Defect Formation Energies without the Band-Gap Problem: Combining Density-Functional Theory and the $GW$ Approach for the Silicon Self-Interstitial

We present an improved method to calculate defect formation energies that overcomes the band-gap problem of Kohn-Sham density-functional theory (DFT) and reduces the self-interaction error of the local-density approximation (LDA) to DFT. We demonstrate for the silicon self-interstitial that combining LDA with quasiparticle energy calculations in the $G_0{W}_0$ approach increases the defect formation energy of the neutral charge state by $\sim 1.1\mathrm{eV}$, which is in good agreement with diffusion Monte Carlo calculations (E.  R. Batista et al., Phys. Rev. B 74, 121102(R) (2006); W.-K. Leung et al. Phys. Rev. Lett. 83, 2351 (1999)). Moreover, the $G_0{W}_0$-corrected charge transition levels agree well with recent measurements.

Figure 1

(a) Split $⟨110⟩$, (b) hexagonal, (c) $C_{3v}$ and (d) tetrahedral configuration of the ${\mathrm{Si}}_i$. Defect atoms are shown in red and nearest neighbors in gray. The middle panel depicts the formation of the neutral ${\mathrm{Si}}_i$ from the $2+$ charge state. $A_{+}$ and $A_{2+}$ are short for the electron affinities $A(+,{\mathbf{R}}_0)$ and $A(2+,{\mathbf{R}}_{2+})$ (see text), respectively, and ${\mathbf{R}}_q$ denotes the atomic positions in charge state $q$.

Figure 2

Vertical electron affinities for different configurations of the ${\mathrm{Si}}_i$: LDA Slater transition states (blue) and $G_0{W}_0$ quasiparticle energies (red).

Figure 3

Formation energies ($E_D^f$) as a function of Fermi energy in LDA (left) and $G_0{W}_0$ (right). The lower panels show the split $⟨110⟩$ as representative configuration and the upper the configuration with the lowest energy for a give Fermi level. The dotted line marks the LDA band gap.