Upper value of thermoelectric figure of merit for isotropic semiconductors

G. Logvinov, Y. Gurevich

Any thermoelectric converter is the thermodynamic heat engine. By this reason, their highest possible efficiency is the Carnot efficiency. Reduction the real efficiency in comparison with the ideal one is caused by the irreversible processes in the quasiparticle subsystems of semiconductor. First of all, it is concerned the thermal flux through the phonon subsystem that does not take part in the thermoelectric energy conversation.

The efficiency of thermoelectric conversation of any semiconductor is determined by the thermoelectric figure of merit **Z **that is depended on the Seebeck coefficient, the bulk and surface electric conductivity, the bulk and surface electron and phonon heat conductivity, and the sample’s size.

Under the quasielastic scattering (for example, electron scattering on the acoustic phonons) the correct description of the heat transport in the electron and phonon subsystem is possible with the help of two different temperatures, namely by the electron and phonon temperature. At that, the characteristic thermal diffusion length (cooling length) is appeared which is measured in submicron scale.

If the sample’s length is less than the cooling length, the electron and phonon temperatures become independent, and are determined by own heat boundary conditions. The perfect isothermal conditions for electrons and the perfect adiabatic conditions for phonons create the ideal situation for the highest **Z. **In this case the phonon temperature is constant and the phonon thermal flux in semiconductor is absent. As a result, the phonon thermal conductivity drops out from the thermoelectric figure of merit. In accordance with these considerations the equation for the upper **Z** is obtained. It has the universal form, and depend on the chemical potential, temperature, and the type of the scattering.