Andreev billiards

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 tau(dwell) between Andreev reflections) if tau(dwell) is sufficiently large. There is a competing quantum time scale, the Ehrenfest time tau(E), which depends logarithmically on h. Two phenomenological theories provide a consistent description of the tau(E)-dependence of the gap, given qualitatively by E-gap similar or equal to min(h/tau(dwell), h/tau(E)). The analytical predictions have been tested by computer simulations but not yet experimentally.