Hopping Transport in Granular Superconductors

We study the conductivity of granular superconductors in the weak coupling insulating regime. We show that it is governed by the hopping of either electrons or Cooper pairs depending on the relation between the superconducting gap and the charging energy of a single granule. Local superconducting pairing plays an important role in both cases. In particular, in the case of the transport via electron hopping the superconducting gap suppresses the inelastic cotunneling processes. We determine transport characteristics of an array in different regimes and construct the transport phase diagram.