Magnetic and Spin Effects in Small Polaron Hopping

Magneto-transport in the hopping regime is an attractive field of research. In particular, quantum-mechanical interference effects in hopping magnetotransport have attracted a great deal of attention.

Holstein [1] was the first to prove that the impact of a magnetic field on phonon-induced hopping transport of localized charge carriers is not due to the Lorentz force but originates from quantum interferences. He pointed out that a magnetic field-dependent contribution to the hopping probability between two sites arises from the interference between the amplitude for a direct transition between the initial and final states and the amplitude for an indirect transition involving intermediate occupancy of a third site. Accordingly, in the presence of a magnetic field H the familiar two-site model for studying hopping transport needs to be extended by including at least a third site. A three-site model allows hopping transitions along paths directed either clockwise or anticlockwise with respect to a magnetic field, with different transition probabilities, which deflects the movement of the charge carrier from the direction of an applied electric field E, therefore giving rise to a Hall effect.

A three-site model was adopted in [2] for studying the hopping Hall effect in polaronic materials with hexagonal structure. It is worth noticing that such a model is only the simplest model for taking into account the interference effects in question. In the case of crystalline materials a three-site model may be used only for crystals with hexagonal structure. In the more complicated case of a cubic crystal, for instance, one needs to consider the more complicated four-site model [3, 4].