Difficulty in predicting shallow defects with hybrid functionals: Implication of the long-range exchange interaction

We identify an important issue in defect studies using hybrid functionals. When modeling a defect, which is supposedly an isolated system, with a finite-size supercell, the inclusion of a fraction of the Hartree-Fock interaction results in a strong cell-size dependence and an extremely slow convergence of the calculated defect properties, especially for shallow defect. These behaviors may give rise to a number of errors in calculated defect properties, including the deepening of transition level and overstabilization of shallow defects. Numerical results from hybrid functional calculations for a diverse array of systems can be understood within the Hartree-Fock theory of an electron-gas model, indicating that the long-range exchange is the main cause for the errors in the calculated defect properties within hybrid functionals.

Figure 1

The defect transition level $\epsilon (+/0)$ of GaAs:Si${}_{\mathrm{Ga}}$ (red solid line) and Si:P${}_{\mathrm{Si}}$ (blue dotted line) as a function of the supercell size. Open and filled dots show the PBE and PBE0 results, respectively. Energy is referenced to the CBM level.

Figure 2

Atomic structure and schematic energy level of (a) Si${}_{\mathrm{d}}^{-1}$ and (b) Si${}_{\mathrm{DX}}^{-1}$ in GaAs. Green and violet balls stand for Si and As atoms. In hybrid functional (PBE0) calculations, the Si${}_{\mathrm{d}}^{-1}$ level is significantly lowered ($-0.29$ eV calculated using a 250-atom supercell) as represented by the red arrow.

Figure 3

The calculated $E_{\mathrm{g}}$ of GaAs as a function of supercell size using the (a) PBE and (b) PBE0 functionals. The dashed line indicates the Kohn-Sham band gap ${\epsilon }_{\mathrm{g}}$.

Figure 4

One-electron levels of neutral (left) and negatively charged (middle) 128-atom GaAs supercell without defect and with the shallow donor GaAs:Si${}_{\mathrm{Ga}}$ (right) calculated using the PBE0 functional, showing surprisingly large level splittings.

Figure 5

Comparison between results calculated using the EG model (solid lines) and PBE0 and HSE functional (symbols) for a range of difference systems, including GaAs, Si, and ZnO, with respect to $r_s$. The Si${}_{\mathrm{Ga}}$ and P${}_{\mathrm{Si}}$ stand for shallow donors Si in GaAs and P in Si, respectively. The HSE results with different mixing and screening parameters are shown in (b) as discussed in the text.