Proximity-Induced Subgaps in Andreev Billiards

Abstract

We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cutoff in the path length distribution P(s) will possess an energy gap on the scale of the Thouless energy. An exact quantum mechanical calculation for different Andreev billiards gives good agreement with the semiclassical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semiclassical Bohr-Sommerfeld approximation of the density of states, we derive a simple formula for the energy gap. We show that the energy gap, in units of Thouless energy, may exceed the value predicted earlier from random matrix theory for chaotic billiards.

Bound states in Andreev billiards with soft walls

“…To construct the wave functions satisfying the Bogoliubovde Gennes equation in the two regions we follow the methods of Beenakker 15,37 and Cserti et al. 21 The wave function in the normal region can be expressed in terms of the scattering matrix S(ε) of the open system in which the superconductor is replaced by a normal lead. This scattering matrix is calculated by the modular recursive Green's function method developed by Rotter et al 31,32 Note that this method to study Andreev billiards is, within the model assumption outlined above, exact.…”

“…16,17,18,19,20,21,22,23,24 Its behavior close to the Fermi energy depends on the classical dynamics found in the isolated normal conducting cavity: In the integrable case, the DOS is proportional to the energy, while for a classically chaotic normal cavity, a minigap (which is smaller than the bulk superconductor gap ∆ 0 ) develops. Moreover, depending on the geometry of the normal cavity one can observe singularities in the DOS.…”

“…25 Recently, similar singular features of the DOS have been found in other Andreev billiards. 20,21,22,24,26 To calculate the DOS for Andreev billiards many researchers have successfully applied the Bohr-Sommerfeld (BS) approximation. 15,16,17,18,19,20,21,22,24,26,27,28 It was shown that the DOS can be related to the purely geometry-dependent path length distribution P (l) which is the classical probability that an electron entering the N region at the N-S interface returns to the interface after a path length l.…”

“…Over the last decade, the Bohr-Sommerfeld approximation for the smoothed density of states has been successfully applied to Andreev billiards. 15,16,17,18,19,20,21,22,24,26,27,28 In the case of a normal dot confined by one N-S interface and infinitely high potential walls, the integrated density of states or smoothed state counting function in the BS approximation reads…”

On the spectral gap in Andreev graphs

“…The negative spectrum describes hole-like quasiparticle excitations (while the spectrum is negative, the physical energy of such an excitation is positive, of course) and the positive spectrum describes electron-like quasiparticle excitations. It has been known for some time that Andreev reflections reduce the density of states near the Fermi energy and various universality classes have been defined [2,3,4,5,6,7,8,9,10] and related to quantum chaos. Spectral gaps have been found in irregularly shaped Andreev billiards for which the classical dynamics of the normal billiard (with specular reflection at the superconducting interface) is chaotic.…”