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In granular superconductors, individual grains can contain bound Cooper pairswhile the system as a whole is strongly insulating. In such cases theconductivity is determined by electron hopping between localized states inindividual grains. Here we examine a model of hopping conductivity in such aninsulating granular superconductor, where disorder is assumed to be provided byrandom charges embedded in the insulating gaps between grains. We use computersimulations to calculate the single-electron and electron pair density ofstates at different values of the superconducting gap $\Delta$, and we identify"triptych" symmetries and scaling relations between them. At a particularcritical value of $\Delta$, one can define an effective charge $\sqrt{2}e$ thatcharacterizes the density of states and the hopping transport. We discuss theimplications of our results for magnetoresistance and tunneling experiments.

Coulomb gap triptychs, $\sqrt{2}$ effective charge, and hopping transport in periodic arrays of superconductor grains