Nonlinear Phonon Interaction in Piezoelectric Semiconductors and Effect on Current Saturation

The nonlinear transport problem for the unstable phonon in piezoelectric semiconductors is described in a picture of electron-phonon interaction; the hydrodynamical approach is followed for the purpose of discussing the collision-frequent regime of electrons ($\mathrm{ql}<\mathrm{}1\mathrm{}$). With the aid of an iteration method for the nonlinear terms which describe the coupling between the electrons and the phonons, the kinetic equation for phonon distribution function $N_{\mathrm{q}}$ is derived; the equation includes a nonlinear collision term due to the three-phonon processes coming from the third order in electron-phonon interaction. The steady-state solution of this equation is discussed. It is found that for phonons of extremely low wave vector the three-phonon process can not effectively limit their growth; but in a restricted wave-vector region, the steady-state solution due to this process is obtained under a certain assumption. The acoustoelectric current is estimated with use of the phonon distribution function determined in this region.