Longitudinal-Dielectric-Permittivity Criterion for Current Instability in a Polar Crystal

Abstract

Electrodynanic phenomena in a spatially dispersive medium are governed by the dielectric permittivity tensor. It appears that this tensor is useful in describing small departures from a steady state as well as small departures from equilibrium. The existence of a simple relationship between the longitudinal parts of the permittivity tensor for certain systems in equilibrium and the same systems in a steady state is noted and then applied in a calculation of the longitudinal dielectric permittivity ${\ensuremath{\epsilon}}^{L}(\mathrm{q}, \ensuremath{\omega}; {\mathrm{v}}_{D})$ of a polar crystal in which there is a stream of degenerate carriers with drift velocity ${\mathrm{v}}_{D}$. Vanishing of ${\ensuremath{\epsilon}}^{L}(\mathrm{q}, \ensuremath{\omega}; {\mathrm{v}}_{D})$ is associated with longitudinal electric waves in the system. Damping of these waves is governed by $\mathrm{Im} {\ensuremath{\epsilon}}^{L}(\mathrm{q}, \ensuremath{\omega}; {\mathrm{v}}_{D})$. It is argued that in some circumstances $\mathrm{Im} {\ensuremath{\epsilon}}^{L}(\mathrm{q}, \ensuremath{\omega}; {\mathrm{v}}_{D})$ can be made to vanish by adjusting ${\mathrm{v}}_{D}$. This general argument is applied to the case of carriers drifting in GaAs, and it is found that the longitudinal electric waves (plasmons coupled to longitudinal optical phonons) become unstable for values of ${v}_{D}$ not much larger than the value reported by Gunn as the threshold for the onset of current oscillations in GaAs. (It is assumed that $\ensuremath{\omega}{\ensuremath{\tau}}_{\mathrm{el}}\ensuremath{\gg}1$, where ${\ensuremath{\tau}}_{\mathrm{el}}$ is the relaxation time for the distribution of electrons.) The significance of this result is discussed.