Hopping Conductivity in Granular Metals

Abstract

We present evidence that in granular metals the observed temperature dependence of the low-field conductivity, $\mathrm{exp}(\ensuremath{-}\frac{b}{{T}^{\ensuremath{\alpha}}})$ with $\ensuremath{\alpha}=\frac{1}{2}$, can be ascribed to a relationship $s{E}_{c}=\mathrm{const}$ between $s$, the separation of neighboring metal grains, and ${E}_{c}$, the electrostatic energy required to create a positive-negative charged pair of grains. This relationship results from simple considerations of the structure of granular metals. The predictions of the theory, for both the high- and the low-field electrical conductivity, are in excellent accord with experimental results in granular Ni-Si${\mathrm{O}}_{2}$ films.