Symmetry Properties of Warm Electron Effects in Cubic Semiconductors

The constant $\beta$ which describes the field dependence of the mobility $\mu$ of warm electrons according to $\mu ={\mu }_0(1+\beta E^2)$ is found to be anisotropic even for crystals of cubic symmetry. Furthermore, the directions of field strength and current density include an angle the tangent of which is given by $E^2\left(\frac{{\gamma }_0}{2}\right)sin\varphi cos\varphi \times (1-3\mathrm{} {cos}^2\varphi )$, where $\varphi$ is the angle between the field situated in the (110) plane and the cubic axis, and ${\gamma }_0$ is a constant. For silicon and germanium one finds the following symmetry relations: ${\beta }_{110}={\beta }_{100}-\left(\frac{{\gamma }_0}{2}\right)$, ${\beta }_{111}={\beta }_{100}-\left(\frac{2{\gamma }_0}{3}\right)$, where the subscripts denote the direction of field strength. The theoretical results have been confirmed experimentally with $n$-type germanium of different impurity content.