Andreev billiards

TL;DR

This review discusses how Andreev reflection affects the excitation spectrum of chaotic quantum dots, highlighting differences from non-superconducting systems and the impact of classical and quantum time scales on the energy gap.

Contribution

It provides a comprehensive overview of recent theoretical advances in understanding Andreev billiards, emphasizing the role of classical chaos and quantum effects on the excitation gap.

Findings

Chaotic billiards exhibit an energy gap around the Fermi level.

The excitation gap depends on classical dwell time and Ehrenfest time.

Theoretical predictions align with computer simulations, awaiting experimental validation.

Abstract

This is a review of recent advances in our understanding of how Andreev reflection at a superconductor modifies the excitation spectrum of a quantum dot. The emphasis is on two-dimensional impurity-free structures in which the classical dynamics is chaotic. Such Andreev billiards differ in a fundamental way from their non-superconducting counterparts. Most notably, the difference between chaotic and integrable classical dynamics shows up already in the level density, instead of only in the level--level correlations. A chaotic billiard has a gap in the spectrum around the Fermi energy, while integrable billiards have a linearly vanishing density of states. The excitation gap E_gap corresponds to a time scale h/E_gap which is classical (h-independent, equal to the mean time t_dwell between Andreev reflections) if t_dwell is sufficiently large. There is a competing quantum time scale, the Ehrenfest time t_E, which depends logarithmically on h. Two phenomenological theories provide a consistent description of the t_E-dependence of the gap, given qualitatively by E_gap min(h/t_dwell,h/t_E). The analytical predictions have been tested by computer simulations but not yet experimentally.