Semiclassical gaps in the density of states of chaotic Andreev billiards

Abstract

The connection of a superconductor to a chaotic ballistic quantum dot leads to interesting phenomena, most notably the appearance of a hard gap in its excitation spectrum. Here we treat such an Andreev billiard semiclassically where the density of states is expressed in terms of the classical trajectories of electrons (and holes) that leave and return to the superconductor. We show how classical orbit correlations lead to the formation of the hard gap, as predicted by random matrix theory in the limit of negligible Ehrenfest time $\tE$, and how the influence of a finite $\tE$ causes the gap to shrink. Furthermore, for intermediate $\tE$ we predict a second gap below $E=π\hbar /2\tE$ which would presumably be the clearest signature yet of $\tE$-effects.

Refereed version. 4 pages, 3 figures

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• Semiclassical Theory of Chaotic Conductors
• Andreev billiards
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• Transport moments beyond the leading order
• Combinatorial theory of the semiclassical evaluation of transport moments I: Equivalence with the random matrix approach
• Ehrenfest-time dependence of counting statistics for chaotic ballistic systems
• A semiclassical matrix model for quantum chaotic transport
• The density of states of chaotic Andreev billiards
• Statistics of time delay and scattering correlation functions in chaotic systems II. Semiclassical Approximation
• Transport moments and Andreev billiards with tunnel barriers
• Combinatorial theory of the semiclassical evaluation of transport moments II: Algorithmic approach for moment generating functions
• Conductance and Thermopower of Ballistic Andreev Cavities
• Energy-dependent correlations in the $S$-matrix of chaotic systems
• Applications of random matrix theory to condensed matter and optical physics
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• Semiclassical approach to $S$ matrix energy correlations and time delay in chaotic systems
• Secondary "Smile"-gap in the density of states of a diffusive Josephson junction for a wide range of contact types
• Combinatorial problems in the semiclassical approach to quantum chaotic transport
• Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-Heisenberg time scales
• Birman-Schwinger and the number of Andreev states in BCS superconductors