Coulomb zero bias anomaly for fractal geometry and conductivity of granular systems near the percolation threshold

A granular system slightly below the percolation threshold is a collection of finite metallic clusters, characterized by wide spectrum of sizes, resistances, and charging energies. Electrons hop from cluster to clusters via short insulating “links” of high resistance. At low temperatures all clusters are Coulomb blockaded and the dc-conductivity $\sigma$ is exponentially suppressed. At lowest $T$ the leading transport mechanism is variable range cotunneling via largest (critical) clusters, leading to the modified Efros-Shklovsky law. At intermediate temperatures the principal suppression of $\sigma$ originates from the Coulomb zero bias anomaly occurring, when electron tunnels between adjacent large clusters with large resistances. Such clusters are essentially extended objects and their internal dynamics should be taken into account. In this regime the $T$-dependence of $\sigma$ is stretched exponential with a nontrivial index, expressed through the indices of percolation theory. Due to the fractal structure of large clusters the anomaly is strongly enhanced: it arises not only in low dimensions, but also in $d=3$ case.