Defect energetics in ZnO: A hybrid Hartree-Fock density functional study

First-principles calculations based on hybrid Hartree-Fock density functionals provide a clear picture of the defect energetics and electronic structure in ZnO. Among the donorlike defects, the oxygen vacancy and hydrogen impurity, which are deep and shallow donors, respectively, are likely to form with a substantial concentration in $n$-type ZnO. The zinc interstitial and zinc antisite, which are both shallow donors, are energetically much less favorable. A strong preference for the oxygen vacancy and hydrogen impurity over the acceptorlike zinc vacancy is found under oxygen-poor conditions, suggesting that the oxygen vacancy contributes to nonstoichiometry and that hydrogen acts as a donor, both of which are without significant compensation by the zinc vacancy. The present results show consistency with the relevant experimental observations.

Figure 1

(Color online) Formation energies of $V_{\text{O}}$ and ${\text{Zn}}_{i\left(o\right)}$ in the neutral and $2+$ charge states obtained by using the GGA as a function of $L^{-1}$ and $L^{-3}$ (where $L$ is the average interdefect distance). The energies are for the oxygen-poor limit and the Fermi energy at the valence-band maximum. The filled and open symbols denote the values before and after the Madelung energy corrections (Refs. 34, 35).

Figure 2

(Color online) (a) Defect formation energies as a function of the Fermi energy at the oxygen-poor and oxygen-rich limits, which were obtained using the $\text{HSE}\left(a=0.375\right)$ hybrid functional with the finite-size corrections. The slope corresponds to the charge state. (b) Defect transition levels equivalent to the filled circles in (a), alongside the relevant charge states.

Figure 3

(Color online) Band structure for the perfect crystal, $V_{\text{O}}$ in the neutral and $2+$ charge states, and ${\text{Zn}}_{i\left(o\right)}$ in the neutral charge state, which was obtained using the $\text{HSE}\left(a=0.375\right)$ hybrid functional. Shown below are the squared wave functions of the states designated by the arrows, which are plotted for the middle of the (0001) Zn and O planes adjacent to the defects. The projected Zn and O atom positions are denoted with green (dark) and yellow (light) circles, respectively.