Quasianalytical simulation of ultrafast relaxation of photoexcited electrons in a semiconductor quantum well

We propose an analytic function approach to study the system of electron and phonon relaxation during an ultrafast process in GaAs-${\mathrm{Al}}_{\mathrm{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As heterojunction. Gauss-type energy functions are used to simulate the peaks of nonthermal electrons, while the background electrons are described by a Boltzmann distribution function. The time variations of parameters describing the amplitudes and widths of Gauss-type functions, and the temperature of background electrons, are determined by solution of solving the Boltzmann equation with electron-electron and electron-phonon interactions. Our choice of analytic distribution function made the calculation simple: only one integral over a two-dimensional momentum bomanq need be numerically handled. After carrier-carrier thermalization, our results automatically reduce to that of the electron-temperature model.