Nonlinear theory of longitudinal oscillations of current plasma in a semiconductor with superlattice

Abstract

It is shown that the behavior of the amplitude of the electric field has a band character, and is determined by its initial value and by the number of the resonance. i. A nonquadratic structure of the dispersion law for electrons in a semiconductor with superlattice leads to a considerable nonlinearity of the condution current, which manifests itself even in relatively weak electric fields. When the superlattice is acted upon simultaneously by electric field with frequency ~ and by a constant electric field Eo, the conduction current behaves in a resonant fashion [1-3]. A consequence of the nonparabolic structure of the energy spectrum of the superlattice is the possibility of resonance at the multiple frequencies nw = ~, where ~ = eEod (d is the period of the superlattice, and h = i), and n is an integer. For the occurrence of nonlinear resonances with n > i, the amplitude of the field must not be too small, so that the conduction current is principally nonlinear. It is known from the linear theory of the Stark resonance with n = 1 [4] that, for ~, the wave is unstable. It is clear that in a small neighborhood of the nonlinear resonance~n~, the corresponding term in the expression for the conduction current determines negative absorption, i.e., amplification of the wave. However, the contribution of the total absorption comes also from a large number of nonresonant terms. Their finite number with n no (in principle, an infinite number) describe positive absorption. It is clear that a steady-state wave field is established as a result of the competition between these groups of terms. In the present work, we take into account consistently the resonant and nonresonant terms in the expression for the conduction current, in the case of small deviation from the Stark resonance. The obtained results are used to find a condition for growth of the longitudinal oscillations, and to determine the steady-state values of the field. 2. It is assumed that electrons exist only in the lower miniband, and the electric fields do not cause transitions to the upper minibands. We consider homogeneous oscillations of the electric field. We assume that the constant (Eo) and variable (E) electric fields are directed along the axis of the superlattice. The starting equation is the nonlinear equation