Proximity-induced interface bound states in superconductor-graphene junctions

We show that interface bound states are formed at isolated graphene-superconductor junctions. These states arise due to the interplay of virtual Andreev and normal reflections taking place at these interfaces. Simple analytical expressions for their dispersion are obtained considering interfaces formed along armchair or zig-zag edges. It is shown that the states are sensitive to a supercurrent flowing on the superconducting electrode. The states provide long-range superconducting correlations on the graphene layer, which may be exploited for the detection of crossed Andreev processes.

Figure 1

(Color online) Simple model for the emergence of IBSs (left panel). It illustrates the scattering processes taking place at a graphene-superconductor interface with an intermediate heavily doped normal graphene region of width $d$. Cases (i) and (ii) correspond to the case $\hslash vq<\left|E\pm E_F\right|$ and $\hslash vq>\left|E\pm E_F\right|$, respectively, with $\Delta >E>E_F$. (right panel) Graphene-superconductor junctions along different edges. On the superconducting side (shaded areas) the on-site order parameter $\Delta$ is finite and the doping level is high $\left(E_F^S⪢\Delta \right)$.

Figure 2

Grayscale plot of the spectral density at a distance $\sim {\xi }_0$ from the interface defined along an armchair edge. The results were obtained using the HDSC model of Ref. 7 for $E_F=0$ (left panel) and $E_F=\Delta /2$ (right panel). The full lines indicate the position of the IBS determined from Eq. (2).

Figure 3

(Color online) Dispersion relation for the IBSs on a zig-zag interface for decreasing parameter $\beta$ controlling the coupling with the superconductor. The parallel momentum $q$ in Eq. (6) is measured from the Dirac points at $K=\pm 2\pi /3a$.

Figure 4

(Color online) Effect of a supercurrent flowing on the superconducting electrode on the dispersion relation (left panel) and on the local density of states at a distance $\sim {\xi }_0/10$ from the interface normalized to ${\rho }_0=\Delta {\left(a/\hslash v\right)}^2/2\pi$ (right panel). The results correspond to $\hslash v{q}_s/\Delta =0.0$, 0.25, 0.5, and 0.75 with $E_F^S=100\Delta$.