Analytical solution of coupled self--consistency and linearised Usadel equations for the dirty superconductors at $T_c$ and with the proximity effect

Abstract

In this manuscript we consider a superconducting film in the vicinity of the critical temperature and presence of the proximity effect. We analytically solve the corresponding linearised Usadel equation and the self-consistency equation, defining the critical temperature. 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 linearised 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.