Deep versus Shallow Behavior of Intrinsic Defects in Rutile and

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Deep versus Shallow Behavior of Intrinsic Defects in Rutile and Anatase TiO2 Polymorphs Giuseppe Mattioli,*,†,‡ Paola Alippi,† Francesco Filippone,† Ruggero Caminiti,‡ and Aldo Amore Bonapasta† Istituto di Struttura della Materia (ISM) del Consiglio Nazionale delle Ricerche, Via Salaria Km 29.5, CP 10, 00016 Monterotondo Stazione, Italy, and Department of Chemistry, UniVersita di Roma “La Sapienza”, P.le A.Moro 2, 00185 Roma, Italy ReceiVed: May 6, 2010; ReVised Manuscript ReceiVed: September 27, 2010

The structural and electronic properties of oxygen vacancies (VOx) and titanium interstitials (Ti(i)) in the bulk of the rutile and anatase forms of TiO2 have been investigated with LSD-GGA+U ab initio simulations. In particular, formation energies of the charged and neutral forms of the VOx and Ti(i) defects as well as the corresponding vertical and thermodynamic transition levels have been estimated. The achieved results can reconcile the apparent inconsistency of experimentally observed deep donor levels with the n-type conductivity observed in reduced TiO2. They show indeed that both defects give rise to vertical transition levels about 1 eV below the conduction band (CB), in agreement with experimental measures, and to thermodynamic transition levels close to the CB. That is, these defects behave as deep donors, when looking at vertical transitions, and as shallow donors, when the effects of the structural relaxations are taken into account. A major part of the explanation of this behavior is played by the polaron-like character of the defect states, which was already noted, but not deepened, in literature. Finally, it is shown that the application of the U correction to both Ti and O species gives qualitatively similar results, but with a better agreement to experimental findings, with respect to the application to Ti only. The former approach gives pretty similar results, for both rutile and anatase bulk properties, to those coming from HSE hybrid functional calculations. I. Introduction Titanium dioxide (TiO2) is one of the most widely investigated metal oxides due to its many applications which include realization of solar cells or photo/electrochromic devices,1 use as a photocatalyst for environmental applications,2,3 and, in perspective, use as a magnetic semiconductor in spintronic devices.4,5 The electron conductivity of TiO2 plays a key role in all such applications. A peculiar property of this metal oxide is the appearance of an appreciable n-type conductivity in the reduced (i.e., O-poor) material.6-8 This feature is common to the two TiO2 polymorphs, rutile and anatase, although a significantly larger conductivity is observed in the latter form.6,9,10 The n-type conductivity of reduced TiO2 is generally related to the formation of intrinsic, donor defects, like oxygen vacancies (VOx) and titanium interstitials (Ti(i)).6 However, such a conductivity is not easily reconciled with an apparent deep character of the electronic levels induced by native defects in both forms of reduced TiO2. VOx and Ti(i) defects are related indeed to the existence of Ti3+ centers coming from the trapping of an exceeding defect electron at a Ti4+ site (in TiO2, Ti and O atoms have formal charges of +4 and -2, respectively). The existence of Ti3+ centers has been demonstrated by electron paramagnetic resonance (EPR) measurements,11,12 while a defect electronic charge localization at Ti3+ sites has been shown by resonant photoelectron diffraction (PED) investigations.13 In reduced TiO2, infrared (IR) absorption,14 photoelectron spectroscopy (PES),15-20 and electron energy loss spectroscopy (EELS)21 measurements indicate the presence of an electronic * To whom correspondence should be addressed, giuseppe.mattioli@ ism.cnr.it. † Istituto di Struttura della Materia (ISM) del Consiglio Nazionale delle Ricerche. ‡ Department of Chemistry, Universita di Roma “La Sapienza”.

level in the energy gap at about 1 eV below the conduction band minimum (CBM). Therefore, the level position in the energy gap and the electronic charge localization at Ti3+ sites suggest that, in reduced TiO2, VOx and Ti(i) defects behave as deep donors. Several density functional theory (DFT) studies have criticized the use of local density (LDA) or generalized gradient approximation (GGA) methods when investigating defects in TiO2.22-34 In this methodological framework, TiO2 defects induce shallow electronic levels, resonant with the CB and strongly delocalized on several Ti atoms, at variance with the experiment, see, e.g., ref 22. In the above studies, founded only on the analysis of the Kohn-Sham electronic spectrum, a better agreement with the experimental findings has been achieved by applying correction schemes to DFT, based on hybrid Hartree-Fock/DFT functionals (HF-DFT) or Hubbard U approaches (GGA+U). For instance, in the case of VOx, both corrections give Kohn-Sham electronic levels located at about 1 eV below the CBM and an appreciable localization of the exceeding electrons at Ti3+ sites in agreement with the experiment. In this framework, a polaronic nature of the charge at these sites is suggested, accompanied by a significant rearrangement of the local structure.22-34 Theoretical investigations have proposed polaron hopping models for the electron transport in order to explain the n-type conductivity of reduced TiO2.35,36 Some of the achieved results agree with the experimental findings by estimating, e.g., a low activation energy for electron transport in rutile. They do not account, however, for the different conductivity of rutile and anatase.36 The present GGA+U study extends previous studies on bulk VOx and Ti(i) defects mainly in two aspects: (i) both defects are investigated in both TiO2 polymorphs by using a same theoretical framework, thus allowing a direct comparison between the

10.1021/jp1041316  2010 American Chemical Society Published on Web 11/19/2010

Defects in Rutile and Anatase TiO2 Polymorphs VOx and Ti(i) properties as well as between their effects on rutile and anatase properties; (ii) both charged and neutral defects are investigated by estimating their formation energies and the corresponding transition energy levels.37 While some previous studies consider only one of the TiO2 forms, or only neutral defects, or only one kind of stoichiometric condition, here defect formation energies have been estimated as a function of both the two limit conditions of stoichiometry, O-poor (i.e., Ti-rich) and O-rich conditions, as well as as a function of different position of the Fermi energy (εF) in the cases of both neutral and charged defects. As expected, O-poor conditions lead to quite high concentrations of both VOx and Ti(i) in both the TiO2 polymorphs. Moreover, the formation of VOx seems favored both in rutile and anatase. The formation energies permit also to estimate transition energy levels, εq/q+1, corresponding to the position of the Fermi energy where the q and q + 1 charge states of the defect have the same energy.37 In particular, we have estimated both vertical q/q+1 , where the final state retains the starting transition states (εvert q/q+1 , state geometry) and thermodynamic transition states (εtherm where both state geometries are fully relaxed). These two different kinds of transition states allow a comparison with estimates of the position of the defect level in the energy gap as given by optical vertical excitations (e.g., PES measurements) and by techniques involving a structural rearrangement, e.g., deep level transient spectroscopy (DLTS) measurements, respectively.37 A novel theoretical picture is proposed for the bulk VOx and Ti(i) defects, as vertical and thermodynamic transition levels result to be quite different. For example, VOx in rutile shows a shallow character, suitable to account for the n-type conductivity +1/+2 0/+1 and εtherm levels are about 0.3 eV of the host, since the εtherm below and 0.1 eV above the CBM, respectively. Moreover, the formation energies also indicate that the V+1 Ox is thermodynami0 and cally favored in the n-type material with respect to VOx +2 VOx and induces a level carrying one electron which can be excited to the CB. At variance with the thermodynamic levels, +1/+2 level is located at about 1 eV below the CBM. the VOx εvert This means that it should be observed as a deep donor level by experimental techniques probing vertical excitations, as actually occurs with PES, EELS and IR results. Similar results are found for both defects in both TiO2 polymorphs. Therefore, both VOx and Ti(i) defects can show a deep or shallow character depending on the way in which they are probed. Such a duality can reconcile the above-mentioned, apparent, inconsistencies in the experimental findings. An explanation to this fact resides in the importance of the polaron-like nature of the defect electrons, actually in the amount of the relaxation energy accompanying the formation of Ti3+ centers. Previous DFT studies with hybrid functionals or the U correction (though used as a tunable parameter) have found cases of nonunivocal pictures for the TiO2 defects, leaving an uncertain localization of the defect electronic charge and claiming for refinements of the theoretical methods.23,25 Here, the Hubbard U correction to be applied to the 3d states of Ti has been calculated self-consistently following refs 38 and 39; this computational setting will be hereafter referred to as U(Ti). In order to attempt some improvement of the theoretical description, we have also performed calculations where an U correction is applied both to the Ti and O atoms (hereafter referred to as U(Ti,O)). U(Ti) and U(Ti,O) results have been carefully checked on rutile and anatase bulk properties against experimental as well as theoretical achievements obtained with the recently proposed Heyd-Scuseria-Ernzerhof (HSE), HF-DFT ap-

J. Phys. Chem. C, Vol. 114, No. 49, 2010 21695 proach.40 These checks revealed the existence of delicate interrelationships between the U correction, the lattice constants, and the convergence parameters, which may affect the results, especially the localization of the defect charge, as it will be discussed in detail in section 3.1. II. Theoretical Methods The properties of O vacancies and Ti interstitials in bulk rutile and anatase have been investigated by using DFT methods inside a (beyond-GGA) LSD-GGA+U (local spin density generalized gradient approximation, plus Hubbard U correction) approach38,39,41 as developed in the Quantum-ESPRESSO package.42 Calculations have been performed in a supercell approach, with ultrasoft pseudopotentials (US).43 The U-corrected functional has been built on the PBE gradient corrected exchange-correlation functional.44 96-atom (108-atom) supercells made up by 2 × 2 × 4 (3 × 3 × 1) unit cells have been used to simulate a rutile (anatase) stoichiometric bulk crystal as well as the properties of defects. Total energies have been calculated by using a 1 × 1 × 1 Monkhorst-Pack k-point mesh (out of Γ), expanding Kohn-Sham orbitals in plane waves up to energy cutoffs of 30 and 180 Ry for the wave functions and the charge density, respectively. Careful convergence tests have been performed by checking the results obtained with larger supercells and/or stricter convergence criteria: the use of 120-atom (2 × 2 × 5 rutile unit cells) supercells, 2 × 2 × 2 k-point mesh, or 35/210 Ry cutoffs gives negligible differences. Convergence has been also achieved with respect to the interaction of a defect with its own replicas (the dispersion of a defect level being below 0.2 eV). Defect equilibrium geometries have been optimized by minimizing the atomic forces of all of the atoms in the supercell. The presence of a compensating uniform background jellium has been assumed in charged supercells.37 The Hubbard U correction for the 3d electrons of rutile (anatase) Ti atoms, U(Ti), has been calculated by using the selfconsistent linear response approach described in refs 38 and 39 with a resulting value of 3.25 eV (3.23 eV). Ti(i) has a different local environment than Ti in the bulk; therefore the U correction for the Ti(i) atom has been recalculated in rutile (anatase), yielding a value of 3.47 eV (3.90 eV). Moreover, in addition to the U(Ti), a Hubbard U correction has been applied to the 2p electrons of O atoms in both anatase and rutile polymorphs, U(O), since Coulomb interactions between these electrons may be comparable to those between Ti 3d electrons.45-47 The U(O) value has been estimated by following two different approaches. First, a value of 10.65 eV (10.59 eV) has been calculated for the rutile (anatase) polymorph by the linear response selfconsistent procedure.38,39 Results from this setting will be hereafter labeled as U(Ti,O)B. Second, a smaller value of U(O) ) 5.0 eV has been tuned on experimental results, as suggested in refs 47 and 48. Results from this setting will be labeled as U(Ti,O)A. We validated the U(Ti), U(Ti,O)A, and U(Ti,O)B settings, by performing several simulations on 6-atom (12-atom) unit cells for rutile (anatase). Satisfactorily converged results were achieved by using a 4 × 4 × 4 (rutile) and a 4 × 4 × 2 (anatase) k-point mesh and the above-mentioned 30/180 Ry cutoffs. Further calculations have been performed by using a 48-atom 2 × 2 × 1 anatase supercell and a 2 × 2 × 2 k-point mesh. As a further check, the results given by the above settings for the rutile and anatase bulk properties have been compared with those from the projector augmented wave method (PAW)49 with the HSE hybrid functional as implemented in the Vienna Ab-Initio Simulation Package (VASP).50 In these calculations,

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we used the mixing parameter R ) 0.25 and the screening parameter µ ) 0.2, as in the prescriptions of ref 40. In all VASP calculations, we have used a cutoff energy of 400 eV (≈30 Ry) together with Γ-centered 8 × 8 × 6 (rutile) and 4 × 4 × 2 (anatase) k-point grids. In the following, we shall mainly discuss results from U(Ti) and U(Ti,O)A settings. The energetics and the electronic properties of VOx and Ti(i) have been investigated by estimating formation energies and transition energy levels. The formation energy of a q-charged defect, Ωf[Dq], is defined in a standard way37 from total energies of bulk and defected TiO2 supercells, E[TiO2] and E[Dq], respectively, as

Ωf[Dq] ) E[Dq] - E[TiO2] -

∑ nνµν + q(εF + εVBM) ν

where nν is the difference between the number of atoms of a given species ν in the defected and the bulk supercells and µν is the chemical potential of the same species, while εF is the Fermi energy and εVBM is the maximum of the valence band (VBM)51,52 taken at the Γ point (whose inclusion in the k-point mesh has actually negligible numerical effects on the formation energy values). Any Makov-Payne-like correction is negligible, given the high dielectric constant in TiO2 rutile and anatase.6 The Fermi energy, referenced to the VBM, represents the chemical potential of the electrons, assumed in a reservoir in contact with the system, and available in order to change the charge state of the defects. Similarly, the chemical potentials µTi and µO correspond to the energy of the reservoirs with which atoms of each species are exchanged. Their values are determined by thermodynamic equilibrium constraints (µTi + 2 µO ) µTiO2), and by their upper bounds, i.e., the chemical potentials of the natural phases of each species, the O2 molecule and Ti-hcp bulk, in order to avoid the formation of such phases. The equilibrium condition allows to consider, in the formation energies, only a single chemical potential. A possible choice is thus to report values of Ωf as a function of µO. Under O-rich conditions, equilibrium with O2 molecule is attained and µO equals half the energy of the O molecule, E[O2]. For the sake of simplicity, in the following, we shall plot formation energies as a function of the parameter µO′ ) µO - 1/2E[O2], ranging from 0 (O-rich conditions) to µ′(B) ) 1/2 ∆Hf(TiO2) (Ti-rich conditions), where ∆Hf is the formation heat of the bulk phase. Under Ti-rich (O-poor) conditions, a further constraint on the range of µO can be considered, i.e., the stability of TiO2 with respect to the other Ti-rich oxide phase, Ti2O3 (2µTi + 3µO e µTi2O3). This sets an upper bound to µO′, µ′ (A) ) 2∆Hf(TiO2) - ∆Hf(Ti2O3).53,54 At equilibrium conditions, the formation energy of a defect permits estimation of its concentration in the host, C[Dq], through the expression C[Dq] ) Nsites exp(-Ωf[Dq]/kBT), where Nsites is the number of available sites for the defect in the crystal. As anticipated, transition energy levels (εq/q+1) correspond to the position of the Fermi energy where the q and q + 1 charge states of the defect have equal formation energy.37,51,52 Differences between vertical and thermodynamic transition levels will be illustrated in the discussion of results. Energy levels of the bulk and defective charged systems are aligned through an electrostatic potential alignment.37 As a final note, in the following section, negative formation energies for the VOx and Ti(i) defects with respect to Fermi level will be also reported. In appearance, this would indicate a release of energy accompanying the defect formation which would eventually lead to an indefinite increase in the defect concentra-

Mattioli et al. tion. However, it has to be kept in mind that Fermi level is not a free parameter, but it is fixed by imposing the condition of charge neutrality. This hinders possible detrimental effects on the stability of the material (see refs 37, 67 and 68). III. Results and Discussion A. Effects of Beyond-GGA Approaches on Rutile and Anatase Bulk Properties. As anticipated, several studies report that DFT-GGA simulations fail to describe the charge localization (and therefore the structural relaxation) of Ti3+ centers in anatase and rutile TiO2 polymorphs, thus suggesting that beyondGGA approaches must be used to achieve a correct description of the above properties of such systems.22-32 Nevertheless, beyond-GGA results need to be carefully checked as well, as shown by the above-mentioned uncertain localization of the electronic charge of TiO2 defects.23,25 Herein, we checked our U(Ti) and U(Ti,O) results on the rutile and anatase bulk properties by using the following procedure: (i) convergence tests on preliminary GGA results of bulk TiO2 properties; (ii) application of U on TiO2, recalculation of bulk properties, comparison with GGA; (iii) further convergence tests on the GGA+U bulk properties. Such a procedure showed the existence of delicate relationships between the U correction, the bulk properties, and the convergence parameters. As an example, the difference of GGA+U and GGA equilibrium lattice constants is numerically meaningful. Thus, the use of GGA lattice constants in GGA+U simulations of TiO2 defects (as performed in some of the above-mentioned studies) could give erroneous results due to spurious strains in the TiO2 host matrix. The use of the U correction also affects energy cutoff (actually higher than in GGA). The results of GGA and GGA+U calculations regarding rutile and anatase bulk properties are reported in Table 1 together with those of HSE calculations and experimental results. Accurately converged PBE-GGA results, which can be also compared with previous results (see, e.g., ref 55), provide a sort of “common ground” to assess different beyond-GGA settings. In the table, PBE-US and PBE-PAW indicate GGA results obtained by using US and PAW pseudopotentials, respectively. Then, we have considered the U(Ti) setting, with a Hubbard U value of 3.23 eV (3.25 eV) for the 3d electrons of anatase (rutile) Ti atoms. We recall that the U(Ti,O)B label in Table 1 indicates a setting with a combination of the U(Ti) and an U(O), operating on O 2p states, of 10.59 eV (10.65 eV) for the anatase (rutile) O atoms, while the U(Ti,O)A label indicates a similar setting with U(O) ) 5.0 eV for both polymorphs. It may be noted that, in the case of the U(Ti,O), such a cautious, 2-fold approach is justified by the novelty of the combined use of U(Ti) and U(O) corrections, never employed previously in simulations of the TiO2 properties, to the best of the authors’ knowledge. A good description of some structural and thermodynamic properties of both TiO2 polymorphs is given already by PBE simulations: lattice parameters, Ti-O bond distances, bulk moduli, and formation heats nicely agree with the available experimental data (see the PBE-US and PBE-PAW lines in Table 1). On the other hand, PBE fails to describe the localization of Ti3+ states and, as it is well-known, provides a Kohn-Sham energy gap (Eg,KS) which underestimates the corresponding experimental value: a value of 2.16 eV (1.78 eV) for anatase (rutile) polymorph has to be compared to the experimental value of 3.2 eV (3.0 eV). PBE results tend also to reduce both the distance from the CBM of the semicore Ti 3p states and the dispersion of the O 2p states, compared to experimental data. In anatase, the main features of the PBE-

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TABLE 1: From Left to Right, Lattice Constants, Atomic Distances, Bulk Moduli, Formation Enthalpies, Energy Gaps Given by Kohn-Sham Eigenvalues (Eg,KS), Distance of Ti 3p Orbitals from the Top of the Valence Band, and Dispersion of O 2p Orbitals in the Valence Band Reported According to Different Methods of Calculations (See the Text) and Experiments (Oap and Oeq Indicate Apical and Equatorial O atoms, Respectively) method

a (Å)

c (Å)

Ti-Oap (Å)

Ti-Oeq (Å)

PBE-US PBE-PAW U(Ti) U(Ti,O)A U(Ti,O)B HSE exptl

3.80 3.81 3.83 3.79 3.76 3.76 3.782a

9.71 9.73 9.75 9.77 9.77 9.63 9.502a

2.00 2.01 2.01 2.00 1.99 1.98 1.979a

1.95 1.95 1.96 1.95 1.93 1.93 1.932a

PBE-US PBE-PAW U(Ti) U(Ti,O)A U(Ti,O)B HSE exptl

4.64 4.64 4.64 4.59 4.56 4.59 4.587a

2.97 2.94 3.01 2.99 3.00 2.95 2.954a

2.00 1.97 2.01 1.98 1.97 1.98 1.976a

1.96 1.95 1.98 1.96 1.96 1.94 1.946a

a

B0 (GPa)

∆Hf (eV)

Anatase 161

-9.0

152 182 214

-9.2 -11.1 -13.2

179 ( 2b Rutile 194

-8.9

185 220 256

-9.2 -11.0 -13.2

211 ( 10e

- 9.90f

Eg,KS (eV)

Ti 3p (eV)

O 2p Bw (eV)

2.16 2.16 2.43 3.23 4.24 3.89 3.2c

34.5 34.7 35.3 35.7 36.0 38.9 38.0d

4.9 4.8 4.8 5.4 6.2 5.3 ≈6.0d

1.78 1.78 2.01 2.69 3.67 3.43 3.0c

34.7 34.8 35.6 35.9 36.2 39.1 38.0d

5.6 5.8 5.4 6.2 7.0 6.3 ≈6.0d

Reference 56. b Reference 57. c Reference 6. d Reference 18. e Reference 58. f Reference 59.

US total density of states (DOS) are shown in Figure 1A; the corresponding projections on Ti 3d and O 2p atomic orbitals are plotted in Figure 1B. With the U(Ti) settings, the significant improvements in the description of Ti3+ properties reported in refs 29-31 (i.e., sizable localization of the defect charge and appearance of an electronic level at about 1 eV below the CBM) seem to be achieved at the expense of a somewhat worse description of the TiO2 structural properties. In both rutile and anatase, U(Ti) results show indeed increased lattice parameters and Ti-O distances and lower bulk moduli with respect to the corresponding PBE and experimental results, that is, the TiO2 lattice becomes “softer”. In order to explain such a “softening”, we have considered the PBE-US/U(Ti) difference density map shown in Figure 2A. The map has been obtained in an anatase 48atoms supercell, by subtracting the electronic density obtained with the application of the U correction only to the 3d orbitals of a single Ti atom, from that estimated by a plain PBE-US calculation.60 Only that Ti atom and its six nearest-neighboring O atoms are shown in Figure 2A. When the U(Ti) is active, some charge occupying nonbonding regions around the Ti atom (red spots near Ti) localizes at Ti 3d orbitals along the Ti-O bonds (blue spots near Ti). Due to the prevailing ionic character of the Ti-O bonds, such a charge displacement induces a small Ti-O repulsive effect, as shown by the charge rearrangement at the O neighbors where charge displaces from bonding to nonbonding directions; therefore, the “softening” of the TiO2 lattice properties should be accounted for. A balancing effect can be achieved by switching on the Hubbard U correction on the 2p shell of O atoms. In fact, in the case of the U(Ti,O)A correction, as shown in Figure 2B, an amount of charge density flows from the Ti neighbors toward the O orbitals and strengthens the ionic contribution to the Ti-O bond. Thus the U(Ti,O)A setting “hardens” back both anatase and rutile, in agreement with both the experimental and the HSE theoretical results. With the U(Ti,O)B setting, the Ti-O bonds are too strong, giving a considerable overestimate of the bulk moduli and of the formation heat. U(Ti) setting shows only little improvement of Eg,KS, basically due to the fact that the Ti4+ 3d orbitals are not totally empty (see Figure 1B) and induce, therefore, a small splitting between occupied and unoccupied levels; see ref 38. Eg,KS values

approaching the experimental ones have been calculated with the U(Ti,O)A settings (3.23 and 2.69 eV for the anatase and rutile polymorphs, respectively), while quite higher values have been estimated both in the U(Ti,O)B and HSE cases, marking a good point in favor of the U(Ti,O)A setting. Similarly, U(Ti,O)A settings give the bandwidth of the O 2p-related states, the uppermost states of the rutile and anatase VB, in agreement with the available experimental data. Moreover, in Figure 1A a comparison of the total DOSs calculated with the above different methods in anatase is possible: both U(Ti,O)A and the HSE approaches show an increase of the Eg,KS, mainly induced by a lowering of the top of the VB, and quite similar shapes of the states close to the top of the VB (the states with prevailing O 2p character). On the grounds of the above results, in particular of the reasonable agreement with hybrid functional HSE simulations, we consider the U(Ti,O)A settings validated in order to perform an extended investigation of TiO2 defects at a lower computational cost. In addition to the widely used U(Ti) correction, the U(Ti,O)A has been used here as a possible step ahead in beyondGGA calculations. One could argue that such U(Ti,O)A parametric approach to the GGA+U method may represent a drawback to its predictive role. In this regard, we point out the following: (i) In the current implementation of the method, an “on-site” projection of Kohn-Sham states on atomic orbitals is well suited in the case of Ti 3d orbitals. This is particularly true in the case of Ti3+ defects, where Kohn-Sham defect states are mostly linear combinations of atomic orbitals of the same atom. (ii) Problems with a self-consistent estimate of the U parameter occur in the case of O 2p atomic orbitals likely because they are strongly hybridized in TiO2 bands. In this case, an “on-site + intersite” (U+V) projection, discussed in ref 61 but not implemented in distributed codes yet, is expected to provide reliable self-consistent U+V parameters. (iii) In the present study, we believe that a parametrical O 2p “on-site” U value, successfully reproducing the TiO2 bulk properties, can give reliable and then predictive, results in the case of the properties of intrinsic defects in both the investigated TiO2 polymorphs. B. Structure and Charge Localization of Vacancies and Interstitials. The formation of an oxygen vacancy in bulk TiO2 leaves two electrons back and induces significant local structural

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Figure 2. Difference electronic density maps obtained by subtracting the GGA+U density from the GGA one in the cases of selected Ti and O atoms belonging to the anatase TiO2 host matrix (see the text). The electronic density displaces from red zones to blue zones when the Hubbard U correction is switched on. (A) U correction applied to the 3d electrons of a single Ti atom, which is shown together with its six O neighbors; (B) U correction applied to the 2p electrons of a single O atom, which is shown together with its three Ti neighbors. For the sake of clarity, only the relevant atoms of the host matrix are shown. Similar results (not shown) are achieved in rutile TiO2 host matrix.

Figure 3. Equilibrium geometry and |ψ|2 plots corresponding to electronic states induced by an oxygen vacancy (VOx) in the rutile TiO2 polymorph, as given by the U(Ti) correction (see the text for details). Atomic distances related to the Tix and Oy species are reported in Table 2. Electronic densities are sampled at 0.14 e/Å 3. Figure 1. Density of states (DOS) of the anatase TiO2 bulk: (A) total and (B) projected on atomic orbitals. All of the curves have been aligned to a common reference. The labels in the upper panel correspond to rows in Table 1 and are discussed in section 3.1. The total density of valence states calculated in the PBE-US case (lower panel) has been projected on a basis of Ti 3d (red) and O 2p (blue) atomic orbitals. PBE-VBM (PBE-CBM) indicates the valence band maximum (conduction band minimum) estimated by the PBE calculations.

rearrangements in the host matrix. Results on the structure of a VOx in anatase and the localization of the corresponding defect electrons have been reported in previous studies using hybrid functionals or a parametric U correction.23,25 The VOx properties in bulk anatase and rutile have been investigated also in two previous studies of ours, using the self-consistent U(Ti) approach.29,30 Previous and present studies substantially agree on the following points: (i) open shell electronic configurations are lower in energy; (ii) electrons localized at Ti3+ sites induce

appreciable deformations of the local structure, that is, they show a polaron-like character; (iii) a significant structural asymmetry characterizes the VOx in anatase, which thoroughly loses the symmetry of the O site, at variance with the same defect in rutile, which retains a C2V site symmetry (see ref 30 for further details on the structural relaxation of VOx defects in both the TiO2 polymorphs). We note also that, both in rutile and in anatase, present results describe the structural rearrangements following the formation of a VOx, in full agreement with our previous findings. The same is true for the defect charge in rutile: the two VOx electrons are accommodated on the d orbitals of two Ti atoms neighboring the vacancy, Ti1 and Ti2 in Figure 3, in a triplet spin configuration. On the other hand, present results in anatase indicate a charge localization of the two defect electrons on two Ti neighboring the VOx, similarly to what is found in rutile, while previous results suggested a full localization of only one of the defect electrons.62



TABLE 2: Atomic Distances (in Å) between the Three Ti Atoms around a VOx in Bulk Rutile and Ti-O Distances Estimated for a Ti Atom Carrying a Vacancy Electron; See Figure 3a defect

method

Ti1-Ti2

Ti1-Ti3

Ti1-O1

Ti1-O2

Ti1-O3

Ti1-O4

Ti3-O6

bulk 2+ VOx 0 VOx 0 VOx bulk

U(Ti) U(Ti) U(Ti) U(Ti,O)A U(Ti,O)A

3.01 3.37 3.31 3.30 3.01

3.61 4.13 4.06 4.03 3.58

1.98 1.79 1.93 1.92 1.96

1.98 1.92 2.01 1.99 1.96

1.98 1.96 2.05 2.04 1.96

2.01 1.98 2.01 2.00 1.98

2.01 1.80 1.79 1.78 1.98

a

The distances are reported for two charge states of the defect together with the corresponding distances calculated for the bulk rutile. O4 and O5 atoms in Figure 3 correspond to apical O atoms having symmetric positions with respect to the Ti1.

Let us focus now on a few structural details which are closely related to the polaron-like nature of the defect electrons induced by the creation of a VOx in bulk rutile and anatase (U(Ti) settings are in use here). Such a polaron-like nature gives the grounds for discussing the defect properties in the following sections. We exemplify the relationships between local structure and charge localization in the case of the rutile vacancies, since similar considerations apply as well to anatase vacancies. Table 2 reports two kinds of atomic distances: Ti-Ti distances estimated for the three Ti atoms around the rutile VOx and Ti-O distances, from which we shall draw the description of the structural changes accompanying the localization of the VOx electrons on Ti3+ atoms (see Figure 3). In order to discriminate between structural effects induced by the Vacancy formation and those produced by the charge localization, we calculated 2+ (removed defect electrons, such distances in the cases of VOx 0 only vacancy formation) and VOx (localized defect electrons, both effects present). Comparing the bond distances of the pristine bulk lattice with those of the V2+ Ox case, the rearrangement following the O removal is shown by a significant displacement of the Ti neighbors away from the vacancy site with shortening Ti-O distances. In the neutral vacancy, the localization of the electronic charge on Ti3+ atoms induces an appreciable increase of the Ti-O distances with respect to the charged vacancy, to give a polaron-like geometry. Such a rearrangement also induces a shortening of the Ti-Ti distances neighboring the vacancy. Ti(i) defects in bulk rutile and anatase have been investigated in a previous study with hybrid DFT calculations.33 Presently obtained local structure and charge localization of these defects substantially agree with those from that study. A Ti(i) in bulk anatase occupies a pseudo-octahedral site63 (see Figure 4) with a formal charge of +3. That is, it carries only one of its four valence electrons, while the remaining electrons are localized on three Ti neighbors. For the sake of clarity, only |ψ|2 distributions corresponding to defect states located on the Ti(i) and on the atom labeled Ti2 have been shown in Figure 4. The remaining two electrons are located on two further Ti3+ species labeled Ti3 and Ti5 in Figure 4. The four defect electrons give rise to an open shell electronic configuration. Similarly to the above discussion on VOx properties, details on the local geometry of an anatase Ti(i) in two different charge states are given in Table 3. Once more, the effects produced on the local structure by the introduction of the interstitial can be distinguished from those produced by the localization of the defect charge by 4+ 0 and Ti(i) , with the comparing different charge geometries, Ti(i) pristine bulk structure. As before, polaron-like geometries are revealed by the appreciable increase of the Ti-O distances accompanying the localization of electronic charge on a Ti atom (e.g., the Ti2 in Figure 4 and the Ti(i) itself). An interesting peculiarity of the interstitial site is that, both in the neutral and charged cases, the distances between the Ti(i) and its Ti neighbors are appreciably shorter than the Ti-Ti distances in the bulk. This effect can be only partially accounted for by a partial

Figure 4. Equilibrium geometry and |ψ|2 plots corresponding to electronic states induced by a Ti interstitial (Ti(i)) in the anatase TiO2 polymorph, as given by the U(Ti) settings (see the text for details). Atomic distances related to the Tix and Oy species are reported in Table 3. Electronic densities are sampled at 0.07 e/Å 3.

screening of the positive charge on the involved Ti atoms.64 Similar results have been found in the case of Ti(i) in bulk rutile. The U(Ti,O)A settings give structural properties and charge localization patterns quite similar to those given by the U(Ti) setting, when considering VOx in rutile and the Ti(i) in anatase. In the former case, the interatomic Ti-Ti and Ti-O distances are generally shorter due to the tightening of the Ti-O bonds; see section 3.1. Relevant distances for the two different defects are reported in the last two rows of Table 2 and Table 3, respectively. Excess electrons induced by the VOx and Ti(i) defects localize on the same Ti3+ centers, showing a similar polaronic nature, both with the U(Ti) and U(Ti,O)A settings. C. Defect Formation Energies and Transition States. As detailed in section II, VOx and Ti(i) formation energies have been calculated as a function of µ′O, i.e., the O chemical potential scaled to the energy of the O2 molecule, and as a function of the Fermi level. The results for the neutral defects (no dependence on εF) are in Figure 5, for both computational settings adopted (left and right panels for U(Ti) and U(Ti,O)A, respectively) and the two TiO2 polymorphs (up and bottom panels for rutile and anatase, respectively). Experimental O-rich conditions correspond to µO ) 1/2Ef[O2]; thus µ′O ) 0. Obviously, these conditions do not favor the formation of VOx or Ti(i); e.g., the concentration of the former defect in anatase would correspond to the neglibile value of 103 cm-3 at 1000 K. Notwithstanding, it may be noted that in both the U(Ti) and U(Ti,O)A schemes and in both TiO2 polymorphs the formation of O vacancies results highly favored

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TABLE 3: Ti-Ti Atomic Distances Estimated for an Interstitial Ti Atom (Ti(i)) and Two Neighboring Ti atoms in Bulk Anatase and Ti-O Distances Estimated for the Ti(i) and a Further, Different Ti Atom Carrying an Excess Defect Electron; See Figure 4a defect

method

Ti(i)-Ti3

Ti(i)-Ti4

Ti(i)-O1

Ti(i)-O2

Ti(i)-O3

Ti(i)-O4

Ti(i)-O5

Ti2-O6

Ti2-O7

Ti2-O8

Ti2-O9

Ti2-O10

bulk 4+ Ti(i) 0 Ti(i) 0 Ti(i) bulk

U(Ti) U(Ti) U(Ti) U(Ti,O)A U(Ti,O)A

2.92 2.72 2.71

2.92 2.85 2.83

1.93 2.03 2.01

1.89 2.01 1.99

1.89 2.01 1.99

1.93 2.02 2.00

1.85 1.96 1.96

1.96 1.93 2.04 2.01 1.95

1.96 1.95 2.03 2.02 1.95

1.96 1.95 2.03 2.02 1.95

1.96 1.93 1.95 1.95 1.95

2.01 1.87 2.05 2.03 2.00

a The distances are reported for different charge states of the defect together with the corresponding distances calculated for the bulk anatase. All distances are given in angstroms.

Figure 5. Formation energies of VOx (red line) and Tii (blue line) as a function of µO′ (see text) for rutile (upper panel) and anatase (lower panel), estimated by applying the Hubbard-U term to the Ti 3d orbitals (U(Ti), left panel), or to both the Ti 3d and O 2p orbitals (U(Ti,O)A, right panel). The two vertical dashed lines represent the Ti-rich limits corresponding to the formation of the Ti2O3 phase (µ′(A)) and to the formation of the Ti metallic phase (µ′(B)), respectively.

Figure 6. Formation energies of VOx (red line) and Ti(i) (blue line) in rutile (left panel) and anatase (right panel) as a function of the Fermi energy under Ti-rich conditions, calculated with the Hubbard-U term applied to Ti 3d orbitals (U(Ti)). The zero of the Fermi energy corresponds to the valence band maximum; the dashed green line indicates the calculated energy gap (see the text).

with respect to that of Ti interstitials: the difference between the formation energies is in fact =4.5 eV in anatase and =4.0 eV in rutile, with Ωf[VOx] lower than Ωf[Ti(i)]. The two formation energies approach each other as µ′O approaches the rightmost limit of the x axis, corresponding to O-poor (Ti-rich) conditions, µ′O ) µ′(A) ) 2∆Hf(TiO2) - ∆Hf(Ti2O3), where ∆Hf is the formation heat of the bulk phases. In this case, the U(Ti) results still indicate a significant difference between the ease of formation of the two defects: in terms of relative concentrations, an estimate for T ) 1000 K of the ratio r ) C[VOx]/C[Ti(i)] under Ti-rich conditions gives r ) 105 in anatase and r ) 103 in rutile where the formation of Ti interstitial defects become competitive with that of O vacancies. In terms of absolute values, C[VOx] reaches the quite high value of =1019 cm-3 in both rutile and anatase. Under the corresponding Ti-rich conditions, the U(Ti,O)A setting further reduces the difference between the formation energies of the two defects. This is related to an increase of the value of the µ′(A) limit.65 In rutile, VOx and Ti(i) defects have almost identical formation energies and C[VOx] = C[Ti(i)] ) 1018 cm-3. In anatase, the two defects are created with a very low formation energy (slightly negative for the VOx66) and spontaneously. Note that these differences between the polymorphs are related to a small increase of the formation energies of both defects occurring only in the case of rutile. Figure 5 also reports the µ′(B) limit for the Ωf values corresponding to equilibrium conditions not involving the formation of the other Ti-rich oxide phase, Ti2O3. An analysis of the formation energies when approaching this limit has been performed by using µ′(B) barely as a speculative tool to give suggestions for experimental procedures where defects are

formed through reducing treatments of TiO2 samples, like annealing at high temperatures in a reducing atmosphere. These treatments should not produce Ti-rich phases like Ti2O3 or metallic Ti. In this condition, U(Ti) results are for spontaneous formation of VOx and Ti(i) defects at µ′(B) limit in both rutile and anatase. In the same condition, U(Ti,O)A results suggest an easier formation of Ti(i)s. Similar considerations were reported in ref 53. The formation energies of the two above defects as a function of εF, resulting from the U(Ti) and U(Ti,O)A settings, under Ti-rich conditions, are shown in Figures 6 and 7, respectively. Here, the different slopes of each line correspond to different charge states of the defect. For the sake of clearness, only segments corresponding to the most stable charge states of a defect are shown. For instance, VOx in rutile data reported in Figure 6, show lines with three different slopes, from the steepest to that parallel to the x axis, correspond to the charge states 2+, 1+, and 0 of the vacancy, respectively. The crossing points between two lines of different slope locate the value of εF where the defect changes its charge state from q + 1 to q, which is the value of the corresponding thermodynamic transition level q/q+1 [VOx]. Note that in some cases, short segments make εtherm difficult the distinction of the crossing points corresponding to the transition levels (εq/q+1 values are more clearly displayed in Figures 8 and 9). In all of these four figures, the reported energy gap (Eg) is estimated by the ε-1/0 transition level calculated for the bulk material, in order to consistently compare transition levels and energy gaps. The indications given by the above defect formation energies can be exemplified considering VOx and Ti(i) in the U(Ti) rutile description; see Figure 6. The neutral state of the VOx defect is


Figure 7. Formation energies of VOx (red line) and Ti(i) (blue line) in rutile (left panel) and anatase (right panel) as a function of the Fermi energy under Ti-rich conditions, calculated with the Hubbard-U term applied to Ti 3d and O 2p orbitals (U(Ti,O)A). The zero of the Fermi energy corresponds to the valence band maximum; the dashed green line indicates the calculated energy gap (see the text).

Figure 8. Locations of vertical and thermodynamic transition levels with respect to the energy gap Eg (see the text) as estimated for VOx and Ti(i) defects in rutile and anatase by applying the Hubbard-U correction to Ti 3d orbitals (U(Ti)).

not stable (the 0/+1 crossing point is above the energy gap) while the +2 charge state could be observed in intrinsic or p-type materials. However, both anatase and rutile samples generally show n-type character, which should favor the +1 charge state for the VOx. Similarly, a Ti(i) is unstable in the neutral state; it could be found in different positive charge states, from +4 to +2, even in n-type TiO2, because the corresponding thermodynamic transition levels are close to the CBM (see also Figure 8). An εF close to the CBM slightly favors the formation of VOx with respect to Ti(i), while slightly lower εF values favor the Ti(i)4+ species (see Figure 6). The same theoretical picture holds for the VOx and Ti(i) formation energies in anatase as well as for U(Ti,O)A results reported in Figure 7 for both the TiO2 polymorphs. Some interesting differences between the properties of the two defects in rutile and anatase are revealed by a detailed analysis of the formation energies as well as of the vertical and thermodynamic transition levels reported in Figures 8 and 9. These two kinds of transition levels have been discussed in detail in ref 37. Their main features can be briefly recalled here by considering the configuration coordinate diagram in Figure 10, where the relationships between the energy of a defect or


Figure 9. Locations of vertical and thermodynamic transition levels with respect to the energy gap Eg (see the text) as estimated for VOx and Ti(i) defects in rutile and anatase by applying the Hubbard-U correction to Ti 3d and O 2p orbitals (U(Ti,O)A).

Figure 10. Schematic configuration coordinate diagram illustrating the difference between thermal (Etherm) and optical (Eopt) ionization + energies for an oxygen vacancy VOx. The curve for the charged VOx is 0 vertically displaced from that for VOx by assuming the presence of an +1 electron in the conduction band (εCBM ) Eg - εVBM). Erel is the relaxation energy (Franck-Condon shift) gained, in the positive charge state, by relaxing from the equilibrium configuration of the neutral vacancy R0 to that of the positively charged vacancy R+. Eg is the energy gap.

impurity and its atomic configuration are shown. As an example, let us assume a dependence of the VOx energy from a generalized coordinate R; Figure 10 reports a sketch of such a dependence for the neutral and the +1 charged states of the vacancy. In the + is drawn by assuming diagram, the curve corresponding to VOx the presence of one electron at the bottom of the conduction band (εCBM). Then Eopt represents an optical (or vertical) ionization energy and E+ rel the Franck-Condon shift, that is, the energy gained, in the positive charge state, when the configuration of the V0Ox is relaxed to that of V+Ox. The thermal ionization energy, Etherm, which is the energy measured in experiments where the final charge state can fully relax to its equilibrium 0/+ , and configuration after the transition, is equal to Eg - εtherm will be clarified in the following. By definition (disregarding possible correction terms), in the case of the VOx

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0/+1 εtherm ) E[VOx]0 - E[VOx]+1 - εVBM

Figure 8. The same theoretical picture holds for the VOx and Ti(i) defects in anatase on the grounds of the corresponding vertical and thermodynamic transition levels reported in Figure 8 too. Some interesting differences between the thermodynamic transition levels in anatase and rutile reported in the same figure are apparent: (i) the only VOx thermodynamic level located in the energy gap is closer to the CB in anatase than in rutile; and (ii) three Ti(i) levels are resonant with the CB in anatase while only two are resonant in rutile. Such differences suggest a stronger donor character of both VOx and Ti(i) defects in anatase, which may contribute to account for the larger conductivity shown by this material with respect to rutile. From results in Figure 9, we note that U(Ti) and U(Ti,O)A behave qualitatively the same. However, U(Ti,O)A give both the energy gap and the location of the vertical transition levels in a better (quantitative) agreement with experiment. Finally, a careful analysis of a recent hybrid functional (HSE) study of the VOx in rutile54 is deserved. At variance with previous studies, this HSE study reports indeed estimates of formation energies and transition energy levels which can be directly compared with present results. Some differences can be found between those results and present ones about the structural rearrangements, the formation energies and the thermodynamic transition levels related to the VOx defect in rutile. The HSE study reports a difference in atomic rearrangements around the 0 2+ site, with respect to the VOx one, much smaller than ours. VOx Note, however, that in the HSE study, no spin polarization effects have been taken into account, at variance with present settings. In fact, herein, we indicate an open shell with two unpaired electrons as the stable electronic configuration of V0Ox, in agreement with other previous studies.23,25 Enforcing a closed 0 leads to results similar shell electronic configuration of the VOx to those reported in the HSE study, as well as to a metastable configuration. The open shell configuration induces appreciable atomic rearrangements around the defect site, more appreciable indeed than those found in the closed shell configuration (the only one sampled, if spin polarization effects are disregarded). In other words, if spin effects are neglected, a different VOx electronic configuration is found, therefore also different formation energies of the neutral and charged states and, finally, different transition states too. The HSE study reports indeed that the VOx defect has a +2 charge for all of the Fermi energy 0/+2 values within the band gap, with both the ε+1/+2 therm and εtherm located above the CBM. On the contrary, here we indicate the possibility of the +1 and +2 charge states for the VOx depending on the position of the Fermi level. We remark that our picture allows to account for the electronic level experimentally observed at about 1 eV below the CBM; an aspect ignored in the HSE study.

and 0/+1 εvert ) E[VOx]0 - E[VOx(R0)]+1 - εVBM

where E[VOx(R0)]+1 represents the energy of a vacancy having the charge +1 and the geometry of a neutral VOx. This implies 0/+1 0/+1 +1 εvert ) εtherm - Erel

Then, if Eg (i.e., ε-1/0) is the energy gap, the relationships between vertical transition level and optical ionization energy, on one hand, and thermodynamic transition level and thermal ionization energy, on the other hand, are illustrated by the expressions 0/+1 Eopt ) Eg - εvert

0/+1 Etherm ) Eg - εtherm q/q+1 Vertical levels, εvert , are to be compared with available results from experimental techniques locating the defect levels in the energy gap, like EELS, PES, and IR, that imply vertical q/q+1 , should be compared transitions. Thermodynamic levels, εtherm instead with results given by, e.g., DLTS (deep levels transient spectroscopy), not available at the moment to the best of the authors’ knowledge. Notwithstanding, thermodynamic transition levels may give useful indications on the tendency of a donor defect to lose electrons when it relaxes to its final equilibrium configuration, tendency correlated to the material conductivity. With the U(Ti) settings, the locations within the rutile and q/q+1 anatase energy gaps of εq/q+1 vert and εtherm levels estimated for VOx and Ti(i) defects are schematically shown in Figure 8. In rutile, +1/+2 only some VOx and Ti(i) Vertical transition states, i.e., εvert +3/+4 +2/+3 [VOx], εvert [Ti(i)], and εvert [Ti(i)], result to be located at about 1 eV from the CBM. These results and the above formation energies are in close agreement with the experimental EELS, PES, and IR positions of the levels with respect to the CBM, all corresponding to vertical excitation (ionization) energies. Note, in addition, that the mentioned vertical transition states correspond to all and only the defect charge states which can have occupied electronic levels, i.e., the +1 state of VOx and the +3 and +2 states of Ti(i). The same figure shows also marked differences between vertical and thermodynamic levels. Both in VOx and in Ti(i) defects, the thermodynamic levels are indeed significantly higher in energy than the vertical ones. Thus, the two defects behave as deep donors when looking at the vertical transitions and as shallow donors when the effects of the structural relaxation are taken into account. The resulting theoretical picture can reconcile the apparent duality of experimental findings, having to do with the deep character of these defects and with the n-type conductivity observed in reduced TiO2. The significant differences between vertical and thermodynamic transition levels are closely related to the polaron-like nature of the defect electronic states; in fact, the structural relaxation accompanying the localization of the defect elec+1 +1 (=0.8 eV for the VOx ). tron(s) implies quite large values of Erel From the present picture, the Ti(i)s are stronger donors than VOxs, q/q+1 levels; see given the higher location of the corresponding εtherm

IV. Conclusions The structural and electronic properties of VOx and Ti(i) defects in the bulk of the rutile and anatase forms of TiO2 have been investigated within a GGA+U approach. As a first step, careful checks of the U values and of the convergence of results have been performed on the rutile and anatase bulk properties and have included a comparison of GGA+U and HSE hybrid functional as well as experimental results. Such a preliminary investigation has shown the existence of delicate relationships between the U correction, the lattice constants, and the convergence parameters, showing the need for some care in order to avoid possible spurious biases in results. VOx and Ti(i) defects are characterized by a polaron-like nature, with charge localization at Ti3+ sites and appreciable structural

Defects in Rutile and Anatase TiO2 Polymorphs rearrangements, both in rutile and anatase. As a major result of the present study, by estimating the formation energies of the charged and neutral forms of both the defects taken into account, together with the corresponding vertical and thermodynamic transition levels, it is shown that both the defects give rise to vertical transition levels about 1 eV below the CBM and, at the same time, to thermodynamic transition levels close to the CB. That is, these defects behave as deep donors, when looking at vertical transitions, and as shallow donors, when the effects of the structural relaxations are taken into account. The resulting theoretical picture can reconcile the experimentally observed deep levels with the n-type conductivity observed in reduced TiO2. A key role in the explanation is played by the polaronlike character of the defect states, which was already noted, but not deepened in literature. In fact, these differences come from the large values of the relaxation energy which, in turn, is related to the structural relaxation accompanying the localization of a defect electron. Thermodynamic transition levels suggest also that VOx and Ti(i) have a stronger donor character in anatase than in rutile, which may contribute to account for the larger conductivity observed in the former material. Finally, it is shown that the application of the U correction to both Ti and O species, here labeled as U(Ti,O)A setting, gives qualitatively similar results but a better (and close) agreement to experimental findings, with respect to the application of the U(Ti) setting. U(Ti,O)A results for both rutile and anatase bulk properties are pretty similar to those coming from HSE hybrid functional. Acknowledgment. We acknowledge the CASPUR consortium for granting computing resources. References and Notes (1) Chen, X.; Mao, S. S. Chem. ReV. 2007, 107, 2891. (2) Carp, O.; Huisman, C. L.; Reller, A. Prog. Solid State Chem. 2004, 32, 33. (3) Thompson, T. L.; Yates, J. T., Jr. Chem. ReV. 2006, 106, 4428. (4) Janisch, R.; Gopal, P.; Spaldin, N. A. J. Phys.: Condens. Matter 2005, 17, R657. (5) Chambers, S. A. Surf. Sci. Rep. 2006, 61, 345. (6) Diebold, U. Surf. Sci. Rep. 2003, 48, 53. (7) Bilmes, S. A.; Mandelbaum, P.; Alvarez, F.; Victoria, N. M. J. Phys. Chem. 2000, 104, 9851. (8) Justicia, I.; Ordejon, P.; Canto, G.; Mozos, J. L.; Fraxedas, J.; Battiston, G. A.; Gerbasi, R.; Figueras, A. AdV. Mater. 2002, 14, 1399. (9) Forro, L.; Chauvet, O.; Emin, D.; Zuppiroli, L.; Berger, H.; Levy, F. J. Appl. Phys. 1994, 75, 633. (10) Yagy, E.; Hasiguti, R. R.; Aono, M. Phys. ReV. B 1996, 54, 7945. (11) Sekiya, T.; Yagisawa, T.; Kamiya, N.; Mulmi, D. D.; Kurita, S.; Murakami, Y.; Kodaira, T. J. J. Phys. Soc. Jpn. 2004, 73, 703. (12) Berger, T.; Sterrer, M.; Diwald, O.; Knozinger, E.; Panayotov, D.; Thompson, T. L.; Yates, J. T., Jr. J. Phys. Chem. B 2005, 109, 6061. (13) Kruger, P.; Bourgeois, S.; Domenichini, B.; Magnan, H.; Chandesris, D.; Le Fevre, P.; Flank, A. M.; Jupille, J.; Floreano, L.; Cossaro, A.; Verdini, A.; Morgante, A. Phys. ReV. Lett. 2008, 100, 055501. (14) Cronemeyer, D. C. Phys. ReV. 1959, 113, 1222. (15) Kurtz, R. L.; StockBauer, R.; Madey, T. E. Surf. Sci. 1989, 218, 178. (16) Thomas, A. G.; Flavell, W. R.; Kumarasinghe, A. R.; Mallick, A. K.; Tsoutsou, D.; Smith, G. C.; Stockbauer, R.; Patel, S.; Gra¨tzel, M.; Hengerer, R. Phys. ReV. B 2003, 67, 035110. (17) Onda, K.; Li, B.; Zhao, J.; Jordan, K. D.; Yang, J.; Petek, H. Science 2005, 308, 1154. (18) Thomas, A. G.; Flavell, W. R.; Mallick, A. K.; Kumarasinghe, A. R.; Tsoutsou, D.; Khan, N.; Stockbauer, R. L.; Warren, S.; Johal, T. K.; Patel, S.; Holland, D.; Taleb, A.; Wiame, F. Phys. ReV. B 2007, 75, 035105. (19) Nolan, M.; Elliot, S. D.; Mulley, J. S.; Bennet, R. A.; Basham, M.; Mulheran, P. Phys. ReV. B 2008, 77, 235424. (20) Wendt, S.; Sprunger, P. T.; Lira, E.; Madsen, G. K. H.; Zheshen, L.; Jonas, O. H.; Matthiesen, J.; Blekinge-Rasmussen, A.; Laegsgaard, E.; Hammer, B.; Besenbacher, F. Science 2008, 320, 1755. (21) Henderson, M. A.; Epling, W. S.; Peden, C. H. F.; Perkins, C. L. J. Phys. Chem. B 2003, 107, 534. (22) Di Valentin, C.; Pacchioni, G.; Selloni, A. Phys. ReV. Lett. 2006, 97, 166803.

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a similar way, in rutile, the localization of the three electrons of the Ti(i) atom cancels the (slightly distorted) D4h site-symmetry of the pseudooctahedral Ti3+ O6 unit. 0 (64) This effect occurs indeed both in the cases of a Ti(i) (where the +4 formal charge of the involved atoms is about +3) and of a Ti(i) (where all the involved atoms have a formal charge of +4). On the other hand, the above screening seems responsible for Ti3+-Ti3+ pairs (e.g., the Ti(i) and Ti3 atoms in the neutral case) closer than Ti4+-Ti3+ (e.g., Ti(i) and Ti4 atoms in the neutral case) and Ti4+-Ti4+ pairs (e.g., Ti(i) and Ti3 atoms in the case of the defect with charge +4). (65) The increase of the µ′ (A) value is related to the strengthening of the Ti-O bonds in the U(Ti,O) approach with respect to the U(Ti) one, which implies different 2∆Hf(TiO2) and ∆Hf(Ti2O3) values.

Mattioli et al. (66) The formation energies depend on the chemical potentials which also fix their limits, like µ′ (A), in Figure 5. At such a limit, the VOx formation energy in anatase is slightly negative. This would possibly require a small change of the above chemical potentials in order to avoid such a drawback. Since the differences are quite small, we have neglected a slight rescaling of the formation energies. (67) Zhang, S. B.; Northrup, J. E. Phys. ReV. Lett. 1991, 67 2339. (68) Van de Walle, C. G.; Laks, D. B.; Neumark, G. F.; Pantelides, S. T. Phys. ReV. B 1993, 47, 9425.

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Deep versus Shallow Behavior of Intrinsic Defects in Rutile and

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