Conductivity of anisotropic inhomogeneous superconductors above the critical temperature

We propose a model and derive analytical expressions for conductivity in heterogeneous fully anisotropic conductors with ellipsoid superconducting inclusions. This model and calculations are useful to analyze the observed temperature dependence of conductivity anisotropy in various anisotropic superconductors, where superconductivity onset happens inhomogeneously in the form of isolated superconducting islands. The results are applied to explain the experimental data on resistivity above the transition temperature $T_c$ in the high-temperature superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ and in the organic superconductor $\beta \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$. The comparison of resistivity data and diamagnetic response in $\beta \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ allows us to estimate the size of superconducting inclusions as $d\sim 1\mu \text{m}$.

Figure 1

Temperature dependence $\varphi \left(T\right)$ of the volume fraction of superconducting inclusions extracted from formula (15) and the experimental data from Ref. [17] for ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$.

Figure 2

Comparison between the proposed theory and the experimental data from Ref. [17] on the temperature dependence of resistivity in ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ along three main axes, ${\rho }_i\left(T\right)$ for $i=1,2,3$ in panel (a), (b), (c). In (a) the high-temperature fit coincides with experimental data and is not shown.

Figure 3

Comparison of the temperature dependence of the SC volume fraction $\varphi$ in $\beta \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ calculated from the resistivity data [21] $[{\varphi }_{\rho }$, solid blue curve, Eq. (15)], and from the magnetic susceptibility data [42] using Eq. (16): with temperature-independent constant $C=2600$ (${\varphi }_{\chi }$, orange dot-dashed curve) and with temperature-dependent coefficient $C\left(T\right)$ given by Eq. (18) (${\varphi }_{\chi }^C$, green dashed curve).

Figure 4

Comparison between the proposed theory and the experimental data from Ref. [21] on the temperature dependence of resistivity in $\beta \text{−}{\left(\mathrm{BEDT}\text{−}\mathrm{TTF}\right)}_2{\mathrm{I}}_3$ along three main axes, ${\rho }_i\left(T\right)$ for $i=1,2,3$. The high-temperature fit is the resistivity extrapolated from $T>9\mathrm{K}$ down to low temperatures and given by the second-order polynomials in $T$: ${\rho }_1^m\left(T\right)=(1.429+0.084{\text{K}}^{-1}T+0.006{\text{K}}^{-2}{T}^2){\rho }_1\left(293\text{K}\right)\times {10}^{-3}$, ${\rho }_2^m\left(T\right)=(2.616+0.063{\text{K}}^{-1}T+0.009{\text{K}}^{-2}{T}^2){\rho }_2\left(293\text{K}\right)\times {10}^{-3}$, and ${\rho }_3^m\left(T\right)=(47.821+0.025{\text{K}}^{-1}T+0.255{\text{K}}^{-2}{T}^2)\mathrm{m}\mathrm{\Omega }\mathrm{cm}$.