Reaction-rate treatment of small-polaron hopping the role of vibrational relaxation

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Abstract
This paper is a sequel to an earlier work (Holstein 1978) in which small-polaron hopping between two sites was treated from the standpoint of a heuristic Eyringtype reaction-rate approach. In the present paper, the relationship of this approach to basic quantum perturbation theory is investigated, and conditions for its validity are established. An essential physical feature has to do with the relaxation of localized vibrational excitation (the ‘activated’ configuration). The relevant time constant associated with this relaxation is determined within the framework of the molecular crystal model (of previous small-polaron studies) by a method which focuses attention on the dynamics of an appropriately denned local ‘reaction’ coordinate, and its coupling to the vibrational degrees of freedom of the host crystal ('reservoir’ system). Agreement with previous calculations based upon an explicitly multiphonon approach is obtained. In particular, the reciprocal of the effective lime-duration of a single hopping event, l/tw , is shown to be proportional to the vibrational bandwidth, and otherwise increases with rising temperature and with increase of the electron—phonon coupling constant. As in the multiphonon approach, the condition for validity of the simple reaction-rate formulation is found to be l/ω0τw > 1 (where ω0 is the central vibrational frequency). An interesting by-product of the treatment, which may be of significance for solid state reactions in general, is the existence of a second relaxation time, which in the high-temperature limit exceeds tw by a factor ∼kT/hω0 .

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