Graphene Andreev billiards

We studied the energy levels of graphene-based Andreev billiards consisting of a superconductor region on top of a monolayer graphene sheet. For the case of Andreev retroreflection we show that the graphene-based Andreev billiard can be mapped to the normal-metal-superconducting billiards with the same geometry. We also derived a semiclassical quantization rule in graphene-based Andreev billiards. The exact and the semiclassically obtained spectrum agree very well both for the case of Andreev retroreflection and specular Andreev reflection.

Figure 1

(Color online) The exact step function $N\left(E\right)$ for circular shape GAB (solid line) and NS (circles) billiards in the case of Andreev retroreflection. The parameters for GAB and NS billiard are $E_{\text{F}}^{\left(G\right)}/{\Delta }^{\left(G\right)}=10$ and $E_{\text{F}}^{\left(N\right)}/{\Delta }^{\left(N\right)}=5$, respectively, and ${\xi }_c^{\left(G\right)}/R=0.12$ for both cases. The trivial factor 2 owing to the valley degeneracy in graphene is not included.

Figure 2

(Color online) The exact (solid line) and the semiclassically (circles) calculated step function $N\left(E\right)$ obtained from Eqs. (1, 2), respectively, for the case of Andreev retroreflection. The parameters are the same as in Fig. 1.

Figure 3

(Color online) The exact (solid line) and the semiclassically (circles) calculated step function $N\left(E\right)$ obtained from Eqs. (1, 3), respectively, for specular Andreev reflection. The parameters are $E_{\text{F}}^{\left(G\right)}=0$ and ${\xi }_c^{\left(G\right)}/R=0.03$.