Analytical solution of coupled self-consistency and linearised Usadel equations for the dirty superconductor at proximity effect

TL;DR

This paper provides an analytical solution to the coupled linearized Usadel and self-consistency equations, determining the critical temperature and spatial profiles of a dirty superconductor under the proximity effect.

Contribution

It introduces an exact analytical approach to solve the coupled equations, reducing them to an eigenvalue problem for the critical temperature in the proximity effect scenario.

Findings

Derived the formal solution for the coupled equations

Reduced the problem to an eigenvalue problem for Tc

Determined spatial distributions of Green function and order parameter

Abstract

In this manuscript we analytically solve the linearised Usadel equation and the self-consistency equation, defining the critical temperature of the superconducting phase transition of a superconducting film under the proximity effect in the dirty limit. This is a system of coupled differential and integral equations for the anomalous Green function and the order parameter of the superconductor. The proximity effect defines the boundary conditions. The formal solution of the system is found for the general case of the linearized boundary conditions defined by the proximity effect, reducing the set of equations to an eigenvalue problem. The latter defines the critical temperature of the superconducting phase transition and the spatial distributions of the anomalous Green function and the superconducting order parameter.