Self-consistent $GW$ calculations for semiconductors and insulators

We present $GW$ calculations for small and large gap systems comprising typical semiconductors (Si, SiC, GaAs, GaN, ZnO, ZnS, CdS, and AlP), small gap semiconductors (PbS, PbSe, and PbTe), insulators (C, BN, MgO, and LiF), and noble gas solids (Ar and Ne). It is shown that the $G_0{W}_0$ approximation always yields too small band gaps. To improve agreement with experiment, the eigenvalues in the Green’s function $G$ $\left(G{W}_0\right)$ and in the Green’s function and the dielectric matrix $\left(GW\right)$ are updated until self-consistency is reached. The first approximation leads to excellent agreement with experiment, whereas an update of the eigenvalues in $G$ and $W$ gives too large band gaps for virtually all materials. From a pragmatic point of view, the $G{W}_0$ approximation thus seems to be an accurate and still reasonably fast method for predicting quasiparticle energies in simple $sp$-bonded systems. We furthermore observe that the band gaps in materials with shallow $d$ states (GaAs, GaN, and ZnO) are systematically underestimated. We propose that an inaccurate description of the static dielectric properties of these materials is responsible for the underestimation of the band gaps in $G{W}_0$, which is itself a result of the incomplete cancellation of the Hartree self-energy within the $d$ shell by local or gradient corrected density functionals.

Figure 1

$G_0{W}_0$ quasiparticle energies determined with respect to the valence-band maximum for Si and ZnS.

Figure 2

(Color online) PBE and quasiparticle energies $G_0{W}_0$ and $G{W}_0$. Logarithmic scale is used for both axes.

Figure 3

(Color online) Band-gap error $r_{G{W}_0}$ versus error in the calculated dielectric constants $r_{ϵ}$. The line shows a linear fit $r_{G{W}_0}=0.01+0.75r_{ϵ}$ to all data points.