$GW$ calculations including spin-orbit coupling: Application to Hg chalcogenides

We present self-energy calculations for Hg chalcogenides (Hg$X$, $X=\text{S}$, Se, and Te) with inverted band structures using an explicit spin-dependent formulation of the $GW$ approximation. Spin-orbit coupling is fully taken into account in calculating the single-particle Green function $G$ and the screened interaction $W$. We have found, apart from an upward shift of the occupied conductionlike ${\Gamma }_6$ state by about $0.7$ eV, an enhancement of spin-orbit splitting by about 0.1 eV, in good agreement with experiment. This renormalization originates mainly from spin-orbit induced changes in $G$ rather than $W$, which is affected only little by spin-orbit coupling.

Figure 1

Band structures of Hg$X$ ($X=\text{S}$, Se, and Te) calculated with LDA (solid lines) and $GW$ (circles). All calculations include spin-orbit coupling nonperturbatively. The vertical bars scale with the projection of the wave functions onto the Hg $6s$ state. The dashed lines are a guide to the eye.

Figure 2

Band structure of Hg$X$ ($X=\text{S}$, Se, and Te) as shown in Fig. 1 magnified around the $\Gamma$ point. Labels indicate $GW$ results.

Figure 3

Self-energy correction for the states ${\Gamma }_6$, ${\Gamma }_7$, and ${\Gamma }_8$: (a) only bare exchange and (b) full $GW$ correction. The lines are a guide to the eye.