Stationary Josephson effect in ballistic graphene junctions: effects of inhomogeneous carrier density

Abstract

We study the stationary Josephson effect in a system of ballistic graphene in planar contact with two superconducting electrodes. Applying a quasi-classical Green's function approach, we derive a general expression of the Josephson current which is valid whenever the chemical potential is away from the Dirac point. In its derivation we treat the mono- and bi-layer cases in a parallel manner, and take account of the fact that the carrier density is higher in the region covered by the superconductors. The behavior of the Josephson critical current is investigated at zero temperature on the basis of the obtained formula.