Self-consistent solution for proximity effect and Josephson current in ballistic graphene SNS Josephson junctions

We use a tight-binding Bogoliubov-de Gennes (BdG) formalism to self-consistently calculate the proximity effect, Josephson current, and local density of states in ballistic graphene superconductor-normal conductor-superconductor (SNS) Josephson junctions. Both short and long junctions, with respect to the superconducting coherence length, are considered, as well as different doping levels of the graphene. We show that self-consistency does not notably change the current-phase relationship derived earlier for short junctions using the non-self-consistent Dirac-BdG formalism [M. Titov and C. W. J. Beenakker, Phys. Rev. B 74, 041401(R) (2006)] but predict a significantly increased critical current with a stronger junction-length dependence. In addition, we show that in junctions with no Fermi-level mismatch between the N and S regions, superconductivity persists even in the longest junctions we can investigate, indicating a diverging Ginzburg-Landau superconducting coherence length in the normal region.