Quantum disorder and quantum chaos in Andreev billiards

We investigate the crossover from the semiclassical to the quantum description of electron energy states in a chaotic metal grain connected to a superconductor. We consider the influence of scattering-off point impurities (quantum disorder) and of quantum diffraction (quantum chaos) on the electron density of states. We show that both the quantum disorder and the quantum chaos open a gap near the Fermi energy. The size of the gap is determined by the mean free time in disordered systems and by the Ehrenfest time in clean chaotic systems. Particularly, if both times become infinitely large, the density of states is gapless, and if either of these times becomes shorter than the electron escape time, the density of states is described by random matrix theory. Using the Usadel equation, we also study the density of states in a grain connected to a superconductor by a diffusive contact.