Weizsacker correction in the Thomas-Fermi and Thomas-Fermi-Dirac models of static dielectric screening in undoped semiconductors: Impurity donor ions in silicon and germanium.

Abstract

Thomas-Fermi (TF-\ensuremath{\lambda}W) and Thomas-Fermi-Dirac (TFD-\ensuremath{\lambda}W) statistical models, including the Weizsacker gradient correction with variable coupling coefficient \ensuremath{\lambda}, 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 \ensuremath{\lambda} 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-\ensuremath{\lambda}W and TFD-\ensuremath{\lambda}W screening radii are continuous, single-valued functions of \ensuremath{\lambda} with a minimum value in the range 0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}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 \ensuremath{\lambda} 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-\ensuremath{\lambda}W 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.