Theory of the Ettingshausen Effect in Semiconductors

The Ettingshausen effect in semiconductors is mainly due to the generation of electron-hole pairs at one side of the sample and their recombination at the other side. The Ettingshausen coefficient is calculated, in agreement with Putley, as $P=\left(\frac{E_g}{\kappa \mathrm{ec}}\right)z{\mathrm{}(1+z)\mathrm{}}^{-2\mathrm{}}({\mu }_e+{\mu }_h)$ where $z=\left(\frac{n_h{\mu }_h}{n_e{\mu }_e}\right)$-ratio of hole conductivity to electron conductivity. $E_g$ is the gap energy, and $\kappa$ the thermal conductivity. We discuss this formula for intrinsic, $p$-type and $n$-type semiconductors. $P$ goes through a maximum for $p$-type semiconductors near the temperature at which the Hall voltage goes through zero. Our results agree reasonably well with the measurements of Mette, Gärtner, and Loscoe of $P$ as a function of temperature for different samples of germanium and silicon.