Propagation and instability of microwaves in extrinsic InSb subject to crossed static electric and magnetic fields

Abstract

The propagation of an electromagnetic wave in a direction perpendicular to both the applied static magnetic field ${\stackrel{\ensuremath{\rightarrow}}{B}}_{0}$ and the static electric field ${\stackrel{\ensuremath{\rightarrow}}{E}}_{0}$ in extrinsic indium antimonide (InSb), and wave instability are studied theoretically. The dispersion equation $D(\ensuremath{\omega},k)$ relating the wave angular frequency $\ensuremath{\omega}$ and the wave number $k$ is derived from Maxwell's equations, the equation of momentum transfer, and the continuity equation, using the magnetohydrodynamic approach and a one-dimensional linearized theory. With the aid of the dispersion relationship, the propagation characteristics of the slow electromagnetic wave in a collision-dominated semiconductor plasma is examined in detail, for both $n$-type and $p$-type materials. The range of parameters considered are $1\ensuremath{\le}f\ensuremath{\le}9$ GHz, $1\ensuremath{\le}{B}_{0}\ensuremath{\le}10$ kG, and $0\ensuremath{\le}{E}_{0}\ensuremath{\le}30$ V/cm, and the variation of the phase velocity of the wave, ${v}_{\ensuremath{\phi}}$, and the amplitude constant of the wave, $\ensuremath{\alpha}\ensuremath{\equiv}\mathrm{Im}(k)$, with the parameters ${B}_{0}$, ${E}_{0}$, and the wave frequency $f$, are investigated. It is shown that under proper conditions wave amplification, defined by $\ensuremath{\beta}\ensuremath{\alpha}>0$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{k}=\ensuremath{\beta}+i\ensuremath{\alpha}$, and the wave instability, defined by ${\ensuremath{\omega}}_{r}{\ensuremath{\omega}}_{i}<0$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}={\ensuremath{\omega}}_{r}+i{\ensuremath{\omega}}_{i}$, is possible. For example, under the conditions ${(\frac{{\ensuremath{\omega}}_{R}}{\ensuremath{\omega}})}^{2}<1+{\ensuremath{\mu}}_{s}^{2}{B}_{0}^{2}$ and $|\frac{\ensuremath{\nabla}{n}_{0}}{{n}_{0}}|\ensuremath{\ll}|k|$, the threshold condition for the wave amplification, the threshold condition for wave instability, the spatial growth rate ($\ensuremath{\alpha}>0$), the phase velocity ${v}_{\ensuremath{\phi}}$, and the threshold oscillation angular frequency ${\ensuremath{\omega}}_{0}={\ensuremath{\omega}}_{r}$ (for which ${\ensuremath{\omega}}_{i}=0$) as functions of ${E}_{0}$, ${B}_{0}$, $f$, and ${n}_{0}$ are derived, where ${\ensuremath{\omega}}_{R}$ and ${n}_{0}$ denote the dielectric-relaxation angular frequency and the carrier density of the material, respectively. ${\ensuremath{\mu}}_{s}$ denotes the carrier drift mobility which takes a negative value for the electron. The effect of the carrier density gradient on ${v}_{\ensuremath{\phi}}$, $\ensuremath{\alpha}$, and ${E}_{\mathrm{th}}$ (the threshold electric field for instability) is also briefly discussed.