An attempt was made to show that a billiard with a finite upper cutoff in the path length distribution P(s) will possess an energy gap in the density of states on the scale of the Thouless energy. By including the energy dependent phase shift of the Andreev reflection, a new expression for the density of the states was given within the framework of the semiclassical Bohr-Sommerfeld approximation. A formula for the energy gap was also derived. To check the results, an exact diagonalization of the Bogoliubov-de Gennes equation for different Andreev billiards was performed. The results of the two methods agree very well both for the integrated density of states and for the energy gap. Finally, it was shown that the energy gap on the scale of the Thouless energy can be much larger than the value 0.6ET predicted from random matrix theory for chaotic billiards.
ASJC Scopus subject areas
- Physics and Astronomy(all)