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Imperfections change essentially the electronic transport properties ofgraphene. Motivated by a recent experiment reporting on the possibleapplication of graphene as junctions, we study transport properties ingraphene-based junctions with single localized defect. We solve theDirac-Bogoliubov-de-Gennes equation with a single localized defectsuperconductor-normal(graphene)-superconductor (SNS) junction. We consider theproperties of tunneling conductance and Josephson current through an undopedstrip of graphene with heavily doped s-wave superconducting electrodes in thedirty limit. We find that spectrum of Andreev bound states are modified in thepresence of single localized defect in the bulk and the minimum tunnelingconductance remains the same. The Josephson junction exhibits signoscillations.

Josephson effect in Graphene SNS Junction with a Single Localized Defect