Electronic structure of representative band-gap materials by all-electron quasiparticle self-consistent $GW$ calculations

In this work, we report results of the quasiparticle self-consistent GW method within the framework of the linearized augmented plane-wave method. The impact of self-consistency on the electronic structure is investigated with the examples of nine semiconductors and insulators, namely, Ar, Si, SiC, C, BN, LiF, MgO, CaO, and GaAs. Possible reasons for discrepancies between different studies and implementations are discussed. For LiF, MgO, and CaO, we assess the charge-density redistribution upon self-consistency. For a representative set of materials, we investigate and confirm the absence of (any) starting-point dependence. The off-diagonal terms in the self-energy matrix are found to considerably impact the electronic structure. For better reproducibility and quality assessment, we describe the implementation of QSGW in the all-electron full-potential code exciting.

Figure 1

Band structure of MgO calculated with QSGW (dark green), $G_0{W}_0$ (light green), and LDA (gray).

Figure 2

Same as Fig. 1 for GaAs. The inset, framed in blue, shows the region around the fundamental (direct) band gap $E_g$.

Figure 3

Charge-density differences between QSGW and LDA results [Eq. (8)] for LiF (top) and MgO (bottom panels). On the left, the isovalue for both is $\pm 2\times {10}^{-2}$ $e/{\mathrm{bohrs}}^3$; on the right, isovalues are $\pm 7\times {10}^{-3}$ $e/{\mathrm{bohrs}}^3$ for LiF and $\pm 2\times {10}^{-3}$ $e/{\mathrm{bohrs}}^3$ for MgO. Positive values are red, negative ones blue. Fluorine atoms are displayed in light blue, lithium in green, magnesium in yellow, oxygen in red (masked by isosurfaces).

Figure 4

Charge-density differences between QSGW and LDA results in CaO at isovalues of $\pm 2\times {10}^{-2}$ $e/{\text{bohrs}}^3$ (top), $\pm 1\times {10}^{-2}$ $e/{\text{bohrs}}^3$ (middle), and $\pm 2\times {10}^{-3}$ $e/{\mathrm{bohrs}}^3$ (bottom), respectively. Orange spheres represent calcium, oxygen (red) sits at the center of the unit cell.

Figure 5

Charge-density differences between QSGW and PBE0 results for LiF (top) and MgO (bottom panels). On the left, the isovalue is $\pm 2\times {10}^{-2}$ $e/{\mathrm{bohrs}}^3$ for both materials; on the right, isovalues of $\pm 7\times {10}^{-3}$ $e/{\mathrm{bohrs}}^3$ for LiF and $\pm 2\times {10}^{-3}$ $e/{\mathrm{bohrs}}^3$ for MgO are chosen. Positive values are red, negative ones blue. F atoms are displayed in light blue, Li in green, Mg in yellow, O in red.

Figure 6

Band structure of argon calculated with QSGW@LDA (green line), QSGW@PBE0 (orange diamonds), and QSGW@HF (blue circles).

Figure 7

QSGW band structure of BN using the full (solid line) and the diagonal self-energy matrix (dashed line). For comparison, the $G_0{W}_0$ results (light green) are included.

Figure 8

Same as Fig. 7 for MgO.

Figure 9

Same as Fig. 7 for Ar.

Figure 10

Flowchart of the QSGW implementation in exciting. Green (white) color refers to operators from group A (B); blue indicates the only representative of group C.

Figure 11

Convergence of the fundamental band gap of MgO with respect to the number of empty states. The production settings are described in Appendix pp2, the reduced settings are summarized in Table 7.

Figure 12

Convergence of the fundamental band gap of MgO with respect to the number of $nl$ terms.