High-electric-field behavior of the magnetophonon resonance in polar semiconductors.

Abstract

A theoretical study of magnetophonon resonance (MPR) in a high electric field is made on the basis of superoperator theory of high-field transport for an electron-phonon system. The system considered consists of nondegenerate electrons in a single parabolic band in a spatially uniform and stationary electric field interacting with longitudinal-optic phonons in polar semiconductors. Analytic calculations are carried out with carefully retaining terms of electric field throughout. The quantity \ensuremath{\Gamma}, probability of decay per unit time given as an imaginary part of the electron irreducible self-energy, is expressed as a function of electric field E. The transverse-electrical conductivity is calculated using the generalized electric-field-dependent conductivity tensor and the MPR condition is derived from its singularity. The cyclotron frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$ where the resonance occurs are obtained from the MPR condition as solutions of quadratic equations, and are expressed in terms of electric field and principal quantum-number difference between Landau levels. The behavior of MPR position with the increase of electric field is examined in detail by geometric considerations of the solutions given as a projective coordinate of intersections of electric-field-independent quadratic curves and a field-dependent straight line in a graphical method. By illustrating geometric configurations of these functions, electric-field behavior of the MPR position can be interpreted clearly. As a result, it is confirmed that with increasing E, the MPR position splits and proceeds in opposite directions from the ordinary (E=0) MPR position ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=${\mathrm{\ensuremath{\omega}}}_{0}$/j, where ${\mathrm{\ensuremath{\omega}}}_{0}$ is the limiting frequency of the optical phonon and j is a natural number. It is found that two MPR positions departed from any ordinary MPR position proceeding in opposite directions with increasing E should coincide with each other not only at the ordinary MPR position, but also at the intermediate position ${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=${\mathrm{\ensuremath{\omega}}}_{0}$/(j+1/2) between the neighboring ordinary MPR positions.