Calculation of the effect of random superfluid density on the temperature dependence of the penetration depth

Microscopic variations in composition or structure can lead to nanoscale inhomogeneity in superconducting properties such as the magnetic penetration depth, but measurements of these properties are usually made on longer length scales. We solve a generalized London equation with a non-uniform penetration depth $\lambda \left(\mathit{r}\right)$, obtaining an approximate solution for the disorder-averaged Meissner screening. We find that the effective penetration depth is different from the average penetration depth and is sensitive to the details of the disorder. These results indicate the need for caution when interpreting measurements of the penetration depth and its temperature dependence in systems which may be inhomogeneous.

Figure 1

Sample realizations of a random penetration depth reveal the influence of the correlation length $l$ and variance $R\left(0\right)$. The variance $R\left(0\right)=\langle {\lambda }^2\rangle /{\langle \lambda \rangle }^2-1$ controls the width the penetration depth distribution, and the correlation length establishes the characteristic length over which $\lambda \left(\mathit{r}\right)$ changes.

Figure 2

The effective penetration depth is a strong function of the parameters that characterize the distribution of local penetration depths. Here we show the value of ${\lambda }_{\mathrm{eff}}$ as the correlation length $l$ and variance $R\left(0\right)$ run across three orders of magnitude. This figure considers the case of squared exponential correlations in the penetration depth; a different case is shown in Fig. 4. The most important features of this color plot are the range of ${\lambda }_{\mathrm{eff}}/\langle \lambda \rangle$ and the appearance of values both above and below 1. The calculation is valid when $R\left(0\right)\ll 1$, but we show the region with $R\left(0\right)>0.1$ to emphasize the trends seen. Any temperature dependence in $l$ or $R\left(0\right)$ will contribute to $\lambda \left(T\right)$. This temperature dependence is not accounted for by the superconducting gap.

Figure 3

The screening can either be enhanced (${\lambda }_{\mathrm{eff}}<\langle \lambda \rangle$) or suppressed (${\lambda }_{\mathrm{eff}}>\langle \lambda \rangle$), depending on the correlation length. The curves for $l=0.1\langle \lambda \rangle$ and $l=0.01\langle \lambda \rangle$ overlap.

Figure 4

The effective penetration depth for Matérn correlations when $\nu =2$ (shown here) has strong similarities to Fig. 2, which represents the limiting case where $\nu \to \infty$. These similarities imply that the smoothness of the random penetration depth does not strongly affect ${\lambda }_{\mathrm{eff}}$.