Spherical harmonics solver for coupled hot-electron — Hot-phonon system

In polar semiconductors with intensive electron-optical-phonon interaction excess energy of non-equilibrium (hot) electrons, supplied by the external electric field, is transferred to longitudinal optical (LO) phonons. LO phonons are displaced from equilibrium with the thermal bath, and hot-phonon effects take place. The kinetics of such a non-equilibrium electron/optical-phonon system is described by two coupled nonlinear Boltzmann-type equations for the non-equilibrium electron and LO phonon distribution functions. Here, to our knowledge for the first time, we present a spherical harmonics expansion (SHE) method for the coupled system for electrons and LO phonons. The nonlinear system of coupled equations is solved with a Gummel type iterative scheme and the Newton-Raphson method. Simulation results are verified against a Monte Carlo model and good agreement is obtained. Hot phonon effects increase the mean energy of electrons and decrease the drift velocity.