Quasiband crystal-field method for calculating the electronic structure of localized defects in solids

Abstract

We introduce an ab initio self-consistent approach—the quasiband crystal-field (QBCF) method—to calculate the electronic structure of localized defect states in solids within a density-functional Green's-function approach. The method is simple, yet it produces very accurate self-consistent solutions both for $s-p$ as well as for the hyperlocalized transitionatom $d$-electron impurities. This is made possible by four ideas: (1) Following the pioneering work (1929) of Bethe and Van Vleck and results of modern computations, it is recognized that whereas the defect and host wave functions may be extended and highly anisotropic in coordinate space, for deep defects the density and potential perturbations $\Delta \rho \left(\overrightarrow{\mathrm{r}}\right)$ and $\Delta V(\overrightarrow{\mathrm{r}}$) are considerably more localized and have a reduced directional anisotropy. We therefore describe the latter in a crystal-field one-center expansion $\Sigma l^{}{F}_l\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{r}}\mathrm{}\left|\right)\mathrm{}{K}_l\left(\hat{r}\right)$, anchored at the defect site, with a separation of radial $F_l\mathrm{}\left(\right|\mathrm{}\overrightarrow{\mathrm{r}}\mathrm{}\left|\right)\mathrm{}$ and angular $K_l\left(\hat{r}\right)$ variables. The defect problem, treated by contemporary Green's-function techniques as a multicenter scattering problem, is then transformed into a far simpler atomic-like problem, characterized by analytic angular integrals (Gaunt coefficients) and simple one-dimensional radial integrals. This permits a simple and highly precise treatment of self-consistency, incorporation of accurate (first-principles) nonlocal pseudopotentials, and the use of variationally flexible and computationally simple single-site basis functions introduced in chemistry in 1933 by Mulliken. (2) The standard Koster-Slater Green's-function approach to defects uses an expansion of the impurity wave functions in terms of (often chemically and physically unrelated) host-crystal Bloch eigenfunctions. It is shown that when the perturbation approaches a characteristic atomic length scale (or when the impurity is chemically sufficiently different from the host atom), such expansions converge exceedingly slowly. We have reformulated the Koster-Slater resolvent problem in terms of quasiband wave functions that incorporate from the outset not only aspects of the host, but also the characteristics of the defect. A large number of conduction-band wave functions, which would have been needed for an adequate representation of localized defect states, are renormalized into a much smaller number of quasiband wave functions. Expansion in terms of quasibands results in a rapidly convergent and efficient description even of very localized defect wave functions. (3) A new Newton-Raphson Jacobian update technique is used to establish self-consistency in the screening potential. It does not require any new information; it "remembers" information from all past iterations, but automatically discounts information from the distant past and is hence not confused by nonlinearities. The method is far more efficient than all standard self-consistency methods and permits a precise assessment of charge-redistribution effects in the system. (4) The lengthy summations over the Brillouin zone encountered in spectral Green's-function methods are transformed into a simpler rapidly convergent series in a supercell representation. This allows one to treat impurities in large supercells (e.g., 2662 atoms per cell) by treating only small matrices (36×36) whose sizes do not depend on the dimensions of the supercell. In this representation the poles of the Green's function are real and can be efficiently located using a new and fast algorithm introduced here. This paper describes these four ideas in physical terms. Full mathematical details are given in a series of Appendices that can be used by the reader to independently reproduce the method. The method is applied to study the electronic structure of the unrelaxed silicon vacancy as well as to the far more difficult problem of a substitutional transition-metal impurity in silicon. The electronic structure of Cu in an extended crystal is described for the first time.