Low-frequency conductivity in disordered systems due to variable-range hopping

The low-frequency hopping conductivity of disordered systems is calculated in a cluster approximation at low temperatures where the temperature dependence of the DC conductivity is described by the Mott law. Calculations show that in the high-frequency region sigma ( omega , T) approximately omega s(T) exp(-p(T 0 /T) p ), where s(T) increases, tending to unity as the temperature T decreases, p= 1 / 4 for a finite density of electronic states at the Fermi level. The results obtained agree with the available experimental data on frequency and temperature dependence of the impurity conductivity in doped semiconductors.