Thomas-Fermi-Dirac statistical theory of dispersive dielectric screening in undoped semiconductors at zero temperature

Abstract

Static dielectric screening in undoped semiconductors at zero temperature is formulated within the framework of the Thomas-Fermi-Dirac (TFD) model of a homogeneous and isotropic solid. At each point in the solid the valence electrons are treated as a degenerate gas in statistical equilibrium in the space-varying self-consistent potential of a point-charge impurity. The theory involves the electrostatic, kinetic, and exchange energies of the electrons in the development of a nonlinear TFD equation for the screened potential. The Thomas-Fermi (TF) theory of dielectric screening is recovered when exchange effects are neglected. Closed analytical expressions for the wave-vector-dependent dielectric function and the spatial dielectric function are obtained by linearization of the TFD equation and the range of validity of approximation investigated. Numerical solutions of the nonlinear TFD equation for point-charge screening show an increasing departure from linear behavior with impurity charge. These properties of the nonlinear TFD theory are already manifest in the TF scheme. A comparison between TFD- and TF-model dielectric functions shows important differences due to exchange. In the linear screening regime, it is found that impurity potentials are more effectively reduced when exchange effects are included. As a result, the TF theory compares more favorably with accurate band-structure calculations of the dielectric functions for silicon and germanium. It is expected that improvement in the TFD dielectric functions depends on extending the treatment to include correlation and/or the quantum correction. In the nonlinear regime, attractive potentials are more effectively screened in the TFD theory, while the opposite is not generally true for repulsive potentials. Finally, it is seen that donor-acceptor asymmetry is stronger in the presence of exchange effects.