Corrections to the impurity-ion potential characterized by a spatially variable dielectric function; corresponding electron-conductivity mobility in silicon and germanium

Abstract

The complete linearized Poisson equation for the potential of impurity ions in a semiconductor, including the spatial variation of the dielectric function, is solved numerically by an equivalent variational principle, incorporating a functional recently proposed by Brownstein. The resulting potential is compared and contrasted with previous formulations which neglect a term involving the derivative of the dielectric function. In the region where this function becomes a constant, the chosen potential approximates that of Dingle, while at the impurity site it is greater than the latter by a multiplicative factor equal to the dielectric constant. Within the linear approximation, the present potential is believed to contain the correct embodiment of spatial variation in the dielectric function in a self-consistent manner. The electron-conductivity mobility, in the Born approximation, is also determined with this potential. Comparison with analogous calculations and with some typical experimental data as reference is made. It is found that the present theory of ionized-impurity-limited mobility yields a significant improvement over the Dingle theory and over prior treatments of this problem.