Effects of optical phonon interaction on dynamical valley polarization in graphene

Abstract

The present report is concerned with the dynamical behavior of [Formula: see text]-electronic valley states, under the interaction with transverse zone-boundary optical phonons, in graphene. It is assumed that the phonons are thermal and obey the Bose–Einstein distribution, while the [Formula: see text]-electrons are initially prepared in an experimentally realizable particular valley state. In our study, we take the view that such a mixture is completely described by a time-dependent density operator which is then determined, to the second-order of perturbation, from the governing Schrödinger equation. Employing the density operator so calculated, an analytical expression for the valley polarization, as a function of time, phonon frequency and temperature, is obtained. The results, accompanying with illustrative figures, reveal that the [Formula: see text]-electrons, through the elastic exchange of energy with phonons, change the valley states periodically with characteristics that strongly depend upon the temperature. It is in particular shown that as the temperature is raised, the time-averaged valley polarization approaches zero, as expected. Our calculations also show that the amplitude of valley oscillations is solely determined by the temperature and phonon frequency: an increase in the temperature enlarges the amplitudes in contrast to the phonon frequency which does the reverse. Along these lines, moreover, we demonstrate that the frequency of valley oscillations is determined by the electronic momentum deviation from the valley states, along with the phonon frequency.