Nonequilibrium electron-phonon scattering in semiconductor heterojunctions

We calculate the energy-loss rate of a quasi-two-dimensional (2D) hot-electron gas at a semiconductor heterojunction due to the emission and reabsorption of nonequilibrium optical phonons. A kinetic equation for the nonequilibrium lattice excitations (hot phonons) in the spatially inhomogeneous field created by the quasi-2D electrons is obtained. The equation is solved by a transformation that introduces the occupation number for wave packets of phonons that are ``localized'' in a spatial region near the electron layer. Our result does not contain the spurious dependence on the size of the sample that results when the nonequilibrium phonons are represented in terms of decoupled 3D plane waves. We apply our results to electrons in a GaAs-Gaₓ{Al}_{1{{-}}x}$As heterojunction. We find that the reabsorption of the emitted hot phonons reduces considerably the electron cooling rate if the optical-phonon lifetime ${{τ}}_{op{≥}}5 psec. Furthermore, the electron-energy relaxation rate, 1/τ, changes from a weakly decreasing function of the electron temperature,{T}ₑ$, when ${{τ}}ₒₚ$=0 to a increasing function of ${T}ₑ$, as ${{τ}}ₒₚ$ increases. This result compares favorably with recent experimental data.