The polylogarithm and the Lambert W functions in thermoelectrics

Abstract

In this work, we determine the conditions for the extremum of the figure of merit, θ2, in a degenerate semiconductor for thermoelectric (TE) applications. We study the variation of the function θ2 with respect to the reduced chemical potential μ* using relations involving polylogarithms of both integral and nonintegral orders. We present the relevant equations for the thermopower, thermal, and electrical conductivities that result in optimizing θ2 and obtaining the extremum equations. We discuss the different cases that arise for various values of r, which depends on the type of carrier scattering mechanism present in the semiconductor. We also present the important extremum conditions for θ2 obtained by extremizing the TE power factor and the thermal conductivity separately. In this case, simple functional equations, which lead to solutions in terms of the Lambert W function, result. We also present some solutions for the zeros of the polylogarithms. Our analysis allows for the possibility of considering the reduced chemical potential and the index r of the polylogarithm as complex variables.