The hall mobility of a small polaron in a square lattice

Abstract

The two-dimensional molecular crystal model of Friedman and Holstein describing the motion of a small polaron in the presence of a magnetic field is utilized as the basis of a study of the small polaron Hall mobility in a square lattice. This work is concerned solely with the high temperature regime where the polaron motion is predominately by means of thermally activated jumps between lattice sites. In addition, the calculation employs the Holstein classical occurrence-probability approach (valid at sufficiently high temperature) which considers the vibrational coordinates to be explicit functions of time. The magnetic field manifests itself solely through the interference of the transition amplitudes associated with the carrier moving between two sites via different paths. Hence in the square lattice (ignoring direct diagonal hops) a minimum of four site are involved in processes associated with the Hall effect. Furthermore, the contributions to the Hall mobility arise from processes in which the initial, final and one intermediate site experience a momentary equality of the electronic energy levels associated with the carrier occupation of these sites, i.e., a triple coincidence event. The distortion of the fourth (noncoincident) site at the time of the triple coincidence event is shown to play an important role in the calculation. It is found that (for reasonable choices of the physical parameters) the temperature dependence of the Hall mobility in a square lattice differs considerably from that found by Friedman and Holstein for the hexagonal lattice structure. In particular, the Hall mobility is shown to be a slowly varying function of temperature which decreases with increasing temperature at temperatures above 2ϵ 2 9k , ϵ 2 being the drift mobility activation energy. The associated Hall coefficient is greater than its “usual” value ( R = −1 nec ) and varies with temperature as exp[ 2ϵ 2 3kT ] .