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

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

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.

*Jc* (*H*) = *Jc*<sup>0</sup>

*P*(*u*) =

Γ(*m*) =

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

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

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,

> 

*wm*−1*e*

*<sup>η</sup>m*Γ(*m*) *u* ≥ 0 0 *u* < 0,

*um*−1*e*(−*u*/*η*)

 ∞ 0

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.

*sin*(*πu*/*u*0) *πu*/*u*<sup>0</sup>

<sup>−</sup>*<sup>w</sup> dw*

 

*du* (20)

 +∞ <sup>−</sup><sup>∞</sup> *<sup>P</sup>*(*u*)

**4. Critical current model**

high-*Tc* ceramic superconductors [24].

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*.

The hyperbolic cotangent has the expansion [36]

$$z \coth(z/2) = 2 \left[ \sum\_{n=0}^{\infty} \frac{b\_{2n}}{2n!} z^{2n} \right] \qquad |z| < \pi \,\tag{22}$$

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 [36]:

$$b\_{2n} = \left[\left(-1\right)^{n-1} 2\binom{2n}{\cdot}\right]/\left[\left(2\pi\right)^{2n}\right] \zeta\left(2n\right)\_{\cdot}$$

where *ζ*(2*n*) is the Zeta Riemann function. The function [1/(*z*<sup>2</sup> + *π*2)] is represented by the Taylor series around zero:

$$\frac{1}{z^2 + \pi^2} = \frac{1}{\pi^2} \sum\_{j=0}^{\infty} (-1)^j \frac{z^{2j}}{\pi^{2j}}. \tag{23}$$

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

$$\frac{z\coth(z/2)}{z^2+\pi^2} = \frac{2}{\pi^2} \left[ 1 + \sum\_{n=1}^{\infty} \left( \sum\_{j=0}^{\infty} \frac{(-1)^{n+j} b\_{2j}}{\pi^{2n-2j} (2j)!} \right) z^{2n} \right]. \tag{24}$$

It is convenient to define

$$\beta\_{\mathbb{N}} = \left[ \sum\_{j=0}^{\infty} \frac{(-1)^{n+j} b\_{2j}}{\pi^{2n-2j} (2j)!} \right] = \frac{2}{\pi^{2n}} \left[ \sum\_{j=0}^{\infty} \frac{(-1)^{n-1}}{(2)^{2j}} \zeta(2j) \right]. \tag{25}$$

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

$$\frac{z\coth(z/2)}{z^2+\pi^2} = \frac{2}{\pi^2} \left[ \frac{1}{z} + \sum\_{n=1}^{\infty} \beta\_n z^{2n-1} \right] \qquad z \neq 0. \tag{26}$$

The critical current density *Jc* (*α*) is calculated by taking *z* = *α* in Eq. (26) and differentiating it (*m* − 2) times, term by term. This yields:

$$J\_c(\mathfrak{a}) = \frac{2J\_{c0}}{\pi^2 (m-1)} \mathfrak{a} \left[ 1 + (-1)^m \sum\_{n\_0}^{\infty} \binom{2n-1}{m-2} \beta\_{ll} \mathfrak{a}^{2n} \right]$$

$$(0 < \mathfrak{a} < \pi/2), \tag{27}$$

#### 12 Will-be-set-by-IN-TECH 376 Superconductors – Materials, Properties and Applications A Description of the Transport Critical Current Behavior of Polycrystalline Superconductors Under the Applied Magnetic Field <sup>13</sup>

where *n*<sup>0</sup> is the lower integer and *n*<sup>0</sup> ≥ (*m* − 1)/2. This expression (27) for *Jc* is valid for the range 0 < *α* < *π*. However, for more efficient calculation, we suggest that it is only used for the range 0 < *α* < *π*/2.

Finally, we can express *Jc* as a function of *H* by substituting the definition of *α*, *α* = *u*0/*η* = *φ*0/(*η*Λ0*H*) in Eq. (27). Thus [37],

$$J\_c(H) = \frac{2J\_{c0}}{\pi^2 (m-1)} 1.02 \sqrt{\frac{H\_0^\*}{H}} \left[ 1 + \right.$$

$$(-1)^m \sum\_{n\_0}^{\infty} (1.02)^{2n} (m)^{2n} \binom{2n-1}{m-2} \beta\_n \left( \frac{H\_0^\*}{H} \right)^{2n} \Big|\_{} \tag{28}$$

where

and

For *z* = *α*

and *fp*(*α*) can be written as

*R*(*α*) as a function of *H*. Thus,

where *ap*<sup>0</sup> <sup>=</sup> (*p*+1)!

written as

*fm*−2(*z*) = *<sup>D</sup>m*−<sup>2</sup>

*R*(*α*) = 2

*fp*(*p*) = (−1)*<sup>p</sup> <sup>p</sup>*!

(*α*<sup>2</sup> + *π*2)*p*+<sup>1</sup>

*fp*(*H*) = (−1)*<sup>p</sup> <sup>p</sup>*! *m*2*p*+<sup>2</sup>

> *ap*<sup>1</sup> � *H*<sup>∗</sup> 0 *H*

> > ∞ ∑ *k*=1

� *<sup>m</sup>*−<sup>2</sup> ∑ *p*=0

*<sup>p</sup>*! *<sup>m</sup>p*, *ap*<sup>1</sup> <sup>=</sup> (*p*+1)!

*R*(*H*) = 2

�� *H*<sup>∗</sup> 0 *H* �2 + *<sup>π</sup>*<sup>2</sup> *m*<sup>2</sup> �*p*+<sup>1</sup>

�*p*−<sup>2</sup>

(*p*−2)!3!*π*2*mp*−2, *ap*<sup>2</sup> <sup>=</sup> (*p*+1)!

commenting again that *fp*(*H*) is finite for all values of *H* and is defined for *p* > 0. *R*(*H*) is

(−1)*m*−*<sup>p</sup>*

*fp*(*H*)*km*−*p*−<sup>2</sup>

+ *ap*<sup>2</sup>

∞ ∑ *k*=1

*<sup>R</sup>*(*z*) = *<sup>D</sup>m*−2[2/(*z*<sup>2</sup> <sup>+</sup> *<sup>π</sup>*<sup>2</sup> <sup>∞</sup>

� *<sup>m</sup>*−<sup>2</sup> ∑ *p*=0

� (*p* + 1)!

�

(−1)*m*−*<sup>p</sup>*

1/(*z*<sup>2</sup> + *π*2)

∑ *k*=1 *e* <sup>−</sup>*kz*)].

� *<sup>m</sup>* <sup>−</sup> <sup>2</sup> *p*

*fp*(*α*)*km*−*p*−<sup>2</sup>

*<sup>p</sup>*! *<sup>α</sup><sup>p</sup>* <sup>−</sup> (*<sup>p</sup>* <sup>+</sup> <sup>1</sup>)!

(*<sup>p</sup>* <sup>−</sup> <sup>4</sup>)!5!*π*4*αp*−<sup>4</sup> <sup>−</sup> ... �

� *ap*<sup>0</sup> � *H*<sup>∗</sup> 0 *H* �*p* −

� *H*<sup>∗</sup> 0 *H*

> ⎛ ⎝

> > � *e*

*m* − 2

⎞ ⎠ ×

*p*

−*km*� *H*∗ 0 *H* �

�*p*−<sup>4</sup>

<sup>−</sup> ... �

(*p* + 1)!

where *p* is greater than or equal to zero. It is advantageous to write *f <sup>p</sup>*(*α*) in this form because it is finite for all *α* > 0. To obtain a expression for *Jc*(*H*) from Eq. (33), we express *fp*(*α*) and

� ×

A Description of the Transport Critical Current Behavior

of Polycrystalline Superconductors Under the Applied Magnetic Field

�

(*<sup>p</sup>* <sup>−</sup> <sup>2</sup>)!3!*π*2*αp*−<sup>2</sup> <sup>+</sup>

*e*−*k<sup>α</sup>* (33)

377

, (34)

, (35)

. (36)

(*p*−2)!5!*π*4*mp*−4, and so on. It is worth

where *H*∗ <sup>0</sup> = *φ*0(�*mη*� Λ0) = *φ*0/*u*¯Λ<sup>0</sup> = *φ*0/*A* is the effective magnetic field characteristic for each polycrystalline superconductor, and *A* is the area perpendicular to the magnetic field direction. It is important to emphasize that the first term of Eq. (28) was determined by González *et al.* [21] for high magnetic fields (*α* � 1).

The estimate errors for the series in Eq. (28) were calculated as [37]:

$$E\_N = \frac{5\pi^2}{12} \left[ \frac{4}{3} \binom{2N-1}{m-2} + \left( 1 + \frac{(-1)^m}{3^{m-1}} \right) \right] (1/2)^{2N} \tag{29}$$

where *EN* is defined as the *N*-order error of the series in Eq. (28).
