Band-edge positions in $GW$: Effects of starting point and self-consistency

We study the effect of starting point and self-consistency within $GW$ on the band-edge positions of semiconductors and insulators. Compared to calculations based on a semilocal starting point, the use of a hybrid-functional starting point shows a larger quasiparticle correction for both band-edge states. When the self-consistent treatment is employed, the band-gap opening is found to result mostly from a shift of the valence-band edge. Within the non-self-consistent methods, we analyse the performance of empirical and nonempirical schemes in which the starting point is optimally tuned. We further assess the accuracy of the band-edge positions through the calculation of ionization potentials of surfaces. The ionization potentials for most systems are reasonably well described by one-shot calculations. However, in the case of ${\mathrm{TiO}}_2$, we find that the use of self-consistency is critical to obtain a good agreement with experiment.

Figure 1

(a) Convergence of the $G_0{W}_0$ QP corrections to the VBM and CBM of GaAs with respect to the energy cutoff of the dielectric matrix ${\varepsilon }^{-1}$, and (b) convergence of the QP corrections with respect to the number of bands used in the self-energy. In (a), we use 750 bands in the self-energy. In (b), the energy cutoff of the dielectric matrix is taken at 30 Ry.

Figure 2

Convergence of the VBM energies with respect to the number of $\mathbf{k}$ points in the PBE0 and the $G_0{W}_0$ calculations. Triangles and disks correspond to the use of Carrier-Görling (CG) and Gygi-Baldereschi (GB) auxiliary functions. For 4H-SiC, the PBE0 values using the anisotropic meshes ($4\times 4\times 4$ and $8\times 8\times 8$) are denoted with open symbols. The eigenvalues are aligned through the local reference potential.

Figure 3

Band gaps (theoretical vs experimental) of various semiconductors and insulators. LiF is not included in here. The band-gap values are collected in Table 4.

Figure 4

The VBM (solid-filled) and CBM (pattern-filled) shifts of the bulk materials relative to PBE references.

Figure 5

The VBM (filled symbols) and the CBM (open symbols) positions obtained by varying $\alpha$ in the PBE0 and the one-shot $G_0{W}_0$ calculations starting from the PBE0($\alpha$). In $G_0{W}_0^{\mathrm{PBE}}$, the screened interaction is fixed at the PBE level.

Figure 6

Discrepancies in the VBM positions between PBE0 and $G_0{W}_0$ calculations when both are empirically tuned to reproduce the experimental band gap. The eigenvalues are aligned through the local reference level. A positive (negative) value indicates that $G_0{W}_0$ predicts a deeper (higher) VBM compared to PBE0.

Figure 7

Ionization potentials (theoretical vs experimental) of various semiconductor surfaces. The data are taken from Table 10. For systems with multiple experimental references, the adopted experimental IP corresponds to the average value.