Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies

We present a first-principles theory of the quasiparticle energies in semiconductors and insulators described in terms of the electron self-energy operator. The full dielectric matrix is used to evaluate the self-energy operator in the GW approximation: the first term in an expansion of the self-energy operator in terms of the dynamically screened Coulomb interaction (W) and the dressed Green’s function (G). Quasiparticle energies are calculated for the homopolar materials diamond, Si, and Ge as well as for the ionic compound LiCl. The results are in excellent agreement with available experimental data. In particular, the indirect band gap is calculated as 5.5, 1.29, and 0.75 eV as compared with experimental gaps of 5.48, 1.17, and 0.744 eV for diamond, Si, and Ge, respectively. The Ge results include relativistic effects. The calculated direct gap for LiCl is within 5% of experiment. Viewed as a correction to the density-functional eigenvalues calculated with the local-density approximation, the present results show a correction dominated by a large jump at the gap. It is found that because of the charge inhomogeneity, the full dielectric screening matrix must be included, i.e., local-field effects are essential. The dynamical effects are also found to be crucial. The required dielectric matrices are obtained within the density-functional approach for the static case and extended to finite frequency with use of a generalized plasmon-pole model based on sum rules. The model reproduces the ω and ${\omega }^{\mathrm{-}1}$ moments of the exact many-body response function. The qualitative features of the electron self-energy operator are discussed. Using the static Coulomb-hole–screened-exchange approximation for illustration, the role of local fields in the self-energy operator are explained. The role of dynamical renormalization is illustrated. The same qualitative features are observed in both the homopolar and ionic materials.