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We study the Josephson coupling of superconducting (SC) islands through thesurface of single-layer (SLG) and bilayer (BLG) graphene, as a function ofdistance between the grains, temperature, chemical potential and external(transverse) gate-voltage. For SLG, we provide a comparison with existingliterature. The proximity effect is analyzed through a Matsubara Green functionapproach. This represents the first step in a discussion of the conditions forthe onset of a granular superconductivity within the film, made possible byJosephson currents flowing between superconductors. To ensure phase coherenceover the 2D sample, a random spatial distribution can be assumed for the SCislands on the SLG sheet (or intercalating the BLG sheets). The tunablegate-voltage-induced band gap of BLG affects the asymptotic decay of theJosephson coupling-distance characteristic for each pair of SC islands in thesample, which results in the end in a qualitatively strong field-dependence ofthe relation between Berezinskii-Kosterlitz-Thouless transition criticaltemperature and gate-voltage.

Josephson coupling between superconducting islands on single and bilayer graphene