Quasiclassical Theory of the Josephson Effect in Ballistic Graphene Junctions

The stationary Josephson effect in a system of ballistic graphene is studied in the framework of quasiclassical Green's function theory. Reflecting the ultimate two-dimensionality of graphene, a Josephson junction involving a graphene sheet embodies what we call a planar Josephson junction, in which superconducting electrodes partially cover the two-dimensional graphene layer, achieving a planar contact with it. For capturing this feature we employ a model of tunneling Hamiltonian that also takes account of the effects of inhomogeneous carrier density. Within the effective mass approximation we derive a general formula for the Josephson current, revealing characteristic features of the superconducting proximity effect in the planar Josephson junction. The same type of analysis has been equally applied to mono-, bi- and arbitrary $N$-layer cases.