Hopping transport in granular superconductors

We investigate hopping transport in granular superconductors in the weak-coupling insulating regime and find that it is governed by the interplay between the granule charging energy E_c and the single granule superconducting gap Δ. At E_c>Δ, the transport is mediated by single electron hopping, while at E_c<Δ, the transport is dominated by hopping of Cooper pairs. We show that in the former case and at low temperatures, T⪡Δ<E_c, superconducting pairing suppresses inelastic cotunneling processes, giving rise to giant negative magnetoresistance. We find the range of intermediate temperatures where transport is dominated by inelastic cotunneling but having an activation temperature behavior; this regime is specific to granular superconductors. We investigate different conductivity regimes and construct a transport phase diagram.