Boundary conditions for quasiclassical equations in the theory of superconductivity

TL;DR

This paper derives improved boundary conditions for quasiclassical equations in superconducting hybrid structures, accounting for finite barrier transparency, and demonstrates their impact on Josephson current behavior.

Contribution

It extends previous models by including higher-order transparency effects, providing more accurate boundary conditions for superconducting-normal hybrid systems.

Findings

Derived boundary conditions valid beyond small transparency limit

Calculated the second harmonic contribution to the Josephson current

Numerically solved integral equations for boundary condition corrections

Abstract

In this paper we derive effective boundary conditions connecting the quasiclassical Green's function through tunnel barriers in superconducting - normal hybrid (S-N or S-S') structures in the dirty limit. Our work extends previous treatments confined to the small transparency limit. This is achieved by an expansion in the small parameter $r^{-1}=T/2(1-T)$ where T is the transparency of the barrier. We calculate the next term in the $r^{-1}$ expansion for both the normal and the superconducting case. In both cases this involves the solution of an integral equation, which we obtain numerically. While in the normal case our treatment only leads to a quantitative change in the barrier resistance $R_b$, in the superconductor case, qualitative different boundary conditions are derived. To illustrate the physical consequences of the modified boundary conditions, we calculate the Josephson current and show that the next term in the $r^{-1}$ expansion gives rise to the second harmonic in the current-phase relation.