Abstract
We study the Josephson coupling of superconducting (SC) islands through the surface of single-layer graphene (SLG) and bilayer graphene (BLG) in the long-junction regime, as a function of the 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’s 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 a qualitatively strong field dependence of the relation between Berezinskii–Kosterlitz–Thouless transition critical temperature and gate voltage.
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