Prediction of subgap states in Zn- and Sn-based oxides using various exchange-correlation functionals

We present a density-functional-theory analysis of crystalline and amorphous Zn- and Sn-based oxide systems which focuses on the electronic defect states within the band gap. A comparison of these electronic levels reveals that the hybrid functionals PBE0, HSE06, or B3LYP agree with a self-interaction corrected (SIC) local-density-approximation functional on occupied defect levels when similar treatments of the self-interaction are considered. However, for unoccupied levels, the hybrid functionals and the SIC approach lead to very different predictions. We show that a prerequisite for the determination of the energetic position of subgap states in these oxides is that a functional needs to predict correctly the electronic band structure over a wide energy range and not just close to the band gap. We conclude that for accurate defect levels, an adequate treatment of the self-interaction problem is required especially in the presence of nearby metal-metal interactions.

Figure 1

Total DOS of ${a\text{−}\text{Zn}}_2{\mathrm{SnO}}_4$ as calculated with SIC-LDA, HSE06, and B3LYP. The DOS of ${c\text{-Zn}}_2{\mathrm{SnO}}_4$ with inverse-spinel $\left({P4}_{1}22\right)$ structure calculated with SIC-LDA is shown for comparison (black dashed line). For HSE06 and B3LYP, the Zn $3d$ bands strongly overlap with the O $2p$ bands whereas SIC-LDA reproduces well the band separation experimentally observed for ZnO [14, 24]. The subgap states extend up to 1 eV for SIC-LDA and to 0.5 eV for HSE06 and B3LYP (see inset). The small arrows on the energy axis indicate the highest occupied levels of the DOS.

Figure 2

Total DOS of ${a\text{-In}}_2{\mathrm{Zn}}_2{\mathrm{O}}_5$ calculated using SIC-LDA, HSE06, and B3LYP. As reference, the DOS of $c$-ZnO calculated using SIC-LDA is shown (black dashed line). The three functionals predict almost the same depth of the subgap states but differ in the position of the Zn $3d$ bands. The small colored arrows on the energy axis mark the highest occupied levels of the DOS.

Figure 3

Total DOS of $c\text{−}{\mathrm{SnO}}_2$ containing a tin vacancy $\left(V{}_{\mathrm{Sn}}\right)$ in (a) charge state $q=-2$ and (b) neutral state $q=0$. The DOS of bulk ${\mathrm{SnO}}_2$ obtained with SIC-LDA is shown for comparison (black dashed line) The different functionals agree quantitatively on the position of the occupied subgap states. However, the hybrid functionals predict unoccupied states distributed over most of the band gap whereas the SIC approach predicts unoccupied states only at the VB tail (see long green arrow). The small colored arrows on the energy axis mark the highest occupied levels of the DOS.

Figure 4

Total DOS of $c\text{−}{\mathrm{SnO}}_2$ containing an oxygen vacancy $\left(V{}_{\mathrm{O}}\right)$ in the charge-neutral state. HSE06, B3LYP, and the reduced SIC with $\alpha =0.25$ (see text) almost agree on the position of the occupied defect level while SIC-LDA puts it about 1 eV higher. The DOS of bulk ${\mathrm{SnO}}_2$ obtained by SIC-LDA is shown for comparison (black dashed line). The small arrows on the energy axis mark the highest occupied levels of the DOS.

Figure 5

Total DOS of $a\text{−}{\mathrm{ZnSnO}}_3:V{}_{\mathrm{O}}$ as calculated with SIC, HSE06, B3LYP, HSE with $a=0.375$ and SIC with $\alpha =0.25$. The DOS of $c\text{−}{\mathrm{ZnSnO}}_3$ ($R3c$ structure) calculated using SIC-LDA is shown for comparison (black dashed line). The hybrid functionals HSE06 and B3LYP predict subgap states in the lower half of the band gap. SIC puts the energetic position of this occupied oxygen-vacancy-like defect significantly higher while the reduced SIC with $\alpha =0.25$ agrees with the hybrid functionals. Increasing the HF contribution moves all occupied and unoccupied levels higher in energy. The small arrows on the energy axis mark the highest occupied levels of the DOS.