$GW$ calculations on post-transition-metal oxides

In order to establish the reliable $GW$ scheme that can be consistently applied to post-transition-metal oxides (post-TMOs), we carry out comprehensive $GW$ calculations on electronic structures of ZnO, ${\text{Ga}}_2$${\text{O}}_3$, ${\text{In}}_2$${\text{O}}_3$, and ${\text{SnO}}_2$, the four representative post-TMOs. Various levels of self-consistency ($G_0{W}_0$, $G{W}_0$, and $QPG{W}_0$) and different starting functionals (GGA, GGA + $U$, and hybrid functional) are tested and their influence on the resulting electronic structure is closely analyzed. It is found that the $G{W}_0$ scheme with GGA + $U$ as the initial functional turns out to give the best agreement with experiment, implying that describing the position of metal-$d$ level precisely in the ground state plays a critical role for the accurate dielectric property and quasiparticle band gap. Nevertheless, the computation on ZnO still suffers from the shallow Zn-$d$ level and we propose a modified approach ($G{W}_0+U_d$) that additionally considers an effective Hubbard $U$ term during $G{W}_0$ iterations and thereby significantly improves the band gap. It is also shown that a GGA + $U$-based $G{W}_0$(+$U_d$) scheme produces an accurate energy gap of crystalline ${\text{InGaZnO}}_4$, implying that this can serve as a standard scheme that can be applied to general structures of post-TMOs.

Figure 1

The primitive cells of (a) wurtzite ZnO, (b) $\beta$-${\text{Ga}}_2$${\text{O}}_3$, (c) cubic ${\text{In}}_2$${\text{O}}_3$, and (d) rutile ${\text{SnO}}_2$. The red (small) balls in each crystal indicates oxygen and other larger ones are metal atoms.

Figure 2

(Color online). Difference between theoretical and experimental macroscopic optical dielectric constant ${\varepsilon }_{\mathrm{RPA}}^{\infty }$ and ${\varepsilon }_{\mathrm{Expt}.}^{\infty }$, respectively. Except for ${\text{In}}_2$${\text{O}}_3$, optical constants are anisotropic, and so we averaged diagonal components of the dielectric tensor. The experimental values are 3.73, 3.57, 4.00, and 3.92 for ZnO [37], ${\text{Ga}}_2$${\text{O}}_3$ [38], ${\text{In}}_2$${\text{O}}_3$ [20], and ${\text{SnO}}_2$ [17], respectively.

Figure 3

The band gaps computed with various computational schemes are plotted with respect to experimental values. See Table 1 for the numerical values.

Figure 4

The $G{W}_0$ band gap obtained with different starting functionals (GGA, GGA$+U$, and HSE06) represented as an optical dielectric constant in the abscissa.

Figure 5

Quasiparticle shifts with respect to the GGA + $U$ eigenvalues. The dashed vertical line indicates the Fermi level.

Figure 6

(a) The metal-$d$ band position with respect to the valence top. The reference experimental values are 7.5, 16.8, 15.0, and 22.9 eV for ZnO [25], ${\text{Ga}}_2$${\text{O}}_3$ [47], ${\text{In}}_2$${\text{O}}_3$ [48], and ${\text{SnO}}_2$ [49] measured from x-ray absorption spectroscopy (XAS) or angle-resolved photoemission spectroscopy (ARPES). (b) The line profile of charge density in ZnO along the line shown in the inset.

Figure 7

Differences in (a) electron density and (b) local potential difference $QPG{W}_0$ and GGA + $U$.