Convergence of density and hybrid functional defect calculations for compound semiconductors

Recent revisions of defect formation energy calculations based on bandgap corrected hybrid functionals have raised concerns about the validity of earlier results based on standard density functionals and about the reliability of the theoretical prediction of electrical properties in semiconductor materials in general. We show here that a close agreement between the two types of functionals can be achieved by determining appropriate values for the electronic and atomic reference energies, thereby mitigating uncertainties associated with the choice of the underlying functional.

Figure 1

Within DFT, the formation energy Δ$H$ for the fully ionized defects should be rather accurately described relative to the average electrostatic potential $V_{av}$, but the single-particle energies of the band edges ($e_{\mathrm{VBM}}^{\mathrm{DFT}}$ and $e_{\mathrm{CBM}}^{\mathrm{DFT}}$) are usually not. The quasiparticle energies $e_{\mathrm{VBM}}^{GW}$ and $e_{\mathrm{CBM}}^{GW}$ define the corrected range of Δ$H$($E_F$). Schematically shown are Δ$H$ for a positively charged (donorlike) defect $D$ and a negatively charged (acceptorlike) defect $A$.

Figure 2

Difference of defect formation energies between the HSE and the GGA($+U$) functionals, based on (a) $\Delta H_{\mathrm{ref}}$, directly calculated in the respective DFT functional, on (b) $\Delta H_{\mathrm{ref}}^{GW}$, taking into account $GW$ band edge shifts, and on (c) $\Delta H_{\mathrm{ref}}^{GW,\mathrm{FERE}}$, taking additionally into account the FERE shifts for the elemental reference energies [cf. Eq. (6)]. The different types of defects are anion vacancies (an-$V$: $V_{\mathrm{N}}$, $V_{\mathrm{O}}$), anion substitutional dopants (an-$S$: O${}_{\mathrm{N}}$, N${}_{\mathrm{O}}$), cation vacancies (cat-$V$: $V_{\mathrm{Cu}}$, $V_{\mathrm{Zn}}$, $V_{\mathrm{Al}}$), cation substitutional dopants (cat-$S$: Al${}_{\mathrm{Zn}}$, Zn${}_{\mathrm{Al}}$), and cation interstitials (cat-$I$: Cu${}_{\mathrm{i}}$).