Figure 4.

Formation energies of the O vacancy in neutral, + and 2+ charge states in ZnO as a function of the Fermi level, obtained using the HSE (a=0.375) hybrid functional in conjunction with finite cell-size corrections [44, 81]. (a) O-poor (Zn-rich) limit. (b) O-rich (Zn-poor) limit. The Fermi level at which lines of two charge states intersect corresponds to the thermodynamic transition level, as given by equation (11). Note that the + charge state is unstable for any value of the Fermi level, and, hence, only the ε(2+/0) level appears, as indicated by the open circle in each panel.