Thermopower, figure of merit and Fermi integrals

The thermoelectric efficiency accounting for the conversion of thermal energy into electricity is usually given by the figure of merit which involves three transport coefficients, with the thermopower, the electrical and the thermal conductivities. These coefficients can be defined at a semi-classical level as a function of Fermi integrals which only allow analytical approximations in either highly degenerate or strongly non-degenerate regimes. Otherwise, the intermediate regime which is of interest in order to describe high thermoelectric performance requires numerical calculations. It is shown that these Fermi integrals can actually be calculated and that the transport coefficients can be reformulated accordingly. This allows for a new definition of the figure of merit which covers all the regimes of interest without numerical calculations. This formulation of the Fermi integrals also provides a good starting point in order to perform a power expansion leading to a new approximation relevant for the intermediate regime. It turns out that the transport coefficients can then be expanded by revealing their high temperatures asymptotic behaviors. These results shed new light on the thermoelectric properties of the materials and point out that the analysis of their high temperatures behaviors allow to characterize experimentally the energy dependence in the transport integrals.