A pedagogical extension of the one-dimensional Schrödinger’s equation to symmetric proximity effect system film sandwiches

B J Luke,P R Broussard

doi:10.1063/5.0074837

Abstract

This study sought to use Schrödigner’s equation to model superconducting proximity effect systems of symmetric forms. As Werthamer noted [Phys. Rev. 132(6), 2440–2445 (1963)], one to one analogies between the standard superconducting proximity effect equation and the one-dimensional, time-independent Schrödinger’s equation can be made, thus allowing one to model the behavior of proximity effect systems of metallic film sandwiches by solving Schrödinger’s equation. In this project, film systems were modeled by infinite square wells with simple potentials. Schrödinger’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 carried out 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 on the type of the film end layer.

Highlights

INTRODUCTIONAs discussed in Ref. 1, the superconducting proximity effect is normally modeled by the de Gennes–Werthamer equation, given by χ(−ξ2
As discussed in Ref. 1, the superconducting proximity effect is normally modeled by the de Gennes–Werthamer equation, given by χ(−ξ2 d2 dx2 )Δ(x) + ln( θD Tc(x) = θD Tc )Δ(x), (1)where χ(z) = ψ(1/2 + z/2) − ψ(1/2) [where ψ(x) is the digamma function], Δ(x) is the self-energy function, ξ depends on the material properties of the superconductor or the metal at point x, θD is the Debye temperature, Tc(x) is the superconducting transition temperature for the specific material at point x in isolation, and Tc is the superconducting transition temperature of the composite system
The main purpose of this paper is to explore how boundary conditions have an effect on proximity effect system film sandwiches of the forms S(NS)M and N(SN)M

Summary

INTRODUCTION

As discussed in Ref. 1, the superconducting proximity effect is normally modeled by the de Gennes–Werthamer equation, given by χ(−ξ2. In Ref. 1, analogies were used (which were first made by Werthamer3) between Eq (1) and the one dimensional, time-independent Schrödinger’s equation, given by h2 d2ψ(x). The point is that if we understand the nature of solutions to Schrödinger’s equation, we can understand the behavior of proximity effect systems. The fact that we are using the one-dimensional Schrödinger’s equation to solve proximity systems makes the study of these systems accessible to students who have completed introductory quantum mechanics courses. Even students with a basic knowledge of quantum theory should be able to make use of this method as the solutions to Schrödinger’s equation for infinite square wells (ISWs). (which we will use here) can be fairly obtained with only a basic knowledge of quantum theory

DETAILS OF THE PROBLEM

SOLUTIONS TO SCHRÖDINGER’S EQUATION

Dirichlet condition

Neumann condition

ANALYSIS

CONCLUSION