Distribution of the order parameter in strongly disordered superconductors: An analytic theory

TL;DR

This paper presents an analytic theory describing how superconducting order parameters distribute in strongly disordered superconductors near the transition, revealing non-Gaussian behaviors influenced by local pairing and neighbor interactions.

Contribution

The authors develop a novel integral equation approach to characterize the distribution of local superconducting order parameters in disordered materials, including non-Gaussian regimes.

Findings

Distribution $P(\Delta)$ depends on effective neighbor number $Z_{eff}$.

Identifies a broad parameter range with non-Gaussian, non-critical distributions.

Analytic results agree well with numerical simulations.

Abstract

We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson-localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized pre-formed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number $Z\gg1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P\left(\Delta\right)$ of local superconducting order parameter $\Delta\left(\boldsymbol{r}\right)$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\lambda\ll1$. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors $Z_{\text{eff}}=2\nu_{0}\Delta_{0}Z$. The solution we provide is valid both at large and small $Z_{\text{eff}}$; the latter case is nontrivial as the function $P\left(\Delta\right)$ is heavily non-Gaussian. The discovery of a broad parameter range where the distribution function $P\left(\Delta\right)$ is non-Gaussian but also non-critical (in the sense of SIT criticality) is one of our key findings. The analytic results are supplemented by numerical data, and good agreement between them is observed.