**4.2. Expression for** *Jc*(*H*) **for** *α* ≥ *π*/2

The hyperbolic cotangent in Eq. (21) can also be written as:

$$\coth(z/2) = 1 + \left(\frac{2e^{-z}}{1 - e^{-z}}\right) = 1 + 2\sum\_{k=1}^{\infty} e^{-kz}$$

$$(\text{Re } z > 0),\tag{30}$$

since the series in (30) is a Dirichlet type and is convergent at all semi-planes Re*z* > 0. Dividing Eq. (30) by (*z*<sup>2</sup> + *π*2), we obtain:

$$\frac{\coth(z/2)}{z^2 + \pi^2} = \frac{1}{z^2 + \pi^2} + \frac{2}{z^2 + \pi^2} \sum\_{k=1}^{\infty} e^{-kz}$$

$$(\text{Re } z > 0). \tag{31}$$

The expression for the critical current density for *α* ≥ *π*/2 is obtained by differentiation of Eq. (31) in Eq. (21) (*m* − 2) times. Thus,

$$f\_{\varepsilon}(z) = \frac{J\_{\varepsilon 0} z^{m} (-1)^{m}}{(m-1)!} \left[ D^{m-2} \left( \frac{1}{z^{2} + \pi^{2}} \right) + D^{m-2} \left( \frac{2}{z^{2} + \pi^{2}} \sum\_{k=1}^{\infty} e^{-kz} \right) \right],$$

$$f\_{\varepsilon}(z) = \frac{J\_{\varepsilon 0} z^{m} (-1)^{m}}{(m-1)!} \left[ f\_{m-2}(z) + \mathbb{R}(z) \right],\tag{32}$$

#### 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> 377 A Description of the Transport Critical Current Behavior of Polycrystalline Superconductors Under the Applied Magnetic Field

where

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

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

Finally, we can express *Jc* as a function of *H* by substituting the definition of *α*, *α* = *u*0/*η* =

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

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

*m* − 2

<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

> +

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

since the series in (30) is a Dirichlet type and is convergent at all semi-planes Re*z* > 0. Dividing

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

 1 *z*<sup>2</sup> + *π*<sup>2</sup>

*fm*−2(*z*) + *R*(*z*)

The expression for the critical current density for *α* ≥ *π*/2 is obtained by differentiation of Eq.

+ *Dm*−<sup>2</sup>

= 1 + 2

2 *z*<sup>2</sup> + *π*<sup>2</sup>

1.02

 *βn H*<sup>∗</sup> 0 *H*

<sup>1</sup> <sup>+</sup> (−1)*<sup>m</sup>* 3*m*−<sup>1</sup>

∞ ∑ *k*=1

∞ ∑ *k*=1 *e* −*kz*

> 2 *z*<sup>2</sup> + *π*<sup>2</sup>

*e*−*kz*

 *H*<sup>∗</sup> 0 *H* 1 +

2*n*

, (28)

(1/2)2*N*, (29)

(Re *z* > 0), (30)

(Re *z* > 0). (31)

∞ ∑ *k*=1 *e* −*kz* ,

, (32)

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

<sup>2</sup>*n*(*m*)2*<sup>n</sup>*

the range 0 < *α* < *π*/2.

where *H*∗

*φ*0/(*η*Λ0*H*) in Eq. (27). Thus [37],

(−1)*<sup>m</sup>* <sup>∞</sup> ∑*n*0 (1.02)

González *et al.* [21] for high magnetic fields (*α* � 1).

*EN* <sup>=</sup> <sup>5</sup>*π*<sup>2</sup> 12 4 3

**4.2. Expression for** *Jc*(*H*) **for** *α* ≥ *π*/2

Eq. (30) by (*z*<sup>2</sup> + *π*2), we obtain:

(31) in Eq. (21) (*m* − 2) times. Thus,

*Jc* (*z*) = *Jc*0*zm*(−1)*<sup>m</sup>*

*Jc* (*z*) = *Jc*0*zm*(−1)*<sup>m</sup>*

(*m* − 1)!

(*m* − 1)!

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

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

coth(*z*/2) = 1 +

coth(*z*/2)

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

 *Dm*−<sup>2</sup>

The hyperbolic cotangent in Eq. (21) can also be written as:

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

$$f\_{m-2}(z) = D^{m-2} \left( 1/(z^2 + \pi^2) \right)$$

and

$$R(z) = D^{m-2}[2/(z^2 + \pi^2 \sum\_{k=1}^{\infty} e^{-kz})].$$

$$R(a) = 2\sum\_{k=1}^{\infty} \left[ \sum\_{p=0}^{m-2} (-1)^{m-p} \binom{m-2}{p} \times \ \right. $$

$$f\_p(a)k^{m-p-2} \left[ e^{-k\alpha} \right] \tag{33}$$

and *fp*(*α*) can be written as

$$f\_p(p) = \frac{(-1)^p p!}{(a^2 + \pi^2)^{p+1}} \left[ \frac{(p+1)!}{p!} a^p - \frac{(p+1)!}{(p-2)!3!} \pi^2 a^{p-2} + \dots \right]$$

$$\frac{(p+1)!}{(p-4)!5!} \pi^4 a^{p-4} - \dots \right],\tag{34}$$

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 *R*(*α*) as a function of *H*. Thus,

$$f\_p(H) = \frac{(-1)^p p!}{m^{2p+2} \left[ \left( \frac{H\_0^\*}{H} \right)^2 + \frac{\pi^2}{m^2} \right]^{p+1}} \left[ a\_{p0} \left( \frac{H\_0^\*}{H} \right)^p - \right]$$

$$a\_{p1} \left( \frac{H\_0^\*}{H} \right)^{p-2} + a\_{p2} \left( \frac{H\_0^\*}{H} \right)^{p-4} - \dots \right],\tag{35}$$

where *ap*<sup>0</sup> <sup>=</sup> (*p*+1)! *<sup>p</sup>*! *<sup>m</sup>p*, *ap*<sup>1</sup> <sup>=</sup> (*p*+1)! (*p*−2)!3!*π*2*mp*−2, *ap*<sup>2</sup> <sup>=</sup> (*p*+1)! (*p*−2)!5!*π*4*mp*−4, and so on. It is worth commenting again that *fp*(*H*) is finite for all values of *H* and is defined for *p* > 0. *R*(*H*) is written as

$$R(H) = 2\sum\_{k=1}^{\infty} \left[ \sum\_{p=0}^{m-2} (-1)^{m-p} \binom{m-2}{p} \times \\\\ \times \\\\ f\_p(H) k^{m-p-2} \bigg| e^{-km\left(\frac{k\_0^{\alpha}}{H}\right)} \right. \tag{36}$$

For *z* = *α*

#### 14 Will-be-set-by-IN-TECH 378 Superconductors – Materials, Properties and Applications

Now Eq. (16) can be expressed as [37]:

$$f\_c(H) = \frac{f\_{c0}(\_1)^m}{(m-1)!} \left(\frac{H\_0^\*}{H}\right)^m \left[f\_{m-2}H + R(H)\right].\tag{37}$$

The series error estimated in *R*(*α*) [Eq. (34)] is [37]:

$$E\_{K+1}(a) \le \frac{e(m-2)!}{\pi} \left[ \sum\_{p=0}^{m-2} \frac{1}{(a^2 + \pi^2)^{(m-p-2)/2}} \times \right]$$

$$\sum\_{l=0}^{p} \frac{K^{p-l}}{(p-l)! \, a^{l+1}} \Big[ e^{-ka} \Big] \tag{38}$$

where *EK*+<sup>1</sup> is defined as the *K* + 1-order error of the series in Eq. (34).

A low applied magnetic field implies that *α* � 1, and Eq. (37) is transformed to:

$$J\_c(H) \approx J\_c(0) \left( 1 - \frac{\pi^2(m+1)}{6m} \frac{H}{H\_0^\*} \right) \tag{39}$$

**6. References**

theoretical fits.

**Author details**

[1] Chen, M., Donzel, L., Lakner, M., Paul, W. (2004). High temperature superconductors for

C. A. C. Passos, M. S. Bolzan, M. T. D. Orlando, H. Belich Jr, J. L. Passamai Jr. and J. A. Ferreira

**Figure 6.** Critical current density as a function of the magnetic field for the *Hg*0.80*Re*0.20*Ba*2*Ca*2*Cu*3*O*<sup>8</sup> sample (*Tc* = 132 K). The measurement was carried out at 125 K. The solid line and dot line are the

A Description of the Transport Critical Current Behavior of Polycrystalline Superconductors Under the Applied Magnetic Field 15

A Description of the Transport Critical Current Behavior

of Polycrystalline Superconductors Under the Applied Magnetic Field

379

[2] Gabovich, A. M. and Moiseev, D. P. (1986). Metal oxide superconductor BaPb1-xBixO3: unusual properties and new applications. *Sov. Phys. Usp.*, Vol. 29 (No. 12), 1135–1150. [3] Babcock S. E. & Vargas J. L. (1995). The nature of grain-boundaries in the high-Tc

[4] Polat, O., Sinclair, J. W., Zuev, Y. L., Thompson, J. R., Christen, D. K., Cook, S. W., Kumar, D., Yimin Chen, and Selvamanickam, V., (2011). Thickness dependence of magnetic relaxation and E-J characteristics in superconducting (Gd-Y)-Ba-Cu-O films with strong

[5] Rosenblatt, J., Peyral, P., Raboutou, A., and Lebeau C. (1988). Coherence in 3D networks:

[6] Raboutou, A., Rosenblatt, and J., Peyral, P., (1980). Coherence and disorder in arrays of

[7] Matsushita, T., Otabe, E.S., Fukunaga, T., Kuga, K., Yamafuji, K., Kimura, K., Hashimoto, M., (1993). Weak link property in superconducting Y-Ba-Cu-O prepared by QMG process,

superconductors, *Annual Review of Materials Science*, Vol. 25 (No. 1) 193–222.

Application to high-Tc supeconductors. *Physica B*, Vol. 152 (No. 1-2), 95–99.

power applications, *J. Eur. Ceram. Soc.*, Vol. 24 (No. 6) 1815–1822.

E. V. L. de Mello, *Physics Department, University Federal Fluminense - Brazil* 

*Physics Department, University Federal of Espirito Santo - Brazil* 

vortex pinning. *Phys. Rev. B*, Vol. 84 (No. 2), 024519-1 – 024519-13.

point contacts. *Phys. Rev. Lett.*, Vol. 45 (No. 12), 1035–1039.

*IEEE Trans. Appl. Supercond.*, Vol. 3 (No. 1) 1045–1048.

where *H*∗ *<sup>o</sup>* <sup>=</sup> *<sup>φ</sup>*<sup>0</sup> �*ηm*� <sup>2</sup> = *<sup>φ</sup>*<sup>0</sup> *<sup>u</sup>*¯2 is a characteristic field that determines the behavior of *Jc* (*H*) in this region. In addition, *u*¯ represents the mean of the width distribution function *P*(*u*) involved in the transport of Cooper pairs through the sample. Eq. (39) reproduces the quasi-linear behavior that was also reported by Gonzalez *et al.* [21].

### **5. Critical current measurement**

Typical *Jc* measurements are performed using the four-probe technique with automatic control of the sample temperature, the applied magnetic field and the bias current [14]. Details of the technique and the experimental setup are in Ref. [14] and of the synthesis and sample characterization were published elsewhere [13].

Figure (6) shows the experimental results of [38] for the critical current as a function of the applied field, together with the theoretical expression derived above for the critical field *H*∗ <sup>0</sup> in the figure and for *m* = 2 and 3. A very close fit to the experimental data is evident for *m* = 2 for many different applied magnetic fields.

Theoretical models of the magnetic field dependence of the transport critical current density for a polycrystalline ceramic superconductor have been studied at last years [37, 39–41]. Here we have described a tunneling critical current between grains follows a Fraunhofer diffraction pattern or a modified pattern . It is important to emphasize that we followed the same approach as in [21] and extended the analytical results to all applied magnetic fields. A characteristic field (*H*∗) was identified and different regimes were considered, leading to analytical expressions for *Jc*(*H*): (i) analysis for low applied magnetic fields (*α* � 1) revealed quasi-linear behavior for *Jc* (*H*) vs. *H*∗; (ii) for high applied magnetic fields (*α* � 1), *Jc* (*H*) is proportional to *H*−0.5, as reported in [21].

A Description of the Transport Critical Current Behavior of Polycrystalline Superconductors Under the Applied Magnetic Field 15 379 A Description of the Transport Critical Current Behavior of Polycrystalline Superconductors Under the Applied Magnetic Field

**Figure 6.** Critical current density as a function of the magnetic field for the *Hg*0.80*Re*0.20*Ba*2*Ca*2*Cu*3*O*<sup>8</sup> sample (*Tc* = 132 K). The measurement was carried out at 125 K. The solid line and dot line are the theoretical fits.
