Josephson effect in Graphene SNS Junction with a Single Localized Defect

Imperfections change essentially the electronic transport properties of graphene. Motivated by a recent experiment reporting on the possible application of graphene as junctions, we study transport properties in graphene-based junctions with single localized defect. We solve the Dirac-Bogoliubov-de-Gennes equation with a single localized defect superconductor-normal(graphene)-superconductor (SNS) junction. We consider the properties of tunneling conductance and Josephson current through an undoped strip of graphene with heavily doped s-wave superconducting electrodes in the dirty limit. We find that spectrum of Andreev bound states are modified in the presence of single localized defect in the bulk and the minimum tunneling conductance remains the same. The Josephson junction exhibits sign oscillations.