Symmetry Properties of Warm Electron Effects in Cubic Semiconductors

Abstract

The constant $\ensuremath{\beta}$ which describes the field dependence of the mobility $\ensuremath{\mu}$ of warm electrons according to $\ensuremath{\mu}={\ensuremath{\mu}}_{0}(1+\ensuremath{\beta}{E}^{2})$ 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 ${E}^{2}(\frac{{\ensuremath{\gamma}}_{0}}{2})sin\ensuremath{\phi}cos\ensuremath{\phi}\ifmmode\times\else\texttimes\fi{}(1\ensuremath{-}3 {cos}^{2}\ensuremath{\phi})$, where $\ensuremath{\phi}$ is the angle between the field situated in the (110) plane and the cubic axis, and ${\ensuremath{\gamma}}_{0}$ is a constant. For silicon and germanium one finds the following symmetry relations: ${\ensuremath{\beta}}_{110}={\ensuremath{\beta}}_{100}\ensuremath{-}(\frac{{\ensuremath{\gamma}}_{0}}{2})$, ${\ensuremath{\beta}}_{111}={\ensuremath{\beta}}_{100}\ensuremath{-}(\frac{2{\ensuremath{\gamma}}_{0}}{3})$, where the subscripts denote the direction of field strength. The theoretical results have been confirmed experimentally with $n$-type germanium of different impurity content.