Effect of electrons and phonons temperature mismatch on thermoelectric properties of semiconductors

A tendency of micro-miniaturisation of thermoelectric generators, coolers and sensors demands an increase of temperature gradients for preserving their power. It is impossible to consider that temperatures of electrons and phonons are equal in these conditions, so an electrons and phonons temperature mismatch begins to realise. A system of nonlinear equations for an energy balance of electrons and phonons and a continuity equation for current density is solved to study this mismatch. It is taken into account an electron-phonon energy interaction and a dependence of all kinetic coefficients on electrons and phonons temperatures. It is shown that the three components of the mismatch are exist under large temperature gradients: the first component is proportional to a temperature gradient squared; the second component is proportional to a product of a temperature gradient and an electric field; and the third to an electric field squared. On the volume of a sample the all components do not depend on coordinates and do not also depend on a type of heat boundary conditions for electrons and phonons. The temperature mismatch is leading to a change of formulae for electric current density and heat flow density, for all kinetic coefficients and for the thermoelectric figure of merit. The figure of merit increase in consequence of the mismatch. This increase is starting to be shown up in thermoelectrics at temperatures gradients ~10⁵ K/cm