Quasi bands in Green's-function defect models

Abstract

A basic difficulty in applying the Green's-function formalism to deep defects in solids is discussed. A cure is provided. It is shown that the conventional Green's-function formalism as applied to point-defect problems is derivable by requiring a dual representation of the defect wave functions both in terms of an expression in "local" basis functions and, independently, in terms of the eigenfunctions of the host-crystal Hamiltonian. It is then shown that this dual representation leads to a fundamental limitation of the method. In contrast to what may have been thought, the defect Green's-function (DGF) method does not become increasingly effective as the defect-induced potential perturbation becomes more localized in coordinate space. In fact, a consequence of this dual representation is that for impurities which are chemically mismatched to the host-crystal atoms a computationally intractable and physically undesirable enormous number of host-crystal eigenfunctions (${10}^{2}$ - ${10}^{4}$) is needed to obtain even modestly accurate defect wave functions, energies, and chemical trends. A new, formally exact approach (the "quasi-band-structure representation") is presented. This approach overcomes the difficulties underlying the dual representation in a simple way. It is based on redefining the zero-order basis set and expanding the defect wave functions in terms of such "quasi band wave functions" rather than by pure host wave functions. The former diagonalize the (finite) matrix of the host-crystal Hamiltonian and include aspects of both host and defect orbitals but need not form eigenstates to the host-crystal Hamiltonian operator. We illustrate this exact method for two analytically solvable models: a parabolic defect potential as well as a transition-atom impurity in a silicon free-electron host crystal. A comparison with the results of the conventional DGF calculation is given. Finally, the method is illustrated for a fully self-consistent calculation for substitutional Cu in silicon using nonlocal pseudopotentials.