Electronic Structure of Point Defects and Impurities in Metals

The scattering of Bloch waves off point defects and impurities in metals is considered, for the following four types of defect potentials: (i) weak scatterers, (ii) strong scatterers, but with slow spatial variations in the solute potentials, (iii) strongly attractive potentials which lead to bound or resonant states, (iv) potentials leading to almost rigid-band behavior. In essence, we develop approximate wave functions or density matrices which should be valuable zero-order solutions in these four regimes. Iteration in the basic integral equation given for the Dirac density matrix can then be employed to refine these solutions when necessary. Thus, in case (1), we determine the linear response of the Bloch-wave system to a weak perturbation in terms of the Dirac density matrix for the periodic-potential problem. The asymptotic form of the displaced charge is discussed in some detail. For the case of a spherical Fermi surface, the Bloch-wave character does not alter the form of the long-range oscillations, though it changes the amplitude and the phase. In (ii), it is shown that the appropriate tool is a generalization of the Thomas-Fermi approximation by the introduction of an energy-dependent potential, into which we build some of the essential wave properties of the problem. By explicit calculation for repulsive potentials, we show that accurate numerical results for the energy-level shifts due to vacancies can be obtained in this way. We then deal, in (iii), with a theory motivated by the Koster-Slater treatment of defects. In this theory, we show that, usually, only the properties at the Fermi level are derivable from a local potential. This potential in the Koster-Slater model is like a square well near the defect, with Wannier-function oscillations in the tail. Systematic improvement of the Koster-Slater theory can be made via an integral equation for the Green's function. A new condition, generalizing the Koster-Slater theory to deal with solute potentials with matrix elements between Bloch functions which are energy-dependent, is proposed. It is pointed out that within this framework, conditions may arise where a rigid-band model could be regained, even for strong potentials.