Rise and fall of shape resonances in thin films of BCS superconductors

TL;DR

This paper investigates how the critical temperature of BCS superconductors in thin films varies with thickness, correcting previous misconceptions about discontinuities and revealing complex behavior influenced by confinement potential.

Contribution

The study provides an exact solution to the BCS gap equation showing that $T_c$ varies smoothly with film thickness and clarifies the effects of confinement potential on superconductivity.

Findings

$T_c$ is a continuous function of film thickness.

Critical confinement potential determines whether $T_c$ increases or decreases with thickness.

Experimental data for lead films can be explained by a thickness-dependent effective mass.

Abstract

The confinement of a superconductor in a thin film changes its Fermi-level density of states and is expected to change its critical temperature $T_c$. Previous calculations have reported large discontinuities of $T_c$ when the chemical potential coincides with a subband edge. By solving the BCS gap equation exactly, we show that such discontinuities are artifacts and that $T_c$ is a continuous function of the film thickness. We also find that $T_c$ is reduced in thin films compared with the bulk if the confinement potential is lower than a critical value, while for stronger confinement $T_c$ increases with decreasing film thickness, reaches a maximum, and eventually drops to zero. Our numerical results are supported by several exact solutions. We finally interpret experimental data for ultrathin lead thin films in terms of a thickness-dependent effective mass.