Effect of temperature-dependent energy-level shifts on a semiconductor's Peltier heat

Abstract

The Peltier heat of a charge carrier in a semiconductor is calculated for the situation in which the electronic energy levels are temperature dependent. The temperature dependences of the electronic energy levels, generally observed optically, arise from their dependences on the vibrational energy of the lattice (e.g., as caused by thermal expansion). It has been suggested that these temperature dependences will typically have a major effect on the Peltier heat. The Peltier heat associated with a given energy level is a thermodynamic quantity; it is the product of the temperature and the change of the entropy of the system when a carrier is added in that level. As such, the energy levels cannot be treated as explicitly temperature dependent. The electron-lattice interaction causing the temperature dependence must be expressly considered. It is found that the carrier's interaction with the atomic vibrations lowers its electronic energy. However, the interaction of the carrier with the atomic vibrations also causes an infinitesimal lowering (approx.1/N) of each of the N vibrational frequencies. As a result, there is a finite carrier-induced increase in the average vibrational energy. Above the Debye temperature, this cancels the lowering of the carrier's electronic energy. Thus, the standard Peltier-heat formula,more » whose derivation generally ignores the temperature dependence of the electronic energy levels, is regained. This explains the apparent success of the standard formula in numerous analyses of electronic transport experiments.« less