Disordered graphene Josephson junctions
  A tight-binding approach based on the Chebyshev-Bogoliubov-de Gennes method is used to describe disordered single-layer graphene Josephson junctions. Scattering by vacancies, ripples or charged impurities is included. We compute the Josephson current and investigate the nature of multiple Andreev reflections, which induce bound states appearing as peaks in the density of states for energies below the superconducting gap. In the presence of single atom vacancies, we observe a strong suppression of the supercurrent that is a consequence of strong inter-valley scattering. Although lattice deformations should not induce inter-valley scattering, we find that the supercurrent is still suppressed, which is due to the presence of pseudo-magnetic barriers. For charged impurities, we consider two cases depending on whether the average doping is zero, i.e. existence of electron-hole puddles, or finite. In both cases, short range impurities strongly affect the supercurrent, similar to the vacancies scenario. 
W.A. Munoz, L. Covaci and F.M. Peeters
cond-mat.supr-con cond-mat.mes-hall