Josephson current for a Graphene nanoribbon using a lattice model

A tight binding approach based on the Bogoliubov de-Genes approach has been used to calculate the DC Josephson current for short junctions with respect to the superconducting coherence length with a lattice model for SGNR- S junctions. We calculate the phase, length, width, and chemical potential dependence at the Josephson junction and discuss the similarities and differences with regard to the theoretical and experimental results obtained for graphene. Regarding the calculations on graphene, using a lattice model, we convert the graphene honeycomb structure to a brick lattice structure that does not change the lattice topology. Then, by removing several atoms from the lattice, we create the simple vacancy defects in the brick lattice. Also we calculate the Josephson current with these vacancies.