Semiclassics of Andreev billiards

W. Ihra, M. Leadbeater, J. L. Vega and K. Richter

The physics of normal metal/superconducting hybrid structures has grown into an important research topic within mesoscopic physics during the last half decade. Recent technological advances in building very clean conductors of mesoscopic size have led us to study the dynamics of ballistic mesoscopic conductors coupled to a superconducting lead.

What are Andreev billiards?

In the framework of the non-interacting electron gas ballistic mesoscopic systems are modelled by billiard systems. The normal conducting billiard is labelled with an "N" in Fig. 1. In a classical picture electrons behave like billiard balls: each time an electron hits the boundaries of the billiard with the vacuum it is mirror-reflected (see also Fig. 2(a)). This is indicated by an electron orbit in the billiard. On the left side the billiard is brought into contact with a superconducting lead labelled with an "S".

Fig. 1

The term "Andreev billiard" has been coined in honor of the Russian physicist A. F. Andreev who pioneered the quasi-classical description of superconductors in the 60ies. The coupling to the superconductor changes the dynamics of electron orbits in an Andreev billiard in a peculiar way: Each time an electron hits the superconducting lead it is retro-reflected (Andreev reflected) into a hole-like particle (a missing electron in the Fermi sea) and vice versa (Fig. 2(b)).

Fig. 2

The physics responsible for electron-hole conversion at the SN-boundary is the pairing interaction in the superconductor (S). Although the pairing interaction vanishes in the billiard (N) itself the influence of the superconductor is still visible there - a typical example of quantum coherence. The particles in the billiard are a coherent superposition of electron- or hole-like quasiparticles depending on which component in the superposition prevails.

Is there quantum chaos in Andreev billiards?

Fig. 3: Average density of states as a function of the energy. The circles are quantum mechanical calculations for a square billiard of side length a=75 and channel width w=25. The solid line is a 20-point average over the set of data points. Dashed line: Semiclassical DOS using the exact length distribution of trajectories. Dotted and dot-dashed line line: Semiclassical DOS using different analytical length distribution functions. Long dashed line: Low energy limit.

Fig. 4: