Density of states of chaotic Andreev billiards

Quantum cavities or dots have markedly different properties depending on whether their classical counterparts are chaotic or not. Connecting a superconductor to such a cavity leads to notable proximity effects, particularly the appearance, predicted by random matrix theory, of a hard gap in the excitation spectrum of quantum chaotic systems. Andreev billiards are interesting examples of such structures built with superconductors connected to a ballistic normal metal billiard since each time an electron hits the superconducting part it is retroreflected as a hole (and vice versa). Using a semiclassical framework for systems with chaotic dynamics, we show how this reflection, along with the interference due to subtle correlations between the classical paths of electrons and holes inside the system, is ultimately responsible for the gap formation. The treatment can be extended to include the effects of a symmetry-breaking magnetic field in the normal part of the billiard or an Andreev billiard connected to two phase-shifted superconductors. Therefore, we are able to see how these effects can remold and eventually suppress the gap. Furthermore, the semiclassical framework is able to cover the effect of a finite Ehrenfest time, which also causes the gap to shrink. However, for intermediate values this leads to the appearance of a second hard gap—a clear signature of the Ehrenfest time.

Figure 1

(a) The Andreev billiard consists of a chaotic normal metal (N) cavity attached to a superconductor (S) via a lead. (b) At the NS interface between the normal metal and the superconductor, electrons are retroreflected as holes.

Figure 2

(a) An Andreev billiard connected to two superconductors (S${}_1$,S${}_2$) at phases $\pm \varphi /2$ via leads carrying $N_1$ and $N_2$ channels, all threaded by a perpendicular magnetic field $B$. (b) The semiclassical treatment involves classical trajectories retroreflected at the superconductors an arbitrary number of times.

Figure 3

(a) The original trajectory structure of the correlation function $C(\epsilon ,2)$, where the incoming channels are drawn on the left, outgoing channels on the right, electrons as solid (blue), and holes as dashed (green) lines. (b) By collapsing the electron trajectories directly onto the hole trajectories, we create a structure where the trajectories only differ in a small region called an encounter. Placed inside the Andreev billiard, this diagram corresponds to Fig. 2b. The encounter can be slid into the incoming channels on the left (c) or the outgoing channels on the right (d) to create diagonal-type pairs.

Figure 4

(a) The original trajectory structure of the correlation function $C(\epsilon ,4)$ where the incoming channels are drawn on the left, outgoing channels on the right, electrons as solid (blue), and holes as dashed (green) lines. (d,g) Equivalent 2D projections of the starting structure as the order is determined by moving along the closed cycle of electron and hole trajectories. (b) By pinching together the electron trajectories (pairwise here), we can create a structure that only differs in three small regions (encounters) and which can have a small action difference. (e) Projection of (b) also created by collapsing the electron trajectories in (g) directly onto the hole trajectories. (c,f) Sliding two of the encounters from (b) together (or originally pinching three electron trajectories together) creates these diagrams. (h,i) Resulting rooted plane tree diagrams of (e,f) or (b,c) defining the top left as the first incoming channel [i.e., the channel ordering as depicted in (e,f)].

Figure 5

Further possibilities arise from moving encounters into the lead(s). Starting from Fig. 4c, we can slide the 2-encounter into the outgoing channels on the right (called “$o$-touching,” see the text) to arrive at (a,d) or the 3-encounter into the incoming channels on the left (called “$i$-touching”) to obtain (b,e). Moving both encounters leads to (c,f), but moving both to the same side means first combining the 3- and 2-encounter in Fig. 4c into a 4-encounter, and it is treated as such.

Figure 6

The tree shown in (a) is cut at its top node (of degree 6) such that the trees (b)–(f) are created. Note that to complete the five new trees, we need to add an additional four new links and leaves and that the trees (c) and (e) in the even positions have the incoming and outgoing channels reversed.

Figure 7

(a) The density of states of a chaotic quantum dot coupled to a single superconductor at $E\ll \Delta$. (b) The density of states with a finite bulk superconducting gap $\Delta =2E_T$ (dashed line) and $\Delta =8E_T$ (solid line) compared to the previous case in (a) with $\Delta \to \infty$ (dotted line).

Figure 8

The effect of a time-reversal symmetry-breaking magnetic field on the density of states of a chaotic Andreev billiard with a single superconducting lead for $b=0$ (dotted line), $b=1/4$ (solid line), $b=1$ (dashed line), and $b=9/4$ (dashed dotted line).

Figure 9

The paths may start and end in either of the two leads as shown. ${\zeta }_4$ as it travels from lead 1 to lead 2 obtains a phase factor $\exp (-\mathrm{i}\varphi )$, ${\zeta }_2$ traveling back contributes $\exp (\mathrm{i}\varphi )$, while the others do not contribute any phase. The encounters are again marked by circles and ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ denote the two superconducting leads at the corresponding superconducting phases $\pm \varphi /2$. This diagram is equivalent to the one in Fig. 4f.

Figure 10

The density of states of a chaotic quantum dot coupled to two superconductors with the same numbers of channels and phase differences $0$ (dotted line), $5\pi /6$ (solid line), $21\pi /22$ (dashed line), and $123\pi /124$ (dashed dotted line).

Figure 11

Magnetic-field dependence of the density of states of a chaotic Andreev billiard with phase difference $\varphi =5\pi /6$ for $b=0$ (dotted line), $b=0.1024$ (solid line), $b=0.4096$ (dashed line), and $b=1$ (dashed dotted line).

Figure 12

Phase dependence of the density of states of a chaotic Andreev billiard with phase difference $\varphi =0$ (dotted line), $\varphi =\pi /2$ (solid line), $\varphi =5\pi /6$ (dashed line), and $\varphi =21\pi /22$ (dashed dotted line). (a) At magnetic field $b=0.1024$, (b) at $b=0.4096$, and (c) at $b=1$.

Figure 13

Dependence of the density of states of an Andreev billiard on the difference $y=(N_1-N_2)/N$ in size of the leads with $y=0$ (dashed dotted line), $y=4/5$ (dashed line), $y=\sqrt{24}/5$, (solid line) and $y=1$ (dotted line). (a) At phase difference $\varphi =2\pi /3$, (b) at $\varphi =5\pi /6$, and (c) at $\varphi =21\pi /22$.

Figure 14

Critical value of the difference in the lead sizes $y$ as a function of the phase difference $\varphi$ between the two leads above which a second gap appears in the density of states.

Figure 15

(a) Width (and end point) of the first gap and (b) width of the second gap as a function of $\tau$.

Figure 16

(a) Density of states for $\tau ={\tau }_{\mathrm{E}}/{\tau }_{\mathrm{d}}=2$ (solid line), along with the BS (dashed) limit $\tau \to \infty$ and the RMT (dotted) limit $\tau =0$, showing a second gap just below $\epsilon \tau =\pi$. (b) Ehrenfest time related $2\pi /\tau$-periodic oscillations in the density of states after subtracting the BS curve.

Figure 17

Density of states as a function of $\epsilon \tau =E/E_{\mathrm{E}}$ for various values of $\tau$ showing the appearance of a second gap below $\epsilon \tau =\pi$. Inset: Density of states for $\tau =20$ (solid line) together with the BS limit (dashed).

Figure 18

Density of states for $\tau =2$ (solid line) along with the $\tau =0$ (dotted) and $\tau =\infty$ (dashed) limits for a chaotic Andreev billiard with phase difference (a) $\varphi =\pi /18$, (b) $\varphi =5\pi /6$, and (c) $\varphi =\pi$.

Figure 19

Density of states for $\tau =1/2$ (dotted line), $\tau =1$ (dashed), and $\tau =2$ (solid) showing the phase-dependent jumps for phase difference (a) $\varphi =\pi /18$ and (b) $\varphi =5\pi /6$.