Analysis of Intrinsic Defects in CeO2 Using a Koopmans-Like GGA+U Approach

Spin-density for a Ce vacancy, with U applied to O and Ti states, to give a defect state localised as four oxygen holes.We have investigated the formation of intrinsic defects in CeO2 using density functional theory with the generalized gradient approximation (GGA) corrected for on-site Coulombic interactions (GGA+U). We employed an ab initio fitting procedure to determine a U {O2p} value that satisfies a Koopmans-like condition and obtained a value of U {O2p} = 5.5 eV. We subsequently demonstrated that by applying GGA+U to the O2p states, in addition to the Ce 4f states, we were able to model localized holes in addition to localized electrons, thus improving the description of p-type defects in CeO2. Our results show that under oxygen-poor conditions the defects with the lowest formation energy are oxygen vacancies, while oxygen interstitials, which form peroxide ions, will be more favorable under oxygen-rich conditions. We carried out temperature and pressure dependence analyses to determine the relative abundance of intrinsic defects under real-world conditions and determined that oxygen vacancies will always be the dominant defect. Furthermore, we determined that at the dilute limit none of the defects studied can account for the intrinsic ferromagnetism that has been observed in nanosized CeO2.

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GGA+U Description of Lithium Intercalation into Anatase TiO2

We have used density-functional theory [generalized gradient approximation (GGA)] to study lithium intercalation at low concentration into anatase TiO2. To describe the defect states produced by Li doping a Hubbard “+U” correction is applied to the Ti d states (GGA+U). Uncorrected GGA calculations predict LixTiO2 to be metallic with the excess charge distributed over all Ti sites, whereas GGA+U predicts a defect state 0.96 eV below the conduction band, in agreement with experimental photoelectron spectra. This occupied defect state corresponds to charge strongly localized at a single Ti 3d site neighboring the intercalated lithium with a magnetization of 1 μB. This polaronic state produces a redshifted optical absorption spectrum, which is compared to those for the native O-vacancy and Ti-interstitial defects. The strong localization of charge at a single Ti center lowers the symmetry of the interstitial geometry relative to that predicted by GGA. The intercalated lithium sits close to the center of the octahedral site, occupying a single potential energy minimum with respect to displacement along the [001] direction. This challenges the previous interpretation of neutron diffraction data that there exist two potential energy minima separated by 1.6 Å along the [001] direction within each octahedron. Nudged elastic band calculations give barriers to interoctahedral diffusion of ~0.6 eV, in good agreement with experimental data. These barrier heights are found to depend only weakly on the position of the donated electron. The intercalation energy is 2.14 eV with GGA and 1.88 eV with GGA+U, compared to the experimental value of ~1.9 eV. Li-electron binding energies have also been calculated. The [Lii-Ti′Ti] complex has a binding energy of 56 meV, and a second electron is predicted to be bound to give [Lii-2Ti′Ti] with a stabilization energy of 30 meV, indicating that intercalated lithium will weakly trap excess electrons produced during photoillumination or introduced by additional n-type doping.

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Intrinsic n-type Defect Formation in TiO2: A Comparison of Rutile and Anatase from GGA+U Calculations

The formations of intrinsic n-type defects, that is, oxygen vacancies and titanium interstitials, in rutile and anatase TiO2 have been compared using GGA+U calculations. In both crystal structures, these defects give rise to states in the band gap, corresponding to electrons localized at Ti3+centers. O vacancy formation in rutile results in two excess electrons occupying 3d orbitals on Ti atoms neighboring the vacancy. Similarly, for anatase, two Ti 3d orbitals are occupied by the excess electrons, with one of these Ti sites neighboring the vacancy, and the second at a next-nearest Ti position. This localization is accompanied by one oxygen moving toward the vacancy site to give a “split vacancy” geometry. A second fully localized solution is also found for anatase, with both occupied Ti sites neighboring the vacancy site. This minimum is 0.05 eV less stable than the split vacancy and is thus expected to be present in experimental samples. A partially delocalized solution corresponding to the split vacancy geometry, with one electron occupying the bottom of the conduction band, is also identified as 0.28 eV less stable. Formation of titanium interstitials donates four electrons to the Ti lattice. In anatase, one of these electrons is located at the interstitial Ti site, and three occupied defect states are hybridized between three nearest neighbor Ti sites. In rutile, these excess electrons are mostly localized at four nearest neighbor Ti sites, with only a small amount of excess charge found on the interstitial Ti atom. This difference in the charge on the interstitial atom is a consequence of the differing interstitial geometries in the two polymorphs. Calculated optical absorption spectra for all defects show significant decreases of the optical band gap, with a larger red shift predicted for titanium interstitials in anatase than in rutile. Defect formation energies have been calculated under oxygen-rich and oxygen-poor conditions for both polymorphs. Under all conditions, O vacancy formation is slightly more favorable in anatase than in rutile, while Ti interstitials form more easily in rutile than anatase. Under O-rich conditions, O vacancies are the favored defect type, but both defect types have high formation energies. Under O-poor conditions, both defect types are stabilized, with Ti interstitials predicted to become the favored defect in rutile samples, particularly at elevated temperatures.

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