Studies of polaron motion: Part III: The Hall mobility of the small polaron

Abstract

The one-dimensional molecular crystal model of polaron motion, developed in parts I and II, is suitably generalized to consider the existence of a Hall effect. As in II, the treatment is confined to the case for which the electronic-overlap term of the total Hamiltonian is a small perturbation. In zeroth order—i.e., in the absence of this term—the electron is localized at a given site. The vibrational state of the system is specified by a set of quantum numbers, N k , giving the degree of excitation of each vibrational mode. The existence of a nonvanishing electronic bandwidth then gives rise to transitions to neighboring sites. Of principal interest in the present paper is the high temperature regime ( T > T t, as defined in II) where polaron motion is predominantly by means of random jumps between neighboring sites. Although the lowest order jump rate is adequate in considering the polaron drift mobility, higher order processes, involving the occupation of (at least) three sites, must be taken into account in treating the Hall effect. In particular, it is demonstrated the relative probability of the electron, initially located on one of the three sites, hopping to one or the other of the remaining two sites, is modified by a contribution which, both in sign and magnitude, is linearly proportional to the applied magnetic field. This effect is shown to arise from the interference between the amplitude for the direct transition from the initial to the final site, and the amplitude for an indirect, second order transition, involving intermediate occupancy of the third site. The (magnetic) field induced components of the jump rates, corresponding to the above processes, are first calculated by a classical occurrence-probability approach which treats the lattice vibrational coordinates as given functions of time. The second approach presents a full quantum mechanical calculation of the jump rates. The results of this treatment agree with those of the occurrence-probability approach in the classical limit ( T ⪢ Θ De bye). The order of magnitude of the calculated Hall Coefficients are found to be greater than or comparable to the “normal” result ( R = −1 nec ) depending on whether or not the three sites involved in the transition are mutually nearest neighbors. A final note corrects an error in a previous paper by one of the authors (T. H.) concerning the sign of the Hall effect in impurity conduction.