Nonlinear balance equations for hot-electron transport with finite phonon-relaxation time

The balance-equation approach developed by Lei and Ting for steady-state hot-electron transport is extended here to include nonequilibrium phonon occupation under the assumption that the effect of phonon-phonon interaction can be represented by an effective phonon relaxation time ${\tau }_p$, which may be mode dependent. Both single heterojunctions and multilayer quantum-well superlattices, as well as three-dimensional (3D) bulk systems, are discussed. A 3D phonon model and a quasi-2D phonon model are employed in describing the various interacting electron-phonon systems. The expressions for the frictional forces and energy transfer rates obtained in steady state are structurally similar to those without hot-phonon effects and the balance equations with finite phonon relaxation can be solved with the same computational effort as in the case of ${\tau }_p$=0. We have examined hot-phonon effects on the energy-transfer rate, the electron temperature, and the linear and nonlinear mobilities. It is shown that finite phonon relaxation generally decreases the phonon-induced Ohmic resistivity at a given lattice temperature. However, it significantly increases the electron temperature so that the nonlinear resistivity of the system is enhanced at large drift velocities. For an n-type GaAs heterosystem the normalized electron-energy-loss rate at ${\tau }_p$=3.5 ps is seen to be a factor of 5–10 smaller than that at ${\tau }_p$=0. The functional dependence of the energy-loss rate on carrier temperature shows considerable difference from the prediction of a carrier temperature-type theory, and is in reasonably good agreement with experiments.