Critique of the tight-binding method: Ideal vacancy and surface states

Publisher Website
Google Scholar
Abstract
We have investigated the dependence of the calculated energy eigenvalues for gap states, introduced by ideal vacancies and surfaces, on the choice of localized basis functions in the tight-binding method. In this method the Hamiltonian describing the system with an ideal vacancy is defined by removing all the Hamiltonian matrix elements between orbitals localized about the central atom with all basis states describing the system; the remaining atoms are assumed to have their positions unaltered, the atomiclike orbitals are retained on these atoms, and their Hamiltonian matrix elements are assumed unaltered. We find, using a Green's-function analysis, that if Wannier functions are employed as a basis there are no ideal vacancy-gap states. In addition, if the atomic orbitals of the isolated atoms are taken as the basis set, no ideal vacancy-gap states exist in the limit as the number of orbitals on the atom to be removed approaches infinity although spurious solutions for gap states can exist in a finite band model. Since the Green's-function analysis is exactly equivalent to finding the solutions of the Schrödinger equation with the Hamiltonian described above, these results obtain for all other techniques of solving the eigenvalue equation when the tight-binding Hamiltonian for an ideal vacancy in a crystal is employed. We find the reason for these surprising results is the fact that the tight-binding method is not equivalent to removing the potential of the removed atom. We demonstrate this by solving a two-atom problem by both the tight-binding method and by the Koster-Slater method, with only the latter method yielding the exact result. Moreover, we show that if the size of the basis set is allowed to increase without limit, the tight-binding method yields the same eigenvalues for the isolated atom as it does for the two-atom Hamiltonian. This result is generalized to the many-atom case and explains why no gap states are found if a complete basis is employed. This result is independent of the method used to solve the eigenvalue equation. The analysis is extended to surface-state calculations where it is shown that no gap states exist in the tight-binding method when Wannier functions are used as the basis set. Finally, using a Green's-function technique, we show how ideal vacancy-gap states may be calculated if the change in the potential, and consequently in the tight-binding matrix elements, is incorporated.