**3. Josephson-type mechanism**

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

*al.* [5] developed an idea to discuss the key concept of granularity and its implications for localization in the normal state and paracoherence in the superconducting state. For arrays formed by niobium grains imbedded in epoxy resin [6] the coherent penetration depth or screening current are influenced by the intergrain regions. In fact, the main obstacles to intergranular critical current flow are weak superconductivity regions between the grains [7], called weak links (WLs) [8]. Ceramic superconductor samples present a random network for the supercurrent path, with the critical current being limited by the weakest links in each path. This Josephson-type mechanism of conduction is responsible to the dependence of the critical current density on the magnetic field *Jc* (*H*), as noted in several experimental studies [9–11]. On the other hand, the intragranular critical current is limited by an activated flux flow at

Considering these factors, Altshuler *et al.* [14] and Muller and Matthews [15] introduced the possibility of calculating the *Jc* (*H*) characteristic under any magnetic history following the proposal of Peterson and Ekin [16]. Basically the model considers that the transport properties of the junctions are determined by an "effective field" resulting from superposition of a external applied field and the field associated with magnetization of the superconducting

Another theoretical approach to the *Jc*(*H*) dependence in a junction took into account the effect of the magnetic field within the grains. This study has revealed that the usual Fraunhofer-like expression for *Jc* (*H*) [17, 18] should be written as *Jc* (*H*) ∝ *sin*(*bH*1/2)/(*bH*1/2), which we call the modified Fraunhofer-like expression [19]. Mezzetti *et al.* [20] and González *et al.* [21] also proposed models to describe *Jc*(*H*) behavior taking into account the latter expression. In both studies the authors concluded that a Gamma-type WL

González *et al.* considered two different regimes [21]: for low applied magnetic field, a linear decrease in *Jc* with the field was observed, whereas for high fields *Jc* (*H*) ∝ (1/*B*)

dependence was found. Here we have decided to follow the same approach and extend the

Usually polycrystalline ceramics samples contain grains of several sizes and the junction length changes from grain to grain. In addition, the granular samples may exhibit electrical, magnetic or other properties which are distinct from those of the material into the grains [5]. The average *Jc* (*H*) is obtained by integrating *Jc*(*H*) for each junction and taking into account a distribution of junction lengths in the sample. It was demonstrated that the WL width follows a Gamma-type distribution [22]. This function yields positive unilateral values and is always used to represent positive physical quantities. Furthermore, this Gamma distribution is the classical distribution used to describe the microstructure of granular samples [23] and satisfactorily reproduces the grain radius distribution in high-*Tc* ceramic superconductors

A superconductor exhibits two interesting properties: the first one is the electrical resistance of the material abruptly drops to zero at critical temperature T*c*. The superconductor is

0.5

high temperature and a high magnetic field [9, 12, 13].

distribution controls the transport critical current density.

analytical results to all applied magnetic fields.

grains.

[24].

**2. Basic properties**

Following the discovery of the electron tunneling (barrier penetration) in semiconductor, Giaever [25] showed that electron can tunnel between two superconductors. Subsequently, Josephson predicted that the Cooper pairs should be able to tunnel through the insulator from one superconductor to the other even zero voltage difference such the supercurrent is given by [26]

$$J = J\_{\mathcal{E}} \sin(\theta\_1 - \theta\_2) \tag{2}$$

where *Jc* is the maximum current in which the junction can support, and *θ<sup>i</sup>* (*i* = 1, 2) is the phase of wave function in *ith* superconductor at the tunnel junctions. This effect takes in account dc current flux in absence of applied electric and magnetic fields, called as the *dc Josephson effect*.

If a constant nonzero voltage *V* is maintained across the Josephson junction (barrier or weak link), an ac supercurrent will flow through the barrier produced by the single electrons tunneling. The frequency of the ac supercurrent is *ν* = 2*eV*/¯*h*. The oscillating current of Cooper pairs is known as the *ac Josephson effect*. These Josephson effects play a special role in superconducting applications.

It was mentioned that the behavior of a superconductor is sensitive to a magnetic field, so that the Josephson junction is also dependent. Therefore another mode of pair tunneling is a tunneling current with an oscillatory dependence on the applied magnetic flux sin(*π*Φ/Φ0), where Φ<sup>0</sup> is the quantum of magnetic flux. This phenomenon is known as *macroscopic quantum interference effect*.

### **3.1. Basic equations of Josephson effect**

As mentioned, the Josephson effect can occur between two superconductors weakly connected. Some types of linkage are possible such as sketched in Figure (1). These configurations depends on the application types in which the weak link can be:


**Figure 1.** Four types of Josephson junctions: (a) SIS with d1 = 10 - 20 Å, (b) SNS where d2 = 102 - 104 Å, (c) Point of contact, and (d) microbridge with d3 ≈ 1*μ*m [27].

Consider that two superconductors are separated from each other by an insulating layer. The junction is of thickness d normal to the y-axis with cross-sectional dimensions a and c along x and z, respectively. A voltage is applied between the superconductors and the junction is thick enough so that one assumes the potential to be zero in the middle of the barrier. Figure (2) displays Josephson junctions corresponding to a SIS junction.

In Feynman approach Ψ<sup>1</sup> and Ψ<sup>2</sup> are the quantum mechanical wavefunction of the superconducting state in the left and the right superconductor, respectively. This system is determined by coupled time-dependent Schroedinger equations:

$$i\hbar\frac{\partial\Psi\_1}{\partial t} = -2eV\_1\Psi\_1 + K\Psi\_2$$

$$i\hbar\frac{\partial\Psi\_2}{\partial t} = -2eV\_2\Psi\_1 + K\Psi\_{1\prime} \tag{3}$$

**Figure 2.** Application of current through the Josephson junctions [28].

*h*¯ *∂ns*<sup>1</sup>

*h*¯ *∂ns*<sup>2</sup>

Ψ<sup>1</sup> may be written as

equation (3) it obtains

be written as

such that the result is

where *Jc* <sup>=</sup> <sup>4</sup>*eK*(*ns*1*ns*2)1/2

a dc current tunnel for this simple case.

where *K* is a coupling constant for the wave functions across the barrier. The functions Ψ<sup>1</sup> and

Ψ<sup>1</sup> = (*ns*1)1/2 exp(*iθ*1)

where *ns*<sup>1</sup> and *ns*<sup>2</sup> are their superelectrons densities. And substituting equation (4) into

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>2</sup>*K*√*ns*1*ns*<sup>2</sup> sin(*θ*<sup>2</sup> <sup>−</sup> *<sup>θ</sup>*1)

Taking the time derivative of the Cooper pair density as being the supercurrent density can

*J* = *Jc* sin(*δ*),

transport, that is, a phase difference *δ* between either side of a superconductor junction causes

(*ns*<sup>1</sup> − *ns*2),

*<sup>h</sup>*¯ and *δ* = *θ*<sup>1</sup> − *θ*2. We interpret these results as describing a charge

*J* = *e ∂ ∂t*

Ψ<sup>2</sup> = (*ns*2)1/2 exp(*iθ*2), (4)

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*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>2</sup>*K*√*ns*1*ns*<sup>2</sup> sin(*θ*<sup>2</sup> <sup>−</sup> *<sup>θ</sup>*1) (5)

**Figure 2.** Application of current through the Josephson junctions [28].

where *K* is a coupling constant for the wave functions across the barrier. The functions Ψ<sup>1</sup> and Ψ<sup>1</sup> may be written as

$$\Psi\_1 = (n\_{s1})^{1/2} \exp(i\theta\_1)$$

$$\Psi\_2 = (n\_{s2})^{1/2} \exp(i\theta\_2),\tag{4}$$

where *ns*<sup>1</sup> and *ns*<sup>2</sup> are their superelectrons densities. And substituting equation (4) into equation (3) it obtains

$$
\hbar \frac{\partial n\_{s1}}{\partial t} = 2K \sqrt{n\_{s1} n\_{s2}} \sin(\theta\_2 - \theta\_1)
$$

$$
\hbar \frac{\partial n\_{s2}}{\partial t} = 2K \sqrt{n\_{s1} n\_{s2}} \sin(\theta\_2 - \theta\_1) \tag{5}
$$

Taking the time derivative of the Cooper pair density as being the supercurrent density can be written as

$$J = e \frac{\partial}{\partial t} (n\_{s1} - n\_{s2})\_\prime$$

such that the result is

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

tunneling current with an oscillatory dependence on the applied magnetic flux sin(*π*Φ/Φ0), where Φ<sup>0</sup> is the quantum of magnetic flux. This phenomenon is known as *macroscopic quantum*

As mentioned, the Josephson effect can occur between two superconductors weakly connected. Some types of linkage are possible such as sketched in Figure (1). These

1. an insulating, corresponding to a SIS junction, in which case the insulate layer can be in

4. a narrow constriction (microbridge) of typical dimensions like the coherence length 1 *μ*m.

**Figure 1.** Four types of Josephson junctions: (a) SIS with d1 = 10 - 20 Å, (b) SNS where d2 = 102 - 104 Å,

Consider that two superconductors are separated from each other by an insulating layer. The junction is of thickness d normal to the y-axis with cross-sectional dimensions a and c along x and z, respectively. A voltage is applied between the superconductors and the junction is thick enough so that one assumes the potential to be zero in the middle of the barrier. Figure

In Feynman approach Ψ<sup>1</sup> and Ψ<sup>2</sup> are the quantum mechanical wavefunction of the superconducting state in the left and the right superconductor, respectively. This system is

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>2*eV*1Ψ<sup>1</sup> <sup>+</sup> *<sup>K</sup>*Ψ<sup>2</sup>

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>2*eV*2Ψ<sup>1</sup> <sup>+</sup> *<sup>K</sup>*Ψ1, (3)

2. a normal metal, corresponding to a SNS junction of typical dimensions 102 - 104 Å;

configurations depends on the application types in which the weak link can be:

3. a very fine superconducting point presses on a flat superconductor;

(c) Point of contact, and (d) microbridge with d3 ≈ 1*μ*m [27].

(2) displays Josephson junctions corresponding to a SIS junction.

determined by coupled time-dependent Schroedinger equations:

*ih*¯ *∂*Ψ<sup>1</sup>

*ih*¯ *∂*Ψ<sup>2</sup>

*interference effect*.

order of 10 - 20 Å;

**3.1. Basic equations of Josephson effect**

$$J = J\_c \sin(\delta)\_\prime$$

where *Jc* <sup>=</sup> <sup>4</sup>*eK*(*ns*1*ns*2)1/2 *<sup>h</sup>*¯ and *δ* = *θ*<sup>1</sup> − *θ*2. We interpret these results as describing a charge transport, that is, a phase difference *δ* between either side of a superconductor junction causes a dc current tunnel for this simple case.

The behavior of *Jc* can also be analyzed the Ambegaokar and Baratoff theory [29]. Ambegaokar and Baratoff generalized the Josephson tunnel theory and derived the tunnelling supercurrent on the basis of the BCS theory for a s-wave homogeneous superconductor. In this approach, the temperature dependence of critical current is given with the following expression:

$$J\_{\mathcal{L}} = \frac{\pi}{2eR\_N\mathcal{S}}\Delta(T)tanh\left[\frac{\Delta(T)}{2k\_BT}\right] \tag{6}$$

Until now it was discussed Josephson junctions independent of magnetic field. However the Josephson contacts exhibit macroscopy quantum effects under magnetic field. In order to examine the effect of applying a magnetic field into the junctions, considering the Josephson

**Figure 3.** Behavior of the magnetic field in a Josephson junction. *A*<sup>1</sup> is potential in the superconductor 1

It is assumed because of symmetry, the magnetic field H(y) has no *x*- or *z*-direction dependence, but it varies in *y*-direction insofar the field penetrates into superconductor.

*H* = *Hz*(*y*)

*A* = *Ax*(*y*)

Consider the equation that relates the gradient of the phase of the wave function of the superconducting state with the magnetic vector potential integration of a closed path where

> Φ<sup>0</sup>

Φ<sup>0</sup>

 *<sup>B</sup> A*

*<sup>A</sup>* · *<sup>d</sup><sup>l</sup>* <sup>+</sup>

 *C B*

<sup>∇</sup> <sup>Θ</sup>(*r*)*<sup>d</sup><sup>l</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup>*

<sup>∇</sup> <sup>Θ</sup>(*r*)*<sup>d</sup><sup>l</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup>*

Φ<sup>0</sup>

The integrals in AB and CD are zero due to orthogonality of the vectors *<sup>A</sup>* · *<sup>d</sup><sup>l</sup>*.

<sup>Θ</sup><sup>1</sup> <sup>−</sup> <sup>Θ</sup><sup>10</sup> <sup>=</sup> <sup>2</sup>*<sup>π</sup>*

It is known that magnetic field is derived from potential vector *<sup>H</sup>* <sup>=</sup> ∇ × *<sup>A</sup>* , such that

*k*.

*i*

<sup>2</sup> . Inside the barrier the material is not superconducting and *Hz* = *H*0. Now it

*k* applied along the vertical z

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of Polycrystalline Superconductors Under the Applied Magnetic Field

371

*<sup>A</sup>* · *<sup>d</sup><sup>l</sup>*. (11)

*<sup>A</sup>* · *<sup>d</sup><sup>l</sup>* <sup>+</sup>

*A*1∞(*x* − *x*0). (13)

 *D C*

*<sup>A</sup>* · *<sup>d</sup><sup>l</sup>* . (12)

junctions as sketched in Figure (3) with a magnetic field *B*<sup>0</sup>

and *A*<sup>2</sup> is potential in the superconductor 2 [28].

must choose an integration contour as shown in Figure (4)

 *D C*

direction,

with <sup>|</sup>*y*<sup>|</sup> <sup>=</sup> *<sup>d</sup>*

 *B A*

the current is zero.

<sup>∇</sup> <sup>Θ</sup>(*r*)*<sup>d</sup><sup>l</sup>* <sup>+</sup>

For the integration path ABCD

 *C B*

<sup>∇</sup> <sup>Θ</sup>(*r*)*<sup>d</sup><sup>l</sup>* <sup>+</sup>

when *<sup>T</sup>* near *Tc*, <sup>Δ</sup>(*T*) � 1.74Δ0(<sup>1</sup> <sup>−</sup> *<sup>T</sup>*/*Tc*)1/2 is the superconducting gap parameter from the BCS theory. *RN* is the normal-state resistance of the junction, *S* is the cross section of a junction, and *e* and *kB* are electron charge and Boltzmann constant, respectively. For temperature relatively close to *Tc*, we can suppose the condition Δ(*T*) � *kBT* and the *tanh*[Δ(*T*)/2*kBT*] ≈ Δ(*T*)/2*kBT*. Taking this into account, Eq. (8) is transformed into [21]

$$J\_{\mathcal{E}} \approx \frac{\pi}{4eR\_N S} \Delta\_0^2 \left[1 - \frac{T}{T\_{\mathcal{E}}}\right]. \tag{7}$$

And in limiting *T* → 0,

$$J\_c \approx \frac{\pi}{4eR\_N S} \frac{\Delta\_0}{e}.\tag{8}$$

To calculate the Josephson coupling energy for cuprate superconductors that have a d-wave order parameter with nodes, we recall the work of Bruder and co-workers [30]. They have found that the tunneling current behaves in a similar fashion of s-wave superconductors junction and the leading behavior is determined by tunneling from a gap node in one side of a junction into the effective gap in the other side. Consequently, as a first approximation to the Josephson coupling energy EJ , we describe the theory of s-wave granular superconductors [29] to an average order parameter Δ in the grains.

In 1974, Rosenblatt [31] also analysed the tunnelling supercurrent through Josephson barriers, but in bulk granular superconductors (BGS). He proposed that the superconducting order parameter of an assembly of superconducting grains in the absence of applied current can be represented by a set of vectors in the complex plane Δ*<sup>α</sup>* = |Δ| exp(*iφα*), where *φα* is the superconducting phase in *α*th grain. He showed the arrays of Josephson junctions become superconducting in two stages. At the bulk transition temperature *To*, the magnitude of the order parameter of each grain becomes nonzero [32]. He considered two neighboring grains along an axis with complex superconducting order parameters Δ<sup>1</sup> and Δ2. Therefore, the Josephson junction can be modelled by [33]

$$H\_{\rm f} = -\sum\_{\langle ij \rangle} f\_{\rm ij} \mathbf{S}\_i^+ \mathbf{S}\_j^- \tag{9}$$

where *Ht* is pair tunnelling Hamilton, *S*± *<sup>i</sup>*;*<sup>j</sup>* is destruction and creation operators,

$$J\_{ij} = \frac{R\_c}{2R\_{ij}} \Delta(T) \tanh\left[\frac{\Delta(T)}{2k\_B T}\right],\tag{10}$$

the Josephson coupling energy between grains *i* and *j*, *Rc* = *πh*¯ /2*e*<sup>2</sup> and *Rij* is normal state resistance of the junctions between grains *i* and *j*.

Until now it was discussed Josephson junctions independent of magnetic field. However the Josephson contacts exhibit macroscopy quantum effects under magnetic field. In order to examine the effect of applying a magnetic field into the junctions, considering the Josephson junctions as sketched in Figure (3) with a magnetic field *B*<sup>0</sup> *k* applied along the vertical z direction,

**Figure 3.** Behavior of the magnetic field in a Josephson junction. *A*<sup>1</sup> is potential in the superconductor 1 and *A*<sup>2</sup> is potential in the superconductor 2 [28].

It is assumed because of symmetry, the magnetic field H(y) has no *x*- or *z*-direction dependence, but it varies in *y*-direction insofar the field penetrates into superconductor.

$$
\vec{H} = H\_z(y)\vec{k}.
$$

It is known that magnetic field is derived from potential vector *<sup>H</sup>* <sup>=</sup> ∇ × *<sup>A</sup>* , such that

$$
\vec{A} = A\_x(y)\vec{i}
$$

with <sup>|</sup>*y*<sup>|</sup> <sup>=</sup> *<sup>d</sup>* <sup>2</sup> . Inside the barrier the material is not superconducting and *Hz* = *H*0. Now it must choose an integration contour as shown in Figure (4)

Consider the equation that relates the gradient of the phase of the wave function of the superconducting state with the magnetic vector potential integration of a closed path where the current is zero.

$$
\oint \vec{\nabla} \Theta(r) d\vec{l} = \frac{2\pi}{\Phi\_0} \oint \vec{A} \cdot d\vec{l}.\tag{11}
$$

For the integration path ABCD

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

The behavior of *Jc* can also be analyzed the Ambegaokar and Baratoff theory [29]. Ambegaokar and Baratoff generalized the Josephson tunnel theory and derived the tunnelling supercurrent on the basis of the BCS theory for a s-wave homogeneous superconductor. In this approach, the temperature dependence of critical current is given with the following

<sup>2</sup>*eRNS*Δ(*T*)*tanh*

when *<sup>T</sup>* near *Tc*, <sup>Δ</sup>(*T*) � 1.74Δ0(<sup>1</sup> <sup>−</sup> *<sup>T</sup>*/*Tc*)1/2 is the superconducting gap parameter from the BCS theory. *RN* is the normal-state resistance of the junction, *S* is the cross section of a junction, and *e* and *kB* are electron charge and Boltzmann constant, respectively. For temperature relatively close to *Tc*, we can suppose the condition Δ(*T*) � *kBT* and the *tanh*[Δ(*T*)/2*kBT*] ≈ Δ(*T*)/2*kBT*. Taking this into account, Eq. (8) is transformed into [21]

Δ(*T*) 2*kBT* 

(6)

. (7)

*<sup>e</sup>* . (8)

*<sup>j</sup>* (9)

, (10)

*Jc* <sup>=</sup> *<sup>π</sup>*

*Jc* <sup>≈</sup> *<sup>π</sup>*

4*eRNS*Δ<sup>2</sup> 0 <sup>1</sup> <sup>−</sup> *<sup>T</sup> Tc* 

*Jc* <sup>≈</sup> *<sup>π</sup>* 4*eRNS*

To calculate the Josephson coupling energy for cuprate superconductors that have a d-wave order parameter with nodes, we recall the work of Bruder and co-workers [30]. They have found that the tunneling current behaves in a similar fashion of s-wave superconductors junction and the leading behavior is determined by tunneling from a gap node in one side of a junction into the effective gap in the other side. Consequently, as a first approximation to the Josephson coupling energy EJ , we describe the theory of s-wave granular superconductors

In 1974, Rosenblatt [31] also analysed the tunnelling supercurrent through Josephson barriers, but in bulk granular superconductors (BGS). He proposed that the superconducting order parameter of an assembly of superconducting grains in the absence of applied current can be represented by a set of vectors in the complex plane Δ*<sup>α</sup>* = |Δ| exp(*iφα*), where *φα* is the superconducting phase in *α*th grain. He showed the arrays of Josephson junctions become superconducting in two stages. At the bulk transition temperature *To*, the magnitude of the order parameter of each grain becomes nonzero [32]. He considered two neighboring grains along an axis with complex superconducting order parameters Δ<sup>1</sup> and Δ2. Therefore, the

*Ht* = − ∑

*Jij* <sup>=</sup> *Rc* 2*Rij* �*ij*�

Δ(*T*)*tanh*

the Josephson coupling energy between grains *i* and *j*, *Rc* = *πh*¯ /2*e*<sup>2</sup> and *Rij* is normal state

*JijS*<sup>+</sup> *<sup>i</sup> S*<sup>−</sup>

*<sup>i</sup>*;*<sup>j</sup>* is destruction and creation operators,

Δ(*T*) 2*kBT* 

Δ0

expression:

And in limiting *T* → 0,

[29] to an average order parameter Δ in the grains.

Josephson junction can be modelled by [33]

where *Ht* is pair tunnelling Hamilton, *S*±

resistance of the junctions between grains *i* and *j*.

$$\int\_{A}^{B} \vec{\nabla} \Theta(r) d\vec{l} + \int\_{B}^{\mathbb{C}} \vec{\nabla} \Theta(r) d\vec{l} + \int\_{\mathbb{C}}^{D} \vec{\nabla} \Theta(r) d\vec{l} = \frac{2\pi}{\Phi\_{0}} \left( \int\_{A}^{B} \vec{A} \cdot d\vec{l} + \int\_{B}^{\mathbb{C}} \vec{A} \cdot d\vec{l} + \int\_{\mathbb{C}}^{D} \vec{A} \cdot d\vec{l} \right). \tag{12}$$

The integrals in AB and CD are zero due to orthogonality of the vectors *<sup>A</sup>* · *<sup>d</sup><sup>l</sup>*.

$$
\Theta\_1 - \Theta\_{10} = \frac{2\pi}{\Phi\_0} A\_{1\infty}(\mathbf{x} - \mathbf{x}\_0). \tag{13}
$$

8 Will-be-set-by-IN-TECH 372 Superconductors – Materials, Properties and Applications A Description of the Transport Critical Current Behavior of Polycrystalline Superconductors Under the Applied Magnetic Field <sup>9</sup>

**Figure 4.** Path of integration around the Josephson junction. One superconductor SC1 is to the above of the insulating barrier and the other superconductor SC2. It was considered a weak-link tunnel short junction. [28]

To Θ2(*x*) the integration result is similar, but it must be to consider the A'B'C'D' path

$$
\Theta\_2 - \Theta\_{20} = -\frac{2\pi}{\Phi\_0} A\_{2\infty}(\mathbf{x} - \mathbf{x}\_0). \tag{14}
$$

Thus the phase difference is given by

of the barrier, the tunneling current is

Taking *u* = *δ*<sup>0</sup> + <sup>2</sup>*π*<sup>Φ</sup>

max/I I c

0

0.2

0.4

0.6

0.8

1

1.2

*I* = *Jc*  *δ*(*x*) = *δ*<sup>0</sup> +

sin(*δ*0)*dxdz* = *cJc*

<sup>Φ</sup><sup>0</sup> , the tunneling current becomes

*I* = *acJc*

When is this current maximum? The answer is for phase difference *δ*<sup>0</sup> = *<sup>π</sup>*

*Imax* = *Ic*

Φ<sup>0</sup>

 

where *Ic* = *acJc* is the critical current. This is named the *Josephson junction diffraction equation* and shows a Fraunhofer-like dependence of the magnetic field as is displayed in Figure (5)


**Figure 5.** Josephson Fraunhofer diffraction pattern dependence of magnetic field [34].

2*π*Φ Φ<sup>0</sup>

 +*a*/2 −*a*/2

*<sup>π</sup>*<sup>Φ</sup> sin(*δ*0) sin(

sin(*π*Φ/Φ0) (*π*Φ/Φ0)

sin(*δ*<sup>0</sup> +

*π*Φ Φ<sup>0</sup>

> 

2*π*Φ Φ<sup>0</sup>

Inserting equation (16) into equation (8) yields after integration over area *S* = *ac* cross section

 *x a* 

. (16)

A Description of the Transport Critical Current Behavior

of Polycrystalline Superconductors Under the Applied Magnetic Field

)*dx*. (17)

373

) (18)

, (19)

<sup>2</sup> . Hence,

Since the aim is to obtain the phase difference *δ*(*x*),

$$\delta(\mathbf{x}) = \Theta\_2(\mathbf{x}) - \Theta\_1(\mathbf{x}) = \delta\_0 + \frac{2\pi}{\Phi\_0} (A\_{1\infty} + A\_{2\infty}) \mathbf{x}\_\prime \tag{15}$$

where *δ*<sup>0</sup> = Θ<sup>20</sup> − Θ<sup>10</sup> is phase difference at *x*0. The total magnetic flux is given by

$$
\Phi = \int \vec{A} \cdot d\vec{l} = \int \vec{H} \cdot dS \vec{n}\_{\nu}
$$

and then

$$
\Phi = a(A\_{1\infty} + A\_{2\infty}).
$$

Thus the phase difference is given by

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

**Figure 4.** Path of integration around the Josephson junction. One superconductor SC1 is to the above of the insulating barrier and the other superconductor SC2. It was considered a weak-link tunnel short

Φ<sup>0</sup>

2*π* Φ<sup>0</sup>

*<sup>H</sup>* · *dS<sup>n</sup>*,

*A*2∞(*x* − *x*0). (14)

(*A*1<sup>∞</sup> + *A*2∞)*x*, (15)

To Θ2(*x*) the integration result is similar, but it must be to consider the A'B'C'D' path

<sup>Θ</sup><sup>2</sup> <sup>−</sup> <sup>Θ</sup><sup>20</sup> <sup>=</sup> <sup>−</sup> <sup>2</sup>*<sup>π</sup>*

*δ*(*x*) = Θ2(*x*) − Θ1(*x*) = *δ*<sup>0</sup> +

Φ = 

where *δ*<sup>0</sup> = Θ<sup>20</sup> − Θ<sup>10</sup> is phase difference at *x*0. The total magnetic flux is given by

*<sup>A</sup>* · *<sup>d</sup><sup>l</sup>* <sup>=</sup>

Φ = *a*(*A*1<sup>∞</sup> + *A*2∞).

Since the aim is to obtain the phase difference *δ*(*x*),

junction. [28]

and then

$$
\delta(\mathbf{x}) = \delta\_0 + \frac{2\pi\Phi}{\Phi\_0} \left(\frac{\mathbf{x}}{a}\right). \tag{16}
$$

Inserting equation (16) into equation (8) yields after integration over area *S* = *ac* cross section of the barrier, the tunneling current is

$$I = J\_{\mathbb{C}} \int \sin(\delta\_0) d\mathbf{x} dz = c f\_{\mathbb{C}} \int\_{-a/2}^{+a/2} \sin(\delta\_0 + \frac{2\pi\Phi}{\Phi\_0}) d\mathbf{x}.\tag{17}$$

Taking *u* = *δ*<sup>0</sup> + <sup>2</sup>*π*<sup>Φ</sup> <sup>Φ</sup><sup>0</sup> , the tunneling current becomes

$$I = acJ\_c \frac{\Phi\_0}{\pi \Phi} \sin(\delta\_0) \sin(\frac{\pi \Phi}{\Phi\_0}) \tag{18}$$

When is this current maximum? The answer is for phase difference *δ*<sup>0</sup> = *<sup>π</sup>* <sup>2</sup> . Hence,

$$I\_{\max} = I\_c \left| \frac{\sin(\pi \Phi / \Phi\_0)}{(\pi \Phi / \Phi\_0)} \right|, \tag{19}$$

where *Ic* = *acJc* is the critical current. This is named the *Josephson junction diffraction equation* and shows a Fraunhofer-like dependence of the magnetic field as is displayed in Figure (5)

**Figure 5.** Josephson Fraunhofer diffraction pattern dependence of magnetic field [34].

### **4. Critical current model**

It is well known which ceramic superconductor samples present a random network for the supercurrent path, with the critical current being limited by the weakest links in each path. Moreover, magneto-optical studies have demonstrated that the magnetic field first penetrates grains associated with these regions, even for very low values of *H*. Consequently, it would be interesting to estimate the influence of the magnetic field on the overall *Jc* of a sample taking into account the previous remarks. There are some general hypotheses about transport properties in polycrystalline ceramic superconductors on application of a magnetic field. (i) The electric current percolates through the material and heating begins to occur in WLs and in channels between them. This means that the critical current measured in the laboratory is an intergranular current. (ii) The junction widths among grains are less than the Josephson length, and the magnetic field penetrates uniformly into the junctions. (iii) The sample temperature during transport measurement must be close to the critical temperature. Under these conditions, the junction widths are less than the bulk coherence length and the Cooper-pairs current is given by Josephson tunneling. (iv) Near the critical temperature the magnetic field first penetrates WLs and, at practically the same time, the grains.

Following the same González *et al.* [21] procedure, we have:

**4.1. Expression for** *Jc*(*H*) **for** 0 < *α* < *π*/2

The hyperbolic cotangent has the expansion [36]

*z* coth(*z*/2) *<sup>z</sup>*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>

> <sup>∞</sup> ∑ *j*=0

*z* coth(*z*/2) *<sup>z</sup>*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>

*Jc* (*α*) = <sup>2</sup>*Jc*<sup>0</sup>

*<sup>π</sup>*2(*<sup>m</sup>* − <sup>1</sup>)

*β<sup>n</sup>* =

*α* ≥ *π*/2 [37].

[36]:

Taylor series around zero:

It is convenient to define

Thus, we can rewrite Eq. (24) as:

(*m* − 2) times, term by term. This yields:

*Jc* (*α*) = *Jc*<sup>0</sup>

*z* coth(*z*/2) = 2

1 *<sup>z</sup>*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup>

Now we can compute the Cauchy product of the series (22) and (23) to obtain:

*π*2 1 + ∞ ∑ *n*=1

*b*2*j*

(2*j*)! <sup>=</sup> <sup>2</sup> *π*2*<sup>n</sup>*

*π*2 1 *z* + ∞ ∑ *n*=1

*α* 

The critical current density *Jc* (*α*) is calculated by taking *z* = *α* in Eq. (26) and differentiating it

<sup>1</sup> + (−1)*<sup>m</sup>* <sup>∞</sup>

(−1)*n*+*<sup>j</sup>*

*π*2*n*−2*<sup>j</sup>*

*<sup>α</sup>m*(−1)*<sup>m</sup>* (*m* − 1)!

*∂m*−<sup>2</sup> *∂αm*−<sup>2</sup>

where the variable *α* is defined as *α* = *u*0/*η* = *φ*0/(*η*Λ0*H*). To obtain a simpler expression for the transport critical current density, we develop Eq. (21) in the ranges 0 < *α* < *π*/2 and

The function *<sup>F</sup>*(*z*)=[coth(*z*/2)] = (*z*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*2) has singular points at *<sup>z</sup>* <sup>=</sup> <sup>±</sup>*iπ*, but is analytical at all remaining points on the disc |*z*| = *π*. Thus, we expanded *F*(*z*) for the disc |*z*| < *π*.

> *b*2*<sup>n</sup>* 2*n*! *z*2*<sup>n</sup>*

where *b*2*<sup>n</sup>* are the Bernoulli numbers (*b*<sup>0</sup> = 1, *b*<sup>2</sup> = 1/6, *b*<sup>4</sup> = −1/30, *b*<sup>6</sup> = 1/42, . . .) given by

*<sup>b</sup>*2*<sup>n</sup>* = [(−1)*n*−12(2*n*)!]/[(2*π*)2*n*]*ζ*(2*n*), where *ζ*(2*n*) is the Zeta Riemann function. The function [1/(*z*<sup>2</sup> + *π*2)] is represented by the

> ∞ ∑ *j*=0

> > <sup>∞</sup> ∑ *j*=0

> > > <sup>∞</sup> ∑ *j*=0

*βnz*2*n*−<sup>1</sup> 

∑*n*0

 <sup>2</sup>*<sup>n</sup>* <sup>−</sup> <sup>1</sup> *m* − 2

 *βnα*2*<sup>n</sup>* 

(−1)*<sup>j</sup> <sup>z</sup>*2*<sup>j</sup> π*2*<sup>j</sup>*

(−1)*n*+*<sup>j</sup>*

*π*2*n*−2*<sup>j</sup>*

*b*2*j*

(2*j*)! *z*2*<sup>n</sup>* 

(−1)*n*−<sup>1</sup> (2)2*<sup>j</sup> <sup>ζ</sup>*(2*j*)

(0 < *α* < *π*/2), (27)

*π*2

 <sup>∞</sup> ∑ *n*=0  *coth*(*α*/2) *α*<sup>2</sup> + *π*<sup>2</sup>

of Polycrystalline Superconductors Under the Applied Magnetic Field

A Description of the Transport Critical Current Behavior

, (21)

375


. (23)

. (24)

. (25)

*z* �= 0. (26)

Normally polycrystalline ceramics samples contain grains of several sizes and the junction length changes from grain to grain. The average *Jc* (*H*) is obtained by integrating *Jc* (*H*) for each junction and taking into account a distribution of junction lengths in the sample. This function yields positive unilateral values and is always used to represent positive physical quantities. Furthermore, this Gamma distribution is the classical distribution used to describe the microstructure of granular samples [23] and reproduces the grain radius distribution in high-*Tc* ceramic superconductors [24].

Following the previous discussion, we can describe *Jc* (*H*) as a statistical average of the critical current density through a grain boundary. In the same way as Mezzetti *et al.* [20] and González *et al.* [21], we consider that the weak-link width fits a Gamma-type distribution [35]. For a magnetic field higher or lower than the first critical field, the usual Fraunhofer diffraction pattern or the modified pattern is used to describe *Jc* (*H*) for each grain boundary. Thus,

$$J\_{\mathcal{E}}(H) = J\_{\mathcal{E}0} \int\_{-\infty}^{+\infty} P(u) \left| \frac{\sin(\pi u/u\_0)}{\pi u/u\_0} \right| du \tag{20}$$

$$P(u) = \begin{cases} \frac{\mu^{m-1} e^{(-u/\eta)}}{\eta^m \Gamma(m)} & u \ge 0 \\ 0 & u < 0, \end{cases}$$

where Γ(*m*) is the Gamma function which is widely tabulated [22, 36]:

$$
\Gamma(m) = \int\_0^\infty w^{m-1} e^{-w} \, dw
$$

when *m* is a real number. Or Γ(*m*)=(*m* − 1)! if *m* is a positive integer. The parameters *m* and *η*, both positive integer, determine the distribution form and scale (width and height), respectively [21]. The variable *u* represents the WL length. The quantity *u*<sup>0</sup> is defined as *u*<sup>0</sup> = *φ*0/Λ0*H*, where *φ*<sup>0</sup> is the quantum flux and Λ<sup>0</sup> is the effective thickness of the WL. Following the same González *et al.* [21] procedure, we have:

$$f\_c(\mathfrak{a}) = f\_{c0} \frac{\mathfrak{a}^m (-1)^m}{(m-1)!} \frac{\partial^{m-2}}{\partial \mathfrak{a}^{m-2}} \left[ \frac{\coth(\mathfrak{a}/2)}{\mathfrak{a}^2 + \pi^2} \right],\tag{21}$$

where the variable *α* is defined as *α* = *u*0/*η* = *φ*0/(*η*Λ0*H*). To obtain a simpler expression for the transport critical current density, we develop Eq. (21) in the ranges 0 < *α* < *π*/2 and *α* ≥ *π*/2 [37].
