Bound states in Andreev billiards with soft walls

The energy spectrum and the eigenstates of a rectangular quantum dot containing soft potential walls in contact with a superconductor are calculated by solving the Bogoliubov–de Gennes equation. We compare the quantum mechanical solutions with a semiclassical analysis using a Bohr-Sommerfeld (BS) quantization of periodic orbits. We propose a simple extension of the BS approximation which is well suited to describe Andreev billiards with parabolic potential walls. The underlying classical periodic electron-hole orbits are directly identified in terms of “scar”-like features engraved in the quantum wave functions of Andreev states which we determine here explicitly.

Figure 1

Andreev billiard with parabolic walls (a) at the wall opposite to the lead and (b) at all sides except the one the superconducting lead is attached to. The superconducting area is shaded.

Figure 2

A rectangular Andreev billiard with one parabolic wall defined by Eq. (1a) (darker shading in the N region corresponds to higher potential). One typical Andreev orbit, consisting of an electron part (dashed line) and a hole part (dotted line), is shown.

Figure 3

Comparison between the geometric path length distribution $P\left(l_g\right)$ (taking into account the length of the curved trajectories) and the modified path length distribution $\tilde{P}\left(\tilde{l}\right)$ [Eq. (12)]. Both $l_g$ and $\tilde{l}$ are given in units of $L$. The inset shows the geometry: Andreev billiard with one parabolic wall, $W=0.7L$, $\alpha =6.7$, $x_0=0.4L$. (b)–(d) denote the lengths $\tilde{l}$ of those bundles of classical orbits, at which the wave functions show enhancement in Fig. 4.

Figure 4

(Color online) (a) Comparison between the quantum mechanical state counting function $N\left(\epsilon \right)$ (solid red staircase) and the BS prediction, $N_{\mathrm{BS}}\left(\epsilon \right)$, [Eq. (8), dashed green line] for a N-S system shown in the upper left inset. The modulus square ${\mid u_N\mid }^2$ and ${\mid v_N\mid }^2$ [see Eq. (6)] of the three eigenstates marked in (a) are shown in (b)–(d) to illustrate the correspondence between continuous families of classical trajectories and the quantum mechanical wave functions. The parameters are $M=15$, $W=0.7L$, $\alpha =6.7$, $x_0=0.4L$, and $\Delta ∕E_{\mathrm{F}}=0.02$.

Figure 5

(Color online) (a) Quantum mechanical state counting function (red solid staircase) and BS approximation (green dashed line) for $\alpha =5.5$. The parameters are $M=30$, $W=L$, $\alpha =125$, $x_0=L$, and $\Delta ∕E_{\mathrm{F}}=0.02$. (b)–(d) show selected eigenstates (see text).

Figure 6

(Color online) (a) Quantum mechanical state counting function (red solid staircase) and BS approximation (green dashed line) for the same system as in Fig. 5 except for a smaller value of $\alpha =5.5$ (curvature of parabolic wall). (b)–(d) show selected eigenstates (see text).