Proximity-induced superconductivity in graphene

We propose a way of making graphene superconductive by putting on it small superconductive islands which cover a tiny fraction of graphene area. We show that the critical temperature, $T_c$, can reach several Kelvins at the experimentally accessible range of parameters. At low temperatures, $T\ll T_c$, and zero magnetic field, the density of states is characterized by a small gap $E_g\leq T_c$ resulting from the collective proximity effect. Transverse magnetic field $H_g(T) \propto E_g$ is expected to destroy the spectral gap driving graphene layer to a kind of a superconductive glass state. Melting of the glass state into a metal occurs at a higher field $H_{g2}(T)$.