Thomas-Fermi (TF-λW) and Thomas-Fermi-Dirac (TFD-λW) statistical models, including the Weizsacker gradient correction with variable coupling coefficient λ, have been applied to the problem of dielectric screening of an impurity donor ion in a homogeneous and isotropic semiconductor. Nonlinear differential Euler-Lagrange equations for the electron density are solved numerically and self-consistently with the impurity potential function under the screening charge constraint, giving microscopic response functions and screening radii with λ and ion-charge state as parameters. Illustrations of the numerical results are given for a monovalent donor point charge in silicon and germanium. It is evident that the TF-λW and TFD-λW screening radii are continuous, single-valued functions of λ with a minimum value in the range 0≤λ≤1. Also, it is found that TF (TFD) and TF-1/9W (TFD-1/9W) spatial dielectric functions are in close agreement, the latter, however, having a smaller screening radius. The gradient correction reduces the nonlinear TF (TFD) screening radius by about 7.5% (9.8%) and 8.4% (12%), respectively, in silicon and germanium. Thus, the theoretical value, 1/9, of λ leads to better results than the original value, 1, of Weizsacker which is associated with less effectively screened impurity potentials in both theories. The TF-1/9W and TFD-1/9W screening radii differ by about 20% for both semiconductors. It is expected that the nonlinear TFD-1/9W screening functions will further improve the nonlinear TFD donor binding energies for silicon and germanium, which are already in quite good agreement with experiment. The TFD-λW theory cannot be regarded as conclusive, since higher-order gradient corrections to the TF kinetic energy and gradient corrections to the Dirac exchange functional and the correlations functional, not to mention self-interaction effects, have not been taken into account here.

• Received 4 November 1991

DOI:https://doi.org/10.1103/PhysRevB.45.8348