Theory of Thermoelectric Power in Semiconductors with Applications to Germanium

The thermoelectric power $Q$ of a semiconductor is found by calculating the Thomson coefficient ${\sigma }_T$ from electrical and thermal current density expressions and then integrating the relation ${\sigma }_T=\frac{T\mathrm{dQ}}{\mathrm{dT}}$. This procedure yields a general expression for $Q$ in terms of the Fermi level, forbidden band width, temperature, ratio of electron to hole mobility, and effective electron and hole masses. In the impurity range the general formula for $Q$ reduces to a simple dependence on the Hall coefficient and temperature if carrier scattering is largely due to the lattice of the semiconductor; the same expression may be used with the addition of a correction term when carrier scattering by impurity ions becomes important at the lower temperatures. When both holes and electrons must be considered as carriers, $Q$ can be evaluated at any temperature from the resistivity and Hall coefficient at that temperature. An expression is also obtained for the thermoelectric power of an intrinsic semiconductor in a form depending on the mobility ratio, forbidden band width at 0°K, and the temperature rate of change of this band width. Hall and resistivity data measured for six polycrystalline germanium samples and two silicon samples have been inserted into the theoretical expressions derived in this paper. The thermoelectric power curves so calculated are found to give generally good agreement with the measured curves.