Andreev levels in a Josephson superconductor graphene superconductor nanostructure

We obtain the bound states in superconductor-graphene-superconductor nanostructure, which are responsible for the Josephson effect. The coupling between graphene and each superconducting region is modeled as two different hopping parameters in the respective SG and GS interfaces. With the purpose of determining the local density of states and the spectrum, the Green function of the junction is calculated resolving the Dyson equation. We obtain that the number of levels depends on the width and doping of graphene region and this occurs for the two types of edge (armchair or zigzag). We investigate the behavior of the bound states as a function of the transparency. In the limit of a transparent junction, the results obtained by the Green׳s function method reproduce those present in the literature. In the tunnel limit the spectrum is different for armchair and zigzag edges.