Superconductor-proximity effect in chaotic and integrable billiards

TL;DR

This paper investigates how the proximity to a superconductor influences the energy level density in billiards with chaotic or integrable classical dynamics, revealing distinct behaviors such as an excitation gap in chaotic systems and a reduced density near the Fermi level in integrable ones.

Contribution

It provides analytical and numerical analysis of the superconductor proximity effect in billiards, highlighting differences between chaotic and integrable geometries and how magnetic field or phase difference affect the excitation gap.

Findings

Chaotic billiards exhibit an excitation gap equal to the Thouless energy.

Integrable billiards show a reduced density of states near the Fermi level without a gap.

The excitation gap in chaotic billiards closes as magnetic field or phase difference varies.

Abstract

We explore the effects of the proximity to a superconductor on the level density of a billiard for the two extreme cases that the classical motion in the billiard is chaotic or integrable. In zero magnetic field and for a uniform phase in the superconductor, a chaotic billiard has an excitation gap equal to the Thouless energy. In contrast, an integrable (rectangular or circular) billiard has a reduced density of states near the Fermi level, but no gap. We present numerical calculations for both cases in support of our analytical results. For the chaotic case, we calculate how the gap closes as a function of magnetic field or phase difference.