Magnetic-field dependence of transport in normal and Andreev billiards: A classical interpretation of the averaged quantum behavior

We perform a comparative study of the quantum and classical transport probabilities of low-energy quasiparticles ballistically traversing normal and Andreev two-dimensional open cavities with a Sinai-billiard shape. We focus on the dependence of the transport on the strength of an applied magnetic field $B$. With increasing field strength the classical dynamics changes from mixed to regular phase space. Averaging out the quantum fluctuations, we find an excellent agreement between the quantum and classical transport coefficients in the complete range of field strengths. This allows an overall description of the nonmonotonic behavior of the average magnetoconductance in terms of the corresponding classical trajectories, thus, establishing a basic tool useful in the design and analysis of experiments.

Figure 1

The open geometry of the Sinai billiard considered in this study.

Figure 2

(Color online) Typical specular reflection (SR) and Andreev reflection (AR) at the circular central “antidot” of Fig. 1. A magnetic field is applied as indicated.

Figure 3

Magnetic-field dependence of the classical (solid line) and quantum (dots) transmission probability for the normal conducting Sinai billiard of Fig. 1. The magnetic field is in units of the strength $B_0$ for which the cyclotron radius is equal to $W$.

Figure 4

Panel (a) shows the transport coefficients of particles escaping the Andreev version of the Sinai billiard of Fig. 1 without particle-to-hole conversion. Panel (b) shows the Andreev reflection and transmission probabilities. In both panels the solid (dashed) line is the classical result of transmission (reflection) and the circles (squares) show the quantum transmission (reflection) coefficients normalized by $N_{\mathrm{ch}}$. The field strength is given in units of $B_0$. Inset: semilogarithmic blow up for $0.75⩽B∕B_0⩽1.25$.

Figure 5

Magnetoconductance for the normal (dots) and the Andreev (solid line) Sinai-shaped billiard of Fig. 1 (in units of the number of open channels $N_{\mathrm{ch}}$ times the conductance quantum $G_0\equiv 2e^2∕h$). The field strength is in units of $B_0$.

Figure 6

(Color online) (a) Classical electron transmission $T_e$ and reflection $R_e$ coefficients for the normal antidot device of Fig. 1 as a function of the applied magnetic field $B$. (b) The mean number of collisions with the boundary of the square cavity (walls 1–4) ${⟨n⟩}_w$ and with the circumference of the antidot ${⟨n⟩}_a$ as a function of $B$. The field strength is given in units of $B_0$.

Figure 7

Relative weight of the chaotic part $w_c$ of the phase space as a function of the applied field $B$ (in units of $B_0$) for the normal conducting Sinai-shaped billiard of Fig. 1.

Figure 8

(Color online) Classical results for the billiard of Fig. 1 with a superconducting disc in the center, as a function of the applied magnetic field $B$ (in units of $B_0$). (a) Classical electron transmission $T_e$ and reflection $\left(R_e\right)$ coefficients. (b) Classical electron-to-hole (Andreev) transmission $T_h$ and reflection $R_h$ probabilities. (c) Mean number of collisions ${⟨n⟩}_w$ with the boundary of the square cavity (walls 1–4) and with the circumference of the antidot ${⟨n⟩}_a$.

Figure 9

Relative weight of the chaotic part of the phase space $w_c$ as a function of the applied field $B$ (in units of $B_0$) for the Andreev version of the billiard in Fig. 1.

Figure 10

Andreev reflected orbit. The mapping from one center to the following one is a point reflection.

Figure 11

(Color online) The shaded area shows all possible locations for centers of orbits that connect the outer walls with the antidot.

Figure 12

$M$ is the center of the first arc before the Andreev reflection. After one Andreev reflection it is mapped to $M^{\prime }$, after another one it is mapped to $M^{″}$, which has the same distance from the center of the antidot as $M$.