Proximity-Induced Subgaps in Andreev Billiards

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.

Figure 1

Geometry.

Figure 2

The integrated density of states $N(E)$ as a function of $E$ (in units of $\Delta$) for $d=0$ and angles $\alpha =80°$, $60°$, and $45°$. For each angle $\alpha$ the stair-type lines correspond to the exact diagonalization of the Bogoliubov–de Gennes equation, while the dotted, solid, and dashed lines are obtained by using Eq. (1). The parameters are $M=55.5$, $\Delta /E_F=0.015$.

Figure 3

The integrated density of states $N(E)$ as a function of $E$ (in units of $\Delta$) for $d\ne 0$ and angles $\alpha =80°$, ${60}^{°}$, and $45°$. In each case, $d$ varies such that the area of the normal dot is fixed at $A=5W^2$ for the angles $\alpha =80°,60°,\text{and}45°$. This choice results in approximately the same number of states below $\Delta$ for different values of $\alpha$. The solid (stair-type) lines correspond to the exact diagonalization, and the dotted lines show the results from Eq. (1). The parameters are the same as those in Fig. 2.

Figure 4

The path length distribution $P(s)$ for the Andreev billiard of Fig. 1 when $d=0$ for $\alpha =80°$, $60°$, and $45°$. The insets show the corresponding semiclassical density of states $n(E)$ (in units of $1/\delta$) using Eq. (3). The parameters used were the same as those in Fig. 2.

Figure 5

The gap $E_{\mathrm{g}\mathrm{a}\mathrm{p}}$ (in units of Thouless energy) as a function of angle $\alpha$ for Andreev billiards with $d=0$. The solid and dashed curves are the semiclassical results from Eq. (7) when $M=55.5$, and from Eq. (9), i.e., when $M\to \infty$, respectively. The circles are the results of the exact diagonalization when $M=55.5$.

Figure 6

A triangle (a) and quadrangle (b) shape of a normal billiard having a finite upper cutoff of the length distribution $P(s)$.