PHYSICAL REVIEW B VOLUME 43, NUMBER 3 15 JANUARY 1991-II Nonlinear dielectric response to a point-donor impurity of an electron-gas-model semiconductor that includes the effect of the Dirac-Slater exchange correlation Leonard M. Scarfone and Ahsan Enver Department of Physics, University of Vermont, Burlington, Vermont 05405 (Received 9 July 1990) The Thomas-Fermi statistical theory, including the Dirac-Slater local-density treatment of ex- change correlation, has been applied to the problem of nonlinear screening of a donor point charge embedded in an electron-gas-model semiconductor. The nonlinear screening equation is solved nu- merically, giving spatial dielectric functions and screening radii with exchange-correlation strength and ion-charge state as parameters. Illustrations and tabulations of these results are given for five semiconductors, four ion charges, and two nonzero values of the exchange-correlation strength cor- responding to the Kohn-Sham and Slater exchange potentials. A variational principle equivalent of the nonlinear equation leads to approximate analytical expressions for the spatial dielectric func- tions which are in close agreement with the exact results. Variational parameters are given for a subset of the semiconductors and charge states. Dielectric functions of silicon are used to illustrate typical comparisons between the two methods of solution. I. INTRODUCTION This paper is concerned with nonlinear screening of an ionized point donor embedded in an electron-gas model semiconductor involving exchange and correlation in the Dirac-Slater Xcx approximation. Two familiar pro- cedures for developing the associated nonlinear equation for dielectric screening of the impurity have been out- lined in a previous paper. ' One of these is based on the original Thomas-Fermi statistical theory but modified to include exchange and correlation. This approach, due to Mott, starts from the uniform-electron-gas relation be- tween the electron density n and the Fermi momentum kF and applies it locally at r to the inhomogeneous sys- tem that results when the point charge is introduced into the semiconductor. A local Fermi energy EF(r) is ex- pressed in terms of kF(r) in the same way that the uniform-gas Fermi energy EI; is written in terms of kF. In the present instance, EF contains kinetic- and potential-energy terms proportional to kF and k~, respec- tively. It then remains to set up the classical equation for the fastest electron moving in the common screened po- tential V(r) with local kinetic and exchange-correlation energy EF(r). This equation is a statement of the con- stancy of the Fermi energy Ez (chemical potential at ab- solute zero) throughout the system, and includes the boundary condition, n(R)=n, on the screening charge density, where R is the finite radius of incomplete screen- ing. The basic nonlinear relation between n (r) and V(r) is given by this classical equation. Self-consistency re- quires that V(r) satisfy Poisson's equation with a charge distribution given in terms of the displacement of n (r) from the unperturbed electron density. This nonlinear differential equation is the fundamental Thomas-Fermi- Dirac screening equation in the Xa approximation. In the following, TF, TFD, and TFDS shall denote Thomas-Fermi (no exchange and correlation), Thomas- Fermi-Dirac (pure exchange), and Thomas-Fermi Dirac- Slater (exchange and correlation in the Xa approxima- tion), respectively. The nonlinear TFDS screening equation may also be obtained from an equivalent variational principle of the modified TF theory, as outlined by March. Minimiza- tion of the total ground-state energy of the system of valence electrons (plus any external potentials) with respect to n (r), and with a constraint on the total num- ber of electrons, leads to an Euler equation which expresses the constancy of the chemical potential and the boundary condition on the screening charge. The latter is incorporated in the theory by using a Lagrange multi- plier (related to the fixed number of electrons) of the form EF+ V(R). The Euler equation embodies the nonlinear relation between n (r) and V(r) as before. The spatial dielectric function F(r) is defined as the ra- tio of the bare Coulomb potential, — Z/r, to the screened potential, V(r), set up around the impurity point charge. Its properties include F(0) =1 and e(r) =e(0), where e(0) is the macroscopic dielectric constant for r ~ R. Lineariza- tion of the TFDS screening equation yields the simple analytical form of F(r) previously derived in the linear- ized TF context. In this approximation, exchange- correlation effects enter the potential V(r) through a pa- rameter, called q [see Eq. (12)j, which reduces to the usu- al TF screening length when exchange and correlation are absent. Reference 1 employed the linearized screen- ing function in a variational calculation of donor ioniza- tion energy with ion charge state Z and exchange- correlation strength a as parameters. It is of interest to extend this application and others to the nonlinear re- gime. The present paper makes a first step in that direc- tion by developing numerical and approximate analytical solutions of the nonlinear screening equation in the Xe scheme. The former is taken up in Sec. II for the poten- tials of donor point charges in diamond, silicon, germani- 43 2272 1991 The American Physical Society