Cooper-pair propagation and superconducting correlations in graphene

We investigate the Cooper-pair propagation and the Josephson effect in graphene under conditions in which the distance L between superconducting electrodes is much larger than the width W of the contacts. In the case of undoped graphene, we show that supercurrents may exist with a spatial decay proportional to W²∕L³, reminiscent of the behavior of the critical current in disordered normal metals. We observe that there is in general a crossover temperature T*~vF∕kBL that marks the onset of the strong decay of the supercurrent and that corresponds to the scale above which the Cooper pairs are disrupted by thermal effects during their propagation. We also show that the spatial decay of the critical current changes upon doping into a 1∕L² behavior, opening the possibility to observe a supercurrent over length scales above 10.3em0exμm at suitable doping levels.