Screened donor impurities in many-valley semiconductors with anisotropic masses

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Abstract
The binding energy of a conduction electron bound to a donor impurity in a doped semiconductor, using a Lindhard dielectric function, is calculated as a function of the free-carrier concentration, taking into full account the effects of the nonisotropic mass of the bound electron on both the kinetic energy and the screened Coulomb potential. A variational approach is adopted in which the trial wave function is chosen with a form similar to the solution of the problem of an electron bound in the Hulthén potential. The wave function is appropriately warped to account for the asymmetry caused by the nonisotropic masses. This calculation, in the limit of zero carrier concentration, is equivalent to work of earlier authors (Kohn and Luttinger and others) where a hydrogenic trial wave function was used. The method thus obtained is specifically applied to silicon and germanium and the results are compared to calculations done in the isotropic-mass approximation. The electron density where the Mott transition takes place is found to be lowered by introducing the nonisotropy. For germanium where the mass ratio is approximately 20, this effect is quite large. A similar calculation using the simpler Thomas-Fermi dielectric function is included for completeness.