First-principles determination of chemical potentials and vacancy formation energies inPbTiO3andBa

Formation energies of different neutral vacancies (${V}_{\mathrm{Pb}}$, ${V}_{\mathrm{Ti}}$, ${V}_{\mathrm{O}}$, and ${V}_{\mathrm{Ba}}$) in tetragonal $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$ and tetragonal $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ are studied by means of first-principles pseudopotential calculations. Using thermodynamic laws, we also analytically determine the constraints that yield valid chemical potentials for atomic reservoirs in equilibrium with ferroelectric solids, and examine how the energies of vacancy formation may be modified by varying reservoir conditions. We further contrast the defect energetics in $\mathrm{Ba}\mathrm{Ti}{\mathrm{O}}_{3}$ with those in $\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_{3}$, revealing notable differences. Detailed findings of this study are rather lengthy and are summarized in Sec. IV.