Electron mobilities based on an exact numerical analysis of the dielectric-function-dependent linearized Poisson's equation for the potential of impurity ions in semiconductors

The complete linearized Poisson's equation for the impurity-ion potential incorporating the spatial variation of the semiconductor dielectric function is treated numerically as a boundary-value problem with the finite-difference method. The impurity-ion potential so derived is shown by direct substitution to be a superior solution to the differential equation than that obtained by an equivalent variational-principle approach, or by the straightforward replacement of the dielectric constant by the dielectric function of the Dingle potential. Theoretical electron mobilities for silicon, germanium, and gallium arsenide based on this potential are compared with the Dingle mobility. It is found that the result of including the spatial variation of the dielectric function in the theory of ionized-impurity scattering is to reduce the electron mobility from that calculated using the Dingle potential. This effect is small, except in the cases of heavy and very heavy doping, where it is found that the difference increases monotonically as the doping density is likewise increased.