Josephson Effect in Graphene-Based Junctions

We study the Josephson effect in graphene based junctions where superconductivity in graphene is induced by the proximity effect from external substrate materials. The electronic properties of the junction is described by the Dirac-Bogoliubov-de-Gennes equations. We consider the junction consisting of two superconductors with different pairing potentials. Using appropriate boundary conditions imposed on the normal region-superconductors interfaces, we calculated the Andreev bound state energy, in the ballistic limit, taking into account two types of reflections namely the retro and specular Andreev reflections.