Ab initio analysis of the defect structure of ceria

We calculated the formation energies of all simple point defects in cubic fluorite structured ${\mathrm{CeO}}_2$ using density functional theory within the GGA+$U$ approximation. All possible defect charge states were considered, and also polarons ${\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }$ and associates of polarons with oxygen vacancies: ${({\mathrm{V}}_{\mathrm{O}}^{··}-{\mathrm{Ce}}_{\mathrm{Ce}}^{\prime })}^{·}$ and ${({\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }-{\mathrm{V}}_{\mathrm{O}}^{··}-{\mathrm{Ce}}_{\mathrm{Ce}}^{\prime })}^{\times }$. From the individual defect energies, we extracted Schottky, Frenkel, and anti-Frenkel energies: we find that anti-Frenkel disorder has the lowest energy in ceria. Energies for the reduction and the hydration of ceria are also computed, and the results are in good agreement with experiment. Finally, point-defect concentrations and conductivities are predicted for undoped and donor-doped systems as a function of oxygen partial pressure and temperature. The characteristic slopes found in experiment are reproduced.

Figure 1

DOS of the supercell with (a) an additional band electron and (b) a polaron. In addition, the charge density of the highest occupied state of the polaron supercell is shown: all charge is localized on one cerium ion.

Figure 2

Partial charge densities of the two highest occupied states of (a) a neutral oxygen vacancy ${\mathrm{V}}_{\mathrm{O}}^{\times }$ and (b) a neutral polaron-vacancy cluster ${({\mathrm{Ce}}_{\mathrm{Ce}}^{\prime }-{\mathrm{V}}_{\mathrm{O}}^{··}-{\mathrm{Ce}}_{\mathrm{Ce}}^{\prime })}^{\times }$. The vacancy position is indicated by the empty square.

Figure 3

GGA+$U$ formation energy of oxygen vacancies at $T=0$ K including the formation energies of polaron-vacancy clusters for O-rich (straight lines) and O-poor (dotted lines) conditions as a function of the Fermi energy, which is set zero at the valence band maximum. The dashed line indicates the position of the lowest Ce 4f states.

Figure 4

GGA+$U$ formation energies of all ionic defects and polarons at $T=0$ K as a function of the Fermi energy under (a) O-poor and (b) O-rich conditions. The dashed line indicates the position of the lowest unoccupied Ce $4f$ states.

Figure 5

Defect formation energies of H interstitials in a $\mathrm{H}$-rich environment at $T=0$ K as a function of the Fermi energy. The configuration with the hydrogen in the octahedral hole is denoted “oct” and the one with a hydrogen close to an oxygen atom “oxy.” The dashed line indicates the position of the lowest Ce $4f$ states.

Figure 6

Calculated defect concentrations in undoped ceria at $T=650$ K as a function of the oxygen partial pressure. At large oxygen partial pressures, the doubly charged oxygen vacancy is the dominant vacancy species. At low $p{\mathrm{O}}_2$, the neutral polaron-vacancy associate is predominant, while ${({\mathrm{V}}_{\mathrm{O}}^{··}-{\mathrm{Ce}}_{\mathrm{Ce}}^{\prime })}^{·}$ compensates the polarons. The dotted line indicates defect concentrations calculated with a Fermi-Dirac approach.

Figure 7

Defect concentrations in 1 ppm donor-doped ceria at $T=650$ K as a function of the oxygen partial pressure. For high oxygen partial pressures, the polaron concentration is fixed by the donor ${\mathrm{D}}^{·}$, while for intermediate partial pressures, ${\mathrm{V}}_{\mathrm{O}}^{··}$ compensate the polarons. At very low $p{\mathrm{O}}_2$, the polarons are compensated by ${({\mathrm{V}}_{\mathrm{O}}^{··}-{\mathrm{Ce}}_{\mathrm{Ce}}^{\prime })}^{·}$. The dotted line indicates defect concentrations calculated with a Fermi-Dirac approach.

Figure 8

Conductivities calculated from the polaron concentrations of undoped ceria at varied temperatures with mobility values from Ref. 24. The dotted lines correspond to the Fermi-Dirac approach [see Eq. (10)], the thin solid lines to the Boltzmann ansatz [see Eq. (7)], and the thick solid lines to the dilute defect regime.

Figure 9

Conductivities calculated from the polaron concentrations for 1000 ppm donor-doped ceria (e.g., 0.1$\%$ U) at various temperatures with mobility values from Ref. 24. In regime I, the polarons are charge compensated by the polaron-vacancy associate ${({\mathrm{V}}_{\mathrm{O}}^{··}-{\mathrm{Ce}}_{\mathrm{Ce}}^{\prime })}^{·}$. In regime II, the polarons are compensated by ${\mathrm{V}}_{\mathrm{O}}^{··}$. In regime III, the polaron concentration is fixed by the donor concentration. The dotted lines correspond to the Fermi-Dirac approach [see Eq. (10)], the thin solid lines to the Boltzmann ansatz [see Eq. (7)], and the thick solid lines to the dilute defect regime.