Hopping Conductivity in Granular Metals

We present evidence that in granular metals the observed temperature dependence of the low-field conductivity, $\exp (-\frac{b}{T^{\alpha }})$ with $\alpha =\frac{1}{2}$, 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.