Inspired by Holstein's work on small polaron hopping, the evolution equations of localized states and extended states in presence of atomic vibrations are derived for an amorphous semiconductor.The transition probabilities are obtained for four types of transitions: from one localized state to another localized state, from a localized state to an extended state, from an extended state to a localized state, and from one extended state to another extended state. At a temperature not too low, any process involving localized state is activated. The computed mobility of the transitions between localized states agrees with the observed 'hopping mobility'. We suggest that the observed 'drift mobility' originates from the transitions from localized states to extended states. Analysis of the transition probability from an extended state to a localized state suggests that there exists a short-lifetime belt of extended states inside conduction band or valence band. It agrees with the fact that photoluminescence lifetime decreases with frequency in a-Si/SiO 2 quantum well while photoluminescence lifetime is not sensitive to frequency in c-Si/SiO 2 structure.

By invoking the microscopic response method in conjunction with a reasonable set of approximations, we obtain new explicit expressions for the electrical conductivity and temperature coefficient of resistivity (TCR) in amorphous semiconductors, especially a-Si:H and a-Ge:H. The predicted TCR for n-doped a-Si:H and a-Ge:H is in agreement with experiments. The conductivity from the transitions from a localized state to an extended state (LE) is comparable to that from the transitions between two localized states (LL). This resolves a long-standing anomaly, a "kink" in the experimental log 10 σ-vs.-T −1 curve.