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

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 preformed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number Z >> 1 of similar neighboring pairs. We derived integral equations for the probability distribution P(Delta) of local superconducting order parameter Delta(r) and analyzed their solutions in the limit of small dimensionless Cooper coupling constant lambda << 1. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors Z(eff) = 2 nu(0)Delta(0)Z, with nu(0) being the single-particle density of states at the Fermi level. The solution we provide is valid both at large and small Z(eff); the latter case is nontrivial as the function P(Delta) is heavily non-Gaussian. One of our key findings is the discovery of a broad range of parameters where the distribution function P(Delta) is non-Gaussian but also noncritical (in the sense of superconductor-insulator transition criticality). The analytic results are supplemented by numerical data and good agreement between them is observed.

Automated summary

We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials close to the superconducting-insulator transition. Our key finding is that the distribution function of the order parameter depends crucially on the effective number of nearest neighbors, with the solution valid for both large and small values of this parameter. The analytic results are supported by numerical data, showing good agreement between them.