Josephson effect in ballistic graphene

We solve the Dirac-Bogoliubov-de Gennes equation in an impurity-free superconductor-normal-metal-superconductor junction, to determine the maximal supercurrent I-c that can flow through an undoped strip of graphene with heavily doped superconducting electrodes. The result I-c similar or equal to(W/L)e Delta(0)/h is determined by the superconducting gap Delta(0) and by the aspect ratio of the junction (length L small relative to the width W and to the superconducting coherence length). Moving away from the Dirac point of zero doping, we recover the usual ballistic result I-c similar or equal to(W/lambda(F))e Delta(0)/h, in which the Fermi wavelength lambda(F) takes over from L. The product IcRN similar or equal to Delta(0)/e of the critical current and normal-state resistance retains its universal value (up to a numerical prefactor) on approaching the Dirac point.