Scattering-theoretic method for defects in semiconductors. II. Self-consistent formulation and application to the vacancy in silicon

Abstract

A self-consistent-field method for calculation of the electronic structure of localized defects in semiconductors is described. The method is based on Green's-function theory and follows the original idea of Koster and Slater and its developments by Callaway and coworkers. The Wannier functions of the original formulations are, however, replaced by a more flexible set of linear combination of atomic orbitals. This choice and an accurate evaluation of the perfect-crystal Green's function bring this method to the level of sophistication, accuracy, and rigor characteristic of state-of-the-art band-structure and surface calculations. The efficiency of the method stems largely from the fact that it exploits both the translational symmetry of the host crystal and the short range of the defect potential. Thus, all bulk properties (e.g., band gaps, bandwidths, etc.) are built in from the start via a band-structure calculation and are preserved. One then focuses on the changes produced by the defect potential, so that the interpretation of the results is straightforward and unambiguous. In this paper, we report an application of this method to an isolated vacancy in Si assuming no lattice relaxation. The unrelaxed vacancy introduces a bound state of ${T}_{2}$ symmetry at 0.7 eV above the valence-band edge and a number of resonances and antiresonances within the valence bands. A detailed analysis of these states in terms of their origin, orbital content, and of state and charge densities is presented. We find that, while many of these states are individually quite extended, they combine destructively to produce a very localized net change in the charge density. We also find that the resulting localized potential can be well approximated by a negative of an atomic silicon potential extracted from a self-consistent bulk calculation. Finally, we compare the relative merits of the three increasingly more sophisticated, but also more costly, approaches to the defect problem, namely, (1) tight-binding, (2) non-self-consistent, and (3) self-consistent calculations.