Non-retracing orbits in Andreev billiards

The validity of the retracing approximation in the semiclassical quantization of Andreev billiards is investigated. The exact energy spectrum and the eigenstates of normal-conducting, ballistic quantum dots in contact with a superconductor are calculated by solving the Bogoliubov-de Gennes equation and compared with the semiclassical Bohr-Sommerfeld quantization for periodic orbits which result from Andreev reflections. We find deviations that are due to the assumption of exact retracing electron-hole orbits rather than the semiclassical approximation, as a concurrently performed Einstein-Brillouin-Keller quantization demonstrates. We identify three different mechanisms producing non-retracing orbits which are directly identified through differences between electron and hole wave functions.

Figure 1

(a) Andreev billiard with a rectangular normal-conducting (N) region. The superconducting (S) lead is assumed to be half-infinite (shaded area). The dashed and dotted lines depict an “almost retracing” electron-hole orbit created by Andreev reflection at the SN interface. For better visibility the starting point of the orbit is marked by a dot. (b) Perfect Andreev reflection, i.e., ${\theta }_e={\theta }_h$. (c) Imperfect Andreev reflection, ${\theta }_e\ne {\theta }_h$, $k_y^e=k_y^h$.

Figure 2

(Color online) The quantum mechanical state counting function $N_{\mathrm{QM}}\left(\epsilon \right)$ (solid red staircase) and its semiclassical BS estimate $N_{\mathrm{BS}}\left(\epsilon \right)$ (dashed green line) for two quadratic cavities with different lead widths as shown in the insets [$k_{\mathrm{F}}=20.5\pi ∕W$, $W=L\left(\mathrm{a}\right)$, $0.8L\left(\mathrm{b}\right)$]. The top left insets show the classical path length distribution in units of the cavity length $L$. Solid triangles mark pronounced cusps in $N\left(\epsilon \right)$ and their classical origin. The quantum number $n$ [Eq. (4)] increases from 0 to 1 at the cusp marked $\mathit{I}$. Open triangles mark the energy positions of states whose wave functions are displayed in Fig. 3. Shading marks the regions where the BS approximation begins to deviate from quantum results.

Figure 3

(Color online) (a)–(d) show the electron and hole wave functions ${\mid u\left(\mathbf{x}\right)\mid }^2$ and ${\mid v\left(\mathbf{x}\right)\mid }^2$ at values of $\epsilon$ indicated by open triangles in Fig. 2b: $\epsilon ∕\Delta$ is (a) 0.565, (b) 0.566, (c) 0.33, and (d) 0.355. The corresponding classical orbits are indicated on the right.

Figure 4

Geometries with tunable boundaries. (a) Stretched N cavities with $L⪢D$ but with $D$ equal to the width of the SN interface $W=D$. (b) Stretched N cavities with $D⪢L$ but constant length $L$ (lower boundary of S and N aligned). (c) As in (b) but SN interface at arbitrary position on the entrance side.

Figure 5

Imperfect Andreev reflection in the fundamental and extended zone scheme. The returning hole hits the SN interface a distance $\delta y$ apart from the starting point of the particle.

Figure 6

(Color online) The DOS calculated quantum mechanically (red solid curve) and by the BS approximation (green dashed curve) for two different geometries with ratios of (a) $L∕D=1$ and (b) $L∕D=6$ (see insets) in units of the Weyl approximation of the DOS per unit area, ${\rho }_W=m_{\mathrm{eff}}∕\left({\hslash }^2\pi \right)$. (c) Error of the semiclassical prediction $\delta \rho$ [Eq. (15)] as a function of an energy-averaged $\overline{r}$ [Eq. (14)] for 300 different ratios of $L∕D∊$ [1,20]. The solid blue line shows a smoothed average (30 adjacent points) of the recorded data.

Figure 7

(Color online) Comparison of the exact quantum mechanical state counting function $N_{\mathrm{QM}}$ (red staircase) with the semiclassical BS (green dashed line) and EBK (blue squares) approximations for a highly elongated cavity $(L∕D=10)$, as shown in the lower inset. Quantum mechanical and EBK solutions are nearly indistinguishable. The upper inset shows a magnification. The energy for which $\delta y={\lambda }_{\mathrm{F}}$ is marked.

Figure 8

(Color online) Selected Andreev eigenstates of nonseparable Andreev billiards stretched parallel to the SN interface displaying a large discrepancy between electron and hole wave function patterns. (a) $L∕D=10$, $\epsilon =0.760\Delta$; (b) $L∕D=10$, $\epsilon =0.766\Delta$, $k_{\mathrm{F}}=21.5\pi ∕W$; and (c) $L∕D=2$, $\epsilon =0.040\Delta$, $k_{\mathrm{F}}=20.5\pi ∕W$;

Figure 9

(Color online) (a) Poincaré surface of section of a stretched nonseparable Andreev billiard, taken at the right vertical side, opposing the SN interface: $W∕D=5$, $W=L$. The light gray area marks regions of classical phase space coupled directly to the entrance lead, with one single reflection at the right wall. Dark red areas show regions of classical phase space decoupled from the superconducting lead. Classical trajectories corresponding to selected areas of phase space are marked by crosses in (a) and shown in (b)–(d).

Figure 10

(Color online) Selected Andreev eigenstates of systems with (a) no diffractive corners ($L∕D=1.6$, $W=D$, $k_{\mathrm{F}}=20.5\pi ∕W$, $\epsilon ∕\Delta =0.48$); (b) one diffractive corner ($D∕L=1.7$, $W=L$, $k_{\mathrm{F}}=21.5\pi ∕W$, $\epsilon ∕\Delta =0.72$); and (c) two diffractive corners ($D∕L=1.7$, $W=L$, $k_{\mathrm{F}}=20.5\pi ∕W$, $\epsilon ∕\Delta =0.82$).

Figure 11

(Color online) The DOS calculated quantum mechanically (red solid curve) and by the BS approximation (green dashed curve) for two different geometries with ratios of (a) $L∕D=1$ and (b) $1∕8$ (see insets), in units of the Weyl approximation of the DOS per unit area, ${\rho }_W=m_{\mathrm{eff}}∕\left({\hslash }^2\pi \right)$. (c) Error of the retracing approximation $\delta \rho$ [Eq. (15)] for transversely elongated cavities (red circles) as a function of $\overline{r}$. The solid blue (dashed red) line shows the average error for longitudinally (transversely) elongated cavities (see also Fig. 6).

Figure 12

(Color online) Error of the retracing approximation $\delta \rho$ as a function of $D∕W$ for geometries with no (blue square, dashed line), one (red circle, solid line), and two (green cross, dashed line) diffractive corners. Different orientations of the cross refer to different positions of the superconducting lead. The lines show a linear root mean square fit to the data.