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

In granular superconductors, individual grains can contain bound Cooper pairs while the system as a whole is strongly insulating. In such cases the conductivity is determined by electron hopping between localized states in individual grains. Here we examine a model of hopping conductivity in such an insulating granular superconductor, where disorder is assumed to be provided by random charges embedded in the insulating gaps between grains. We use computer simulations to calculate the single-electron and electron pair density of states at different values of the superconducting gap $Δ$, and we identify "triptych" symmetries and scaling relations between them. At a particular critical value of $Δ$, one can define an effective charge $\sqrt{2}e$ that characterizes the density of states and the hopping transport. We discuss the implications of our results for magnetoresistance and tunneling experiments.