Hopping conductivity in granular disordered systems

We study the temperature dependence of conductivity for high-resistivity granular disordered systems in terms of the critical-path method. It is shown that the low-field conductivity exp(-AT^α) with α=1/2 is obeyed over large temperature ranges with possible crossovers to α=1/4 at low temperatures and to α>1/2 at high temperatures. The temperatures at which the crossovers occur depend on the distribution of grain sizes. Comparisons are made with experiments and with other theoretical approaches.