Proximity-induced sub-gaps in Andreev billiards
  We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cut-off 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 semi-classical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semi-classical 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. 
J. Cserti, A. Korm\'anyos, Z. Kaufmann, J. Koltai, C. J. Lambert