Density of states of chaotic Andreev billiards

Abstract

Quantum cavities or dots have markedly different properties depending on whether their classical counterparts are chaotic or not. Connecting a superconductor to such a cavity leads to notable proximity effects, particularly the appearance, predicted by random matrix theory, of a hard gap in the excitation spectrum of quantum chaotic systems. Andreev billiards are interesting examples of such structures built with superconductors connected to a ballistic normal metal billiard since each time an electron hits the superconducting part it is retroreflected as a hole (and vice versa). Using a semiclassical framework for systems with chaotic dynamics, we show how this reflection, along with the interference due to subtle correlations between the classical paths of electrons and holes inside the system, is ultimately responsible for the gap formation. The treatment can be extended to include the effects of a symmetry-breaking magnetic field in the normal part of the billiard or an Andreev billiard connected to two phase-shifted superconductors. Therefore, we are able to see how these effects can remold and eventually suppress the gap. Furthermore, the semiclassical framework is able to cover the effect of a finite Ehrenfest time, which also causes the gap to shrink. However, for intermediate values this leads to the appearance of a second hard gap-a clear signature of the Ehrenfest time.

Conductance and thermopower of ballistic Andreev cavities

“…This allows us to approximate exp[−i arccos( / )] ≈ −i such that the scattering matrix of Andreev reflection becomes independent of the energy. 25 Thus, the diagrammatic rules for the ζ side trees read 13,24 (i) an e-h path pair contributes [N (…”

“…Andreev reflection and interference between quasiparticles with slightly different paths lead to a hard gap in the density of states of such chaotic ballistic conductors coupled to a superconductor. [11][12][13] In Ref. 10 such a chaotic Andreev quantum dot is coupled to two normal conducting and one superconducting lead and its transport characteristics was studied.…”

“Smile” Gap in the Density of States of a Cavity between Superconductors

“…The Thouless energy is given by E Th ¼ ðG Σ =G Q Þδ S , G Q ¼ e 2 =πℏ being the conductance quantum, G Σ ≫ G Q being the total conductance of the contacts, and δ S being the level spacing in the normal metal provided the contacts are closed. The DOS in chaotic cavities has been studied for years [19][20][21][22].The DOS depends on the ratio of E Th and the superconducting energy gap Δ, and on the superconducting phase difference. If the dwell time exceeds the Ehrenfest time, qualitative features of the DOS do not seem to depend much on the contact nature and are the same for ballistic, diffusive, and tunnel contacts.…”

“…The Thouless energy is given by E Th ¼ ðG Σ =G Q Þδ S , G Q ¼ e 2 =πℏ being the conductance quantum, G Σ ≫ G Q being the total conductance of the contacts, and δ S being the level spacing in the normal metal provided the contacts are closed. The DOS in chaotic cavities has been studied for years [19][20][21][22].…”

Energy-dependent correlations in the S-matrix of chaotic systems

“…These results, restricted to leading order 1/M expansions, were later used to obtain the density of states of chaotic Andreev billiards. [39,40] The first few 1/M corrections have also been treated. [41] In a recent development, [42] restricted to broken TRS, some correlation functions were expressed as power series in ǫ with coefficients that are rational functions of M .…”