**1. Introduction**

30 Will-be-set-by-IN-TECH

[262] Gabovich AM, Voitenko AI (2000) Nonstationary Josephson tunneling involving

[263] Koren G, Levy N (2002) Experimental evidence for a small *s*-wave component in the order parameter of underdoped YBa2Cu3O6+*x*. Europhys. Lett. 59: 121–127. [264] Babcock SE, Vargas JL (1995) The nature of grain boundaries in the high-*Tc*

[265] Hilgenkamp H, Mannhart J, Mayer B (1996) Implications of *dx*<sup>2</sup>−*y*<sup>2</sup> symmetry and faceting for the transport properties of grain boundaries in high-*Tc* superconductors.

[266] Neils WK, Van Harlingen DJ, Oh S, Eckstein JN, Hammerl G, Mannhart J, Schmehl A, Schneider CW, Schulz RR (2002) Probing unconventional superconducting symmetries

[267] Aligia AA, Kampf AP, Mannhart J (2005) Quartet formation at (100)/(110) interfaces of

[268] Graser S, Hirschfeld PJ, Kopp T, Gutser R, Andersen BM, Mannhart J (2010) How grain boundaries limit supercurrents in high-temperature superconductors. Nature Phys. 6:

[269] Aswathy PM, Anooja JB, Sarun PM, Syamaprasad U (2010) An overview on iron based

[270] Seidel P (2011) Josephson effects in iron based superconductors. Supercond. Sci.

[271] Johrendt D (2011) Structure-property relationships of iron arsenide superconductors.

[272] Wen J, Xu G, Gu G, Tranquada JM, Birgeneau RJ (2011) Interplay between magnetism and superconductivity in iron-chalcogenide superconductors: crystal growth and

[273] Hirschfeld PJ, Korshunov MM, Mazin II (2011) Gap symmetry and structure of Fe-based

[274] Richard P, Sato T, Nakayama K, Takahashi T, Ding H (2011) Fe-based superconductors: an angle-resolved photoemission spectroscopy perspective. Rep. Prog. Phys. 74: 124512. [275] Hoffman JE (2011) Spectroscopic scanning tunneling microscopy insights into Fe-based

superconductors with spin-density waves. Physica C 329: 198–230.

superconductors. Annu. Rev. Mater. Sci. 25: 193–222.

using Josephson interferometry. Physica C 368: 261–266.

*d*-wave superconductors. Phys. Rev. Lett. 94: 247004.

superconductors. Supercond. Sci. Technol. 23: 073001.

characterizations. Rep. Prog. Phys. 74: 124503.

superconductors. Rep. Prog. Phys. 74: 124508.

superconductors. Rep. Prog. Phys. 74: 124513.

Phys. Rev. B 53: 14586–14593.

609–614.

Technol. 24: 043001.

J. Mater. Chem. 21: 13726–13736.

Superconductivity is a macroscopic quantum phenomenon [1]. Therefore it shows quite interesting properties because of its quantum nature. Such properties are described by a macroscopic complex wave function of the superconductivity. Especially, a phase of the macroscopic wave function play an important role in these properties. For example, superconducting devices, such as, superconducting charge, flux and phase qubits, superconducting single flux quantum device, and intrinsic Josephson junction Terahertz emitter of high Tc cuprate superconductors, use such quantum nature of superconductivity. They have attracted much attention recently.

In conventional superconductors, there is only single phase φ of the superconducting order parameter or the macroscopic wave function φ Δ= Δ *<sup>i</sup> e* . This phase causes interference effect, such as, Josephson effect, and quantization of vortices in the superconductor. But unconventional and anisotropic superconductivity have different phase that comes from internal degree of freedom of the superconducting order parameter. Because in the superconductors, electrons are paired and if their paring symmetry is an s-wave, as in the conventional superconductors, the order parameter is just a single complex number. But if the symmetry is other one such as p-wave or d-wave, then the order parameter has an internal phase [2,3]. For example, d-wave superconductors, especially 2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* -wave superconductors have a symmetry that is shown in Fig. 1. This symmetry is internal and it appears in momentum space that means the wave function of the Cooper pair moving along the x-axis has + sign and that moving along y-axis has – sign. This is also another phase of superconducting order parameter and it affects the interference phenomena in the superconductors.

© 2012 Kato et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Kato et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Composite Structures of d-Wave and s-Wave Superconductors (d-Dot):

of the Cooper pairs in high-Tc superconductors was determined as d-wave, especially

In addition, this experiment showed that the critical current become smax when the total magnetic flux through the corner junction was <sup>0</sup> Φ 2 . Here, <sup>0</sup> Φ = *hc e*2 is the flux quantum. Because the phase difference π between two Josephson currents through junctions A and B,

22 2

2 2

<sup>Φ</sup> = ⋅ = ⋅ = Φ= <sup>Φ</sup> *C s*

where C is a contour around the corner junction and *S* is the area surrounded by the C and Φ is the total magnetic flux through the junctions. When total magnetic flux <sup>0</sup> Φ=Φ 2 , this

This result shows, the stable superconducting state under zero external current and zero external field becomes nontrivial. For such stable state, the free energy of whole system, especially the superconducting condensation energies of both d- and s-wave superconductors should be low. Therefore the order parameter should become continuous across both of junctions. And then the phase of order parameter cannot be uniform because of the phase difference π between two junctions. This phase difference π should be compensated by changing the phase of the order parameter spatially in both s- and d-wave superconducting regions because of single valuedness of the order parameter. This spatial variation of the order parameter causes the supercurrent around the corner junction and then spontaneous magnetic flux is created at the corner without an external field. Because the associated phase change of this supercurrent is not 2π but only π, the spontaneous magnetic flux is not singly quantized magnetic flux Φ<sup>0</sup> , but half-flux quantum magnetic

An experiment showing this property was done by Higenkamp et al., who made a zigzag junction between conventional s-wave superconductor Nb and high-Tc d-wave superconductor YBCO, which consists of successive corner junctions [6]. And they observed spontaneous magnetic fluxes at every corner under a zero external field, using scanning SQUID microscope. The spontaneous magnetic fluxes aligned antiferromagnetically, because of the attractive vortex-anti-vortex interaction. They also made small ring with two junctions between Nb and YBCO, which is called π-rings, and controlled the spontaneous

When spontaneous magnetic flux appears in the zero external magnetic field, this state does not have the time reversal symmetry, which the original system has. Therefore there are always two degenerate stable states. In these states, supercurrent directions are opposite and henceforth directions of spontaneous magnetic fluxes are opposite. This spontaneously appeared magnetic flux is useful and can be used as a spin or a bit by it self. Ioffe et al. [8] proposed a quantum bit using this half-quantum flux. Using the spontaneous magnetic flux

π

. With this phase, total phase difference becomes 0 or 2π. Therefore

2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* .

phase becomes

flux ( <sup>0</sup> Φ 2 ).

is compensated by additional phase,

χ = π

half-quantum magnetic fluxes [7].

χ

two Josephson currents now are reinforced with each other.

Analysis Using Two-Component Ginzburg-Landau Equations 321

0

 π

*ee e A ds curlA dS c c ch* (1)

**Figure 1.** The symmetry of 2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* -wave superconductivity. The color shows the sign of wave function (order parameter) and the shape shows the amplitude of the wave function (order parameter).

The high-Tc cuprate superconductors are typical example of d-wave superconductors. Since the discovery of the high-Tc superconductors, the pairing symmetry, as well as its origin, has been controversial, but phase-sensitive experiment is crucial for determining the symmetry of the Cooper pairs. One such phase-sensitive experiment is a corner junction experiment [4,5]. In this experiment, a square-shaped high-Tc superconductor is connected with a conventional s-wave superconductor with two Josephson junctions A and B, around a corner and each junction is perpendicular to either of the x or the y axes of the high-Tc superconductor, as shown in Fig. 2.

**Figure 2.** Configuration of corner junction between d- and s-wave superconductors. There are two junctions, A and B.

In such geometry if the Cooper pair tunnels through junction A has phase 0, then the Cooper pair tunnels through junction B has phase π. The critical current between the s- and the d-wave superconductors through these two junctions is zero under zero external magnetic field because each supercurrent cancels with each other. This is in apparent contrast with a corner junction between both s-wave superconductors for which the critical current is largest under zero external field. Therefore, from this experiment, the symmetry of the Cooper pairs in high-Tc superconductors was determined as d-wave, especially 2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* .

320 Superconductors – Materials, Properties and Applications

superconductor, as shown in Fig. 2.

junctions, A and B.

**Figure 1.** The symmetry of 2 2 *<sup>x</sup>* <sup>−</sup> *<sup>y</sup> d* -wave superconductivity. The color shows the sign of wave function

The high-Tc cuprate superconductors are typical example of d-wave superconductors. Since the discovery of the high-Tc superconductors, the pairing symmetry, as well as its origin, has been controversial, but phase-sensitive experiment is crucial for determining the symmetry of the Cooper pairs. One such phase-sensitive experiment is a corner junction experiment [4,5]. In this experiment, a square-shaped high-Tc superconductor is connected with a conventional s-wave superconductor with two Josephson junctions A and B, around a corner and each junction is perpendicular to either of the x or the y axes of the high-Tc

**Figure 2.** Configuration of corner junction between d- and s-wave superconductors. There are two

In such geometry if the Cooper pair tunnels through junction A has phase 0, then the Cooper pair tunnels through junction B has phase π. The critical current between the s- and the d-wave superconductors through these two junctions is zero under zero external magnetic field because each supercurrent cancels with each other. This is in apparent contrast with a corner junction between both s-wave superconductors for which the critical current is largest under zero external field. Therefore, from this experiment, the symmetry

(order parameter) and the shape shows the amplitude of the wave function (order parameter).

In addition, this experiment showed that the critical current become smax when the total magnetic flux through the corner junction was <sup>0</sup> Φ 2 . Here, <sup>0</sup> Φ = *hc e*2 is the flux quantum. Because the phase difference π between two Josephson currents through junctions A and B, is compensated by additional phase,

$$\mathcal{X} = \int\_{\mathcal{C}} \frac{2e}{c\hbar} A \cdot d\mathbf{s} = \int\_{\mathcal{C}} \frac{2e}{c\hbar}curl A \cdot dS = 2\pi \frac{2e}{c\hbar} \Phi = 2\pi \frac{\Phi}{\Phi\_0} \tag{1}$$

where C is a contour around the corner junction and *S* is the area surrounded by the C and Φ is the total magnetic flux through the junctions. When total magnetic flux <sup>0</sup> Φ=Φ 2 , this phase becomes χ = π . With this phase, total phase difference becomes 0 or 2π. Therefore two Josephson currents now are reinforced with each other.

This result shows, the stable superconducting state under zero external current and zero external field becomes nontrivial. For such stable state, the free energy of whole system, especially the superconducting condensation energies of both d- and s-wave superconductors should be low. Therefore the order parameter should become continuous across both of junctions. And then the phase of order parameter cannot be uniform because of the phase difference π between two junctions. This phase difference π should be compensated by changing the phase of the order parameter spatially in both s- and d-wave superconducting regions because of single valuedness of the order parameter. This spatial variation of the order parameter causes the supercurrent around the corner junction and then spontaneous magnetic flux is created at the corner without an external field. Because the associated phase change of this supercurrent is not 2π but only π, the spontaneous magnetic flux is not singly quantized magnetic flux Φ<sup>0</sup> , but half-flux quantum magnetic flux ( <sup>0</sup> Φ 2 ).

An experiment showing this property was done by Higenkamp et al., who made a zigzag junction between conventional s-wave superconductor Nb and high-Tc d-wave superconductor YBCO, which consists of successive corner junctions [6]. And they observed spontaneous magnetic fluxes at every corner under a zero external field, using scanning SQUID microscope. The spontaneous magnetic fluxes aligned antiferromagnetically, because of the attractive vortex-anti-vortex interaction. They also made small ring with two junctions between Nb and YBCO, which is called π-rings, and controlled the spontaneous half-quantum magnetic fluxes [7].

When spontaneous magnetic flux appears in the zero external magnetic field, this state does not have the time reversal symmetry, which the original system has. Therefore there are always two degenerate stable states. In these states, supercurrent directions are opposite and henceforth directions of spontaneous magnetic fluxes are opposite. This spontaneously appeared magnetic flux is useful and can be used as a spin or a bit by it self. Ioffe et al. [8] proposed a quantum bit using this half-quantum flux. Using the spontaneous magnetic flux

as an Ising spin system, Kirtley et al. made a frustrated triangular lattice of π-rings [7]. In their systems, the π-rings were isolated and interacted with each other purely by the electromagnetic force.

Composite Structures of d-Wave and s-Wave Superconductors (d-Dot):

superconductors, their anisotropy cannot be treated by the simple GL theory. This is because using only up to quartic term and quadratic term of gradients of the single order parameters in free energy, there are no anisotropic terms. Anisotropy of the vortices in dwave superconductors within the phenomenological theory was treated by Ren et al., who used a two-component GL theory [12-15]. Here, two components mean the two components of the order parameter of with s and d symmetries. They derived the two-component GL

Multi-components GL equations were used for exotic superconductors, e.g. heavy fermion superconductors[16-18] and Sr2RuO4[19,20]. Also, recently, two components GL equations

The model by Ren.et al. especially emphasize the anisotropy of d-wave superconductivity. Therefore, we use the following two-component Ginzburg-Landau (GL) free energy for d-

<sup>3</sup> 4ln , , <sup>2</sup>

*d sd d d d s s*

∗ ∗ ∗∗ ∗

+ Δ Δ + Δ Δ +Δ Δ + − + Ω

8 3

<sup>1</sup> <sup>2</sup> H.c.

+ Δ + Δ + Π Δ Π Δ −Π Δ Π Δ +

*dF d s xs xd ys yd*

1 1 <sup>2</sup> div 2 8

∗ ∗

( )

**A** is a generalized momentum operator that is gauge invariant and

= ( ) 0 *d d V N* is the strengths of the coupling constants for the d-wave

Π Π (2)

2 2 4 2

αλ

**hH A**

*<sup>V</sup>* . Here, *<sup>N</sup>* ( ) <sup>0</sup> is the density of states of electrons at the

( ) ( )

π

2 2 2 2 22 2 2

<sup>=</sup> *<sup>T</sup>* . Δ*<sup>d</sup>* and <sup>Δ</sup>*<sup>s</sup>* are the d-wave and the s-wave components of the order parameter,

Fermi energy and *Vd* and *Vs* are interaction constants between electrons for d- and s-wave channels, respectively. We assume attractive and repulsive interactions for the d- and the swave channels, respectively. *Tcd* is the transition temperature of the d-wave superconductivity under zero-external field. **H** is an external field and **h A** = curl . We take the London gauge i.e. ∇⋅ = **A** 0 and **A n**⋅ = 0 at the surface of superconductor. The term

Also, for s-wave superconductors, we use the following two-component GL equation with attractive and repulsive interactions for the s-wave and the d-wave channels,

α

*cd <sup>s</sup>*

*T T*

ΔΔ = Δ − + Δ+ Δ

are studied for two-band or two-gap superconductors, such as MgB2 [21,22].

( ) ( )

λ

α

<sup>2</sup> <sup>2</sup> <sup>2</sup>

*d s d s d sd*

1

+ =

*s d*

*V*

λ

∇ ⋅ **A** in the integrand of Eq. 2 is added for fixing the gauge.

*d*

α

*s*

Ω

free energy from the Gor'kov equations.

wave superconductors;

where <sup>2</sup> Π= ∇− *<sup>e</sup>*

( ) ( )<sup>2</sup> 7 3 8 ζ

π

respectively.

( )<sup>2</sup>

respectively:

α

4

F

αλ

*i c*

λ

interaction channel and

αλ

*v*

**A**

Analysis Using Two-Component Ginzburg-Landau Equations 323

 α

> α

> > *d*

**Figure 3.** Schematic diagram of a d-dot.

In contrast to the previous approach for using the spontaneous half-flux quantum, we considered nano-sized d-wave superconductor embedded in an s-wave matrix, as shown in Fig. 3 [9-11]. We want to consider the whole d-dot system as a single element, not the individual half-quantum fluxes. As in the single half-quantum system, our d-dot has two degenerate states if the spontaneous magnetic fluxes appear, because the state with spontaneous magnetic fluxes under zero external field also breaks time-reversal symmetry. This property is independent from the shape, and the d-dot in any shape always has two degenerate stable states. Therefore, the d-dot as a whole can be considered as a single element with two level states and it might be used as a spin or a bit also as a qubit. It has better properties than those of single flux quantum element, which will be shown in following sections.

In the following, we first show a phenomenological superconducting theory, which describes the spontaneous magnetic fluxes in these composite structures, especially in d-dot systems and then we discuss the basic properties of this d-dot, based on this phenomenological theory. Also, we discuss the difference between a single half-quantum flux system and our d-dot system.
