We examine the density of states of an Andreev billiard and show that any billiard with a finite upper cutoff 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 semiclassical predictions when the energy dependent phase shift for Andreev reflections is properly taken into account. Based on this new semiclassical 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.

We present a numerical study of the universal gap fluctuations and the ensemble averaged density of states (DOS) of chaotic two-dimensional Andreev billiards for finite Ehrenfest time E . We show that the distribution function of the gap fluctuation for small enough Ehrenfest time can be related to that derived earlier for zero Ehrenfest time. An effective description based on the random matrix theory is proposed giving a good agreement with the numerical results. A systematic linear decrease of the mean field gap with increasing Ehrenfest time E is observed but its derivative with respect to E is in between two competing theoretical predictions and close to that of the recent numerical calculations for Andreev map. The exponential tail of the density of states is interpreted semiclassically. PACS number(s): 74.45.ϩc, 75.45.ϩj, 03.65.Sq Recently, mesoscopic ballistic two dimensional normal (N) dots in contact with a superconductor(S) have been extensively studied. Such hybrid systems are commonly called Andreev billiards. [1][2][3][4][5] In the most recent works, interest has shifted to mesoscopic fluctuations of the excitation spectrum of these systems. [6][7][8][9] Since the subgap spectrum determines the tunneling conductance of an N-S contact this is an essential question both experimentally and theoretically.Based on the semiclassical treatments and random matrix theory (RMT), it was shown by Melsen et al. 2 that integrable Andreev billiards are gapless, whereas systems with classically chaotic dots possess an energy gap on the scale of the Thouless energy E T = ប / ͑2 D ͒, where D = A / ͑Wv F ͒ is the mean dwell time in the normal dot (here A is the area of the normal dot, W is the width of the superconducting region, and v F is the Fermi velocity). For such systems, it is assumed that ␦ N Ӷ E T Ӷ⌬, where ␦ N =2ប 2 / ͑mA͒ is the mean level spacing of the isolated normal dot with effective mass m of the electrons and ⌬ is the bulk order parameter of the superconductor. 4 In further studies 6,7 it was concluded that in chaotic cases, the lowest energy level E 1 of the system varies from sample to sample with a universal probability distribution P͑x͒ given in Ref. 6 if the energy levels E 1 are rescaled as͑1b͒Here ␥ = 1 2 ͑ ͱ 5−1͒ is the golden ratio, cЈ = ͓͑15− 6 ͱ 5͒ /20͔ 1/3 /2, M = Int͓k F W / ͔ is the number of open channels in the S region and k F is the Fermi wave number (Int͓.͔ stands for the integer part). The resulting distribution P͑x͒ yields the RMT values 6 ͗x͘Ϸ1.21 and ␦x = ͱ͗x 2 ͘ − ͗x͘ 2 Ϸ 1.27.Equations (1) are strictly valid only in the RMT limit, i.e., when the Ehrenfest time E = ͑1/͒ln͑L / F ͒ tends to zero ( E is the time needed for a wave packet of minimal size F =2 / k F to spread to the characteristic size L of the classically chaotic normal dot with Lyapunov exponent ). For finite but small enough Ehrenfest time, Silvestrov et al., 10 and Vavilov and Larkin 11 predicted that to lowest order in E / D the mean-field gap E g decreases linearly by increasing the ratio E / D . The first ...

We study a transverse electron-hole focusing effect in a normal-superconductor system. The spectrum of the quasiparticles is calculated both quantum mechanically and in semiclassical approximation, showing an excellent agreement. A semiclassical conductance formula is derived, which takes into account the effect of electronlike as well as holelike quasiparticles. At weak magnetic fields, the semiclassical conductance shows characteristic oscillations due to the Andreev reflection, while for stronger fields it goes to zero. These findings are in line with the results of previous quantum calculations and with the expectations based on the classical dynamics of the quasiparticles.

We demonstrate that the exact quantum mechanical calculations are in good agreement with the semiclassical predictions for rectangular Andreev billiards and therefore for a large number of open channels it is sufficient to investigate the Bohr-Sommerfeld approximation of the density of states. We present exact calculations of the classical path length distribution P (s) which is a nondifferentiable function of s, but whose integral is a smooth function with logarithmically dependent asymptotic behavior. Consequently, the density of states of rectangular Andreev billiards has two contributions on the scale of the Thouless energy: one which is well-known and it is proportional to the energy, and the other which shows a logarithmic energy dependence. It is shown that the prefactors of both contributions depend on the geometry of the billiards but they have universal limiting values when the width of the superconductor tends to zero.

The energy spectrum of cake shape normal -superconducting systems is calculated by solving the Bogoliubov-de Gennes equation. We take into account the mismatch in the effective masses and Fermi energies of the normal and superconducting regions as well as the potential barrier at the interface. In the case of a perfect interface and without mismatch, the energy levels are treated by semi-classics. Analytical expressions for the density of states and its integral, the step function, are derived and compared with that obtained from exact numerics. We find a very good agreement between the two calculations. It is shown that the spectrum possesses an energy gap and the density of states is singular at the edge of the gap. The effect of the mismatch and the potential barrier on the gap is also investigated.