Universal low-frequency behaviour in the a.c. hopping conductivity of disordered systems

The empirical scaling law σ(ω, T)/σ(0, T)=f[Aω/σ(0, T)] is implied for a.c. conductivities calculated in the extended pair approximation (EPA) for a variety of models in parameter ranges that are typical experimentally. The number A depends upon the model considered; ω and σ represent frequency and conductivity, respectively, in reduced units. It is shown that, for energy-independent hopping in the EPA, the law becomes more accurate when ω→0 and σ(0)→0. Numerical results for an energy-dependent model imply similar behaviour. The same law gives a good account of experimental data on conduction in amorphous germanium, impurity bands, and polyacetylene. A quasi-universal behaviour of the frequency-dependent conductivity is suggested, in which the specific nature of a given material is manifest only in its d.c. conductivity and in the value of A for a given f(x). In consequence the ωs law is reinterpreted; s depends upon σ(ω)/σ(0) and is not related to the distribution of hopping rates via the pair approximation to σ(ω).