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

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 $Z\gg 1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P\left(\Delta \right)$ of local superconducting order parameter $\Delta \left(\mathbf{r}\right)$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\lambda \ll 1$. 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.

20 pages with 7 Figures of the Main Text and extra 43 pages with 7 Appendices and 7 more Figures of Supplementary Materials

References in corpus (5)

• Localization of preformed Cooper-pairs in disordered superconductors
• The Higgs Mode in Disordered Superconductors Close to a Quantum Phase Transition
• Quantum breakdown of superconductivity in low-dimensional materials
• Superconductivity at the three-dimensional Anderson metal-insulator transition
• Microwave properties of superconductors close to SIT