SUPERCONDUCTOR-PROXIMITY EFFECT IN CHAOTIC AND INTEGRABLE BILLIARDS

We explore the effects of the proximity to a superconductor on the lev el density of a billiard for the two extreme cases that the classical motion in the billiard is chaotic or integrable. In zero magnetic fiel d and for a uniform phase in the superconductor, a chaotic billiard ha s an excitation gap equal to the Thouless energy. In contrast, an inte grable (rectangular or circular) billiard has a reduced density of sta tes near the Fermi level, but no gap. We present numerical calculation s for both cases in support of our analytical results. For the chaotic case, we calculate how the gap closes as a function of magnetic field or phase difference.