Self-consistent solution of Hedin's equations: Semiconductors and insulators

Abstract

The band gaps of a few selected semiconductors/insulators are obtained from the self-consistent solution of the Hedin's equations. Two different schemes to include the vertex corrections are studied: (i) the vertex function of the first order (in the screened interaction $W$) is applied in both polarizability $P$ and self-energy $\mathrm{\ensuremath{\Sigma}}$, and (ii) the vertex function obtained from the Bethe-Salpeter equation is used in $P$ whereas the vertex of the first order is used in $\mathrm{\ensuremath{\Sigma}}$. Both schemes show considerable improvement in the accuracy of the calculated band gaps as compared to the self-consistent $GW$ approach ($\mathrm{sc}GW$) and to the self-consistent quasiparticle $GW$ approach ($\mathrm{QS}GW$). To further distinguish between the performances of two vertex-corrected schemes one has to properly take into account the effect of the electron-phonon interaction on the calculated band gaps which appears to be of the same magnitude as the difference between schemes (i) and (ii).