Semiclassical theory of integrable and rough Andreev billiards

Abstract

We study the effect on the density of states in mesoscopic ballistic billiards to which a superconducting lead is attached. The expression for the density of states is derived in the semiclassical S-matrix formalism shedding insight into the origin of the differences between the semiclassical theory and the corresponding result derived from random matrix models. Applications to a square billiard geometry and billiards with boundary roughness are discussed. The saturation of the quasiparticle excitation spectrum is related to the classical dynamics of the billiard. The influence of weak magnetic fields on the proximity effect in rough Andreev billiards is discussed and an analytical formula is derived. The semiclassical theory provides an interpretation for the suppression of the proximity effect in the presence of magnetic fields as a coherence effect of time reversed trajectories, similar to the weak localisation correction of the magneto-resistance in chaotic mesoscopic systems. The semiclassical theory is shown to be in good agreement with quantum mechanical calculations.

Magnetic-field dependence of transport in normal and Andreev billiards: A classical interpretation of the averaged quantum behavior

“…The main signatures of classical integrability ͑or lack of it͒ on the statistics of energy levels and properties of the transport coefficients for closed and open systems, respectively, have been discussed in detail in various reviews. [1][2][3][4] Discussions on modifications owing to the possibility of Andreev reflection appear in more recent studies, 5,6,[9][10][11][12][13][14][15] mostly focusing on the features of the quantum mechanical level density.…”

Non-retracing orbits in Andreev billiards

“…13 This phenomenon is commonly referred to as Andreev reflection. 14 In the frequently used and remarkably successful semiclassical Bohr-Sommerfeld (BS) method 15,16,17 it is assumed that the Andreev reflection is perfect, i.e. that the path of the backscattered hole will exactly trace that of the incoming electron ( v e = − v h ) [see Fig.…”

“…The BS approximation has been found to be suprisingly accurate, 15,22 even in the presence of a magnetic field 16 or a soft wall potential. 23 However, some difficulties have emerged: for example, the excitation gap in the DOS for billiards with a chaotic N cavity cannot be properly accounted for.…”

Semiclassical theory in Andreev billiards: beyond the diagonal approximation

“…This process of Andreev reflection [2] generates a new kind of dynamics in comparison to conventional (normal) billiards. The excitation spectrum of an Andreev billiard depends on the shape of its boundary [3,4]. Using the random-matrix theory, it was shown [3] that the density of states (DoS) d(E) in the chaotic billiard has a minigap around the Fermi energy E F , while in the integrable billiard it is proportional to E, the energy counted from E F .…”