The ac conductivity of disordered and dielectric solids. II. Distributions of relaxation times for clusters

For the calculation of the contribution to the ac conductivity σ(ω), or to the dielectric loss ε{lunate}″(ω), from transitions of charge carriers between states in a small cluster, the rate equations provide the most convenient formalism. These equations are derived for carriers obeying either classical or Fermi-Dirac statistics, and their formal solution is presented in terms of a distribution of relaxation times. The application of our results to amorphous semiconductors reveals, inter alia, an error in previous applications of the pair approximation in the case of uncorrelated charge carriers obeying Fermi-Dirac statistics. Our method also has considerable advantages over previous methods for the numerical calculation of σ(ω) for any given real or model system. For dielectrics, our analysis permits some general predictions as to the number of relaxation times that will be observed, and their dependence on temperature in some cases.