Microstructural and electrical transport properties of uniaxially pressed $\text {Bi}_{1.65}\text {Pb}_{0.35}\text {Sr}_2\text {Ca}_{2.5}\text {Cu}_{3.5}\text {O}_{10+\delta }$ ceramic superconductors

Abstract

We have studied the effect of the pelletization pressure on microstructural and electrical transport properties of superconducting ceramics with starting composition given by the formula $\text {Bi}_{1.65}\text {Pb}_{0.35}\text {Sr}_2\text {Ca}_{2.5}\text {Cu}_{3.5}\text {O}_{10+\delta }$. The experimental data of electrical measurements was processed in order to obtain the weak-link resistivity, the orientation probability of the grains’ a-axes along a certain preferential direction, the slope of the linear part in the temperature dependence of the ab-planes resistivity, and the intrinsic effective anisotropy of the grains, of each sample. In contrast with the behaviour of $\text {Bi}_{1.65}\text {Pb}_{0.35}\text {Sr}_2\text {Ca}_{2}\text {Cu}_{3}\text {O}_{10+\delta }$ ceramics, the Ca, Cu enriched samples exhibit a reduction of their effective anisotropy at sample level and weak links resistivity with increasing compacting pressures. In addition, a compacting pressure of around 488 MPa may affect considerably the electrical and structural parameters of the material. The results suggest that a combined effect of the pelletization pressure and the doping with Ca and Cu can be used to improve the electrical transport properties of these materials for technological applications.

Appendices

Appendix 1: A model to anisotropic granular superconductors

In this Appendix 1 we summarize the main features of the model previously published in references [14, 15]. The model is based on two main ideas. The goal here is to make the article more self-consistent. First, the resistivity of the polycrystalline sample can be written as

$$\begin{aligned} \rho =\frac{1}{\alpha _{n}}(\rho _{ab}+\rho _{wl}). \end{aligned}$$

(10)

Eq. (10) expresses that the resistivity of the sample depends on the in-plane resistivity of the crystallites plus the resistivity of the Josephson junctions or weak links. This sum is amplified by a factor $1/\alpha _{n}$ that represents the combined effects of the grains’ misalignment and the defects in the polycystalline sample. Here, $\rho _{ab}=A_{ab}T$ and $\alpha _{n}=f\alpha _{str}$ where $\alpha _{str}$ represents the effects of the defects on the resistivity of the sample [26]. When we compare (10) with the linear part of experimental dependence $\rho (T),$

$$\begin{aligned} \rho =AT+\rho (0), \end{aligned}$$

(11)

$A_{ab}/\alpha _{n}=A$ and $\rho _{wl}/\alpha _{n}=\rho (0)$ are obtained easily [26, 41, 42].

In the case of a sample with two main directions, Eqs. (10) and (11) should be written for each of them [14]. If $\mu$ does not depend on the temperature or has a weak dependence with it, then one may take $\mu =\alpha _{nx}/\alpha _{nz}$ where x and z are the two main directions of the samples. Here, z is taken along the compacting direction and x is perpendicular to it. For high $T_{c}$ superconductors, like YBCO and Bi-based samples, have a very high intragranular anisotropy parameter in the range $10^{2}-10^{4}$ [39, 43, 44]

The orientation factor of the grains, $f_{x},$ has been expressed as a function of the orientation probability along the x-axis, $\gamma _{xa},$ the shape anisotropy of the grains, $L_{a}/L_{c},$ and the intrinsic effective anisotropy of the grains, t, as already was reported elsewhere [14]. The calculations are based on EMA [13], which constitutes the second pillar of the model: here it is assumed the pollicrystalline sample made up of two different phases represented by the grains oriented with their a or b axes parallel to the x-axis of the sample and those with their c axes oriented along the x-axis of the sample, respectively. Thus, the orientation probability, $\gamma _{xa},$ can be also interpreted as the probability of finding the first phase in the polycrystalline sample.

In this paper we will use a similar description to that presented before [14], but modified in some aspects. Here, we are including a new phase in the calculations of the effective conductivity, which accounts for the pores. These will be assumed spherical with conductivity zero, probability of appearance, p, and randomly distributed inside the polycrystalline sample. Moreover, all the contributions of the defects to the resistivity of the polycrystalline sample will be included in this new phase. Finally, in the present approach, the effects of the grains’ disorientation and defects are not separated into two factors, but contained in the same fraction $\alpha _{n}$.

In Appendix 1 of the reference [15] the calculations to obtain the functions $\mu =F(\mu ,\gamma _{xa},L_a/L_c,p,t)$ and $\alpha _{nx}=\alpha _{nx}(\mu ,\gamma _{xa},L_a/L_c,p,t)$ of the polycrystalline sample based on EMA [13, 14] are presented in detail. Notice that once having $\mu$ and $\alpha _{nx}$ it is easy to find $\alpha _{nz}$ since $\alpha _{nz}=\alpha _{nx}/\mu.$

Now it is described how the experimental data is processed by means of the model.

From the Eq. (10), the tensor resistivity component of the paracoherent resistivity along the x-axis is given by the equation

$$\begin{aligned} \rho _{px}=\frac{1}{\alpha _{nx}(\mu ,\gamma _{xa},L_{a}/L_{c},p,1)}(\rho _{wl_{x}}). \end{aligned}$$

(12)

Here, $t=1,$ because the grains are in the superconducting state. On the other hand, the extrapolation of the normal-state resistivity to zero temperature is given by a similar expression, but in this case $t\ne 1$ if the grains are anisotropic:

$$\begin{aligned} \rho _{x}(0)=\frac{1}{\alpha _{nx}(\mu ,\gamma _{xa},L_{a}/L_{c},p,t)}(\rho _{wl_{x}}). \end{aligned}$$

(13)

The ratio of Eqs. (12) and (13) gives

$$\begin{aligned} f_{x}=\frac{\rho _{px}}{\rho _{x}(0)}= \frac{\alpha _{nx}(\mu ,\gamma _{xa},L_{a}/L_{c},p,t)}{\alpha _{nx}(\mu ,\gamma _{xa},L_{a}/L_{c},p,1)}, \end{aligned}$$

(14)

which was experimentally determined. If we add to the Eq. (14) the equation of the anisotropy parameter $\mu,$ which was also experimentally determined

$$\begin{aligned} \mu =\frac{\alpha _{nx}(\mu ,\gamma _{xa},L_{a}/L_{c},p,t)}{\alpha _{nz}(\mu ,\gamma _{xa},L_{a}/L_{c},p,t)}, \end{aligned}$$

(15)

a system of equations with $\gamma _{xa}$ and t as unknown quantities is obtained. After finding $\gamma _{xa}$ and t it is easy to obtain $A_{ab}=A_{x}\alpha _{nx}$ where $\alpha _{nx}$ is calculated and $A_{x}$ is experimentally determined [15].

As an example of how the model works the dependence of $\alpha _{nx}$ and $\alpha _{nz}$ as functions of $\gamma _{xa}$ are shown in reference [15]. The variation provoked by a 15 $\%$ or 28 $\%$ of pores is quite small when 0.4 $<\gamma _{xa}<$ 0.48. The significant differences appear in the region of very low and very high values of $\gamma _{xa}$. The values of the intrinsic anisotropy parameter and shape anisotropy of the pollycrystalline sample were taken similar to those that will be used in this work.

Appendix 2: Analytical procedure for Levin-Gonzalez’ method

We have applied the measurement model proposed by González et. al. [25] for measuring the different components of the resistivity tensor in $\text {Bi}_{1.65}\text {Pb}_{0.35}\text {Sr}_2\text {Ca}_{2.5}\text {Cu}_{3.5}\text {O}_{10+\delta }$ polycrystalline superconductors. For the $\rho _z$ experimental determination, the current is injected onto the top surface of the sample and extracted at the bottom. Two aligned contacts recording the voltage are located according to Fig. 10b.

Fig. 10

Measurement configurations. a Experimental configuration used for the determination of the x-axis effective resistivity, $\rho _x$. b Experimental configuration used in the determination of the z-axis effective resistivity, $\rho _z$

In the quasistatic limit, the expression which satisfies the Laplace’s differential equation with the first boundary condition is [25]:

$$\begin{aligned} V(x,z)= & {} \displaystyle V_0z+\sum _{n=1}^\infty \!\!V_n\cos \left( \frac{n\pi x}{L}\right) \times \nonumber \\&\displaystyle \times \sinh \left[ \frac{n\pi \sqrt{\rho _z}}{\sqrt{\rho _x}L}(D/2-z)\right] \end{aligned}$$

(16)

Hence, the $V_0$ and $V_n$ coefficients can be found through the Fourier methods using the last boundary condition on the transport current applied. Here, we just give the explicit coefficients, leaving the detailed development for Appendix 1 in reference [25]: $V_0=I\rho _z/bL$ and $V_n=-2I/n\pi b \cosh (n\pi \sqrt{\rho _z}D/2\sqrt{\rho _x}L)$. Establishing that the voltage signal in a measurement is $\varDelta V=V(x_0,D)-V(x_0,0)$ and taking in account Eq. (16) where it has been substituted the $V_0$ and $V_n$ dependence, it can be deduced an expression for the measured resistance as a function of the parameters of the sample $\sqrt{\rho _x},\sqrt{\rho _z},D, b$ and L:

$$\begin{aligned} \displaystyle R_z=\frac{\rho _zD}{bL}\left[ 1+\sum _{n=1}^\infty \!\!\frac{4L\cos \left( \frac{n\pi x_0}{L}\right) \tanh \left( \frac{n\pi \sqrt{\mu } D}{2L}\right) }{n\pi \sqrt{\mu }D} \right] \end{aligned}$$

(17)

where $\mu _=\rho _z/\rho _x$. An important point here is that the series for $R_z$ is a slowly (conditionally) converging one, which cannot be truncated for any value of anisotropy $\sqrt{\mu, }$ large or small. This is a common feature of the potential distributions due to arrays of charges [45]. A way to solve this problem is to separate the slowly converging series of $R_z$ into slowly and rapidly converging parts in such a way that would allow to carry out the summation of the slowly converging part exactly (analytically or numerically). For example, Eq. (17) can be rewritten as follows:

$$\begin{aligned} \displaystyle R_z=\frac{4\sqrt{\rho _z\rho _x}}{b\pi }\left[ \frac{\pi \sqrt{\mu }D}{4L}+Y_z(x_0)+S_z(\sqrt{\mu }) \right] \end{aligned}$$

(18)

where

$$\begin{aligned} \displaystyle Y_z(x_0)=\sum _{n=1}^\infty \frac{\cos (n\pi x_0/L)}{n}=\ln \left[ 2\sin \left( \frac{\pi x_0}{2L} \right) \right] \end{aligned}$$

(19)

and

$$\begin{aligned} \displaystyle S_z(\sqrt{\mu })=\sum _{n=1}^\infty \frac{\cos (\frac{n\pi x_0}{L})}{n}\left[ \tanh \left( \frac{n\pi \sqrt{\mu }D}{2L} \right) -1 \right] \end{aligned}$$

(20)

The term $Y_z(x_0)$ is reported in reference [25] and characterizes the sample’s geometry, but does not depend on the resistivity. On the other hand, $S_z(\sqrt{\mu })$ is a rapidly (exponentially) converging series, which can be truncated and therefore lends itself easily to numerical evaluation even for relatively small values of anisotropy such that $\frac{\pi \sqrt{\mu }D}{2L}\sim 1$.

The other configuration in which the contacts are located on the face perpendicular to the z-axis, as in Fig. 10a, was analyzed by reference [24] in the single harmonic approximation and for a multi-terminal contact configuration. By using analogous boundary conditions on the current like in the paragraphs above, it is easy to show that the final expression for the resistance in this configuration is [24]:

$$\begin{aligned} \displaystyle R_x=\frac{8\sqrt{\rho _z\rho _x}}{b\pi }\left[ Y_x(l)+S_x(\sqrt{\mu }) \right] \end{aligned}$$

(21)

where

$$\begin{aligned} Y_x(l)= & {} \displaystyle \sum _{n=1}^\infty \!\!(-1)^{n-1}\frac{\cos [\pi l(2n-1)/2L]}{2n-1}\nonumber \\= & {} \frac{1}{2}\ln \left[ \tan \left( \frac{\pi }{4}+\frac{\pi l}{4L} \right) \right] \end{aligned}$$

(22)

and

$$\begin{aligned} \displaystyle S_x(\sqrt{\mu })= & {} \displaystyle \sum _{n=1}^\infty \!\!(-1)^{n-1}\frac{\sin \left[ \frac{(2n-1)\pi l}{2L}\right] }{2n-1}\times \nonumber \\&\displaystyle \times \left[ \coth \left[ \frac{(2n-1)\pi \sqrt{\mu }D}{L}\right] -1 \right] \end{aligned}$$

(23)

In reference [24] was reported than even for the samples with the lowest anisotropy ($\frac{\pi \sqrt{\mu }D}{2L}\sim 1$) it was sufficient to retain only the first three terms, $n=1, 2, 3,$ to obtain convergence of the results better than 0.1% [24]. For that reason in the series given to Eqs. (18) and (21) only the first three terms will be taken into consideration. From the ratio $R_z/R_x,$ they can be obtained all values of anisotropy $\sqrt{\mu }$ after solving a transcendental equation. The result may be substituted into Eqs. (18) and (21) and both resolved later to give the true resistivities $\rho _x$ and $\rho _z,$ through the following expressions:

$$\begin{aligned} \rho _x=\frac{\displaystyle R_xb\pi /8\sqrt{\mu }}{\displaystyle \left\{ \frac{1}{2}\ln \left[ \tan \left( \frac{\pi }{4}+\frac{\pi l}{4L}\right) \right] +S_x(\sqrt{\mu })\right\} ^{-1}} \end{aligned}$$

(24)

and

$$\begin{aligned} \rho _z=\frac{\displaystyle R_z\sqrt{\mu }b\pi /4}{\displaystyle \left\{ \frac{\sqrt{\mu }\pi D}{4L}+\ln \left[ 2\sin \left( \frac{\pi x_0}{2L}\right) \right] +S_z\left( \sqrt{\mu }\right) \right\} ^{-1}} \end{aligned}$$

(25)