Wigner-function formulation of nonlinear electron-hole transport in a quantum well and analysis of the linear transient and steady state

This work is concerned with the determination of nonlinear coupled quasi-two-dimensional transport equations for the electron and hole Wigner functions of a quantum well subject to a time-dependent electric field, based upon our earlier development of a generalized shielded-potential approximation for the corresponding nonequilibrium Green’s functions and the application of the generalized Kadanoff-Baym ansatz. In this work we take account of the electron-hole interaction, electron-phonon, and hole-phonon interactions, as well as treating impurity-scattering effects. Furthermore, the electron-electron and hole-hole Coulomb repulsions and associated dynamic, nonlocal screening effects are also incorporated. This formal development is applied to a linear analysis of electron-hole drag effects in a GaAs-${\mathrm{Al}}_{1\mathrm{-}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$As quantum well, treating both the transient and steady-state regimes. We examine the electron and hole mobilities in the ballistic regime, followed in time by their overshoot phenomena and ultimately saturation. At low temperature, our computations trace out in detail the time development of negative absolute minority electron mobility for this system. In the steady-state dc limit, we find that a phonon-mediated electron-hole interaction induces a small irregularity or oscillation in the knee of the electron-mobility curve as a function of temperature at about 60 K, which nicely parallels the data of Höpfel, Wolff, and Gossard.