Non-retracing orbits in Andreev billiards

TL;DR

This paper examines the limitations of the retracing approximation in semiclassical quantization of Andreev billiards by comparing exact quantum calculations with semiclassical predictions, revealing mechanisms for non-retracing orbits.

Contribution

It identifies three mechanisms causing non-retracing electron-hole orbits in Andreev billiards, challenging the assumption of perfect retracing in semiclassical models.

Findings

Deviations from retracing approximation are observed.

Differences between electron and hole wave functions are identified.

Three mechanisms for non-retracing orbits are characterized.

Abstract

The validity of the retracing approximation in the semiclassical quantization of Andreev billiards is investigated. The exact energy spectrum and the eigenstates of normal-conducting, ballistic quantum dots in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes equation and compared with the semiclassical Bohr-Sommerfeld quantization for periodic orbits which result from Andreev reflections. We find deviations that are due to the assumption of exact retracing electron-hole orbits rather than the semiclassical approximation, as a concurrently performed Einstein-Brillouin-Keller quantization demonstrates. We identify three different mechanisms producing non-retracing orbits which are directly identified through differences between electron and hole wave functions.