Quantum Transport Theory of Impurity-Scattering-Limited Mobility inn-Type Semiconductors Including Electron-Electron Scattering

The Kubo formula is employed to obtain an expression for ionized-impurity mobility, including higher-order impurity scattering effects and electron-electron scattering effects for non-degenerate semiconductors. Without electron-electron scattering, the lowest-order term of the mobility, as has been previously demonstrated, is found to be the same as the well-known Brooks-Herring formula for ionized-impurity mobility. The form of the higher-order impurity scattering terms has also been previously given. We have evaluated these terms and found them to be negligibly small for the range of impurity concentration and temperature considered here. Electron-electron scattering is calculated by a single-particle approximation from the time-independent Hartree-Fock theory, taking the electron-impurity scattering as the basic mechanism and treating the electron-electron interactions as perturbations. As impurity-scattered electron in our model is found to interact only with other independent electrons not interacting with impurities. Taking into account the correction of electron-electron scattering effects, the Brooks-Herring formula is found to be reduced by a factor which can be expressed in closed form as $1-e^{-1\mathrm{}}$. This factor of 0.632 agrees well with other theoretical calculations using the Boltzmann transport theory.