Polarizing Oxygen Vacancies in Insulating Metal Oxides under a High Electric Field

We demonstrate a thermodynamic formulation to quantify defect formation energetics in an insulator under a 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. The 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 a high electric field.

Figure 1

(a) Relative electric Gibbs free energy of formation and (b) relative formation energy of $V_{\mathrm{O}}^{\times }$ as a function of electric field in the studied oxides.

Figure 2

(a) The field dependence of the dipole moment of $V_{\mathrm{O}}^{\times }$, $|{\overrightarrow{\mathbf{p}}}_{V_{\mathrm{O}}^{\times }}|$. For comparison, the zero-field dipole moment of the gas-phase water molecule $|{\overrightarrow{\mathbf{p}}}_{{\mathrm{H}}_2\mathrm{O}}^0|=1.86\mathrm{D}$ [68] is indicated by the black horizontal line. (b) Field-dependent polarizability of $V_{\mathrm{O}}^{\times }$, ${\alpha }_{V_{\mathrm{O}}^{\times }}$. The inset focuses on the low-field polarizability in the case of BaO.

Figure 3

Visualizations of the charge density of the two electrons trapped in $V_{\mathrm{O}}^{\times }$ at zero field in (a) MgO, (b) CaO, (c) SrO, and (d) BaO. Similar visualizations at a field of $21.8\mathrm{MV}/\mathrm{cm}$ in $+x$ direction are shown for (e) MgO, (f), CaO, (g) SrO, and (h) BaO. Red, blue, cyan, green, and gray spheres represent O, Mg, Ca, Sr, and Ba ions, respectively. The yellow isosurfaces in (a)–(h) represent the electronic charge density and are taken at 15% of the maximum value in each plot. These visualizations were generated using the software xcrysden [74]. (i)–(l) High-field site-decomposed polarizability ${\alpha }_i$ as a function of distance $r$ from the defect site, in the case of (i) MgO, (j) CaO, (k) SrO, and (l) BaO. ${\alpha }_i$‘s were calculated by finite difference between field values of 18.2 and $21.8\mathrm{MV}/\mathrm{cm}$.

Figure 4

Field-dependent static permittivity of (a) the perfect crystal and (b) the defective crystal containing $V_{\mathrm{O}}^{\times }$ for the studied oxides. The inset in (b) focuses on the ${ϵ}^{\mathrm{def}}$ of BaO at low fields.