Excitation gap of a graphene channel with superconducting boundaries

We calculate the density of states of electron-hole excitations in a superconductor-normal-metal-superconductor (SNS) junction in graphene, in the long-junction regime that the superconducting gap Delta(0) is much larger than the Thouless energy E-T=hv/d (with v the carrier velocity in graphene and d the separation of the NS boundaries). If the normal region is undoped, the excitation spectrum consists of neutral modes that propagate along the boundaries-transporting energy but no charge. These Andreev modes are a coherent superposition of electron states from the conduction band and hole states from the valence band, coupled by specular Andreev reflection at the superconductor. The lowest Andreev mode has an excitation gap of E-0=1/2(pi-parallel to phi parallel to)E-T, with phi is an element of(-pi,pi) the superconducting phase difference. At high doping (Fermi energy mu > E-T) the excitation gap vanishes proportional to E-0(E-T/mu)(2), and the usual gapless density of states of Andreev levels is recovered. We use our results to calculate the phi dependence of the thermal conductance of the graphene channel.