Dispersion of Defects in TiO 2 Semiconductor: Oxygen Vacancies in the Bulk and Surface of Rutile and Anatase

: Oxygen deﬁciency (O-vacancy) contributes to the photoe ﬃ ciency of TiO 2 semiconductors by generating electron rich active sites. In this paper, the dispersion of O-vacancies in both bulk and surface of anatase and rutile phases was computationally investigated. The results showed that the O-vacancies dispersed in single- and double-cluster forms in the anatase and rutile phases, respectively, in both bulk and surface. The distribution of the O-vacancies was (roughly) homogeneous in anatase, and heterogenous in rutile bulk. The O-vacancy formation energy, width of defect band, and charge distribution indicated the overlap of the defect states in the rutile phase and thus eased the formation of clusters. Removal of the ﬁrst and the second oxygen atoms from the rutile surface took less energy than the anatase one, which resulted in a higher deﬁciency concentration on the rutile surface. However, these deﬁciencies formed one active site per unit cell of rutile. On the other hand, the ﬁrst O-vacancy formed on the surface and the second one formed in the subsurface of anatase (per unit cell). Supported by previous studies, we argue that this distribution of O-vacancies in anatase (surface and subsurface) could potentially create more active sites on its surface.


Introduction
Nonstoichiometric titanium dioxide (TiO 2 ) has drawn broad interest due to its important role in photocatalysis, solar cells, and environmental procedures [1][2][3][4][5][6][7][8][9]. Under various experimental conditions, a broad range of O-vacancies can be formed in TiO 2 (bulk and surface), i.e., 5-10% [10]. Inducing large effects on the electronic, photonic, and photocatalytic properties of TiO 2 , oxygen deficiencies are critical features [11][12][13]. Rutile and anatase are two widely used phases of TiO 2 , anatase being the more photocatalytically efficient for reasons that are not fully understood [14,15]. In addition to the intrinsic natures of the aforementioned phases, energetics, concentration, and location of O-vacancies could be significant factors in their photocatalytic activity. While surficial oxygen vacancies directly introduce active sites for the photocatalytic process, such as water dissociation and oxygen adsorption, the bulk ones could also affect surface activities [10].
Having a looser and more flexible structure, anatase has been reported to have higher O-vacancy concentration and thus higher photocatalytic activity based on ultraviolet photoemission spectroscopy (UPS) results [16]. However, many other experimental results have indicated a lower concentration of O-vacancy on the surface of anatase compared to rutile [17,18]. Elsewhere, similar concentrations of O-vacancy defects in the two polymorphs has been reported, based on elemental analysis [19]. A majority of experimental studies, however, suggest a higher surface/bulk ratio of O-vacancy in rutile [10,20]. Inferred from calorimetric measurements, it has been claimed that the lower density of surficial O-vacancies in anatase is consistent with its higher surface stability [21].
To shed more light on the existing controversy, this study investigated the dispersion of O-vacancies in the bulk and surface of anatase and rutile. To this end, single O-vacancies and their effects were computationally studied and compared with those of double O-vacancies. Being methodologically consistent, all geometrically unique arrangements were simulated for single/double O-vacancy structures using full potential DFT calculations.

Computational Details
Anatase and rutile TiO 2 have tetragonal structures with the space groups of D 14 4h − p4 2 /mnm and C 19 4h − I4 1 /amd, respectively. The optimized lattice constants of the bulk structures are a = b = 4.594 Å and c = 2.96 Å for rutile and a = b = 3.785 Å and c = 9.51 Å for anatase phase, with a margin of less than 1% compared to the experimental values. For both perfect and defective states, 2 × 2 × 1 and 2 × 2 × 2 supercells for anatase and rutile were used. First-principle calculations using full potential periodic DFT and generalized gradient approximations (GGA) were employed to find the most stable structures. The revised form of Pardew, Burke, and Ernzerh (rPBE) level of theory, implemented in FHI aims program [42], was used to optimize pure and defective supercells. As an alternative, hybrid functionals such as Heyd-Scuseria-Ernzerhof (HSE) were used, along with plain DFT, to reproduce accurate defect states and band gaps, and to compensate for the lack of electron self-interaction and derivative discontinuity. The criteria for the convergence of energy and force were set to 5 × 10 −6 eV and 0.01 eV/Å, respectively. K-grids of 4 × 4 × 6 and 6 × 6 × 2 were used to model the first Brillouin zone for rutile and anatase unit cells, respectively. Modeling was performed for different defective structures to study the size effects (Ti 8 30 were used to simulate the perfect and single/double O-vacancy structures, respectively. For the defective structures, one and two O atoms were eliminated from the super cell, resulting in 6.25% at. and 12.5% at. concentrations of the O-vacancy. To study the effects of O-vacancy on the electronic structure of the aforementioned structures, partial density of states pDOS was calculated using the hybrid method HSE06 (with 25% admixture of HF exchange) and rPBE. Grids of 12 × 12 × 18 and 18 × 18 × 6 k-points were used for integrating the first Brillouin zone during density of states (DOS) calculations of rutile and anatase, respectively. A periodic slab model of 2 × 2 and 2 × 3 surface unit cells of rutile (110) and anatase (101), as their most stable surfaces, with a vacuum of 15 Å was considered in all simulations. The rutile slab, including 64 oxygen and 32 titanium atoms, was made of four layers of O-Ti-O, and therefore 12 atomic layers. The anatase slab was made of 48 oxygen and 24 titanium atoms.

O-Vacancy in the Bulk Structure of Rutile and Anatase
Single O-vacancy TiO 2 anatase and rutile bulk structures have been frequently studied both theoretically and experimentally [25][26][27][28][29][30][31][32]. It has been demonstrated that oxygen deficiency induces some charge on the two Ti atoms next to the vacancy position, and that the resulting occupied states appeared near the conduction band [33,34]. This has been evaluated here in detail via pDOS and charge dispersion calculations. Figure 1 shows the pDOS patterns of pure and single O-vacancy of anatase and rutile structures, which clearly represented the main contribution of O 2p and Ti 3d orbitals in the valance and conduction bands, respectively. This was in a close agreement with previous results [38][39][40][41], for the defect states of anatase and rutile. The band gap, regardless of defect states, increased by generation of oxygen defect from 3 to 3.4 eV in rutile, and from 3.2 to 3.6 eV in anatase structures, similar to our previous study [43]. However, the calculated band gap energies were 0.12, 0.63, and 1.13 eV for rutile and 0.44 and 1.1 eV for anatase, with respect to defect states ( Figure 1). The charge transfer from oxygen deficiency was visualized by taking the corresponding difference in electron densities. Figure 2 exhibits the main released charge from a single O-vacancy that was localized on the two and three nearest Ti atoms to the defect position in rutile and in anatase, respectively. Finally, the single O-vacancy formation energy was calculated using Equation (1).
where E formation , E (O-vacancy) , and E (prefect) are the formation energy, super cell energy with a single O-vacancy, and the perfect super cell energy, respectively. The parameter n is the number of vacancies and µ O is the chemical potential of O that is half of the total energy of an isolated oxygen molecule.
The results give formation energies of 6.42 and 6.49 eV for 48-atom supercells of anatase and rutile phases with a single O-vacancy, respectively. Lower defect formation energy of anatase could be justified by its more flexible structure [22]. Removal of one oxygen molecule from 24, 48, 96, and 192 atom supercells of anatase and rutile bulk created concentrations of 12.5%, 6.25%, 3.12%, and 1.56% at. O-vacancy, respectively. The formation energy of the single O-vacancy structure diminished by increasing the size of the unit cell in the bulk of anatase and rutile, e.g., 6.6 eV (12.5%), 6.4 eV (6.25%), 6.2 eV (3.12%), and 5.2 eV (1.56%) for anatase phase, similar to previous studies [23]. Based on our previous study, a metal-metal impurity prefers a heterogeneous dispersion in the bulk of anatase and rutile [44]. In addition, large defect points in sub-stoichiometric TiO 2 have been observed experimentally, representing the presence of cluster defects structures [10,20].  Table 1. Accordingly, there was no significant correlation between the O-O distance and the crystal structure stability. The most stable configurations of O-O in anatase and rutile bulks, with a respective distance of 3.8 and 2.8 Å, are shown in Figure 3.  These defects were located around the same Ti atom; one at the equatorial position and the other at the axial position in the octahedral arrangement of O atoms around the Ti atom in the rutile phase ( Figure 3). These results were also doubly confirmed by the ABINIT code [45] in this study.
Formation energy of the double was expected to be more than twice that of the single O-vacancy of rutile; however, the elimination energy of the second O (6.1 eV) was easier than the first one (6.5 eV). This indicated an overlap between the defect states generated by the O-vacancies in TiO 2 rutile structure. In contrast, O-vacancies behaved independently in the anatase bulk and had larger distances, whereas the formation energy of the double O-vacancy (12.65 eV) was around twice that of the single one (6.4 eV). This could have been due to the fact that the local structure of rutile (D 4 h) was more stable than that of anatase (D 2 d), which provided shorter Ti-Ti distances and a denser phase, leading to an easier overlap between orbitals. Accordingly, based on the pDOS calculations, the lowest energy states lay at the bottom of the conduction band, consisting of plain Ti-d xy orbitals in anatase with non-bonding character ( Figure 1) [46]. However, the main contributors to the lowest region of the conduction band in rutile consisted of Ti-d xy and O-2 p orbitals with anti-bonding character [47]. Therefore, a shorter Ti-Ti distance in the case of double O-vacancy in the rutile phase may have provided a better overlap between neighboring Ti-3d orbitals, which lessened the ground state energy.
The hybrid methods were also used to explore the effect of increasing the concentration of the O-vacancies and the interactions between them in the TiO 2 super cells. Based on the HS06 results, the presence of two O-vacancies increased the occupied 3d defect states in the band gap for both anatase and rutile phases ( Figure 1). As is shown, the defect states of the two O-vacancies occupied a width of 1.2 eV and 1.4 eV for anatase and rutile, respectively, inside the gap bellow the conduction band. A comparison of the defect bands of the single defect structures, with a width of 1.1 eV (Figure 1), showed an overlap between the O-vacancy defect states in the rutile phase, where molecules arranged tightly. In the anatase phase, the transferred charges from each defect were independently localized on the three nearest Ti atoms, while two O-vacancies in rutile acted as a large defect point, their charges were localized on the three nearest Ti atoms as well (Figure 4).

O-Vacancy on the Surface of Rutile and Anatase
While O 2c (bridge) atom has been widely reported as the easiest atom to remove for the formation of O-vacancy@rutile (110) surface, contradictory results exist for O-vacancy@anatase (101) [16,[22][23][24].   It has been claimed that larger defect sites can impair quantum yields by acting as recombination centers for photogenerated electron-hole pairs [10]. This may also have been a reason for the lower activity of the rutile compared to anatase phase. The O-vacancies of the anatase phase distributed on the surface and sub-surface, which has been shown to increase its photocatalytic activity in water adsorption and dissociation, playing the role of active sites [24]. Thus, higher and smaller active points on the anatase surface could have played a critical role in its higher activity in comparison with rutile ( Figure 6).

Conclusions
Plain DFT and Hybrid methods were used to study defective anatase and rutile TiO 2 with single and double O-vacancy defects using the FHI aims code (and ABINIT code for confirmation). Using hybrid methods, band structure calculations resulted in band gap energies in close agreement with experimental values for rutile (3.0 eV) and anatase (3.2 eV). The most stable O-O-vacancy configuration in the rutile crystal structure was where the two O-vacancies were next to each other on the same Ti atom. However, a homogeneous dispersion in the anatase morphology was preferred. The elimination of the second O-vacancy turned out to be easier than the first one in the defective anatase and rutile TiO 2 . While oxygen deficiency had similar results on the electronic structures of anatase and rutile structures, their charge distributions differed; the transferred charge from two defects were localized on three nearest Ti atoms in rutile, whereas defect positions disperse their charges independently in anatase phase.
In the surface studies, a single O-vacancy was placed in an O 2c -bridge atom on both (101) anatase and (110) rutile surfaces, but unlike the bulk results, the single O-vacancy formation energy value on anatase surface was higher than that of rutile. While a double cluster of O-vacancy was preferred on the rutile (110) surface, as a result of omitting the two nearest O 2c -bridge atoms, the O-vacancies dispersed in the first layer (surface) and the sub-layer in anatase (101) structure, resulting in cluster-and singledefect features of O-vacancies in the rutile and anatase structures, respectively. The elimination of the first oxygen atom had a synergistic effect for the second O-vacancy for both anatase and rutile, yet these defects generated higher active sites on the surface of anatase.

Conflicts of Interest:
The authors declare no conflict of interest.