First-principles study of charged vacancies in PbTiO3

First-principles calculations within local density approximation (LDA) are performed to study the vacancy formation energies for charged vacancies in PbTiO$_3$. Within the constraint of thermodynamic limit, the chemical potential and Fermi energy dependence of the vacancy formation energy, as well as the transition energy levels, are studied. We found that: (i)the Formation energy of $V_{O}^{2+}$ is much lower than that of the neutral oxygen vacancy, irrespective of $\mu_O$; (ii) Under oxygen rich condition, the main defect is $V_{Pb}^{2-}$; (iii) When oxygen is deficient, either $V_{Pb}^{2-}$ or $V_{O}^{2+}$ can form, depending on the Fermi energy level; (iv) At certain condition, both $V_{Pb}^{2-}$ and $V_{O}^{2+}$ have similar formation energy, and can coexist.