Electronic structure and optical properties of $F$ centers in $\alpha$-alumina

We use a state-of-the-art $GW$ Bethe-Salpeter equation (BSE) formalism to study electronic structure and optical properties of oxygen vacancies ($F$ centers) in $\alpha$-alumina. The density functional theory (DFT) + $GW$ formalism has been employed to compute the charge transition levels (CTLs) for oxygen vacancies. We propose a reformulation of the DFT+GW approach to calculate these CTLs. Our new approach allows for transparent application of electrostatic corrections required in finite supercell calculations using periodic boundary conditions. We find that $F$ centers in this material introduce deep donor levels, (+2/+1) at 2.4 eV, and a (+1/0) level at 3.9 eV above the valence band maximum. We also study $F$-center absorption and emission processes using constrained DFT and BSE. Our calculated absorption and emission energies are in excellent agreement with experiments and provide an unambiguous interpretation of the same.

Figure 1

Paths that can take a system at equilibrium with a defect at charge state $q$ to one where it is at charge state $q-1$. Paths that are equivalent to using (a) Eq. (3) (red), Eq. (4) (blue), and (b) Eq. (5) are shown.

Figure 2

Variation of band gap with lattice parameter, calculated using both GGA (blue circle) and LDA (black squares), the equilibrium lattice parameter in both cases and the corresponding band gap have been marked. In these calculations the $c/a$ ratio was fixed to the equilibrium value. $GW$ quasiparticle gaps starting from a DFT calculation with GGA at two different lattice parameter values have been marked with red triangles.

Figure 3

Defect level positions and their occupations for different charge states from DFT as well as $GW$ calculations.

Figure 4

Cell size dependence of the relaxation energy ($E^{\text{relax}}$) for defect at charge state +1 (a) and +2 (b). $E^{\text{relax}}$ values for four different cell sizes has been used to obtain the value at infinite cell size limit ($E^{\text{relax}}\left[\infty \right]$).

Figure 5

Energy-level scheme for absorption and emission processes associated with the $F$ center in $\alpha$-alumina