An important characteristic of thermoelectric materials, i.e., the power factor (PF σS², σ—specific electrical conductivity, S—Seebeck coefficient), often has extreme values ​​depending on the absolute temperature, composition, concentration of charge carriers, Fermi energy, etc. In this work, PF–T dependence is considered. In some cases, it has a minimum in others, a maximum, and in some cases, both. For practical purposes, it is of interest to determine the maxima of this thermoelectric characteristic, which is included in the expression of the figure of merit, ZT.

A mathematical approach to the issue has been made. In particular, it turned out that despite the presence of several variable quantities in the formulas used, there was no need for the use of the Lagrange multiplier method; the usual calculation could be applied instead.

To compile formulas from the literature relating the absolute temperature, effective mass, and concentration of charge carriers, the following expression for the Seebeck coefficient can be obtained: Using f(S), we will get PF AT^3/2, where A= and S_r is the reduced Seebeck coefficient.

For SiGe samples under study, the empirical result is (PF)_max 1.42∙10⁸A_maxT^3/2.

The dependence (ZT)_max– B is then investigated (B=ZT/B_S,

B_S=[( e^2−Sr)/(1+e^−5(Sr−1))]+[3.29S/(1+e^5(Sr−1))] – scaled power factor, S_r=(q_e/k_B)│S│ – reduced Seebeck coefficient). It can be described by the expression (ZT)_max≅3.2B+0.05.

The dependence of (ZT)_max - B* (generalized parameter of material) is also investigated. A formula is used to calculate the values of B*, which contains the quantities of band gap (E_g), specific electrical conductivity, a scaled parameter, and thermal conductivity (k): B* 7.755·10^-4E_g k^-1. By calculating B and B* parameters, the maximum value of the figure of merit for almost anythermoelectric material can be approximately estimated.