Coulomb interactions in semiconductors and insulators

In tight-binding theory, the electron-electron interaction enters (1) through an intra-atomic repulsion U, which is tabulated for nontransition elements, (2) through an interatomic repulsion $e^2$/d, and (3) through relaxation effects, which are included through a dielectric constant ε. For self-consistent band-structure calculations an effective repulsion $U^{\mathrm{*}}$=U-α$e^2$/d, with α the Madelung constant, enters. For semiconductors, $U^{\mathrm{*}}$ is ordinarily less than zero, suggesting that Coulomb shifts should be neglected. In insulators, $U^{\mathrm{*}}$ is small compared with U, but not negligible. The photothreshold in covalent solids is predicted to be reduced, in comparison to the tight-binding valence-band maximum, by U/2 because of dielectric relaxation. The band gap is predicted to be enhanced, in comparison to tight-binding theory (or density-functional theory) by U/ε. Both predictions are in rough accord with experiment. Similar shifts are expected for ionic crystals. The cohesive energy is predicted to be given by the change in the eigenvalues of occupied states, but reduced in homopolar covalent solids by $U^{\mathrm{*}}$/(2ε) (with a Madelung constant α of unity for this correction). In alkali halides the cohesion is reduced, in comparison to the difference in alkali and halogen free-atom electron energies, by $U^{\mathrm{*}}$ (with the halogen U and the crystalline Madelung constant). In the divalent compounds the cohesion is predicted to be reduced, in comparison to twice the difference in free-atom energy levels, by 3 times the corresponding $U^{\mathrm{*}}$. The intra-atomic repulsion can also lead to formation of a correlated state, as in the Mott transition. This is treated in the unrestricted Hartree-Fock approximation. A condition on the ratio of the appropriate $U^{\mathrm{*}}$ and the interatomic level coupling V is obtained for the formation of the correlated state, that ($U^{\mathrm{*}}$/V${)}^2$ exceed 4 times the number of coupled nearest neighbors. This is applied also to the ideal silicon (111) surface, confirming that it should be antiferromagnetic. The prevention of buckling of this surface by Coulomb effects is also discussed.