Converged $GW$ quasiparticle energies for transition metal oxide perovskites

The ab initio calculation of quasiparticle (QP) energies is a technically and computationally challenging problem. In condensed matter physics, the most widely used approach to determine QP energies is the $GW$ approximation. Although the $GW$ method has been widely applied to many typical semiconductors and insulators, its application to more complex compounds such as transition metal oxide perovskites has been comparatively rare, and its proper use is not well established from a technical point of view. In this work, we have applied the single-shot $G_0{W}_0$ method to a representative set of transition metal oxide perovskites including $3d$ (${\mathrm{SrTiO}}_3, {\mathrm{LaScO}}_3, {\mathrm{SrMnO}}_3, {\mathrm{LaTiO}}_3, {\mathrm{LaVO}}_3, {\mathrm{LaCrO}}_3, {\mathrm{LaMnO}}_3$, and ${\mathrm{LaFeO}}_3$), $4d$ (${\mathrm{SrZrO}}_3, {\mathrm{SrTcO}}_3$, and ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$), and $5d$ (${\mathrm{SrHfO}}_3, {\mathrm{KTaO}}_3$, and ${\mathrm{NaOsO}}_3$) compounds with different electronic configurations, magnetic orderings, structural characteristics, and band gaps ranging from 0.1 to 6.1 eV. We discuss the proper procedure to obtain well-converged QP energies and accurate band gaps within single-shot $G_0{W}_0$ by comparing the conventional approach based on an incremental variation of a specific set of parameters (number of bands, energy cutoff for the plane-wave expansion and number of k points) and the basis-set extrapolation scheme [J. Klimeš et al., Phys. Rev. B 90, 075125 (2014)]. Although the conventional scheme is not supported by a formal proof of convergence, for most cases it delivers QP energies in reasonably good agreement with those obtained by the basis-set correction procedure and it is by construction more useful for calculating band structures. In addition, we have inspected the difference between the adoption of norm-conserving and ultrasoft potentials in $GW$ calculations and found that the norm violation for the $d$ shell can lead to less accurate results in particular for charge-transfer systems and late transition metals. A minimal statistical analysis indicates that the correlation of the $GW$ data with the density functional theory gap is more robust than the correlation with the experimental gaps; moreover, we identify the static dielectric constant as alternative useful parameter for the approximation of $GW$ gap in high-throughput automatic procedures. Finally, we compute the QP band structure and spectra within the random phase approximation and compare the results with available experimental data.

Figure 1

Different types of lattice distortions (a)–(d) and TM $d$ orbitals (e) in the oxide perovskites studied in this paper. (a) $P_{m\overline{3}m}$ for $\mathrm{Sr}M{\mathrm{O}}_3$ ($M=$ Sr, Hf, Zr), ${\mathrm{SrMnO}}_3$ and ${\mathrm{KTaO}}_3$; (b) $P_{nma}$ for ${\mathrm{LaScO}}_3, {\mathrm{LaTiO}}_3, {\mathrm{LaCrO}}_3, {\mathrm{LaMnO}}_3, {\mathrm{LaFeO}}_3, {\mathrm{SrTcO}}_3, {\mathrm{NaOsO}}_3$; (c) $P_{21/b}$ for ${\mathrm{LaVO}}_3$; (d) $P_{bca}$ for ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$. The blue and red balls represent the TM and O ions, respectively. (e) Different degree of spatial extension in $3d, 4d$, and $5d$ orbitals (derived from atomic calculations).

Figure 2

Schematic representation of the conventional nonextrapolated method. Convergence of the QP energy gap as a function of $N$ and $E_{\text{pw}}$. The convergence inspected as a function (a) $N$ (at fixed k points and $E_{\text{pw}}$) and $E_{\text{pw}}$ (for fixed k points and $N$) where $E_{\text{pw}}^3>E_{\text{pw}}^2>E_{\text{pw}}^1$, and (b) shows the convergence of the QP gap with respect to the number of k points (for fixed $N$ and $E_{\text{pw}}$).

Figure 3

The schematic representation of the basis-set correction and the k-points correction for the QP gap $E_g$. The labels indicate the contributions in Eq. (2) and are defined in the text.

Figure 4

Conventional nonextrapolated method applied to ${\mathrm{SrTiO}}_3$. Convergence of the indirect QP band gap $E_g^i$ of ${\mathrm{SrTiO}}_3$ with respect to (a) number of bands $N$ and plane-wave cutoff energy $E_{\text{pw}}$ (k-point mesh fixed to $4\times 4\times 4$), and (b) size of the k-point mesh for $N$ and $E_{\text{pw}}$ fixed to the optimum values for US and NC potentials.

Figure 5

Basis-set correction data for ${\mathrm{SrTiO}}_3$ using US (a)–(c) and NC (d)–(f) PAWs. For each type of PAW three different graphs are shown: (a), (d) convergence of the QP band gap $E_g^i$ with respect to the inverse of the number of bands (1000/$N$); (b), (e) k-point correction ${\mathrm{\Delta }}_k$ as a function of 1000/$N$; (c), (f) basis-set correction ${\mathrm{\Delta }}_N$ as a function of k points.

Figure 6

$E_g^i$ and QP corrections to Kohn-Sham eigenvalues ($E^{\text{QP}}-E^{\text{KS}}$) for the CBM and VBM at $\mathrm{\Gamma }$ as a function of $N=N_{\text{pw}}$ for ${\mathrm{SrTiO}}_3$ (a), (d), ${\mathrm{SrZrO}}_3$ (b), (e), and ${\mathrm{SrHfO}}_3$ (c), (f), computed within the basis-set correction scheme using a $2\times 2\times 2$ k-point mesh.

Figure 7

Band gap $E_g^{\mathrm{\Gamma }}$ and QP correction $E^{\text{QP}}-E^{\text{KS}}$; similar to Fig. 6 but for ${\mathrm{SrMnO}}_3, {\mathrm{SrTcO}}_3$, and ${\mathrm{NaOsO}}_3$ [without (wo) SOC].

Figure 8

Comparison between the DFT and $G_0{W}_0$ band gaps and available experimental measurements (reference given in Table 4). The $G_0{W}_0$ refer to three different sets: nonextrapolated results obtained using US potentials, and basis-set corrected values using both type of potentials, US and NC (not for ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$).

Figure 9

Statistical interpretation of the $GW$ data by means of a linear regression. (a) Comparison between the calculated and experimental gaps. The calculated values include $G_0{W}_0$ results obtained following the different schemes discussed in the main text: conventional nonextrapolated scheme using US PAW and the basis-set correction procedure using both NC and US PAW. (b) Correlation between the $G_0{W}_0$ and the DFT gap. (c) Correlation between the $G_0{W}_0$ gap and the calculated static dielectric constant ${\epsilon }_{\infty }$. (d) Comparison between the DFT gap and the QP shift at $\mathrm{\Gamma }$. In each panel, the linear relation and the ${\mathrm{R}}^2$ factors are given in the insets.

Figure 10

Comparison between calculated QP shift, band gap (indirect, down triangles, and direct at $\mathrm{\Gamma }$, up triangles), and average of the diagonal component of the static dielectric tensor using the conventional nonextrapolated scheme and adopting US PAWs. The QP shift is defined as the difference between the $G_0{W}_0$ QP energy and DFT eigenvalue.

Figure 11

Collection of the calculated band structures for DFT (gray lines) and $GW$ (black) together with the $GW$ density of TM-$d$ (shadow, cyan line) and O-$p$ (full line, red) states. The filled circles indicate the calculated $GW$ QP energies (used for the Wannier interpolation). As mentioned in the main text, the DFT calculations for ${\mathrm{LaTiO}}_3$ and ${\mathrm{LaVO}}_3$ were performed with the addition of a small effective $U$.

Figure 12

Collection of calculated (black line) optical spectra along with available experimental curves (gray/red line); for ${\mathrm{SrTiO}}_3, {\mathrm{SrZrO}}_3$, and ${\mathrm{SrHfO}}_3$ the experimental data are taken from Ref. [93], for ${\mathrm{KTaO}}_3$ from Ref. [94], for ${\mathrm{NaOsO}}_3$ from Ref. [83], for the La series from Ref. [9] (${\mathrm{LaMnO}}_3$ from Ref. [95]), and for ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ from Ref. [96]. For ${\mathrm{Ca}}_2{\mathrm{RuO}}_4$ the inset shows a zoom of the low-energy region. The calculated dielectric functions from which the optical conductivity spectra have been derived are shown in Fig. 13.

Figure 13

Average of diagonal component of the real (${\epsilon }_1$) and imaginary (${\epsilon }_2$) parts of the dielectric function obtained by $G_0{W}_0$ for the entire materials data set studied in this work. The values of the static ion-clamped dielectric function ${\epsilon }_{\infty }$ are listed in Table 7.