Accurate all-electron $G_0{W}_0$ quasiparticle energies employing the full-potential augmented plane-wave method

The $GW$ approach of many-body perturbation theory has become a common tool for calculating the electronic structure of materials. However, with increasing number of published results, discrepancies between the values obtained by different methods and codes become more and more apparent. For a test set of small- and wide-gap semiconductors, we demonstrate how to reach the numerically best electronic structure within the framework of the full-potential linearized augmented plane-wave (FLAPW) method. We first evaluate the impact of local orbitals in the Kohn-Sham eigenvalue spectrum of the underlying starting point. The role of the basis-set quality is then further analyzed when calculating the $G_0{W}_0$ quasiparticle energies. Our results, computed with the exciting code, are compared to those obtained using the projector-augmented plane-wave formalism, finding overall good agreement between both methods. We also provide data produced with a typical FLAPW basis set as a benchmark for other $G_0{W}_0$ implementations.

Figure 1

Wurtzite ZnO KS eigenenergies at the $\mathrm{\Gamma }$ point. The black curve corresponds to a calculation with the set of local orbitals defined in Table 3, while the red one is obtained combining the latter basis with local orbitals for unoccupied states as defined in Table 4.

Figure 2

Convergence of the QP band gap of wurtzite ZnO for different configurations of the basis set using $2\times 2\times 2$ k/q grids. Each configuration is characterized by a maximum angular momentum $l_{\max }$ up to which additional local orbitals for representing unoccupied states are added (see Table 4).

Figure 3

Uncertainty of the extrapolation procedure with respect to the interval of empty bands (specified in parentheses) used for fitting Eq. (12). The dashed lines on top represent the extrapolated values.

Figure 4

Top: $G_0{W}_0$ gap as a function of the number of unoccupied bands used in the calculations ($2\times 2\times 2$ k/q grids). The blue, green, and orange curves correspond to calculations with $R_{\mathrm{MT}}{G}_{\max }$ values of 8, 10, and 12, respectively. In all cases, local orbitals up to $l_{\max }=8$ were employed. Bottom: Corresponding KS energies of unoccupied bands at the $\mathrm{\Gamma }$ point. Vertical dashed lines indicate the threshold values, $N_0$, at which $E_{\mathrm{g}}^{\mathrm{QP}}\left(N\right)$ stops following the asymptotic behavior Eq. (11).

Figure 5

$G_0{W}_0$ gap ($2\times 2\times 2$ k/q grids) as a function of $l_{\max }$ as used to construct additional local orbitals for unoccupied states.

Figure 6

Mean absolute deviations ($\mathrm{\Delta }E$) of results obtained with the optimized basis set from those of the default one (orange bars); results for the direct band gap at $\mathrm{\Gamma }$ (left) and the ionization potential shift (right) based on the extrapolation scheme. The blue bars indicate differences between this work and Ref. [12]. Black bars depict differences between the extrapolation procedure and all-states calculations.

Figure 7

Convergence of the QP gap (top) and absolute shift of the VB maximum compared to LDA calculations (bottom) in GaAs as a function of the number of empty states for different $\mathbf{k}/\mathbf{q}$-point grids.