Conduction in Granular Metals by Virtual Tunneling on the Fractal Percolation Cluster

Abstract

We discuss the calculation of the dc conductivity of granular metals on the insulating side of the metal-insulator transition. The granular structure is represented by a fractal percolation cluster and the problem of virtual tunneling-assisted conductivity is shown to be related to estimating the number of minimal paths for the problem of chemical distance metric on the fractal. The resulting conductivity temperature dependence has the form ln σ const - (T 0 /T) x , where x = ζ/(D B + ζ), D B is the fractal dimensionality of the backbone cluster and ζ is the superlocalization exponent. This gives the value of x close to 0.4 both in two and three dimensions that agrees fairly well with the experimental value x 1/2 for many granular conductors.