Semiclassical Theory of Integrable and Rough Andreev Billiards

TL;DR

This paper develops a semiclassical theory to analyze the density of states in mesoscopic Andreev billiards, explaining the effects of boundary roughness and magnetic fields, and aligning well with quantum calculations.

Contribution

It introduces a semiclassical S-matrix formalism for Andreev billiards, providing new insights into the proximity effect and magnetic field influence, bridging classical dynamics and quantum results.

Findings

Semiclassical theory matches quantum calculations.

Magnetic fields suppress the proximity effect via time-reversed trajectories.

Boundary roughness affects the density of states and spectrum saturation.

Abstract

We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding insight into the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magnetic fields on the proximity effect in rough Andreev billiards is discussed and an analytical formula is derived. The semiclassical theory provides an interpretation for the suppression of the proximity effect in the presence of magnetic fields as a coherence effect of time reversed trajectories, similar to the weak localisation correction of the magneto-resistance in chaotic mesoscopic systems. The semiclassical theory is shown to be in good agreement with quantum mechanical calculations.