Energy-loss rate of hot carriers in semiconductors with nonequilibrium phonon distribution in the extreme quantum limit at low temperatures

A theoretical model has been developed to investigate the effect of nonequilibrium phonons or hot phonons on the energy-loss rate of hot carriers in semiconductors in the extreme quantum limit at low temperatures. The acoustic-phonon scattering via the deformation potential and piezoelectric scattering are assumed to be the dominant scattering mechanisms at low temperatures. The model includes band nonparabolicity, energy nonequipartition of phonons, Landau-level broadening, and classical free-carrier screening. The energy-loss rates of hot electrons with nonequilibrium phonons as well as the thermal phonon distribution have been calculated using the above-mentioned model for n-type InSb. These theoretical results have also been compared with experimental results for n-type InSb at B=3 T and ${\mathit{T}}_{\mathit{L}}$=4.2 K. The energy-loss rate with the thermal phonon distribution is found to be much higher than the experimental result. The energy-loss rate of hot electrons calculated with use of the nonequilibrium phonon distribution with phonon boundary scattering is reduced compared with the values of the energy-loss rate obtained with use of the equilibrium phonon distribution. The incorporation of nonequilibrium phonons brings the theoretical results into agreement with the experimental data, giving a reasonable value for the phonon lifetime. However, the phonon lifetime required to fit the experimental data is found to be much higher than the values obtained from phonon boundary scattering. This discrepancy may be attributed to the acoustic-phonon mismatch factor.