Brillouin instability inn-type piezoelectric semiconductors

Abstract

The paper aims at the detailed analytical investigation of Brillouin instability in a magnetoactive $n$-type cubic piezoelectric semiconducting crystal belonging to class $\overline{4}3m$ under a geometrical configuration which can also be employed in analyzing the phenomenon under either Voigt or Faraday orientation. The electric vector ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}}_{0}$ of the spatially uniform pump electromagnetic wave (applied along the $y$ axis) is normal to the magnetostatic field ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}}_{0}$ (along the $z$ axis) as well as to the plane of propagation $x\ensuremath{-}z$ plane) of the internally generated low-frequency transverse-acoustic wave ($\ensuremath{\omega},\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$) and the scattered electromagnetic wave (${\ensuremath{\omega}}_{1},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$). The propagation vectors $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ (antiparallel to each other) are in the $x\ensuremath{-}z$ plane making an angle $\ensuremath{\theta}$ with the $x$ axis. The dispersion relation has been obtained by using a hydrodynamic model of the homogeneous, piezoelectric, one-component (electron) semiconductor plasma, and the critical value of the pump electric field (necessary to achieve physically reasonable growth of an unstable wave) and the growth rate of the unstable Brillouin mode well above the critical field have been obtained for isotropic (${B}_{0}=0$) and magnetoactive (${B}_{0}\ensuremath{\ne}0$) plasmas. We have applied our analysis to a specific semiconductor, $n$-InSb at 77 K duly irradiated by a pulsed 10.6-\ensuremath{\mu}m C${\mathrm{O}}_{2}$ laser for numerical estimation. Qualitative agreement between the analytical results and the numerical analysis has been noticed. The laser wave intensities used here are in the range of ${10}^{9}$ to ${10}^{12}$ W ${\mathrm{m}}^{\ensuremath{-}2}$ which is assumed to be less than the damage threshold of the InSb crystal. The phase velocity of the growing unstable mode is found to be constant over the whole range of system parameters and equal to the acoustic velocity in the crystal. The magnitude of the critical field decreases with increasing magnetostatic field and decreasing $\ensuremath{\theta}$. The growth rate increases and attains a maximum value at a certain value of the pump intensity, magnetostatic field, and $\ensuremath{\theta}$, and if these are raised further, growth rate starts decreasing. The magnitude of the growth rate is found to be \ensuremath{\sim} ${10}^{8}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ at laser intensities of the order of ${10}^{10}$ W ${\mathrm{m}}^{\ensuremath{-}2}$. When the analysis is extended to Voigt and Faraday configurations, the results are not very encouraging.