  We demonstrate a thermodynamic formulation to quantify defect formation
energetics in an insulator under high electric field. As a model system, we
analyzed neutral oxygen vacancies (color centers) in alkaline-earth-metal
binary oxides using density functional theory, Berry phase calculations, and
maximally localized Wannier functions. Work of polarization lowers the field
dependent electric Gibbs energy of formation of this defect. This is attributed
mainly to the ease of polarizing the two electrons trapped in the vacant site,
and secondarily to the defect induced reduction in bond stiffness and softening
of phonon modes. The formulation and analysis have implications for
understanding the behavior of insulating oxides in electronic, magnetic,
catalytic, and electrocaloric devices under high electric field.
