTS and the magnetic field from the PM. The critical current density ( ) *CJ H* is closely related with the flux pinning center density and flux pinning force of the sample, and also closely related with the flux pinning, depinning process and the magnetic flux creep rate during the levitation force measurement. The ( ) *CJ H* is also a fixed parameter (if we consider the process based on

Fig. 1. The levitation forces evolution of the distance of between HTS and PM with

magnetization depends on the prehistory of the external magnetic fields.

levitation force studies are very poor for single crystalline materials.

experimental and different theoretical models (Chun & Lee, 2002).

3. A finite levitation force is required non uniform magnetic field.

systems, generators or motors and permanent magnets.

**3. Levitation experimental system and data analaysis** 

Experimental data Lorentz equation Maxwell stress tensor Virtual work method

313


Field expression in cylindrical coordinates (John David Jackson, 1998), is

$$
\vec{H}(\vec{r}) = \vec{\nabla} \times \vec{A} = \frac{\mu\_0}{4\pi} \int \int\_V \frac{\vec{J}(\vec{r}) \times (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|} dv' \tag{8}
$$

Numerical methods have been developed towards to solve the ( ) *CJ H* and the levitation force. Analysis and simulation can be carried out based on above models and various parameters.

The levitation forces evaluation of distance between HTS and PM (or air gap) are seen in Fig. 1. As can be seen from this figure, levitation forces in a magnetic media for experimental and theoretical calculation decreases when distance between HTS and PM increases.

There are several methods for the calculation of levitation force in magnetic field. Some of these methods are Lorentz force equation (Chun &Lee, 2002), Maxwell's stress tensor (Amrhein & Krein, 2009) and virtual work methods (Vandevelde& Melkebeek, 2001). The Lorentz force equation method is important to know the exact distribution of the critical current density and the magnetic field. The Maxwell's stress tensor method is calculated by the integration of the Maxwell's stress tensor on an arbitrary surface surrounding the object.

The magnetic force as the virtual work method is given by the derivation of the co-energy with respect to a virtual displacement.

The magnetic levitation forces evolution of the distance of between HTS and PM with experimental and different theoretical models are seen in Fig. 1.

From this figure, levitation force nearly exponentially increase while distance of between HTS and PM decreases. But, the experimental results nearly agree with the Lorentz force equation methods. In other words, from the comparison, the result by the Lorentz force methods is in strict correspondence with the experimental results quantitatively according to the others simulation methods. Agreement between the Lorentz force methods with the experimental results originated perhaps from the penetration of the applied field into HTSs. Because of the penetration of the applied field into the HTS occurs where the vortex state is. This field passes through the HTSs produce a flux tubes. This states in this type studied so called known repulsive magnetic field in general. Finally, compatible Lorentz force method with the experimental result is not surprising.

Since the discovery of high temperature superconductors (HTSs) this interaction has achieved considerable attention, mainly due to its capability of producing stable noncontacting conditions for such (HTS-PM) systems. The magnetic levitation forces can be demonstrated very easily in experimentally. But, their theoretical models are far more difficult. The reason for these difficulties are the below.

 <sup>0</sup> () ' ( ) ' 4 ' *<sup>V</sup> Jr r r Hr A dv*

Numerical methods have been developed towards to solve the ( ) *CJ H* and the levitation force. Analysis and simulation can be carried out based on above models and various

The levitation forces evaluation of distance between HTS and PM (or air gap) are seen in Fig. 1. As can be seen from this figure, levitation forces in a magnetic media for experimental

There are several methods for the calculation of levitation force in magnetic field. Some of these methods are Lorentz force equation (Chun &Lee, 2002), Maxwell's stress tensor (Amrhein & Krein, 2009) and virtual work methods (Vandevelde& Melkebeek, 2001). The Lorentz force equation method is important to know the exact distribution of the critical current density and the magnetic field. The Maxwell's stress tensor method is calculated by the integration of the Maxwell's stress tensor on an arbitrary surface surrounding the

The magnetic force as the virtual work method is given by the derivation of the co-energy

The magnetic levitation forces evolution of the distance of between HTS and PM with

From this figure, levitation force nearly exponentially increase while distance of between HTS and PM decreases. But, the experimental results nearly agree with the Lorentz force equation methods. In other words, from the comparison, the result by the Lorentz force methods is in strict correspondence with the experimental results quantitatively according to the others simulation methods. Agreement between the Lorentz force methods with the experimental results originated perhaps from the penetration of the applied field into HTSs. Because of the penetration of the applied field into the HTS occurs where the vortex state is. This field passes through the HTSs produce a flux tubes. This states in this type studied so called known repulsive magnetic field in general. Finally, compatible Lorentz force method

Since the discovery of high temperature superconductors (HTSs) this interaction has achieved considerable attention, mainly due to its capability of producing stable noncontacting conditions for such (HTS-PM) systems. The magnetic levitation forces can be demonstrated very easily in experimentally. But, their theoretical models are far more

and theoretical calculation decreases when distance between HTS and PM increases.

iii. Vector potential method: Calculating vector potential *A PM*( ), *H* is the curl of *A* .

Field expression in cylindrical coordinates (John David Jackson, 1998), is

(8)

*r r*

in current-free area, *HPM* is the

ii. Scalar potential method: Calculating scalar potential

gradient of

parameters.

object.

.

with respect to a virtual displacement.

with the experimental result is not surprising.

difficult. The reason for these difficulties are the below.

experimental and different theoretical models are seen in Fig. 1.

Fig. 1. The levitation forces evolution of the distance of between HTS and PM with experimental and different theoretical models (Chun & Lee, 2002).


During the past years, many research groups have done extensive studies of levitation forces using HTS bulk materials of various kinds (Takao *et al.* 2011; Lu *et al.,* 2011; Krabbes *et al.*, 2000). The higher levitation force and stable levitation or suspension between a PM and a bulk superconductor, makes superconducting bulks possible for various applications such as superconducting magnetic bearings, flywheel, magnetic levitation transportation systems, generators or motors and permanent magnets.

## **3. Levitation experimental system and data analaysis**

The levitation forces measure between permanent magnet and superconducting high temperature, bulk or film in general. The critical current has the crucial role in the levitation force measuring system. The levitation force between HTS and PM originated from by the interaction between the induced electric current in the HTS and the magnetic field from the PM. The critical current density ( ) *CJ H* is closely related with the flux pinning center density and flux pinning force of the sample, and also closely related with the flux pinning, depinning process and the magnetic flux creep rate during the levitation force measurement. The ( ) *CJ H* is also a fixed parameter (if we consider the process based on

Measurement of Levitation Forces in High Temperature Superconductors

Fig. 2. Schematic representation of the preparing of HTS sample.

Fig. 3. Schematic representation of the annealing process of the HTS disk.

Magnetic properties were measured by using the magnetic levitation force-distance (between HTS and PM) system at the temperature 77 K in Fig. 4. Measurements were performed in the zero field cooled (ZFC) regimes in applied magnetic field 0.15 *H T <sup>r</sup>* .

315

Bean's model) because the sample is the same, so the levitation force is only related with the total induced electric current in the sample which is depending on magnet moving speed, this is the only variable during the measurement. Superconductors is different from perfect conductor, that if a superconductor is cooled below certain temperature (is called critical temperature, *TC* ) and an external magnetic field is applied then the superconductor repel the magnetic field because of the supercurrent. This super-current is made of the inside the superconductor via of the induced of the external field as the same Lenz law. This situation, the superconductor has the supercurrent as defined superconductor. Firstly grains effected and a current circulate interior of the grain, which is called internor current, *CI* . Secondly all of the interior current sums to surface of the superconductor and they called screening current, *SI* ; which is repulsive and screening to external magnetic field.

The screening currents produce a mutually repulsing behavior, so the external magnetic field do not penetrate interior of the superconductor, which is called Meissner effect in the superconductor. The German physicists Meissner and Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting thin and lead samples. These samples, in the presence of an applied magnetic field, were cooled below what is called their superconducting transition temperature. Below the transition temperature the samples canceled nearly all magnetic fields inside. They detected this effect only indirectly; because the magnetic flux is conserved by a superconductor, when the interior field decreased the exterior field increased. The experiment demonstrated for the first time that superconductors were more than just perfect conductors and provided a uniquely defining property of the superconducting state.

Macroscopic transport properties mechanism of the grain boundary effect on HTS is not entirely understood yet, though on this mechanism several theoretical models have been suggested (Luine & Kresin, 1998; Coble, 2009). In particular, it is assumed that the grain boundary effects are describing as that to following factors: (i) a decrease of electron free path in vicinities of grain boundaries; (ii) stress-field-induced structural disorder and pinnig of magnetron vortices at grain boundaries; (iii) formation of the antiferromagnetic phase within grain cores; (iv) faceting of grain boundaries and d-symmetry of the superconducting order parameter, (v) deviations from bulk stoichiometry in vicinities of grain boundaries.

Figures 2-3 shows a schematic representation of the preparing of the HTS sample with a wet-technique (or ammonium nitrate method) and annealing process of the HTS disk, respectively. The starting material was the powder of nominal Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10 compound, which was synthesized by a wet-technique using high-purity (>99.99%) Bi2O3, PbO, SrCO3, CaCO3 and CuO, We produce our powder by using this method at room temperature. Then, the powder was calcined at 700 °C for 10h, twice as much, and pressed into pellets with 225 MPa. The pellets were placed in a preheated furnace at 850 °C for 100 h then directly cooled to room temperature for quench. After this sintering process pellets regrinding and the powders at various pressures and time were pressed in to cylindrical superconductor disk (CSD: Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10**)** with diameter ( 2*r )* of 13 mm and height ( *h* ) of 13.52 mm. The CSD was placed in a preheated furnace at 815 °C at a rate of 0.1 °C/min for 100 h then directly cooled to room temperature. The details of producing of HTS material was given in previously our work (Karaca, 2009).

Bean's model) because the sample is the same, so the levitation force is only related with the total induced electric current in the sample which is depending on magnet moving speed, this is the only variable during the measurement. Superconductors is different from perfect conductor, that if a superconductor is cooled below certain temperature (is called critical temperature, *TC* ) and an external magnetic field is applied then the superconductor repel the magnetic field because of the supercurrent. This super-current is made of the inside the superconductor via of the induced of the external field as the same Lenz law. This situation, the superconductor has the supercurrent as defined superconductor. Firstly grains effected and a current circulate interior of the grain, which is called internor current, *CI* . Secondly all of the interior current sums to surface of the superconductor and they called screening

The screening currents produce a mutually repulsing behavior, so the external magnetic field do not penetrate interior of the superconductor, which is called Meissner effect in the superconductor. The German physicists Meissner and Ochsenfeld discovered the phenomenon in 1933 by measuring the magnetic field distribution outside superconducting thin and lead samples. These samples, in the presence of an applied magnetic field, were cooled below what is called their superconducting transition temperature. Below the transition temperature the samples canceled nearly all magnetic fields inside. They detected this effect only indirectly; because the magnetic flux is conserved by a superconductor, when the interior field decreased the exterior field increased. The experiment demonstrated for the first time that superconductors were more than just perfect conductors and provided

Macroscopic transport properties mechanism of the grain boundary effect on HTS is not entirely understood yet, though on this mechanism several theoretical models have been suggested (Luine & Kresin, 1998; Coble, 2009). In particular, it is assumed that the grain boundary effects are describing as that to following factors: (i) a decrease of electron free path in vicinities of grain boundaries; (ii) stress-field-induced structural disorder and pinnig of magnetron vortices at grain boundaries; (iii) formation of the antiferromagnetic phase within grain cores; (iv) faceting of grain boundaries and d-symmetry of the superconducting order parameter, (v) deviations from bulk stoichiometry in vicinities of

Figures 2-3 shows a schematic representation of the preparing of the HTS sample with a wet-technique (or ammonium nitrate method) and annealing process of the HTS disk, respectively. The starting material was the powder of nominal Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10 compound, which was synthesized by a wet-technique using high-purity (>99.99%) Bi2O3, PbO, SrCO3, CaCO3 and CuO, We produce our powder by using this method at room temperature. Then, the powder was calcined at 700 °C for 10h, twice as much, and pressed into pellets with 225 MPa. The pellets were placed in a preheated furnace at 850 °C for 100 h then directly cooled to room temperature for quench. After this sintering process pellets regrinding and the powders at various pressures and time were pressed in to cylindrical superconductor disk (CSD: Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10**)** with diameter ( 2*r )* of 13 mm and height ( *h* ) of 13.52 mm. The CSD was placed in a preheated furnace at 815 °C at a rate of 0.1 °C/min for 100 h then directly cooled to room temperature. The details of producing of HTS

current, *SI* ; which is repulsive and screening to external magnetic field.

a uniquely defining property of the superconducting state.

material was given in previously our work (Karaca, 2009).

grain boundaries.

Fig. 2. Schematic representation of the preparing of HTS sample.

Fig. 3. Schematic representation of the annealing process of the HTS disk.

Magnetic properties were measured by using the magnetic levitation force-distance (between HTS and PM) system at the temperature 77 K in Fig. 4. Measurements were performed in the zero field cooled (ZFC) regimes in applied magnetic field 0.15 *H T <sup>r</sup>* .

Measurement of Levitation Forces in High Temperature Superconductors

space (0-6 mm), because, there are no hysteretic behavior in this area. The maximum hysteretic phenomena are seen in the distance range 6-20 mm. This phenomena attributed that the vortex state. The results of the levitation force have generally been measured in a ZFC process, a few works were done in the field cooled (FC) process (Yang *et al.,*2003)**.** 

gradient produced by the magnet. The trapped field in HTS originated from the PM, an attractive force occurs when the sample is moved away from the PM. This result can be attributed to the number of pinning centers in the HTS sample, which results in an increase of trapped magnetic field inside the samples. In addition, the levitation force is related to the

The force-distance ( ) *F x* between CSD (Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10) and PM always hysteresis loop as an example for this chapter during the descending and ascending process

> descending ascending

**Distance, x (mm) 0 10 20 30 40**

Fig. 5. The interaction force between a CSD and a PM always shows a force-distance ( *F x* ) hysteresis loop during the descending (dark green full circle) and ascending (light green full

As can be seen from Fig.5, the levitation force while distance increasing smaller than levitation force while distance decreasing. The maximum values between of two levitation forces are seen near the 10 mm. These values are smaller than expectation values for HTS.

<sup>n</sup>

. Here,

<sup>i</sup> <sup>M</sup> i 1<sup>μ</sup> . *dH dx* is the field

The force between HTS and PM system can be written as *F dH d x*

magnetic moment related with the bulk magnetization *M*

grain size and crystallographic orientation.

are seen in Fig. 5.

**F**

circle) process.

**0,0**

**0,2**

**0,4**

**0,6**

**0,8**

**1,0**

**1,2**

**x / F0 (reduce force)**

317

is the

Magnetic repulsive and attractive forces between the HTS disk and PM were measured using a homemade levitation measuring device (Fig.4). It is figure out in Fig.4 an Nd–B–Fe permanent magnet (1.6x1.4x0.34 cm3 and 0.15 *H T <sup>r</sup>* ) was put on the plate of a sensitive electronic scale. CSD sample (13 mm in diameter) was immobilized at the bottom of a liquid nitrogen-filled receptacle which is movable in the vertical direction (CSD and PM are symmetrically fixed on the same line). This receptacle was then placed just above the plate of the balance without touching it. Thus, the observed weight change of the magnet directly reflected the magnetic force induced in this handmade system (Benlhachemi, *et al.,* 1993, Karaca, 2009)**.** 

Fig. 4. Schematic representation of the magnetic levitation force-distance (between HTS and PM) hysteresis loop measurement experimental system.

The magnetic levitation force-distance (between HTS and PM) ( ) *F x* hysteresis loop ZFC process during the descending and ascending process is seen in Fig.5. Similarly, the full light green and full dark green circles correspond to the levitation force curves ascending and descending with respectively. The hysteresis curves of magnetic levitation are not the same. In other words, magnetic levitation force when distance ascending faster than magnetic levitation force while distance descending. However, the hysteresis loops strongly depend on the size of grains and their connectivity in the bulk HTS. The magnetic levitation forces nearly the same when distance increases from 20 mm to 36 mm. this behavior corresponds to Meissner effects. The levitation force linearly decreases when distance increases from 0 mm to 6 mm. The external magnetic field fully penetrated HTS disk in this

Magnetic repulsive and attractive forces between the HTS disk and PM were measured using a homemade levitation measuring device (Fig.4). It is figure out in Fig.4 an Nd–B–Fe permanent magnet (1.6x1.4x0.34 cm3 and 0.15 *H T <sup>r</sup>* ) was put on the plate of a sensitive electronic scale. CSD sample (13 mm in diameter) was immobilized at the bottom of a liquid nitrogen-filled receptacle which is movable in the vertical direction (CSD and PM are symmetrically fixed on the same line). This receptacle was then placed just above the plate of the balance without touching it. Thus, the observed weight change of the magnet directly reflected the magnetic force induced in this handmade system (Benlhachemi, *et al.,* 1993,

Fig. 4. Schematic representation of the magnetic levitation force-distance (between HTS and

The magnetic levitation force-distance (between HTS and PM) ( ) *F x* hysteresis loop ZFC process during the descending and ascending process is seen in Fig.5. Similarly, the full light green and full dark green circles correspond to the levitation force curves ascending and descending with respectively. The hysteresis curves of magnetic levitation are not the same. In other words, magnetic levitation force when distance ascending faster than magnetic levitation force while distance descending. However, the hysteresis loops strongly depend on the size of grains and their connectivity in the bulk HTS. The magnetic levitation forces nearly the same when distance increases from 20 mm to 36 mm. this behavior corresponds to Meissner effects. The levitation force linearly decreases when distance increases from 0 mm to 6 mm. The external magnetic field fully penetrated HTS disk in this

PM) hysteresis loop measurement experimental system.

Karaca, 2009)**.** 

space (0-6 mm), because, there are no hysteretic behavior in this area. The maximum hysteretic phenomena are seen in the distance range 6-20 mm. This phenomena attributed that the vortex state. The results of the levitation force have generally been measured in a ZFC process, a few works were done in the field cooled (FC) process (Yang *et al.,*2003)**.** 

The force between HTS and PM system can be written as *F dH d x* . Here, is the magnetic moment related with the bulk magnetization *M* <sup>n</sup> <sup>i</sup> <sup>M</sup> i 1<sup>μ</sup> . *dH dx* is the field gradient produced by the magnet. The trapped field in HTS originated from the PM, an attractive force occurs when the sample is moved away from the PM. This result can be attributed to the number of pinning centers in the HTS sample, which results in an increase of trapped magnetic field inside the samples. In addition, the levitation force is related to the grain size and crystallographic orientation.

The force-distance ( ) *F x* between CSD (Bi1.84Pb0.34Sr1.91Ca2.03Cu3.06O10) and PM always hysteresis loop as an example for this chapter during the descending and ascending process are seen in Fig. 5.

Fig. 5. The interaction force between a CSD and a PM always shows a force-distance ( *F x* ) hysteresis loop during the descending (dark green full circle) and ascending (light green full circle) process.

As can be seen from Fig.5, the levitation force while distance increasing smaller than levitation force while distance decreasing. The maximum values between of two levitation forces are seen near the 10 mm. These values are smaller than expectation values for HTS.

Measurement of Levitation Forces in High Temperature Superconductors

for detail, www.superpower-inc.com).

will be need to this investigation results.

I would like to thanks Mehmet Ali Güzel for help.

*Appl. Supercond.*, Vol.9, pp. 1864.

*Compounds*, Vol.215, pp. 263

Vol.24 Issue: 3, pp. 587 – 593.

Bean, C. P. (1962). *Phys. Rev. Lett.,* Vol.8, pp. 250.

Vol.215, pp. 263.

Abrikosov, A. A. (1957). *Zh. Eksp. Teor. Fiz*., Vol.32, pp. 1442.

**6. Acknowledgements** 

**7. References** 

**5. Summary** 

HTS generators are substantially smaller and lighter than copper based machines. These advantages are very attractive in a ship environment. HTS power cables will enable utilities to upgrade power density by 2 to 8 times power corridors because HTS cables transmit power with essential no electrical resistance. Superpower's SFCL technology is enable scaling to high voltage and current with customization to the user's requirements, losses or voltage drop across the device during normal operation, and etc. The HTS technology can be used to make all superconducting magnets with field that will soon exceed 30 Tesla (see

In the scope of this chapter, we have focus on the magnetic levitation force-distance loops for HTS samples. The inter CSD and PM exchange type interaction are determined magnetic levitation forces. The magnetic levitation forces are heavily dependence on grain size, grain connectivity, critical current density, history of applied magnetic field and etc. Combination with hysteresis ( ) *F x* results, these indicate that the Meissner effects, vortex state and metallic behavior are seen the same. The CSD with ZFC process showed that the sample has low quality with weak-link characteristics. A typical hysteresis loop of the axial levitation force between the permanent magnet and the CSCD showed the lower repulsive force in this chapter. This phenomenon is attributed to the fact that the wet technique causes the production of lower dense samples**,** and is caused by a lower critical current density (Altunbaş et al., 1994; Karaca et al., 1998) and by a lower levitation force. It is considered that the changes in the levitation force are related to the grain orientation, the homogeneity, and the number of the pinning centers in the samples (Yanmaz et al., 2002)**.** Levitation force measurements need to investigate several factors end different materials. In the future life

Alamgir, A. K., Yamada, M. H., Harada, N., Osaki, K. & Tada, N. (1999). *IEEE Transaction on* 

Altunbaş, M., Yanmaz, E., Nezir, S., Karal H. & Vidadi Yu A. (1994). *J. Alloys and Compounds*,

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Amrhein, M., Krein, P.T. (2009). Force Calculation in 3-D Magnetic Equivalent Circuit

Anderson, P. W. & Kim, Y. B. (1964). Hard superconductivity: theory of the motion of

abrikosov flux lines, *Rev. Mod. Phys.*, Vol. 36, No. 1, pp. 3943.

Networks With a Maxwell Stress Tensor", *Energy Conversion, IEEE Transactions on,*

319

The levitation force can be related from the interaction between CSD and PM materials in general. This interaction between CSD and PM are mainly dependent on the properties of the CSD and the magnetic field distribution of the magnet. For a bulk superconductor, the levitation force is dependent on many parameters, such as the critical current density and grain radius, grain-orientation, thickness of the sample, and the cooling temperature. For a magnet, the levitation force is closely related with the magnetic ux density, magnetic field distribution, etc. But for a given pair of bulk superconductor and PM, it is known that the same levitation force can be obtained at a gap distance between the superconductor and the magnet, because of the magnetization history of the superconductor, by a mechanical descending and ascending process of the magnet during the axial levitation force measurement state (Yang *et al.,* 2003)**.** 

The low levitation force of sintered superconductor materials can be attributed to two intrinsic material problems of a superconductor in this chapter (Murakami, 1993). The first is the grain boundary weak-link problem and the second is the weak flux pinning problem. In order to resolve these two problems, different material processing techniques have been developed from other workers (Chen *et al.,* 1998)**.** 

## **4. Technological applications of levitation forces in HTS**

Temperatures in the range near the 4-30 K and near the 30-180 K are called the low temperature superconductors (LTS) and high temperature superconductors (HTS) regime with respectively. In the scope of this chapter, we have studied levitation force-distance loops for HTS in 77 K. In the recent 30 years, LTS and especially HTS are in a class of superconductor. HTS structures have potential applications in a broad range topical areas, superconducting magnetic energy storage (SMES), power devices including transmission cable, HTS motors and HTS generators, superconducting fault current limiters (SFCL), and high field magnets. See the Fig.6 in below for an example of some application of these type HTS samples.

Fig. 6. Some application of the HTS

(www.superpower-inc.com; http://yahoo.brand.edgar-online.com).

The levitation force hysteretic loops applied to new technologies as medical and space technologies in general.

HTS generators are substantially smaller and lighter than copper based machines. These advantages are very attractive in a ship environment. HTS power cables will enable utilities to upgrade power density by 2 to 8 times power corridors because HTS cables transmit power with essential no electrical resistance. Superpower's SFCL technology is enable scaling to high voltage and current with customization to the user's requirements, losses or voltage drop across the device during normal operation, and etc. The HTS technology can be used to make all superconducting magnets with field that will soon exceed 30 Tesla (see for detail, www.superpower-inc.com).

## **5. Summary**

318 Superconductors – Properties, Technology, and Applications

The levitation force can be related from the interaction between CSD and PM materials in general. This interaction between CSD and PM are mainly dependent on the properties of the CSD and the magnetic field distribution of the magnet. For a bulk superconductor, the levitation force is dependent on many parameters, such as the critical current density and grain radius, grain-orientation, thickness of the sample, and the cooling temperature. For a magnet, the levitation force is closely related with the magnetic ux density, magnetic field distribution, etc. But for a given pair of bulk superconductor and PM, it is known that the same levitation force can be obtained at a gap distance between the superconductor and the magnet, because of the magnetization history of the superconductor, by a mechanical descending and ascending process of the magnet during the axial levitation force

The low levitation force of sintered superconductor materials can be attributed to two intrinsic material problems of a superconductor in this chapter (Murakami, 1993). The first is the grain boundary weak-link problem and the second is the weak flux pinning problem. In order to resolve these two problems, different material processing techniques have been

Temperatures in the range near the 4-30 K and near the 30-180 K are called the low temperature superconductors (LTS) and high temperature superconductors (HTS) regime with respectively. In the scope of this chapter, we have studied levitation force-distance loops for HTS in 77 K. In the recent 30 years, LTS and especially HTS are in a class of superconductor. HTS structures have potential applications in a broad range topical areas, superconducting magnetic energy storage (SMES), power devices including transmission cable, HTS motors and HTS generators, superconducting fault current limiters (SFCL), and high field magnets. See the Fig.6 in below for an example of some application of these type

measurement state (Yang *et al.,* 2003)**.** 

HTS samples.

Fig. 6. Some application of the HTS

technologies in general.

developed from other workers (Chen *et al.,* 1998)**.** 

**4. Technological applications of levitation forces in HTS** 

(www.superpower-inc.com; http://yahoo.brand.edgar-online.com).

The levitation force hysteretic loops applied to new technologies as medical and space

In the scope of this chapter, we have focus on the magnetic levitation force-distance loops for HTS samples. The inter CSD and PM exchange type interaction are determined magnetic levitation forces. The magnetic levitation forces are heavily dependence on grain size, grain connectivity, critical current density, history of applied magnetic field and etc. Combination with hysteresis ( ) *F x* results, these indicate that the Meissner effects, vortex state and metallic behavior are seen the same. The CSD with ZFC process showed that the sample has low quality with weak-link characteristics. A typical hysteresis loop of the axial levitation force between the permanent magnet and the CSCD showed the lower repulsive force in this chapter. This phenomenon is attributed to the fact that the wet technique causes the production of lower dense samples**,** and is caused by a lower critical current density (Altunbaş et al., 1994; Karaca et al., 1998) and by a lower levitation force. It is considered that the changes in the levitation force are related to the grain orientation, the homogeneity, and the number of the pinning centers in the samples (Yanmaz et al., 2002)**.** Levitation force measurements need to investigate several factors end different materials. In the future life will be need to this investigation results.

## **6. Acknowledgements**

I would like to thanks Mehmet Ali Güzel for help.

## **7. References**

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Measurement of Levitation Forces in High Temperature Superconductors

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**15** 

*Ukraine* 

**Applications of** 

*Transcarpathian State University* 

**Superconductor/Photosemiconductor** 

The integration of high temperature superconductors(HTSC's) with conventional semiconductor(SeC) - based technology would have important consequences for micro- and cryophotoelectronics, with the promise of high performance hybrid circuits incorporating the best of what superconductors and semiconductors have to offer as well as the possibility for novel devices (Bunda, 1991). The rapid progress in the fabrication of high quality HTSC thin films on oxygen containing substratum such as SrTiO3, MgO, and LaAlO3 has been observed. However, the results on semiconductors like Si, Ge and GaAs has been also

The high temperature superconductors (HTSC's) are considered to be low carrier density materials. Therefore, the light can penetrate the superconductor and can effectively excite the quasiparticles in it. The study of light detection by a "HTSC – photosemiconductor" hybrid contact structures (HCS's) is very perspective for fabrication of multifunctional photonic circuits - high speed detectors with reasonable sensitivity covering a broad electromagnetic spectrum (Bunda et al., 1996). Therefore, for deeper investigation of such type compounds we need fabrication of the HCS's based on oxygen containing

Superconductors (SC) have a wide field of applications due to their charge carrier transport properties without energy loss. The charge carriers in the superconducting state are correlated charge carriers (Cooper pairs) as well as uncorrelated ones (quasiparticles). The special magnetic and transport properties of Cooper pairs provide new applications in the field of tunnel contacts, electronic devices and the generation of very high magnetic fields (Akani et al., 1995). Compared to classical ultrasonic sources, superconducting tunnel contacts emit phonons with higher frequency. Therefore, the properties of different HTSC/photosemiconductor heterojunctions are of great interest for the development of new electronic devices such as diodes with very high rectifying parameters (Bunda et al., 2010a)

In recent years, two-dimensional (2D) nanostructured materials, such as nanoplates and nanosheets, have attracted much attention because of not only their unique electronic,

and transistors based on the Josephson field effect (Akani et al., 1995).

**1. Introduction** 

reported.

photosemiconductor.

**Contact Structures in Electronics** 

Viktor V. Bunda and Svitlana O. Bunda

Sanchez, A. & Navau, C. (1997). *Physica C,* Vol.275, pp. 322.


## **Applications of Superconductor/Photosemiconductor Contact Structures in Electronics**

Viktor V. Bunda and Svitlana O. Bunda *Transcarpathian State University Ukraine* 

## **1. Introduction**

322 Superconductors – Properties, Technology, and Applications

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249.

The integration of high temperature superconductors(HTSC's) with conventional semiconductor(SeC) - based technology would have important consequences for micro- and cryophotoelectronics, with the promise of high performance hybrid circuits incorporating the best of what superconductors and semiconductors have to offer as well as the possibility for novel devices (Bunda, 1991). The rapid progress in the fabrication of high quality HTSC thin films on oxygen containing substratum such as SrTiO3, MgO, and LaAlO3 has been observed. However, the results on semiconductors like Si, Ge and GaAs has been also reported.

The high temperature superconductors (HTSC's) are considered to be low carrier density materials. Therefore, the light can penetrate the superconductor and can effectively excite the quasiparticles in it. The study of light detection by a "HTSC – photosemiconductor" hybrid contact structures (HCS's) is very perspective for fabrication of multifunctional photonic circuits - high speed detectors with reasonable sensitivity covering a broad electromagnetic spectrum (Bunda et al., 1996). Therefore, for deeper investigation of such type compounds we need fabrication of the HCS's based on oxygen containing photosemiconductor.

Superconductors (SC) have a wide field of applications due to their charge carrier transport properties without energy loss. The charge carriers in the superconducting state are correlated charge carriers (Cooper pairs) as well as uncorrelated ones (quasiparticles). The special magnetic and transport properties of Cooper pairs provide new applications in the field of tunnel contacts, electronic devices and the generation of very high magnetic fields (Akani et al., 1995). Compared to classical ultrasonic sources, superconducting tunnel contacts emit phonons with higher frequency. Therefore, the properties of different HTSC/photosemiconductor heterojunctions are of great interest for the development of new electronic devices such as diodes with very high rectifying parameters (Bunda et al., 2010a) and transistors based on the Josephson field effect (Akani et al., 1995).

In recent years, two-dimensional (2D) nanostructured materials, such as nanoplates and nanosheets, have attracted much attention because of not only their unique electronic,

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 325

transport reaction method in closed volume. The compounds BiOX of 99,99% purity was loaded into a high quality polished quartz ampoule (150-165 mm length and 15 mm diameter; combination of cylinder and cone). The ampoule was evacuated under vacuum of 10-4 Torr. After pumping the ampoule was filled with a (H2O and HX) transport agent (TA).

�����(�) � ���(�) � �����(�) � ���(�).

During transport involved three gaseous compounds: H2O, HX and BiX3. In the solid state exists a bismuth oxyhalides in polycrystalline and single crystal form. Reaction (1) describes the dissociation process of bismuth oxyhalides. The reaction (2) is a direct gas transport reaction with the water vapour as a transport agent. Reaction (3) describes the halogenation of bismuth oxide to bismuth oxyhalides. The (1-3) reactions cycle is closed and carried to the final transport in a BiOX single crystal of polycrystalline bismuth oxyhalide. The decomposition (1) and hydrolysis (2) reactions of BiOX are endothermic. Therefore, transport is possible only in the T2→T1 direction (T2>T1). A two-zone resistance heated furnace was used for crystal growth. The axis of the ampoule was kept slightly inclined to the horizontal to enhance the transport of the vapour from the source zone to the growth zone during growth. We selected the pressure of transport agent in the ampoule on the basis of optimal mass transfer calculation by the convection method according to the equation (2). The best bismuth oxyhalide crystals were obtained at the 8-10 Torr pressure of transport agent. The some growth parameters of the BiOX crystals obtained are given in table 1.

(Torr)

Source Growth period (days)

Table 1. Temperature of source and growth zones, transport agent pressure and size of BiOX

The crystallinity and morphology of the BiOX crystals was examined by X-ray diffraction (XRD), and field effect-scanning electron microscopy (FE-SEM). The lattice parameters were refined by X-ray powder diffraction data with a Si internal standard (high purity) and by using a least square method. All of the sharp diffraction peaks in the XRD patterns were perfectly indexed as pure phase of BiOCl, in good agreement with the standard JCPDS file (No. 06-0249) of BiOCl. A Laue photograph taken parallel to the large face of the platelets showed clearly a fourfold symmetry axis characterizing (001) planes of a matlockite

�����(�) � �����(�) � ����(�); (1)

Morphology and size

� �� (No. 129) (Beck & Beyer, 1997) with two

7x7x3 mm3 <sup>5</sup>

6x7x2 mm3 <sup>6</sup>

5x5x1 mm3 <sup>7</sup>

(3)

Growth

����(�) � ���(�) � ����(�������) � ���(�) (2)

The system of chemical gas-transport reactions are as follows (Bunda, 1991):

Substance Zone temperature (oC) TA pressure

crystals obtained

tetragonal PbFCl--like structure ���

BiOCl 720 600 8-10 Platelets

BiOBr 700 600 8-10 Platelets

BiOI 580 450 8-10 Platelets

� � � �

formula units in the unit cell and the following atom position (Bercha at al., 1973):

magnetic, optical, and catalytic properties, which mainly arise from their large surface areas, nearly perfect crystallinity, structural anisotropy, and quantum confinement effects in the thickness (Deng et al., 2008). The potential of 2D nanostructured materials uses for building blocks for advanced materials and devices with designed functions in areas as diverse as lasers, transistors, catalysis, solar cells, light emission diodes, and chemical and biological sensors (Bunda et al., 2010b).

## **2. Grown and crystal structure of bismuth oxyhalides single crystals**

Oxyhalides of bismuth BiOX (X = Cl, Br, I) are very interesting materials which find various applications as X-ray luminescent screens, as anti-Stokes converters, photocatalist, usual luminophors and as photoconductive analyzer of linear polarized radiation in the 0.24 - 1.2μm spectral region. Bismuth oxyhalides provides a very convenient matrix for activation by various rare-earth and 3d-elements: by doping this matrix with different, is possible to obtain a wide variety of optical, luminescent and photoconductive properties. Moreover, these crystals are of essential interest in connection with the research and development work on laser materials. (Bunda et al., 2010b). The great interest for these materials is strongly related to the influence of dimensionality on the behaviour of physical properties (they are 2D structured materials). Bismuth oxyhalides are one of the V-VI-VII group compound semiconductors belonging to the tetragonal system. The structure of BiOX is known to have a layered structure, which is constructed by the combination of the halide ion layer and the bismuth oxygen layer (Bletskan et al., 1973).

The BiOX single crystals are layered 2D structured materials. The crystal structure of BiOX is very similar to the symmetry of space groups P4/݉݉݉ and/or I4/mmm, where are usual for the YBa2Cu3O7- and Bi(Tl,Hg)2Sr(Ba)2Can-1CunO2n+4 HTSC's. Besides, the BiOX crystals, as well as a HTSC's contain oxygen. The thermal expansion and the lattice constants mishmatch in the (001) base plane are in good agreement with the same parameters of the HTSC's. Also, the BiOX crystals are good "sparring partners" for the fabrication of "HTSC – photosensitive semiconductor" hybrid contact structure.

Bismuth oxychloride crystals exhibit many intriguing and interesting properties such as photoluminescence, photoconductivity, and thermally stimulated conductivity (Bunda, 1989, 1991, 2005). The luminescent band at 1.6 - 2.2 eV is a result of capture of c-band free electrons by r-centres of recombination. The shape of the absorption edge has been explained by the effect of the laminarity of these crystals. Previous investigations on the crystal structure and morphology of BiOX crystallites showed that they grow usually in the form of platelets with the c-axis normal to the platelets. Investigations on the influence of uniaxial compression on the crystals revealed that they are isotropic along the normal to the layers (Ganesha et al., 1993). Heat conductivity measurements as a function of temperature showed that the electronic component of heat conductivity is negligibly small, since the crystals are nearly insulators in the temperature range 80 -300 K (Bunda, 1991).

The BiOHal samples were prepared in several ways. BiOX pure(starting material for a BiOX single crystals growth) was prepared by dissolving 99,99 pure bismuth oxides in hydrohalides acid, evaporating to form the hydrated halides, dehydrating the latter under vacuum, heating at ~380-700oC in a X2 atmosphere, cooling to room temperature, and washing the product to remove BiX3. The BiOX crystals were grown by the vapour gas

magnetic, optical, and catalytic properties, which mainly arise from their large surface areas, nearly perfect crystallinity, structural anisotropy, and quantum confinement effects in the thickness (Deng et al., 2008). The potential of 2D nanostructured materials uses for building blocks for advanced materials and devices with designed functions in areas as diverse as lasers, transistors, catalysis, solar cells, light emission diodes, and chemical and biological

Oxyhalides of bismuth BiOX (X = Cl, Br, I) are very interesting materials which find various applications as X-ray luminescent screens, as anti-Stokes converters, photocatalist, usual luminophors and as photoconductive analyzer of linear polarized radiation in the 0.24 - 1.2μm spectral region. Bismuth oxyhalides provides a very convenient matrix for activation by various rare-earth and 3d-elements: by doping this matrix with different, is possible to obtain a wide variety of optical, luminescent and photoconductive properties. Moreover, these crystals are of essential interest in connection with the research and development work on laser materials. (Bunda et al., 2010b). The great interest for these materials is strongly related to the influence of dimensionality on the behaviour of physical properties (they are 2D structured materials). Bismuth oxyhalides are one of the V-VI-VII group compound semiconductors belonging to the tetragonal system. The structure of BiOX is known to have a layered structure, which is constructed by the combination of the halide ion layer and the

The BiOX single crystals are layered 2D structured materials. The crystal structure of BiOX is very similar to the symmetry of space groups P4/݉݉݉ and/or I4/mmm, where are usual for the YBa2Cu3O7- and Bi(Tl,Hg)2Sr(Ba)2Can-1CunO2n+4 HTSC's. Besides, the BiOX crystals, as well as a HTSC's contain oxygen. The thermal expansion and the lattice constants mishmatch in the (001) base plane are in good agreement with the same parameters of the HTSC's. Also, the BiOX crystals are good "sparring partners" for the fabrication of "HTSC –

Bismuth oxychloride crystals exhibit many intriguing and interesting properties such as photoluminescence, photoconductivity, and thermally stimulated conductivity (Bunda, 1989, 1991, 2005). The luminescent band at 1.6 - 2.2 eV is a result of capture of c-band free electrons by r-centres of recombination. The shape of the absorption edge has been explained by the effect of the laminarity of these crystals. Previous investigations on the crystal structure and morphology of BiOX crystallites showed that they grow usually in the form of platelets with the c-axis normal to the platelets. Investigations on the influence of uniaxial compression on the crystals revealed that they are isotropic along the normal to the layers (Ganesha et al., 1993). Heat conductivity measurements as a function of temperature showed that the electronic component of heat conductivity is negligibly small, since the

The BiOHal samples were prepared in several ways. BiOX pure(starting material for a BiOX single crystals growth) was prepared by dissolving 99,99 pure bismuth oxides in hydrohalides acid, evaporating to form the hydrated halides, dehydrating the latter under vacuum, heating at ~380-700oC in a X2 atmosphere, cooling to room temperature, and washing the product to remove BiX3. The BiOX crystals were grown by the vapour gas

crystals are nearly insulators in the temperature range 80 -300 K (Bunda, 1991).

**2. Grown and crystal structure of bismuth oxyhalides single crystals** 

sensors (Bunda et al., 2010b).

bismuth oxygen layer (Bletskan et al., 1973).

photosensitive semiconductor" hybrid contact structure.

transport reaction method in closed volume. The compounds BiOX of 99,99% purity was loaded into a high quality polished quartz ampoule (150-165 mm length and 15 mm diameter; combination of cylinder and cone). The ampoule was evacuated under vacuum of 10-4 Torr. After pumping the ampoule was filled with a (H2O and HX) transport agent (TA). The system of chemical gas-transport reactions are as follows (Bunda, 1991):

$$3BiOX\_{\text{(s)}} \to Bi\_2O\_{3\text{(s)}} + BiX\_{3\text{(g)}};\tag{1}$$

$$BiX\_{3(g)} + H\_2O\_{(g)} \to BiOX\_{(crystal)} + 2HX\_{(g)}\tag{2}$$

$$2Bi\_2O\_{3(s)} + 2HX\_{(g)} \to 2BiOX\_{(s)} + H\_2O\_{(g)}.\tag{3}$$

During transport involved three gaseous compounds: H2O, HX and BiX3. In the solid state exists a bismuth oxyhalides in polycrystalline and single crystal form. Reaction (1) describes the dissociation process of bismuth oxyhalides. The reaction (2) is a direct gas transport reaction with the water vapour as a transport agent. Reaction (3) describes the halogenation of bismuth oxide to bismuth oxyhalides. The (1-3) reactions cycle is closed and carried to the final transport in a BiOX single crystal of polycrystalline bismuth oxyhalide. The decomposition (1) and hydrolysis (2) reactions of BiOX are endothermic. Therefore, transport is possible only in the T2→T1 direction (T2>T1). A two-zone resistance heated furnace was used for crystal growth. The axis of the ampoule was kept slightly inclined to the horizontal to enhance the transport of the vapour from the source zone to the growth zone during growth. We selected the pressure of transport agent in the ampoule on the basis of optimal mass transfer calculation by the convection method according to the equation (2). The best bismuth oxyhalide crystals were obtained at the 8-10 Torr pressure of transport agent. The some growth parameters of the BiOX crystals obtained are given in table 1.


Table 1. Temperature of source and growth zones, transport agent pressure and size of BiOX crystals obtained

The crystallinity and morphology of the BiOX crystals was examined by X-ray diffraction (XRD), and field effect-scanning electron microscopy (FE-SEM). The lattice parameters were refined by X-ray powder diffraction data with a Si internal standard (high purity) and by using a least square method. All of the sharp diffraction peaks in the XRD patterns were perfectly indexed as pure phase of BiOCl, in good agreement with the standard JCPDS file (No. 06-0249) of BiOCl. A Laue photograph taken parallel to the large face of the platelets showed clearly a fourfold symmetry axis characterizing (001) planes of a matlockite tetragonal PbFCl--like structure ��� � � � � � �� (No. 129) (Beck & Beyer, 1997) with two formula units in the unit cell and the following atom position (Bercha at al., 1973):

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 327

multiplicity 2. The faces of the decahedron are 2 rectangles (O-O-O-O, X-X-X-X, which are parallel to the (110) plane and 8 isosceles triangles (four X-O-X and four O-X-O). The decahedra are linked to each other by a common O-X edge along the a and b axes forming infinite layers. The coordination of O and X are follows. Each O atom is linked to four Bi atoms and forming a tetragonal pyramid with the O atom at its apex. Also each X atoms forms with the neighbouring Bi atoms tetragonal pyramid with the X atom at its apex. Neighbouring decahedra form layers along (001) which are connected by common O-X edges. Neighbouring layers of decahedra are connected by common O-O or X-X edges.

The X atom is bonded with four Bi atoms in a planar square to form a pyramid and with its nonbonding (lone pair) electrons pointing to the other side of the square. These nonbonding electrons convert the three-dimensional fluorite-like structure into a two-dimensional layered structure. In the three-dimensional Bi2O3, the Bi atoms are coordinated to six O atoms. This structural difference between them is the major reason for BiOX to have a wider optical band-gap. As shown in the fig. 1, the [BiOX] layers are stacked together by the nonbonding (van der Waals) interaction through the X atoms along the c-axis. Therefore, the structure is not closely packed in this direction. When one photon excites zone electron from X np states (n = 3, 4, 5 for Cl, Br, I respectively) to Bi 6p states in BiOX, one pair of a hole and an excited electron appear. The layered structure BiOX can provide the space large enough to polarize the related atoms and orbitals (Zhang et al., 2006). The induced dipole can separate the hole–electron pair efficiently, enhancing photocatalytic activities. BiOX has an indirect-transition band-gap so that the excited electron has to travel certain k-space distance to be emitted to valence band (Table 2). Therefore, the BiOX single crystals can be

Substance Lattice constants, Å Relaxed atomic position Ɵ�� � ��� ��

BiOCl 3.883 7.347 0.645 0.170 205 3.455 BiOBr 3.915 8.076 0.653 0.154 168 2.924 BiOI 3.984 9.128 0.668 0.132 146 1.89

**3. Fabrication and classification of HTSC/photosemiconductor hybrid contact** 

In this capture we report the formation process of "ceramics HTSC(YBa2Cu3O7-)/thin film SeC (BiOX)" hybrid contact structures(HCS). The choice of the structure "HTSC-ceramic - SC-thin film" was due to the nonstability of the HTSC- film parameters. The layers of bismuth oxyhalogenides were prepared in the three ways (Bunda, 1991): a) the thermal discrete evaporation in the oxygen atmosphere (p=7-10 Torr); b) the laser impulse evaporation (the laser type of LTYPTSH-500; =0.53 m; =50 Hz); c) the deposition by the method of gas chemical transport reactions in the quasiclosed volume. The films were characterized by qualitative X-ray analysis (powder diffraction measurements, using CuK radiation), and by scanning electron microscopy with energy disperse chemical analysis SEM-EDS (for phase composition information). Preliminary YBa2Cu3O7- ceramic were overreached in oxygen ions plasma atmosphere for reconstruction of the superconducting

), Debye temperature ( Ɵ�)

*acz* ��

Table 2. Lattice parameter (*a, c*) , relaxed atomic position (z, ��

and band gap ( ��) of bismuth oxyhalide crystals (Bunda et al., 2011)

described as 2D structured materials.

**structures** 

2�� �� �������� (2�) (0, 1 2, ⁄ �)(1 2, ⁄ 0, �̅ ) �� ��� (���)�������� 

2� �� �������� (2�) (0,0,0) (1 2, 1 2, ⁄⁄ 0) �� ��2� (���) �������� (4)

2� �� �������� (2�) (0, 1 2, �, ) (1 2,0, �̅ , ⁄ ) ⁄ �� ��� (���)���������.

The position (2a) is always occupied by the smaller anion which is coordinated tetrahedrally by the bismuth cations. The Bi3+ ion itself is coordinated ninefold capped quadratic antiprism) and site symmetry has C4*v*. Fig.1 gives the coordination polyhedron of the cation (d) and a projection of the structure along the [010] direction (b).

An examination of the platelets using a polarizing microscope shows no extinction, thus indicating that the tetragonal *c*-axis is perpendicular to the plane of the platelets. In optical micrograph of a BiOCl crystal multiple growth spiral (two on the same sign) can be seen originating from a growth defect. The growth defect is responsible for the formation of the growth spirals. The observed spirals illustrate once again working of the Burton-Cabrera-Frank (BCF) theory of the crystal growth mechanism (Burton et al., 1951). On the other hand, the layers observed under different magnifications in the SEM of BiOBr and BiOI crystals indicate the layer growth mechanism of these crystals (Ganesha et al., 1993).

Fig. 1. Crystal structure of bismuth oxyhalides: (a) projection to (001) plane; (b) projection to (100) or (101) plane; (c) clinographic projection of the unit cell; (d) coordination polyhedron of the bismuth

Fig.1a shows the clinographic projection of the unit cell of BiOX crystals. Each Bi atoms is eight-coordinated by four O atoms and four X (halogen) atoms in the form of an asymmetric decahedron (Keramidas et al., 1993). Each atom (Bi, O, X) occupies a special position with

) (1 2,0, �̅

The position (2a) is always occupied by the smaller anion which is coordinated tetrahedrally by the bismuth cations. The Bi3+ ion itself is coordinated ninefold capped quadratic antiprism) and site symmetry has C4*v*. Fig.1 gives the coordination polyhedron of the cation

An examination of the platelets using a polarizing microscope shows no extinction, thus indicating that the tetragonal *c*-axis is perpendicular to the plane of the platelets. In optical micrograph of a BiOCl crystal multiple growth spiral (two on the same sign) can be seen originating from a growth defect. The growth defect is responsible for the formation of the growth spirals. The observed spirals illustrate once again working of the Burton-Cabrera-Frank (BCF) theory of the crystal growth mechanism (Burton et al., 1951). On the other hand, the layers observed under different magnifications in the SEM of BiOBr and BiOI

crystals indicate the layer growth mechanism of these crystals (Ganesha et al., 1993).

Fig. 1. Crystal structure of bismuth oxyhalides: (a) projection to (001) plane; (b) projection to (100) or (101) plane; (c) clinographic projection of the unit cell; (d) coordination polyhedron

Fig.1a shows the clinographic projection of the unit cell of BiOX crystals. Each Bi atoms is eight-coordinated by four O atoms and four X (halogen) atoms in the form of an asymmetric decahedron (Keramidas et al., 1993). Each atom (Bi, O, X) occupies a special position with

of the bismuth

2� �� �������� (2�) (0,0,0) (1 2, 1 2, ⁄⁄ 0) �� ��2� (���) �������� (4)

, ⁄ ) ⁄ �� ��� (���)���������.

) �� ��� (���)�������� 

2�� �� �������� (2�) (0, 1 2, ⁄ �)(1 2, ⁄ 0, �̅

2� �� �������� (2�) (0, 1 2, �,

(d) and a projection of the structure along the [010] direction (b).

multiplicity 2. The faces of the decahedron are 2 rectangles (O-O-O-O, X-X-X-X, which are parallel to the (110) plane and 8 isosceles triangles (four X-O-X and four O-X-O). The decahedra are linked to each other by a common O-X edge along the a and b axes forming infinite layers. The coordination of O and X are follows. Each O atom is linked to four Bi atoms and forming a tetragonal pyramid with the O atom at its apex. Also each X atoms forms with the neighbouring Bi atoms tetragonal pyramid with the X atom at its apex. Neighbouring decahedra form layers along (001) which are connected by common O-X edges. Neighbouring layers of decahedra are connected by common O-O or X-X edges.

The X atom is bonded with four Bi atoms in a planar square to form a pyramid and with its nonbonding (lone pair) electrons pointing to the other side of the square. These nonbonding electrons convert the three-dimensional fluorite-like structure into a two-dimensional layered structure. In the three-dimensional Bi2O3, the Bi atoms are coordinated to six O atoms. This structural difference between them is the major reason for BiOX to have a wider optical band-gap. As shown in the fig. 1, the [BiOX] layers are stacked together by the nonbonding (van der Waals) interaction through the X atoms along the c-axis. Therefore, the structure is not closely packed in this direction. When one photon excites zone electron from X np states (n = 3, 4, 5 for Cl, Br, I respectively) to Bi 6p states in BiOX, one pair of a hole and an excited electron appear. The layered structure BiOX can provide the space large enough to polarize the related atoms and orbitals (Zhang et al., 2006). The induced dipole can separate the hole–electron pair efficiently, enhancing photocatalytic activities. BiOX has an indirect-transition band-gap so that the excited electron has to travel certain k-space distance to be emitted to valence band (Table 2). Therefore, the BiOX single crystals can be described as 2D structured materials.


Table 2. Lattice parameter (*a, c*) , relaxed atomic position (z, �� ), Debye temperature ( Ɵ�) and band gap ( ��) of bismuth oxyhalide crystals (Bunda et al., 2011)

## **3. Fabrication and classification of HTSC/photosemiconductor hybrid contact structures**

In this capture we report the formation process of "ceramics HTSC(YBa2Cu3O7-)/thin film SeC (BiOX)" hybrid contact structures(HCS). The choice of the structure "HTSC-ceramic - SC-thin film" was due to the nonstability of the HTSC- film parameters. The layers of bismuth oxyhalogenides were prepared in the three ways (Bunda, 1991): a) the thermal discrete evaporation in the oxygen atmosphere (p=7-10 Torr); b) the laser impulse evaporation (the laser type of LTYPTSH-500; =0.53 m; =50 Hz); c) the deposition by the method of gas chemical transport reactions in the quasiclosed volume. The films were characterized by qualitative X-ray analysis (powder diffraction measurements, using CuK radiation), and by scanning electron microscopy with energy disperse chemical analysis SEM-EDS (for phase composition information). Preliminary YBa2Cu3O7- ceramic were overreached in oxygen ions plasma atmosphere for reconstruction of the superconducting

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 329

Some progress has been made in the growth HTSC thin films on Si and on GaAs (Shiwagan et al., 2002). All successful growths have used intermediate buffer layer between the HTSC films and semiconductor substrates. The major problem in using the semiconductor substrates is the substrate-film inter-diffusion during the in-situ depositions rn addition to the formation of micro cracks in the films due to a relatively large difference rn thermal expansion coefficients between the semiconductor and HTSC. After the deposition of CdSe the critical transition temperature of HgBaCaCuO films was found be increased from 115 K with *J*c = 1.7 x 103 A/cm2 to 117.2 K with *J*c = 1.91 x 103 A/cm2. When the heterostructure was irradiated with red He-Ne laser (2 mW), the *T*c was further enhanced to 120.3 K with *J*c = 3.7 x 103 A/cm2 (Shiwagan et al., 2002, 2004). The resistivity is decreased after the laser irradiation, but the nature of variations in resistivity with temperature remains similar to that of without laser irradiated samples. The decrease rn resistivity is attributed to increase in charge carriers in the heterostructure during laser irradiation. The CdSe has the direct band gap 1.74 eV and hence the laser source with energy 1.95 eV greater than band gap of CdSe was selected. Hence during the irradiations, the electron-hole pair generates in the CdSe sample individually at the junction of the heterostructure. As authors (Shiwagan et al., 2004) of work specify, the built-in-junction potential (n) CdSe and (p) HgBaCaCuO heterojunction the electron-hole pairs generated at interface could not recombine, rather holes are transferred to the superconductor region where they can be further trapped into CuO2 layer and hence there is increase in

**4. Mechanism of photoconductivity in BiOX (X=Cl, Br, I) single crystals** 

The original specimens had a dark resistivity of ρd = 1010 to 1011 Ω.

maximum is shifted towards the long-wave side at a rate of ���⁄��= -6.3.

was done by a platinum reducer (Bunda, 1989, 2005).

The spectral dependences of the steady-state photocurrent (photoconductivity) were recorded using a LOMO MDR3 monochromator. Low temperatures were obtained by mounting the sample in an UTREKS continuous flow cryostat. The decay time measurements were performed using pulse UV- laser LGI-21 (pulse width 8-9 ns; excitation wavelength λ=0.337 nm; filters UFS-2 and UFS-6 were used). The change in the intensity

photosensitivity (Bletskan ey al., 1973). At a temperature 293 K and an illumination of 104 1x, the ratio ρ�⁄ρ�� (ρ�� is the resistivity in the light) reached 103; cooling of the specimen led to an increase of 3-4 orders of magnitude in the steady-state photocurrent such that at 90 K, ρ�⁄ρ�� > 107. Thus, there was an intense temperature (T) suppression of the photocurrent

The spectral dependence of the photoconductivity ������� for a typical specimen of BiOCl recorded at 95 K (curve 1) and 293 K (curve 2) is illustrated in Fig. 2b. In addition to the intrinsic maximum ���= 3.54 eV (350 nm) which corresponds to the band gap of �� = 3.5 eV at T = 100 K a significant impurity photoconductivity is observed due to the transfer of an electron from the *r* centre to the conduction band (Bunda, 1989). On heating, the main

the photoconductivity in both the impurity and intrinsic regions falls sharply. The depth of

the level which produces the impurity photocurrent maximum was found to be ���

cm and significant

10-4 eV/K, while

� =2.2 eV

conductivity.

(see fig. 2a).

Bunda, 2005)..

properties of disturbed surface layer. The hybrid contact structures were finally annealed at a temperature T=230-250oC (recrystallization and thermal relaxation procedure). Surface factor of prepared samples was controlled by scanning electron microscopy (SEM).

In the "HTSC - SeC" system can be formed three principal types of HCS's (Bunda et al., 1996): tunneling(1), proximity(2) and combined(3). In the case ���� (��- the value of coherence length in the HTSC; d- the thickness of insulator(I) layer between super- and semiconductor) the electrical properties of "HTSC - SC" junction are mainly determinate by the parameters of the transition layer d. At d ~ �� take place both the HTSC-I-SC junction (HCS of type 1) with high (�� > 103 - 105 Ω. cm) values of the differential resistance �� (at the bias voltage Vtr→0) and proximity contact(HCS of type 2) with the ���0. In the case d ~ �� and 0< ��< 103 - 105 Ω. cm take place HCS of type 3 - combined "proximity- tunnelling" junction (Bunda et al., 1996). The classification of principal possible types of "HTSC-SC" hybrid contact structures is show in the table 3.


Table 3. Classification and methods of fabrications HTSC/SeC hybrid contact structures

The realization of "HTSC-SC" HCS's assumes the following conditions (Bunda et al., 1996):


properties of disturbed surface layer. The hybrid contact structures were finally annealed at a temperature T=230-250oC (recrystallization and thermal relaxation procedure). Surface

In the "HTSC - SeC" system can be formed three principal types of HCS's (Bunda et al., 1996): tunneling(1), proximity(2) and combined(3). In the case ���� (��- the value of coherence length in the HTSC; d- the thickness of insulator(I) layer between super- and semiconductor) the electrical properties of "HTSC - SC" junction are mainly determinate by the parameters of the transition layer d. At d ~ �� take place both the HTSC-I-SC junction

bias voltage Vtr→0) and proximity contact(HCS of type 2) with the ���0. In the case d ~ ��

junction (Bunda et al., 1996). The classification of principal possible types of "HTSC-SC"

Type of structure Modification Method of fabrication �� (Ω.

(type 1, 3) Ceramics Thin films Thermal discrete

(type 1, 3) Ceramics Crystals Chemical gas transport

(type 2,3) Films Films Laser ablation; magnetron

(type 2) Films Crystals Laser ablation; magnetron

Table 3. Classification and methods of fabrications HTSC/SeC hybrid contact structures

The realization of "HTSC-SC" HCS's assumes the following conditions (Bunda et al., 1996): - semiconducting layers should be hole conductivity type (p-type; only for the HCS's of type 1). For the semiconductors with the compensation type of conductivity the



HTSC ceramics with oxygen concentra-tion gradient

HTSC single crystals with oxygen concentration gradient

should not exceed the values of 15% and 2-5%, respectively;

cm) values of the differential resistance �� (at the

the polished ceramics 103 ÷ 1012

reactions 101 ÷ 105

10-1 ÷ 102

10-3 ÷ 101

evaporation; laser ablation 102 ÷ 105

Oxygen ions implantation 10-2 ÷ 103

Oxygen ions implantation 10-3 ÷ 102

cm take place HCS of type 3 - combined "proximity- tunnelling"

cm) HTSC SeC

Press optical contact with

sputtering

sputtering

factor of prepared samples was controlled by scanning electron microscopy (SEM).

(HCS of type 1) with high (�� > 103 - 105 Ω.

hybrid contact structures is show in the table 3.

contakt (type 1) Ceramics Crystals

degeneracy(p=EF-Ev < 0) is necessary;

Bi(Tl,Hg)2Sr(Ba)2Can-1CunO2n+4 HTSCs.

and 0< ��< 103 - 105 Ω.

1. Van der Waals press

2. Heterojunction

3. Heterojunction

4. Heterojunction

5. Heterojunction

6. Homogeneous pp+-junction in HTSC ceramics (type 1-3)

7. Homogeneous pp+-junction in HTSC single crystals (type

1-3)

Some progress has been made in the growth HTSC thin films on Si and on GaAs (Shiwagan et al., 2002). All successful growths have used intermediate buffer layer between the HTSC films and semiconductor substrates. The major problem in using the semiconductor substrates is the substrate-film inter-diffusion during the in-situ depositions rn addition to the formation of micro cracks in the films due to a relatively large difference rn thermal expansion coefficients between the semiconductor and HTSC. After the deposition of CdSe the critical transition temperature of HgBaCaCuO films was

found be increased from 115 K with *J*c = 1.7 x 103 A/cm2 to 117.2 K with *J*c = 1.91 x 103 A/cm2. When the heterostructure was irradiated with red He-Ne laser (2 mW), the *T*c was further enhanced to 120.3 K with *J*c = 3.7 x 103 A/cm2 (Shiwagan et al., 2002, 2004). The resistivity is decreased after the laser irradiation, but the nature of variations in resistivity with temperature remains similar to that of without laser irradiated samples. The decrease rn resistivity is attributed to increase in charge carriers in the heterostructure during laser irradiation. The CdSe has the direct band gap 1.74 eV and hence the laser source with energy 1.95 eV greater than band gap of CdSe was selected. Hence during the irradiations, the electron-hole pair generates in the CdSe sample individually at the junction of the heterostructure. As authors (Shiwagan et al., 2004) of work specify, the built-in-junction potential (n) CdSe and (p) HgBaCaCuO heterojunction the electron-hole pairs generated at interface could not recombine, rather holes are transferred to the superconductor region where they can be further trapped into CuO2 layer and hence there is increase in conductivity.

## **4. Mechanism of photoconductivity in BiOX (X=Cl, Br, I) single crystals**

The spectral dependences of the steady-state photocurrent (photoconductivity) were recorded using a LOMO MDR3 monochromator. Low temperatures were obtained by mounting the sample in an UTREKS continuous flow cryostat. The decay time measurements were performed using pulse UV- laser LGI-21 (pulse width 8-9 ns; excitation wavelength λ=0.337 nm; filters UFS-2 and UFS-6 were used). The change in the intensity was done by a platinum reducer (Bunda, 1989, 2005).

The original specimens had a dark resistivity of ρd = 1010 to 1011 Ω. cm and significant photosensitivity (Bletskan ey al., 1973). At a temperature 293 K and an illumination of 104 1x, the ratio ρ�⁄ρ�� (ρ�� is the resistivity in the light) reached 103; cooling of the specimen led to an increase of 3-4 orders of magnitude in the steady-state photocurrent such that at 90 K, ρ�⁄ρ�� > 107. Thus, there was an intense temperature (T) suppression of the photocurrent (see fig. 2a).

The spectral dependence of the photoconductivity ������� for a typical specimen of BiOCl recorded at 95 K (curve 1) and 293 K (curve 2) is illustrated in Fig. 2b. In addition to the intrinsic maximum ���= 3.54 eV (350 nm) which corresponds to the band gap of �� = 3.5 eV at T = 100 K a significant impurity photoconductivity is observed due to the transfer of an electron from the *r* centre to the conduction band (Bunda, 1989). On heating, the main maximum is shifted towards the long-wave side at a rate of ���⁄��= -6.3. 10-4 eV/K, while the photoconductivity in both the impurity and intrinsic regions falls sharply. The depth of the level which produces the impurity photocurrent maximum was found to be ��� � =2.2 eV Bunda, 2005)..

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 331

The lux-ampere characteristics of the photocurrent (fig.3) at low temperatures up to the

������]. In the T-suppression region the lux-ampere characteristics become superlinear and the value of α depends on the concentration of photocurrent carriers *n*, at a fixed temperature. The maximum value of α observed in the experiment reached α = up 3 to 4. At low temperatures, an optical IR suppression of the photocurrent was observed (Bletskan et

A study of the photoconductivity kinetics was made in the linear regime: �� � �, where �� is the change in n caused by a short pulse of light in the intrinsic excitation region. The curve of the decay of the photocurrent during the illumination by a pulse of intrinsic light consisted, of course, of two sections with clear by different times : �� = (2.5 − 6.6) ∙ 10�� and

The presence of temperature and IR suppression of the photocurrent, the superlinearity of the lux-ampere characteristics, and the two sections observed on the relaxation curves of the photoconductivity, indicate unequivocally that in these particular crystals, the recombination processes are determined by two types of centres: "sensitizing" *r* centres and

determined from the dependence on the carrier concentration of the temperature at which

Another more accurate method of determining E is to measure the dependence of �� �⁄ on 1/T, where ΔL is the change in intensity of the exciting light L which is necessary for maintaining the electron concentration *n*, with a change in the temperature in the T-

> ��� ����

where �� is the portion of the total carrier-recombination current passing through the *s* centres; �� is the portion passing through the *r* centres; ��� and ��� are the probabilities of hole or electron trapping at an r centre; and �� is the static factor of the valence band. From

Next, we present results of the study of photoconducting spectra anisotropy of the BiOX single crystals, obtained with the polarization of electrical vector ��� of the electromagnetic wave parallel and perpendicular to each of the two axes ([100] or [010]) of the unit cell. The BiOHal compounds are broad-band (1.80 - 3.50 eV) photo-semiconductors. The maxima of their photoconductivity are in the interval of 0.24-1.2 μm and theirs photosensitivity's are �=���⁄�� = 10� − 10� arbitrary units at the temperature 80K (Fig. 4). The anisotropy of photoconducting spectra is believed to be the results of the layer matlockite-type (PbFCl)

exp (− ��� �

� = 1.3 eV (Bunda, 1989, 2005).

∙ 100% , (6)

the thermal suppression of photocurrent begins as in (Bletskan et al., 1973): ���

���� = ����

� = ��� ���

The energy distance of the *r* centres from the valence band is ���

�� � <sup>=</sup> ���

the curve of �� (�� �) ⁄ (Fig. 3) was determined ���

crystal structure of all bismuth oxyhalide compounds. The coefficient of photopleochroism was calculated as

�� ���� � ��� );

� and the thermal value was

�� ), (5)

� =1.3 eV.

region of T-suppression are linear in the experimental illumination range α=1 [� =

al., 1973).

�� = (0.8 − 4) ∙ 10�� s.

fast *s* centres.

suppression region,

Fig. 2. The temperature dependence (a) and the spectral dependence (b) of photoconductivity BiOCl single crystal

Fig. 3. Temperature dependence of the ������ �⁄ ) value

(a) (b)

Fig. 2. The temperature dependence (a) and the spectral dependence (b) of

photoconductivity BiOCl single crystal

Fig. 3. Temperature dependence of the ������ �⁄ ) value

The lux-ampere characteristics of the photocurrent (fig.3) at low temperatures up to the region of T-suppression are linear in the experimental illumination range α=1 [� = �� ���� � ��� ); ������]. In the T-suppression region the lux-ampere characteristics become superlinear and the value of α depends on the concentration of photocurrent carriers *n*, at a fixed temperature. The maximum value of α observed in the experiment reached α = up 3 to 4. At low temperatures, an optical IR suppression of the photocurrent was observed (Bletskan et al., 1973).

A study of the photoconductivity kinetics was made in the linear regime: �� � �, where �� is the change in n caused by a short pulse of light in the intrinsic excitation region. The curve of the decay of the photocurrent during the illumination by a pulse of intrinsic light consisted, of course, of two sections with clear by different times : �� = (2.5 − 6.6) ∙ 10�� and �� = (0.8 − 4) ∙ 10�� s.

The presence of temperature and IR suppression of the photocurrent, the superlinearity of the lux-ampere characteristics, and the two sections observed on the relaxation curves of the photoconductivity, indicate unequivocally that in these particular crystals, the recombination processes are determined by two types of centres: "sensitizing" *r* centres and fast *s* centres.

The energy distance of the *r* centres from the valence band is ��� � and the thermal value was determined from the dependence on the carrier concentration of the temperature at which the thermal suppression of photocurrent begins as in (Bletskan et al., 1973): ��� � =1.3 eV.

Another more accurate method of determining E is to measure the dependence of �� �⁄ on 1/T, where ΔL is the change in intensity of the exciting light L which is necessary for maintaining the electron concentration *n*, with a change in the temperature in the Tsuppression region,

$$\frac{\Delta L}{L} = \frac{g\_s B}{c\_{rn} n} = g\_s Q\_v \frac{c\_{rp}}{n c\_{rn}} \exp(-\frac{E\_{vr}^T}{kT}) \,\tag{5}$$

where �� is the portion of the total carrier-recombination current passing through the *s* centres; �� is the portion passing through the *r* centres; ��� and ��� are the probabilities of hole or electron trapping at an r centre; and �� is the static factor of the valence band. From the curve of �� (�� �) ⁄ (Fig. 3) was determined ��� � = 1.3 eV (Bunda, 1989, 2005).

Next, we present results of the study of photoconducting spectra anisotropy of the BiOX single crystals, obtained with the polarization of electrical vector ��� of the electromagnetic wave parallel and perpendicular to each of the two axes ([100] or [010]) of the unit cell. The BiOHal compounds are broad-band (1.80 - 3.50 eV) photo-semiconductors. The maxima of their photoconductivity are in the interval of 0.24-1.2 μm and theirs photosensitivity's are �=���⁄�� = 10� − 10� arbitrary units at the temperature 80K (Fig. 4). The anisotropy of photoconducting spectra is believed to be the results of the layer matlockite-type (PbFCl) crystal structure of all bismuth oxyhalide compounds.

The coefficient of photopleochroism was calculated as

$$P = \frac{I\_{\parallel} - I\_{\perp}}{I\_{\parallel} + I\_{\perp}} \cdot 100\% \,\, \tag{6}$$

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 333

cause changes in the photoelectric properties. Crystal structure of the sample reduces the intrinsic noise of the radiation detector. The functional photoresistors based on BiOX single crystals, included in electronic circuits as a two-terminal networks, what is their great advantage. They can follow the rotation of polarization plane or work in schemes of the maximum signal finding. Besides, the BiOX crystals can be used as photo-potentiometers. The selectivity of BiOX spectral characteristics make them good radiation detectors of the nitrogen, argon and helium-cadmium lasers, as well as some of the semiconducting lightemitting diodes (LEDs). Therefore, they can be used in optoelectronic circuits in conjunction

**5. Energy structure and tunnelling effects of HTSC/BiOX hybrid contact** 

HTSC/BiOX hybrid contact structures. This is for the following reasons:

photosensitivity's are �=���⁄�� = 10� − 10� at the temperature 80K.

The bismuth oxyhalide single crystals are very good "sparring partners" to create a

1. The good thermal expansion and lattice constants mishmash (15% and 3% respectively)

2. The BiOX compounds have a layered structure and can be easily crystallized as an ultra-thin (0.25-0.5 μm) single crystalline layers on the HTSC substratum's using the

3. The BiOX crystals are broad-band (��=1.85-3.65 eV) photosemiconductors. The maximum of their photoconductivity are in the 0. 24 - 1. 0 mkm interval and theirs

4. The bismuth oxyhalide single crystals can be easily intercalated (5-7 weigh %) and doped (0. 01-3%) by the 3d- and 4f- elements. That leads to the stable *p*-type conductivity, and the level of degeneration �=�� − �� (�� – valence band; �� – Fermi

5. The BiOX layered compounds contain not less than two elements - the cation (Bi3+) and anion (O2-), like in superconductors YBa2Cu3O7- or Bi2-xPbxSr2Can-1CunO2n+4. Therefore the contamination process of the heterojunctions during the formation procedure are

The unique temperature behaviour of the HCS's resistance consists in an N-shape anomaly around the superconducting transition temperature (interval 85-105K; ��=92K; fig.6). The temperature dependence of the BiOCl:Ti semiconductor resistance can be described by the exponential function �(�) = ����� (−��⁄��), where ��=2.8 eV and ��=102-107 Ω.

The detailed consideration of the temperature dependence of the heterojunction resistance in both directions (curves 1 and 2 on fig.6) shows the anomaly of heterojunction resistance in the narrow temperature interval above �� (ΔT=5K). The resistance first decreases down to ���� at T=��+2K and then rapidly increases to the maximum at T=��+5K. Subsequently, the resistance of heterojunction decreases according to the exponential dependence up to the room temperature. So the temperature dependence of YBa2Cu3O7-/BiOCl:Ti heterojunction resistance were characterized strongly pronounced N-shaped character in the narrow temperature interval above the superconducting transition. Fig. 6 shows the

cm,

with these lasers and LEDs.

for the base planes (001) is observed.

level position) can be controlled.

depending on the activation regimes.

method of chemical gas transport reactions.

**structures** 

absent.

considerable

where � and � is the photocurrent values at the ��� �100� and ��� �100� light polarization, respectively.

The position of electrical vector ��� can be varied either by rotating of the crystal. Anisotropy in 3 - 5 times was observed for the photosensitive value S. The change in light polarization at the 90 degrees (from [100] to [010] crystallographic direct) leads to the change in the photoconductivity of 50-100 units (Fig. 5).

Fig. 4. Photoconductivity and photopleochroism spectra of (a) BiOCl, (b) BiOBr and (c) BiOI single crystals at different light polarization and temperature 80K

Fig. 5. Angular dependencies of photoconductivity of BiOX single crystals

The temperature region of BiOX based photoresistors is in the range 80-400K. They are stable in air, and their parameters do not depend on the environment, which eliminates the need for sealing. It was established experimentally that the adsorption of oxygen leads to a slight increase even values of photosensitivity (Bunda, 1989, 2005). Other adsorbents do not

The position of electrical vector ��� can be varied either by rotating of the crystal. Anisotropy in 3 - 5 times was observed for the photosensitive value S. The change in light polarization at the 90 degrees (from [100] to [010] crystallographic direct) leads to the change in the

(a) (b) (c) Fig. 4. Photoconductivity and photopleochroism spectra of (a) BiOCl, (b) BiOBr and (c) BiOI

�100� and ���

�100� light polarization,

is the photocurrent values at the ���

single crystals at different light polarization and temperature 80K

Fig. 5. Angular dependencies of photoconductivity of BiOX single crystals

The temperature region of BiOX based photoresistors is in the range 80-400K. They are stable in air, and their parameters do not depend on the environment, which eliminates the need for sealing. It was established experimentally that the adsorption of oxygen leads to a slight increase even values of photosensitivity (Bunda, 1989, 2005). Other adsorbents do not

where � and �

respectively.

photoconductivity of 50-100 units (Fig. 5).

cause changes in the photoelectric properties. Crystal structure of the sample reduces the intrinsic noise of the radiation detector. The functional photoresistors based on BiOX single crystals, included in electronic circuits as a two-terminal networks, what is their great advantage. They can follow the rotation of polarization plane or work in schemes of the maximum signal finding. Besides, the BiOX crystals can be used as photo-potentiometers. The selectivity of BiOX spectral characteristics make them good radiation detectors of the nitrogen, argon and helium-cadmium lasers, as well as some of the semiconducting lightemitting diodes (LEDs). Therefore, they can be used in optoelectronic circuits in conjunction with these lasers and LEDs.

## **5. Energy structure and tunnelling effects of HTSC/BiOX hybrid contact structures**

The bismuth oxyhalide single crystals are very good "sparring partners" to create a HTSC/BiOX hybrid contact structures. This is for the following reasons:


The unique temperature behaviour of the HCS's resistance consists in an N-shape anomaly around the superconducting transition temperature (interval 85-105K; ��=92K; fig.6). The temperature dependence of the BiOCl:Ti semiconductor resistance can be described by the exponential function �(�) = ����� (−��⁄��), where ��=2.8 eV and ��=102-107 Ω. cm, depending on the activation regimes.

The detailed consideration of the temperature dependence of the heterojunction resistance in both directions (curves 1 and 2 on fig.6) shows the anomaly of heterojunction resistance in the narrow temperature interval above �� (ΔT=5K). The resistance first decreases down to ���� at T=��+2K and then rapidly increases to the maximum at T=��+5K. Subsequently, the resistance of heterojunction decreases according to the exponential dependence up to the room temperature. So the temperature dependence of YBa2Cu3O7-/BiOCl:Ti heterojunction resistance were characterized strongly pronounced N-shaped character in the narrow temperature interval above the superconducting transition. Fig. 6 shows the considerable

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 335

Fig. 7 show the dependencies of resistance versus temperature for the hybrid contact structure YBa2Cu3O7- (ceramics)/BiOI (single crystals) along the (001) plane of semiconductor at the different values of the bias voltage ܸ௧. We observed very similar to YBa2Cu3O7-/BiOCl:Ti heterojunction "resistivity - temperature" behaviour (see Fig.6). Note the characteristic features of these curves. The first, in Tc vicinity the N-like anomaly of electrical resistance is observed (these anomaly is not dependent on the geometry of the measurements). Second, the amplitude jumps of the resistance *R(T)* (and width of derivate negative value *dR(T)/dT*) was strongly dependent on the applied bias voltage. Third, exist the areas of bias voltage where the resistance jumps have maximum amplitude. For planar and through geometry them are 90-100 V and 220-255 V, respectively. Fourth, in the *Tc*

The red He-Ne laser (with λ = 632.8 nm, E = 1.95 eV and power P = 1.5 mW) was irradiated onto BiOI surface of YBa2Cu3O7-/BiOI heterostructure, for the 1 hour. The variation in the *GT/G300K* normalized conductance during laser irradiation was measured as a function of temperature and is shown in figure 8. It can be excitingly observed that the *Tc* is further increased from 92K to 94.5 K and critical current value *Jc* measured at 80 K is 1.85x103 A/cm2. Here, the increase superconducting parameters can only be attributed to the increase in carrier concentration when the sample was irradiated by laser having the energy greater than the band gap of semiconductor (Shiwagan el al.,

Fig. 8. Variation in normalized conductance of YBa2Cu3O7-/BiOI heterostructure in (1) dark;

The increase in superconducting parameters after laser irradiation is attributed to the microlevel dislocations due to lattice mismatch, attachment to the nonstoichiometric oxygen present on the BiOI surface, development of a diffusion region in semiconductor

vicinity the anomalies of thermopower are observed.

2002).

and (2) under laser irradiation

side of the interface (Shiwagan et al., 2004).

Fig. 6. Dependencies of the resistance R versus temperature for the YBa2Cu3O7-/BiOCl:Ti HCS along the semiconductor layer in (100) plane (curve 1) and trough the junction (perpendicular to (001) plane; curve 2). The curve 3 characterize the temperature dependence of YBa2Cu3O7- ceramics resistance

Fig. 7. Temperature dependencies of YBa2Cu3O7-/BiOI hybrid contact structure resistivity at the different values of bias voltage along the (100) plane. The curve 6 characterize the temperature dependence of YBa2Cu3O7- ceramics resistance difference in the values of maximally resistance's (amplitudes of the peaks) and N-anomalous temperature width for the curves 2 and 3 (see fig.6). It is interesting that these anomalies in the ߩሺܶሻ dependence appear at the T>ܶ, and the appreciable deviation (decreasing) of the differential resistance appeared long before the approach to ܶ (at T=85K).

Fig. 6. Dependencies of the resistance R versus temperature for the YBa2Cu3O7-/BiOCl:Ti HCS along the semiconductor layer in (100) plane (curve 1) and trough the junction (perpendicular to (001) plane; curve 2). The curve 3 characterize the temperature

Fig. 7. Temperature dependencies of YBa2Cu3O7-/BiOI hybrid contact structure resistivity at the different values of bias voltage along the (100) plane. The curve 6 characterize the temperature dependence of YBa2Cu3O7- ceramics resistance difference in the values of maximally resistance's (amplitudes of the peaks) and N-anomalous temperature width for the curves 2 and 3 (see fig.6). It is interesting that these anomalies in the ߩሺܶሻ dependence appear at the T>ܶ, and the appreciable deviation (decreasing) of the differential resistance

dependence of YBa2Cu3O7- ceramics resistance

appeared long before the approach to ܶ (at T=85K).

Fig. 7 show the dependencies of resistance versus temperature for the hybrid contact structure YBa2Cu3O7- (ceramics)/BiOI (single crystals) along the (001) plane of semiconductor at the different values of the bias voltage ܸ௧. We observed very similar to YBa2Cu3O7-/BiOCl:Ti heterojunction "resistivity - temperature" behaviour (see Fig.6). Note the characteristic features of these curves. The first, in Tc vicinity the N-like anomaly of electrical resistance is observed (these anomaly is not dependent on the geometry of the measurements). Second, the amplitude jumps of the resistance *R(T)* (and width of derivate negative value *dR(T)/dT*) was strongly dependent on the applied bias voltage. Third, exist the areas of bias voltage where the resistance jumps have maximum amplitude. For planar and through geometry them are 90-100 V and 220-255 V, respectively. Fourth, in the *Tc* vicinity the anomalies of thermopower are observed.

The red He-Ne laser (with λ = 632.8 nm, E = 1.95 eV and power P = 1.5 mW) was irradiated onto BiOI surface of YBa2Cu3O7-/BiOI heterostructure, for the 1 hour. The variation in the *GT/G300K* normalized conductance during laser irradiation was measured as a function of temperature and is shown in figure 8. It can be excitingly observed that the *Tc* is further increased from 92K to 94.5 K and critical current value *Jc* measured at 80 K is 1.85x103 A/cm2. Here, the increase superconducting parameters can only be attributed to the increase in carrier concentration when the sample was irradiated by laser having the energy greater than the band gap of semiconductor (Shiwagan el al., 2002).

Fig. 8. Variation in normalized conductance of YBa2Cu3O7-/BiOI heterostructure in (1) dark; and (2) under laser irradiation

The increase in superconducting parameters after laser irradiation is attributed to the microlevel dislocations due to lattice mismatch, attachment to the nonstoichiometric oxygen present on the BiOI surface, development of a diffusion region in semiconductor side of the interface (Shiwagan et al., 2004).

$$H\_T = \Sigma\_{\sigma kq} T\_{kq} c\_{kq}^\* c\_{kq} + h.c.\tag{7}$$

$$I(U) = A \left| T\_{kq} \right|^2 \int\_{-\infty}^{+\infty} N\_1(E) N\_2(E + eU) [f(E) - f(E + eU)] dE \tag{8}$$

$$
\Delta(T) \approx 1.74 \Delta(0) \sqrt{1 - \frac{T}{\tau\_c}}.\tag{9}
$$

$$N\_S(E) = \begin{cases} N\_n(E\_F) \cdot \frac{|E - E\_F|}{\sqrt{(E - E\_F)^2 - \Delta^2}} & \text{for } |E - E\_F| \ge \Delta \\ 0 & \text{for } |E - E\_F| \le \Delta \end{cases} \tag{10}$$

$$I\{U\} = A\left|T\_{kq}\right|^2 N\_1(0) \int\_{-\infty}^{+\alpha} N\_{2s}(E) [f(E) - f(E + eU)] dE.\tag{11}$$

$$G\_{ns} = \frac{d\mathbf{l}\_{ns}}{dU} = G\_{nn} \int\_{-\infty}^{+\infty} \frac{N\_{2s}(E)}{N\_2(0)} \left[ -\frac{\partial f(E+eU)}{\partial (eU)} \right] dE \tag{12}$$

Applications of Superconductor/Photosemiconductor Contact Structures in Electronics 339

Bunda, V.V., Bunda, S., Krivsky, V., Butorin, A. & Zihika, V. (1996). Physical properties of

*Polonica A*, Vol. 118, No.5 (November 2010), pp. 1040-1041, ISSN 0587-4254 Bunda, V., Bunda, S., Kajnakova, M., Feher, A., Syrkin, E. & Feodosyev, S. (2010). Low

*Polonica A*, Vol. 118, No.5 (November 2010), pp. 986-987, ISSN 0587-4254 Bunda, V., Bunda, S., Kajnakova, M., Feher, A., Syrkin, E. & Feodosyev, S. (2011). Low-

Bletskan, D.I., Kopinets, I.F., Rubish, I.D., Turyanitsa, I.I. & Shtilikha, M.V. (1973).

Bunda, V.V. (1989). Optical anisotropy of deep recombination centers in the layered crystals

Bunda, V.V. (2005). Functional detectors of radiation for the middle ultraviolet area based on

Burton, V.K., Cabrera, N. & Frank, F.C. (1951). The growth of crystals and the equilibrium

Beck, H.P. & Beyer, T. (1997). Lattice potentials as an instrument in crystal chemistry. II.

Keramidas, K. G., Voutsas, G.P. & Rentzeperis, P. I. (1993). The crystal structure of BiOCl.

Bercha, D.M., Puga, G.D. & Borets, A.N. (1973). Two-phonon processes in crystals of

Zhang, Ke-Lei, Liu, Cun-Ming, Huang, Fu-Qiang, Zheng, Chong & Wang, Wen-Deng (2006).

Shivagan, D.D., Shirage, P.M. & Pawar, S.H. (2004). Studies on the fabrication of

*Conference Proceedings. Kaluga. Russia. May 24-27,* Vol.2, pp. 3940-3942 Ganesha, R., Arivuoli, D. & Ramasamy, P. (1993). Growth of some group V-VI-VII

*physics*, Vol.37, No. 4 (July 2010), pp. 326-328, ISSN 1063-777X

Vol.16, No.5 (May 1973), pp.646-648 ISSN 1064-8887

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No.8(December 1996), pp. 565-571, ISSN 0044-2968

ISSN 0022-0248

0044-2968

0370-1972

pp.125–129, ISSN 0926-3373

323-332, ISSN 0268-1242

���

*Physics*, Vol. 46, No. 3 (November 1995), pp. 1361-1362, ISSN 0011-4626 Bunda, V., Bunda, S., Vashchuk, F., Feher, A. & Kovač, J. (2010). Physical properties of

"HTSC-photosemiconductor" hybrid contact structures. *Czechoslovak Journal of* 

"HTSC/ Photosemiconductor" micro- and nanoplates based junctions. *Acta Physica* 

temperature specific heat of BiOX (X=Cl, Br and I) single crystals. *Acta Physica* 

temperature specific heat of single crystal bismuth oxyhalides. *Low temperature* 

Photoconductivity and photoluminescence in BiOCl crystals. *Soviet Physics Journal*,

of the matlokyte family. *Ukrainskii Fizicheskii Zhurnal (Russian Edition),* Vol.34, No.5

bismuth oxyhalide single crystals. *In: Physics of Electronic Materials. 2nd International* 

compounds from the vapour. *Journal of crystal growth*, Vol.128, No.1-4 ,pp.1081-1085

structure of their surfaces. Philosophical transaction of the royal society, Vol.243,

Applications to the crystal bulk. *Zeitschrift für Kristallographie*, Vol. 221,

*Zeitschrift für Kristallographie*, Vol. 205, No. Part-1 (January 1992), pp. 35-40, ISSN

� symmtry. Physica Status Solidi (B), Vol.55, No.1 (January 1973), pp.69-78 ISSN

Study of the electronic structure and photocatalytic activity of the BiOCl photocatalist. Applied Catalysis B: Environmental, Vol.68, No.3-4 (August 2006),

Ag/Hg1Ba2Ca1Cu2O6+δ/CdSe heterostructures using the pulse electrodeposition technique. *Semiconductor Science and Technology*, Vol.19, No.3 (September 2003), pp.

which directly gives �� for ���. Temperature dependent STS measurements, as shown in Figures 6 and 7, exhibit smeared features with increasing T, because the energy gap shrinks with T and even with an applied bias |�| �� �⁄ some quasiparticles can tunnel due to thermal energy (Kuznetsov, 2001).

When a superconductor (SC) and a semiconductor (SeC) are in electrical contact and form an SeCSC interface, a variety of different phenomena and transport processes can happen. In a device containing multiple SeCSC interfaces, both local (Andreev reflection) and nonlocal (charge imbalance) processes can occur (Klein, 2010).

## **6. Conclusion**

In the "HTSC - SeC" system can be formed three principal types of HCS's (Bunda et al., 1996): tunneling(1), proximity(2) and combined(3).

The bismuth oxyhalide single crystals are layered 2D structured materials. The crystal structure of BiOX is very similar to the symmetry of space groups P4/��� and/or I4/mmm, where are usual for the YBa2Cu3O7- and Bi(Tl,Hg)2Sr(Ba)2Can-1CunO2n+4 HTSC's. Besides, the BiOX crystals, as well as a HTSC's contain oxygen. The thermal expansion and the lattice constants mish-match in the (001) base plane are in good agreement with the same parameters of the HTSC's. Also, the BiOX crystals are good "sparring partners" for the fabrication of "HTSC – photosensitive semiconductor" hybrid contact structure.

The transition of a high temperature superconductor in the superconducting state leads to a significant increase in the photosensitivity of the semiconductor layer of the heterostructure. This is due to single spin quantum tunneling processes of the Cooper pairs inside the semiconductor layer. Also observed the N - shaped anomalies in the temperature dependences of the conductivity (resistivity) in the vicinity of the superconducting transition.

Thus, we believe that based on "high-temperature superconductor - photosensitive semiconductor" hybrid contact structures can be created entirely new device of cryophotoelectronics.

## **7. References**


which directly gives �� for ���. Temperature dependent STS measurements, as shown in Figures 6 and 7, exhibit smeared features with increasing T, because the energy gap shrinks with T and even with an applied bias |�| �� �⁄ some quasiparticles can tunnel due to

When a superconductor (SC) and a semiconductor (SeC) are in electrical contact and form an SeCSC interface, a variety of different phenomena and transport processes can happen. In a device containing multiple SeCSC interfaces, both local (Andreev reflection) and

In the "HTSC - SeC" system can be formed three principal types of HCS's (Bunda et al.,

The bismuth oxyhalide single crystals are layered 2D structured materials. The crystal structure of BiOX is very similar to the symmetry of space groups P4/��� and/or I4/mmm, where are usual for the YBa2Cu3O7- and Bi(Tl,Hg)2Sr(Ba)2Can-1CunO2n+4 HTSC's. Besides, the BiOX crystals, as well as a HTSC's contain oxygen. The thermal expansion and the lattice constants mish-match in the (001) base plane are in good agreement with the same parameters of the HTSC's. Also, the BiOX crystals are good "sparring partners" for the

The transition of a high temperature superconductor in the superconducting state leads to a significant increase in the photosensitivity of the semiconductor layer of the heterostructure. This is due to single spin quantum tunneling processes of the Cooper pairs inside the semiconductor layer. Also observed the N - shaped anomalies in the temperature dependences of the conductivity (resistivity) in the vicinity of the superconducting

Thus, we believe that based on "high-temperature superconductor - photosensitive semiconductor" hybrid contact structures can be created entirely new device of

Bunda, V.V. (1991). Specification of the temperature behavior of the resistance of heterojunc-

Akani, M., Froese, A., Staikov, G., Lorenz, W.J.,Röhner, M., Hopfengärter, R. & Saemann-

Deng, Zhengtao, Chen,Dong, Peng, Bo & Tang, Fangqiong. (2008). From bulk metal Bi to

(December 1990), pp. 15-18, ISSN 0320-0116

131-138, ISSN 0921-4534

2995-3003, ISSN 1528-7483

tions from high-temperature superconductors with degenerate semiconductors near Tc. *Pis'ma v Zhurnal Tekhnicheskoi Fiziki (written in Russian)*, Vol. 17, No.22

Ischenko, G. (1995). Low-temperature behaviour of high-Tc superconductor / semiconductor heterojunctions. *Physica C,* Vol.245, No.1-2, (December 1994), pp.

two-dimensional well-crystallized BiOX (X = Cl, Br) micro- and nanostructures: synthesis and characterization. *Crystal growth & design*, Vol.8, No.8 (May 2008), pp.

fabrication of "HTSC – photosensitive semiconductor" hybrid contact structure.

thermal energy (Kuznetsov, 2001).

**6. Conclusion** 

transition.

cryophotoelectronics.

**7. References** 

nonlocal (charge imbalance) processes can occur (Klein, 2010).

1996): tunneling(1), proximity(2) and combined(3).


**16** 

Janos Kosa

*Hungary* 

**Novel Application of YBCO Ring and Closed** 

The chapter presents the theoretical and experimental results of DC and AC magnetic flux transfer with one closed superconducting loop. This method may open a new path for advanced applications by using superconducting ring and perfect closed superconducting loops. I examined a novel arrangement of YBCO ring and continuous loops made of HTS wire. With this solution we can develop a self-limiting transformer. At the same time I also tested the operation of the YBCO ring made from bulk and continuous YBCO loop developed with my own technology. I could carry out these experiments because earlier I

In the first experiment I used YBCO superconducting rings made from a bulk because I managed to drill the YBCO bulk just in some minutes. This technology is low price (Kosa & Vajda, 2007; Kosa et al., 2009). Fig. 1 shows one bulk and two rings drilled from the same

Later I applied superconducting wire because the diameter of the superconducting ring from bulk is small. It is a positive fact there is possibility to soft-solder the YBCO wires to

had elaborated the machining of the YBCO bulk and wire.

bulk. The used YBCO bulk was produced at IPHT Jena, Germany.

Fig. 1. Rings from YBCO bulk with economical and fast technology

create closed loop, but this method is not optimal solution.

**1. Introduction** 

*Kecskemet College, Faculty of Mechanical Engineering and Automation* 

**Loop with DC and AC Magnetic** 

**Flux Transformation** 


## **Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation**

Janos Kosa *Kecskemet College, Faculty of Mechanical Engineering and Automation Hungary* 

## **1. Introduction**

340 Superconductors – Properties, Technology, and Applications

Shivagan, D.D., Shirage, P.M. & Pawar, S.H. (2002). Studies on photoinduced effects in

Alfeev, V.N., Men'shov, V.N. & Neustroev L.N. (1991). Photoelectric amplification in the

Kuznetsov, G.V. (2004). Transport of the charge carrier through superconductor –

Kuznetsov, G.V. (2002) Charge carrier passing through the metal-superconducting semicon-

Kuznetsov, G.V. (2001) Thermoelectrical current through the metals – superconducting

*Russian)*, vol.30, No.10 (April 2004), pp. 1038-1044, ISSN 0132-6414

*of physics*, Vol.58, No.5&6 (May 2002) pp. 1183-1190, ISSN 0304-4289 Alfeev, V.N. & Neustroev, L.N. (1990). The photo-current in the tunnel contact: high-tempe-

pp. 1106-1110, ISSN 0002-3264

No.9 (February 2002), ISSN 0015-3222

(October 2001), pp. 556-559, ISSN 0021-3640

108 p.

pulse-electrodeposited Ag/Hg-1212/CdSe hetero-nanostructures. *Pramana–Journal* 

rature superconductor-semiconductor-semiconductor. *Doclady Akademii Nauk SSSR (Reports of the Soviet Academy of Sciences -in Russian)*, Vol.311, No. 5 (Yanuary 1990),

tunneling structure degenerate semiconductor/high-temperature superconductor/insulator/metal. *Fizika Nizkich Temperatur (Low Temperature Physics - in Russian)*, vol.17, No.1 (November 1990), pp. 128-130, ISSN 0132-6414 Kleine, A. (2010). *Experiments on nonlocal processes in NS devices.* Inauguraldissertation zur

Erlangung der Würde eines Doktors der Philosophie vorgelegt der Philosophisch-Naturwissenschaftlichen Fakultät der Universität Basel, UNI, Basel, Schwizerland,

semiconductor contact. *Fizika Nizkich Temperatur (Low Temperature Physics - in* 

ductor contact. *Fizika i technika poluprovodnikov (Semiconductors – in Russian)*, Vol.39,

semiconductor contact. Pisma v ZETF (JETP Letters - in Russian), Vol.74, No.10

The chapter presents the theoretical and experimental results of DC and AC magnetic flux transfer with one closed superconducting loop. This method may open a new path for advanced applications by using superconducting ring and perfect closed superconducting loops. I examined a novel arrangement of YBCO ring and continuous loops made of HTS wire. With this solution we can develop a self-limiting transformer. At the same time I also tested the operation of the YBCO ring made from bulk and continuous YBCO loop developed with my own technology. I could carry out these experiments because earlier I had elaborated the machining of the YBCO bulk and wire.

In the first experiment I used YBCO superconducting rings made from a bulk because I managed to drill the YBCO bulk just in some minutes. This technology is low price (Kosa & Vajda, 2007; Kosa et al., 2009). Fig. 1 shows one bulk and two rings drilled from the same bulk. The used YBCO bulk was produced at IPHT Jena, Germany.

Fig. 1. Rings from YBCO bulk with economical and fast technology

Later I applied superconducting wire because the diameter of the superconducting ring from bulk is small. It is a positive fact there is possibility to soft-solder the YBCO wires to create closed loop, but this method is not optimal solution.

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 343

possibility to transform the DC and AC magnetic field. This arrangement can result in novel applications in the field of transfer both the energy of DC magnetic field and AC electrical energy. In Fig . 3 we can see the scheme of the connection of the theory and applications in

Fig. 3. The scheme of the connection of the theory and applications in this chapter

**2. Theoretical and measurement examination of the transfer of the DC** 

Fig. 4. Scheme of the transformation of the DC magnetic flux with one loop

this chapter.

**magnetic flux with one loop** 

We can see the experimental arrangement in Fig. 4.


This is the reason why I did not apply the closed loops with soldering for my applications. However, we must consider the fact, that even in the case of a really good soldering, the superconducting join will not exist.

A Korean group produced 100 % BSCCO loops for the first time (Hee-Gyoun Lee et al.,2006). In 2009 the other Korean group wrote that the perfectly closed YBCO wire making process needs further development. I modified their technology and managed to develop a different cutting method (Kosa et al., 2010). After development of the technology of the cut I examine the arrangement of the continuous YBCO superconducting ring and loop (100% YBCO join). Fig. 2 shows the photograph of an eleven-turn serial loop. The advantage of this winding is that there is no twist along the longitudinal axis that could cause cracks in the YBCO layer at operating temperature, i.e. at 77 K or a lower temperature. It must be mentioned that with this method we can only produce coils with odd number of turns. The odd number comes from topology of the surface of the cut wire. The applied superconductor wire (2G HTS wire) was produced by SuperPower, Inc., New York, USA. Type: SF 12050.

Fig. 2. Eleven-turn serial loop from YBCO wire without soldering and twists along its longitudinal axis (loops with layering)

With used technology we can create parallel and serial loops. This is the task of planning yet. We can apply the cut wires in DC and AC developments.

For example:


With these rings and loops we can transfer the magnetic flux between two independent iron cores inductively from one closed iron core to the other one. This method gives a new possibility to transform the DC and AC magnetic field. This arrangement can result in novel applications in the field of transfer both the energy of DC magnetic field and AC electrical energy. In Fig . 3 we can see the scheme of the connection of the theory and applications in this chapter.

Fig. 3. The scheme of the connection of the theory and applications in this chapter

## **2. Theoretical and measurement examination of the transfer of the DC magnetic flux with one loop**

We can see the experimental arrangement in Fig. 4.

342 Superconductors – Properties, Technology, and Applications

For the beginning the transfer of the magnetic flux is impossible with DC current

 Secondly it is established fact we can use the soldered wire with AC current in different applications but with soldering we cannot ensure appropriate uniform temperature in the superconducting wire when the YBCO loop will change to normal state from superconducting state. In soldered spot the Joule loss is considerably higher than in the

This is the reason why I did not apply the closed loops with soldering for my applications. However, we must consider the fact, that even in the case of a really good soldering, the

A Korean group produced 100 % BSCCO loops for the first time (Hee-Gyoun Lee et al.,2006). In 2009 the other Korean group wrote that the perfectly closed YBCO wire making process needs further development. I modified their technology and managed to develop a different cutting method (Kosa et al., 2010). After development of the technology of the cut I examine the arrangement of the continuous YBCO superconducting ring and loop (100% YBCO join). Fig. 2 shows the photograph of an eleven-turn serial loop. The advantage of this winding is that there is no twist along the longitudinal axis that could cause cracks in the YBCO layer at operating temperature, i.e. at 77 K or a lower temperature. It must be mentioned that with this method we can only produce coils with odd number of turns. The odd number comes from topology of the surface of the cut wire. The applied superconductor wire (2G HTS wire) was produced by SuperPower, Inc., New York, USA.

Fig. 2. Eleven-turn serial loop from YBCO wire without soldering and twists along its

 self-limiting transformer and inductive fault current limiter (AC application) different solutions with active magnetic short circuit (DC and AC applications), etc.

With used technology we can create parallel and serial loops. This is the task of planning

With these rings and loops we can transfer the magnetic flux between two independent iron cores inductively from one closed iron core to the other one. This method gives a new

between independent iron cores.

superconducting join will not exist.

longitudinal axis (loops with layering)

yet. We can apply the cut wires in DC and AC developments.

DC flux transformer with one loop (Kosa & Vajda, 2009)

wire itself.

Type: SF 12050.

For example:

Fig. 4. Scheme of the transformation of the DC magnetic flux with one loop

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 345

The operation is based on well known physical law. The possibility of the DC and AC flux transformation with single loop is based on the principle of the magnetic flux constancy in the time in a superconducting closed loop. The mentioned arrangement gives possibilities for the realization of novel industrial applications. The temporary constancy of the resultant flux of the superconductor ring is realized with less ring electrical current if we use

We can see from Eq. 6 that the current of the superconducting ring decreases if there is a secondary iron core. We can see from equation (6), that the electrical current of the

If the current of superconducting ring is bigger than the critical current of superconductor, there is not magnetic flux in the secondary iron core. In this case the coupling ends between the iron cores. The magnetic flux constancy depends on the current of the superconducting ring or loop because the characteristic (voltage–current) of superconductor. Non-linear effect of the characteristic is bigger in the case of the DC current, than at application of AC

> 2 *SUP m I R*

2 2 1 12

*NIRR RR RR RR R*

*m m E E air mm mm mm air air m*

*m E E air mm mm mm air air NIR RR RR RR*

1 2

*m air*

1 2

*air*

*R*

*R*

2

2 1 12

*E E m m m m*

> 2 *E E m eq N I R*

*N I R R R R*

(7)

(11)

(12)

(9)

(10)

(8)

1 2

2

21 2 *m m m mm eq m*

*R R R RR*

superconductor depends on the geometric data and magnetic permeability.

**2.2 Determination of the equivalent reluctance on the secondary side** 

2

2

2

secondary iron core.

The starting equation

The result is as follows

2

current.

### **2.1 Mathematical description of DC flux transfer**

We can find a detailed theoretical description of the DC flux transformer with two coils (Orlando & Delin, 1990). In my case I describe a different solution. My DC flux transformer is implemented by using one YBCO ring. First I describe the theoretical examination. The Table 1 shows the nomenclature.


Table 1. Nomenclature

$$
\rho \varrho\_E = \varrho\_{SLP} \mathbf{1}\_1 + \varrho\_{SLP} \mathbf{2}\_2 + \varrho\_{SLPair} \tag{1}
$$

$$\frac{N\_E I\_E}{R\_1^m} = I\_{SLIP} \left( \frac{1}{R\_1^m} + \frac{1}{R\_2^m} + \frac{1}{R\_{air}^m} \right) \quad \text{where} \quad R = \text{relative} \tag{2}$$

$$I\_{SIIP} = \frac{N\_E I\_E R\_2^m R\_{air}^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m} \tag{3}$$

If 2 *<sup>m</sup> R* , there is not secondary iron core, then:

$$I\_{SUP} = \lim\_{R\_2^{m} \to \infty} \frac{N\_E I\_E R\_2^m R\_{air}^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m} = \frac{N\_E I\_E R\_{air}^m}{R\_{air}^m + R\_1^m} \tag{4}$$

If 2 *<sup>m</sup> <sup>R</sup>* and *<sup>m</sup> Rair* , then

$$I\_{SLP} = \lim\_{R\_{air}^{m} \to \cdots \to \infty} \frac{N\_E I\_E R\_{air}^m}{R\_{air}^m + R\_1^m} = \lim\_{R\_{air}^{m} \to \cdots} \frac{N\_E I\_E}{R\_{air}} = N\_E I\_E \tag{5}$$

If *<sup>m</sup> Rair* and és 2 *<sup>m</sup> R* then

$$I\_{\rm SLP} = \lim\_{R\_{\rm air}^{\rm m} \to \infty} \frac{N\_E I\_E R\_2^{\rm m} R\_{\rm air}^{\rm m}}{R\_2^{\rm m} R\_{\rm air}^{\rm m} + R\_1^{\rm m} R\_{\rm air}^{\rm m} + R\_1^{\rm m} R\_2^{\rm m}} = \frac{N\_E I\_E R\_2^{\rm m}}{R\_2^{\rm m} + R\_1^{\rm m}} \tag{6}$$

The operation is based on well known physical law. The possibility of the DC and AC flux transformation with single loop is based on the principle of the magnetic flux constancy in the time in a superconducting closed loop. The mentioned arrangement gives possibilities for the realization of novel industrial applications. The temporary constancy of the resultant flux of the superconductor ring is realized with less ring electrical current if we use secondary iron core.

We can see from Eq. 6 that the current of the superconducting ring decreases if there is a secondary iron core. We can see from equation (6), that the electrical current of the superconductor depends on the geometric data and magnetic permeability.

If the current of superconducting ring is bigger than the critical current of superconductor, there is not magnetic flux in the secondary iron core. In this case the coupling ends between the iron cores. The magnetic flux constancy depends on the current of the superconducting ring or loop because the characteristic (voltage–current) of superconductor. Non-linear effect of the characteristic is bigger in the case of the DC current, than at application of AC current.

### **2.2 Determination of the equivalent reluctance on the secondary side**

The starting equation

344 Superconductors – Properties, Technology, and Applications

We can find a detailed theoretical description of the DC flux transformer with two coils (Orlando & Delin, 1990). In my case I describe a different solution. My DC flux transformer is implemented by using one YBCO ring. First I describe the theoretical examination. The

magnetic flux of the exciter coil in the primary iron core without

magnetic flux of the superconducting loop in the primary and

magnetic flux of the superconducting loop inside the closed

*<sup>m</sup> R* reluctance (magnetic resistance) of the primary and secondary iron cores

*<sup>m</sup> Rair* reluctance (magnetic resistance) of the air around superconducting loop

 *E SUP SUP SUP air* 

*air*

*RR RR RR*

2 2 1 12 1 2

*SUP mm mm mm m m mR air air air NIR R NIR <sup>I</sup>*

*NIR R <sup>I</sup>*

lim lim 

*m*

2 2 1 12

*RR RR RR R R*

1 1

2 1 12 2 1

*RR RR RR R R*

*E E air E E SUP m m <sup>m</sup> E E <sup>m</sup> <sup>R</sup><sup>m</sup> <sup>R</sup> air air air air m m*

*NIR N I <sup>I</sup> N I R R R R*

*SUP mm mm mm m m mR air air air NIRR NIR <sup>I</sup>*

*m m E E air SUP mm mm mm air air*

1 2 (1)

(3)

(4)

(5)

(6)

, where R = reluctance (2)

*m m m E E air E E air*

*R R*

2 2

*m m m E E air E E*

*air air*

**2.1 Mathematical description of DC flux transfer** 

NE number of the primary turn (exciter coil)

superconductor loop

secondary iron cores

ISUP current of the superconducting loop

1 12

lim 

*<sup>m</sup> R* , there is not secondary iron core, then:

*<sup>m</sup> R* then

lim 

*SUP R RRR*

<sup>111</sup> ( ) *E E m mmm*

superconducting loop in air

Table 1 shows the nomenclature.

φ<sup>E</sup>

φSUP1; φSUP2

φSUP air

1 *<sup>m</sup> <sup>R</sup>* ; 2

If 2

If 2

**Parameters Nomenclature** 

IE exciting current

*<sup>m</sup> <sup>R</sup>* and *<sup>m</sup> Rair* , then

If *<sup>m</sup> Rair* and és 2

*N I <sup>I</sup>*

Table 1. Nomenclature

$$
\varphi\_2 = \frac{I\_{SLIP}}{R\_2'''} \tag{7}
$$

$$\varphi\_2 = \frac{\frac{N\_E I\_E R\_2^m R\_{air}^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m}}{R\_2^m} \tag{8}$$

$$\varphi\_2 = \frac{N\_E I\_E R\_{air}^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m} \tag{9}$$

$$\varphi\_2 = \frac{N\_E I\_E}{R\_1^m + R\_2^m + \frac{R\_1^m R\_2^m}{R\_{air}^m}} \tag{10}$$

$$
\rho\_2 = \frac{N\_E I\_E}{R\_{eq2}^m} \tag{11}
$$

The result is as follows

$$R\_{eq2}^{m} = R\_1^m + R\_2^m + \frac{R\_1^m R\_2^m}{R\_{air}^m} \tag{12}$$

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 347

*E E air mm mm mm air air NI R R <sup>φ</sup> RR RR RR* 

> 1 2 *E E m mm*

1

11 2

1 12

*m m R R air* then

1 21 2

1 2 *EQ*

1 2

*EE EE mm m*

*NI NI RR R*

*EQ*

2 1 12

1 *E E m eq N I R*

*N I RRR*

( ) *m m*

1

Fig. 6. Equivalent magnetic circuit on the primary side

*m m m m*

*R R R R*

*m air*

and 2

1 2

 

*R*

If 2

In this case

*m m R R air* , then

We can see, if 1 2 1 2

1

2

*air*

(18)

(19)

(20)

*m m mm R RRR eq air* (21)

*m mm R RR eq* (22)

*m m mm R R RR eq eq* (23)

*mmm RRR eq eq* (24)

(25)

$$\text{If } \quad R\_1^m + R\_2^m \gg \frac{R\_1^m R\_2^m}{R\_{air}^m} \text{, then }$$

$$\mathbf{R\_{eq}^{\rm m}}\_{\mathbf{eq},2} \to \mathbf{R\_1^m} + \mathbf{R\_2^m} \tag{13}$$

In Fig. 5 we can see the equivalent circuit. Consequently so the magnetic flux in the secondary side is with ease definable.

Fig. 5. Equivalent magnetic circuit on the secondary side

The scheme shows importance of proportion of the reluctances for the calculus of the magnetic flux in the secondary side.

#### **2.3 Determination of the equivalent reluctance on the primary side**

$$\rho\_1 = \frac{N\_E I\_E - I\_{SUP}}{R\_1^m} \tag{14}$$

$$\varphi\_1 = \frac{N\_E I\_E \cdot \frac{N\_E I\_E R\_2^m R\_{air}^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m}}{R\_1^m} \tag{15}$$

$$\varphi\_1 = \frac{\frac{N\_E I\_E R\_2^m R\_{air}^m + N\_E I\_E R\_1^m R\_{air}^m + N\_E I\_E R\_1^m R\_2^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m} - \frac{N\_E I\_E R\_2^m R\_{air}^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m}}{R\_1^m} \tag{16}$$

$$\varphi\_1 = \frac{\frac{N\_E I\_E R\_1^m R\_{air}^m + N\_E I\_E R\_1^m R\_2^m}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m}}{R\_1^m} \tag{17}$$

$$\varphi p\_1 = \frac{N\_E I\_E \left(R\_{air}^m + R\_2^m\right)}{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m} \tag{18}$$

$$\rho\_1 = \frac{N\_E I\_E}{R\_1^m + R\_2^m \times R\_{air}^m} \tag{19}$$

$$\varphi\_1 = \frac{N\_E I\_E}{R\_{eq1}^m} \tag{20}$$

$$R\_{eq}^{m} = R\_1^m + R\_2^m \times R\_{air}^m \tag{21}$$

#### Fig. 6. Equivalent magnetic circuit on the primary side

If 2 *m m R R air* , then

346 Superconductors – Properties, Technology, and Applications

In Fig. 5 we can see the equivalent circuit. Consequently so the magnetic flux in the

The scheme shows importance of proportion of the reluctances for the calculus of the

1 *E E SUP m NI I R*

1

*mm mm mm m m E E air E E air E E E E air*

1

1

2 1 12

2 1 12 2 1 12

*RR RR RR RR RR RR*

1 1 2 2 1 12

*mm mm*

*mm mm mm mm mm mm air air air air m*

(16)

*RR RR RR*

*m m E E air E E mm mm mm air air m*



*E E air E E mm mm mm air air m*

*NIRR NIRR RR RR RR*

*NIRR NIRR NIRR NIRR*

*NIRR N I*

m <sup>2</sup> <sup>m</sup>

<sup>1</sup> <sup>m</sup> <sup>R</sup> 2eq <sup>→</sup> R+R (13)

(14)

(15)

(17)

If 1 2 1 2

*R R R R*

*m m m m*

*m air*

secondary side is with ease definable.

Fig. 5. Equivalent magnetic circuit on the secondary side

1

*<sup>φ</sup> <sup>R</sup>*

1

**2.3 Determination of the equivalent reluctance on the primary side** 

1

*<sup>φ</sup> <sup>R</sup>*

*<sup>φ</sup> <sup>R</sup>*

magnetic flux in the secondary side.

1

*R* , then

$$R\_{eq1}^{m} \to R\_1^{m} + R\_2^{m} \tag{22}$$

We can see, if 1 2 1 2 *m m m m m air R R R R R* and 2 *m m R R air* then

$$R\_{eq1}^{\;m} = R\_{eq2}^{\;m} = R\_1^{\;m} + R\_2^{\;m} \tag{23}$$

$$R\_{eq\ 1}^{m} = R\_{eq\ 2}^{m} = R\_{\ 
eq}^{m} \tag{24}$$

In this case

$$\varphi\_1 = \varphi\_2 = \frac{N\_E I\_E}{R\_1^m + R\_2^m} = \frac{N\_E I\_E}{R\_{EQ}^m} \tag{25}$$

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 349

First I changed the DC current of the coil of the primary side with rapid switches and then

It can be seen in Fig. 8 that with this solution the change of the magnetic flux of the primary iron core generates opposed flux in the secondary iron core. This is natural because the

In Fig. 9 and 10 we can see the magnetic flux density of the secondary iron core does not

**2.5 Experimental results of DC magnetic flux transfer with one loop** 

**2.5.1 Measurement experiments with ZFC method** 

decrease during 2200 seconds.

Fig. 8. DC results with one loop

resultant flux of a closed superconducting loop does not change.

slowly. I measured the flux density in the primary and secondary iron core.

#### **2.4 General solution in the case of "M" pieces of closed iron core**

As long as M pieces of iron core loops the superconductor ring and number of the electrical excitation are K pieces then we can describe the current of the superconducting ring. Fig. 6 shows the arrangement.

Fig. 7. The arrangement in the case of general solution

We can define numerically the magnetic flux in the iron cores from Ampere's Law.

$$\sum\_{i=1}^{K} N\_i I\_i \Lambda\_i = I\_{\text{SLP}}(\sum\_{1}^{M} \Lambda\_i) \quad \text{where} \quad \Lambda\_i = \frac{1}{R\_i^m} \quad \text{and} \quad \mathbf{K} \le \mathbf{M} \tag{26}$$

$$I\_{SLP} = \frac{\sum\_{i=1}^{K} N\_i I\_i \Lambda\_i}{\sum\_{i=1}^{M} \Lambda\_i} = \frac{\sum\_{i=1}^{M} N\_i I\_i \Lambda\_i}{\sum\_{i=1}^{M} \Lambda\_i}, \text{ as } \ I\_{K+1} = 0 \tag{27}$$

In optional iron core where there is not exciting current:

$$\rho\_X = I\_{SUP} \Lambda\_X = \frac{\sum\_{i=1}^{M} N\_i I\_i \Lambda\_i}{\sum\_{i=1}^{M} \Lambda\_i} \Lambda\_X \tag{28}$$

In optional iron core where there is exciting current:

$$\log\_y = \text{(N}\_y \text{I}\_y - \text{I}\_{SLP}) \text{\AA}\_y = \text{(N}\_y \text{I}\_y - \frac{\sum\_{i=1}^{M} \text{N}\_i \text{\AA}\_i}{\sum\_{i=1}^{M} \text{\AA}\_i} \text{\AA}\_y \tag{29}$$

Equation (29) is the general solution.

## **2.5 Experimental results of DC magnetic flux transfer with one loop**

First I changed the DC current of the coil of the primary side with rapid switches and then slowly. I measured the flux density in the primary and secondary iron core.

## **2.5.1 Measurement experiments with ZFC method**

348 Superconductors – Properties, Technology, and Applications

As long as M pieces of iron core loops the superconductor ring and number of the electrical excitation are K pieces then we can describe the current of the superconducting ring. Fig. 6

**2.4 General solution in the case of "M" pieces of closed iron core** 

Fig. 7. The arrangement in the case of general solution

1 1

In optional iron core where there is not exciting current:

1

In optional iron core where there is exciting current:

Equation (29) is the general solution.

*i*

*K M i i i SUP i*

*NI I*

We can define numerically the magnetic flux in the iron cores from Ampere's Law.

*<sup>i</sup> <sup>m</sup> Ri*

and K ≤ M (26)

, as 1 0 *KI* (27)

(28)

(29)

( )

where <sup>1</sup>

1 1

*i i SUP M M*

*I*

*K M*

1 1

*I*

*NI I NI*

*NI NI*

*ii i ii i*

*i i i i*

1

*N I*

*ii i*

*i i*

1

*i*

*M*

1

*N I*

*ii i*

*i i*

*M*

*i X SUP X M X*

( -)( - )

*y yy SUP y yy M y*

shows the arrangement.

It can be seen in Fig. 8 that with this solution the change of the magnetic flux of the primary iron core generates opposed flux in the secondary iron core. This is natural because the resultant flux of a closed superconducting loop does not change.

In Fig. 9 and 10 we can see the magnetic flux density of the secondary iron core does not decrease during 2200 seconds.

Fig. 8. DC results with one loop

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 351

The value of the critical current depends on definitions and even the producers give

Fig. 11. The decrease of the current of the ring (effect of the non-linear characteristic)

Fig. 12. Measuring result with a thick copper-ring in liquid nitrogen

In Fig. 12 we can see a measuring result with a thick copper-ring (instead of superconductor) in liquid nitrogen. Surface of ring was 1.5 cm2. The flux density of the

*cm* 

; another one gives 0.5 *<sup>V</sup>*

*cm* 

as the

different definitions. One of the producers gives <sup>1</sup> *<sup>V</sup>*

secondary side decreases very fast.

critical value. This is also influenced by business purposes.

Fig. 9. Measurement result during 2000 seconds I.

Fig. 10. Measurement result during 2000 seconds II.

The practical application of the transformation of DC magnetic field cannot be carried out with the critical current of superconductor during long time. We have to use a lower superconducting current.

Fig. 9. Measurement result during 2000 seconds I.

Fig. 10. Measurement result during 2000 seconds II.

superconducting current.

The practical application of the transformation of DC magnetic field cannot be carried out with the critical current of superconductor during long time. We have to use a lower The value of the critical current depends on definitions and even the producers give different definitions. One of the producers gives <sup>1</sup> *<sup>V</sup> cm* ; another one gives 0.5 *<sup>V</sup> cm* as the critical value. This is also influenced by business purposes.

Fig. 11. The decrease of the current of the ring (effect of the non-linear characteristic)

In Fig. 12 we can see a measuring result with a thick copper-ring (instead of superconductor) in liquid nitrogen. Surface of ring was 1.5 cm2. The flux density of the secondary side decreases very fast.

Fig. 12. Measuring result with a thick copper-ring in liquid nitrogen

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 353

In the next experiment I have generated magnetic flux in primary and secondary iron core with the same direction when the ring was not yet in superconducting state. The flux density of both iron cores was 0.9 T. After cooling I changed the flux of the primary iron

Fig. 15. Exciting current is on both iron cores and after this is cooling (77 K). After this is

Fig. 16. Scheme of the single-phase self-limiting transformer using by AC flux transformer

**3. Theoretical and measurement examination of the transfer of the AC magnetic flux with one loop for single-phase self-limiting transformer** 

core. We can see the result in Fig. 15. The system is also stable.

decrease and increase of the primary current.

We can see the experimental arrangement in Fig. 16.

### **2.5.2 Measurement experiments with FC method**

In the next experiment I have generated magnetic flux only in the primary iron core when the ring was not yet in superconducting state. Then I cooled it down to 77 K and reduced exciting current on the primary iron core to 0 ampere. In parallel in the secondary iron core the flux increased in the opposite direction. After this I changed the polarity of the primary exciting current and the flux increased in the secondary iron core. We can see a measurement result in Fig. 13.

Fig. 13. Measurement result with FC method

Fig. 14 shows a measurement during 5200 seconds.

Fig. 14. Measurement result with FC method during 5200 seconds

In the next experiment I have generated magnetic flux in primary and secondary iron core with the same direction when the ring was not yet in superconducting state. The flux density of both iron cores was 0.9 T. After cooling I changed the flux of the primary iron core. We can see the result in Fig. 15. The system is also stable.

Fig. 15. Exciting current is on both iron cores and after this is cooling (77 K). After this is decrease and increase of the primary current.

## **3. Theoretical and measurement examination of the transfer of the AC magnetic flux with one loop for single-phase self-limiting transformer**

We can see the experimental arrangement in Fig. 16.

352 Superconductors – Properties, Technology, and Applications

In the next experiment I have generated magnetic flux only in the primary iron core when the ring was not yet in superconducting state. Then I cooled it down to 77 K and reduced exciting current on the primary iron core to 0 ampere. In parallel in the secondary iron core the flux increased in the opposite direction. After this I changed the polarity of the primary exciting current and the flux increased in the secondary iron core. We can see a

**2.5.2 Measurement experiments with FC method** 

measurement result in Fig. 13.

Fig. 13. Measurement result with FC method

Fig. 14 shows a measurement during 5200 seconds.

Fig. 14. Measurement result with FC method during 5200 seconds

Fig. 16. Scheme of the single-phase self-limiting transformer using by AC flux transformer

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 355

( ) ( )

*SUP m m Ni t R i t R R*

Fig. 17 shows a static characteristic of the single-phase self-limiting transformer. This was my first experiment in the case of self-limiting transformer using by flux transformer (Kosa

**3.2 Measurement results of single-phase self-limiting transformer using by flux** 

We can see a theoretical scheme of the arrangement for dynamic examination in Fig 18.

et. al., 2010). I have measured the primary and secondary voltages.

**transformer** 

Fig. 17. Static characteristic

Fig. 18. Measurement arrangement

11 2 2 1

*m*

(37)

The experimental arrangement is a single-phase self-limiting transformer by using AC flux transformer (Kosa et al.) In the case of AC application we can use closed YBCO wire. The superconducting YBCO loop creates the coupling between the primary and secondary coils. This case occurs if the current of superconducting loop is bigger than the critical current of the superconducting loop. The advantage of this solution is that the transformer is able to break the coupling between the primary and secondary coils even if there is not load current in the secondary side, only the primary current increases.

#### **3.1 Mathematical description of single-phase AC flux transfer**

$$
\varphi\_1(t) + \varphi\_2(t) = \varphi\_{SLP1}(t) + \varphi\_{SLP2}(t) + \varphi\_{air}(t)
$$

$$\frac{N\_1 i\_1(t)}{R\_1^{\rm m}} + \frac{N\_2 i\_2(t)}{R\_2^{\rm m}} = \frac{i\_{SUP}(t)}{R\_1^{\rm m}} + \frac{i\_{SUP}(t)}{R\_2^{\rm m}} + \frac{i\_{SUP}(t)}{R\_{air}} = (i\_{SLP}(t))(\frac{1}{R\_1^{\rm m}} + \frac{1}{R\_2^{\rm m}} + \frac{1}{R\_{air}})\tag{30}$$

The following functions contain the effect of the inductivities of all transformer elements: ( ) *SUP i t* , 1*i t*( ) , 2*i t*( ) .

$$i\_{SLP}(t) = \frac{N\_1 i\_1(t)}{R\_1^m(\frac{1}{R\_1^m} + \frac{1}{R\_2^m} + \frac{1}{R\_{air}^m})} + \frac{N\_2 i\_2(t)}{R\_2(\frac{1}{R\_1^m} + \frac{1}{R\_2^m} + \frac{1}{R\_{air}^m})}\tag{31}$$

$$i\_{SLP}(t) = \frac{N\_1 i\_1(t)}{1 + \frac{R\_1^m}{R\_2^m} + \frac{R\_1}{R\_{air}^m}} + \frac{N\_2 i\_2(t)}{1 + \frac{R\_2^m}{R\_1^m} + \frac{R\_2^m}{R\_{air}^m}} \tag{32}$$

$$i\_3(t) = \frac{N\_1 i\_1(t)}{1 + \frac{R\_1^m}{R\_2^m} + \frac{R\_1^m}{R\_{air}^m}} + \frac{N\_2 I\_2 i\_2(t)}{1 + \frac{R\_2^m}{R\_1^m} + \frac{R\_2^m}{R\_{air}^m}} = \tag{33}$$

$$\dot{q}\_{SID}(t) = \frac{N\_1 \dot{i}\_1(t)}{\frac{R\_2^m R\_{air} + R\_1^m R\_{air}^m + R\_1^m R\_2^m}{R\_2^m R\_{air}^m}} + \frac{N\_2 \dot{i}\_2(t)}{\frac{R\_2^m R\_{air}^m + R\_1^m R\_{air}^m + R\_1^m R\_2^m}{R\_1^m R\_{air}^m}}\tag{34}$$

$$\dot{\mathbf{q}}\_{\text{SLP}}(t) = \frac{\mathbf{N}\_1 \dot{\mathbf{i}}\_1(t) \; \mathbf{R}\_2^m \mathbf{R}\_{air}^m}{\mathbf{R}\_2^m \mathbf{R}\_{air}^m + \mathbf{R}\_1^m \mathbf{R}\_{air}^m + \mathbf{R}\_1^m \mathbf{R}\_2^m} + \frac{\mathbf{N}\_2 \dot{\mathbf{i}}\_2(t) \; \mathbf{R}\_1^m \mathbf{R}\_{air}^m}{\mathbf{R}\_2^m \mathbf{R}\_{air}^m + \mathbf{R}\_1^m \mathbf{R}\_{air}^m + \mathbf{R}\_1^m \mathbf{R}\_2^m} \tag{35}$$

If *<sup>m</sup> Rair* , then

$$i\_{SLP}(t) = \frac{N\_1 i\_1(t) \ R\_2^m}{R\_2^m + R\_1^m} + \frac{N\_2 i\_2(t) \ R\_1^m}{R\_2^m + R\_1^m} \tag{36}$$

If there is not current in the secondary side, then

$$i\_{SUP}(t) = \frac{N\_1 i\_1(t) \ R\_2^m}{R\_2^m + R\_1^m} \tag{37}$$

#### **3.2 Measurement results of single-phase self-limiting transformer using by flux transformer**

Fig. 17 shows a static characteristic of the single-phase self-limiting transformer. This was my first experiment in the case of self-limiting transformer using by flux transformer (Kosa et. al., 2010). I have measured the primary and secondary voltages.

Fig. 17. Static characteristic

354 Superconductors – Properties, Technology, and Applications

The experimental arrangement is a single-phase self-limiting transformer by using AC flux transformer (Kosa et al.) In the case of AC application we can use closed YBCO wire. The superconducting YBCO loop creates the coupling between the primary and secondary coils. This case occurs if the current of superconducting loop is bigger than the critical current of the superconducting loop. The advantage of this solution is that the transformer is able to break the coupling between the primary and secondary coils even if there is not load current

12 1 2 () () () () () *SUP SUP air*

1 2 12 1 2

1 2

*Ni t Ni t i t R R*

*R R RR R R R R*

The following functions contain the effect of the inductivities of all transformer elements:

*tt t t t*

() () () () () 11 1 ( ( ))( ) *SUP SUP SUP m m mm SUP m m air air*

 

1 1 2 2

*mmm mmm*

*mm mm air air*

*RRR RRR*

*air air*

(31)

(32)

(33)

(36)

(34)

(35)

1 2 1 2

1 1 2 2 1 1 22 2 1

1 1 2 22

 

*RR RR RR RR*

11 22 2 1

1 1 2 2 2 1 12 2 1 12 2 1

*RR RR RR RR RR RR R R R R*

11 2 22 1 2 1 12 2 1 12

> 11 2 22 1 21 21

*RR RR RR RR RR RR* 

*m m m m air air*

*m m*

*mm mm mm mm air air*

*R R RR RR RR*

( ) ( ) ( ) 1 1 *SUP m m m*

*Ni t Ni t i t*

( ) ( ) ( )

( ) ( ) ( ) *SUP m mm mm mm mm mm air air air air m m m m air air*

*SUP mm mm mm mm mm mm air air air air*

() () ( )

*SUP mm mm Ni t R Ni t R i t RR RR* 

*Ni t NIi t i t*

1 1

*Ni t Ni t i t*

( ) ( ) ( )

*Ni t R R Ni t R R i t*

( ) ( ) ( ) 111 111 ( )( )

(30)

 

in the secondary side, only the primary current increases.

*SUP <sup>m</sup>*

3

If there is not current in the secondary side, then

11 22

( ) *SUP i t* , 1*i t*( ) , 2*i t*( ) .

If *<sup>m</sup> Rair* , then

**3.1 Mathematical description of single-phase AC flux transfer** 

 

*Ni t Ni t i ti ti t i t*

We can see a theoretical scheme of the arrangement for dynamic examination in Fig 18.

Fig. 18. Measurement arrangement

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 357

**Parameters Data**  Srated 400 VA Uprimary 3 x 400/230 V Usecondary 3 x 42/24 V Nprimary 697 Nsecondary 73 YBCO loops 3 x 7 loops

We can see the cutaway view of the device (Fig. 21) and the electrical connection (Fig. 22).

Table 2 shows the parameters of the experimental arrangement.

Table 2. Parameters of transformer

Fig. 21. Cutaway view of arrangement

Fig. 22. Electrical connection

I generated artificial fault from the operating current of the secondary coil and measured its current. In the experiment, the fault time was 5 periods. The sample rate was 10 kHz.

The current limitation occurs within 3.3 ms (Kosa et al.). This time is less than a quarter of a period (Fig. 19).

Fig. 19. Dynamic characteristic

## **3.3 Arrangement of the three-phase self-limiting transformer with flux transfer**

The possibility of three-phase application is also based on the principle of magnetic flux constancy in the closed superconducting loop. The solution includes two independent iron cores (Kosa & Vajda, 2011). As I used two iron cores, consequently the loss, the weight, the size, and the price are higher compared to the conventional transformers but this solution has several advantages. For example, the fault power is less, switching is fast. In the case, when there is no load current on the secondary coil and primary voltage increases, the arrangement is able to break coupling between the primary and secondary sides. The developed new type three-phase self-limiting transformer was 400 VA. I realised the primary and secondary windings of the three-phase transformer for this experiment with copper wires. I used SF 12050 wire. This wire (2G HTS wire) was produced 2G HTS wire) was produced by SuperPower, Inc., New York, USA. We can see a photo from device during fitting (Fig. 20).

Fig. 20. Photo from the self-limiting transformer during fitting


Table 2 shows the parameters of the experimental arrangement.

## Table 2. Parameters of transformer

356 Superconductors – Properties, Technology, and Applications

I generated artificial fault from the operating current of the secondary coil and measured its

The current limitation occurs within 3.3 ms (Kosa et al.). This time is less than a quarter of a

**<sup>950</sup> <sup>1000</sup> <sup>1050</sup> <sup>1100</sup> <sup>1150</sup> <sup>1200</sup> <sup>1250</sup> t [ms]**

**0.5 A**

**5 periods fault**

**operating after fault**

**3.3 Arrangement of the three-phase self-limiting transformer with flux transfer** 

The possibility of three-phase application is also based on the principle of magnetic flux constancy in the closed superconducting loop. The solution includes two independent iron cores (Kosa & Vajda, 2011). As I used two iron cores, consequently the loss, the weight, the size, and the price are higher compared to the conventional transformers but this solution has several advantages. For example, the fault power is less, switching is fast. In the case, when there is no load current on the secondary coil and primary voltage increases, the arrangement is able to break coupling between the primary and secondary sides. The developed new type three-phase self-limiting transformer was 400 VA. I realised the primary and secondary windings of the three-phase transformer for this experiment with copper wires. I used SF 12050 wire. This wire (2G HTS wire) was produced 2G HTS wire) was produced by SuperPower, Inc., New York, USA. We can see a photo from device during fitting (Fig. 20).

**-1.33 A**

current. In the experiment, the fault time was 5 periods. The sample rate was 10 kHz.

**1.12 A**

period (Fig. 19).

**-3**

Fig. 19. Dynamic characteristic

**current**

**-2**

**-1**

**0**

**1**

**2**

**3**

**I [A]**

**1.61 A**

**2.46 A**

**1.58 A**

**-2.21 A fault**

**-1.71 A**

Fig. 20. Photo from the self-limiting transformer during fitting

We can see the cutaway view of the device (Fig. 21) and the electrical connection (Fig. 22).

Fig. 21. Cutaway view of arrangement

Fig. 22. Electrical connection

Novel Application of YBCO Ring and Closed Loop with DC and AC Magnetic Flux Transformation 359

In Fig. 25 we can see a fault. There is load-current on the secondary side (Rload = 3 x 4.5 Ω)

The advantage of the arrangement is that in the case of single-phase short circuit the current will decrease in all the three phases. This can be an appropriate solution for high power machines. In the case of high-power electric motor if there is a single-phase breaking or a single phase short circuit, we can decrease the current in the three phases with this solution.

The continuous closed loop made from YBCO tape is suitable for use in such equipment. It is possible to create the novel type 3-phase self-limiting transformer with flux transfer. The short circuit current is less than the operational current within some periods. This is very advantageous for high-power machines (motor). It seems that the arrangement is good also in the case of the single-phase breaking of high-power electric motor. If there is a breaking in one phase, the current increases in both the other phases. And the superconductor will be in the normal state. This means that the current will decrease in an electric motor. The two iron cores' greater size, cost and loss compared to the conventional transformers are facts but the transformer has several advantages e. g. fast switching, the ability of limiting even if the

The author thanks the SuperPower, Inc. in New York for YBCO wire, facts and figures of the

Kosa, J. & Vajda, I. (2007). Environmentally friendly machining of ceramic based YBCO bulk superconductor, Journal of Materials Processing Technology, 181, pp. 48-51

primary current is increased without loading the secondary side.

wire and thanks the German colleagues at IPHT, Jena for YBCO bulk.

and single-phase short circuit in L1 phase.

Fig. 25. Single-phase short circuit in L1 phase

**4. Conclusion** 

**5. Acknowledgment** 

**6. References** 

### **3.4 Measurement results**

Measurements were carried out using 4-channel data logger. Sample rate = 50 kS/s.

You can see in Fig. 23 and Fig. 24 the secondary voltage of the self-limiting transformer without load-current and with maximum load-current. What is the difference? Hardly anything.

Fig. 23. Measurement result without load-current

Fig. 24. Measurement result with maximum load-current (3 x 4.5 Ω, P = 400 Watt)

In Fig. 25 we can see a fault. There is load-current on the secondary side (Rload = 3 x 4.5 Ω) and single-phase short circuit in L1 phase.

Fig. 25. Single-phase short circuit in L1 phase

The advantage of the arrangement is that in the case of single-phase short circuit the current will decrease in all the three phases. This can be an appropriate solution for high power machines. In the case of high-power electric motor if there is a single-phase breaking or a single phase short circuit, we can decrease the current in the three phases with this solution.

## **4. Conclusion**

358 Superconductors – Properties, Technology, and Applications

You can see in Fig. 23 and Fig. 24 the secondary voltage of the self-limiting transformer without

Measurements were carried out using 4-channel data logger. Sample rate = 50 kS/s.

load-current and with maximum load-current. What is the difference? Hardly anything.

Fig. 24. Measurement result with maximum load-current (3 x 4.5 Ω, P = 400 Watt)

**3.4 Measurement results** 

Fig. 23. Measurement result without load-current

The continuous closed loop made from YBCO tape is suitable for use in such equipment. It is possible to create the novel type 3-phase self-limiting transformer with flux transfer. The short circuit current is less than the operational current within some periods. This is very advantageous for high-power machines (motor). It seems that the arrangement is good also in the case of the single-phase breaking of high-power electric motor. If there is a breaking in one phase, the current increases in both the other phases. And the superconductor will be in the normal state. This means that the current will decrease in an electric motor. The two iron cores' greater size, cost and loss compared to the conventional transformers are facts but the transformer has several advantages e. g. fast switching, the ability of limiting even if the primary current is increased without loading the secondary side.

## **5. Acknowledgment**

The author thanks the SuperPower, Inc. in New York for YBCO wire, facts and figures of the wire and thanks the German colleagues at IPHT, Jena for YBCO bulk.

## **6. References**

Kosa, J. & Vajda, I. (2007). Environmentally friendly machining of ceramic based YBCO bulk superconductor, Journal of Materials Processing Technology, 181, pp. 48-51

**17** 

*Republic of Korea* 

**Superconductor Application to the Magnetic** 

**Fusion Devices for the Steady-State Plasma** 

Yeong-Kook Oh\*, Keeman Kim, Kap-Rai Park and Young-Min Park

The exploitation of the new energy source is necessary in upcoming century due to the rapid increasing of energy consumption and the shortage of the fossil energy sources in the world. There have been a lot of scientific and engineering efforts to make realization of the fusion energy production as a clean and limitless energy source to the mankind. The fusion energy has great advantages in the points of the energy density and the amount of resources, and it is a dominant energy source in the universe like the sun and stars. To achieve a reliable fusion reaction, a good confinement technology is required to overcome the repulsive force between nucleuses of light atoms like hydrogen isotopes as shown in Fig. 1. Among the confinement technologies two kinds of methods are dominant in the fusion researches. The one is the inertial confinement which focusing the high power laser beams in to the small size fuel pellet. The other is the magnetic confinement which confining the fuel gas in

Fig. 1. A representative fusion reaction. The confinement of hydrogen isotopes, deuterium and tritium produces neutron and helium atom with total energy of about 17.6 MeV

**1.1 History of the superconductor application into fusion devices** 

plasma state in vacuum by applying the high magnetic fields.

**1. Introduction** 

**Confinement Achievement** 

*National Fusion Research Institute (NFRI), Daejeon* 


## **Superconductor Application to the Magnetic Fusion Devices for the Steady-State Plasma Confinement Achievement**

Yeong-Kook Oh\*, Keeman Kim, Kap-Rai Park and Young-Min Park *National Fusion Research Institute (NFRI), Daejeon Republic of Korea* 

## **1. Introduction**

360 Superconductors – Properties, Technology, and Applications

Kosa, J.; Vajda, I. & Farkas, L. (2009). Qualification of the Machining and Fitting Precision of

*IEEE Transactions on Applied Superconductivity*, vol. 19, No. 3, pp. 2182-2185 Hee-Gyoun Lee; Jae-Geun Kim, Sun-Wang Lee, Woo-Seak Kim, Seung-Wook Lee, Kyeong-

Kosa, J.; Vajda, I. & Gyore, A. (2010) Application Possibilities with Continuous YBCO Loops Made of HTS Wire, *Journal of Physics-Conference Series* 234:(3) Paper 032030, http://www.ewh.ieee.org/tc/csc/europe/newsforum/Contents11.html, ST 153 Kosa, J. & vajda, I. (2009). Transformation of the DC and AC Magnetic Field with Novel

www.ewh.ieee.org/tc/csc/europe/newsforum/technicalnews.html, ST 74 Orlando, T. & Delin K. (1990). *Foundations of Applied Superconductivity*, Addison-Wesley

Kosa, J. & Vajda I. (2011). Novel 3-Phase Self-Limiting Transformer With Magnetic Flux

1102

vol. 19, No. 3, pp. 2186-2189,

Publishing Company, pp. 88-89

*Superconductivity*, vol. 21, No. 3, pp. 1388-1392

YBCO Bulks and Rings Joined Together via the Examination of the Trapped Flux,

Dal Chai, Gye-Wan Hong, Tae-Kuk Ko. (2006). Design and fabrication of permanent mode magnet by using coated conductor, *Physica C*, 445-448, pp. 1099-

Application of the YBCO HTS ring, *IEEE Transactions on Applied Superconductivity*,

Applied by Perfect Closed YBCO Wire Loops, *IEEE Transactions on Applied* 

## **1.1 History of the superconductor application into fusion devices**

The exploitation of the new energy source is necessary in upcoming century due to the rapid increasing of energy consumption and the shortage of the fossil energy sources in the world. There have been a lot of scientific and engineering efforts to make realization of the fusion energy production as a clean and limitless energy source to the mankind. The fusion energy has great advantages in the points of the energy density and the amount of resources, and it is a dominant energy source in the universe like the sun and stars. To achieve a reliable fusion reaction, a good confinement technology is required to overcome the repulsive force between nucleuses of light atoms like hydrogen isotopes as shown in Fig. 1. Among the confinement technologies two kinds of methods are dominant in the fusion researches. The one is the inertial confinement which focusing the high power laser beams in to the small size fuel pellet. The other is the magnetic confinement which confining the fuel gas in plasma state in vacuum by applying the high magnetic fields.

Fig. 1. A representative fusion reaction. The confinement of hydrogen isotopes, deuterium and tritium produces neutron and helium atom with total energy of about 17.6 MeV

Superconductor Application to the Magnetic

double-null or single-null configurations.

superconducting fusion devices.

Fusion Devices for the Steady-State Plasma Confinement Achievement 363

KSTAR device has lots of technical similarities with the ITER device such as using the same material of superconducting conductor, Nb3Sn CICC, and targeting the steady-state high performance plasma confinement over 300s. KSTAR adopted the most outstanding research results from the present devices and will exploit high performance steady-state operation which will provide the core technology for ITER and for future reactors. The designed parameters of KSTAR, as shown in Table 1, are major radius 1.8m, minor radius 0.5 m, toroidal field 3.5 T, and plasma current 2 MA for 300 s. The specific features in the KSTAR design are full superconducting magnets, passive stabilizers and segmented 3-dimentional in-vessel coils for the fast plasma stabilizations, various kinds of heating and current drive systems for the high-beta and non-inductive current drive, and strong plasma shaping for

Fig. 2. The progress of the superconductor application in the magnetic fusion devices. The comparison of the operating current and peak field of the conductors in the worldwide

Fig. 3. The design comparison between KSTAR and ITER devices. The dimension is much

different but the components and their engineering are similar together.

The first use of superconducting coils in experimental fusion devices dates back to the midseventies. Three main kinds of fusion devices have been studied for plasma confinement using superconducting coils: the mirror machines, the stellerators and the tokamaks. In the initial stage, the application of the superconductor was only for DC or slowly changing in magnetic field like toroidal field (TF) coils in tokamaks or stellerator magnets. The efforts of the last 10 years concentrated on the tokamaks with limited stellerator. The stored magnetic energy and the size of the superconducting fusion devices grew by up to three orders of magnitude in 30 years, from the 20 MJ in the first superconducting tokamak, T-7 to the international thermonuclear experimental reactor (ITER).

There has been a lot of development in the superconducting magnet technologies for the application to the fusion devices, in the superconducting technologies related with the cooling, the superconductor, and the operation mode (Bruzzone, P., 2010). In the cooling method to keep cold a superconducting magnet, the liquid helium bath cooling was the only option in the very first applications. As alternative cooling options, superfluid helium bath cooling at 1.8 K was applied into Tore-Supra in France and forced-flow supercritical helium cooling at about 4.5 K was applied into the recent tokamaks (SST-1, EAST, KSTAR, JT60SA, and ITER) and stellerators (LHD and W7-X). The requirement of mechanical stiffness and reliability of the high voltage electric insulation has ruled out the pool cooling option from the fusion devices of present and future generations. In large devices, with total mass flow rate in the range of several kg/s, the forced flow cooling of supercritical helium is assisted by cold circulation pumps and heat exchangers. Extensive activities on conductor development led to a large variety of NbTi and Nb3Sn conductor design. With increasing coil size and stored energy, the operating current must be increased to keep low values in the number of turns and the inductance. At operating current over 5 kA, the single multi-filamentary composite is no longer an option because of the ac loss and stability issues. Cable-in-conduit conductor (CICC), both NbTi and Nb3Sn, became a dominant option in superconductor for large current over 10 kA in fusion device. The advantages of the CICC are direct contact of the conductor with forced-flow helium, structural rigidity is guaranteed by steel jacket surrounding the cable, and the electric insulation by the turn insulation and vacuum pressure impregnation (VPI). In the operation mode of the superconductor, only the DC coils were considered an application target for superconductors. The T7, T15, and Tore Supra tokamaks had copper coil for plasma start-up and shaping. By appearance of the CICC, the superconductor application was expanded into the fast varying central solenoid (CS) or poloidal field (PF) coils in the recent tokamaks, the EAST (Wu, S. et al., 2002), KSTAR (Lee, G.S. et al., 2001), JT-60SA (Yoshida, K. et al., 2008), and ITER (Mitchell, N., 2003). Figure 2 shows the peak field and operation current of the superconducting coils in the fusion devices. NbTi conductor is used in the coils operated at lower field less than 7 T, except Tore-Supra device which is operated at 9 T by cooling with 1.8 K superfluid helium. Nb3Sn conductor is used in the coils operated higher field. KSTAR TF conductor is operated at 35 kA and is the highest current carrying conductor before ITER operation.

## **1.2 A recent superconducting tokamak, KSTAR**

The operation of the KSTAR device has been started after the integrated commissioning of all superconducting magnets in 2008 (Oh, Y.K. et al., 2009). The mission of the KSTAR project is to develop a steady-state-capable advanced superconducting tokamak, and to establish a scientific and technological basis for an attractive fusion reactor (Lee, G.S. et al., 2001). Figure 3 shows the design of the KSTAR device comparing with ITER device. The

The first use of superconducting coils in experimental fusion devices dates back to the midseventies. Three main kinds of fusion devices have been studied for plasma confinement using superconducting coils: the mirror machines, the stellerators and the tokamaks. In the initial stage, the application of the superconductor was only for DC or slowly changing in magnetic field like toroidal field (TF) coils in tokamaks or stellerator magnets. The efforts of the last 10 years concentrated on the tokamaks with limited stellerator. The stored magnetic energy and the size of the superconducting fusion devices grew by up to three orders of magnitude in 30 years, from the 20 MJ in the first superconducting tokamak, T-7 to the

There has been a lot of development in the superconducting magnet technologies for the application to the fusion devices, in the superconducting technologies related with the cooling, the superconductor, and the operation mode (Bruzzone, P., 2010). In the cooling method to keep cold a superconducting magnet, the liquid helium bath cooling was the only option in the very first applications. As alternative cooling options, superfluid helium bath cooling at 1.8 K was applied into Tore-Supra in France and forced-flow supercritical helium cooling at about 4.5 K was applied into the recent tokamaks (SST-1, EAST, KSTAR, JT60SA, and ITER) and stellerators (LHD and W7-X). The requirement of mechanical stiffness and reliability of the high voltage electric insulation has ruled out the pool cooling option from the fusion devices of present and future generations. In large devices, with total mass flow rate in the range of several kg/s, the forced flow cooling of supercritical helium is assisted by cold circulation pumps and heat exchangers. Extensive activities on conductor development led to a large variety of NbTi and Nb3Sn conductor design. With increasing coil size and stored energy, the operating current must be increased to keep low values in the number of turns and the inductance. At operating current over 5 kA, the single multi-filamentary composite is no longer an option because of the ac loss and stability issues. Cable-in-conduit conductor (CICC), both NbTi and Nb3Sn, became a dominant option in superconductor for large current over 10 kA in fusion device. The advantages of the CICC are direct contact of the conductor with forced-flow helium, structural rigidity is guaranteed by steel jacket surrounding the cable, and the electric insulation by the turn insulation and vacuum pressure impregnation (VPI). In the operation mode of the superconductor, only the DC coils were considered an application target for superconductors. The T7, T15, and Tore Supra tokamaks had copper coil for plasma start-up and shaping. By appearance of the CICC, the superconductor application was expanded into the fast varying central solenoid (CS) or poloidal field (PF) coils in the recent tokamaks, the EAST (Wu, S. et al., 2002), KSTAR (Lee, G.S. et al., 2001), JT-60SA (Yoshida, K. et al., 2008), and ITER (Mitchell, N., 2003). Figure 2 shows the peak field and operation current of the superconducting coils in the fusion devices. NbTi conductor is used in the coils operated at lower field less than 7 T, except Tore-Supra device which is operated at 9 T by cooling with 1.8 K superfluid helium. Nb3Sn conductor is used in the coils operated higher field. KSTAR TF conductor is operated at 35 kA

international thermonuclear experimental reactor (ITER).

and is the highest current carrying conductor before ITER operation.

The operation of the KSTAR device has been started after the integrated commissioning of all superconducting magnets in 2008 (Oh, Y.K. et al., 2009). The mission of the KSTAR project is to develop a steady-state-capable advanced superconducting tokamak, and to establish a scientific and technological basis for an attractive fusion reactor (Lee, G.S. et al., 2001). Figure 3 shows the design of the KSTAR device comparing with ITER device. The

**1.2 A recent superconducting tokamak, KSTAR** 

KSTAR device has lots of technical similarities with the ITER device such as using the same material of superconducting conductor, Nb3Sn CICC, and targeting the steady-state high performance plasma confinement over 300s. KSTAR adopted the most outstanding research results from the present devices and will exploit high performance steady-state operation which will provide the core technology for ITER and for future reactors. The designed parameters of KSTAR, as shown in Table 1, are major radius 1.8m, minor radius 0.5 m, toroidal field 3.5 T, and plasma current 2 MA for 300 s. The specific features in the KSTAR design are full superconducting magnets, passive stabilizers and segmented 3-dimentional in-vessel coils for the fast plasma stabilizations, various kinds of heating and current drive systems for the high-beta and non-inductive current drive, and strong plasma shaping for double-null or single-null configurations.

Fig. 2. The progress of the superconductor application in the magnetic fusion devices. The comparison of the operating current and peak field of the conductors in the worldwide superconducting fusion devices.

Fig. 3. The design comparison between KSTAR and ITER devices. The dimension is much different but the components and their engineering are similar together.

Superconductor Application to the Magnetic

superconductor (a) and CICC (b).

final dimension of the CICC.

Fusion Devices for the Steady-State Plasma Confinement Achievement 365

shown in the Fig. 4(b). The cable patterns of the TF and PF conductors are 3x3x3x3x6 of 486 strands and 3x4x5x6 of 360 strands, respectively. Two superconducting strands and one OFHC copper strand are cabled together to become a triplet in the first cabling stage. The cabling pitch of the TF and PF conductors are 40-73-157-227-355 mm and 40-80-145-237 mm, respectively. At the final stage of cable fabrication, the cable is wrapped with a thin stainless-steel strip, 30 mm wide and 0.05 mm thick, with 20 % overlap at each side. The selection of the jacket material is also important factor deciding the Nb3Sn conductor performance, because the operating current limit is decreased according to the strain applied to the conductor and the strain is mainly come from the difference in coefficient of the thermal expansion (COE) between superconductor and jacket material. In KSTAR,

(a) (b)

The length of the conductor is also an important factor in the magnet design. If a longer CICC is available then the number of internal joint could be reduced or excluded. It could guarantee more stable operation of the magnet because lots of troubles come from the joints. In KSTAR, tube-mill process was developed to produce long CICC with various lengths up to 1.8 km to exclude the internal joint. The tube-mill process consists of forming, welding, sizing and squaring procedures. A strip is wrapped around the superconducting cable through a series of progressive roller dies and welded using gas tungsten arc welding (GTAW). The welded sheath is cooled immediately by water and the face-bead of the weld is ground by a bead grinding machine. The shape of the welding back-bead is controlled to be smooth not to damage the superconducting cable. Then, the conductors are formed to the

The leak tightness is an important qualification of the conductor, because the helium leak from the conductor could hinder the cooling the magnet to the cryogenic temperature due to the increased convective heat transfer. The CICC fabricated by tube-mill process have more possibility of the leak due to the longitudinal welding line. Leak detection of CICC spool was performed in the water chamber by pressurizing the CICC spool with helium at about 50 bars as shown in Fig. 5(a). The locations of the leak were identified by detecting bubbles generated in the water chamber. After machining the leak point with a special grinder,

Fig. 4. The picture of the KSTAR superconductor. Cross sectional view of the Nb3Sn

Incoloy 908 is selected as the jacket material for Nb3Sn conductor.


Table 1. Key design parameters of the KSTAR and ITER devices.

In this chapter, the generals about the superconducting technology used in the magnetic fusion devices are described with an example of the KSTAR device, which is the most recently constructed superconducting fusion device and is in operation. The development of a CICC conductor for the fusion application is described in section 2, and Superconducting magnet fabrication and test is in section 3, magnet assembly and interfaces are in section 4, integrated commissioning of the superconducting tokamak is in section 5, and the plasma experiments and consideration of the magnet operation are in section 6.

## **2. Cable-in-conduit conductor development**

Most of the superconducting coils in KSTAR are made of Nb3Sn superconductor with Incoloy908 jacket, but large PF coils (PF6-7), which operated at lower field, are made of NbTi superconductor with STS316LN jacket.

Usually TF magnet operated in static mode but PF magnet in pulsed mode. So in ITER design, two different type of superconductor was considered. The superconductor for the TF magnet should satisfy the HP-I criteria that is high current density capacity over 750 A/mm2 at 12 T, 4.2 K and the superconductor for PF magnet should satisfy the HP-II criteria that is low hysteresis loss generation less than 250 mJ/cc at of 3 T cycle. But in KSTAR a new superconductor has been developed to satisfy both criteria simultaneously (HP-III), and constructed TF and PF magnet using same type of superconductor. The cross sectional view of the Nb3Sn superconductor strand is seen in Fig. 4(a). In the superconductor configuration, the number Nb3Sn filaments are over 3,000 to get the current density and low hysteresis loss. To reduce the ac loss at the fast varying field environment, each strand is coated with high resistance chrome with a thickness of 1 ± 0.2 μm. The chrome coating is used to survive after the long period of heat treatment of the Nb3Sn coil.

Large number of strands are cabled together and overwrapped with steel jacket in the form of CICC to have structural rigidity at the large current in the range of several tens of kA and high field over 7 T. KSTAR CICC is shaped into square cross section with round corner as

In this chapter, the generals about the superconducting technology used in the magnetic fusion devices are described with an example of the KSTAR device, which is the most recently constructed superconducting fusion device and is in operation. The development of a CICC conductor for the fusion application is described in section 2, and Superconducting magnet fabrication and test is in section 3, magnet assembly and interfaces are in section 4, integrated commissioning of the superconducting tokamak is in section 5, and the plasma

Most of the superconducting coils in KSTAR are made of Nb3Sn superconductor with Incoloy908 jacket, but large PF coils (PF6-7), which operated at lower field, are made of NbTi

Usually TF magnet operated in static mode but PF magnet in pulsed mode. So in ITER design, two different type of superconductor was considered. The superconductor for the TF magnet should satisfy the HP-I criteria that is high current density capacity over 750 A/mm2 at 12 T, 4.2 K and the superconductor for PF magnet should satisfy the HP-II criteria that is low hysteresis loss generation less than 250 mJ/cc at of 3 T cycle. But in KSTAR a new superconductor has been developed to satisfy both criteria simultaneously (HP-III), and constructed TF and PF magnet using same type of superconductor. The cross sectional view of the Nb3Sn superconductor strand is seen in Fig. 4(a). In the superconductor configuration, the number Nb3Sn filaments are over 3,000 to get the current density and low hysteresis loss. To reduce the ac loss at the fast varying field environment, each strand is coated with high resistance chrome with a thickness of 1 ± 0.2 μm. The chrome coating is used to survive

Large number of strands are cabled together and overwrapped with steel jacket in the form of CICC to have structural rigidity at the large current in the range of several tens of kA and high field over 7 T. KSTAR CICC is shaped into square cross section with round corner as

Major radius, R0 [m] Minor radius, a [m] Plasma current, Ip [MA]

Elongation, Triangularity, Toroidal field, B0 [T] Pulse length [s] Plasma volume [m3] Plasma cross section [m2]

Plasma shape Normalized beta Plasma fuel Superconductor

Auxiliary heating /CD [MW]

Table 1. Key design parameters of the KSTAR and ITER devices.

experiments and consideration of the magnet operation are in section 6.

**2. Cable-in-conduit conductor development** 

after the long period of heat treatment of the Nb3Sn coil.

superconductor with STS316LN jacket.

 **KSTAR ITER** 

1.8 0.5 2.0 2.0 0.8 3.5 300 17.8 1.6 DN, SN 5.0 H, D Nb3Sn, NbTi 28

6.2 2.0 15 (17) 1.7 0.33 5.3 400 830 22 SN 1.8 (2.5) H, D, T Nb3Sn, NbTi 73 (110)

shown in the Fig. 4(b). The cable patterns of the TF and PF conductors are 3x3x3x3x6 of 486 strands and 3x4x5x6 of 360 strands, respectively. Two superconducting strands and one OFHC copper strand are cabled together to become a triplet in the first cabling stage. The cabling pitch of the TF and PF conductors are 40-73-157-227-355 mm and 40-80-145-237 mm, respectively. At the final stage of cable fabrication, the cable is wrapped with a thin stainless-steel strip, 30 mm wide and 0.05 mm thick, with 20 % overlap at each side. The selection of the jacket material is also important factor deciding the Nb3Sn conductor performance, because the operating current limit is decreased according to the strain applied to the conductor and the strain is mainly come from the difference in coefficient of the thermal expansion (COE) between superconductor and jacket material. In KSTAR, Incoloy 908 is selected as the jacket material for Nb3Sn conductor.

Fig. 4. The picture of the KSTAR superconductor. Cross sectional view of the Nb3Sn superconductor (a) and CICC (b).

The length of the conductor is also an important factor in the magnet design. If a longer CICC is available then the number of internal joint could be reduced or excluded. It could guarantee more stable operation of the magnet because lots of troubles come from the joints. In KSTAR, tube-mill process was developed to produce long CICC with various lengths up to 1.8 km to exclude the internal joint. The tube-mill process consists of forming, welding, sizing and squaring procedures. A strip is wrapped around the superconducting cable through a series of progressive roller dies and welded using gas tungsten arc welding (GTAW). The welded sheath is cooled immediately by water and the face-bead of the weld is ground by a bead grinding machine. The shape of the welding back-bead is controlled to be smooth not to damage the superconducting cable. Then, the conductors are formed to the final dimension of the CICC.

The leak tightness is an important qualification of the conductor, because the helium leak from the conductor could hinder the cooling the magnet to the cryogenic temperature due to the increased convective heat transfer. The CICC fabricated by tube-mill process have more possibility of the leak due to the longitudinal welding line. Leak detection of CICC spool was performed in the water chamber by pressurizing the CICC spool with helium at about 50 bars as shown in Fig. 5(a). The locations of the leak were identified by detecting bubbles generated in the water chamber. After machining the leak point with a special grinder,

Superconductor Application to the Magnetic

Superconductor

Peak field in conductor [T] Operating current [kA] Stored magnetic energy [Mj] Centering force [MN] No. of turns per coil No. of pancakes

No. of turns per pancake Conductor length per coil [m] No. of cooling channels Overall height [m] Overall width [m]

Table 2. Major parameters of KSTAR TF coils.

Conduit No. of coils Toroidal field [T]

Fusion Devices for the Steady-State Plasma Confinement Achievement 367

Fig. 6. The elevation view of the KSTAR device and the location of the magnets.

**Parameters TF coils** 

Nb3Sn Incoloy 908 16 3.5 7.2 35.2 470 15 56 8 7 610 4 4.2 3.0

repairing welding was performed carefully with high purity helium supplying inside the CICC as shown in Fig. 5(b). Leak tightness of each CICC spool was guaranteed by repeating the leak detection and repairing process (Lim, B.S. et al., 2005).

Fig. 5. Picture of qualification of the KSTAR CICC. The leak detection of the CICC spool in the water chamber by pressurizing the conductor with helium gas (a) and grinding the leak point for the repair welding (b).

## **3. Superconducting magnet fabrication and qualification test**

## **3.1 Fabrication procedure of superconducting magnet**

The superconducting magnet system of the KSTAR device consists of 16 TF coils and 14 PF coils as shown in Fig. 6. The design parameters of the TF coils are listed in table 2. The TF magnet system provides a magnetic field of 3.5 T at the plasma centre, with a peak flux density at the TF coils of 7.2 T and the stored energy of 470 MJ. The nominal current of the TF coils is 35.2 kA with all the coils connected in a series. The total cold mass of the TF magnet system is about 150 tons. The coolant used in TF coils is supercritical helium with an inlet temperature of 4.5K and an inlet pressure of 5 bar. There are four cooling channels per TF coil and the design value of the total helium mass flow rate in the 16 TF coils is 300 g/s.

The design parameters of the CS and PF coils are listed in table 3. The PF magnet system, which consists of 8 coils in the central solenoid coil system (PF1–4) and 6 outer PF coils (PF5–7), sustains the plasma current of 2 MA for 20 s, inductively (Kim, K. et al., 2005). The designed peak currents are 25 kA and 20 kA for the Nb3Sn conductor and the NbTi conductor, respectively. The upper and lower coils of PF1, PF2 and PF7 are connected in a series inside the cryostat and the other coils can be operated separately for single–null and double-null configurations. The CS coils are segmented into four pairs of solenoid coils with different number of turns for the string shaping achievement. The total helium mass flow rate in CS and PF coils is about 300 g/s.

The coil fabrication procedure is as follows: (i) coil winding with numerical dimension control and zirconia bead grit blasting, (ii) attachment of the He feed-throughs and joint terminations, (iii) heat treatment for A15 reaction of Nb3Sn superconductor, (iv) insulation taping and ground wrapping, (v) vacuum pressure impregnation (VPI), (vi) encasing in magnet structure, and (vii) test and delivery.

repairing welding was performed carefully with high purity helium supplying inside the CICC as shown in Fig. 5(b). Leak tightness of each CICC spool was guaranteed by repeating

(a) (b) Fig. 5. Picture of qualification of the KSTAR CICC. The leak detection of the CICC spool in the water chamber by pressurizing the conductor with helium gas (a) and grinding the leak

The superconducting magnet system of the KSTAR device consists of 16 TF coils and 14 PF coils as shown in Fig. 6. The design parameters of the TF coils are listed in table 2. The TF magnet system provides a magnetic field of 3.5 T at the plasma centre, with a peak flux density at the TF coils of 7.2 T and the stored energy of 470 MJ. The nominal current of the TF coils is 35.2 kA with all the coils connected in a series. The total cold mass of the TF magnet system is about 150 tons. The coolant used in TF coils is supercritical helium with an inlet temperature of 4.5K and an inlet pressure of 5 bar. There are four cooling channels per TF coil and the design value of the total helium mass flow rate in the 16 TF coils is 300 g/s. The design parameters of the CS and PF coils are listed in table 3. The PF magnet system, which consists of 8 coils in the central solenoid coil system (PF1–4) and 6 outer PF coils (PF5–7), sustains the plasma current of 2 MA for 20 s, inductively (Kim, K. et al., 2005). The designed peak currents are 25 kA and 20 kA for the Nb3Sn conductor and the NbTi conductor, respectively. The upper and lower coils of PF1, PF2 and PF7 are connected in a series inside the cryostat and the other coils can be operated separately for single–null and double-null configurations. The CS coils are segmented into four pairs of solenoid coils with different number of turns for the string shaping achievement. The total helium mass flow

The coil fabrication procedure is as follows: (i) coil winding with numerical dimension control and zirconia bead grit blasting, (ii) attachment of the He feed-throughs and joint terminations, (iii) heat treatment for A15 reaction of Nb3Sn superconductor, (iv) insulation taping and ground wrapping, (v) vacuum pressure impregnation (VPI), (vi) encasing in

**3. Superconducting magnet fabrication and qualification test** 

**3.1 Fabrication procedure of superconducting magnet** 

the leak detection and repairing process (Lim, B.S. et al., 2005).

point for the repair welding (b).

rate in CS and PF coils is about 300 g/s.

magnet structure, and (vii) test and delivery.

Fig. 6. The elevation view of the KSTAR device and the location of the magnets.


Table 2. Major parameters of KSTAR TF coils.

Superconductor Application to the Magnetic

in a molding frame.

**3.2 Qualification test of superconducting magnet** 

Fusion Devices for the Steady-State Plasma Confinement Achievement 369

460 0C for 100 hours to remove oxygen and oil contaminants from the cable, second plateau at 570 0C for 200 hours to enhance the diffusion of Sn to Nb filament and 660 0C for 240 hours for the A15 reaction of Nb3Sn. An Argon gas purging system is being operated during the baking process to prevent the Stress Accelerated Grain Boundary Oxidation (SAGBO) of Incoloy 908 and the oxygen content is maintained below 0.1 ppm (Kim, K. et al., 2006).

After the heat treatment, each turn of the coil is individually separated and the CICC is insulated with 50 % overlapped layers of Kapton and S2-glass tapes. S2-glass roving filler is applied at the corner of CICC to minimize the resin rich area. G10 pieces, which are shaped to fill the empty space of layer transition area, are also inserted and the coil bundle is ground insulation wrapped using S2-glass tape. As a quench detection sensor of PF1-5 coils, an externally co-wound voltage tap, which is made of a conducting polymer tape, is wound along the CICC during the turn insulation process. The co-wound voltage tap has been developed to minimize the induced voltage in quench detection line by cancellation of selfinductance. Finally vacuum pressure impregnation (VPI) is performed after placing the coil

After completion of VPI, acceptance test of each coil is followed including visual and dimensional checks, Jc value, helium flow balance and high voltage insulation test (HIPOT). In the visual check and dimensional inspection, any defect or dimensional deformation is checked as shown in Fig. 8. The results show that all coils are fabricated within the allowable dimensional tolerance of ±2 mm. To qualify the heat treatment of the Nb3Sn coil, several standard sample strands are located around each coil in vacuum furnace during the heat treatment and the critical current density (Jc) of the strand samples is measured to evaluate the Nb3Sn magnet performance. This qualification method is very effectively compared to the full current test at cryogenic temperature. Most of the sample showed the current density is over the design criteria of 750 A/mm2 at 12 T, 4.2 K as shown in Fig. 9. Pressure drop of each channel is measured to assess the flow unbalance among channels during operation. From the measurement, the TF07 coil has the largest flow imbalance among channels, which was about

(a) (b)

Fig. 8. Visual and dimensional inspection of the TF coil (a) and PF coils (b).

20%. The averaged values are within approximately 10% (Park, K.R. et al., 2009).


Table 3. Major parameters of KSTAR PF coils.

The continuous winding scheme is developed to exclude the internal joints. In other words, each coil is wound using a single CICC except PF6U and PF6L coils, which have one internal joint for each coil. Helium feed-throughs are attached per each double pancake. The winding stations for the TF and PF coils are shown in Fig. 7. The TF and PF1-5 coils use Nb3Sn strand and require the reaction heat treatment process. Since PF6 and PF7 coils use NbTi CICC, which does not require the reaction heat treatment process, the helium feedthrough attachment and Kapton and S2-glass insulation taping are carried out during the winding process (Park, K.R. et al., 2005).

Fig. 7. The picture of coil winding stations of the TF coil (a) and PF coil (b).

After the winding, the coils are placed in a structure for the heat treatment and the preparation for heat treatment including the installation of magnet lead holding fixtures, the attachment of feed-throughs and the preparation of joint termination is carried out. In order to minimize the deformation of the winding pack during the heat treatment process, 3 dimensionally machined stainless steel pieces are also installed to the empty space of layer transition area. Heat treatment is conducted in three steps of temperature: first plateau at

PF1 PF2 PF3 PF4 PF5 PF6 PF7

**Parameters CS coils PF coils** 

Number of coils

Radial position, R [mm] Axial position, Z [mm] Coil width, dR [mm] Coil height, dZ [mm] No. turns per coil No. pancakes

No. turns per pancake CICC length /coil [m] No. cooling channels Weight per coil [ton]

Table 3. Major parameters of KSTAR PF coils.

winding process (Park, K.R. et al., 2005).

Superconductor Nb3Sn Nb3Sn NbTi Conduit Incoloy908 Incoloy908 STS316LN

The continuous winding scheme is developed to exclude the internal joints. In other words, each coil is wound using a single CICC except PF6U and PF6L coils, which have one internal joint for each coil. Helium feed-throughs are attached per each double pancake. The winding stations for the TF and PF coils are shown in Fig. 7. The TF and PF1-5 coils use Nb3Sn strand and require the reaction heat treatment process. Since PF6 and PF7 coils use NbTi CICC, which does not require the reaction heat treatment process, the helium feedthrough attachment and Kapton and S2-glass insulation taping are carried out during the

(a) (b)

After the winding, the coils are placed in a structure for the heat treatment and the preparation for heat treatment including the installation of magnet lead holding fixtures, the attachment of feed-throughs and the preparation of joint termination is carried out. In order to minimize the deformation of the winding pack during the heat treatment process, 3 dimensionally machined stainless steel pieces are also installed to the empty space of layer transition area. Heat treatment is conducted in three steps of temperature: first plateau at

Fig. 7. The picture of coil winding stations of the TF coil (a) and PF coil (b).

460 0C for 100 hours to remove oxygen and oil contaminants from the cable, second plateau at 570 0C for 200 hours to enhance the diffusion of Sn to Nb filament and 660 0C for 240 hours for the A15 reaction of Nb3Sn. An Argon gas purging system is being operated during the baking process to prevent the Stress Accelerated Grain Boundary Oxidation (SAGBO) of Incoloy 908 and the oxygen content is maintained below 0.1 ppm (Kim, K. et al., 2006).

After the heat treatment, each turn of the coil is individually separated and the CICC is insulated with 50 % overlapped layers of Kapton and S2-glass tapes. S2-glass roving filler is applied at the corner of CICC to minimize the resin rich area. G10 pieces, which are shaped to fill the empty space of layer transition area, are also inserted and the coil bundle is ground insulation wrapped using S2-glass tape. As a quench detection sensor of PF1-5 coils, an externally co-wound voltage tap, which is made of a conducting polymer tape, is wound along the CICC during the turn insulation process. The co-wound voltage tap has been developed to minimize the induced voltage in quench detection line by cancellation of selfinductance. Finally vacuum pressure impregnation (VPI) is performed after placing the coil in a molding frame.

## **3.2 Qualification test of superconducting magnet**

After completion of VPI, acceptance test of each coil is followed including visual and dimensional checks, Jc value, helium flow balance and high voltage insulation test (HIPOT). In the visual check and dimensional inspection, any defect or dimensional deformation is checked as shown in Fig. 8. The results show that all coils are fabricated within the allowable dimensional tolerance of ±2 mm. To qualify the heat treatment of the Nb3Sn coil, several standard sample strands are located around each coil in vacuum furnace during the heat treatment and the critical current density (Jc) of the strand samples is measured to evaluate the Nb3Sn magnet performance. This qualification method is very effectively compared to the full current test at cryogenic temperature. Most of the sample showed the current density is over the design criteria of 750 A/mm2 at 12 T, 4.2 K as shown in Fig. 9. Pressure drop of each channel is measured to assess the flow unbalance among channels during operation. From the measurement, the TF07 coil has the largest flow imbalance among channels, which was about 20%. The averaged values are within approximately 10% (Park, K.R. et al., 2009).

Fig. 8. Visual and dimensional inspection of the TF coil (a) and PF coils (b).

Superconductor Application to the Magnetic

2004).

Fusion Devices for the Steady-State Plasma Confinement Achievement 371

operating condition of the KSTAR CS coil at 25 kA. The ac loss measurement was conducted with several cycles of triangular current waveform and with a long pulse sinusoidal waveform with offset to exclude nonlinear control error of the power supply at low current. However, the prototype AC power supply showed a logical error during the crossing of zero current and the AC test was performed with an offset current, which prevents the control failure of the AC power supply near zero current. The coupling time constant, nτ, which was measured by changing the frequency and the amplitude of sinusoidal current as shown in Fig. 10 (b) (Oh, Y.K. et al. 2006, & Lee, S. et al., 2006). It was less than 30 ms, which is approximately half of the design specification of 60 ms. The result of lower coupling loss time constant have a positive expectation of the KSTAR CS coil operation and it could guarantee the fast field variance required for the plasma startup and feedback control of the KSTAR CS magnet during plasma operation. The CS model coil could be applied for the sample conductor test of ITER or for conductor R&D for the fusion reactor (Oh, D.K. et al.,

(a) (b) Fig. 10. CS model coil performance test. Coil installation in the vacuum cryostat (a) and

The KSTAR magnet structure consists of 16 TF structures, one CS structure, and 80 PF structures. Most of structures are made of strengthened authentic stainless steel STS316LN. Key design factors are mechanical stiffness under the large amount of Lorentz forces, electric insulation along the toroidal direction to prevent eddy current by the plasma startup and disruption, and effective cooling schemes. TF coils are under large amount of in-plane force up to 15 MN when TF coil is charged at the nominal operating current of 35.2. And there are also out-of-plane forces up to 2.6 MN/m due to by the CS, PF, and the plasma current (Choi, C.H. et al., 2002). To sustain the electromagnetic load on the TF coils, TF structure is designed to have a wedged shape at inboard leg and to locate inter-coil structures which are strongly connected to TF case outboard legs and containing shear keys and bolts. The shape of the inter-coil structure is optimized to reduce the peak stress within allowable according to the analysis (Ahn, H.J. et al., 2002). To prevent eddy current into the toroidal direction at the plasma startup or disruptions, insulation spacer made of G-10 is attached on the entire surface

Sinusoidal current waveform for ac loss measurement (b).

**4. Assembly of the magnets and interfaces** 

**4.1 TF Magnet structure and assembly** 

Fig. 9. The measured Jc values of the all Nb3Sn coil after heat treatment.

The HIPOT tests were carried out up to DC 15 kV and AC 10 kV@60 Hz to ensure the insulation integrity of the coils after the VPI. All coils satisfied insulation requirement of DC insulation resistance above 100 M for the ground insulation. The impulse test was carried out to check the turn-to-turn insulation by applying the pulsed 2.4 kV at a double pancake in a coil. The measured voltage waveforms of all pancakes were compared with each other because the waveform difference is a sign of the turn-to-turn insulation breakage. There was no observed turn-to-turn insulation failure.

Performance test of the full size TF prototype coil, TF00, and a pair of CS model coil were carried at cryogenic temperature in the test facility as shown in Fig. 10(a). The major objective of the test is to confirm the validity of the design and the fabrication process. TF00 was cooled down in 10 days within a maximum temperature variation of 50 K. The residual resistivity ratio (RRR) of the coil was measured to be over 200. The superconducting phase transition of TF00 occurred around 18 K. TF00 was repeatedly charged and discharged with various scenarios. The coil was operated well without any quench up to 30 kA (Oh, Y.K. et al., 2004). The result of repeated current charge tests shows that TF00 is fabricated robustly. But the test was interrupted by the arc occurrence on one of joint due to weakness of the busline supporters and movement of the busline. We could learn from the TF00 experience that the supporting structures of the magnet system should be robust enough to sustain any kinds of force and movement. A pair of CS model coils, which has been designed for the background coil using the same conductor as KSTAR CS coil for the conductor test at 8T with a ramp rate of 3T/s, has been installed in the test facility. The supporting structure of the main background magnetic field coil system was made of glass-fiber reinforced plastic (GFRP) material for the reduction of the eddy current during pulsed operations. CS model coils were cooled down to 5 K in 9 days. However, the temperature of the G10 coil support structure was approximately 30 K due to the low thermal conductivity. The total helium flow rate was ~45 g/s and the pressure drops ~1 bar. The coil was successfully excited to 20 kA, where the peak field is 8.62 T, using the DC power supply for the KSTAR TF coil. In the view point of temperature margin, the operating condition of the coil at 20 kA is a similar

Fig. 9. The measured Jc values of the all Nb3Sn coil after heat treatment.

no observed turn-to-turn insulation failure.

The HIPOT tests were carried out up to DC 15 kV and AC 10 kV@60 Hz to ensure the insulation integrity of the coils after the VPI. All coils satisfied insulation requirement of DC insulation resistance above 100 M for the ground insulation. The impulse test was carried out to check the turn-to-turn insulation by applying the pulsed 2.4 kV at a double pancake in a coil. The measured voltage waveforms of all pancakes were compared with each other because the waveform difference is a sign of the turn-to-turn insulation breakage. There was

Performance test of the full size TF prototype coil, TF00, and a pair of CS model coil were carried at cryogenic temperature in the test facility as shown in Fig. 10(a). The major objective of the test is to confirm the validity of the design and the fabrication process. TF00 was cooled down in 10 days within a maximum temperature variation of 50 K. The residual resistivity ratio (RRR) of the coil was measured to be over 200. The superconducting phase transition of TF00 occurred around 18 K. TF00 was repeatedly charged and discharged with various scenarios. The coil was operated well without any quench up to 30 kA (Oh, Y.K. et al., 2004). The result of repeated current charge tests shows that TF00 is fabricated robustly. But the test was interrupted by the arc occurrence on one of joint due to weakness of the busline supporters and movement of the busline. We could learn from the TF00 experience that the supporting structures of the magnet system should be robust enough to sustain any kinds of force and movement. A pair of CS model coils, which has been designed for the background coil using the same conductor as KSTAR CS coil for the conductor test at 8T with a ramp rate of 3T/s, has been installed in the test facility. The supporting structure of the main background magnetic field coil system was made of glass-fiber reinforced plastic (GFRP) material for the reduction of the eddy current during pulsed operations. CS model coils were cooled down to 5 K in 9 days. However, the temperature of the G10 coil support structure was approximately 30 K due to the low thermal conductivity. The total helium flow rate was ~45 g/s and the pressure drops ~1 bar. The coil was successfully excited to 20 kA, where the peak field is 8.62 T, using the DC power supply for the KSTAR TF coil. In the view point of temperature margin, the operating condition of the coil at 20 kA is a similar operating condition of the KSTAR CS coil at 25 kA. The ac loss measurement was conducted with several cycles of triangular current waveform and with a long pulse sinusoidal waveform with offset to exclude nonlinear control error of the power supply at low current. However, the prototype AC power supply showed a logical error during the crossing of zero current and the AC test was performed with an offset current, which prevents the control failure of the AC power supply near zero current. The coupling time constant, nτ, which was measured by changing the frequency and the amplitude of sinusoidal current as shown in Fig. 10 (b) (Oh, Y.K. et al. 2006, & Lee, S. et al., 2006). It was less than 30 ms, which is approximately half of the design specification of 60 ms. The result of lower coupling loss time constant have a positive expectation of the KSTAR CS coil operation and it could guarantee the fast field variance required for the plasma startup and feedback control of the KSTAR CS magnet during plasma operation. The CS model coil could be applied for the sample conductor test of ITER or for conductor R&D for the fusion reactor (Oh, D.K. et al., 2004).

Fig. 10. CS model coil performance test. Coil installation in the vacuum cryostat (a) and Sinusoidal current waveform for ac loss measurement (b).

## **4. Assembly of the magnets and interfaces**

## **4.1 TF Magnet structure and assembly**

The KSTAR magnet structure consists of 16 TF structures, one CS structure, and 80 PF structures. Most of structures are made of strengthened authentic stainless steel STS316LN. Key design factors are mechanical stiffness under the large amount of Lorentz forces, electric insulation along the toroidal direction to prevent eddy current by the plasma startup and disruption, and effective cooling schemes. TF coils are under large amount of in-plane force up to 15 MN when TF coil is charged at the nominal operating current of 35.2. And there are also out-of-plane forces up to 2.6 MN/m due to by the CS, PF, and the plasma current (Choi, C.H. et al., 2002). To sustain the electromagnetic load on the TF coils, TF structure is designed to have a wedged shape at inboard leg and to locate inter-coil structures which are strongly connected to TF case outboard legs and containing shear keys and bolts. The shape of the inter-coil structure is optimized to reduce the peak stress within allowable according to the analysis (Ahn, H.J. et al., 2002). To prevent eddy current into the toroidal direction at the plasma startup or disruptions, insulation spacer made of G-10 is attached on the entire surface

Superconductor Application to the Magnetic

**4.2 CS/PF magnet structure and assembly** 

TF structure shoulder as shown in Fig. 13(a).

design.

Fusion Devices for the Steady-State Plasma Confinement Achievement 373

The major function of the CS structure is to apply a vertical compression force on the CS coil assembly to prevent a relative motion of CS coils and to sustain the repulsive forces between CS coils during operation (Oh, Y.K. et al., 2002). The CS coils are under compressive force during the reference scenario. However, the peak separation force of 12 MN occurs between PF1 and PF2 coils under several operation conditions, and it is called as maximum repulsive force (MRF) condition (Choi, C.H. et al., 2002; Sa, J.W. et al., 2002). The designed axial compression force is about 15 MN at 5 K. It will be applied partly by assembling the precompression structure at room temperature up to 13.4 MN and partly by the thermal contraction difference between CS coils and pre-compression structure during cool down. During the subassembly of the CS magnet structire, the pre-compress structures are heated to apply the pre-compression at room temperature. When heating applied on the structure, the temperature of the epoxy-rich area of the coil also has possibility of damage due to overheat. So, the temperature of the structure was limited at 140 0C, and the precompression on CS coils are estimated to be about 8 MN at room temperature. The sub assembled CS magnet assembly is inserted into the center of KSTAR device and hang on the

The PF coil structures to place the 6 PF coils on the TF coil with a vertically symmetry to the machine mid-plane at 80 locations. All PF coil structures should absorb the difference in the thermal contraction between TF coil structures and PF coils during cool-down to cryogenic temperature, and endure the vertical and radial magnetic forces due to current charging. In order to satisfy these design requirements, the hinges or flexible plates are applied in the

(a) (b) Fig. 13. Pictures of the KSTAR magnet system. The installation of the CS coil assembly into

the center of the KSTAR device (a) and the completion of magnet assembly (b).

at the gap between TF structure modules. There are 18 cooling tubes embedded inside of the structure around the coil and an additional cooling tube on the TF structure surface to limit the heat influx into the TF coil in case of the hot spot generation in the TF structure due to plasma disruption or PF coil quench. The major fabrication procedure of the TF structures are (i) fabrication of C-shaped coil case welded with inter-coil structure, (ii) fabrication of flat cover plate welded with inter-coil structure, (iii) coil encasing, (iv) final enclosure welding, (v) second vacuum pressure impregnation, (vi) final machining and delivery, and (vii) toroidal insulation attachment (Bak, J.S. et al., 2006). Figure 11 shows the final machining of TF coil case after the coil encasing. A special assembly tools has been developed to locate each magnet at the exact position. Each TF magnet is launched carefully into the 22.5 degree gap of the vacuum vessel using the loading vehicle. It is rotated in the toroidal direction and is aligned at the final position within installation error less than 1 mm as shown in Fig. 12 (Yang, H.L. et al., 2006). To compensate the thermal contraction of the TF magnet structure, whole TF structure was installed with a vertical offset of 5 mm at room temperature.

Fig. 11. TF magnet structure fabrication and assembly. Final machining after coil encasing (a) and the schematic drawing of a TF magnet installation into torus of the vacuum vessel (b).

Fig. 12. TF magnet assembly qualification. Completion of whole TF magnet assembly (a) and installation error of TF magnet within 1 mm in the parallel shift (b).

## **4.2 CS/PF magnet structure and assembly**

372 Superconductors – Properties, Technology, and Applications

at the gap between TF structure modules. There are 18 cooling tubes embedded inside of the structure around the coil and an additional cooling tube on the TF structure surface to limit the heat influx into the TF coil in case of the hot spot generation in the TF structure due to plasma disruption or PF coil quench. The major fabrication procedure of the TF structures are (i) fabrication of C-shaped coil case welded with inter-coil structure, (ii) fabrication of flat cover plate welded with inter-coil structure, (iii) coil encasing, (iv) final enclosure welding, (v) second vacuum pressure impregnation, (vi) final machining and delivery, and (vii) toroidal insulation attachment (Bak, J.S. et al., 2006). Figure 11 shows the final machining of TF coil case after the coil encasing. A special assembly tools has been developed to locate each magnet at the exact position. Each TF magnet is launched carefully into the 22.5 degree gap of the vacuum vessel using the loading vehicle. It is rotated in the toroidal direction and is aligned at the final position within installation error less than 1 mm as shown in Fig. 12 (Yang, H.L. et al., 2006). To compensate the thermal contraction of the TF magnet structure, whole TF structure

(a) (b) Fig. 11. TF magnet structure fabrication and assembly. Final machining after coil encasing (a) and the schematic drawing of a TF magnet installation into torus of the vacuum vessel (b).

(a) (b)

Fig. 12. TF magnet assembly qualification. Completion of whole TF magnet assembly (a) and

installation error of TF magnet within 1 mm in the parallel shift (b).

was installed with a vertical offset of 5 mm at room temperature.

The major function of the CS structure is to apply a vertical compression force on the CS coil assembly to prevent a relative motion of CS coils and to sustain the repulsive forces between CS coils during operation (Oh, Y.K. et al., 2002). The CS coils are under compressive force during the reference scenario. However, the peak separation force of 12 MN occurs between PF1 and PF2 coils under several operation conditions, and it is called as maximum repulsive force (MRF) condition (Choi, C.H. et al., 2002; Sa, J.W. et al., 2002). The designed axial compression force is about 15 MN at 5 K. It will be applied partly by assembling the precompression structure at room temperature up to 13.4 MN and partly by the thermal contraction difference between CS coils and pre-compression structure during cool down. During the subassembly of the CS magnet structire, the pre-compress structures are heated to apply the pre-compression at room temperature. When heating applied on the structure, the temperature of the epoxy-rich area of the coil also has possibility of damage due to overheat. So, the temperature of the structure was limited at 140 0C, and the precompression on CS coils are estimated to be about 8 MN at room temperature. The sub assembled CS magnet assembly is inserted into the center of KSTAR device and hang on the TF structure shoulder as shown in Fig. 13(a).

The PF coil structures to place the 6 PF coils on the TF coil with a vertically symmetry to the machine mid-plane at 80 locations. All PF coil structures should absorb the difference in the thermal contraction between TF coil structures and PF coils during cool-down to cryogenic temperature, and endure the vertical and radial magnetic forces due to current charging. In order to satisfy these design requirements, the hinges or flexible plates are applied in the design.

Fig. 13. Pictures of the KSTAR magnet system. The installation of the CS coil assembly into the center of the KSTAR device (a) and the completion of magnet assembly (b).

Superconductor Application to the Magnetic

into external dump resistors.

Fusion Devices for the Steady-State Plasma Confinement Achievement 375

from the voltage spike to trigger the interlock. As the TF QDS holding time is set to be 2 s, the voltage noise due to blip operation was therefore not a problem The PF current abrupt change which is called as the blip operation during the short time span is necessary for making the loop voltages in the toroidal direction and initiating the plasma current. Hence, the noises of the quench detection system should be carefully analyzed and compensated below the proper levels. Two quench detectors were used for detecting the resistive voltage drop across the winding. The primary one uses the co-wound voltage sensing cable, which traces the surface of the CICC closely. If the magnetic coupling The PF current abrupt change which is called as the blip operation during the short time span is necessary for making the loop voltages in the toroidal direction and initiating the plasma current. Hence, the noises of the quench detection system should be carefully analyzed and compensated below the proper levels. Two quench detectors were used for detecting the resistive voltage drop across the winding. The primary one uses the cowound voltage sensing cable, which traces the surface of the CICC closely. If the magnetic coupling between the coil and the co-wound tape is sufficiently strong, the measured voltage at the co-wound voltage sensor can eliminate the most part of the inductive voltages in a coil. A conductive fabric tape was selected as a co-wound voltage sensing cable. The secondary one is the most commonly used method as shown. However the inductive voltage is detected in both conventional and co-wound sensors due to the field profile in each coil. By adding the differential comparator to subtract unbalanced signals in PF coils, most of the inductive voltage signal could be expelled as shown in the Fig. 15. Beside of the noise handling scheme, the quench detection holding time is introduced. That is, if the measured voltage stays above the threshold voltage over the predefined holding time, then the quench interlock is activated and the magnet energies are dumped

Fig. 14. Process flow diagram of the 9 kW cryo-facility to cool the KSTAR magnet system.

## **4.3 Magnet interfaces**

Magnet interface components are to supply electric power and cryogen and to monitor the operational status. Figure 13(b) shows the whole magnet system after installation of whole magnet system and its interface components.

SC buslines, which is components to carry the current from the cryostat boundary to the coil terminal joints, are made of NbTi CICC with circular cross section to allow bending in any angle. Each busline was bended before installation according to the 3-dimensional assembly drawing and was insulated by wrapping with pre-pregnant glass–fiber and Kapton film. Two types of joints are installed between magnet and buslines, one is lap type joints which are installed at 136 locations and the other type is strand-to-strand joint which are installed at TF coil only (Park, Y.M. et al., 2009).

Current lead system is an interface component to deliver the large current from the power supply at room temperature to the superconducting magnet at the cryogenic temperature. KSTAR requires 7 pairs of PF leads having the current carrying capacity up to 25 kA for more than 350 s and 2 pairs of TF leads having the current capacity over 17.5 kA for steady state. Usually the heat load on the current lead is major portion of the heat load on the magnet system. So there are lots of researches to reduce the heat load of the current lead. In KSTAR, heat load reduction was achieved by replacing the copper conductor in the current lead material with brass conductor due to the low thermal conduction.

For the cooling of the magnet system and its interface, a cryogenic facility was constructed with a cooling capacity of 9 kW at 4.5 K equivalent. The cryogenic facility consists of gas management and compress station at room temperature, cold box to cool-down the cryogen, and helium distribution box to match the heat load in KSTAR. Figure 14 shows the process flow diagram of the KSTAR cryo-facility. The cryogenic helium is supplied to the magnet system in three phases. Supercritical helium is supplied to the all SC coils and magnet structures at 4.5 K, 5.5 bar with mass flow rate of 600 g/s, liquid helium into current lead at 4.2 K, 1.3 bar, and gaseous helium into thermal shield at about 55 K, 18 bar. (Chang, H.S. et al., 2008; Kim, Y.S. et al., 2009). A thermal damper system to moderate the pulsed heat from PF magnet consists of several heat exchangers immersed in the liquid helium bath and several supercritical helium circulators.

In the magnet operation with large stored energy, normal regions arising in the superconducting coil can lead to permanent damage due to excessive heating. Reliable quench detection and protection is very important for the safe operation of the superconducting magnets. A total of 83 quench voltage detectors are installed and operated independently to detect quenches in the TF and PF coil systems (Chu, Y. et al., 2009). The quench voltage detection system of the KSTAR TF coils uses a conventional balanced bridge method detecting and comparing the voltage signal across each two TF coils with balanced inductance. For the fail-safe operation and to anticipate the malfunction of the detector, two detectors monitor the same section constantly in a simultaneous manner. Most of the self-inductance of each module could be compensated easily. There is some amount of non-linear voltage at low current due to Incoloy908 ferromagnetic material in the Nb3Sn superconductors. When PF coil is operated together, the induced voltage is detected for about 100ms due to the coupling of the inductance between TF and PF coils. Generally, the holding time was set to prevent the false signal

Magnet interface components are to supply electric power and cryogen and to monitor the operational status. Figure 13(b) shows the whole magnet system after installation of whole

SC buslines, which is components to carry the current from the cryostat boundary to the coil terminal joints, are made of NbTi CICC with circular cross section to allow bending in any angle. Each busline was bended before installation according to the 3-dimensional assembly drawing and was insulated by wrapping with pre-pregnant glass–fiber and Kapton film. Two types of joints are installed between magnet and buslines, one is lap type joints which are installed at 136 locations and the other type is strand-to-strand joint which are installed

Current lead system is an interface component to deliver the large current from the power supply at room temperature to the superconducting magnet at the cryogenic temperature. KSTAR requires 7 pairs of PF leads having the current carrying capacity up to 25 kA for more than 350 s and 2 pairs of TF leads having the current capacity over 17.5 kA for steady state. Usually the heat load on the current lead is major portion of the heat load on the magnet system. So there are lots of researches to reduce the heat load of the current lead. In KSTAR, heat load reduction was achieved by replacing the copper conductor in the current

For the cooling of the magnet system and its interface, a cryogenic facility was constructed with a cooling capacity of 9 kW at 4.5 K equivalent. The cryogenic facility consists of gas management and compress station at room temperature, cold box to cool-down the cryogen, and helium distribution box to match the heat load in KSTAR. Figure 14 shows the process flow diagram of the KSTAR cryo-facility. The cryogenic helium is supplied to the magnet system in three phases. Supercritical helium is supplied to the all SC coils and magnet structures at 4.5 K, 5.5 bar with mass flow rate of 600 g/s, liquid helium into current lead at 4.2 K, 1.3 bar, and gaseous helium into thermal shield at about 55 K, 18 bar. (Chang, H.S. et al., 2008; Kim, Y.S. et al., 2009). A thermal damper system to moderate the pulsed heat from PF magnet consists of several heat exchangers immersed in the liquid helium bath and

In the magnet operation with large stored energy, normal regions arising in the superconducting coil can lead to permanent damage due to excessive heating. Reliable quench detection and protection is very important for the safe operation of the superconducting magnets. A total of 83 quench voltage detectors are installed and operated independently to detect quenches in the TF and PF coil systems (Chu, Y. et al., 2009). The quench voltage detection system of the KSTAR TF coils uses a conventional balanced bridge method detecting and comparing the voltage signal across each two TF coils with balanced inductance. For the fail-safe operation and to anticipate the malfunction of the detector, two detectors monitor the same section constantly in a simultaneous manner. Most of the self-inductance of each module could be compensated easily. There is some amount of non-linear voltage at low current due to Incoloy908 ferromagnetic material in the Nb3Sn superconductors. When PF coil is operated together, the induced voltage is detected for about 100ms due to the coupling of the inductance between TF and PF coils. Generally, the holding time was set to prevent the false signal

lead material with brass conductor due to the low thermal conduction.

**4.3 Magnet interfaces** 

magnet system and its interface components.

at TF coil only (Park, Y.M. et al., 2009).

several supercritical helium circulators.

from the voltage spike to trigger the interlock. As the TF QDS holding time is set to be 2 s, the voltage noise due to blip operation was therefore not a problem The PF current abrupt change which is called as the blip operation during the short time span is necessary for making the loop voltages in the toroidal direction and initiating the plasma current. Hence, the noises of the quench detection system should be carefully analyzed and compensated below the proper levels. Two quench detectors were used for detecting the resistive voltage drop across the winding. The primary one uses the co-wound voltage sensing cable, which traces the surface of the CICC closely. If the magnetic coupling The PF current abrupt change which is called as the blip operation during the short time span is necessary for making the loop voltages in the toroidal direction and initiating the plasma current. Hence, the noises of the quench detection system should be carefully analyzed and compensated below the proper levels. Two quench detectors were used for detecting the resistive voltage drop across the winding. The primary one uses the cowound voltage sensing cable, which traces the surface of the CICC closely. If the magnetic coupling between the coil and the co-wound tape is sufficiently strong, the measured voltage at the co-wound voltage sensor can eliminate the most part of the inductive voltages in a coil. A conductive fabric tape was selected as a co-wound voltage sensing cable. The secondary one is the most commonly used method as shown. However the inductive voltage is detected in both conventional and co-wound sensors due to the field profile in each coil. By adding the differential comparator to subtract unbalanced signals in PF coils, most of the inductive voltage signal could be expelled as shown in the Fig. 15. Beside of the noise handling scheme, the quench detection holding time is introduced. That is, if the measured voltage stays above the threshold voltage over the predefined holding time, then the quench interlock is activated and the magnet energies are dumped into external dump resistors.

Fig. 14. Process flow diagram of the 9 kW cryo-facility to cool the KSTAR magnet system.

Superconductor Application to the Magnetic

operation sequence of the plasma experiments (b).

**5.1 Inspection at room temperature** 

coil (Park, K.R. et al., 2009).

Fusion Devices for the Steady-State Plasma Confinement Achievement 377

(a) (b)

After completion of the KSTAR device assembly, a detailed inspection of each component was followed before the cooling down of the magnet system. The major inspection activities at room temperature are leak detection, electric insulation check, and magnetic field survey. The leak detection is one of the critical activities in the superconducting magnet system. Although most of gas except helium are condensed on the magnet surface after the magnet cool-down, so the leak from the helium circulation line should be identified and repaired for the stable magnet cool-down and long-time operation for several month. The careful inspection of the leak from the magnet cooling lines is conducted by pressurizing the magnet cooling line with helium up to about 20 bar. The leak from the magnet system in KSTAR is not so high and could be repaired well. It is due to the tight quality control of each component before its installation in KSTAR. Usually most of the leaks come from the joint or welding point between different materials, and electrical breaker is the components of high leak possibility due to the connection of stainless steel tube to Ceramic insulator or GFRP insulator. So the leak inspection was conducted of ach electric breakers under high pressure helium after the several tens of thermal cycles from room temperature and liquid nitrogen temperature. To ensure the integrity of insulation of the superconducting system after all sub-systems were completely installed, the dc HIPOT test was carried out up to 10 kV in each coil to diagnose the coil insulation before the cool-down. Conclusively, the insulation resistances of all coils were more than 100 M which is the same criterion for the individual

The magnetic field distribution in the vacuum vessel is measured after a low current less than 100 A is applied to each coil at the room temperature environment and compared with the calculated data. When the TF current is 100 A, the B-field at the plasma center is approximately 100 gauss, and the B-field distribution is consistent with the calculated data although there was some discrepancy of the field detection close to the vacuum vessel inboard surface. There was a large discrepancy between measurements and calculations for the PF coils as shown in Fig. 17 (Yonekawa, H. et al., 2009). For instance, PF7 measurements at all the radial positions were approximately 30% lower than the calculated values

Fig. 16. Bird-eye view of the KSTAR device in during the 2010 operation (a), and the

Fig. 15. Quench detection of KSTAR PF magnets. Quench detection circuit of the PF1 by balanced bridge and co-wound voltage tap (a), and comparison of the inductive voltage signals on the quench detection sensors during the regular PF coil operation (shot #5199). Most of the inductive signal is cancelled at Vc3 signal lines. (b)

## **5. Integrated commissioning of the magnet system in fusion device**

The operation of the superconducting fusion device could be classified into two phases, the integrated commissioning phase to inspect the quality of each component and to reduce unexpected risk occurrences and the normal operation phase to achieve the mission of the fusion device by the integrated operation of all system. There were several cases that the occurrence of damage or problem in the superconducting magnet system during its commissioning or initial operation made delay or prohibit the normal operation in superconducting fusion device or particle accelerator. The commissioning and the initial operation for the first plasma in the KSTAR device have been accomplished successfully without any severe failure preventing the device operation and plasma experiments. All the superconducting magnets operated in stable without quench for long-time dc operation and with synchronized pulse operation by the plasma control system . The first plasma target in the range of 100 kA was achieved by assisted startup using an 84GHz electron cyclotron resonance heating (ECH) system and real-time feedback control (Oh, Y.K. et al., 2009). Figure 16 shows KSTAR device and the overall sequence of the KSTAR commissioning and the first plasma operation and the picture of control room during the commissioning. The commissioning of the KSTAR magnet system is classified into several steps: (i) inspection at room temperature after tokamak assembly, (ii) inspection during and after magnet cooldown, and (iii) stability assessment under operation condition of static or pulsed operation of magnet system.

Fig. 16. Bird-eye view of the KSTAR device in during the 2010 operation (a), and the operation sequence of the plasma experiments (b).

## **5.1 Inspection at room temperature**

376 Superconductors – Properties, Technology, and Applications

(a) (b)

The operation of the superconducting fusion device could be classified into two phases, the integrated commissioning phase to inspect the quality of each component and to reduce unexpected risk occurrences and the normal operation phase to achieve the mission of the fusion device by the integrated operation of all system. There were several cases that the occurrence of damage or problem in the superconducting magnet system during its commissioning or initial operation made delay or prohibit the normal operation in superconducting fusion device or particle accelerator. The commissioning and the initial operation for the first plasma in the KSTAR device have been accomplished successfully without any severe failure preventing the device operation and plasma experiments. All the superconducting magnets operated in stable without quench for long-time dc operation and with synchronized pulse operation by the plasma control system . The first plasma target in the range of 100 kA was achieved by assisted startup using an 84GHz electron cyclotron resonance heating (ECH) system and real-time feedback control (Oh, Y.K. et al., 2009). Figure 16 shows KSTAR device and the overall sequence of the KSTAR commissioning and the first plasma operation and the picture of control room during the commissioning. The commissioning of the KSTAR magnet system is classified into several steps: (i) inspection at room temperature after tokamak assembly, (ii) inspection during and after magnet cooldown, and (iii) stability assessment under operation condition of static or pulsed operation

Fig. 15. Quench detection of KSTAR PF magnets. Quench detection circuit of the PF1 by balanced bridge and co-wound voltage tap (a), and comparison of the inductive voltage signals on the quench detection sensors during the regular PF coil operation (shot #5199).

**5. Integrated commissioning of the magnet system in fusion device** 

Most of the inductive signal is cancelled at Vc3 signal lines. (b)

of magnet system.

After completion of the KSTAR device assembly, a detailed inspection of each component was followed before the cooling down of the magnet system. The major inspection activities at room temperature are leak detection, electric insulation check, and magnetic field survey.

The leak detection is one of the critical activities in the superconducting magnet system. Although most of gas except helium are condensed on the magnet surface after the magnet cool-down, so the leak from the helium circulation line should be identified and repaired for the stable magnet cool-down and long-time operation for several month. The careful inspection of the leak from the magnet cooling lines is conducted by pressurizing the magnet cooling line with helium up to about 20 bar. The leak from the magnet system in KSTAR is not so high and could be repaired well. It is due to the tight quality control of each component before its installation in KSTAR. Usually most of the leaks come from the joint or welding point between different materials, and electrical breaker is the components of high leak possibility due to the connection of stainless steel tube to Ceramic insulator or GFRP insulator. So the leak inspection was conducted of ach electric breakers under high pressure helium after the several tens of thermal cycles from room temperature and liquid nitrogen temperature. To ensure the integrity of insulation of the superconducting system after all sub-systems were completely installed, the dc HIPOT test was carried out up to 10 kV in each coil to diagnose the coil insulation before the cool-down. Conclusively, the insulation resistances of all coils were more than 100 M which is the same criterion for the individual coil (Park, K.R. et al., 2009).

The magnetic field distribution in the vacuum vessel is measured after a low current less than 100 A is applied to each coil at the room temperature environment and compared with the calculated data. When the TF current is 100 A, the B-field at the plasma center is approximately 100 gauss, and the B-field distribution is consistent with the calculated data although there was some discrepancy of the field detection close to the vacuum vessel inboard surface. There was a large discrepancy between measurements and calculations for the PF coils as shown in Fig. 17 (Yonekawa, H. et al., 2009). For instance, PF7 measurements at all the radial positions were approximately 30% lower than the calculated values

Superconductor Application to the Magnetic

Fusion Devices for the Steady-State Plasma Confinement Achievement 379

(a) (b)

Fig. 18. The cool down characteristics of the magnet system. The change of overall mass flow

After finishing the cool-down, the HIPOT test was carried out up to 6 kV for the final insulation check. As the operating currents were lower than the designed ones for all the coils in the first operation, the maximum possible voltage generation is less than 2.5 kV between coil terminals (Park, K.R. et al., 2009). Therefore, the insulation test voltage of 6 kV

Before the high current operation, it is mandatory to check the electrical joints for high currents. The designed resistance value is less than 5 n per one lap joint. As the voltage was measured across the bus line which has the 3–4 lap joints, the averaged joint resistance was used to validate the joint resistances. The joint resistances were measured by V-I characteristic curve by scanning the current in the range of 900 A. The resistance was obtained by the slope of the V-I curve. The measured joint resistance is less than 2.5 n per

The objectives of inspection of the superconducting magnet under the high current operation are to check operational stability of magnets at high current operation conditions and to check the controllability of the coil currents and magnetic field profile for plasma discharges and confinement. The TF system was tested by increasing the current level in steps. The thermo-hydraulic and magnetic parameters were measured for the operating conditions. Figure 19(a) shows that TF magnet operated stable 30 kA for more than 5 hours with the temperature rising less than 0.1 K during the current charge and discharge periods. The quench detection system also operated reliably. During the current charging and discharging time, the detected signal was less than 25 mV and well below the trigger

rate of helium (a), and phase transition to superconductor at around 18 K (b).

joint and it satisfied the design value of less than 5 n (Park, Y.M. et al., 2009).

was considered to be sufficient for ensuring the insulation integrity.

**5.3 Inspection under the high current operation** 

condition of 50 mV and 2 s (Chu, Y. et al., 2009).

dismissing Incoloy 908. On the other hand, the PF1 discrepancy gradually increased from 3% to 36% as the measurement point approached the VV inside. The PF coils appeared to be shielded by some magnetic component surrounding the VV as their magnetic field was reduced in the VV whereas the TF coils seemed to be embedded in some magnetic material as their magnetic field rose suddenly at the VV inside close to the TF's windings. Magnet system cool-down was launched after passing the all required room temperature inspection for magnet system and other components.

Fig. 17. The positions of the hall proves to measure the field profile in vacuum vessel (a), and the measured field strength at five radial positions according to the TF, PF1 or PF7 coil current charging up to 100 A (b).

#### **5.2 Inspection during and after cool-down**

The total cold mass is about 300 tons. In order to cool-down this massive structure, the 9 kW helium refrigerators, the helium distribution valve box and in-cryostat cooling piping were installed. The cool-down period was about 23 days as shown in Fig. 18 (Kim, Y.S. et al, 2009). The major parameters such as the vacuum, the temperature, and the mass flow rate, were carefully checked during the cool-down. The structural behavior was also monitored because the different thermal shrinkage between the structurally connected bodies may give harmful effects on the structural integrity. The maximum difference of the temperature distribution inside the cryostat was set to be 50 K, which is the value obtained from the experience in many large scale superconducting magnet applications. This criterion is not applied to the thermal shield because the thermal shield is an independent structure without any structural connection with the superconducting magnet system. As the much possibility of the cold leak exists, the residual gas inside the cryostat was continuously analyzed. The superconducting to normal transitions of the KSTAR magnet system were observed during the cool-down. As expected, the TF coil and the PF1-5 coils showed the transition around 18 K. The measured RRR values were about 200 which satisfied the design value of more than 100. The PF6-7 and bus lines which are made of NbTi superconductor showed the transition around 9 K with RRR of about 100.

dismissing Incoloy 908. On the other hand, the PF1 discrepancy gradually increased from 3% to 36% as the measurement point approached the VV inside. The PF coils appeared to be shielded by some magnetic component surrounding the VV as their magnetic field was reduced in the VV whereas the TF coils seemed to be embedded in some magnetic material as their magnetic field rose suddenly at the VV inside close to the TF's windings. Magnet system cool-down was launched after passing the all required room temperature inspection

(a) (b) Fig. 17. The positions of the hall proves to measure the field profile in vacuum vessel (a), and the measured field strength at five radial positions according to the TF, PF1 or PF7 coil

The total cold mass is about 300 tons. In order to cool-down this massive structure, the 9 kW helium refrigerators, the helium distribution valve box and in-cryostat cooling piping were installed. The cool-down period was about 23 days as shown in Fig. 18 (Kim, Y.S. et al, 2009). The major parameters such as the vacuum, the temperature, and the mass flow rate, were carefully checked during the cool-down. The structural behavior was also monitored because the different thermal shrinkage between the structurally connected bodies may give harmful effects on the structural integrity. The maximum difference of the temperature distribution inside the cryostat was set to be 50 K, which is the value obtained from the experience in many large scale superconducting magnet applications. This criterion is not applied to the thermal shield because the thermal shield is an independent structure without any structural connection with the superconducting magnet system. As the much possibility of the cold leak exists, the residual gas inside the cryostat was continuously analyzed. The superconducting to normal transitions of the KSTAR magnet system were observed during the cool-down. As expected, the TF coil and the PF1-5 coils showed the transition around 18 K. The measured RRR values were about 200 which satisfied the design value of more than 100. The PF6-7 and bus lines which are made of NbTi superconductor showed the transition

for magnet system and other components.

current charging up to 100 A (b).

around 9 K with RRR of about 100.

**5.2 Inspection during and after cool-down** 

Fig. 18. The cool down characteristics of the magnet system. The change of overall mass flow rate of helium (a), and phase transition to superconductor at around 18 K (b).

After finishing the cool-down, the HIPOT test was carried out up to 6 kV for the final insulation check. As the operating currents were lower than the designed ones for all the coils in the first operation, the maximum possible voltage generation is less than 2.5 kV between coil terminals (Park, K.R. et al., 2009). Therefore, the insulation test voltage of 6 kV was considered to be sufficient for ensuring the insulation integrity.

Before the high current operation, it is mandatory to check the electrical joints for high currents. The designed resistance value is less than 5 n per one lap joint. As the voltage was measured across the bus line which has the 3–4 lap joints, the averaged joint resistance was used to validate the joint resistances. The joint resistances were measured by V-I characteristic curve by scanning the current in the range of 900 A. The resistance was obtained by the slope of the V-I curve. The measured joint resistance is less than 2.5 n per joint and it satisfied the design value of less than 5 n (Park, Y.M. et al., 2009).

## **5.3 Inspection under the high current operation**

The objectives of inspection of the superconducting magnet under the high current operation are to check operational stability of magnets at high current operation conditions and to check the controllability of the coil currents and magnetic field profile for plasma discharges and confinement. The TF system was tested by increasing the current level in steps. The thermo-hydraulic and magnetic parameters were measured for the operating conditions. Figure 19(a) shows that TF magnet operated stable 30 kA for more than 5 hours with the temperature rising less than 0.1 K during the current charge and discharge periods. The quench detection system also operated reliably. During the current charging and discharging time, the detected signal was less than 25 mV and well below the trigger condition of 50 mV and 2 s (Chu, Y. et al., 2009).

Superconductor Application to the Magnetic

components.

**6.1 Plasma confinement and experiments** 

**6. Plasma experiments and magnet operation** 

Fusion Devices for the Steady-State Plasma Confinement Achievement 381

After the successful integrated commissioning in 2008, KSTAR device has been operated to meet the project mission to establish a scientific and technological basis for the attractive fusion reactor. The operating performance of the device was upgraded every year. In the first campaign, the first plasma was achieved with a plasma current over 100 kA just after the integrated commissioning in 2008 (Oh, Y.K. et al., 2009). In the second and third campaigns, by the reliable control of the plasma current and shape, high performance confinement (H-mode) plasma could be achieved at the plasma current over 500 kA as shown in the Fig. 20. In the 4th campaign, the plasma current reached up to 1 MA and various researches could be conducted such as suppression of the edge local mode (ELM).

Fig. 20. High performance plasma confinement (H-mode) achievement in KSTAR. Plasma

Usually the plasma operation could be classified into several phases; plasma startup to generate a plasma current up to about 100 kA, feedback plasma control for the current ramp up and shaping, plasma heating and experimental research under various operational modes, and plasma current ramp-down or disruption. The plasma current and shape are controlled by the plasma control system which is a feedback control loop with the

The basic criteria for plasma startup are formation of the field null and the loop voltage to drive the plasma current rise (Leuer, J.A. et al., 2010). However the startup in the KSTAR was complicated due to the nonlinear magnetic behavior associated with the Incoloy 908 jacket material used in the construction of the Nb3Sn superconductors. The field distortion was investigated by scanning the field profile using the hall probe array and monitoring the e-beam trajectories according to the TF and PF coil current (England, A.C. et al., 2011). The

profile reconstruction using EFIT, CCD visual image, and the picture of in-vessel

diagnostics sensors and PF magnet power supplies (Hahn, S.H. et al., 2009).

Fig. 19. The operational characteristic of the magnets. Stable operation of TF magnet at 30 kA over 5.5 hours with temperature rising of about 0.1 K (a), and ac loss measurement of PF1 coil by applying sinusoidal current waveform up to 600 s (b).

For KSTAR operation at currents up to 2 MA and long pulses up to 300 s, the ac losses of PF magnets can be a very important factor in deciding the operational capability. Precise measurements of the ac losses of the PF magnet system in KSTAR are difficult due to the limited number of sensors installed in KSTAR. AC loss was measured at the current waveforms of the typical single pulse waveform and of long pulse sinusoidal waveforms which were adopted in the KSTAR CS model coil experiments (Oh, Y.K. et al., 2009). Figure 19(b) shows the current waveform for ac loss measurement of the PF1 coil. The applied current was dc biased 2 kA and sinusoidal ac current with amplitude of 0.5 kA for 600 s according to changing the frequency from 0.1 to 0.2 Hz. The figure shows that thermal parameters reached a steady-state condition after 350 s. The coupling loss of PF1 showed different results according to the waveforms (Lee, H.J. et al., 2010; Oh, D.K. et al., 2010). The coupling loss time constant, n of PF1 measured by the single trapezoidal waveform was about 62.5 ms at the initial operation time and reduced to about 50ms after two months operation. It could be expected that the cyclic electromagnetic force applied to the strands tends to increase the inter-strand contact resistance, and resulted in the reduced coupling loss. When compared with sinusoidal method, the measured time constant was about 35 ms. The difference of the n could come from the non-linear current control at low current less than 500 A. Another interesting result is that the coupling time constant of the PF6 coil is also about 33ms even though it has NbTi conductor. These results could come from the adoption of chrome coating on all three kinds of strands, Nb3Sn, NbTi, and copper. These results show that the KSTAR construction has advantages for various operational capabilities especially high speed PF6 coil operation as in the first plasma operation. The superconducting magnet commissioning was completed without any severe problems in spite of the fact that individual cool-down tests were not conducted prior to assembly.

## **6. Plasma experiments and magnet operation**

## **6.1 Plasma confinement and experiments**

380 Superconductors – Properties, Technology, and Applications

(a) (b)

For KSTAR operation at currents up to 2 MA and long pulses up to 300 s, the ac losses of PF magnets can be a very important factor in deciding the operational capability. Precise measurements of the ac losses of the PF magnet system in KSTAR are difficult due to the limited number of sensors installed in KSTAR. AC loss was measured at the current waveforms of the typical single pulse waveform and of long pulse sinusoidal waveforms which were adopted in the KSTAR CS model coil experiments (Oh, Y.K. et al., 2009). Figure 19(b) shows the current waveform for ac loss measurement of the PF1 coil. The applied current was dc biased 2 kA and sinusoidal ac current with amplitude of 0.5 kA for 600 s according to changing the frequency from 0.1 to 0.2 Hz. The figure shows that thermal parameters reached a steady-state condition after 350 s. The coupling loss of PF1 showed different results according to the waveforms (Lee, H.J. et al., 2010; Oh, D.K. et al., 2010). The coupling loss time constant, n of PF1 measured by the single trapezoidal waveform was about 62.5 ms at the initial operation time and reduced to about 50ms after two months operation. It could be expected that the cyclic electromagnetic force applied to the strands tends to increase the inter-strand contact resistance, and resulted in the reduced coupling loss. When compared with sinusoidal method, the measured time constant was about 35 ms. The difference of the n could come from the non-linear current control at low current less than 500 A. Another interesting result is that the coupling time constant of the PF6 coil is also about 33ms even though it has NbTi conductor. These results could come from the adoption of chrome coating on all three kinds of strands, Nb3Sn, NbTi, and copper. These results show that the KSTAR construction has advantages for various operational capabilities especially high speed PF6 coil operation as in the first plasma operation. The superconducting magnet commissioning was completed without any severe problems in spite of the fact that individual cool-down tests were not

Fig. 19. The operational characteristic of the magnets. Stable operation of TF magnet at 30 kA over 5.5 hours with temperature rising of about 0.1 K (a), and ac loss measurement of

PF1 coil by applying sinusoidal current waveform up to 600 s (b).

conducted prior to assembly.

After the successful integrated commissioning in 2008, KSTAR device has been operated to meet the project mission to establish a scientific and technological basis for the attractive fusion reactor. The operating performance of the device was upgraded every year. In the first campaign, the first plasma was achieved with a plasma current over 100 kA just after the integrated commissioning in 2008 (Oh, Y.K. et al., 2009). In the second and third campaigns, by the reliable control of the plasma current and shape, high performance confinement (H-mode) plasma could be achieved at the plasma current over 500 kA as shown in the Fig. 20. In the 4th campaign, the plasma current reached up to 1 MA and various researches could be conducted such as suppression of the edge local mode (ELM).

Fig. 20. High performance plasma confinement (H-mode) achievement in KSTAR. Plasma profile reconstruction using EFIT, CCD visual image, and the picture of in-vessel components.

Usually the plasma operation could be classified into several phases; plasma startup to generate a plasma current up to about 100 kA, feedback plasma control for the current ramp up and shaping, plasma heating and experimental research under various operational modes, and plasma current ramp-down or disruption. The plasma current and shape are controlled by the plasma control system which is a feedback control loop with the diagnostics sensors and PF magnet power supplies (Hahn, S.H. et al., 2009).

The basic criteria for plasma startup are formation of the field null and the loop voltage to drive the plasma current rise (Leuer, J.A. et al., 2010). However the startup in the KSTAR was complicated due to the nonlinear magnetic behavior associated with the Incoloy 908 jacket material used in the construction of the Nb3Sn superconductors. The field distortion was investigated by scanning the field profile using the hall probe array and monitoring the e-beam trajectories according to the TF and PF coil current (England, A.C. et al., 2011). The

Superconductor Application to the Magnetic

Toroidal field, B0 [T] Peak field, Bpeak [T) Coil current, ITF [kA] Strain on Nb3Sn [%]

Current sharing Temp, TCS [K] Temp. margin, Tmargin [K]

extended performance conditions.

cryogenic circulator as well as magnet system itself.

increasing overall capacity of the cryo-facility.

Fusion Devices for the Steady-State Plasma Confinement Achievement 383

K. The superconductor of the KSTAR TF magnet has an operational capability of toroidal

**Parameters Designed operation Extended operation** 

Table 4. Estimation of the temperature margin of the TF magnet at designed condition and

The PF magnets are designed to be operated for the plasma current startup and ramp up to 2 MA inductively, and plasma of 1 MA current has been achieved in 2011 operation in spite of the limited electric power supply. The designed current of 2 MA is expected to be achieved after the completion of a motor-generator system. Nominal operational scenario of the PF magnets is started from the initial magnetization to store a maximum flux in each magnet and is followed by a synchronized fast flux swing for the plasma startup and finally is continued by the fast feedback control of plasma current, shape, and position. When the PF magnet operated in pulse, large amount of pulsed heat is generated in the magnet due to

The cryogenic system of the PF magnet has been designed to be operated under the pulsed heat load by adopting the thermal damper concept. A cryogenic circulator will be operated in static to supply the supercritical helium in to PF magnets with a total mass flow rate of about 300 g/s with pressure drop of about 2 bar. The transient heat from the PF magnet is extracted by the heat exchangers which are immersed at the liquid helium bath as shown in Fig. 22(a). Most of the PF shot was operated stably within the However the operation range of the circulator. However in some scenario with fast current changing scenario, a large thermal load causes abrupt flow and pressure increase of the helium. The supply and return pressure of the circulator are increased simultaneously with shot. The mass flow rate in the PF 1 magnet was maintained the value of zero for several seconds as shown in Fig. 22(b). And the operation point of the cryogenic circulator moved into the unstable zone and transiently (Lee, H.J. et al., 2011; park, Y.M. et al., 2010). Although the PF magnets are made of with low as loss conductor, additional efforts are required for the stable operation of the

One of the approaches is optimizing the PF magnet operation scenario sharing the burden of ac loss with other magnets. There were similar approaches in the conventional tokamaks to reduce the burden on central solenoid by solenoid-free startup in DIII-D (Leuer, J.A. et al., 2010) or by coaxial helicity injection in NSTX (Raman, R. et al., 2001). Another way is to limit the maximum field change rate within certain range because the ac loss is proportion to the square of the field change rate. The modification the cryogenic circuits could be considered to clamp the cryogenic parameter fluctuation in the cryogenic circulator under the large thermal load in PF magnet. It could maximize the operation capability of PF magnet without

the ac loss which is deeply related with conductor design and the field swing rate.

3.5 7.2 35.2 **0.7**  9.16 4.4

3.5 7.2 35.2 **0.3**  10.5 5.7

**4.5**  9.3 45.3 **0.7**  5.86 1.1

**5.0**  10.3 50.3 **0.3**  5.85 1.1

field up to 5.0 T with a temperature margin of 1.1 K as shown in Table 4.

field deformation due to the ferromagnetic effect was deliberately compensated by including the control of field gradients since the field gradients determined the positional stability of the plasma current channel. As a consequence, plasma start-up became much more robust against the deviation from the expected plasma initial ramp-up. And the field null formation was intentionally delayed until the toroidal electric field reached a sufficient level to make inductive breakdown of the neutral gas. By doing so, we could obtain better breakdown conditions with the same level of loop voltage. Hence purely ohmic discharges were routinely achieved under various toroidal magnetic fields (Kim, J. et al., 2011).

ELMy H-mode discharges have been observed from the third campaign with combined auxiliary heating of neutral beam injection (PNBI 1.4 MW) and electron cyclotron resonant heating (PECRH 0.2 MW) in a relatively low density regime. The L–H transitions occurred typically right after the end of the plasma current (Ip) ramp-up to 0.6 MA and after the fast shaping to a highly elongated double-null (DN) divertor configuration (Yoon, S.W. et al., 2011). The ELM is an evidence of the H-mode confinement but it is an issue to be prevented or mitigated to protect the plasma facing components or divertor from the damage by the energetic particles flux. The ELM is one of the critical issues in the ITER device. By applying three-dimensional field on the plasma, ELM suppression or mitigation could be achieved in the 4th campaign. As shown in Fig. 21. KSTAR is targeted to achieve the reliable plasma control at the plasma current of 2 MA, in-inductive steady-state operation up to 300s, and plasma heating using by 28 MW auxiliary heating systems.

Fig. 21. Edge localized mode (ELM) occurrence in H-mode plasma and its suppression by applying 3D field (n=1 RMP) using in-vessel control coils.

## **6.2 Cryogenic stability of magnet system**

In the KSTAR operation, the TF magnet has been operated up to the designed operating current of 35.2 kA in stable with temperature rising of 0.1 K. The toroidal field was 3.5 T at plasma center and peak field inside conductor was 7.2 T with temperature margin of about 4

field deformation due to the ferromagnetic effect was deliberately compensated by including the control of field gradients since the field gradients determined the positional stability of the plasma current channel. As a consequence, plasma start-up became much more robust against the deviation from the expected plasma initial ramp-up. And the field null formation was intentionally delayed until the toroidal electric field reached a sufficient level to make inductive breakdown of the neutral gas. By doing so, we could obtain better breakdown conditions with the same level of loop voltage. Hence purely ohmic discharges

ELMy H-mode discharges have been observed from the third campaign with combined auxiliary heating of neutral beam injection (PNBI 1.4 MW) and electron cyclotron resonant heating (PECRH 0.2 MW) in a relatively low density regime. The L–H transitions occurred typically right after the end of the plasma current (Ip) ramp-up to 0.6 MA and after the fast shaping to a highly elongated double-null (DN) divertor configuration (Yoon, S.W. et al., 2011). The ELM is an evidence of the H-mode confinement but it is an issue to be prevented or mitigated to protect the plasma facing components or divertor from the damage by the energetic particles flux. The ELM is one of the critical issues in the ITER device. By applying three-dimensional field on the plasma, ELM suppression or mitigation could be achieved in the 4th campaign. As shown in Fig. 21. KSTAR is targeted to achieve the reliable plasma control at the plasma current of 2 MA, in-inductive steady-state operation up to 300s, and

Fig. 21. Edge localized mode (ELM) occurrence in H-mode plasma and its suppression by

In the KSTAR operation, the TF magnet has been operated up to the designed operating current of 35.2 kA in stable with temperature rising of 0.1 K. The toroidal field was 3.5 T at plasma center and peak field inside conductor was 7.2 T with temperature margin of about 4

were routinely achieved under various toroidal magnetic fields (Kim, J. et al., 2011).

plasma heating using by 28 MW auxiliary heating systems.

applying 3D field (n=1 RMP) using in-vessel control coils.

**6.2 Cryogenic stability of magnet system** 


K. The superconductor of the KSTAR TF magnet has an operational capability of toroidal field up to 5.0 T with a temperature margin of 1.1 K as shown in Table 4.

Table 4. Estimation of the temperature margin of the TF magnet at designed condition and extended performance conditions.

The PF magnets are designed to be operated for the plasma current startup and ramp up to 2 MA inductively, and plasma of 1 MA current has been achieved in 2011 operation in spite of the limited electric power supply. The designed current of 2 MA is expected to be achieved after the completion of a motor-generator system. Nominal operational scenario of the PF magnets is started from the initial magnetization to store a maximum flux in each magnet and is followed by a synchronized fast flux swing for the plasma startup and finally is continued by the fast feedback control of plasma current, shape, and position. When the PF magnet operated in pulse, large amount of pulsed heat is generated in the magnet due to the ac loss which is deeply related with conductor design and the field swing rate.

The cryogenic system of the PF magnet has been designed to be operated under the pulsed heat load by adopting the thermal damper concept. A cryogenic circulator will be operated in static to supply the supercritical helium in to PF magnets with a total mass flow rate of about 300 g/s with pressure drop of about 2 bar. The transient heat from the PF magnet is extracted by the heat exchangers which are immersed at the liquid helium bath as shown in Fig. 22(a). Most of the PF shot was operated stably within the However the operation range of the circulator. However in some scenario with fast current changing scenario, a large thermal load causes abrupt flow and pressure increase of the helium. The supply and return pressure of the circulator are increased simultaneously with shot. The mass flow rate in the PF 1 magnet was maintained the value of zero for several seconds as shown in Fig. 22(b). And the operation point of the cryogenic circulator moved into the unstable zone and transiently (Lee, H.J. et al., 2011; park, Y.M. et al., 2010). Although the PF magnets are made of with low as loss conductor, additional efforts are required for the stable operation of the cryogenic circulator as well as magnet system itself.

One of the approaches is optimizing the PF magnet operation scenario sharing the burden of ac loss with other magnets. There were similar approaches in the conventional tokamaks to reduce the burden on central solenoid by solenoid-free startup in DIII-D (Leuer, J.A. et al., 2010) or by coaxial helicity injection in NSTX (Raman, R. et al., 2001). Another way is to limit the maximum field change rate within certain range because the ac loss is proportion to the square of the field change rate. The modification the cryogenic circuits could be considered to clamp the cryogenic parameter fluctuation in the cryogenic circulator under the large thermal load in PF magnet. It could maximize the operation capability of PF magnet without increasing overall capacity of the cryo-facility.

Superconductor Application to the Magnetic

superconducting fusion devices.

experimental fusion device, ITER.

**7. Conclusion** 

in technologies.

range.

**8. Acknowledgement** 

Fusion Devices for the Steady-State Plasma Confinement Achievement 385

Those kinds of issues will be explored according to the operation of the present

The application of the superconductor to the magnetic confinement fusion devices is inevitable to achieve a steady-state plasma confinement under the high field environment. There has been a lot of progress in the superconducting magnet technology in fusion application. Initially it was only to the static field magnets by using the Rutherford-type conductor cooled in a liquid helium bath, but now it is applied to the pulsed field magnets also by using the CICC type conductor which is cooled with forced flow supercritical helium and is robust for the large electromagnetic and high voltage conditions. The KSTAR device has been constructed using the fully superconducting magnets which are up-to-date

The representative outstanding technologies of the magnet systems in the KSTAR devices are as follows; high performance Nb3Sn conductor development to be used commonly in TF and PF magnets, reliable chrome coating on strand surface to reduce the ac coupling loss, long-length CICC production up to 1.8 km by tube-mill welding process of jacket, high reliability in the leak detection and repair welding of CICC, continuous magnet winding without internal joint by numerical shape control, successful heat treatment of Nb3Sn magnet with Incoloy908 jacket without failure, turn insulation taping and full penetration VPI technology, new method of TF magnet installation by rotating each one along the surface of the vacuum vessel, applying pre-compression on the central solenoids structure by heating the structure and wedge adjustment, and stable operation and protection of the magnet system. The successful construction and operation of the KSTAR device using the Nb3Sn superconductor brought a great positive impact in the worldwide fusion reactor preparation including the international

There are several remained issues in the superconducting magnet application in the world-wide superconductor application into the fusion reactors, especially for the large scale fusion reactors. Large fluctuations of the thermo-hydraulic parameters in cryofacility are mainly due to the ac loss in the PF magnet limit the operational capability of the PF magnets. And the performance degradation of the Nb3Sn conductor was found in the development for the ITER magnet conductor such as allowable current density decrease under the large amount of the transverse force In the CICC and the current sharing temperature is also decreased after the large number of operation cycle. Those kinds of issues will be explored according to the operation of the present superconducting fusion devices. New optimum operational scenarios of the PF magnets should be developed also to reduce the ac loss and to maximize the static operation

The authors appreciate all the participants and researchers who contributed to the construction, operation and experiments of the KSTAR device, especially to the

Fig. 22. Flow diagram of the PF magnet cooling circuits (a), and flow reversal occurrence in some extreme operation of the PF magnet at bipolar operation with 10 kA/s (b)

## **6.3 Other issues in superconductor application to fusion devices**

For superconductor application to the future fusion reactor, more R&D activities are required to develop the conductor designs having stable operation capability under the extreme environments such as higher magnetic field and higher current density operation. However there are some technical issues which hinder the fabrication of the large current Nb3Sn conductor for ITER.

One of the issues is irreversible degradation in Nb3Sn CICC (Bruzzone, P. 2010), usually the longitudinal strain applied to the Nb3Sn makes the decrease the allowable limit of the operation current density. The longitudinal strain mainly comes when the CICC was cooled down from the heat treatment temperature about 600 C to operating temperature of 5 K due to the difference of coefficient of thermal expansion. And it is regarded as reversible phenomena. Another load applied on the conductor is the transverse load by the Lorentz force from the conductor current and self-field. In many sample test of the ITER conductor showed the degradation of the performance of Nb3Sn CICC by the number of load cycle. The possible interpretation of the degradation is plastic deformation of the superconductor filament by the bending of strand when the cyclic transverse load is applied. The cycled load applied on the Nb3Sn conductor is helpful for the ac loss reduction but reduction of the dc performance.

ITER TF magnet is designed with a concept of insertion of circular conductor in the premachined groove in the radial plate to prevent the accumulation of the transverse force on the conductor. But the conductor length changed after heat treatment mainly due to the residual stress during the jacket compaction and coil winding. So it is not easy to control the gap between TF CICC and the grove in the radial plate.

Those kinds of issues will be explored according to the operation of the present superconducting fusion devices.

## **7. Conclusion**

384 Superconductors – Properties, Technology, and Applications

(a) (b)

Fig. 22. Flow diagram of the PF magnet cooling circuits (a), and flow reversal occurrence in

For superconductor application to the future fusion reactor, more R&D activities are required to develop the conductor designs having stable operation capability under the extreme environments such as higher magnetic field and higher current density operation. However there are some technical issues which hinder the fabrication of the large current

One of the issues is irreversible degradation in Nb3Sn CICC (Bruzzone, P. 2010), usually the longitudinal strain applied to the Nb3Sn makes the decrease the allowable limit of the operation current density. The longitudinal strain mainly comes when the CICC was cooled down from the heat treatment temperature about 600 C to operating temperature of 5 K due to the difference of coefficient of thermal expansion. And it is regarded as reversible phenomena. Another load applied on the conductor is the transverse load by the Lorentz force from the conductor current and self-field. In many sample test of the ITER conductor showed the degradation of the performance of Nb3Sn CICC by the number of load cycle. The possible interpretation of the degradation is plastic deformation of the superconductor filament by the bending of strand when the cyclic transverse load is applied. The cycled load applied on the Nb3Sn conductor is helpful for the ac loss reduction but reduction of the dc

ITER TF magnet is designed with a concept of insertion of circular conductor in the premachined groove in the radial plate to prevent the accumulation of the transverse force on the conductor. But the conductor length changed after heat treatment mainly due to the residual stress during the jacket compaction and coil winding. So it is not easy to control the

gap between TF CICC and the grove in the radial plate.

some extreme operation of the PF magnet at bipolar operation with 10 kA/s (b)

**6.3 Other issues in superconductor application to fusion devices** 

Nb3Sn conductor for ITER.

performance.

The application of the superconductor to the magnetic confinement fusion devices is inevitable to achieve a steady-state plasma confinement under the high field environment. There has been a lot of progress in the superconducting magnet technology in fusion application. Initially it was only to the static field magnets by using the Rutherford-type conductor cooled in a liquid helium bath, but now it is applied to the pulsed field magnets also by using the CICC type conductor which is cooled with forced flow supercritical helium and is robust for the large electromagnetic and high voltage conditions. The KSTAR device has been constructed using the fully superconducting magnets which are up-to-date in technologies.

The representative outstanding technologies of the magnet systems in the KSTAR devices are as follows; high performance Nb3Sn conductor development to be used commonly in TF and PF magnets, reliable chrome coating on strand surface to reduce the ac coupling loss, long-length CICC production up to 1.8 km by tube-mill welding process of jacket, high reliability in the leak detection and repair welding of CICC, continuous magnet winding without internal joint by numerical shape control, successful heat treatment of Nb3Sn magnet with Incoloy908 jacket without failure, turn insulation taping and full penetration VPI technology, new method of TF magnet installation by rotating each one along the surface of the vacuum vessel, applying pre-compression on the central solenoids structure by heating the structure and wedge adjustment, and stable operation and protection of the magnet system. The successful construction and operation of the KSTAR device using the Nb3Sn superconductor brought a great positive impact in the worldwide fusion reactor preparation including the international experimental fusion device, ITER.

There are several remained issues in the superconducting magnet application in the world-wide superconductor application into the fusion reactors, especially for the large scale fusion reactors. Large fluctuations of the thermo-hydraulic parameters in cryofacility are mainly due to the ac loss in the PF magnet limit the operational capability of the PF magnets. And the performance degradation of the Nb3Sn conductor was found in the development for the ITER magnet conductor such as allowable current density decrease under the large amount of the transverse force In the CICC and the current sharing temperature is also decreased after the large number of operation cycle. Those kinds of issues will be explored according to the operation of the present superconducting fusion devices. New optimum operational scenarios of the PF magnets should be developed also to reduce the ac loss and to maximize the static operation range.

## **8. Acknowledgement**

The authors appreciate all the participants and researchers who contributed to the construction, operation and experiments of the KSTAR device, especially to the

Superconductor Application to the Magnetic

*Superconductivity,* Vol. 15, pp. 1415-1418

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The discovery of the superconductivity at a transition temperature (*T*C) of 39 K in magnesium diboride (MgB2) has attracted much attention from many researchers for scientific as well as technical reasons [1]. Compared with Cu-based superconductors (cuprates), MgB2 has lower anisotropy and larger coherence length, in addition to high *T*<sup>c</sup> [2]. These characteristics of MgB2 give rise to new applications for superconductor devices that can operate in the temperature range between 20 and 30 K; examples of such devices are Josephson junctions and integrated circuits. This temperature range can be easily achieved by using economical and compact cryocoolers or liquid hydrogen. The use of cryocoolers may transform the superconductor from being specialized and advanced technology into common usage in consumer devices. In the future, hydrogen gas may be widely used for carbon-free power generation such as in fuel cells. Liquid hydrogen would be available for these purposes, and may be utilized for the cooling of low-temperature devices. In addition, MgB2 is considered to be a clean superconducting material, using

Thus far, we have developed SQUID (Superconducting Quantum Interference Device) apparatuses, such as magnetocardiography (MCG) and non-destructive evaluation (NDE) systems, by using Nb and cuprates. Since the discovery of superconductivity in MgB2 and its promising potential, we have focused our research on developing SQUID as well as a highfrequency filter made of MgB2. As a first step, we developed a synthesis method for highquality films, which is the basis of fabrication of such devices. Next, since 2004, we have been working on developing a SQUID and a high-frequency filter device. Finally, we succeeded in measuring the magnetic signal from a human being (MCG signal) by using MgB2 SQUID with a specially developed control circuit (digital FLL). In this article, we discuss our achievements

Synthesis of high-quality film is a key technology in developing superconductor devices. We would first like to introduce various synthesis methods for MgB2 films. There have already been many reports on the techniques that have been used far for the preparation of MgB2

and progress on the development of superconducting devices using MgB2 films.

**1. Introduction** 

neither toxic nor rare earth elements.

**2. Synthesis of high-quality films** 

Koichiro Kobayashi2 and Masahito Yoshizawa2 *1National Institute for Materials Science, Tsukuba,* 

*2Graduate School of Engineering, Iwate University, Morioka,* 

Yoshitomo Harada1,

*Japan* 

Yoshida, K.; et al. (2008). Conceptual design of superconducting magnet system for JT-60SA. *IEEE Trans. on Applied Superconductivity*, Vol. 18, pp. 441-446 **18** 

Yoshitomo Harada1,

Koichiro Kobayashi2 and Masahito Yoshizawa2 *1National Institute for Materials Science, Tsukuba, 2Graduate School of Engineering, Iwate University, Morioka, Japan* 

## **1. Introduction**

388 Superconductors – Properties, Technology, and Applications

Yoshida, K.; et al. (2008). Conceptual design of superconducting magnet system for JT-60SA.

The discovery of the superconductivity at a transition temperature (*T*C) of 39 K in magnesium diboride (MgB2) has attracted much attention from many researchers for scientific as well as technical reasons [1]. Compared with Cu-based superconductors (cuprates), MgB2 has lower anisotropy and larger coherence length, in addition to high *T*<sup>c</sup> [2]. These characteristics of MgB2 give rise to new applications for superconductor devices that can operate in the temperature range between 20 and 30 K; examples of such devices are Josephson junctions and integrated circuits. This temperature range can be easily achieved by using economical and compact cryocoolers or liquid hydrogen. The use of cryocoolers may transform the superconductor from being specialized and advanced technology into common usage in consumer devices. In the future, hydrogen gas may be widely used for carbon-free power generation such as in fuel cells. Liquid hydrogen would be available for these purposes, and may be utilized for the cooling of low-temperature devices. In addition, MgB2 is considered to be a clean superconducting material, using neither toxic nor rare earth elements.

Thus far, we have developed SQUID (Superconducting Quantum Interference Device) apparatuses, such as magnetocardiography (MCG) and non-destructive evaluation (NDE) systems, by using Nb and cuprates. Since the discovery of superconductivity in MgB2 and its promising potential, we have focused our research on developing SQUID as well as a highfrequency filter made of MgB2. As a first step, we developed a synthesis method for highquality films, which is the basis of fabrication of such devices. Next, since 2004, we have been working on developing a SQUID and a high-frequency filter device. Finally, we succeeded in measuring the magnetic signal from a human being (MCG signal) by using MgB2 SQUID with a specially developed control circuit (digital FLL). In this article, we discuss our achievements and progress on the development of superconducting devices using MgB2 films.

## **2. Synthesis of high-quality films**

Synthesis of high-quality film is a key technology in developing superconductor devices. We would first like to introduce various synthesis methods for MgB2 films. There have already been many reports on the techniques that have been used far for the preparation of MgB2

Figure 2 shows the critical current density (*J*C) of MgB2 films as a function of temperature. The measurement was carried out using patterned micro-bridges, which were made by standard photo-lithography and the Ar ion beam milling method. The inset of Fig. 2 is an optical microscope image of the MgB2 micro-bridge. The dimensions of the micro-bridge were 10 m (width) 30 m (length). The electrical contact was made by gold wire directly bonded to a Cu/MgB2 contact pad using silver paste. The highest value of *J*c is 11.6 MA/cm2 at 4.2 K on an MgO (100) substrate. The *J*C(0) value is 12.5 MA/cm2. This value is considered

Fig. 2. Temperature dependence of the critical current density (*J*C) of MgB2 micro-bridge. *J*<sup>C</sup>

Fig. 1. Temperature dependence of the resistivity of MgB2 films.

to be very high for MgB2 films.

is calculated from the critical current (*I*C).

thin films. These preparation methods can be categorized into two types: post-annealing methods and as-grown methods.

We will now introduce the post-annealing method. First, Mg and B together or MgBX alone are evaporated on the substrates at room temperature in this method. Second, these films are annealed inside an electric furnace in an atmosphere of Mg gas. This process improves the physical properties so that and we can obtain superconducting behavior [3,4]. This method is characterized by achieving a high *T*C through high-temperature annealing. However, this method is not preferred for the synthesis of the films used in the devices due to the occurrence of the interfacial reactions and the degradation of the surface flatness by high-temperature annealing.

On the other hand, the as-grown method does not need an annealing process. The as-grown method is suitable for the fabrication of superconducting devices because it provides a flat surface. To date, several groups have reported the synthesis of as-grown superconducting MgB2 films. In particular, the HPCVD (hybrid physical chemical vapor deposition) [5] method and the MBE (molecular beam epitaxy) method [6-9] have achieved high *T*c values that are over 30 K. The HPCVD method, which enables a high-temperature growth process of the film at about 1000 K, succeeded in synthesizing the film with a high *T*C value of 39 K by using Mg that has a high degree of volatility.

In contrast to HPCVD, the MBE method adopts a low synthesis temperature of about 573 K and is carried out in a high vacuum. The *T*C of MgB2 films grown by the MBE method is relatively lower than that obtained from other methods and is limited to be around 34 K. The growth conditions for the MBE method adopted by the research groups other than our group are, however, high growth rate and high substrate temperature. The high growth rate compensates for the deficiency of Mg through re-evaporation due to its volatility and prevents the oxidation of Mg. This concept is very similar to that of the HPCVD method. High deposition rate and high growth temperature have been a mainstay in the fabrication of the high-quality MgB2 films.

We adopted a low substrate temperature, a low deposition rate, and an ultra-high vacuum to obtain high-quality MgB2 films. The characteristic feature of the MBE method is lowtemperature synthesis. According to Liu *et al*., MgB2 is synthesized in a wider temperature range, particularly at low temperatures, as the vacuum increases [10]. The growth at low temperature reduces the re-evaporation of Mg. The deposition rate and the supply of Mg and B have to be reduced due to the super-saturation of Mg. Under these conditions, Mg may be oxidized. However, the ultra-high vacuum in an MBE apparatus prevents the oxidation and permits the growth of high-quality films.

In terms of synthesis conditions, our method is conceptually opposite to the conventional fabrication methods for MgB2 films. Figure 1 shows the typical temperature dependence of the resistivity curves of MgB2 films fabricated by our method. The growth temperature was set as 473 K, which is about 100 K lower than that in other reports. In our method, the range of the deposition temperature was between 373 K and 523 K. This is the same range as that in the phase-diagram of Lie *et al.* and may be obtained by extrapolating the pressure range to 10-7 Pa and 10-8 Pa. The *T*c of the sample in Fig. 1 was 32 K, and the transition width is smaller than 1.0 K. In our method, *T*c can reach a value of up to 37 K by the adoption of various additional process improvements, which will be presented later.

thin films. These preparation methods can be categorized into two types: post-annealing

We will now introduce the post-annealing method. First, Mg and B together or MgBX alone are evaporated on the substrates at room temperature in this method. Second, these films are annealed inside an electric furnace in an atmosphere of Mg gas. This process improves the physical properties so that and we can obtain superconducting behavior [3,4]. This method is characterized by achieving a high *T*C through high-temperature annealing. However, this method is not preferred for the synthesis of the films used in the devices due to the occurrence of the interfacial reactions and the degradation of the surface flatness by

On the other hand, the as-grown method does not need an annealing process. The as-grown method is suitable for the fabrication of superconducting devices because it provides a flat surface. To date, several groups have reported the synthesis of as-grown superconducting MgB2 films. In particular, the HPCVD (hybrid physical chemical vapor deposition) [5] method and the MBE (molecular beam epitaxy) method [6-9] have achieved high *T*c values that are over 30 K. The HPCVD method, which enables a high-temperature growth process of the film at about 1000 K, succeeded in synthesizing the film with a high *T*C value of 39 K

In contrast to HPCVD, the MBE method adopts a low synthesis temperature of about 573 K and is carried out in a high vacuum. The *T*C of MgB2 films grown by the MBE method is relatively lower than that obtained from other methods and is limited to be around 34 K. The growth conditions for the MBE method adopted by the research groups other than our group are, however, high growth rate and high substrate temperature. The high growth rate compensates for the deficiency of Mg through re-evaporation due to its volatility and prevents the oxidation of Mg. This concept is very similar to that of the HPCVD method. High deposition rate and high growth temperature have been a mainstay in the fabrication

We adopted a low substrate temperature, a low deposition rate, and an ultra-high vacuum to obtain high-quality MgB2 films. The characteristic feature of the MBE method is lowtemperature synthesis. According to Liu *et al*., MgB2 is synthesized in a wider temperature range, particularly at low temperatures, as the vacuum increases [10]. The growth at low temperature reduces the re-evaporation of Mg. The deposition rate and the supply of Mg and B have to be reduced due to the super-saturation of Mg. Under these conditions, Mg may be oxidized. However, the ultra-high vacuum in an MBE apparatus prevents the

In terms of synthesis conditions, our method is conceptually opposite to the conventional fabrication methods for MgB2 films. Figure 1 shows the typical temperature dependence of the resistivity curves of MgB2 films fabricated by our method. The growth temperature was set as 473 K, which is about 100 K lower than that in other reports. In our method, the range of the deposition temperature was between 373 K and 523 K. This is the same range as that in the phase-diagram of Lie *et al.* and may be obtained by extrapolating the pressure range to 10-7 Pa and 10-8 Pa. The *T*c of the sample in Fig. 1 was 32 K, and the transition width is smaller than 1.0 K. In our method, *T*c can reach a value of up to 37 K by the adoption of

methods and as-grown methods.

high-temperature annealing.

of the high-quality MgB2 films.

by using Mg that has a high degree of volatility.

oxidation and permits the growth of high-quality films.

various additional process improvements, which will be presented later.

Fig. 1. Temperature dependence of the resistivity of MgB2 films.

Figure 2 shows the critical current density (*J*C) of MgB2 films as a function of temperature. The measurement was carried out using patterned micro-bridges, which were made by standard photo-lithography and the Ar ion beam milling method. The inset of Fig. 2 is an optical microscope image of the MgB2 micro-bridge. The dimensions of the micro-bridge were 10 m (width) 30 m (length). The electrical contact was made by gold wire directly bonded to a Cu/MgB2 contact pad using silver paste. The highest value of *J*c is 11.6 MA/cm2 at 4.2 K on an MgO (100) substrate. The *J*C(0) value is 12.5 MA/cm2. This value is considered to be very high for MgB2 films.

Fig. 2. Temperature dependence of the critical current density (*J*C) of MgB2 micro-bridge. *J*<sup>C</sup> is calculated from the critical current (*I*C).

Fig. 3. Temperature dependence of the magnetization of MgB2 film on the ZnO and Ti/ZnO

Fig. 4. XRD scan image of MgB2 1122 peaks from MgB2/Ti films deposited on ZnO

We achieved a value of *T*C = 37 K by the MBE method, which is the same as that with HPCVD. Fujiyoshi *et al.* report that the *J*C of the MgB2 films on the Ti buffer layer is higher than that of MgB2 on the substrates [16]. Therefore, the pinning force due to the grain boundaries at the MgB2 film on the Ti buffer layer is stronger than that of other

substrate.

substrate.

films.

We will discuss the reason why the *J*C of our sample synthesized on an MgO substrate is so high. From AFM and SEM measurements, the grain size of MgB2 films on MgO substrates is 20-50 nm. This value is smaller than that on an Al2O3 or Si substrate. The mismatch between the in-plane lattice constant of MgB2 and MgO is 26.7 %. MgO (100) plane has a square surface structure, whereas MgB2 has a hexagonal surface structure. These conditions led to a smaller grain size of MgB2 when it is deposited on MgO. The grain shape of MgB2 is of the columnartype, and the grain boundaries work as effective pinning centers. Therefore, the smaller grain size provides more pinning centers and leads to high values of *J*C in the perpendicular field.

Next, we tried to further improve the quality of the grown MgB2 films such that they can be used for the fabrication of superconducting devices. According to the works of Shimane University [11] and NICT [12], the in-plane lattice matching between MgB2 and the substrate plays an important role in the growth of high-quality MgB2 films using the MBE method.

The structural quality of MgB2 films was improved by using AlN ( ∆d = 1.9%) and TiZr ( ∆d = 3.6%) buffer layers having lattice constants close to that of MgB2. We initially employed ZnO (0001) as the substrate as its substrate closely matched that of MgB2. The in-plane lattice spacing of the hexagonal ZnO lattice (a = 0.3522 nm) is close to that of the MgB2 lattice (∆d = 5.4%; ∆d of Al2O3, for example, is 35.2 %) [13]. However, we could not do an in-plane alignment for these films, although their *T*c is relatively high at 35 K. The cross-sectional transmission electron microscope (TEM) image showed a large amount of reaction products near the MgB2/ZnO interface. In addition, intermixing between Zn and MgO was observed near the MgB2/ZnO boundary, which is ascribed to the free energy difference between the substrate and the film. The free energy of MgO is -547.1 kJ/mol and is lower than the value of - 303.3 kJ/mol for ZnO. Therefore, Zn in ZnO is easily replaced by Mg, and MgO is formed.

The best-quality MgB2 film was obtained by adopting titanium (Ti) film as a buffer layer on a ZnO substrate, which was fabricated by evaporation using an electron-beam gun cell in an MBE chamber [14]. Figure 3 shows the superconducting properties of MgB2 films when a 15 nm Ti buffer layer is adopted. When the MgB2 film was deposited on a ZnO (0001) substrate, the *T*C increased from 33 K to 36K with the insertion of a Ti buffer layer. By using *in situ* reflection high-energy electron diffraction (RHEED), Ti buffer layers were grown epitaxially in the configurations Ti (0001) || ZnO (0001) and Ti 1100 || ZnO 1120 . The in-plane

orientation () scans of the 1122 reflections around the c-axis of the MgB2 films are shown in Fig. 4, wherein two sets of 6-fold symmetric peaks appear at intervals of every 60º. The peak sets corresponds to two types of domains rotated by 30º from each other about the caxis. Furthermore, we found that one set of the peaks increased in highest and became predominant with increasing thickness of the Ti buffer layer. This result indicates that the in-plane alignment of the MgB2 lattice is improved by the presence of the Ti buffer layer.

Nishidate *et al.* investigated the effect of the Ti buffer layer on ZnO substrates prior to the deposition of Mg and B by studying and calculating its molecular dynamics (MD) [15]. Because Ti atoms occupy a particular adsorption point, Mg adsorption was limited on a metal oxide substrate and Mg is used in the formation of MgB2. In addition, the lattice constant of Ti, a = 0.295 nm, matches that of MgB2, and the free energy of -849.89 kJ/mol for Ti-O is lower than the corresponding values for both MgO and ZnO. Ti also hinders the formation of MgO.

We will discuss the reason why the *J*C of our sample synthesized on an MgO substrate is so high. From AFM and SEM measurements, the grain size of MgB2 films on MgO substrates is 20-50 nm. This value is smaller than that on an Al2O3 or Si substrate. The mismatch between the in-plane lattice constant of MgB2 and MgO is 26.7 %. MgO (100) plane has a square surface structure, whereas MgB2 has a hexagonal surface structure. These conditions led to a smaller grain size of MgB2 when it is deposited on MgO. The grain shape of MgB2 is of the columnartype, and the grain boundaries work as effective pinning centers. Therefore, the smaller grain size provides more pinning centers and leads to high values of *J*C in the perpendicular field.

Next, we tried to further improve the quality of the grown MgB2 films such that they can be used for the fabrication of superconducting devices. According to the works of Shimane University [11] and NICT [12], the in-plane lattice matching between MgB2 and the substrate plays an important role in the growth of high-quality MgB2 films using the MBE method.

The structural quality of MgB2 films was improved by using AlN ( ∆d = 1.9%) and TiZr ( ∆d = 3.6%) buffer layers having lattice constants close to that of MgB2. We initially employed ZnO (0001) as the substrate as its substrate closely matched that of MgB2. The in-plane lattice spacing of the hexagonal ZnO lattice (a = 0.3522 nm) is close to that of the MgB2 lattice (∆d = 5.4%; ∆d of Al2O3, for example, is 35.2 %) [13]. However, we could not do an in-plane alignment for these films, although their *T*c is relatively high at 35 K. The cross-sectional transmission electron microscope (TEM) image showed a large amount of reaction products near the MgB2/ZnO interface. In addition, intermixing between Zn and MgO was observed near the MgB2/ZnO boundary, which is ascribed to the free energy difference between the substrate and the film. The free energy of MgO is -547.1 kJ/mol and is lower than the value of - 303.3 kJ/mol for ZnO. Therefore, Zn in ZnO is easily replaced by Mg, and MgO is formed.

The best-quality MgB2 film was obtained by adopting titanium (Ti) film as a buffer layer on a ZnO substrate, which was fabricated by evaporation using an electron-beam gun cell in an MBE chamber [14]. Figure 3 shows the superconducting properties of MgB2 films when a 15 nm Ti buffer layer is adopted. When the MgB2 film was deposited on a ZnO (0001) substrate, the *T*C increased from 33 K to 36K with the insertion of a Ti buffer layer. By using *in situ* reflection high-energy electron diffraction (RHEED), Ti buffer layers were grown epitaxially in the configurations Ti (0001) || ZnO (0001) and Ti 1100 || ZnO 1120 . The in-plane orientation () scans of the 1122 reflections around the c-axis of the MgB2 films are shown in Fig. 4, wherein two sets of 6-fold symmetric peaks appear at intervals of every 60º. The peak sets corresponds to two types of domains rotated by 30º from each other about the caxis. Furthermore, we found that one set of the peaks increased in highest and became predominant with increasing thickness of the Ti buffer layer. This result indicates that the in-plane alignment of the MgB2 lattice is improved by the presence of the Ti buffer layer.

Nishidate *et al.* investigated the effect of the Ti buffer layer on ZnO substrates prior to the deposition of Mg and B by studying and calculating its molecular dynamics (MD) [15]. Because Ti atoms occupy a particular adsorption point, Mg adsorption was limited on a metal oxide substrate and Mg is used in the formation of MgB2. In addition, the lattice constant of Ti, a = 0.295 nm, matches that of MgB2, and the free energy of -849.89 kJ/mol for Ti-O is lower than the corresponding values for both MgO and ZnO. Ti also hinders the

formation of MgO.

Fig. 3. Temperature dependence of the magnetization of MgB2 film on the ZnO and Ti/ZnO substrate.

Fig. 4. XRD scan image of MgB2 1122 peaks from MgB2/Ti films deposited on ZnO substrate.

We achieved a value of *T*C = 37 K by the MBE method, which is the same as that with HPCVD. Fujiyoshi *et al.* report that the *J*C of the MgB2 films on the Ti buffer layer is higher than that of MgB2 on the substrates [16]. Therefore, the pinning force due to the grain boundaries at the MgB2 film on the Ti buffer layer is stronger than that of other films.

(a) (b)

Fig. 7. Magnetic flux dependence of voltage modulation of an MgB2-SQUID with various

nano-bridge by decreasing film thickness. At present, we retain a high yield of about 90 % for the fabrication of 65 nm thick film and the 400 nm wide nano-bridges. The second problem was resolved by placing a resistor in parallel to the MgB2 nano-bridges. We were

*-V* properties and even expand the width of the

We encountered two problems during the development of SQUID. One is the reduction of the yield ratio, due to the small size of the nano-bridge, which is close to the size limitation of the FIB fabrication process. Another is the hysteretic behavior of the *I-V* curve at the transition edge, which appears at low temperatures. This causes instability in the bias current and makes measurements very difficult. The first problem was solved by adopting

Fig. 6. SQUID characteristics of 200 nm nano-bridge type MgB2-SQUID. (a)Voltage

modulation, (b) *I-V* property.

bias currents.

thinner films. We are able to obtain *I-V* and

able to eliminate the effect of hysteresis by inserting the resistor.

## **3. Implementation of Josephson junction and SQUID by lithography technique**

There are three types of Josephson junction fabrication methods for SQUID and they are as follows: SIS tunneling junctions, nano-bridge junctions and SNS junctions. The first tunneling MgB2 junctions were fabricated by the NTT group [17]. We have developed and used an SIS junction [18]. Subsequently, both the Twente University and European groups succeeded in the fabrication of nano-bridge MgB2 junctions [19]. We have developed nano-bridge MgB2 SQUIDs devices that are optimized for MCG measurement, which are based on the design of the Twente University group. Nano-bridge junctions were made using focused ion beam (FIB) milling apparatus. Detailed fabrication conditions are reported elsewhere [20]. Figure 5(a) shows the layout of the MgB2-SQUID device in this experiment. Figure 5(b) shows the scanning ion microscope (SIM) image of the nanobridge junctions. We designed the size of junction to be 200 nm wide and 200 nm in length. From the SIM image, we observed that the actual junction size matched the design expectations very close.

Figure 6 (a) shows the voltage modulation (*–V* curve) of the nano-bridge MgB2 SQUIDs at 24.4 K. Figure 6 (b) is the *I-V* curve for this SQUID. The voltage modulation due to the applied magnetic field can be seen to increase with decreasing temperature. An approximate magnitude of 22 V is recorded at 20 K. The bias current applied was tuned to get the largest voltage modulation; the bias current value was 2280 A for this sample, giving rise to the maximum modulation of 18.47 V.

Figure 7 shows the *-V* curve of other SQUID with the same nano-bridge size. The measurement temperature is set to 20.81 K. The voltage modulation was measured for the bias current from 2550 to 2585 V. The maximum modulation voltage was 33.55 V when the bias current was set to be 2577 A. The external magnetic field density corresponding to the magnetic flux quantum was 11.4 nT. The measured effective SQUID area (*A*eff) was estimated to be 0.18 mm2.

Fig. 5. (a) Layout of an MgB2-SQUID device. (b) FIB-SIM image of nano-bridge junctions. The size of nano-bridge is 200 nm wide and 200 nm long.

There are three types of Josephson junction fabrication methods for SQUID and they are as follows: SIS tunneling junctions, nano-bridge junctions and SNS junctions. The first tunneling MgB2 junctions were fabricated by the NTT group [17]. We have developed and used an SIS junction [18]. Subsequently, both the Twente University and European groups succeeded in the fabrication of nano-bridge MgB2 junctions [19]. We have developed nano-bridge MgB2 SQUIDs devices that are optimized for MCG measurement, which are based on the design of the Twente University group. Nano-bridge junctions were made using focused ion beam (FIB) milling apparatus. Detailed fabrication conditions are reported elsewhere [20]. Figure 5(a) shows the layout of the MgB2-SQUID device in this experiment. Figure 5(b) shows the scanning ion microscope (SIM) image of the nanobridge junctions. We designed the size of junction to be 200 nm wide and 200 nm in length. From the SIM image, we observed that the actual junction size matched the design

24.4 K. Figure 6 (b) is the *I-V* curve for this SQUID. The voltage modulation due to the applied magnetic field can be seen to increase with decreasing temperature. An approximate magnitude of 22 V is recorded at 20 K. The bias current applied was tuned to get the largest voltage modulation; the bias current value was 2280 A for this sample, giving rise

measurement temperature is set to 20.81 K. The voltage modulation was measured for the bias current from 2550 to 2585 V. The maximum modulation voltage was 33.55 V when the bias current was set to be 2577 A. The external magnetic field density corresponding to the magnetic flux quantum was 11.4 nT. The measured effective SQUID area (*A*eff) was

6 m

SQUID loop

Fig. 5. (a) Layout of an MgB2-SQUID device. (b) FIB-SIM image of nano-bridge junctions.

The size of nano-bridge is 200 nm wide and 200 nm long.

(a) (b)

5 m 5 m 51 m

*–V* curve) of the nano-bridge MgB2 SQUIDs at

*-V* curve of other SQUID with the same nano-bridge size. The

**3. Implementation of Josephson junction and SQUID by lithography** 

**technique** 

expectations very close.

Figure 7 shows the

estimated to be 0.18 mm2.

2.4 mm

3.8 mm

Figure 6 (a) shows the voltage modulation (

to the maximum modulation of 18.47 V.

Fig. 6. SQUID characteristics of 200 nm nano-bridge type MgB2-SQUID. (a)Voltage modulation, (b) *I-V* property.

Fig. 7. Magnetic flux dependence of voltage modulation of an MgB2-SQUID with various bias currents.

We encountered two problems during the development of SQUID. One is the reduction of the yield ratio, due to the small size of the nano-bridge, which is close to the size limitation of the FIB fabrication process. Another is the hysteretic behavior of the *I-V* curve at the transition edge, which appears at low temperatures. This causes instability in the bias current and makes measurements very difficult. The first problem was solved by adopting thinner films. We are able to obtain *I-V* and *-V* properties and even expand the width of the nano-bridge by decreasing film thickness. At present, we retain a high yield of about 90 % for the fabrication of 65 nm thick film and the 400 nm wide nano-bridges. The second problem was resolved by placing a resistor in parallel to the MgB2 nano-bridges. We were able to eliminate the effect of hysteresis by inserting the resistor.

integrator of the D-FLL is reset every time the feedback flux exceeds ±10 and the corresponding 0 steps are counted. Usually, the original input flux is reconstructed on a digital signal processor that is a part of the FLL circuit. The dynamic range depends on the architecture of the digital signal processor, which is typically 32-bit. Furthermore, highspeed signal processing makes a digital feedback loop delay shorter. A high-performance digital signal processor is required to increase the dynamic range and slew rate. So far, it has proven difficult to make a high performance system with the typical single-chip

We have developed a double-counter equipped D-FLL system in Fig. 10. The doublecounter method has two counters on a single-chip microcontroller and a host computer. The digital integrator unit calculates feedback flux data. If feedback flux exceeds ±0.50, the integrator unit subtracts 1 0 from the digital integrator data or adds 1 0 to it, instead of resetting the FLL. The feedback loop process works more quickly than the data transmission from the one-chip microcontroller into a host computer. The integrator data and Couner1 data is sent at periodic time intervals defined by a timer. Counter1 is reset every time after this transmission is complete and commences the count of 0 steps during the next interval. Counter2 integrates the data transmitted from Counter1. The host computer reconstructs the measured flux data using Counter2 as well as integrator data. Because a host computer reconstructs input flux, the workload of the single-chip microcontroller is decreased, leading to a quicker operation of the feedback loop than that of the system without a double-counter method. The high-performance D-FLL system using a single-chip microcontroller is

A/D **∫**dt compare <sup>±</sup>1<sup>0</sup>

**Counter2**

Integrated data

Our latest D-FLL system is composed of a commonly used 8-bit single-chip microcontroller, a 12-bit A/D converter, two 16-bit D/A converters and some other inexpensive and easily

**Counter1**

combine

separate

up/down

**One-chip microcontroller**

**<sup>0</sup> counting**

Timer

**Host computer**

**Integrator unit**

microcontroller that is generally used in such circuits.

achieved by a method that employs two counters.

Preamp

**SQUID** Feedback

Data acquisition memory

**flux data**

**Reconstructed**

D/A

loop

Merge

Fig. 10. Block diagram of the double-counter D-FLL system.

## **4. SQUID electronics**

The input-output (*- V* ) characteristic of SQUID is nonlinear as shown in Fig. 8. A sinusoidal output voltage with the appropriate periodicity is generated, when a magnetic flux is applied in SQUID. A control circuit, known as the FLL (flux locked loop) circuit, is necessary to linearize the input-output characteristic so that the SQUID can be utilized as a magnetic sensor. An example of such an FLL circuit is shown in Fig. 9. The FLL circuit provides the feedback necessary to fix the output at the defined value (the lock point) in a sinusoidal SQUID output. Then, by using only the small area in the vicinity of the lock point, we get the desired linear output. The external magnetic field applied to a pick-up coil produces the voltage in the SQUID. The EMF-voltage is amplified, integrated and then fed back to SQUID in order to maintain to the lock point. Then the applied magnetic field is nullified. This integrated value becomes the output signal of the FLL circuit, and it is equal to the input magnetic signal applied to the pick-up coil.

Fig. 8. Input-output ( *- V*) characteristic of SQUID.

Fig. 9. Example of the FLL circuit

In recent years, it has been shown by several groups that the electronics within the digital SQUID, where the flux-locked loop system is controlled by a digital circuit, can achieve a wide dynamic range [21-23]. These FLL systems can measure biomagnetic signals in a magnetically unshielded environment. We call a digitally controlled flux-locked loop system (D-FLL).

A D-FLL system satisfies unshielded conditions: its dynamic range is expanded by jumping to a lock point on the SQUID's periodic *-V* characteristic and by keeping a count of the number of such events. This method was named flux-quanta counting (FQC) [24, 25]. The

sinusoidal output voltage with the appropriate periodicity is generated, when a magnetic flux is applied in SQUID. A control circuit, known as the FLL (flux locked loop) circuit, is necessary to linearize the input-output characteristic so that the SQUID can be utilized as a magnetic sensor. An example of such an FLL circuit is shown in Fig. 9. The FLL circuit provides the feedback necessary to fix the output at the defined value (the lock point) in a sinusoidal SQUID output. Then, by using only the small area in the vicinity of the lock point, we get the desired linear output. The external magnetic field applied to a pick-up coil produces the voltage in the SQUID. The EMF-voltage is amplified, integrated and then fed back to SQUID in order to maintain to the lock point. Then the applied magnetic field is nullified. This integrated value becomes the output signal of the FLL circuit, and it is equal

**Φ(t)**

Head amp

In recent years, it has been shown by several groups that the electronics within the digital SQUID, where the flux-locked loop system is controlled by a digital circuit, can achieve a wide dynamic range [21-23]. These FLL systems can measure biomagnetic signals in a magnetically unshielded environment. We call a digitally controlled flux-locked loop system

A D-FLL system satisfies unshielded conditions: its dynamic range is expanded by jumping

number of such events. This method was named flux-quanta counting (FQC) [24, 25]. The

 *- V*) characteristic of SQUID.

**V(t)**

*- V* ) characteristic of SQUID is nonlinear as shown in Fig. 8. A

Integrator <sup>C</sup>

R


*-V* characteristic and by keeping a count of the

Vout

**4. SQUID electronics** 

to the input magnetic signal applied to the pick-up coil.

Feed back coil Rf

to a lock point on the SQUID's periodic

Fig. 9. Example of the FLL circuit

**Working Point**

bias current

SQUID

The input-output (

Φ

Fig. 8. Input-output (

V

(D-FLL).

integrator of the D-FLL is reset every time the feedback flux exceeds ±10 and the corresponding 0 steps are counted. Usually, the original input flux is reconstructed on a digital signal processor that is a part of the FLL circuit. The dynamic range depends on the architecture of the digital signal processor, which is typically 32-bit. Furthermore, highspeed signal processing makes a digital feedback loop delay shorter. A high-performance digital signal processor is required to increase the dynamic range and slew rate. So far, it has proven difficult to make a high performance system with the typical single-chip microcontroller that is generally used in such circuits.

We have developed a double-counter equipped D-FLL system in Fig. 10. The doublecounter method has two counters on a single-chip microcontroller and a host computer. The digital integrator unit calculates feedback flux data. If feedback flux exceeds ±0.50, the integrator unit subtracts 1 0 from the digital integrator data or adds 1 0 to it, instead of resetting the FLL. The feedback loop process works more quickly than the data transmission from the one-chip microcontroller into a host computer. The integrator data and Couner1 data is sent at periodic time intervals defined by a timer. Counter1 is reset every time after this transmission is complete and commences the count of 0 steps during the next interval. Counter2 integrates the data transmitted from Counter1. The host computer reconstructs the measured flux data using Counter2 as well as integrator data. Because a host computer reconstructs input flux, the workload of the single-chip microcontroller is decreased, leading to a quicker operation of the feedback loop than that of the system without a double-counter method. The high-performance D-FLL system using a single-chip microcontroller is achieved by a method that employs two counters.

Fig. 10. Block diagram of the double-counter D-FLL system.

Our latest D-FLL system is composed of a commonly used 8-bit single-chip microcontroller, a 12-bit A/D converter, two 16-bit D/A converters and some other inexpensive and easily

10-2 10-1 100 101 102 103 Frequency (Hz)

Fig. 12. Magnetic noise spectra measured in an MSR. The lower line is the measurement in

SQUIDs have been utilized for the detection of weak magnetic fields in our daily life from non-distraction evaluation to bio-magnetic field detection. Recently, the magnetocardiography (MCG) has attracted much attention as a potential application of SQUID, because MCG systems with SQUID provide higher sensitivity than other systems and because cardiac diagnosis plays an important role in the prevention of heart disease. Thus far, we have developed a multi-channel MCG system by using Nb SQUID [27].

Figure 13 shows the block diagram of a cryocooler system and the close-up of a cold cylinder. A commercially available pulse-tube cooler (Sumitomo RP-052D) was used. The cold head of the cryocooler is connected to the compressor via a valve unit. Because the valve unit produces magnetic noise, the cold head and the refrigerator component are installed in an MSR, and the valve unit is set outside of it. They are connected by a long

The mechanical vibration of the refrigerator is ±3 m at 1.2 Hz. The cold stage is connected to the refrigerator by a copper column to enable refrigeration. The distance between them is

This refrigerator can achieve a temperature of about 5 K without a load. Our cryocooler has two modes. The first mode is the SQUIDs characterization mode. This is used for characterization of MgB2 SQUIDs. The cold stage and the cold cylinder are covered by a radiation shield, which is made of copper and has a thickness of 3.4 mm. The MgB2 SQUID

is set underneath the cold stage. The lowest temperature in this mode was 5.6 K.

Following that effort, we developed a conventional MCG system made of MgB2.

**Shielded**

an MSR and the upper line is from an unshielded environment.

**Unshielded**

100

**5. Magnetocardiogram measurements** 

copper tube.

608 mm.

101

102

103

Magnetic noise rms

 (fT/Hz 1/2 )

104

105

106

obtained devices [26]. The measured magnetic flux data is recorded at the rate of 1 kHz via an RS-232 interface. The time delay of the digital feedback loop is 8.4 s. The first-order gradiometer is made of a low-*Tc* dc SQUID (niobium:Nb), has a pickup coil with a diameter of 17.8 mm and a baseline of 50 mm, and was used for the characteristic evaluations of the D-FLL system. The sensitivity which is calculated using the relation *B/* = 0.6 nT/0, where *B* is the flux density in the pickup coil, gives a measure of the flux resolution of 11.35 fT/digit (19 0/digit) for this system.

Figure 11 shows an active area of the D-FLL system. The terminology "active area" means that the system can measure magnetic signals just in this area. The right vertical axis is shown as a dynamic range when a 16-bit D/A converter is used for about 1 <sup>0</sup> feedback flux. The measured dynamic range is 218 0 (141 dB) at 1 Hz, 22 0 (121 dB) at 10 Hz, and 2.1 0 (100 dB) at 100 Hz. In addition, the measured slew rate is 1.3 k0/s. The D-FLL system can operate in a stable manner if power line noise that is usually the main problem encountered with unshielded operations is less than 4.1 0 in amplitude at 50 Hz.

Figure 12 shows the measured D-FLL output noise spectra both inside and outside a magnetically shielded room (MSR). The lower line is measured in the MSR and the upper line is measured in an unshielded environment. The white noise level used was 36.1 fT/Hz1/2 and the 1/*f* corner frequency was 2 Hz in an MSR. The noise spectrum measured outside an MSR contained some unidentified noises beside power line noise that exhibited 465 pT/Hz1/2 (0.78 0/Hz1/2) at 50 Hz.

Fig. 11. Active area of the D-FLL system. An "active area" implies that the system can measure magnetic signal just in this area. The right vertical axis is shown as a dynamic range when 16 bit D/A converter is used for 1 0 feedback flux.

obtained devices [26]. The measured magnetic flux data is recorded at the rate of 1 kHz via an RS-232 interface. The time delay of the digital feedback loop is 8.4 s. The first-order gradiometer is made of a low-*Tc* dc SQUID (niobium:Nb), has a pickup coil with a diameter of 17.8 mm and a baseline of 50 mm, and was used for the characteristic evaluations of the D-FLL system. The sensitivity which is calculated using the relation *B/* = 0.6 nT/0, where *B* is the flux density in the pickup coil, gives a measure of the flux resolution of 11.35

Figure 11 shows an active area of the D-FLL system. The terminology "active area" means that the system can measure magnetic signals just in this area. The right vertical axis is shown as a dynamic range when a 16-bit D/A converter is used for about 1 <sup>0</sup> feedback flux. The measured dynamic range is 218 0 (141 dB) at 1 Hz, 22 0 (121 dB) at 10 Hz, and 2.1 0 (100 dB) at 100 Hz. In addition, the measured slew rate is 1.3 k0/s. The D-FLL system can operate in a stable manner if power line noise that is usually the main problem encountered with unshielded operations is less than 4.1 0 in amplitude at

Figure 12 shows the measured D-FLL output noise spectra both inside and outside a magnetically shielded room (MSR). The lower line is measured in the MSR and the upper line is measured in an unshielded environment. The white noise level used was 36.1 fT/Hz1/2 and the 1/*f* corner frequency was 2 Hz in an MSR. The noise spectrum measured outside an MSR contained some unidentified noises beside power line noise that exhibited

> 101 10-2 10-1 100 102 103 Frequency (Hz)

Fig. 11. Active area of the D-FLL system. An "active area" implies that the system can measure magnetic signal just in this area. The right vertical axis is shown as a dynamic

54

74

94

114

134

Dynamic range (dB)

154

174

194

fT/digit (19 0/digit) for this system.

465 pT/Hz1/2 (0.78 0/Hz1/2) at 50 Hz.

10-2

10-1

**Active area**

range when 16 bit D/A converter is used for 1 0 feedback flux.

100

101

102

Flux amplitude (

0)

103

104

105

50 Hz.

Fig. 12. Magnetic noise spectra measured in an MSR. The lower line is the measurement in an MSR and the upper line is from an unshielded environment.

## **5. Magnetocardiogram measurements**

SQUIDs have been utilized for the detection of weak magnetic fields in our daily life from non-distraction evaluation to bio-magnetic field detection. Recently, the magnetocardiography (MCG) has attracted much attention as a potential application of SQUID, because MCG systems with SQUID provide higher sensitivity than other systems and because cardiac diagnosis plays an important role in the prevention of heart disease. Thus far, we have developed a multi-channel MCG system by using Nb SQUID [27]. Following that effort, we developed a conventional MCG system made of MgB2.

Figure 13 shows the block diagram of a cryocooler system and the close-up of a cold cylinder. A commercially available pulse-tube cooler (Sumitomo RP-052D) was used. The cold head of the cryocooler is connected to the compressor via a valve unit. Because the valve unit produces magnetic noise, the cold head and the refrigerator component are installed in an MSR, and the valve unit is set outside of it. They are connected by a long copper tube.

The mechanical vibration of the refrigerator is ±3 m at 1.2 Hz. The cold stage is connected to the refrigerator by a copper column to enable refrigeration. The distance between them is 608 mm.

This refrigerator can achieve a temperature of about 5 K without a load. Our cryocooler has two modes. The first mode is the SQUIDs characterization mode. This is used for characterization of MgB2 SQUIDs. The cold stage and the cold cylinder are covered by a radiation shield, which is made of copper and has a thickness of 3.4 mm. The MgB2 SQUID is set underneath the cold stage. The lowest temperature in this mode was 5.6 K.

10-2 10-1 100 101 102 103 104 Frequency (Hz)

(a) (b)

Fig. 15. Photographs of the MgB2-SQUID device and magnetocardiogram measurements.

above 20 K without liquid cryogen, although the system is still under development.

Figure 16 shows a magnetocardiogram waveform of a healthy volunteer. This waveform was obtained by applying a digital low-pass filter whose cutoff frequency was 40 Hz and the waveform was averaged over 536 measurements [28]. We have succeeded in measuring the QRS complex and the T wave which is characterized in the activity of the human heart. This result shows a possible practical use of the MgB2-SQUID MCG system cooled by refrigeration

Figure 15 shows the photograph of the actual measurement and the MgB2 magnetometer that was used in the experiments. The magnetocardiogram was measured by our MgB2 SQUID system inside the MSR. This measuring apparatus was cooled by a pulse tube

Magnetic flux density (pT/Hz1/2)

refrigerator instead of liquid helium.

10-1

Fig. 14. Magnetic noise spectrum of the system measured inside an MSR.

100

101

102

103

104

105

106

Fig. 13. Block diagram of a cryocooler system and the close-up of a cold cylinder. The MgB2 SQUID is set under the cold stage or on the bottom plate of the cold cylinder when the system is used in either the SQUIDs characterization mode or the measurement mode.

The second mode is the measurement mode. This mode is used for measurement of an external magnetic field, as in the case of MCG. The cold stage and the cold cylinder are covered by super insulation films instead of a radiation shield. The MgB2 SQUID is set on the bottom plate of the cold cylinder and has a thickness of 2.0 mm. The lowest temperature achieved in this mode was 11.9 K. A temperature sensor and a heater are placed on the cold stage and the column, respectively. They are connected to a PID controller, with which the temperature of the cold stage is controlled. Both the SQUID characterization mode and the measurement mode are controlled and operated by a computer through the D-FLL electronics.

Figure 14 shows the magnetic flux noise spectrum measured inside a single layer MSR. The white magnetic flux noise was 6.8 pT/Hz1/2. The largest peak was caused by power lines of 50 Hz. Other peaks were observed at 1.2 Hz and its harmonics of that frequency. They were caused by the refrigerator and had a magnitude of 13.8 pT at 1.2 Hz.

**Digital FLL electronics**

**From refrigerator**

Fig. 13. Block diagram of a cryocooler system and the close-up of a cold cylinder. The MgB2 SQUID is set under the cold stage or on the bottom plate of the cold cylinder when the system is used in either the SQUIDs characterization mode or the measurement mode.

The second mode is the measurement mode. This mode is used for measurement of an external magnetic field, as in the case of MCG. The cold stage and the cold cylinder are covered by super insulation films instead of a radiation shield. The MgB2 SQUID is set on the bottom plate of the cold cylinder and has a thickness of 2.0 mm. The lowest temperature achieved in this mode was 11.9 K. A temperature sensor and a heater are placed on the cold stage and the column, respectively. They are connected to a PID controller, with which the temperature of the cold stage is controlled. Both the SQUID characterization mode and the measurement mode are controlled and operated by a computer through the D-FLL

Figure 14 shows the magnetic flux noise spectrum measured inside a single layer MSR. The white magnetic flux noise was 6.8 pT/Hz1/2. The largest peak was caused by power lines of 50 Hz. Other peaks were observed at 1.2 Hz and its harmonics of that frequency. They were

caused by the refrigerator and had a magnitude of 13.8 pT at 1.2 Hz.

**Temperature sensor Cold stage**

**Heater**

**Temperature sensor and heater controller**

**MgB2 SQUID chip** 

**MgB2 SQUID chip (the measurement mode)**

**(the SQUIDs characterization mode)**

**Compressor**

**Computer**

**Magnetically shielded room Valve**

**Refrigerator**

**FRP chamber**

**Cold cylinder**

electronics.

**Radiation shield**

Fig. 14. Magnetic noise spectrum of the system measured inside an MSR.

Figure 15 shows the photograph of the actual measurement and the MgB2 magnetometer that was used in the experiments. The magnetocardiogram was measured by our MgB2 SQUID system inside the MSR. This measuring apparatus was cooled by a pulse tube refrigerator instead of liquid helium.

Fig. 15. Photographs of the MgB2-SQUID device and magnetocardiogram measurements.

Figure 16 shows a magnetocardiogram waveform of a healthy volunteer. This waveform was obtained by applying a digital low-pass filter whose cutoff frequency was 40 Hz and the waveform was averaged over 536 measurements [28]. We have succeeded in measuring the QRS complex and the T wave which is characterized in the activity of the human heart. This result shows a possible practical use of the MgB2-SQUID MCG system cooled by refrigeration above 20 K without liquid cryogen, although the system is still under development.

The authors thank H. Yamaguchi, Y. Fujine, D. Oyama, K. Ikeda, S. Goto, T. Nakajima, T. Takahashi, H. Iriuda, T. Oba, A. Okubo, J. Araaki, K. Meguro, T. Abe, H. Endo, Y. Uchikawa and K. Fujisawa for the cooperation of MgB2 SQUID project, and M. Iitake for his technical assistance of FIB process. This work was supported by the Japan Science and Technology Agency. A part of this work was conducted at the AIST Nano-Processing Facility, supported by "Nanotechnology Network Japan" of the Ministry of Education, Culture, Sports, Science

[1] Nagamatsu, J., Nakagawa, N., Muranaka, T., Zenitani, Y. & Akimitsu, J. (2001).

[2] Buzea, C. & Yamashita, T. (2001). Review of the superconducting properties of MgB2,

[3] Kang, W. N., Kim, H. J., Choi, E. M., Jung, C. U. & Lee, S. I. (2001). MgB2 superconducting

[4] Eom, C. B., Lee, M. K., Choi, J. H., Belenky, J. J., Song, X., Colley, L. D., Naus, M. T.,

[5] Zeng, X., Pogrebnyakov, A. V., Kotcharov, A., Jones, J. E., Xi, X. X., Lysczek, E. M., Redwing,

epitaxial MgB2 thin films for superconducting electronics, *Nature Mater.* 1: 35. [6] Ueda, K. & Naito, M. (2003). In situ growth of superconducting MgB2 thin films by

[7] Jo, W., Huh, J.-H., Ohnishi, T., Marshall, A. F., Beasley, M. R. & Hammond, R. H. (2002).

[8] Erven, A. J., Kim, T. H., Muenzenberg, M. & Moodera, J. S. (2002). Highly crystallized as-

[9] Kobayashi, Y., Doi, T., Kitaguchi, H., Okuzono, M., Nagatomo, K., Hamada, S. & Hakuraku,

[12] Tsujimoto, K., Shimakage, H., Wang, Z. & Kaya, N. (2005). Crystallinity and superconductivity of as-grown MgB2 thin films with AlN buffer layers, *Physica C* 426-431: 1464. [13] Harada, Y., Takahashi, T., Kuroha, M., Iriuda, H., Nakanishi, Y., Izumida, H., Endo, H.

substrates by molecular beam epitaxy, *Physica C* 445-448: 884.

Patnaik, S., Jiang, J. J., Rikel, M., Polyanskii, A., Gurevich, A., Cai, X. Y., Bu, S. D., Babcock, S. E., Hellstrom, E. E., Larbalestier, D. C., Rogado, N., Regan, K. A., Hayward, M. A., He, T., Slusky, J. S., Inumaru, K., Haas, M. K. & Cava, R. J. (2001). Thin film magnesium boride superconductor with very high critical current density

J. M., Xu, S., Li, Q., Lettieri, J., Schlom, D. G., Tian, W., Pan,X. & Liu, Z. K. (2002). In situ

In situ growth of superconducting MgB2 thin films with preferential orientation by

grown smooth and superconducting MgB2 films by molecular beam epitaxy, *Appl.* 

Y. (2005). Preparation of MgB2 thin films by an electron-beam evaporation technique and post-annealing effect on the as-grown films, *TEION KOGAKU* 40: 7 (in Japanese). [10] Liu, Z.-K., Schlom, D. G., Li, Q. & Xi, X. X. (2001). As-grown superconducting MgB2 thin films prepared by molecular beam epitaxy, *Appl. Phys. Lett.* 78: 3678. [11] Sakata, O., Kimura, S., Tanaka, M., Yata, S., Kato, T., Yamanaka, K., Yamada, Y.,

Matsushita, A. & Kubo, S. (2004). High-quality as-grown MgB2 thin-films fabrication at a low substrate temperature using an in-plane-lattice near-matched

& Yoshizawa, M. (2006). Fabrication of as-grown MgB2 films on ZnO (0001)

Superconductivity at 39K in magnesium diboride, *Nature* 410: 53.

thin films with a transition temperature of 39 Kelvin, *Science* 292: 1521.

and enhanced irreversibility field, *Nature* 411: 558.

molecular beam epitaxy, *J. Appl. Phys.* 93: 2113.

molecular beam epitaxy, *Appl. Phys. Lett.* 80: 3563.

epitaxial-buffer-layer. *J. Appl. Phys.* 96: 3580.

**7. Acknowledgments** 

and Technology (MEXT), Japan.

*Phys. Lett.* 81: 4982.

*Supercond. Sci. Technol.* 14: R115.

**8. References** 

Fig. 16. Magnetocardiogram waveform measured by the MgB2 SQUID system inside an MSR. The waveform was obtained by applying a software low-pass filter (40Hz) and 536 times averaging.

## **6. Summary**

We built a prototype of a nano-bridge type MgB2 SQUID device and optimized it for the measurement of magnetocardiogram. The results indicate that our method for the fabrication of SQUID of an MgB2 magnetometer optimized for MCG measurement is effective. We are currently conducting research to reduce the noise from the refrigerator and to optimize the MgB2 SQUID MCG system. A multi-channel type MCG system is also under development. The crystalline characteristics of MgB2 films should be further improved to obtain better device performance. In the near future, the MgB2 SQUID device is expected to have excellent performance. In addition to the SQUID device, we developed a highfrequency filter device. The detailed results of the filter device are shown in the papers of coworkers at Yamagata University [29]. Figure 17 shows the photographs of the filter device that was developed and studied.

Fig. 17. Photographs of the high-frequency filter device. The filter device was mounted on a Cu cavity.

## **7. Acknowledgments**

402 Superconductors – Properties, Technology, and Applications

R wave T wave

Magnetic flux density (pT)

40

30

20

10

0



times averaging.

**6. Summary** 

Cu cavity.

that was developed and studied.

Time (s) 0 0.2 0.4 0.6 0.8

We built a prototype of a nano-bridge type MgB2 SQUID device and optimized it for the measurement of magnetocardiogram. The results indicate that our method for the fabrication of SQUID of an MgB2 magnetometer optimized for MCG measurement is effective. We are currently conducting research to reduce the noise from the refrigerator and to optimize the MgB2 SQUID MCG system. A multi-channel type MCG system is also under development. The crystalline characteristics of MgB2 films should be further improved to obtain better device performance. In the near future, the MgB2 SQUID device is expected to have excellent performance. In addition to the SQUID device, we developed a highfrequency filter device. The detailed results of the filter device are shown in the papers of coworkers at Yamagata University [29]. Figure 17 shows the photographs of the filter device

(a) (b)

Fig. 17. Photographs of the high-frequency filter device. The filter device was mounted on a

Fig. 16. Magnetocardiogram waveform measured by the MgB2 SQUID system inside an MSR. The waveform was obtained by applying a software low-pass filter (40Hz) and 536 The authors thank H. Yamaguchi, Y. Fujine, D. Oyama, K. Ikeda, S. Goto, T. Nakajima, T. Takahashi, H. Iriuda, T. Oba, A. Okubo, J. Araaki, K. Meguro, T. Abe, H. Endo, Y. Uchikawa and K. Fujisawa for the cooperation of MgB2 SQUID project, and M. Iitake for his technical assistance of FIB process. This work was supported by the Japan Science and Technology Agency. A part of this work was conducted at the AIST Nano-Processing Facility, supported by "Nanotechnology Network Japan" of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

## **8. References**


**1. Introduction**

is cleared, etc.).

Universidad Autónoma de Madrid, Spain.

Superconducting fault current limiters (SFCL) based on high temperature superconductors (HTSC) of different types, inductive, resistive or hybrid, have being extendedly studied (Noe & Steurer, 2007; Sokolovsky et al., 2004; Paul et al., 2001) and tested in power plants (Paul et al., 1997; Leung et al., 1997; Zueger et al., 1998). Inductive SFCL, intended to operate at very high powers, are in general based on bulk HTSC whereas resistive or hybrid types are based on both bulk (mainly in the shape of bars) and thin films grown on different types of substrates and mainly patterned in meandered paths. HTSC based limiters represent one of the more promising applications of such materials but both bulk and thin film HTSC present different drawbacks which make difficult the practical implantation of these devices. Important problems concern, in particular, to the thermal behaviour of samples under current faults. In bulk samples, which are quite inhomogeneous, specially the huge elements required for limiters operating at high powers, overcurrents under fault generate hot spots in weak zones which endanger their thermal stability and lead to an irreversible damage or to an unsatisfactory performance of the device (low impedance, long recovery times once the fault

**Thermal Behaviour and Refrigeration of** 

**Current Limiters and Microlimiters** 

*Universidade de Santiago de Compostela, Santiago de Compostela* 

*LBTS, Departamento de Física da Materia Condensada,* 

J.A. Veira, M.R. Osorio\* and F. Vidal

*Spain* 

**19**

**High-Temperature Superconducting Fault** 

In terms of thermal behaviour, thin films could show a better performance than bulk samples. Their critical current density and electric field above their normal-superconducting transition are around two orders of magnitude higher than those for bulk samples. Accordingly, the heat developed per unit volume under a current fault is also orders of magnitude higher. This dissipation, together with the small thickness of films, should provoke a homogeneous and full transition of the overall sample showing, therefore, a very fast and abrupt increase of the resistance (Antognazza et al., 2002). In addition, the large surface-to-volume ratio of thin film samples, the low boundary thermal impedance film-substrate and the great thermal conductivity of substrates determine a very good heat evacuation once the current fault is cleared. Hence the recovery times observed in limiters based on thin films are much shorter than for limiters based on bulk samples (Osorio et al., 2006). In what concerns the transition to the normal state it may occur, however, that indeed only some parts of the film meander

\*Present address: Laboratorio de Bajas Temperaturas, Departamento de Física de la Materia Condensada,


## **Thermal Behaviour and Refrigeration of High-Temperature Superconducting Fault Current Limiters and Microlimiters**

J.A. Veira, M.R. Osorio\* and F. Vidal *LBTS, Departamento de Física da Materia Condensada, Universidade de Santiago de Compostela, Santiago de Compostela Spain* 

#### **1. Introduction**

404 Superconductors – Properties, Technology, and Applications

[14] Harada, Y., Yamaguchi, H., Oba, Y., Fujine, Y., Goto, S., Iriuda, H. & Yoshizawa, M.

[15] Nishidate, K., Yoshizawa, M. & Hasegawa, M. (2008). Energetics of Mg and B adsorption on

[16] Haruta, M., Fujiyoshi, T., Kihara, S., Sueyoshi, T., Miyahara, K., Harada, Y., Yoshizawa,

[17] Ueda, K. & Naito, M. (2004). Tunnel junctions on as-grown MgB2 films, *Physica C* 408–

[18] Oba, T., Goto, S., Sasaki, S., Nakanishi, Y., Fujino, Y., Harada, Y., Nakamura, M., Saito,

[19] Gonnelli, R. S., Daghere, D., Calzolari, A., Ummarino, G. A., Tortello, M., Stepanov, V.

[20] Harada, Y., Oyama, D., Fujine, Y., Goto, S., Ikeda, K., Nakashima, T., Meguro, K.,

[21] Drung, D. (2003). High- Tc and low-Tc dc SQUID electronics, *Supercond. Sci. Technol.* 16:

[22] Ludwig, C., Kessler, C., Steinfort, A. J. & Ludwig, W. (2001). Versatile high performance

[23] Schurig, T., Drung, D., Bechstein, S., Beyer, J. & Ludwig, F. (2002). High-Tc superconductor dc SQUIDs for unshielded operation and their applications, *Physica C*, 378-381: 1378. [24] Vrba, J., Betts, K., Burbank, M., Cheung, T., Fife, A. A., Haid, G., Kubik, P. R. &

[25] Vrba, J. (1996). SQUID gradiometers in real environments SQUID Sensors:

[26] Oyama, D., Kobayashi, K., Yoshizawa, M. and Uchikawa, Y. (2006). Development of

[27] Yoshizawa, M., He, D. F., Nakai, K., Kobayashi, K., Nakamura, Y., Yaegashi, M., Ito, M.,

[28] Oyama, D., Harada, Y., Fujine, Y., Kobayashi, K., Uchikawa, Y. & Yoshizawa, M. (2009).

[29] Ono, S., Kimura, M., Sekiya, N., Saito, A., Hirano, S., Ohshima, S., Harada, Y.,

SQUIDs in Iwate CREATE project, *Physica C* 426-431: 1572.

resonators, *IEEE. Trans. Appl. Supercond.* 17: 89.

digital SQUID electronics, *IEEE Trans. Appl. Supercond.* 11: 1122.

polar zinc oxide surfaces from first principles, *Phys. Rev. B* 77: 035330.

insulator layer, *Physica C: Superconductivity* 468: 1892.

MgB2 films, *TEION KOGAKU* 43: 354 (in Japanese).

*IEEE Trans. Appl. Supercond.* 3: 878.

Weinstock (Dordrecht: Kluwer) 117.

*Magnetics* 42: 3539.

*Europe* GO-04.

*Physica C* 463-465: 945.

410: 134.

*Physica C* 435: 59.

1320.

(2007). Novel stage in fabrication of as-grown MgB2 films by adopting Ti seed layer,

M., Takahashi, T., Iriuda, H., Oba, T., Awaji, S., Watanabe, K. & Miyagawa, R. (2007). High critical current density under magnetic fields in as-grown MgB2 thin films deposited by molecular-beam epitaxy," Supercond. Sci. Technol. 20 (2007) L1.

A. & Yoshizawa, M. (2008). Fabrication of all-MgB2 Josephson junctions using MgO

A., Zhigadlo, N. D., Rogacki, K., Portesi, C., Monticone, E., Mijatovic, D., Veldhuis, D. & Brinkman, A. (2006). Recent achievements in MgB2 physics and applications: A large-area SQUID magnetometer and point-contact spectroscopy measurements,

Kobayashi, K. & Yoshizawa, M. (2008). Fabrication of DC-SQUIDs of As-grown

Weinberg, H. (1993). Whole cortex, 64 channel SQUID biomagnetometer system,

Fundamentals, Fabrication and Applications, (NATO ASI Series E329) ed H.

digital FLL system for SQUID using double counter method, *IEEE Transactions on* 

Yashiro, H., Daibo, M., Simizu, T., Uchikawa, Y. & Noto, K. (2005). Application of

Detection of MCG signal by using MgB2 SQUID on pulse-tube cryocooler, *Intermag* 

Takahashi, T., Iriuda, H. & Yoshizawa, M. (2007). Design and fabrication of 5GHz miniaturized bandpass filters using superconducting microstrip quasi-spiral Superconducting fault current limiters (SFCL) based on high temperature superconductors (HTSC) of different types, inductive, resistive or hybrid, have being extendedly studied (Noe & Steurer, 2007; Sokolovsky et al., 2004; Paul et al., 2001) and tested in power plants (Paul et al., 1997; Leung et al., 1997; Zueger et al., 1998). Inductive SFCL, intended to operate at very high powers, are in general based on bulk HTSC whereas resistive or hybrid types are based on both bulk (mainly in the shape of bars) and thin films grown on different types of substrates and mainly patterned in meandered paths. HTSC based limiters represent one of the more promising applications of such materials but both bulk and thin film HTSC present different drawbacks which make difficult the practical implantation of these devices. Important problems concern, in particular, to the thermal behaviour of samples under current faults. In bulk samples, which are quite inhomogeneous, specially the huge elements required for limiters operating at high powers, overcurrents under fault generate hot spots in weak zones which endanger their thermal stability and lead to an irreversible damage or to an unsatisfactory performance of the device (low impedance, long recovery times once the fault is cleared, etc.).

In terms of thermal behaviour, thin films could show a better performance than bulk samples. Their critical current density and electric field above their normal-superconducting transition are around two orders of magnitude higher than those for bulk samples. Accordingly, the heat developed per unit volume under a current fault is also orders of magnitude higher. This dissipation, together with the small thickness of films, should provoke a homogeneous and full transition of the overall sample showing, therefore, a very fast and abrupt increase of the resistance (Antognazza et al., 2002). In addition, the large surface-to-volume ratio of thin film samples, the low boundary thermal impedance film-substrate and the great thermal conductivity of substrates determine a very good heat evacuation once the current fault is cleared. Hence the recovery times observed in limiters based on thin films are much shorter than for limiters based on bulk samples (Osorio et al., 2006). In what concerns the transition to the normal state it may occur, however, that indeed only some parts of the film meander

<sup>\*</sup>Present address: Laboratorio de Bajas Temperaturas, Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, Spain.

100 150 200 250 300

*T* (K)

*T*cI

<sup>407</sup> Thermal Behaviour and Refrigeration

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

0.0 0.1 0.2 0.3 0.4

ρ (mΩ-cm)

Fig. 1. Resistivity as a function of temperature for a piece of the material employed as secondary of the SFCL. Inset: detailed view of the superconducting transition. The solid line represents the temperature derivative (on arbitrary units). The critical temperatures *T*cI and *T*c0 are defined as the ones corresponding to the maximum of the derivative and the

width of the derivative at half height. Figure from Ref. (Osorio et al., 2004).

(FC) of the sample and squares to its zero field cooling (ZFC).

**2.1.2 Samples' behaviour under a current fault**

minimum value for which *ρ* 0, respectively. The full width half maximum (FWHM) is the

transition, defined as the width of the derivative *dρ*/*dT* (solid line) at medium height. *Tc I* the temperature at which *dρ*/*dT* presents its maximum, and *Tc*0, the minimum temperature for which *ρ* 0 are indicated. In the CVC shown in figure 2, symbols represent the experimental data measured at the indicated temperatures and the solid lines the fits to the usual power law function (Prester, 1998; Osorio et al., 2004). In figure 3 circles correspond to the field cooling

X-ray diffraction analyses (Osorio et al., 2004) show the presence in our samples of two phases: around 15% of Bi2212 and 85% of Bi2223. In *ρ*(*T*) or CVC it cannot be distinguished any effect provokated by this inhomogeneity, as may be expected because of the percolative nature of electrical transport properties. It is, however, well visible in the *χ*(*T*) curves. As can be seen in the FC and ZFC measurements of figure 3, the normal-superconducting transition occurs in two steps: one around 108 K (corresponding to Bi2223 phase) and the the other around 98 K (corresponding to Bi2212 phase). We will se below that to numerically account for the electrical resistance and temperature of our samples operating as secondaries in our inductive

The electrical resistance and temperature evolution under a current fault of cylinders or rings when acting as secondaries in an inductive limiter was measured by attaching to the samples some voltage contacts and thermocouples along their perimeter as indicated in figure 4. Details on the limiter configuration and global experimental set up can be seen in reference (Osorio et al., 2004). In figures 5 and 6 we show typical results for this type of test. In this example data were measured by using a 5 mm in height ring with five couples of voltage

SFCL under current fault, the bi-phase nature of the material plays a central role.

105 110 115 120

*T*c0

FWHM

0.0 0.3 0.6 0.9 1.2

(dρ/d*T*) (arbitrary units)

*T* (K)

0.0

0.2

0.4

ρ (m

Ω-cm)

0.6

0.8

are triggered to the normal state by the current fault (Paul et al., 2001). In that case dissipation can be so concentrated in some of these parts that films could also be irreversibly damaged. To prevent that some weak parts of the long meander thin film paths be burning out, it has been proposed to fabricate these elements by patterning a set of constrictions alternated with wider portions. With this design, the normal transition would formerly occur in the narrower segments and the heat arising from this overheated sections will rapidly propagate along the whole meander path (i.e. including also the wider parts), hence leading to a complete and more homogeneous transition (Decroux et al., 2003; Antognazza et al., 2005; Duron et al., 2005; 2007; Antognazza et al., 2007; Duron et al., 2007).

For both bulk and thin film based SFCL the thermal behaviour and refrigeration of the superconducting elements determine, therefore, the overall performance of these devices. In this work we will summarize some of our results on this issue including the thermal recovery of inductive SFCL operating with bulk cylinders of different sizes (Osorio et al., 2006) or with artificial weak zones (Osorio et al., 2005; 2008), the improvement obtained by using stacks of bulk rings or thin film washers as secondary (Osorio et al., 2004; 2006), the use of thermoacoustic refrigerators (Osorio et al., 2008) and the possibility of using thin film microbridges as very efficient microlimiters in very low power applications (Lorenzo et al., 2009).

## **2. Bulk cylinders as secondary of inductive SFCL**

Despite the advances in the fabrication of HTSC materials the presence in the samples of inhomogeneities which act as weak zones still represents an important drawback for their use in SFCL. The presence of weak zones provokes an earlier transition to a dissipative state of only a part of the samples that, as indicated above, can cause a great degradation of the performance of the overall SFCL device. In the first part of this section we will summarize our results on the non-homogeneous quench of bulk Bi-2223 cylinders acting as secondaries of inductive SFCL, in particular, on the current limitation capacity under fault of the limiters and their thermal recovery once the current fault is cleared. In the second part we will outline our numerical approaches to account for the effects of the bi-phase character of Bi-2223 samples on the quench under a current fault.

### **2.1 Non-homogeneous quench of the superconducting secondary of inductive SFCL**

#### **2.1.1 Samples' characteristics**

In our studies of inductive SFCL based on bulk HTSC we have used samples of nominal composition Bi1.8Pb0.26Sr2Ca2Cu3O10+*x*. This samples were obtained from cylinders of 120 mm in height, 21 mm in internal diameter and 2.4 mm in wall thickness supplied by Can Superconductors (Czech Republic). Commercial samples were cut in shorted cylinders and rings, around 40 mm and 5 mm in height, respectively, to carry out our measurements. Also, other small pieces were properly cut for material characterization: electrical resistivity, characteristic voltage-current curves (CVC), magnetic susceptibility and x-ray diffraction. The results of our characterization tests show a very good reproducibility for different pieces of the same commercial cylinders and also for pieces obtained from different cylinders.

Typical examples of electrical resistivity versus temperature curves, *ρ*(*T*), CVC and magnetic susceptibility *χ*(*T*) are displayed in figures 1 to 3. In figure 1 the dotted line represents the linear fir to the normal state resistivity data. In the inset it is shown the width of the 2 Will-be-set-by-IN-TECH

are triggered to the normal state by the current fault (Paul et al., 2001). In that case dissipation can be so concentrated in some of these parts that films could also be irreversibly damaged. To prevent that some weak parts of the long meander thin film paths be burning out, it has been proposed to fabricate these elements by patterning a set of constrictions alternated with wider portions. With this design, the normal transition would formerly occur in the narrower segments and the heat arising from this overheated sections will rapidly propagate along the whole meander path (i.e. including also the wider parts), hence leading to a complete and more homogeneous transition (Decroux et al., 2003; Antognazza et al., 2005; Duron et al.,

For both bulk and thin film based SFCL the thermal behaviour and refrigeration of the superconducting elements determine, therefore, the overall performance of these devices. In this work we will summarize some of our results on this issue including the thermal recovery of inductive SFCL operating with bulk cylinders of different sizes (Osorio et al., 2006) or with artificial weak zones (Osorio et al., 2005; 2008), the improvement obtained by using stacks of bulk rings or thin film washers as secondary (Osorio et al., 2004; 2006), the use of thermoacoustic refrigerators (Osorio et al., 2008) and the possibility of using thin film microbridges as very efficient microlimiters in very low power applications (Lorenzo et al.,

Despite the advances in the fabrication of HTSC materials the presence in the samples of inhomogeneities which act as weak zones still represents an important drawback for their use in SFCL. The presence of weak zones provokes an earlier transition to a dissipative state of only a part of the samples that, as indicated above, can cause a great degradation of the performance of the overall SFCL device. In the first part of this section we will summarize our results on the non-homogeneous quench of bulk Bi-2223 cylinders acting as secondaries of inductive SFCL, in particular, on the current limitation capacity under fault of the limiters and their thermal recovery once the current fault is cleared. In the second part we will outline our numerical approaches to account for the effects of the bi-phase character of Bi-2223 samples

**2.1 Non-homogeneous quench of the superconducting secondary of inductive SFCL**

same commercial cylinders and also for pieces obtained from different cylinders.

In our studies of inductive SFCL based on bulk HTSC we have used samples of nominal composition Bi1.8Pb0.26Sr2Ca2Cu3O10+*x*. This samples were obtained from cylinders of 120 mm in height, 21 mm in internal diameter and 2.4 mm in wall thickness supplied by Can Superconductors (Czech Republic). Commercial samples were cut in shorted cylinders and rings, around 40 mm and 5 mm in height, respectively, to carry out our measurements. Also, other small pieces were properly cut for material characterization: electrical resistivity, characteristic voltage-current curves (CVC), magnetic susceptibility and x-ray diffraction. The results of our characterization tests show a very good reproducibility for different pieces of the

Typical examples of electrical resistivity versus temperature curves, *ρ*(*T*), CVC and magnetic susceptibility *χ*(*T*) are displayed in figures 1 to 3. In figure 1 the dotted line represents the linear fir to the normal state resistivity data. In the inset it is shown the width of the

2005; 2007; Antognazza et al., 2007; Duron et al., 2007).

**2. Bulk cylinders as secondary of inductive SFCL**

on the quench under a current fault.

**2.1.1 Samples' characteristics**

2009).

Fig. 1. Resistivity as a function of temperature for a piece of the material employed as secondary of the SFCL. Inset: detailed view of the superconducting transition. The solid line represents the temperature derivative (on arbitrary units). The critical temperatures *T*cI and *T*c0 are defined as the ones corresponding to the maximum of the derivative and the minimum value for which *ρ* 0, respectively. The full width half maximum (FWHM) is the width of the derivative at half height. Figure from Ref. (Osorio et al., 2004).

transition, defined as the width of the derivative *dρ*/*dT* (solid line) at medium height. *Tc I* the temperature at which *dρ*/*dT* presents its maximum, and *Tc*0, the minimum temperature for which *ρ* 0 are indicated. In the CVC shown in figure 2, symbols represent the experimental data measured at the indicated temperatures and the solid lines the fits to the usual power law function (Prester, 1998; Osorio et al., 2004). In figure 3 circles correspond to the field cooling (FC) of the sample and squares to its zero field cooling (ZFC).

X-ray diffraction analyses (Osorio et al., 2004) show the presence in our samples of two phases: around 15% of Bi2212 and 85% of Bi2223. In *ρ*(*T*) or CVC it cannot be distinguished any effect provokated by this inhomogeneity, as may be expected because of the percolative nature of electrical transport properties. It is, however, well visible in the *χ*(*T*) curves. As can be seen in the FC and ZFC measurements of figure 3, the normal-superconducting transition occurs in two steps: one around 108 K (corresponding to Bi2223 phase) and the the other around 98 K (corresponding to Bi2212 phase). We will se below that to numerically account for the electrical resistance and temperature of our samples operating as secondaries in our inductive SFCL under current fault, the bi-phase nature of the material plays a central role.

#### **2.1.2 Samples' behaviour under a current fault**

The electrical resistance and temperature evolution under a current fault of cylinders or rings when acting as secondaries in an inductive limiter was measured by attaching to the samples some voltage contacts and thermocouples along their perimeter as indicated in figure 4. Details on the limiter configuration and global experimental set up can be seen in reference (Osorio et al., 2004). In figures 5 and 6 we show typical results for this type of test. In this example data were measured by using a 5 mm in height ring with five couples of voltage

*Vab Vbc*

*Vde Vcd*

<sup>409</sup> Thermal Behaviour and Refrigeration

Voltage paths

Fig. 4. Scheme of division into regions (five in this example) of a ring with voltage paths and thermocouples attached on each region. The dashed lines indicate the artificial weak zone made by reducing the cross section of the ring in a small portion of its perimeter. Figure from

contacts and five thermocouples. Figure 5 shows the measured wave-forms. For clarity we

0.00 3.45 3.50 3.55 3.60 3.65

*t* (s)

Fig. 5. Voltage measured during a fault with the arrangement depicted in figure 4. The different regions do not behave in the same way, as one of them cools down while others

In figure 6(a) it is shown the rms values of the five voltage signals and in figure 6(b) the corresponding temperature readings. We can observe that the voltage signals of some zones grow continuously during the fault but others experience the inverse tendency, so the voltage is progressively reduced up to typical values of the flux-flow regime and even up to zero indicating that these zones return to the superconducting state. The corresponding temperatures, displayed in figure 6(b), also indicate this tendency: the parts at high voltage heat up to temperatures well above *Tc I* whereas the parts at low voltage scarcely heat up remaining at temperatures below *Tc*0. In this sample, for the region denoted *cd*, the temperature reading is lower that the one that could be expected by inspecting the rms

Thermocouples

use lines instead of symbols and represent only three of the five signals.

*Vab Vbc*

*Vcd*



Fault starts

enter in the ohmic state. Figure from Ref. (Osorio et al., 2004).

0

*V*sup (mV)

20

40

*Vef*

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

AWZ

Ref. (Osorio et al., 2004).

Fig. 2. *E* − *J* curves measured in a piece of the superconducting elements employed in the limiter. The solid lines represent the fittings to the usual power law function (Prester, 1998). Figure from Ref. (Osorio et al., 2004).

Fig. 3. Variation with temperature of the magnetic susceptibility during field cooling and zero field cooling for a piece of the cylinder used as secondary of the limiter. Notice the presence of two superconducting phases, both identified by their own critical temperatures. Figure from Ref. (Osorio et al., 2004).

4 Will-be-set-by-IN-TECH

0 750 1500 2250 3000

 98 K 102 K 104 K

108 K

*J* (A/cm2 )

40 60 80 100 120

Fig. 3. Variation with temperature of the magnetic susceptibility during field cooling and zero field cooling for a piece of the cylinder used as secondary of the limiter. Notice the presence of two superconducting phases, both identified by their own critical temperatures.

T (K)

98 K

Fig. 2. *E* − *J* curves measured in a piece of the superconducting elements employed in the limiter. The solid lines represent the fittings to the usual power law function (Prester, 1998).

0




χ (emu)

Figure from Ref. (Osorio et al., 2004).


Bi-lim-3

 FC ZFC

0.000

20

*E* (mV/cm)

Figure from Ref. (Osorio et al., 2004).

40

60

 77 K 79 K 82 K 85 K 88 K 90 K 94 K

Fig. 4. Scheme of division into regions (five in this example) of a ring with voltage paths and thermocouples attached on each region. The dashed lines indicate the artificial weak zone made by reducing the cross section of the ring in a small portion of its perimeter. Figure from Ref. (Osorio et al., 2004).

contacts and five thermocouples. Figure 5 shows the measured wave-forms. For clarity we use lines instead of symbols and represent only three of the five signals.

Fig. 5. Voltage measured during a fault with the arrangement depicted in figure 4. The different regions do not behave in the same way, as one of them cools down while others enter in the ohmic state. Figure from Ref. (Osorio et al., 2004).

In figure 6(a) it is shown the rms values of the five voltage signals and in figure 6(b) the corresponding temperature readings. We can observe that the voltage signals of some zones grow continuously during the fault but others experience the inverse tendency, so the voltage is progressively reduced up to typical values of the flux-flow regime and even up to zero indicating that these zones return to the superconducting state. The corresponding temperatures, displayed in figure 6(b), also indicate this tendency: the parts at high voltage heat up to temperatures well above *Tc I* whereas the parts at low voltage scarcely heat up remaining at temperatures below *Tc*0. In this sample, for the region denoted *cd*, the temperature reading is lower that the one that could be expected by inspecting the rms

0 3 6 9 12 15

*x* = 1 cm

*x* = 2 cm

*t* (s)

Fig. 7. Temperature excursions when provoking the transition (by exceeding the critical current) of the ring of figure 4 (once the AWZ has been made). The readings of the thermocouples located 1 cm away from the hot spot are denoted by down-triangles and squares, and the ones for 2 cm away by up-triangles and diamonds. Notice that, while the majority of the ring heats up scarcely, and it remains very far from the ohmic state, the dissipative zone (circles) warms up till around the critical temperature. Then it is reached a stable plateau till the end of the fault at *t* = 30 s. The fact that, during the first seconds, the temperature increment is slightly greater at *x* = 2 cm than at *x* = 1 cm can be due to the local

increment in the majority of the ring is quite low, as the thermocouples indicate, and only in the weak zone it is reached the critical value, *Tc*. In fact, the temperature increment far away from the weak zone should be due to heat coming from the dissipative zone, instead of being generated in the corresponding parts of the sample. This fact would explain why at 1 cm away from the centre of the weak zone the temperature increases slowly when it has been already reached a stable plateau in the hottest region. Notice that the showed behaviour, with the decreasing temperature in the further parts of the ring, means that the warm zone does not tend to expand itself, but remains essentially still till the end of the fault (at about *t* = 30 s). This fact is in part a consequence of the reduction of the induced current, due to the gain in impedance originated by the very resistive zone. As the current that circulates in the ring diminishes, the dissipation becomes lower, and so the thermal excursion in the majority of the sample. This fact provokes that the resistance of the superconductor, and so the total

The behaviour of inductive SFCL is mainly determined by the overheating of the weak zone in the superconducting cylinder. In turn, the behavior of this zone seems to be mainly dominated by the minority phase Bi-2212, that has the lower critical temperature *Tc*<sup>0</sup> ≈ 98 K, as indicates the change of slope observed in the voltage curves of figure 6(a), which corresponds to Δ*T* ≈

20 K, i.e. at *T* ≈ 98 K, in the temperature versus time curves plotted in figure 6(b).

Weak zone

<sup>411</sup> Thermal Behaviour and Refrigeration

0

inhomogeneities. Figure from Ref. (Osorio et al., 2004).

impedance of the limiter, increase quite smooth.

**2.2 The effects of the bi-phase character of samples on the quench**

10

Fault starts

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

20

Δ*T* (K)

30

40

Fig. 6. (a) Rms voltages for all the considered regions of the ring of figure 4. The major part of the sample recovers the superconducting state as the current is limited by the dissipative zones. (b) Temperature excursions obtained for an identical fault. As it can be seen, it exists a remarkable coherence between these results and the voltage measurements. See text for details. Figure from Ref. (Osorio et al., 2004).

voltage. This disagreement could be due to the presence of a relatively small thermal inhomogeneity in this zone, or to an inhomogeneity located in depth, relatively far away from the point at which the thermocouple was attached. In all the studied rings it was found a voltage signal behaviour like the one followed by the voltages denoted *ab*, clearly due to the thermal evolution during the fault of the more dissipative zone in each sample.

The results obtained in this type of test are very similar to those of figure 7 measured by using a ring in which a weak zone was created by reducing its height somewhere (as indicated by dotted lines in figure 4). The symbols represent the temperatures measured with thermocouples attached at the weak zone and at points at ±1 cm and ±2 cm far away along the ring perimeter. When the fault is provoked, so the induced current overtakes the critical current, *I*c, the whole superconductor enters in the dissipative state, but the temperature 6 Will-be-set-by-IN-TECH

*Vab Vbc Vcd Vde Vea*

> *Tab Tbc Tcd Tde Tea*

02468

Fig. 6. (a) Rms voltages for all the considered regions of the ring of figure 4. The major part of the sample recovers the superconducting state as the current is limited by the dissipative zones. (b) Temperature excursions obtained for an identical fault. As it can be seen, it exists a remarkable coherence between these results and the voltage measurements. See text for

voltage. This disagreement could be due to the presence of a relatively small thermal inhomogeneity in this zone, or to an inhomogeneity located in depth, relatively far away from the point at which the thermocouple was attached. In all the studied rings it was found a voltage signal behaviour like the one followed by the voltages denoted *ab*, clearly due to the

The results obtained in this type of test are very similar to those of figure 7 measured by using a ring in which a weak zone was created by reducing its height somewhere (as indicated by dotted lines in figure 4). The symbols represent the temperatures measured with thermocouples attached at the weak zone and at points at ±1 cm and ±2 cm far away along the ring perimeter. When the fault is provoked, so the induced current overtakes the critical current, *I*c, the whole superconductor enters in the dissipative state, but the temperature

thermal evolution during the fault of the more dissipative zone in each sample.

*t* (s)

(b)

Heating of the weak zone beyond *T*c0

Return to the superconducting state

(a)

Transition to the ohmic state

0

10

20

Fault starts

Δ

details. Figure from Ref. (Osorio et al., 2004).

*T* (K ) 30

40

0

10

20

*V*suprms (mV)

30

Fig. 7. Temperature excursions when provoking the transition (by exceeding the critical current) of the ring of figure 4 (once the AWZ has been made). The readings of the thermocouples located 1 cm away from the hot spot are denoted by down-triangles and squares, and the ones for 2 cm away by up-triangles and diamonds. Notice that, while the majority of the ring heats up scarcely, and it remains very far from the ohmic state, the dissipative zone (circles) warms up till around the critical temperature. Then it is reached a stable plateau till the end of the fault at *t* = 30 s. The fact that, during the first seconds, the temperature increment is slightly greater at *x* = 2 cm than at *x* = 1 cm can be due to the local inhomogeneities. Figure from Ref. (Osorio et al., 2004).

increment in the majority of the ring is quite low, as the thermocouples indicate, and only in the weak zone it is reached the critical value, *Tc*. In fact, the temperature increment far away from the weak zone should be due to heat coming from the dissipative zone, instead of being generated in the corresponding parts of the sample. This fact would explain why at 1 cm away from the centre of the weak zone the temperature increases slowly when it has been already reached a stable plateau in the hottest region. Notice that the showed behaviour, with the decreasing temperature in the further parts of the ring, means that the warm zone does not tend to expand itself, but remains essentially still till the end of the fault (at about *t* = 30 s). This fact is in part a consequence of the reduction of the induced current, due to the gain in impedance originated by the very resistive zone. As the current that circulates in the ring diminishes, the dissipation becomes lower, and so the thermal excursion in the majority of the sample. This fact provokes that the resistance of the superconductor, and so the total impedance of the limiter, increase quite smooth.

#### **2.2 The effects of the bi-phase character of samples on the quench**

The behaviour of inductive SFCL is mainly determined by the overheating of the weak zone in the superconducting cylinder. In turn, the behavior of this zone seems to be mainly dominated by the minority phase Bi-2212, that has the lower critical temperature *Tc*<sup>0</sup> ≈ 98 K, as indicates the change of slope observed in the voltage curves of figure 6(a), which corresponds to Δ*T* ≈ 20 K, i.e. at *T* ≈ 98 K, in the temperature versus time curves plotted in figure 6(b).

and

relations

corresponding phase.

(Osorio et al., 2004).

<sup>Q</sup>˙ <sup>=</sup> *hP*�(*c*,*h*)

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

current, *I*sup, depends on the resistances, *R*(*c*,*h*)

80

90

100

*T* (K)

110

Cold part{

{

Change of slope around *T*c0 = 98 K

also considered as a Bi-2223 matrix with Bi-2212 inclusions of spherical shape

the model (i.e. the resistivity and the length of the weak zone). Figure from Ref.

Hot part

Fault starts

120

(*T*(*c*,*h*) <sup>−</sup> *<sup>T</sup>*b),

, and so on the length of the hot domain, �*h*,

*<sup>J</sup>* for *<sup>T</sup>* <sup>&</sup>lt; *<sup>T</sup>*c0 (2)

*ρ*normal(*T*) for *T* ≥ *T*c0, (3)

where *�* is the density of the material, *C*p(*T*) its heat capacity (Xiao et al., 1999), *h* the convective factor, which can be taken as temperature independent with a small error, *T*<sup>b</sup> is the environmental temperature and *P* is the cross section perimeter of the cylinder. The circulating

<sup>413</sup> Thermal Behaviour and Refrigeration

which is set as a free parameter in our calculations. It has been also supposed, for the sake of simplicity, that during the fault the heat flow along the cylinder is negligible, in agreement

The lines plotted in figure 9 represent the temperatures determined in this scenario. The experimental data are those corresponding to the *a*b region in figure 6. The dashed and short-dashed lines were obtained in the framework of the effective-medium-theory (Osorio et al., 2004; Davidson & Tinkhan, 1976), considering that the effective electrical resistivity, *ρ*e, of each domain can be approached according to the concentrations of the majority and the minority phases (*C*<sup>1</sup> and *C*2, respectively), the depolarization factor for the minority phase inclusions (*X*) and the corresponding resistivities, for which it is used the

and with *ρ*normal(*T*) = 103 + 2.3*T μ*Ω − *cm* (i.e., the same normal resistivity displayed in figure 1). The value of the critical temperature, *T*c0, must be chosen depending on the

> Experiment al Two phases One phase

 Two phases One phase

0246

Fig. 9. Temperature excursions simulated when it is supposed that it exists a hot spot created by a slightly different resistivity in a 10% of the perimeter of the cylinder. It is considered two cases: in the first one the superconductor is a pure Bi-2223 phase, while in the second one it is

*t* (s)

homogeneously distributed in the sample. It is also included the experimental data for the temperature excursion of the *a*b region of figure 6, which allows fixing the free parameters of

End of fault

with the low thermal conductivity of this material and with our experimental results.

*<sup>ρ</sup>*(*J*, *<sup>T</sup>*) = *<sup>E</sup>*(*J*)

The role of the minority phase in the thermal evolution of the superconducting element during a fault can be illustrated by using a very simple approach for the temperature distribution in the superconductor. The calculations are done by using an iterative numerical routine in Matlab (The MathWorks, Inc, USA). The magnetic circuit is represented through field variable inductance coefficients (Osorio et al., 2004; MITStaff, 1961; Paul et al., 1995) and the superconducting element is modeled by taking into account the experimental characterization data, as it will be described later. The input variable is the applied voltage, kept as constant, and the fault is simulated by reducing the impedance of the circuit. During the iterative process, the circulating currents (*I*p and *I*sup) are first calculated at a time *t*, and then the resistivity and the temperature increment of the superconducting element. These steps are repeated till it is reached a set of self-consistent values, so the routine advances to time *t* + *δt* and so on.

We suppose that the cylinder is divided into two temperature regions, as indicated in figure 8. The region around the weak zone is considered to be at a higher temperature (hot domain) than the remainder part of the cylinder (cold domain). The length, temperature and resistance of each domain are denoted, respectively, by *<sup>h</sup>*, *<sup>T</sup>h*, *<sup>R</sup><sup>h</sup>* <sup>≡</sup> (*ρh*(*I*sup, *<sup>T</sup>h*)*<sup>h</sup>*)(*A*⊥)−<sup>1</sup> and *<sup>c</sup>* (<sup>=</sup> <sup>2</sup>*π<sup>r</sup>* <sup>−</sup> *<sup>h</sup>*, being *<sup>r</sup>* the radius of the cylinder), *<sup>T</sup>c*, *<sup>R</sup><sup>c</sup>* <sup>≡</sup> (*ρ<sup>c</sup>* (*I*sup, *<sup>T</sup><sup>c</sup>* )*<sup>c</sup>*)(*A*⊥)−1, where *<sup>ρ</sup>*(*c*,*h*) represents the resistivity of each domain and *<sup>A</sup>*<sup>⊥</sup> the cylinder cross section.

The temperature evolution was calculated by applying to both regions the heat balance

$$\log A\_{\perp} \ell^{(c,h)} \mathcal{C}\_{\mathbb{P}}(T^{(c,h)}) \frac{\partial T^{(c,h)}}{\partial t} = \dot{\mathcal{W}} - \dot{\mathcal{Q}} \tag{1}$$

<sup>W</sup>˙ and <sup>Q</sup>˙ being the power dissipated inside each superconducting region and the convective flux through their surfaces, respectively, which can be expressed as

$$\dot{\mathcal{W}} = I\_{\text{sup}}^2 \mathcal{R}^{(c,h)} (I\_{\text{sup}}, T^{(c,h)})$$

Fig. 8. (a) Scheme of the model supposed for an inhomogeneous superconducting cylinder, which is divided into two regions. The dark one is the hot domain, denoted by the index *h* (for the cold one stands *c*), which it is supposed to have a defect that generates a more dissipative zone. Notice that this region occupies only a small portion of the cylinder perimeter, but its whole height, so the current that goes through both parts is the same. (b) Equivalent circuit of the inhomogeneous superconductor. The resistances of both parts are joint in series. Figure from Ref. (Osorio et al., 2004).

and

8 Will-be-set-by-IN-TECH

The role of the minority phase in the thermal evolution of the superconducting element during a fault can be illustrated by using a very simple approach for the temperature distribution in the superconductor. The calculations are done by using an iterative numerical routine in Matlab (The MathWorks, Inc, USA). The magnetic circuit is represented through field variable inductance coefficients (Osorio et al., 2004; MITStaff, 1961; Paul et al., 1995) and the superconducting element is modeled by taking into account the experimental characterization data, as it will be described later. The input variable is the applied voltage, kept as constant, and the fault is simulated by reducing the impedance of the circuit. During the iterative process, the circulating currents (*I*p and *I*sup) are first calculated at a time *t*, and then the resistivity and the temperature increment of the superconducting element. These steps are repeated till it is reached a set of self-consistent values, so the routine advances to time *t* + *δt*

We suppose that the cylinder is divided into two temperature regions, as indicated in figure 8. The region around the weak zone is considered to be at a higher temperature (hot domain) than the remainder part of the cylinder (cold domain). The length, temperature and resistance of each domain are denoted, respectively, by *<sup>h</sup>*, *<sup>T</sup>h*, *<sup>R</sup><sup>h</sup>* <sup>≡</sup> (*ρh*(*I*sup, *<sup>T</sup>h*)*<sup>h</sup>*)(*A*⊥)−<sup>1</sup> and *<sup>c</sup>* (<sup>=</sup> <sup>2</sup>*π<sup>r</sup>* <sup>−</sup> *<sup>h</sup>*, being *<sup>r</sup>* the radius of the cylinder), *<sup>T</sup>c*, *<sup>R</sup><sup>c</sup>* <sup>≡</sup> (*ρ<sup>c</sup>* (*I*sup, *<sup>T</sup><sup>c</sup>* )*<sup>c</sup>*)(*A*⊥)−1, where *<sup>ρ</sup>*(*c*,*h*)

represents the resistivity of each domain and *<sup>A</sup>*<sup>⊥</sup> the cylinder cross section.

*�A*⊥(*c*,*h*)

flux through their surfaces, respectively, which can be expressed as

*I* sup

(a)

joint in series. Figure from Ref. (Osorio et al., 2004).

<sup>W</sup>˙ <sup>=</sup> *<sup>I</sup>* 2 sup*R*(*c*,*h*)

The temperature evolution was calculated by applying to both regions the heat balance

) *∂T*(*c*,*h*)

<sup>W</sup>˙ and <sup>Q</sup>˙ being the power dissipated inside each superconducting region and the convective

(*I*sup, *T*(*c*,*h*)

)

*R c c* (, )

*R*

*I* sup

*h h* (, )

*J T*

*J T*

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> W −˙ <sup>Q</sup>˙ (1)

*C*p(*T*(*c*,*h*)

Weak zone

<sup>A</sup> (b) *<sup>h</sup>*

Fig. 8. (a) Scheme of the model supposed for an inhomogeneous superconducting cylinder, which is divided into two regions. The dark one is the hot domain, denoted by the index *h* (for the cold one stands *c*), which it is supposed to have a defect that generates a more dissipative zone. Notice that this region occupies only a small portion of the cylinder perimeter, but its whole height, so the current that goes through both parts is the same. (b) Equivalent circuit of the inhomogeneous superconductor. The resistances of both parts are

and so on.

$$\dot{\mathcal{Q}} = hP\ell^{(c,h)}(T^{(c,h)} - T\_{\mathbf{b}})\_{\lambda}$$

where *�* is the density of the material, *C*p(*T*) its heat capacity (Xiao et al., 1999), *h* the convective factor, which can be taken as temperature independent with a small error, *T*<sup>b</sup> is the environmental temperature and *P* is the cross section perimeter of the cylinder. The circulating current, *I*sup, depends on the resistances, *R*(*c*,*h*) , and so on the length of the hot domain, �*h*, which is set as a free parameter in our calculations. It has been also supposed, for the sake of simplicity, that during the fault the heat flow along the cylinder is negligible, in agreement with the low thermal conductivity of this material and with our experimental results.

The lines plotted in figure 9 represent the temperatures determined in this scenario. The experimental data are those corresponding to the *a*b region in figure 6. The dashed and short-dashed lines were obtained in the framework of the effective-medium-theory (Osorio et al., 2004; Davidson & Tinkhan, 1976), considering that the effective electrical resistivity, *ρ*e, of each domain can be approached according to the concentrations of the majority and the minority phases (*C*<sup>1</sup> and *C*2, respectively), the depolarization factor for the minority phase inclusions (*X*) and the corresponding resistivities, for which it is used the relations

$$\rho(f, T) = \frac{E(f)}{f} \text{ for } T < T\_{\text{c0}} \tag{2}$$

$$\rho\_{\text{normal}}(T) \text{ for } \ T \ge T\_{\text{0}\prime} \tag{3}$$

and with *ρ*normal(*T*) = 103 + 2.3*T μ*Ω − *cm* (i.e., the same normal resistivity displayed in figure 1). The value of the critical temperature, *T*c0, must be chosen depending on the corresponding phase.

Fig. 9. Temperature excursions simulated when it is supposed that it exists a hot spot created by a slightly different resistivity in a 10% of the perimeter of the cylinder. It is considered two cases: in the first one the superconductor is a pure Bi-2223 phase, while in the second one it is also considered as a Bi-2223 matrix with Bi-2212 inclusions of spherical shape homogeneously distributed in the sample. It is also included the experimental data for the temperature excursion of the *a*b region of figure 6, which allows fixing the free parameters of the model (i.e. the resistivity and the length of the weak zone). Figure from Ref. (Osorio et al., 2004).

012

of the cylinder plays a major role. Notice that this would entail that this weak zone would refrigerate faster than a homogeneous cylinder, provided that the same final temperature is attained in both cases. This suggests that if the cylinder was made up from weak zones separated by cold segments, after the fault clearance the excess heat could be removed from the hot parts to the colder ones, till the cylinder was thermalized at a low temperature, maybe higher than that of the coolant, but much lower than the maximum value obtained during the fault. It can be argued that, as only the weak parts would enter into the normal state, the impedance of the limiter should be much lower. Nevertheless, and despite this drawback, it could be found a configuration of made-to-measure weak zones, properly spaced, which allow obtaining an acceptable impedance but, in turn, would reduce the recovery time to a

The above ideas were proposed in previous works (Osorio et al., 2005; 2008; 2006), where two possible procedures where stated; in the first one (Osorio et al., 2005), the set of artificial weak zones (AWZ) could be made from a different material, with a lower *Jc* and a higher resistivity (so somewhat balancing the loss of impedance of the limiter), as depicted in figure 11(a), for a set of *n* AWZ of length *<sup>n</sup>*, *<sup>h</sup>* being the total weak length. Figure 11(b) shows the equivalent electrical circuit for the cylinder, with resistances *R<sup>c</sup>* and *R<sup>h</sup>* for the *cold* and *hot* parts (i.e. the

In the second procedure (Osorio et al., 2008; 2006), each AWZ would be made as a constriction of the cross-section of the cylinder as it is shown in figure 14(a). This configuration allows using the same material for the whole sample, and the reduction in the critical current and the gain in the total impedance is easily yielded by the diminution of the cross-section. For convenience, this idea was initially tested by means of a 1-D numerical simulation routine. This implies assuming that the vertical axis of the cylinder is long enough to neglect

Fig. 10. Temperature measured in the weak zone during the thermal recovery of the bulk cylinder under study. The dashed line shows the calculation for the pure convection model (exponential decrease), while the solid line takes into account not only the convection term, but also a conductive exchange between the weak zone and the rest of the cylinder. Figure

Convection

( )

<sup>415</sup> Thermal Behaviour and Refrigeration

*t* (min)

Convection + conduction

(b)

0

competitive value, even in non-liquid environments.

15

Δ

from Ref. (Osorio et al., 2005).

AWZ), respectively.

*T* (K)

30

45 End of fault

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

In our calculations (Osorio et al., 2004) we have used the values *X* = <sup>1</sup> <sup>3</sup> (i.e., we assume that the inclusions are spherical), *C*<sup>1</sup> = 0.85 and *C*<sup>2</sup> = 0.15, in accordance with the x-ray analysis already presented. We have also left the resistivity of the hot domain to be slightly higher than the one of the cold part at each temperature. For that, for *ρ<sup>h</sup>* <sup>e</sup> , we have used *ρ<sup>h</sup>* <sup>e</sup> (*Th*) = *αρ<sup>c</sup>* <sup>e</sup>(*Th*) (*<sup>α</sup>* <sup>≥</sup> 1) and carried out the calculations with *<sup>α</sup>*, altogether with the length *<sup>h</sup>*, as free parameters. The dashed and short-dashed lines of figure 9 correspond to the best fit with *<sup>α</sup>* <sup>=</sup> 1.01 and *<sup>h</sup>* <sup>=</sup> 0.1 <sup>×</sup> <sup>2</sup>*πr*. We must note that the precise values of *<sup>C</sup>*1, *<sup>C</sup>*<sup>2</sup> and *<sup>X</sup>* do not change the main results from our approach. Small variations of these parameters should be balanced by the free ones used in our calculations.

As it can be seen, the general behaviour of the thermal evolution of the hot part is roughly accounted for, in particular the slope changes of the *T*(*t*) curve during the fault. The thermal excursion calculated for the cold part of the cylinder is also in good agreement with the data displayed in figures 6 and 7 measured in the parts far away from the weak zones. The dotted lines in figure 9 represent the thermal evolution of the hot (short-dotted) and cold (dotted) parts, calculated by assuming a cylinder composed of only Bi-2223 phase. Clearly, in this scenario the experimental data are not properly accounted for.

Despite the crudeness of our approach, our results indicate that the thermal evolution of the superconducting cylinder during a fault is mainly determined by the minority phase. This behaviour represents one of the most important drawbacks of the inhomogeneous materials to be used for fault current limiter devices, as the limitation can be carried out by only a small part of the superconducting element. This would imply a useless waste of material and, for high values of the critical current density, a huge dissipation in the surroundings of the weakest parts which could lead to a further degradation or even to a local melting. Therefore, work should be done in order to improve the homogeneity of the superconducting samples employed in SFCLs or by designing special configurations which allow to reduce the importance of this problem.

We must finally note that the global resistance of the cylinder, obtained according to the scheme of figure 8(b) by using the expression

$$R(f, T^c, T^h) = \rho\_\text{e}^c(f, T^c) \left(\frac{\ell^c}{A\_\perp}\right) + \rho\_\text{e}^h(f, T^h) \left(\frac{\ell^h}{A\_\perp}\right)'\tag{4}$$

roughly reproduces the experimental data (Osorio et al., 2004).

In figure 10, symbols represent the temperature measured in the weak zone during the thermal recovery. The lines represent the temperature determined with our numerical routine, which in this case is, in principle, very simple. The current is set to zero and the temperature is let to vary according to the convective exchange of heat, dashed line, or to a convective plus conductive model (to the neighbouring cold parts), solid line. It is well noticeable that a conductive exchange term between the weak zone and the rest of the cylinder must be included to account for the temperature diminution.

#### **3. Thermal behavior of an inductive SFCL whose secondary has artificial weak zones**

The results shown in figure 10 indicate that the convection mechanism is not enough to explain the cooling down of the weak zone once the fault is cleared, but conduction to the cold parts

10 Will-be-set-by-IN-TECH

the inclusions are spherical), *C*<sup>1</sup> = 0.85 and *C*<sup>2</sup> = 0.15, in accordance with the x-ray analysis already presented. We have also left the resistivity of the hot domain to be slightly higher

<sup>e</sup>(*Th*) (*<sup>α</sup>* <sup>≥</sup> 1) and carried out the calculations with *<sup>α</sup>*, altogether with the length *<sup>h</sup>*, as free parameters. The dashed and short-dashed lines of figure 9 correspond to the best fit with *<sup>α</sup>* <sup>=</sup> 1.01 and *<sup>h</sup>* <sup>=</sup> 0.1 <sup>×</sup> <sup>2</sup>*πr*. We must note that the precise values of *<sup>C</sup>*1, *<sup>C</sup>*<sup>2</sup> and *<sup>X</sup>* do not change the main results from our approach. Small variations of these parameters should be

As it can be seen, the general behaviour of the thermal evolution of the hot part is roughly accounted for, in particular the slope changes of the *T*(*t*) curve during the fault. The thermal excursion calculated for the cold part of the cylinder is also in good agreement with the data displayed in figures 6 and 7 measured in the parts far away from the weak zones. The dotted lines in figure 9 represent the thermal evolution of the hot (short-dotted) and cold (dotted) parts, calculated by assuming a cylinder composed of only Bi-2223 phase. Clearly, in this

Despite the crudeness of our approach, our results indicate that the thermal evolution of the superconducting cylinder during a fault is mainly determined by the minority phase. This behaviour represents one of the most important drawbacks of the inhomogeneous materials to be used for fault current limiter devices, as the limitation can be carried out by only a small part of the superconducting element. This would imply a useless waste of material and, for high values of the critical current density, a huge dissipation in the surroundings of the weakest parts which could lead to a further degradation or even to a local melting. Therefore, work should be done in order to improve the homogeneity of the superconducting samples employed in SFCLs or by designing special configurations which allow to reduce the

We must finally note that the global resistance of the cylinder, obtained according to the

In figure 10, symbols represent the temperature measured in the weak zone during the thermal recovery. The lines represent the temperature determined with our numerical routine, which in this case is, in principle, very simple. The current is set to zero and the temperature is let to vary according to the convective exchange of heat, dashed line, or to a convective plus conductive model (to the neighbouring cold parts), solid line. It is well noticeable that a conductive exchange term between the weak zone and the rest of the cylinder must be

**3. Thermal behavior of an inductive SFCL whose secondary has artificial weak**

The results shown in figure 10 indicate that the convection mechanism is not enough to explain the cooling down of the weak zone once the fault is cleared, but conduction to the cold parts

 +*ρ<sup>h</sup>* <sup>e</sup>(*J*, *<sup>T</sup>h*)

 *h A*⊥ , (4)

<sup>3</sup> (i.e., we assume that

<sup>e</sup> (*Th*) =

<sup>e</sup> , we have used *ρ<sup>h</sup>*

In our calculations (Osorio et al., 2004) we have used the values *X* = <sup>1</sup>

than the one of the cold part at each temperature. For that, for *ρ<sup>h</sup>*

scenario the experimental data are not properly accounted for.

balanced by the free ones used in our calculations.

importance of this problem.

**zones**

scheme of figure 8(b) by using the expression

*R*(*J*, *T<sup>c</sup>*

included to account for the temperature diminution.

, *Th*)=*ρ<sup>c</sup>*

roughly reproduces the experimental data (Osorio et al., 2004).

<sup>e</sup>(*J*, *<sup>T</sup><sup>c</sup>* ) *<sup>c</sup> A*⊥

*αρ<sup>c</sup>*

Fig. 10. Temperature measured in the weak zone during the thermal recovery of the bulk cylinder under study. The dashed line shows the calculation for the pure convection model (exponential decrease), while the solid line takes into account not only the convection term, but also a conductive exchange between the weak zone and the rest of the cylinder. Figure from Ref. (Osorio et al., 2005).

of the cylinder plays a major role. Notice that this would entail that this weak zone would refrigerate faster than a homogeneous cylinder, provided that the same final temperature is attained in both cases. This suggests that if the cylinder was made up from weak zones separated by cold segments, after the fault clearance the excess heat could be removed from the hot parts to the colder ones, till the cylinder was thermalized at a low temperature, maybe higher than that of the coolant, but much lower than the maximum value obtained during the fault. It can be argued that, as only the weak parts would enter into the normal state, the impedance of the limiter should be much lower. Nevertheless, and despite this drawback, it could be found a configuration of made-to-measure weak zones, properly spaced, which allow obtaining an acceptable impedance but, in turn, would reduce the recovery time to a competitive value, even in non-liquid environments.

The above ideas were proposed in previous works (Osorio et al., 2005; 2008; 2006), where two possible procedures where stated; in the first one (Osorio et al., 2005), the set of artificial weak zones (AWZ) could be made from a different material, with a lower *Jc* and a higher resistivity (so somewhat balancing the loss of impedance of the limiter), as depicted in figure 11(a), for a set of *n* AWZ of length *<sup>n</sup>*, *<sup>h</sup>* being the total weak length. Figure 11(b) shows the equivalent electrical circuit for the cylinder, with resistances *R<sup>c</sup>* and *R<sup>h</sup>* for the *cold* and *hot* parts (i.e. the AWZ), respectively.

In the second procedure (Osorio et al., 2008; 2006), each AWZ would be made as a constriction of the cross-section of the cylinder as it is shown in figure 14(a). This configuration allows using the same material for the whole sample, and the reduction in the critical current and the gain in the total impedance is easily yielded by the diminution of the cross-section. For convenience, this idea was initially tested by means of a 1-D numerical simulation routine. This implies assuming that the vertical axis of the cylinder is long enough to neglect

0 5 10 15 20

0.0 0.2 0.4 0.6 0.8 1.0

Working point

<sup>417</sup> Thermal Behaviour and Refrigeration

λ

*n*

Fig. 12. (a) Recovery time calculated for a set of values of *λ* normalized to the value obtained for a homogeneous cylinder (circles). The right axis shows the normalized impedance (triangles). Both solid and dashed lines are given as guides for the eyes. (b) Recovery time as

weak zone characterized by the same relative length and *n* = 10. In figure 13(a) it can be seen the heating and thermal recovery for a fault of about 0.2 s in duration. Notice that the final temperature difference is about 30 K in the case of the homogeneous cylinder, while the AWZ of the second and third cylinders reach the same maximum value, about two times higher. Due to this, the recovery of the connected weak zone is even slower than that of the homogeneous cylinder. However, when the AWZ is split into 10 parts the thermal recovery is noticeably enhanced. Figure 13(b) shows the impedance obtained with these cylinders. The homogeneous cylinder yields a much higher impedance, as the transited length is 80% higher. This drawback can be somewhat balanced by using AWZ made form a material with a higher resistivity (see dashed line, for a material with a normal resistivity two times higher than that

Further tests were realized by means of a finite element routine which allows exploring the effects associated with thick walls. Figure 14 (a) and (b) shows an scheme of the top and the lateral view, respectively, of the cylinder with constrictions equally spaced used in the

a function of the number of slices in which it is divided a weak zone with *λ* = 0.2, normalized to the value when the cylinder is fully homogeneous. The solid line is again a

(b)

λ = 0.2

(a)

0.0

0.5

1.0

1.5

2.0

*Z* wz /

*Z* hom

0

guide for the eyes. Figure from Ref. (Osorio et al., 2005).

of the rest of the superconducting element).

1

*t*rec split(*n*) / *t* hom rec

2

3

0

1

*t* wz rec / *t*rechom 2

3

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

Fig. 11. (a) Scheme of the superconducting cylinder when several thin weak zones are considered instead of a greater size one. The length of each weak zone, *<sup>n</sup>*, is the total one, h, divided by the number of slices, *n*. (b) Equivalent electrical circuit of the inhomogeneous superconducting cylinder. The resistances are both dependent on the current density and the temperature of the corresponding zone: *R*(c,h)(*J*, *T*(c,h)). Figure from Ref. (Osorio et al., 2005).

any possible temperature gradient along this direction. In addition, it is assumed that the temperature across the wall thickness is approximately the same no matter the point under consideration, i.e., the surface of the cylinder is not at a lower temperature than the inner parts. This is the rougher approximation of the model, but for cylinders of wall thickness of the order of 2 − 3 mm it can be quite acceptable. The AWZ are parametrized by the relative length *λ*, which is the ratio of the length of the weak zone to the whole perimeter of the cylinder.

Figure 12(a) shows the recovery time calculated for different values of *λ* normalized to that obtained for a homogeneous cylinder (left axis) as well as the normalized impedance (right axis). Notice two important facts; first of all, the recovery time is not necessarily shorter for a cylinder with a weak zone, unless the AWZ is short enough to warranty a fast evacuation of the heat by conduction. Secondly, the impedance yielded by the AWZ can be lower than that offered by the homogeneous cylinder. This second drawback can be removed, in principle, by using a more resistive material or by making AWZ in the shape of grooves (the lower the cross-section the higher the impedance). Even if the resulting impedance is not as high as that offered by the homogeneous cylinder, this inconvenient could be balanced by a much faster thermal recovery. The solution to the first drawback appears to be splitting the AWZ in short slices, so that the heat can be easily removed by conduction from the cold parts to the colder segments which act as heat reservoirs. This effect is shown in 12(b), where an AWZ of length *λ* = 2 (corresponding to the working point chosen in (a), for which the weak zone represents 20% of the perimeter of the cylinder) is split into *n* slices. Notice that, although the weak zones are individually shorter, the total "weak length" is the same, so the impedance can be considered as invariable with respect to the case of *n* = 1 (i.e. a non-split weak zone). Maybe a better refrigeration of these slices could lead to a slightly lower impedance, but this effect is neglected here.

Figure 13 shows the numerical calculations for the performances of different cylinders acting as secondary of a SFCL. The first one is a homogeneous element, the second a cylinder with a connected weak zone of relative length *λ* = 2, and the last one is a cylinder with a split 12 Will-be-set-by-IN-TECH

*R* c

S

*I* sup

*R* h

> *R* c

*R* h

lh

any possible temperature gradient along this direction. In addition, it is assumed that the temperature across the wall thickness is approximately the same no matter the point under consideration, i.e., the surface of the cylinder is not at a lower temperature than the inner parts. This is the rougher approximation of the model, but for cylinders of wall thickness of the order of 2 − 3 mm it can be quite acceptable. The AWZ are parametrized by the relative length *λ*, which is the ratio of the length of the weak zone to the whole perimeter of the

Figure 12(a) shows the recovery time calculated for different values of *λ* normalized to that obtained for a homogeneous cylinder (left axis) as well as the normalized impedance (right axis). Notice two important facts; first of all, the recovery time is not necessarily shorter for a cylinder with a weak zone, unless the AWZ is short enough to warranty a fast evacuation of the heat by conduction. Secondly, the impedance yielded by the AWZ can be lower than that offered by the homogeneous cylinder. This second drawback can be removed, in principle, by using a more resistive material or by making AWZ in the shape of grooves (the lower the cross-section the higher the impedance). Even if the resulting impedance is not as high as that offered by the homogeneous cylinder, this inconvenient could be balanced by a much faster thermal recovery. The solution to the first drawback appears to be splitting the AWZ in short slices, so that the heat can be easily removed by conduction from the cold parts to the colder segments which act as heat reservoirs. This effect is shown in 12(b), where an AWZ of length *λ* = 2 (corresponding to the working point chosen in (a), for which the weak zone represents 20% of the perimeter of the cylinder) is split into *n* slices. Notice that, although the weak zones are individually shorter, the total "weak length" is the same, so the impedance can be considered as invariable with respect to the case of *n* = 1 (i.e. a non-split weak zone). Maybe a better refrigeration of these slices could lead to a slightly lower impedance, but this effect is

Figure 13 shows the numerical calculations for the performances of different cylinders acting as secondary of a SFCL. The first one is a homogeneous element, the second a cylinder with a connected weak zone of relative length *λ* = 2, and the last one is a cylinder with a split

l*n*<sup>=</sup> *n* \_\_\_

Fig. 11. (a) Scheme of the superconducting cylinder when several thin weak zones are considered instead of a greater size one. The length of each weak zone, *<sup>n</sup>*, is the total one, h, divided by the number of slices, *n*. (b) Equivalent electrical circuit of the inhomogeneous superconducting cylinder. The resistances are both dependent on the current density and the temperature of the corresponding zone: *R*(c,h)(*J*, *T*(c,h)). Figure from Ref. (Osorio et al., 2005).

Weak zones

(a) (b)

*I* sup

cylinder.

neglected here.

Fig. 12. (a) Recovery time calculated for a set of values of *λ* normalized to the value obtained for a homogeneous cylinder (circles). The right axis shows the normalized impedance (triangles). Both solid and dashed lines are given as guides for the eyes. (b) Recovery time as a function of the number of slices in which it is divided a weak zone with *λ* = 0.2, normalized to the value when the cylinder is fully homogeneous. The solid line is again a guide for the eyes. Figure from Ref. (Osorio et al., 2005).

weak zone characterized by the same relative length and *n* = 10. In figure 13(a) it can be seen the heating and thermal recovery for a fault of about 0.2 s in duration. Notice that the final temperature difference is about 30 K in the case of the homogeneous cylinder, while the AWZ of the second and third cylinders reach the same maximum value, about two times higher. Due to this, the recovery of the connected weak zone is even slower than that of the homogeneous cylinder. However, when the AWZ is split into 10 parts the thermal recovery is noticeably enhanced. Figure 13(b) shows the impedance obtained with these cylinders. The homogeneous cylinder yields a much higher impedance, as the transited length is 80% higher. This drawback can be somewhat balanced by using AWZ made form a material with a higher resistivity (see dashed line, for a material with a normal resistivity two times higher than that of the rest of the superconducting element).

Further tests were realized by means of a finite element routine which allows exploring the effects associated with thick walls. Figure 14 (a) and (b) shows an scheme of the top and the lateral view, respectively, of the cylinder with constrictions equally spaced used in the

(a)

*e*-2

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

*e*

<sup>419</sup> Thermal Behaviour and Refrigeration

<sup>h</sup>

c

(c) *<sup>x</sup>* (mm)

superconducting element can be found elsewhere (Rettelbach & Schmitz, 2003).


When the fault occurs, the circulating current overtakes the critical value *Ic* and dissipation arises. Following the temperature and current dependent equations for the electric field, the dissipation is calculated and so the temperature at each node of the mesh. Heat exchanges by convection with the coolant and conduction between hot and cold segments are modeled by taking into account experimental values for the temperature dependent convective coefficient for liquid nitrogen (Duron et al., 2007) and helium gas (Chapman, 1984), and the thermal conductivity (Fujishiro et al., 2003; 2006). As we are mainly interested on the thermal recovery, the condition that the circulating current is constant (not realistic for an inductive SFCL) does not preclude at all the validity of our conclusions about the recovery time. More sophisticated models, where the current changes according to the resistance developed in the

Figure 15 displays the temperature distribution within the cylinder's wall after a fault of 100 ms. We can see that, despite the high temperature attained in the inner of the wall, the heating

Fig. 14. (a) Schematic top view of the grooved cylinder under study. (b) Lateral view of a segment of the cylinder's perimeter used in the simulation. (c) Top view of the mesh used in FEM calculations. Due to symmetry, only a quarter of a groove is used. Figure from Ref.

Weak zone

*x*

(b)

*y*

(Osorio et al., 2008).

5

(mm)

10

*z*

Fig. 13. (a) Temperature excursions calculated for a homogeneous monolith-type cylinder (solid line), a cylinder with a connected weak zone (dotted line) and a cylinder with a weak zone split into 10 small parts (dashed line). The recovery time of the sample with the split weak zone is noticeable shorter than that for the homogeneous sample. In this example, the resistivity of the weak zone is assumed to be 2 times the resistivity of the reminder part. (b) Impedance under a current fault calculated for the device operating with each one of the samples considered in (a). The impedance of the device with the inhomogeneous sample (the same for the connected and the split weak zone) is around a 40% lower than with the homogenous one, except for the first cycles. The dotted line should be the impedance if the resistivity is the same for both the weak zone and the other part of the cylinder. Figure from Ref. (Osorio et al., 2005).

simulation. It is assumed that a constant current circulates along the circumferential perimeter of a cylinder. For the sake of simplicity, the cylinder is considered to be infinite along its axis, hence the problem is 2-D. Due to symmetry, only a quarter of a groove has to be considered. The mesh is denser in the AWZ and its vicinity and coarser far away, where important changes are less likely to occur, as it is shown in figure 14 (c). The details of the numerical routine can be found in reference (Osorio et al., 2008). We just indicate here the main steps of our calculations.

14 Will-be-set-by-IN-TECH

 Homogenous monolite Weak zone (*n* = 1)

Weak zone (*<sup>n</sup>* = 10)}λ = 0.2

(a)

(b)

0.0 0.1 0.2 200

<sup>ρ</sup>normal

Fault starts

Homogeneous monolite

2 x ρnormal}Weak zone

(λ= 0.2)

0.00 0.05 0.10 0.15 0.20

*t* (s)

Fig. 13. (a) Temperature excursions calculated for a homogeneous monolith-type cylinder (solid line), a cylinder with a connected weak zone (dotted line) and a cylinder with a weak zone split into 10 small parts (dashed line). The recovery time of the sample with the split weak zone is noticeable shorter than that for the homogeneous sample. In this example, the resistivity of the weak zone is assumed to be 2 times the resistivity of the reminder part. (b) Impedance under a current fault calculated for the device operating with each one of the samples considered in (a). The impedance of the device with the inhomogeneous sample (the

same for the connected and the split weak zone) is around a 40% lower than with the homogenous one, except for the first cycles. The dotted line should be the impedance if the resistivity is the same for both the weak zone and the other part of the cylinder. Figure from

simulation. It is assumed that a constant current circulates along the circumferential perimeter of a cylinder. For the sake of simplicity, the cylinder is considered to be infinite along its axis, hence the problem is 2-D. Due to symmetry, only a quarter of a groove has to be considered. The mesh is denser in the AWZ and its vicinity and coarser far away, where important changes are less likely to occur, as it is shown in figure 14 (c). The details of the numerical routine can be found in reference (Osorio et al., 2008). We just indicate here the main steps of our calculations.

0

300

Fault starts

600

*Z* (

Ref. (Osorio et al., 2005).

Ω)

900

0

20

40

Δ

*T* (K)

End of fault

60

80

Fig. 14. (a) Schematic top view of the grooved cylinder under study. (b) Lateral view of a segment of the cylinder's perimeter used in the simulation. (c) Top view of the mesh used in FEM calculations. Due to symmetry, only a quarter of a groove is used. Figure from Ref. (Osorio et al., 2008).

When the fault occurs, the circulating current overtakes the critical value *Ic* and dissipation arises. Following the temperature and current dependent equations for the electric field, the dissipation is calculated and so the temperature at each node of the mesh. Heat exchanges by convection with the coolant and conduction between hot and cold segments are modeled by taking into account experimental values for the temperature dependent convective coefficient for liquid nitrogen (Duron et al., 2007) and helium gas (Chapman, 1984), and the thermal conductivity (Fujishiro et al., 2003; 2006). As we are mainly interested on the thermal recovery, the condition that the circulating current is constant (not realistic for an inductive SFCL) does not preclude at all the validity of our conclusions about the recovery time. More sophisticated models, where the current changes according to the resistance developed in the superconducting element can be found elsewhere (Rettelbach & Schmitz, 2003).

Figure 15 displays the temperature distribution within the cylinder's wall after a fault of 100 ms. We can see that, despite the high temperature attained in the inner of the wall, the heating

redistribution of the excess heat, confined at the beginning in the close vicinity of the weak

<sup>421</sup> Thermal Behaviour and Refrigeration

Figure 17 shows an example of the variation of the primary current before and during a fault when the secondary is an ungrooved cylinder (i.e. without AWZ) and a grooved one, UC and GC1, respectively. Both samples are supposed to be made from YBCO, and GC1 is characterized by a ratio of grooved to ungrooved cross-sections *a* = 0.8, and by the lengths of the hot and cold parts being *<sup>h</sup>* = 0.5 and *<sup>c</sup>* = 0.5 cm. Notice that the limited current is the same no matter the secondary used, so making grooves does not necessarily affect the

Fig. 17. Variation of the primary current for an ungrooved cylinder (UGC) and a grooved cylinder (GC1). Notice that the limitation performance is identical. Figure from Ref.

(further details of these studies can be found in (Osorio et al., 2008)).

As the temperature of the grooves can be very high and the thermal conductivity of YBCO within the *a-b* planes is quite low, about 2.5 Wm−1K−1, it was proposed to use cylinders of YBCO grown in such a way that the current circulates along the *c* axis (Osorio et al., 2008). This entails that the normal state resistivity is 15 times higher (Vanderbemden et al., 1999), and that the thermal conductivity along the perimeter is 10 Wm−1K−<sup>1</sup> (and 3 Wm−1K−<sup>1</sup> along the cross-section) (Fujishiro et al., 2003; 2006). With these new parameters, the AWZ can be made thinner without reducing the normal state resistance, while the removal of heat is made easier due to the higher thermal conductivity and to the reduction of the path the heat must travel to be exchanged between hot and cold parts. Figure 18 shows our best result for a grooved cylinder characterized by *a* = 0.8, *<sup>c</sup>* = 0.8 cm and *<sup>h</sup>* = 0.2 cm, when the limiter works in a liquid nitrogen bath (a) and a helium gas atmosphere (b). The thin dotted line shows the temperature at which the recovery ends. Notice that the recovery times for the grooved sample are about 1 s, as demanded in practical applications. The solid lines correspond to the ungrooved cylinder. When a liquid nitrogen bath is used, the recovery for the ungrooved cylinder occurs at about 3.5 s, despite the initial temperature is much lower. For the helium gas atmosphere, its temperature does not even change within the time interval under study

zone, leads to a temperature map which is not so far above 77 K.

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

limitation capacity.

(Osorio et al., 2008).

Fig. 15. Temperature map of a superconducting element after a fault of 100 ms in duration. Figure from Ref. (Osorio et al., 2008).

Fig. 16. Thermal recovery for the example of Figure 15 after 1 s of zero-current recovery (a) and after 5 s (b). Figure from Ref. (Osorio et al., 2008).

is quite confined to the AWZ. This effect of "non-propagation" of the hot spot during the fault was one of the main assumptions of the 1-D model, and this more refined model confirms its validity. The evolution of the hot spot once the fault is removed and the current is set to zero is shown in figures 16(a) and 16(b) for *t* = 1 s and *t* = 5 s, respectively. Notice that the 16 Will-be-set-by-IN-TECH

*t I* = 100 ms, = 0.2 MA


*t I* = 1s, = 0 A


*t I* = 5s, = 0 A


Fig. 16. Thermal recovery for the example of Figure 15 after 1 s of zero-current recovery (a)

is quite confined to the AWZ. This effect of "non-propagation" of the hot spot during the fault was one of the main assumptions of the 1-D model, and this more refined model confirms its validity. The evolution of the hot spot once the fault is removed and the current is set to zero is shown in figures 16(a) and 16(b) for *t* = 1 s and *t* = 5 s, respectively. Notice that the

(b) *x* (cm)

and after 5 s (b). Figure from Ref. (Osorio et al., 2008).

Fig. 15. Temperature map of a superconducting element after a fault of 100 ms in duration.

*y*

Figure from Ref. (Osorio et al., 2008).

*y*

(a)

*y* 0


1

(cm)

0


1

(cm)

0


1

(cm)

Temperature

 map (K)

170

150

130 110

90

100

90

95

80

85

*Tmax*= 84.3 K

82

80

78

84

Temperature

 map (K)

*Tmax*= 102.3 K

*Tmax*= 180.1 K

redistribution of the excess heat, confined at the beginning in the close vicinity of the weak zone, leads to a temperature map which is not so far above 77 K.

Figure 17 shows an example of the variation of the primary current before and during a fault when the secondary is an ungrooved cylinder (i.e. without AWZ) and a grooved one, UC and GC1, respectively. Both samples are supposed to be made from YBCO, and GC1 is characterized by a ratio of grooved to ungrooved cross-sections *a* = 0.8, and by the lengths of the hot and cold parts being *<sup>h</sup>* = 0.5 and *<sup>c</sup>* = 0.5 cm. Notice that the limited current is the same no matter the secondary used, so making grooves does not necessarily affect the limitation capacity.

Fig. 17. Variation of the primary current for an ungrooved cylinder (UGC) and a grooved cylinder (GC1). Notice that the limitation performance is identical. Figure from Ref. (Osorio et al., 2008).

As the temperature of the grooves can be very high and the thermal conductivity of YBCO within the *a-b* planes is quite low, about 2.5 Wm−1K−1, it was proposed to use cylinders of YBCO grown in such a way that the current circulates along the *c* axis (Osorio et al., 2008). This entails that the normal state resistivity is 15 times higher (Vanderbemden et al., 1999), and that the thermal conductivity along the perimeter is 10 Wm−1K−<sup>1</sup> (and 3 Wm−1K−<sup>1</sup> along the cross-section) (Fujishiro et al., 2003; 2006). With these new parameters, the AWZ can be made thinner without reducing the normal state resistance, while the removal of heat is made easier due to the higher thermal conductivity and to the reduction of the path the heat must travel to be exchanged between hot and cold parts. Figure 18 shows our best result for a grooved cylinder characterized by *a* = 0.8, *<sup>c</sup>* = 0.8 cm and *<sup>h</sup>* = 0.2 cm, when the limiter works in a liquid nitrogen bath (a) and a helium gas atmosphere (b). The thin dotted line shows the temperature at which the recovery ends. Notice that the recovery times for the grooved sample are about 1 s, as demanded in practical applications. The solid lines correspond to the ungrooved cylinder. When a liquid nitrogen bath is used, the recovery for the ungrooved cylinder occurs at about 3.5 s, despite the initial temperature is much lower. For the helium gas atmosphere, its temperature does not even change within the time interval under study (further details of these studies can be found in (Osorio et al., 2008)).

Fig. 19. Temperature excursions under current fault of cold parts and grooves for: (a) All of grooves with the same ratio *a*; (b) and (c) One of the grooves with greater and smaller *a*,

<sup>423</sup> Thermal Behaviour and Refrigeration

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

surface-to-volume ratio of the secondary by using short bulk rings or thin film washers. This idea was proposed in a previous work (Osorio et al., 2006), when it was presented a comparison between the performance of both types of secondaries. To balance the reduction in the effective critical current of the limiter (due to the lower cross-section of rings and washers when compared to bulk cylinders), it was proposed to stack these elements up to add up their

In figure 20 it is displayed the impedance under current fault for three secondaries: a 40 mm-high Bi-2223 cylinder (labeled 1*R*40) a stack of three 2 mm-high Bi-2223 rings (labeled 3*R*2), and a stack of three 4.5 mm-wide Y123/Au washers (labeled 3*W*4.5). Y123/Au washers were trepanned from wafers made from a Y1Ba2Cu3O7−*<sup>δ</sup>* thin film (300 nm thick) shunted by a gold layer (100 nm thick), deposited on a AlO2 substrate provided by Theva (Germany). It can be seen that these secondaries present a very different behaviour under a current fault; with 1*R*40 the increase of impedance is very smooth, whereas it is somewhat abrupter with 3*R*2 and much more with 3*W*4.5. The differences are even more remarkable after the current fault. With 1*R*40 the impedance for the highest voltage continues to increase. On the contrary, with 3*R*2 the thermal recovery (i.e. when the circulating current is the same than before the

respectively, than the other. Figure from Ref. (Osorio et al, 2010).

critical current values.

Fig. 18. Thermal recovery for an ungrooved cylinder (solid line) and for a sample with grooves characterized by *a* = 0.8, *<sup>c</sup>* = 0.8 cm and *<sup>h</sup>* = 0.2 cm in liquid nitrogen (a) and helium gas (b). The thin dotted lines indicate the temperature at which no dissipation arises in the superconducting elements. Figure from Ref. (Osorio et al., 2008).

Our results show that the use of grooves samples can improve the performance of bulk superconductors working in SFCL. However such improvement requires a set of grooves with very close behaviour. We illustrate the importance of this requirement with the example displayed in Fig. 19. When all grooves are identical, which is the case discussed above, the temperature increase of these parts under current fault should reach only up to around 100 K (Fig. 19(a)). If one of the grooves presents a slightly greater ratio *a* (around 1%) the temperature map changes very little (Fig. 19(b)) and the main efect would be a small decrease of the fault impedance. One major drawback however occur if any of the grooves is smaller than the other: the temperature of the groove with *a* around 1% smaller should increase up to around 150 K (Fig. 19(c)) which would cause a very long thermal recovery time after fault and therefore a poorer overall performance of the limiter device (a deeper study of this type of drawback can be found in (Osorio et al, 2010)).

#### **4. Stacks of bulk rings and thin film washers as secondary**

In the above section we have seen a way to improve the cooling of bulk superconducting samples by making a set of artificial weak zones. Another procedure could be increase the 18 Will-be-set-by-IN-TECH

L iquid nitrogen

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Gas atmosphere (b)

 Ungrooved cylinder Cylinder with grooves: *a* = 0.8 l h

> l c

= 0.2 cm

(a)

= 0.8 cm

*t* (s)

Our results show that the use of grooves samples can improve the performance of bulk superconductors working in SFCL. However such improvement requires a set of grooves with very close behaviour. We illustrate the importance of this requirement with the example displayed in Fig. 19. When all grooves are identical, which is the case discussed above, the temperature increase of these parts under current fault should reach only up to around 100 K (Fig. 19(a)). If one of the grooves presents a slightly greater ratio *a* (around 1%) the temperature map changes very little (Fig. 19(b)) and the main efect would be a small decrease of the fault impedance. One major drawback however occur if any of the grooves is smaller than the other: the temperature of the groove with *a* around 1% smaller should increase up to around 150 K (Fig. 19(c)) which would cause a very long thermal recovery time after fault and therefore a poorer overall performance of the limiter device (a deeper study of this type

In the above section we have seen a way to improve the cooling of bulk superconducting samples by making a set of artificial weak zones. Another procedure could be increase the

Fig. 18. Thermal recovery for an ungrooved cylinder (solid line) and for a sample with grooves characterized by *a* = 0.8, *<sup>c</sup>* = 0.8 cm and *<sup>h</sup>* = 0.2 cm in liquid nitrogen (a) and helium gas (b). The thin dotted lines indicate the temperature at which no dissipation arises

in the superconducting elements. Figure from Ref. (Osorio et al., 2008).

80

of drawback can be found in (Osorio et al, 2010)).

**4. Stacks of bulk rings and thin film washers as secondary**

90

100

*T* (K ) 110

80

90

100

*T* (K ) 110

120

Fig. 19. Temperature excursions under current fault of cold parts and grooves for: (a) All of grooves with the same ratio *a*; (b) and (c) One of the grooves with greater and smaller *a*, respectively, than the other. Figure from Ref. (Osorio et al, 2010).

surface-to-volume ratio of the secondary by using short bulk rings or thin film washers. This idea was proposed in a previous work (Osorio et al., 2006), when it was presented a comparison between the performance of both types of secondaries. To balance the reduction in the effective critical current of the limiter (due to the lower cross-section of rings and washers when compared to bulk cylinders), it was proposed to stack these elements up to add up their critical current values.

In figure 20 it is displayed the impedance under current fault for three secondaries: a 40 mm-high Bi-2223 cylinder (labeled 1*R*40) a stack of three 2 mm-high Bi-2223 rings (labeled 3*R*2), and a stack of three 4.5 mm-wide Y123/Au washers (labeled 3*W*4.5). Y123/Au washers were trepanned from wafers made from a Y1Ba2Cu3O7−*<sup>δ</sup>* thin film (300 nm thick) shunted by a gold layer (100 nm thick), deposited on a AlO2 substrate provided by Theva (Germany).

It can be seen that these secondaries present a very different behaviour under a current fault; with 1*R*40 the increase of impedance is very smooth, whereas it is somewhat abrupter with 3*R*2 and much more with 3*W*4.5. The differences are even more remarkable after the current fault. With 1*R*40 the impedance for the highest voltage continues to increase. On the contrary, with 3*R*2 the thermal recovery (i.e. when the circulating current is the same than before the

Fig. 21. Recovery time versus applied voltage normalized to the threshold value for the limiter working with a stack of three bulk Bi2223 rings of 2 mm in height (triangles) and a stack of three thin-film Y123/Au washers of 4.5 mm in width (diamonds). It is also included

<sup>425</sup> Thermal Behaviour and Refrigeration

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

times one and, respectively, two orders of magnitude lower than the one for 1*R*40. Note that the lowest times obtained for 3*R*2 correspond to trivial examples of low fault voltages for

Furthermore, although 3*R*2 and 3*W*4.5 operate at a lower power rating than 1*R*40, the recovery time will remain unchanged if more rings or washers were added to the stack. For instance, with about 25 of our thin films or short rings one should obtain the same operation currents than for the 1*R*40 ring, the secondary being of similar size, with similar or higher impedance but with recovery time strongly shortened, specially with thin films. With this procedure it should be also possible to fabricate limiters able to operate at the high currents withstood, for instance, by many resistive type limiters based on meander thin films, which also present recovery time values in the range of tenths of seconds even when submerged in liquid

In case of a bulk cylinder (1*R*40) of the same diameter as the bulk rings and height 4 cm, a low current (even below the nominal one) circulating after the fault is enough, as discussed above, to provoke appreciable dissipation in the sample. Therefore, the recovery time is extremely enlarged, even for the lowest applied voltages. So, for this secondary the recovery times were obtained by opening the circuit just after the end of the fault, i.e., we measure the *zero current recovery time*. With this procedure, the temperature at which the application of the nominal current does not generate dissipation is measured with the thermocouples attached on the rings. For 40-mm-high rings we found that the nominal current does not produce any appreciable heating for temperatures around 5 *K* above the working value (77 *K*). The circles plotted in figure 21 represent the interval between the end of the fault and the moment at

the recovery time when using as secondary a bulk Bi2223 ring of 40 mm in height (circles)(*V*threshold = 7 *V* for 1*R*40 and 0.9 *V* for 3*R*2 and 3*W*4.5). Figure from Ref.

(Osorio et al., 2006).

nitrogen.

which the samples develop very low resistance.

Fig. 20. Impedance at different fault voltages versus time for a limiter working with (a) 1 bulk Bi2223 ring of 40 mm in height, (b) a stack of 3 bulk Bi2223 rings of 2 mm in height and (c) a stack of 3 thin-film Y123/Au washers of 4.5 mm in width. Figure from Ref. (Osorio et al., 2006).

fault) is accomplished within a few seconds, and it is very fast for 3*W*4.5. The recovery times obtained for different faults and these three secondaries are plotted in figure 21. For the secondaries 3*R*2 and 3*W*4.5 the recovery time values plotted were directly obtained from the measured current waveforms, assuming that the limiter was recovered when the amplitude of the current is the same as before the fault. The scattering of the data points around the guide lines, about 10%, represents the precision of our measurements. With this procedure, the recovery times obtained could be somewhat overestimated because the circuit was not opened after each fault. Therefore some slight dissipation in the samples could still remain during the recovery process. Despite these small uncertainties, our results clearly show the better performance of the thiner samples. The secondaries 3*R*2 and 3*W*4.5 show recovery 20 Will-be-set-by-IN-TECH

Fig. 20. Impedance at different fault voltages versus time for a limiter working with (a) 1 bulk Bi2223 ring of 40 mm in height, (b) a stack of 3 bulk Bi2223 rings of 2 mm in height and

fault) is accomplished within a few seconds, and it is very fast for 3*W*4.5. The recovery times obtained for different faults and these three secondaries are plotted in figure 21. For the secondaries 3*R*2 and 3*W*4.5 the recovery time values plotted were directly obtained from the measured current waveforms, assuming that the limiter was recovered when the amplitude of the current is the same as before the fault. The scattering of the data points around the guide lines, about 10%, represents the precision of our measurements. With this procedure, the recovery times obtained could be somewhat overestimated because the circuit was not opened after each fault. Therefore some slight dissipation in the samples could still remain during the recovery process. Despite these small uncertainties, our results clearly show the better performance of the thiner samples. The secondaries 3*R*2 and 3*W*4.5 show recovery

(c) a stack of 3 thin-film Y123/Au washers of 4.5 mm in width. Figure from Ref.

(Osorio et al., 2006).

Fig. 21. Recovery time versus applied voltage normalized to the threshold value for the limiter working with a stack of three bulk Bi2223 rings of 2 mm in height (triangles) and a stack of three thin-film Y123/Au washers of 4.5 mm in width (diamonds). It is also included the recovery time when using as secondary a bulk Bi2223 ring of 40 mm in height (circles)(*V*threshold = 7 *V* for 1*R*40 and 0.9 *V* for 3*R*2 and 3*W*4.5). Figure from Ref. (Osorio et al., 2006).

times one and, respectively, two orders of magnitude lower than the one for 1*R*40. Note that the lowest times obtained for 3*R*2 correspond to trivial examples of low fault voltages for which the samples develop very low resistance.

Furthermore, although 3*R*2 and 3*W*4.5 operate at a lower power rating than 1*R*40, the recovery time will remain unchanged if more rings or washers were added to the stack. For instance, with about 25 of our thin films or short rings one should obtain the same operation currents than for the 1*R*40 ring, the secondary being of similar size, with similar or higher impedance but with recovery time strongly shortened, specially with thin films. With this procedure it should be also possible to fabricate limiters able to operate at the high currents withstood, for instance, by many resistive type limiters based on meander thin films, which also present recovery time values in the range of tenths of seconds even when submerged in liquid nitrogen.

In case of a bulk cylinder (1*R*40) of the same diameter as the bulk rings and height 4 cm, a low current (even below the nominal one) circulating after the fault is enough, as discussed above, to provoke appreciable dissipation in the sample. Therefore, the recovery time is extremely enlarged, even for the lowest applied voltages. So, for this secondary the recovery times were obtained by opening the circuit just after the end of the fault, i.e., we measure the *zero current recovery time*. With this procedure, the temperature at which the application of the nominal current does not generate dissipation is measured with the thermocouples attached on the rings. For 40-mm-high rings we found that the nominal current does not produce any appreciable heating for temperatures around 5 *K* above the working value (77 *K*). The circles plotted in figure 21 represent the interval between the end of the fault and the moment at

Acoustic power

*Tambient*

**Resonator Engine**

*T*hot

Stack Ambient H-E

Hot H-E

*x* (a.u.)

Hot H-E Cold H-E Ambient H-E Regenerator

**Refrigerator**

<sup>427</sup> Thermal Behaviour and Refrigeration

*Tcold*

Fig. 22. (a) Schematic view of a thermoacoustic machine. On the right it can be seen the thermoacoustic engine. The thermoacoustic refrigerator, of pulse tube type, is comprised of three heat exchangers (HE), a regenerator and the buffer tube, which limits the entrance in the cold part of the heat generated by the dissipation of acoustic power in the charge. (b) Temperature profile along the TA machine. A strong gradient is imposed on the engine, so the wave can be generated. It is assumed that the dissipation increases the temperature in the

(produced in a photovoltaic system, for instance) or burnt gas, as has already been done in some natural gas plants, where a conventional liquefying system were not cost effective.

In a previous work (Osorio et al., 2008) we proposed a 2.2 kVA prototype of hybrid fault current limiter based on a YBCO film meander path which was directly attached to the cold heat exchanger of a thermoacoustic refrigerator, as it is schematically depicted in figure 23. The main advantage of this configuration is that the close contact between the superconducting path and the cold heat exchanger allows a very efficient removal of the excess heat produced during a fault, thus greatly improving the performance of refrigerators in which the element to be cooled is inside an isolated cavity connected to the cooler through some kind of pipe. The thermoacoustic refrigerator was intented to remove about 50 W at 80 K so the recovery time of the whole SFCL does not exceed 1 s. The final design was tested by using a numerical routine (Osorio et al., 2008) and it was found that the SFCL was able to reduce the fault current in about 3 orders of magnitude, while the superconducting path heated about 130 K during a fault of 100 ms in duration. The total heat deposited in the cold

heat exchanger was around 30 J, low enough to be removed by the refrigerator in 1 s.

Resistive configurations were not considered as the length of the superconducting path necessary to get a suitable impedance would be excessive to allow attaching it on the cold heat exchanger of a medium-size thermoacoustic refrigerator. In addition, all the fault power would be dissipated just in the meander path, and so the temperature increase could be very high. A hybrid configuration offers the advantages of a lower dissipation in the superconducting element (part of the energy is required to get the magnetic field inside the core) and also a high impedance which does not demand for a very long meander path. A pure inductive limiter was immediately rejected as the core would occupy too much room

Buffer tube

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

**Acoustic charge**

(a)

*T T ambient* +

acoustic charge. Figure from Ref. (Osorio et al., 2008).

*T* (a.u.)

inside the refrigerator.

(b)

which this temperature is reached. The precise value of the temperature threshold and the differences from sample to sample do not change the values of the recovery time in more than 20%.

We must note that using stacks of bulk rings provokes a loss of impedance, as the superconducting elements of the stack would behave as resistances connected in parallel (Osorio et al., 2004). Although this drawback could be balanced by joining together several cores (Osorio et al., 2004), stacks of thin films show a better thermal performance and there is no reduction in the impedance or this is negligible (Lorenzo et al., 2006; 2007). Therefore, thin films offer important advantages concerning their refrigeration and the impedance does not diminish even when many of them are stacked to increase the effective critical current of the limiter.

## **5. SFCL based on superconducting thin film microbridges and meander paths**

Thin film samples present several advantages to be used in current limiting applications, as we have discussed above. We have probed the use of this kind of samples in other two up to now nearly unexplored configurations. One is a hybrid limiting device (i.e inductive-resistive) refrigerated by using a thermoacoustic refrigerator. The other is the microlimiter, a resistive FCL based on "thermally small" (Ferro et al., 2008; 2009) superconducting microbridges (Lorenzo et al., 2009; 2010) intended to operate at very low powers (SQUID based electronics, infrared detectors, etc).

### **5.1 Hybrid SFCL integrated in a thermoacoustic refrigerator**

Thermoacoustic refrigerators, heat engines which are based on the ability of a sound wave to play the role of the mechanical compressor and expander (Swift, 2002), have gained attention in the last decades and the state of art has allowed reaching very low temperatures (Yang & Thummes, 2005; Qiu et al., 2005). Thermoacoustic devices are for the time being slightly less efficient than conventional refrigerators. Nevertheless, their simplicity, low cost and lack of moving parts (which entails a strong diminution of maintenance costs, the main trouble with commercial refrigerators) and contaminating lubricants offer advantages over certain other mechanical devices like mechanical compressors.

A scheme of a thermoacoustic refrigerator is shown in figure 22. It consists of a resonant tube filled with an inert gas (usually Helium at a high pressure, about tens of bars) in which a standing sound wave is generated by using an engine (which can be also of thermoacoustic nature, or just a compressor). Two heat exchangers are located at precise locations inside the tube and they are separated by a porous medium (regenerator) which acts as a heat pump, transferring heat from the cold heat exchanger to the ambient one (kept at room temperature by a water stream), by means of thermoacoustic processes (Swift, 2002). The proper phase between the acoustic pressure and velocity inside the regenerator (where a local traveling wave must exist to verify the heat pumping) is set by an acoustic charge, usually made up from a constriction, a long and thin tube and a huge cavity which behave as a resistor, an inductor and a capacitor in electrical circuits. A third heat exchanger (the "hot" one) is used to prevent the entrance of heat from the acoustic charge into the cold heat exchanger. When the engine is of thermoacoustic nature, the acoustic wave can be generated by a thermal gradient in an appropriately designed porous element and the heating source can be electric power

22 Will-be-set-by-IN-TECH

which this temperature is reached. The precise value of the temperature threshold and the differences from sample to sample do not change the values of the recovery time in more than

We must note that using stacks of bulk rings provokes a loss of impedance, as the superconducting elements of the stack would behave as resistances connected in parallel (Osorio et al., 2004). Although this drawback could be balanced by joining together several cores (Osorio et al., 2004), stacks of thin films show a better thermal performance and there is no reduction in the impedance or this is negligible (Lorenzo et al., 2006; 2007). Therefore, thin films offer important advantages concerning their refrigeration and the impedance does not diminish even when many of them are stacked to increase the effective critical current of the

**5. SFCL based on superconducting thin film microbridges and meander paths**

**5.1 Hybrid SFCL integrated in a thermoacoustic refrigerator**

certain other mechanical devices like mechanical compressors.

Thin film samples present several advantages to be used in current limiting applications, as we have discussed above. We have probed the use of this kind of samples in other two up to now nearly unexplored configurations. One is a hybrid limiting device (i.e inductive-resistive) refrigerated by using a thermoacoustic refrigerator. The other is the microlimiter, a resistive FCL based on "thermally small" (Ferro et al., 2008; 2009) superconducting microbridges (Lorenzo et al., 2009; 2010) intended to operate at very low powers (SQUID based electronics,

Thermoacoustic refrigerators, heat engines which are based on the ability of a sound wave to play the role of the mechanical compressor and expander (Swift, 2002), have gained attention in the last decades and the state of art has allowed reaching very low temperatures (Yang & Thummes, 2005; Qiu et al., 2005). Thermoacoustic devices are for the time being slightly less efficient than conventional refrigerators. Nevertheless, their simplicity, low cost and lack of moving parts (which entails a strong diminution of maintenance costs, the main trouble with commercial refrigerators) and contaminating lubricants offer advantages over

A scheme of a thermoacoustic refrigerator is shown in figure 22. It consists of a resonant tube filled with an inert gas (usually Helium at a high pressure, about tens of bars) in which a standing sound wave is generated by using an engine (which can be also of thermoacoustic nature, or just a compressor). Two heat exchangers are located at precise locations inside the tube and they are separated by a porous medium (regenerator) which acts as a heat pump, transferring heat from the cold heat exchanger to the ambient one (kept at room temperature by a water stream), by means of thermoacoustic processes (Swift, 2002). The proper phase between the acoustic pressure and velocity inside the regenerator (where a local traveling wave must exist to verify the heat pumping) is set by an acoustic charge, usually made up from a constriction, a long and thin tube and a huge cavity which behave as a resistor, an inductor and a capacitor in electrical circuits. A third heat exchanger (the "hot" one) is used to prevent the entrance of heat from the acoustic charge into the cold heat exchanger. When the engine is of thermoacoustic nature, the acoustic wave can be generated by a thermal gradient in an appropriately designed porous element and the heating source can be electric power

20%.

limiter.

infrared detectors, etc).

Fig. 22. (a) Schematic view of a thermoacoustic machine. On the right it can be seen the thermoacoustic engine. The thermoacoustic refrigerator, of pulse tube type, is comprised of three heat exchangers (HE), a regenerator and the buffer tube, which limits the entrance in the cold part of the heat generated by the dissipation of acoustic power in the charge. (b) Temperature profile along the TA machine. A strong gradient is imposed on the engine, so the wave can be generated. It is assumed that the dissipation increases the temperature in the acoustic charge. Figure from Ref. (Osorio et al., 2008).

(produced in a photovoltaic system, for instance) or burnt gas, as has already been done in some natural gas plants, where a conventional liquefying system were not cost effective.

In a previous work (Osorio et al., 2008) we proposed a 2.2 kVA prototype of hybrid fault current limiter based on a YBCO film meander path which was directly attached to the cold heat exchanger of a thermoacoustic refrigerator, as it is schematically depicted in figure 23. The main advantage of this configuration is that the close contact between the superconducting path and the cold heat exchanger allows a very efficient removal of the excess heat produced during a fault, thus greatly improving the performance of refrigerators in which the element to be cooled is inside an isolated cavity connected to the cooler through some kind of pipe. The thermoacoustic refrigerator was intented to remove about 50 W at 80 K so the recovery time of the whole SFCL does not exceed 1 s. The final design was tested by using a numerical routine (Osorio et al., 2008) and it was found that the SFCL was able to reduce the fault current in about 3 orders of magnitude, while the superconducting path heated about 130 K during a fault of 100 ms in duration. The total heat deposited in the cold heat exchanger was around 30 J, low enough to be removed by the refrigerator in 1 s.

Resistive configurations were not considered as the length of the superconducting path necessary to get a suitable impedance would be excessive to allow attaching it on the cold heat exchanger of a medium-size thermoacoustic refrigerator. In addition, all the fault power would be dissipated just in the meander path, and so the temperature increase could be very high. A hybrid configuration offers the advantages of a lower dissipation in the superconducting element (part of the energy is required to get the magnetic field inside the core) and also a high impedance which does not demand for a very long meander path. A pure inductive limiter was immediately rejected as the core would occupy too much room inside the refrigerator.

a good thermal behaviour before, during and after the current fault, as we will see below. In addition, the widths of our microbridges are well above the threshold at which *J*∗ is sample-width dependent (Ruibal et al., 2007). The strong increase of *E* for current densities around *J*∗ make this type of samples very useful for FCL. However, the current is effectively limited only for electric fields or, equivalently, applied voltages well above that at which the limiter is triggered from normal operation (sample in the superconducting state) to current fault mode. In addition, once *J*∗ is attained, a thermal runaway (Viña et al., 2003; Maza et al., 2008) can be provoked which causes the reduction of the circulating current well below the nominal value (i.e., the current in normal operation without a fault) or very important damage

<sup>429</sup> Thermal Behaviour and Refrigeration

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

The superconducting microbridge exchanges heat mainly with its substrate, because at the operation temperatures, around 90 K, the heat transfer coefficient between YBCO films, for instance, and their substrate is *hbs* <sup>≈</sup> 103 Wcm−2K, whereas through liquid nitrogen or between the substrate and the copper holder they are 1000 times less (Duron et al., 2007; Lorenzo et al., 2009; Mosqueira et al., 1993). Hence, the conditions for a good refrigeration and, therefore, an optimal operation of the limiting device, depend very much on the relative dimensions of the microbridge and their substrate. The optimal refrigeration should allow us to operate with the minimum microbridge temperature increment relative to that of the bath, Δ*Tb*. If the Biot number (Lorenzo et al., 2009; Chapman, 1984) of both the microbridge and its substrate are *Bi* � 1 and, simultaneously, the thermal diffusion length, *Lth*, is longer than the

on the microbridge (that can be even burnt out).

corresponding thickness, Δ*Tb* may be crudely approached by

condition of "thermal smallness" being,

1993).

<sup>Δ</sup>*Tb* <sup>=</sup> *eb JE* <sup>1</sup>

*hbs* + *Ab As* 1 *hsr*

where *eb* is the microbridge thickness, *Ab* and *As* are the surface areas of the superconductor bridge and, respectively, its substrate. As *hbs* � *hsr*, Δ*Tb* will directly depend on *Ab*/*As*, the

under which Δ*Tb* approaches its "intrinsic" value, which is the temperature increase just associated with the effective thermal resistance of the interface bridge-substrate. For our microbridges similar to BS7 and under faults with characteristic times above 10 ms (of the order of commercial ac current periods), the conditions, *Bi* � 1 and *Lth* > *e* fully apply. In addition, as *Ab*/*As* <sup>≈</sup> <sup>4</sup><sup>×</sup> <sup>10</sup>−<sup>4</sup> whereas *hsr*/*hbs* <sup>≈</sup> <sup>10</sup>−3, this microbridge is "thermally small". One may use equation 5 to estimate Δ*Tb* when the microbridge is in stationary conditions under currents just below *<sup>J</sup>*∗. From figure 24, the power density involved is around 5 <sup>×</sup> 106 W/cm3 which leads, by also using the appropriate parameter values1, to <sup>Δ</sup>*Tb* <sup>≈</sup> 0.1 K. Even for faults involving powers ten times higher than those considered above, Δ*Tb* will remain below 1 K. One may also use equation 5 to roughly estimate that if the superconducting bridge had *Ab*/*As* one thousand times larger, as it is the case of those currently proposed for high power applications (Noe & Steurer, 2007; Weinstock, 2000), Δ*Tb* would take values at least

<sup>1</sup> The substrate thermal conductivities (*Ks*) and heat capacities (*Cs*) are of the order of 10 W/cm K and 0.6 J/cm<sup>3</sup> K for sapphire, and 0.2 W/cm K and 1 J/cm<sup>3</sup> K for SrTiO3, whereas for YBCO bridges *Kb* <sup>≈</sup> 0.05 W/cm K and *Cb* <sup>≈</sup> 0.7 J/cm<sup>3</sup> K. See, e. g., Refs. (Duron et al., 2007; Chapman, 1984; Mosqueira et al.,

*Ab As* < *hsr hbs* , (5)

, (6)

Fig. 23. Axial view of a hybrid limiter attached to the cold heat exchanger. The superconducting element is represented as a meander path. Figure from Ref. (Osorio et al., 2008).

#### **5.2 Superconducting fault current microlimiters**

In figure 24 we show a typical electric field versus current density (*E* − *J*) curve, corresponding to one of the microbridges (denoted BS7) used in our experiments. This microbridge, whose length, width and thickness are, respectively, 385 *μ*m, 28 *μ*m and 300 nm, and with *Tc* = 88.6 K, has been grown on a sapphire substrate. The two characteristic current densities are indicated in this figure: The critical, *Jc*, at which dissipation first appears, and the so-called supercritical, *J*∗, at which the microbridge is triggered into highly dissipative states. Let us note here that we have chosen for our studies microbridges of these dimensions to guarantee

Fig. 24. A typical *E* − *J* curve obtained at *T* = 77.3 K in one of the microbridges used in our experiments (BS7). The inset shows a schematic diagram of the circuit used to probe how the microbridge acts as a FCL, protecting the load resistance, *RL*, from a voltage fault. Figure from Ref. (Lorenzo et al., 2009).

24 Will-be-set-by-IN-TECH

Superconducting path

**5.2 Superconducting fault current microlimiters**

from Ref. (Lorenzo et al., 2009).

2008).

V

superconducting element is represented as a meander path. Figure from Ref. (Osorio et al.,

In figure 24 we show a typical electric field versus current density (*E* − *J*) curve, corresponding to one of the microbridges (denoted BS7) used in our experiments. This microbridge, whose length, width and thickness are, respectively, 385 *μ*m, 28 *μ*m and 300 nm, and with *Tc* = 88.6 K, has been grown on a sapphire substrate. The two characteristic current densities are indicated in this figure: The critical, *Jc*, at which dissipation first appears, and the so-called supercritical, *J*∗, at which the microbridge is triggered into highly dissipative states. Let us note here that we have chosen for our studies microbridges of these dimensions to guarantee

Fig. 24. A typical *E* − *J* curve obtained at *T* = 77.3 K in one of the microbridges used in our experiments (BS7). The inset shows a schematic diagram of the circuit used to probe how the microbridge acts as a FCL, protecting the load resistance, *RL*, from a voltage fault. Figure

Fig. 23. Axial view of a hybrid limiter attached to the cold heat exchanger. The

Heat exchanger

Magnetic core

a good thermal behaviour before, during and after the current fault, as we will see below. In addition, the widths of our microbridges are well above the threshold at which *J*∗ is sample-width dependent (Ruibal et al., 2007). The strong increase of *E* for current densities around *J*∗ make this type of samples very useful for FCL. However, the current is effectively limited only for electric fields or, equivalently, applied voltages well above that at which the limiter is triggered from normal operation (sample in the superconducting state) to current fault mode. In addition, once *J*∗ is attained, a thermal runaway (Viña et al., 2003; Maza et al., 2008) can be provoked which causes the reduction of the circulating current well below the nominal value (i.e., the current in normal operation without a fault) or very important damage on the microbridge (that can be even burnt out).

The superconducting microbridge exchanges heat mainly with its substrate, because at the operation temperatures, around 90 K, the heat transfer coefficient between YBCO films, for instance, and their substrate is *hbs* <sup>≈</sup> 103 Wcm−2K, whereas through liquid nitrogen or between the substrate and the copper holder they are 1000 times less (Duron et al., 2007; Lorenzo et al., 2009; Mosqueira et al., 1993). Hence, the conditions for a good refrigeration and, therefore, an optimal operation of the limiting device, depend very much on the relative dimensions of the microbridge and their substrate. The optimal refrigeration should allow us to operate with the minimum microbridge temperature increment relative to that of the bath, Δ*Tb*. If the Biot number (Lorenzo et al., 2009; Chapman, 1984) of both the microbridge and its substrate are *Bi* � 1 and, simultaneously, the thermal diffusion length, *Lth*, is longer than the corresponding thickness, Δ*Tb* may be crudely approached by

$$
\Delta T\_b = e\_b f E \left( \frac{1}{h\_{bs}} + \frac{A\_b}{A\_s} \frac{1}{h\_{sr}} \right),
\tag{5}
$$

where *eb* is the microbridge thickness, *Ab* and *As* are the surface areas of the superconductor bridge and, respectively, its substrate. As *hbs* � *hsr*, Δ*Tb* will directly depend on *Ab*/*As*, the condition of "thermal smallness" being,

$$\frac{A\_b}{A\_s} < \frac{h\_{sr}}{h\_{bs}},\tag{6}$$

under which Δ*Tb* approaches its "intrinsic" value, which is the temperature increase just associated with the effective thermal resistance of the interface bridge-substrate. For our microbridges similar to BS7 and under faults with characteristic times above 10 ms (of the order of commercial ac current periods), the conditions, *Bi* � 1 and *Lth* > *e* fully apply. In addition, as *Ab*/*As* <sup>≈</sup> <sup>4</sup><sup>×</sup> <sup>10</sup>−<sup>4</sup> whereas *hsr*/*hbs* <sup>≈</sup> <sup>10</sup>−3, this microbridge is "thermally small". One may use equation 5 to estimate Δ*Tb* when the microbridge is in stationary conditions under currents just below *<sup>J</sup>*∗. From figure 24, the power density involved is around 5 <sup>×</sup> 106 W/cm3 which leads, by also using the appropriate parameter values1, to <sup>Δ</sup>*Tb* <sup>≈</sup> 0.1 K. Even for faults involving powers ten times higher than those considered above, Δ*Tb* will remain below 1 K. One may also use equation 5 to roughly estimate that if the superconducting bridge had *Ab*/*As* one thousand times larger, as it is the case of those currently proposed for high power applications (Noe & Steurer, 2007; Weinstock, 2000), Δ*Tb* would take values at least

<sup>1</sup> The substrate thermal conductivities (*Ks*) and heat capacities (*Cs*) are of the order of 10 W/cm K and 0.6 J/cm<sup>3</sup> K for sapphire, and 0.2 W/cm K and 1 J/cm<sup>3</sup> K for SrTiO3, whereas for YBCO bridges *Kb* <sup>≈</sup> 0.05 W/cm K and *Cb* <sup>≈</sup> 0.7 J/cm<sup>3</sup> K. See, e. g., Refs. (Duron et al., 2007; Chapman, 1984; Mosqueira et al., 1993).

Fig. 26. Two examples of the time evolution of the current measured in presence of voltage

<sup>431</sup> Thermal Behaviour and Refrigeration

of High-Temperature Superconducting Fault Current Limiters and Microlimiters

The results presented in figure 25 also illustrate two other central aspects of the microbridges when working as FCL. Note first that, once the source voltage overcomes *V*∗, the current in the circuit varies quite slowly even up to voltage faults as important as four times *V*∗, the current taking a minimum value, *Imin*, at some (temperature-dependent) voltage. Moreover, the sharp drop at *V*∗ of the current is also temperature dependent, being almost absent in the curve at 85.0 K. Both aspects are related and may be explained in terms of the approaches based on the propagation of self-heating hot spots (Skocpol et al., 1974; Gurevich & Mints, 1987; Paulin et al., 1995): Above *V*∗ part of the microbridge becomes normal and then, as the total voltage of the circuit is fixed, the resistance increase originates a current decrease up to the minimum current, *Imin*, capable of sustaining the normal zone. If the fault voltage increases, the hotspot length will grow accordingly, keeping the current roughly constant. At a quantitative level, both aspects may be easily explained by just taking into account the

faults between *t*<sup>1</sup> and *t*2. Figure from Ref. (Lorenzo et al., 2009).

reduced temperature (*T*/*Tc*) dependence of *I*∗ and *Imin*,

*I*(*T*) = *I*(0)

 <sup>1</sup> <sup>−</sup> *<sup>T</sup> Tc <sup>n</sup>*

with *n* = 3/2 for *I*<sup>∗</sup> (Ruibal et al., 2007; Viña et al., 2003) and 1/2 for *Imin* (Skocpol et al., 1974; Gurevich & Mints, 1987; Paulin et al., 1995). Therefore, if the reduced temperature increases both the discontinuity at *V*<sup>∗</sup> and the ratio *I*∗/*Imin* will decrease, in agreement with the results of figure 25 The "optimal" reduced temperature for the microlimiter operation, *To p*/*Tc*, will

<sup>=</sup> <sup>1</sup> <sup>−</sup> *Imin*(0)

At *To p* the current limited during the voltage fault will be roughly equal to the nominal one. As *Imin*(0) < *I*∗(0), *To p* will be near, but below enough, *Tc* to make *I*∗(*To p*) adequate for the

The above results, experimentally confirmed with our studies (Lorenzo et al., 2009; 2010), allow us to "thermally tune" a superconducting microlimiter for an optimal protection. The

be then given by the condition *I*∗(*To p*) = *Imin*(*To p*). By using equation 7, this leads to,

*To p Tc*

practical operation of the microlimiter under such an optimal temperature.

, (7)

*<sup>I</sup>*∗(0) . (8)

two orders of magnitude higher. To confirm these values at a quantitative level, Δ*Tb* has been calculated by using a finite element method similar to the one described in (Maza et al., 2008). Under the same conditions as above, for the BS7 microbridge we found again Δ*Tb* ≈ 0.1 K.

In the case of SrTiO3 substrates, which have relatively poor thermal conductivities, the condition *Bi* � 1 does not apply anymore. Therefore, a term proportional to 1/*ks* (i.e. to the thermal conductivity of the substrate) must be added in equation 5. For microbridges under the same conditions as before, this leads to Δ*Tb* ≈ 2 K, a value that is confirmed by using the finite element method commented above. Let us stress, finally, that for a bridge on sapphire but having a surface relative to that of its substrate one thousand times larger, the finite element method yields Δ*Tb* ≈ 50 K.

To probe a microlimiter with low thermal dimensions, we have implemented the electrical circuit schematized in the inset of figure 24, with the microbridge BS7 as *Rb* connected in series to the variable load resistance *RL*, this last one representing the impedance of the circuit to be protected. The measurements were made in a cryostat with the sample submerged in a forced flow of helium gas. The temperature of the copper holder of the microlimiter was measured with a platinum thermometer and regulated with an electronic system which ensures a temperature stabilization better than 0.05 K. Two examples of the *I* − *V* curves obtained in this *RL* − *Rb* circuit (with *RL* = 4.9 Ω, this value taking already into account the resistance of the circuit electrical wires, of the order of 1.9 Ω) by using the electronic system described elsewhere (Ruibal et al., 2007; Viña et al., 2003) are shown in figure 25. In these curves the voltage was imposed and acquired during pulses of 1 s, a time much longer than the one needed by the microlimiter to reach the stationary state. The bath temperatures were 81.9 K (circles) and 85.0 K (triangles). As below *V*∗ the flux-flow resistance of the microbridge remains much lower than *RL* , both curves are almost linear up to *V*∗. Together with the low heating estimated above, this quasi-ohmic behavior is crucial to allow the microlimiter to work just below *V*∗ under stationary conditions and without disturbing the circuit to be protected.

Fig. 25. Two *I* − *V* curves, showing their strong bath temperature dependence, of the *RL* − *Rb* circuit schematized in the inset of figure 25 In these examples *Rb* is the microbridge BS7 and *RL* = 4.9 Ω. Figure from Ref. (Lorenzo et al., 2009).

26 Will-be-set-by-IN-TECH

two orders of magnitude higher. To confirm these values at a quantitative level, Δ*Tb* has been calculated by using a finite element method similar to the one described in (Maza et al., 2008). Under the same conditions as above, for the BS7 microbridge we found again Δ*Tb* ≈ 0.1 K. In the case of SrTiO3 substrates, which have relatively poor thermal conductivities, the condition *Bi* � 1 does not apply anymore. Therefore, a term proportional to 1/*ks* (i.e. to the thermal conductivity of the substrate) must be added in equation 5. For microbridges under the same conditions as before, this leads to Δ*Tb* ≈ 2 K, a value that is confirmed by using the finite element method commented above. Let us stress, finally, that for a bridge on sapphire but having a surface relative to that of its substrate one thousand times larger, the

To probe a microlimiter with low thermal dimensions, we have implemented the electrical circuit schematized in the inset of figure 24, with the microbridge BS7 as *Rb* connected in series to the variable load resistance *RL*, this last one representing the impedance of the circuit to be protected. The measurements were made in a cryostat with the sample submerged in a forced flow of helium gas. The temperature of the copper holder of the microlimiter was measured with a platinum thermometer and regulated with an electronic system which ensures a temperature stabilization better than 0.05 K. Two examples of the *I* − *V* curves obtained in this *RL* − *Rb* circuit (with *RL* = 4.9 Ω, this value taking already into account the resistance of the circuit electrical wires, of the order of 1.9 Ω) by using the electronic system described elsewhere (Ruibal et al., 2007; Viña et al., 2003) are shown in figure 25. In these curves the voltage was imposed and acquired during pulses of 1 s, a time much longer than the one needed by the microlimiter to reach the stationary state. The bath temperatures were 81.9 K (circles) and 85.0 K (triangles). As below *V*∗ the flux-flow resistance of the microbridge remains much lower than *RL* , both curves are almost linear up to *V*∗. Together with the low heating estimated above, this quasi-ohmic behavior is crucial to allow the microlimiter to work just below *V*∗ under stationary conditions and without disturbing the circuit to be

Fig. 25. Two *I* − *V* curves, showing their strong bath temperature dependence, of the *RL* − *Rb* circuit schematized in the inset of figure 25 In these examples *Rb* is the microbridge

BS7 and *RL* = 4.9 Ω. Figure from Ref. (Lorenzo et al., 2009).

finite element method yields Δ*Tb* ≈ 50 K.

protected.

Fig. 26. Two examples of the time evolution of the current measured in presence of voltage faults between *t*<sup>1</sup> and *t*2. Figure from Ref. (Lorenzo et al., 2009).

The results presented in figure 25 also illustrate two other central aspects of the microbridges when working as FCL. Note first that, once the source voltage overcomes *V*∗, the current in the circuit varies quite slowly even up to voltage faults as important as four times *V*∗, the current taking a minimum value, *Imin*, at some (temperature-dependent) voltage. Moreover, the sharp drop at *V*∗ of the current is also temperature dependent, being almost absent in the curve at 85.0 K. Both aspects are related and may be explained in terms of the approaches based on the propagation of self-heating hot spots (Skocpol et al., 1974; Gurevich & Mints, 1987; Paulin et al., 1995): Above *V*∗ part of the microbridge becomes normal and then, as the total voltage of the circuit is fixed, the resistance increase originates a current decrease up to the minimum current, *Imin*, capable of sustaining the normal zone. If the fault voltage increases, the hotspot length will grow accordingly, keeping the current roughly constant. At a quantitative level, both aspects may be easily explained by just taking into account the reduced temperature (*T*/*Tc*) dependence of *I*∗ and *Imin*,

*I*(*T*) = *I*(0) <sup>1</sup> <sup>−</sup> *<sup>T</sup> Tc <sup>n</sup>* , (7)

with *n* = 3/2 for *I*<sup>∗</sup> (Ruibal et al., 2007; Viña et al., 2003) and 1/2 for *Imin* (Skocpol et al., 1974; Gurevich & Mints, 1987; Paulin et al., 1995). Therefore, if the reduced temperature increases both the discontinuity at *V*<sup>∗</sup> and the ratio *I*∗/*Imin* will decrease, in agreement with the results of figure 25 The "optimal" reduced temperature for the microlimiter operation, *To p*/*Tc*, will be then given by the condition *I*∗(*To p*) = *Imin*(*To p*). By using equation 7, this leads to,

$$\frac{T\_{op}}{T\_c} = 1 - \frac{I\_{\rm min}(0)}{I^\*(0)}\,. \tag{8}$$

At *To p* the current limited during the voltage fault will be roughly equal to the nominal one. As *Imin*(0) < *I*∗(0), *To p* will be near, but below enough, *Tc* to make *I*∗(*To p*) adequate for the practical operation of the microlimiter under such an optimal temperature.

The above results, experimentally confirmed with our studies (Lorenzo et al., 2009; 2010), allow us to "thermally tune" a superconducting microlimiter for an optimal protection. The

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results summarized in figure 26 were measured by using a circuit similar to that of the inset of figure 24. In these examples, *RS* is again the BS7 microbridge and *RL* = 51.2 Ω (taking into account the resistance of the circuit electrical wires). For the curve measured at 83.0 K, the applied voltage was 5.8 V before (*t*1) and after (*t*2) a fault regime of 9.5 V. For the curve at 85.0 K, which is near *To p*, these values were 3.5 V and 7.5 V, respectively. As expected, the protection is excellent, whereas there is an overprotection when working well below *To p*. In both cases the recovery after the fault is achieved under current. This is another considerable practical advantage when compared to the superconducting limiters used in high power applications (Noe & Steurer, 2007; Weinstock, 2000). These results suggest that optimal current limitation in superconducting electronics could be accomplished by the own conductive pathways after a proper design (e. g. by decreasing the width of the pathway at well refrigerated selected locations), thus improving compactness.

## **6. Conclusion**

We have summarized some of our works on the thermal behaviour and the refrigeration of fault current limiters based on HTSC. Our results were experimentally or numerically obtained for the three basic SFCL configurations (resistive, inductive and hybrid types). For inductive devices we report on the effects on the limiter's performance of the inhomogeneous nature of HTSC, on the advantages of using this type of samples with artificially created weak zones or with stacks of elements of great surface-to-volume ratio, in particular thin film washers, which present a extremely large value of this ratio. The good refrigeration conditions of the superconducting elements obtained with these designs allow us to operate with limiters of improved thermal stability under a current fault and shorter recovery times after the fault removal. For bulk cylinders with AWZ, the recovery time values are several times lower than with homogeneous cylinders, while the impedance remains unchanged. By using stacks of bulk rings, with a large surface-volume ratio, the recovery time is about one order of magnitude shorter and without impedance loss. By using stacks of thin film washers, the recovery time is two orders of magnitude lower than with bulk cylinders and the impedance is even better.

For hybrid and resistive configurations, we have presented results for designs scarcely explored. On one hand, a limiter based on a thin film meander path integrated in the cold heat exchanger of a thermoacoustic refrigerator. The hybrid design of this device would allow us to operate at a power about 2.2 kVA with a thermoacoustic refrigerator capable of removing about 50 W at 80 K. Finally, we have probed the use of thin film microbridges to operate in very low power applications. Our results with these samples indicate that, by using the appropriate substrates and the right ratio microbridge-to-substrate area, it is possible to operate in a regime where a relative small fault provokes their transition to highly dissipative states that strongly increase the limitation efficiency.

## **7. Acknowledgements**

The earlier phases of the research summarized here were partially supported by the electrical company UNION-FENOSA under Grants No. 0666-98 and 0220-0085. This work has been supported by Ministerio de Ciencia e Innovación ERDF FI2010-19807 and by Xunta de Galicia ERDF 2010/XA043 and 10TMT206012PR.

### **8. References**

28 Will-be-set-by-IN-TECH

results summarized in figure 26 were measured by using a circuit similar to that of the inset of figure 24. In these examples, *RS* is again the BS7 microbridge and *RL* = 51.2 Ω (taking into account the resistance of the circuit electrical wires). For the curve measured at 83.0 K, the applied voltage was 5.8 V before (*t*1) and after (*t*2) a fault regime of 9.5 V. For the curve at 85.0 K, which is near *To p*, these values were 3.5 V and 7.5 V, respectively. As expected, the protection is excellent, whereas there is an overprotection when working well below *To p*. In both cases the recovery after the fault is achieved under current. This is another considerable practical advantage when compared to the superconducting limiters used in high power applications (Noe & Steurer, 2007; Weinstock, 2000). These results suggest that optimal current limitation in superconducting electronics could be accomplished by the own conductive pathways after a proper design (e. g. by decreasing the width of the pathway at

We have summarized some of our works on the thermal behaviour and the refrigeration of fault current limiters based on HTSC. Our results were experimentally or numerically obtained for the three basic SFCL configurations (resistive, inductive and hybrid types). For inductive devices we report on the effects on the limiter's performance of the inhomogeneous nature of HTSC, on the advantages of using this type of samples with artificially created weak zones or with stacks of elements of great surface-to-volume ratio, in particular thin film washers, which present a extremely large value of this ratio. The good refrigeration conditions of the superconducting elements obtained with these designs allow us to operate with limiters of improved thermal stability under a current fault and shorter recovery times after the fault removal. For bulk cylinders with AWZ, the recovery time values are several times lower than with homogeneous cylinders, while the impedance remains unchanged. By using stacks of bulk rings, with a large surface-volume ratio, the recovery time is about one order of magnitude shorter and without impedance loss. By using stacks of thin film washers, the recovery time is two orders of magnitude lower than with bulk cylinders and the

For hybrid and resistive configurations, we have presented results for designs scarcely explored. On one hand, a limiter based on a thin film meander path integrated in the cold heat exchanger of a thermoacoustic refrigerator. The hybrid design of this device would allow us to operate at a power about 2.2 kVA with a thermoacoustic refrigerator capable of removing about 50 W at 80 K. Finally, we have probed the use of thin film microbridges to operate in very low power applications. Our results with these samples indicate that, by using the appropriate substrates and the right ratio microbridge-to-substrate area, it is possible to operate in a regime where a relative small fault provokes their transition to highly dissipative states that strongly

The earlier phases of the research summarized here were partially supported by the electrical company UNION-FENOSA under Grants No. 0666-98 and 0220-0085. This work has been supported by Ministerio de Ciencia e Innovación ERDF FI2010-19807 and by Xunta de Galicia

well refrigerated selected locations), thus improving compactness.

**6. Conclusion**

impedance is even better.

increase the limitation efficiency.

ERDF 2010/XA043 and 10TMT206012PR.

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## *Edited by Yury Grigorashvili*

Book "Superconductors - Properties, Technology, and Applications" gives an overview of major problems encountered in this field of study. Most of the material presented in this book is the result of authors' own research that has been carried out over a long period of time. A number of chapters thoroughly describe the fundamental electrical and structural properties of the superconductors as well as the methods researching those properties. The sourcebook comprehensively covers the advanced techniques and concepts of superconductivity. It's intended for a wide range of readers.

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Superconductors - Properties, Technology, and Applications

Superconductors

Properties, Technology, and Applications

*Edited by Yury Grigorashvili*