First-principles study of intrinsic point defects in ZnO: Role of band structure, volume relaxation, and finite-size effects

Density-functional theory (DFT) calculations of intrinsic point defect properties in zinc oxide were performed in order to remedy the influence of finite-size effects and the improper description of the band structure. The generalized gradient approximation (GGA) with empirical self-interaction corrections $(\mathrm{GGA}+U)$ was applied to correct for the overestimation of covalency intrinsic to GGA-DFT calculations. Elastic as well as electrostatic image interactions were accounted for by application of extensive finite-size scaling and compensating charge corrections. Size-corrected formation enthalpies and volumes as well as their charge state dependence have been deduced. Our results partly confirm earlier calculations but reveal a larger number of transition levels: (1) For both the zinc interstitial as well as the oxygen vacancy, transition levels are close to the conduction band minimum. (2) The zinc vacancy shows a transition rather close to the valence band maximum and another one near the middle of the calculated band gap. (3) For the oxygen interstitials, transition levels occur both near the valence band maximum and the conduction band minimum.

Figure 1

Band structures obtained from density-functional theory calculations within the generalized-gradient approximation (GGA) (top) and using the $\mathrm{GGA}+U$ method with $\overline{U}-\overline{J}=7.5\mathrm{eV}$ (bottom). The conduction band states have been rigidly shifted to the experimental band gap. The small black circles represent data from self-interaction and relaxation corrected (SIRC) pseudopotential calculations (Ref. 27). In the plots on the right the solid and dashed lines show the calculated and experimental (Ref. 19) density of states, respectively. The gray stripe indicates the calculated band gap.

Figure 2

Scaling behavior of formation enthalpies with concentration (equivalent to the inverse cell volume) exemplified for the case of the zinc interstitial $\left({\mathrm{Zn}}_{\mathrm{i},\mathrm{oct}}\right)$. For the charged defects the small circles show the data before monopole-monopole corrections are applied (Ref. 50). Large closed circles correspond to formation enthalpies after volume relaxation while large open circles show the data obtained at fixed volume. The data presented here have been computed by the $\mathrm{GGA}+U$ method.

Figure 3

Variation of defect formation volumes with charge state. Open and closed circles indicate defects on the zinc and oxygen sublattices, respectively.

Figure 4

Variation of defect formation enthalpies with Fermi level under zinc (left) and oxygen-rich (right) conditions as obtained from $\mathrm{GGA}+U$ calculations. The gray shaded area indicates the difference between the calculated and the experimental band gap. The numbers in the plot indicate the defect charge state; parallel lines imply equal charge states. Open and closed circles correspond to defects on the zinc and oxygen sublattices, respectively.

Figure 5

Variation of dominant defect as a function of chemical potential and Fermi level as obtained from $\mathrm{GGA}+U$ calculations. In regions where the most stable defect is neutral, the transitions for the most stable charged defect are indicated by arrows and dotted lines.

Figure 6

Transition levels in the band gap calculated within GGA, $\mathrm{GGA}+U$ and using the extrapolation formula (4). The dark gray shaded areas indicate the error bars resulting from the errors given in Table .