Correspondence of defect energy levels in hybrid density functional theory and many-body perturbation theory

We demonstrate the correspondence between charge transition levels of localized point defects in hybrid density functional theory and in $G_0{W}_0$ many-body perturbation theory. To achieve this correspondence, it is necessary to properly combine the treatments of the finite-size effect, the delocalization error, and the path dependency in the $G_0{W}_0$ scheme. In particular, we introduce a beyond-monopole finite-size electrostatic correction for the eigenvalue which is fully consistent with the analogous correction for the total energy. We illustrate how the comparison with experiment is affected by the calculated band-edge positions, which unlike the defect levels are sensitive to the adopted electronic-structure method.

Figure 1

The convergence of the quasiparticle (QP) correction with respect to the number of bands $N_{\mathrm{b}}$ used in the self-energy for an $F^{+}$ center in LiF. QP corrections are given for the VBM and the $a_{1g}$ defect level. The extrapolation limits are shown as horizontal dashed lines. A 63-atom supercell is used and the neutral defect geometry is assumed.

Figure 2

$G_0{W}_0$ defect energy levels $\mu (0/+)$ of the $F$ center in LiF as calculated within PAW and NCPP schemes, in comparison to PBE results. The calculations are aligned with respect to the electrostatic potential of the pristine bulk. Values in parentheses show the shifts with respect to the PBE defect levels. Energies are in eV.

Figure 3

Finite-size scaling of the KS (open circles) and quasiparticle (closed circles) levels of the $a_{1g}$ defect level for (a) the neutral $F^0$ and (b) the charged $F^{+}$ centers in LiF. In (b), the corrected KS eigenvalues are given with (closed squares) and without (open squares) the alignment term. The evolution of quasiparticle corrections of the $a_{1g}$ level and of the band edges are shown for (c) $F^0$ and (d) $F^{+}$. The band-edge quasiparticle corrections as obtained from a pristine bulk calculation are also shown (dashed). Both charge states are considered in the geometry of the neutral defect $\left(R_0\right)$. The numbers of atomic sites in the supercells are indicated.

Figure 4

Thermodynamic charge transition levels of (a) LiF:$V_{\mathrm{F}}$, (b) SiO${}_2$:O${}_{\mathrm{int}}$, and (c) 3C-SiC:C${}_{\mathrm{sp}\langle 100\rangle }$, as determined within PBE, PBE+$G_0{W}_0$, PBE0, and HSE. The energy scales are aligned with respect to the electrostatic potential in the pristine bulk. The PBE VBM is set to zero. The values in parentheses show the shifts from the PBE charge transition levels.

Figure 5

Left: Configuration coordinate diagram of the oxygen interstitial in $\alpha$-quartz depicting two possible transition paths (I and II), as obtained in the PBE+$G_0{W}_0$ scheme. Right: Sketch of the electronic structures of the neutral defect in PBE and in $G_0{W}_0$, showing the occupied (closed circles) and unoccupied (open circles) defect levels.

Figure 6

Vertical charge transition levels ${\mu }^{\mathrm{opt}}(0/+)$ of LiF:$F^0$ (at the neutral defect geometry) calculated within PBE, $G_0{W}_0$, PBE0, and HSE. The $G_0{W}_0$ and hybrid-functional calculations reproduce the experimental band gap (14.2 eV). The energy levels are aligned with respect to the electrostatic potential of the pristine bulk. Energies are in eV.

Figure 7

The electrostatic-potential profile for an $F^{+}$ center in LiF. The vacancy is at the origin.

Figure 8

Optical absorption spectrum of LiF:$F^0$ calculated by solving the Bethe-Salpeter equation. The absorption peak at 1.9 eV is assigned to the $F^0$ band. The peak at 7.7 eV is assigned to bulk absorption.