Extended analytical BCS theory of superconductivity in thin films

TL;DR

This paper develops an analytical BCS theory for superconductivity in thin films, accurately predicting how the critical temperature varies with film thickness and revealing a topological Fermi surface transition that explains non-monotonic $T_c$ behavior.

Contribution

It introduces a solvable analytical model for superconductivity in confined geometries, linking Fermi surface topology changes to $T_c$ variations without adjustable parameters.

Findings

Quantitative agreement with experimental $T_c$ data

Identification of a topological Fermi surface transition

Explanation of non-monotonic $T_c$ dependence on film thickness

Abstract

We present an analytically solvable theory of BCS-type superconductivity in good metals which are confined along one of the three spatial directions, such as thin films. Closed-form expressions for the dependence of the superconducting critical temperature $T_c$ as a function of the confinement size $L$ are obtained, in quantitative agreement with experimental data with no adjustable parameters. Upon increasing the confinement, a crossover from a spherical Fermi surface, which contains two growing hollow spheres corresponding to states forbidden by confinement, to a strongly deformed Fermi surface, is predicted. This crossover represents a new topological transition, driven by confinement, between two Fermi surfaces belonging to two different homotopy classes. This topological transition provides a mechanistic explanation of the commonly observed non-monotonic dependence of $T_c$ upon film thickness with a maximum which, according to our theory, coincides with the topological transition.