**1. Introduction**

22 Superconductors

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364 Superconductors – Materials, Properties and Applications

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Since their discovery in 1986 the high-T*c* superconductors (HTSC) have been employed in several applications.The expectation with the discover of new devices sparked the beginning of an intense research to understand the parameters which control the physical properties of these materials. With the goal to the practical applications, the critical current density (J*c*) is one of the crucial parameters that must be optimized for HTSC [1]. Thus the aim of this chapter is to describe he transport critical current behavior of polycrystalline superconductors under the applied magnetic field.

According to Gabovich and Mosieev [2], there is a dependence of the superconducting properties on the macrostructure of ceramic. They studied the BaPb1−*x*Bi*x*O3 metal oxide superconductor properties which are a consequence of the granularity of the ceramic macrostructure and the existence of weak Josephson links between the grains. In this case, the superconductivity depends strongly on the presence of grain boundaries and on the properties of the electronic states at the grain boundaries. This determines the kinetic characteristics of the material. For instance, the temperature dependence of the electrical conductivity of oxide superconductor is related to complex Josephson medium.

Nowadays it is well known that the J*c* in polycrystalline superconductors is determined by two factors: the first is related to the defects within the grains (intragrain regions) such as point defects, dislocations, stacking faults, cracks, film thickness, and others [3, 4]. When polycrystalline samples are submitted to magnetic field, the intragranular critical current can be limited by the thermally activated flux flow at high magnetic fields. Secondly the critical current depends on the grain connectivity, that is, intergrain regions. Rosenblatt *et*

©2012 Passos 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 Passos 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.

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

able to carry electrical current without resistance. This phenomenon is related to the perfect diamagnetism. The second feature of superconductivity is also known as Meissner effect. In

This struggle between superconductivity and magnetic field penetration select two important behaviors. If a superconductor does not permit any applied magnetic flux, it is known as Type I superconductor. In this case, if the superconducting state is put in the presence of a too high magnetic field, the superconductivity is destroyed when the magnetic field magnitude exceeds the critical value H*c*. Other superconductor category is the Type II material in which the magnetic properties are more complex. For this material the superconductor switches from the Meissner state to a state of partial magnetic flux penetration. The penetration of

In addition to the two limiting parameters T*c* and H*c*, the superconductivity is also broken down when the material carries an electrical current density that exceeds the critical current density J*c*. In the Ginzburg-Landau theory, the superconducting critical current density can

The current density given by Eq. (1) is sometimes called the *Ginzburg-Landau depairing current*

Once into the superconductor state, it is possible to cross the superconductor surface changing only the current. In this case, even for *T* < *Tc* and *H* < *Hc* with the material reaching its

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

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*

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

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

2/3 *Hc*

*<sup>λ</sup>* . (1)

A Description of the Transport Critical Current Behavior

of Polycrystalline Superconductors Under the Applied Magnetic Field

367

*J* = *Jc* sin(*θ*<sup>1</sup> − *θ*2) (2)

this case a superconductor expels an external applied magnetic field into its interior.

magnetic flux starts at a lower field H*c*<sup>1</sup> to reach at an upper a higher field H*c*2.

*Jc* = 2 3

normal state, and with loss of its superconductor properties.

**3. Josephson-type mechanism**

be written as

*density*.

by [26]

*Josephson effect*.

superconducting applications.

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 grains.

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 distribution controls the transport critical current density.

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*) 0.5 dependence was found. Here we have decided to follow the same approach and extend the analytical results to all applied magnetic fields.

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 [24].
