Conduction in Granular Metals: Virtual State-Assisted Hopping

We discuss the calculation of the d.c. 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 a form ln(sigma)=const-(T0/T)^x, where x=zeta/(D_B+zeta), D_B is the fractal dimensionality of the backbone cluster and zeta 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 about 0.5 for many granular conductors.