Continued fractions in the kinetic theory of supersonic amplification in anisotropic semiconductors

Abstract

The method previously developed by the author (Fiz. Tekh. Poluprovodn.,9, 360, 1975;10, 808, 1976) is applied to evaluate the electron coefficient of supersonic absorption (or of amplification) with arbitrary ql (Γ_e) and of acoustic electric current (j^ae) in a single-pole “one-valley” semiconductor in the case of no external electric field E₀. The anisotropy of the acoustic-electronic interaction (the piezoelectrical one and the deformation potential) is taken into account as well as the anisotropy of scattering and of the effective mass. In contrast to the Zil'berman-Mishin results (Zh. Éksp. Tekh. Fiz.,65, 1473, 1973; Fiz. Tekh. Poluprovodn.,10, 1882, 1976) which were obtained for the isotropic case, compact expressions are obtained for Γ_e and j^ae in the form of an infinite continuous fraction of simple structure with all the spherical harmonics and all their relaxation times (τ_n) taken into account. The summation of the fraction in the Zil'berman-Mishkin approximation converts the latter into a finite continuous fraction yielding at once any order of this approximation which simplifies very considerably the analysis of its convergence. The taking into account of anisotropy does not really complicate the result and maintains its compact form. Another method of fraction summation enables one to estimate the contribution of individual harmonics.