Thermoelectric power of small polarons in magnetic semiconductors

The thermoelectric power (Seebeck coefficient) $\alpha$ of a small polaron in both ferromagnetic and antiferromagnetic semiconductors and insulators is calculated for the first time. In particular, we obtain the contribution to the Seebeck coefficient arising from exchange interactions between the severely localized carrier (i.e., small polaron) of charge $q$ and the spins of the host lattice. In essence, we study the heat transported along with a carrier. This heat, the Peltier heat, II, is related to the Seebeck coefficient by the Kelvin relation: $\Pi =qT\alpha$, where $T$ is the temperature. The heat per carrier is simply the product of the temperature and the change of the entropy of the system when a small polaron is added to it. The magnetic contribution to the Seebeck coefficient is therefore directly related to the change of the magnetic entropy of the system upon introduction of a charge carrier. We explicitly treat the intrasite and intersite exchange interactions between a small polaron and the spins of a spin-½ system. These magnetic interactions produce two competing contributions to the Seebeck coefficient. First, adding the carrier tends to provide extra spin freedom (e.g., spin up or spin down of the carrier). This effect augments the entropy of the system, thereby producing a positive contribution to the Peltier heat. Second, however, the additional exchange between the carrier and the sites about it enhances the exchange binding among these sites. This generally reduces the energetically allowable spin configurations. The concomitant reduction of the system's entropy provides a negative contribution to the Peltier heat. At the highest of temperatures, when $\mathrm{kT}$ exceeds the intrasite exchange energy, the first effect dominates. Then, the Peltier heat is simply augmented by $\mathrm{kT}$ ln2. Alternatively, at temperatures well below the magnetic transition temperature, the second effect dominates. The Peltier heat then garners a negative contribution. In the experimentally accessible range between these limits, both effects are comparable. There the magnetic contribution to the Seebeck coefficient is generally sizable, ~ 100 μV/K. Furthermore, this magnetic contribution to the Seebeck coefficient is distinguished from the usual nonmagnetic contribution by its temperature dependence; it rises with temperature. Thus, the exchange interactions between a small polaron and its magnetic environment produce a significant and distinctive contribution to the carrier's Seebeck coefficient.