Self-energy operators and exchange-correlation potentials in semiconductors

We show how the density-functional theory (DFT) exchange-correlation potential ${V}_{\mathrm{xc}(\mathrm{r}}$) of a semiconductor is calculated from the self-energy operator \ensuremath{\Sigma}(r,r',\ensuremath{\omega}), and how \ensuremath{\Sigma} is obtained using the one-particle Green's function and the screened Coulomb interaction (the GW approximation). We discuss the nature of ${V}_{\mathrm{xc}}$ and the self-energy in real space, and investigate features and trends found in Si, GaAs, AlAs, and diamond. In each case the calculated quasiparticle band structure is in good agreement with experiment, while the DFT band structure is surprisingly similar to that with the common local-density approximation (LDA); in particular, about 80% of the severe LDA band-gap error is shown to be inherent in DFT calculations, being accounted for by the discontinuity \ensuremath{\Delta} in ${V}_{\mathrm{xc}}$ upon addition of an electron. The relationship of the calculated ${V}_{\mathrm{xc}}$ to the LDA and its extensions is also examined.