Charge Transfer between a Superconductor and a Hopping Insulator

We develop a theory of the low-temperature charge transfer between a superconductor and a hopping insulator. We show that the charge transfer is governed by the coherent two-electron–Cooper pair conversion process time-reversal reflection, where electrons tunnel into a superconductor from the localized states in the hopping insulator located near the interface, and calculate the corresponding interface resistance. A specific feature of this problem is the interplay between the time-reversal reflection at the interface and transport through the percolation cluster. To allow for this interplay, we have generalized the connectivity criterion of the percolation theory to include surface effects. We show that the time-reversal interface resistance is accessible experimentally, and that in mesoscopic structures it can exceed the bulk hopping resistance.