Quasiparticle Excitations and Charge Transition Levels of Oxygen Vacancies in Hafnia

We calculate the quasiparticle defect states and charge transition levels (CTLs) of oxygen vacancies in monoclinic hafnia using density functional theory (DFT) and the $GW$ method. We introduce the criterion that the quality and reliability of CTLs may be evaluated by calculating the same CTL via two physical paths and show that it is necessary to include important electrostatic corrections previously neglected within the supercell $\mathrm{DFT}+GW$ approach. Contrary to previous reports, the oxygen vacancies in hafnia are large positive $U$ centers, where $U$ is the defect charging energy.

Figure 1

Formation energies evaluated with $\mu =E_v$ vs generalized coordinate, illustrating the terms in the $\mathrm{DFT}+GW$ formalism for the CTL ${\epsilon }^{q/q-1}$.

Figure 2

(a) Schematic diagram showing oxygen vacancy induced defect levels in the band gap of hafnia for various charge states. (b) Effect of the spurious potential on the defect-state eigenvalues. The plot shows the PBE eigenvalue vs supercell size with respect to the top of the valence band in the same supercell calculation for defect states in $V_{{\mathrm{O}}_3}$ in different charge states. Empty (filled) symbols represent unoccupied (occupied) defect level as per (a).

Figure 3

Relative formation energy (relative to that of $V_{{\mathrm{O}}_3}^0$) vs chemical potential (or Fermi energy) for charged oxygen vacancies in monoclinic hafnia. The shaded region denotes the placement of the Si band gap at a hafnia-Si interface. The formation energy of $V_{{\mathrm{O}}_4}^0$ is lower than $V_{{\mathrm{O}}_3}^0$ by 0.14 eV.