Thermally assisted hopping transport in disordered systems

The response to an external field of localized electrons coupled to phonons is investigated. The low frequency (ω<T) linear response function is shown to obey a kinetic equation. The transition probabilities (including multiphonon contributions) can be expressed in terms of the dynamical correlation functions(k, ?) of the phonons. The low temperature d.c. conductivity in three dimensions obeys a law σ(0)=σ₀ · exp(? (T ₀/T)^1/4). By a combined variational and “nearest neighbor” approximation upper limits for the exponential as well as the pre-exponential factor are obtained. In two dimensions the 1/4 in the exponent has to be replaced by 1/3. The one-dimensional case requires separate considerations which do not simply lead to an exponent 1/2. An expression for the thermopower in the hopping regime is derived and evaluated.