Percolation model for the superconductor-insulator transition in granular films

We study the temperature dependence of the superconductor-insulator transition in granular superconductors. Empirically, these systems are characterized by very broad resistance tails, which depend exponentially on the temperature, and the normal state resistance. We model these systems by two-dimensional random resistor percolation networks in which the resistance between two grains is governed either by Josephson junction coupling (Cooper pair's tunneling) or by quasiparticle tunneling. Our numerical simulations as well as an effective medium evaluation explain the experimental results over a wide range of temperatures and resistances. Using effective medium approximation we find an analytical expression for the effective resistance of the system and the value of the critical resistance separating conducting from insulating branches.