Diffusion of Excess Carriers in a Many-Valley Band Structure

By means of simple kinetic-theory considerations, an expression is derived for the diffusion constant $D^{\mathrm{}\left(i\right)\mathrm{}}$ of thermal carriers in the $i\mathrm{th}$ valley of a many-valley semiconductor. $D^{\mathrm{}\left(i\right)\mathrm{}}$ is shown to be a tensor, the $\alpha \beta$ component of which is $〈\tau v_{\alpha }{v}_{\beta }〉$, where $\tau$ is the relaxation time and $v_{\alpha }$ the $\alpha \mathrm{th}$ component of the velocity. The Einstein relation is shown to hold between $D^{\mathrm{}\left(i\right)\mathrm{}}$ and ${\mu }^{\mathrm{}\left(i\right)\mathrm{}}$, the mobility tensor for the $i\mathrm{th}$ valley. The anisotropy of $D^{\mathrm{}\left(i\right)\mathrm{}}$, which may be quite considerable, should produce observable effects if the intervalley scattering rate is low enough. When the latter condition is not satisfied, effects may still be found in situations where the equivalence of the valleys is destroyed.