Phonon induced Superconductivity of High Temperatures in Electrical Graphene Superlattices

We discuss the BCS theory for electrons in graphene with a superimposed electrical unidirectional superlattice potential (SL). New Dirac points emerge together with van Hove singularities (VHS) linking them. We obtain a superconducting transition temperature $T_c$ for chemical potentials close to the VHS assuming that acoustic phonon coupling should be the dominant mechanism. Pairing of two onsite electrons with one electron close to the ${\bf K}$ and the other close to the $-{\bf K}$ point is the most stable pair formation. The resulting order parameter is almost constant over the entire SL.