Density functional theory has been used to study lithium intercalation into TiO_{2}(B) at low to moderate concentrations (0 < *x*(Li) ≤ 0.25) with a range of density functionals: LDA, GGA (PW91, PBE, PBEsol), and GGA+*U* (PBE+*U*, PBEsol+*U*), with the GGA+*U* calculations employing a Hubbard +*U* correction to the Ti d states. LDA and GGA functionals give the same general behaviour, whereas qualitatively different behaviour is predicted by GGA+*U* for electronic structure and the order of stability of occupied intercalation sites. LDA/GGA functionals predict Li_{x}TiO_{2}(B) to be metallic, with the excess charge distributed over all the Ti sites. In contrast, GGA+*U* predicts defect states in the band gap corresponding to charge strongly localised at specific Ti sites. All the considered functionals predict A1 and/or A2 site occupation at *x*(Li)=0.25, which challenges the interpretation of previous neutron data that at this composition the C site is preferentially occupied.

# Lithium intercalation into TiO

# The Role of Lithium Ordering in the LixTiO2 Anatase → Titanate Phase Transition

The mechanism of the tetragonal to orthorhombic phase separation of Li-intercalated anatase TiO_{2} has previously been proposed to be a cooperative Jahn-Teller distortion due to occupation of low-lying Ti 3d_{xz,yz} orbitals.^{
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Using density functional calculations we show that the orthorhombic distortion of Li_{0.5}TiO_{2} is not a purely electronic phenomenon, and that intercalated Li plays a critical role. For a 2×1×1 expanded supercell for 0 ≤ *x*(Li) ≤ 1, the intercalation voltage is minimized for *x*(Li) = 0.5. The low energy structures display a common structural motif of edge-sharing pairs of LiO_{6} octahedra, that allows all Li to adopt favourable oxygen coordination. Long-ranged disorder of these sub-units explains the apparent random Li distribution seen in experimental diffraction data.

# 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 TiO_{2}. 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 Li_{x}TiO_{2} 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 μ_{B}. 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 [Li^{•}_{i}-Ti′_{Ti}] complex has a binding energy of 56 meV, and a second electron is predicted to be bound to give [Li^{•}_{i}-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.

# 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 TiO_{2} 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 Ti^{3+}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.