A Pedagogical Extension of the One Dimensional Schr\"odinger's Equation to Symmetric Proximity Effect System Film Sandwiches

TL;DR

This paper extends the use of Schr"odinger's equation to model symmetric superconducting proximity effect systems in film sandwiches, analyzing boundary condition impacts and eigenvalue convergence.

Contribution

It introduces a pedagogical extension of Schr"odinger's equation for modeling symmetric proximity effect systems and compares boundary condition effects on eigenvalues.

Findings

Dirichlet boundary conditions lead to converging eigenvalues with increasing layers.

Neumann boundary conditions produce eigenvalues approaching two different limits.

Eigenvalue behavior depends on the type of film end layer.

Abstract

This study sought to use Schr\"odigner's equation to model superconducting proximity effect systems of symmetric forms. As N. R. Werthamer noted, [Phys. Rev. \textbf{132} (6), 2441 (1963)] one to one analogies between the standard superconducting proximity effect equation and the one-dimensional, time-independent Schr\"odinger's equation can be made, thus allowing one to model the behavior of proximity effect systems of metallic film sandwiches by solving Schr\"odinger's equation. In this project, film systems were modeled by infinite square wells with simple potentials. Schr\"odinger's equation was solved for sandwiches of the form $S(NS)_M$ and $N(SN)_M$, where $S$ and $N$ represent superconducting and nonsuperconducting metal films, respectively, and $M$ is the number of repeated bilayers, or the period. A comparison of Neumann and Dirichlet boundary conditions was done in order to explore their effects. The Dirichlet type produced eigenvalues for $S(NS)_M$ and $N(SN)_M$ sandwiches that converged for increasing $M$, but the Neumann type produced eigenvalues for the same structures that approached two different limits as $M$ increased. This last behavior is unexpected, as it implies a dependence upon the type of film end layer.