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