Electronic Structure of Partially Reduced Rutile TiO2 (110) Surface

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Electronic Structure of Partially Reduced Rutile TiO2(110) Surface: Where Are the Unpaired Electrons Located? Steeve Chretien and Horia Metiu* Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, United States ABSTRACT: When an oxygen atom is removed from the surface of rutile TiO2(110), to make an oxygen vacancy, two unpaired electrons are left in the oxide. We perform density functional calculations with on-site repulsion (DFTþU) to find where these electrons are located. If U e 2.5 eV, they are delocalized. If 3.0 e U e 6.0 eV, they are both localized on different Ti atoms, reducing them (formally) from Ti4þ to Ti3þ. The energy of vacancy formation depends on the location of this pair of reduced Ti atoms. Three kinds of states have low energies that are very close to each other. In these states, the electrons are located on the five-coordinated Ti atoms at the surface and on Ti atoms below the surface. Previous calculations proposed that the unpaired electrons reduced two Ti atoms located near the vacancy. We find that this state has higher energy than all other states examined here.

1. INTRODUCTION Rutile TiO2 is used as a support in catalysis, as a photocatalyst, and as a material for photoelectrochemical water splitting.1-7 Unless it is carefully oxidized, the surface of TiO2 has oxygen vacancies that are chemically active and chemisorb a variety of molecules. Moreover, the formation and the annihilation of oxygen vacancies are essential steps in the Mars-van Krevelen mechanism8-10 for catalytic oxidation. In spite of extensive studies,2-4,11-66 there are some unresolved issues regarding the formation of oxygen vacancies on the surface of rutile TiO2(110), which are addressed here. The nomenclature used for describing the structure of the TiO2(110) surface is defined in Figure 1. The outermost stoichiometric surface layer has two kinds of oxygen atoms, the bridging oxygen (labeled 1) and the in-plane oxygen (labeled 2), and two kinds of Ti atoms, the 5-fold-coordinated titanium (denoted 5c-Ti and labeled 3) and the 6-fold-coordinated titanium (denoted 6c-Ti and labeled 4). The oxygen vacancy examined here is made by removing one bridging oxygen atom and leaving behind the void marked in Figure 1 by a green sphere. The two 6c-Ti atoms that used to be bonded to the removed oxygen atom are denoted by 6c-Ti*; one of them is labeled 5 in Figure 1 and the other is located between the atoms labeled 4 and 5. The removal of a neutral oxygen atom leaves behind two electrons that had been engaged in two Ti-O bonds that were broken when the vacancy was formed. In what follows we call these the unpaired electrons (UPEs). Much of the recent discussion regarding oxygen-vacancy formation revolves around the location of these electrons and the energy of the orbital that they occupy. Density functional theory (DFT) calculations based on the generalized gradient approximation (GGA) predict that both UPEs occupy Kohn-Sham molecular orbitals delocalized over r 2011 American Chemical Society

several Ti atoms located in the subsurface layers.36,39,67-69 It is believed that this prediction of DFT-GGA is incorrect and that better results are obtained by using DFTþU or hybrid functionals.23-25,30,31,33,35,54,67 A recent calculation using PBEþU (U = 4.2 eV)24 predicted that the two UPEs are localized on the 6c-Ti* atoms adjacent to the vacancy and the formation of the vacancy creates states in the band gap. Forming the vacancy reduces the two 6c-Ti* atoms from a formal charge of 4þ to a formal charge of 3þ. However, B3LYP calculations found that one electron is located on a 6c-Ti* atom, as predicted by PBEþU (U = 4.2 eV), but the second one is on a 5c-Ti atom located next to the vacancy site.67 In this article we use DFTþU calculations to examine in detail the fate of the UPEs created when an oxygen vacancy is formed on the surface of rutile TiO2(110). Since there are many procedures for choosing the value of U,24,25,27-32,66,70-75 and none is clearly superior to the others, we decided to examine the dependence of our conclusions on the value of U by performing calculations with 0 e U e 6 eV. We find that if U e 2.5 eV, the two UPEs are delocalized on Ti atoms below the surface. The situation is much more complicated when 3.0 e U e 6.0 eV: the two UPEs are always localized on two different Ti atoms, reducing them. The localization is stabilized by a displacement of the oxygen atoms surrounding the reduced Ti atom. We call this complex—formed by the reduced Ti ion, the unpaired electron, and the displaced oxygen atoms—a polaron. There are many minima on the potential energy surface of the reduced oxide (i.e., the oxide with one oxygen vacancy in the surface layer of the supercell). Each minimum corresponds to two polarons Received: November 24, 2010 Revised: January 31, 2011 Published: March 03, 2011 4696

dx.doi.org/10.1021/jp111209a | J. Phys. Chem. C 2011, 115, 4696–4705

The Journal of Physical Chemistry C

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the vacancy site (or to a 5c-Ti atom) varies by as much as 0.7 eV, depending on the location of the unpaired electron. The most stable location of the remaining UPE is on a subsurface Ti atom and the least stable is on a 6c-Ti* atom. Similar results have been reported for the formation of a hydroxyl, on a reduced TiO2(110) surface, by adding an H atom to a bridging oxygen atom.79 It is likely that polaron formation and its location needs to be taken into account whenever a chemical process gives electrons to a reducible oxide.

Figure 1. (a) Side view and (b) top view of the top stoichiometric layer of a partially reduced rutile TiO2(110) surface showing a [3 2] supercell. The green sphere indicates the position of a bridging oxygen vacancy created by removing one of the protruding bridging oxygen atoms. The atoms are labeled as follows: (1) bridging oxygen; (2) inplane oxygen; (3) 5-fold-coordinated Ti (5c-Ti); (4) 6-fold-coordinated Ti (6c-Ti); (5) 6-fold-coordinated Ti at the vacancy site (6c-Ti*).

located at different lattice sites. We group these states into six categories, classified according to the location of the polarons. Three of these categories represent states whose energies are very close to each other but differ through the location of the polarons. For example, one category contains all states in which the two polarons are subsurface and another in which one polaron is located on a 5c-Ti atom and one subsurface. These three categories of states have the lowest energy, which means that they are all present in the reduced oxide at room temperature. The states of the other three categories have higher energy and contain at least one electron located on a 6c-Ti* atom. The state in which the two UPEs are located on the two 6c-Ti* has the highest energy. Recent experiments22,55 suggest that the unpaired electrons produced by O removal are delocalized on the subsurface Ti atoms. This is in agreement with our results for 3.0 e U e 6.0 eV. Previous work76-78 has shown that the presence of oxygen vacancies substantially modifies the adsorption of electrophilic adsorbates. Because the adsorption of an electrophile on the reduced TiO2 surface uses one of the two UPEs created when the oxygen vacancy was formed, we expect that the adsorption energy of the electrophile will depend on the position of the remaining UPE. Since a Au atom is such an electrophile,69,77 we calculated how the adsorption energy of Au atom on the reduced rutile surface depends on the position of this remaining UPE. We found that, for U = 4.2 eV, the binding strength of the Au atom to

2. COMPUTATIONAL DETAILS The electronic structure of rutile TiO2 has been studied with the spin-polarized density functional theory provided by the Vienna ab initio simulation package (VASP) programs.80-83 We use the Perdew-Burke-Ernzerhof (PBE) functional84 and describe the ionic cores by relativistic, scalar, projected augmented wave (PAW) pseudopotentials.85,86 Ten and six valence electrons for Ti and O atoms, respectively, were explicitly taken into account. We use a plane-wave basis set with an energy cutoff of 400 eV. The on-site correction has been applied to the 3d electron of Ti atoms by the approach due to Dudarev et al.87 and the PBE functional (hereafter called PBEþU). The values of U ranged from 0.0 to 6.0 eV in increments of 0.5 eV. The spin-polarization Ns (the absolute value of the difference between the number of electrons with spin up and those with spin down) was fixed during the geometry optimization. For the reduced rutile we report only the results of calculations performed with Ns = 2, because the presence of oxygen vacancies on the rutile TiO2(110) surface or on a TiO2 thin film has been shown to create a small magnetic moment.11,17,88-90 The Harris-Foulkes correction was applied when we calculated forces. The Kohn-Sham matrix was diagonalized iteratively by use of a Davidson block iteration scheme. Fractional occupancies of the bands were allowed with a window of 0.05 eV and the Gaussian smearing method. The crystal structure of stoichiometric (i.e., unreduced) rutile TiO2 was determined by performing an automatic relaxation of the cell shape and volume. To minimize the Pulay stress, the energy cutoff for the plane-wave expansion was 520 eV and the “ACCURATE” option of the “PREC” keyword was used. Starting from the experimental structure,91,92 the positions of the atoms were varied, by using quasi-Newton optimization, until the x-, y-, and z-components of the forces acting on the atoms were smaller than 0.001 eV/Å. No symmetry was imposed during the optimization procedure. The Brillouin zone was sampled with an automatically generated Γ-centered 7 7 11 MonkhorstPack mesh,93 giving a total of 270 irreducible k-points. The convergence criterion was 10-6 eV for the self-consistent electronic minimization. Only the value of Ns equal to zero has been considered for the stoichiometric oxide. Rutile TiO2 belongs to the tetragonal space group P42/mnm. The unit cell contains two TiO2 units. Ti occupies sites 2(a) located at (0, 0, 0), (1/2, 1/2, 1/2), while oxygen occupies sites 4(f) located at (u, u, 0), (1 - u, 1 - u, 0), (1/2 - u, 1/2 þ u, 1/2), and (1/2 þ u, 1/2 - u, 1/2). Table 1 shows the experimental and calculated structural parameters of the rutile TiO2 crystal for 0 e U e6 . Fitting an equation of state to a set of energies calculated for different unitcell volumes produces the same values for the structural parameters. The crystal structure is best reproduced by the PBE 4697

dx.doi.org/10.1021/jp111209a |J. Phys. Chem. C 2011, 115, 4696–4705


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Table 1. Structural Parameters of the Rutile TiO2 Crystala method

U (eV) a (Å) c (Å)

u

V (Å3) B (GPa)

PBE

0.0

4.651 2.970 0.304 93

64.3

201.7

PBEþU

1.0

4.658 2.984 0.304 96

64.7

201.0

PBEþU

1.5

4.661 2.992 0.304 98

65.0

200.9

PBEþU

2.0

4.664 2.999 0.305 01

65.3

200.6

PBEþU

2.5

4.668 3.007 0.305 03

65.5

200.3

PBEþU

3.0

4.671 3.014 0.305 06

65.8

200.0

PBEþU

3.5

4.675 3.021 0.305 11

66.0

199.7

PBEþU PBEþU

4.0 4.2

4.678 3.028 0.305 13 4.680 3.031 0.305 16

66.3 66.4

199.4 199.2

PBEþU

4.5

4.682 3.036 0.305 19

66.6

199.2

PBEþU

5.0

4.686 3.043 0.305 22

66.8

199.0

PBEþU

5.5

4.690 3.045 0.305 27

67.1

198.6

PBEþU

6.0

4.693 3.056 0.305 31

67.3

198.2

expt (T = 15 K)91

4.587 2.956 0.304 69

62.1

expt (T = 295 K)92

4.585 2.953 0.304 94

62.4

expt (T = 298 K)108

215.5

a

The unit cell dimensions (a and c) are in angstroms and the volumes (V) are in cubic angstroms. The internal parameter u is dimensionless. The bulk modulus (B in GPa) was obtained from fitting a series of energy-volume data by use of the 4th-order natural strain equation of state107 implemented in the Exciting program (http://exciting-code. org). The calculated values were obtained from an automatic relaxation of the volume and shape of the unit cell by use of the PBEþU functional, the PAW pseudopotentials, and an energy cutoff of 520 eV.

functional (i.e., U = 0 eV). The difference between the experimental and the computed values increases as U increases. Slabs composed of four and six stoichiometric layers (a stoichiometric layer has three atomic layers as shown in Figure 1a) were used to study the TiO2 surface. The atoms in the bottom stoichiometric layer of TiO2(110) were kept fixed at their bulk position, while the positions of the remaining atoms were varied until the x-, y-, and z-components of the atomic forces were smaller than 0.02 eV/Å. The convergence criterion was 10-5 eV for the self-consistent electronic minimization. A vacuum space of 15 Å was inserted between the slab and its periodic replicas, along the direction perpendicular to the surface. The electrostatic interaction between the slab and its periodic images in the direction perpendicular to the slab was canceled by applying monopole, dipole, and quadrupole corrections to the energy using a modified version of the method proposed by Makov and Payne.94 Except when we explicitly mention it, the Brillouin zone was sampled at the Γ-point only. The TiO2(110) surface was modeled by [3 2] and [5 2] slab supercells. Removing one bridging oxygen atom from a supercell produces a surface having bridging oxygen vacancy concentrations of 17% (1/6) for the [3 2] supercell and 10% (1/10) for the [5 2] supercell. The oxygen-vacancy formation energy (Evf) was calculated as Evf ¼ E½TiO2 ðRÞ þ

1 E½O2 - E½TiO2 ðSÞ 2

ð1Þ

where E[O2] is the total energy of O2 in gas phase, E[TiO2(S)] is the total energy of the stoichiometric slab, and E[TiO2(R)] is the total energy of a relaxed, partially reduced TiO2 slab having one bridging O vacancy on the surface of the supercell. Determining whether an electron is localized or not is somewhat arbitrary. The most popular definition is based on density

Figure 2. Density plots of the Kohn-Sham molecular orbitals associated with creation of a protruding bridging oxygen vacancy on a clean rutile TiO2(110) surface. (a, b) Defect states of the lowest energy structure obtained by use of the PBE functional. (c, d) Defect states of the lowest energy structure obtained with PBEþU (U = 4.2 eV). Density plots show equal density surfaces of 0.02 e/Å3. The analysis was performed at the Γ-point by use of the PAW pseudopotential, a [3 2] supercell, and a slab composed of four stoichiometric layers.

plots of the Kohn-Sham molecular orbitals (KSMO) created when an oxygen atom is removed to make an oxygen vacancy (defect states). This definition has several shortcomings. For instance, the density plots shown in Figure 2c,d are clearly more localized than those shown in Figure 2a,b, but some electron density always spreads over the neighboring O and Ti atoms. Another problem is that the shape of the orbital, and hence the degree of localization, depends on the value of the isosurface chosen when the electron density in an KSMO is plotted. Here, we perform a Bader charge analysis95-98 on the spin density (i.e., the difference between the spin-up and spin-down densities). We consider an electron to be localized when the “spin density” on a Ti atom is larger than 0.7e and delocalized otherwise. According this definition, the KSMO are localized in Figure 2c,d because the atomic spin density on a Ti atom is 0.9e, while they are delocalized in Figure 2a,b because the largest atomic spin density on a Ti atom is 0.25e. We have optimized more than 50 different initial structures in which the two UPEs, created by the formation of the oxygen vacancy, are localized on different pairs of Ti atoms. To create states in which the UPEs are located at a preselected pair of Ti atoms we do the following. First, we remove a bridging oxygen atom from a stoichiometric rutile TiO2(110) surface. Then, we create a polaronic distortion around the two Ti atoms on which we want to localize the UPEs. We do that by stretching, by 0.1 Å, all the bonds between those two Ti atoms and the neighboring 4698



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O atoms, while keeping the other Ti-O bonds fixed. Finally, we set up the initial spin polarization on the atoms as follows. We use the symbol Ns for the total number of unpaired electrons (i.e., the difference between the total number of electron with spin up and the total number of the electrons with spin down) in the supercell. The symbol ns is the atomic spin polarization (i.e., the number of electrons with spin up minus the number of electrons with spin down, on a specific atom). This corresponds to integrating the spin density over some atomic volume. Summing the values of ns for all atoms has to be equal to Ns. In the input file we define the value of Ns (this fixes the spin multiplicity of the total electronic wave function) and the values of ns for each atom. For a given atom, ns can be any number, as long as the sum of ns over all atoms equals Ns. We set ns = 0 for all atoms except the two Ti atoms on which we want to localize the two UPEs, for which we use ns = 1 (by using the keyword MAGMOM). The values of the ns are optimized by VASP so, in principle, we do not need these initial settings. However, we found that this choice speeds up convergence and sometimes when we do not make it the computer fails to localize the electrons at the desired position, missing thus some of the minimum-energy states.

3. LOW-ENERGY STRUCTURES OF REDUCED RUTILE TIO2(110) SURFACE Due to the large number of structures we need to examine, the first calculations were performed at the Γ-point only, using the PBEþU (U = 4.2 eV) functional, a [3 2] supercell, and a slab composed of four stoichiometric layers. We have chosen U = 4.2 eV because this value was used in previous work.24 The dependence of the results on U, the number of layers, and the size of the supercell will be discussed in section 4. Because we have found a very large number of local minima, practically one for each pair of Ti3þ atoms, and because the energies of many of these minima differ from each other by less than 0.1 eV, we decided to group the structures corresponding to these minima into six categories. All the states in a given category have the same kind of UPE localization and the structure having the lowest energy in each category is shown in Figure 3. The structures in the first three categories (Figure 3a-c) are very close in energy. In addition, each structure shown for a given category is representative of a number of structures of the same category having similar energies. For example, in the case of the second category (i.e., Figure 3b), the subsurface Ti3þ atom can be located at other subsurface positions and the energy would change by a very small amount. In the last three categories (Figure 3d-f), at least one electron is located on a 6c-Ti* atom. These states have higher energies than the first three, and they are unlikely to be observed at room temperature. In particular, the state in which both UPEs are located on the two 6c-Ti* atoms has the highest energy. The low-energy structures shown in Figure 3a,b are similar to the ones obtained by use of LSDAþU (U = 5.5 eV).25 In contrast, the lowest energy structure obtained by Morgan and Watson,24 using the same DFT program and the same functional (PBEþU, U = 4.2 eV) as we do, corresponds to the structure displayed in Figure 3f. This structure is unstable by 1.1 eV compared to the lowest energy structure (see Figure 3a). In the next section, we show that the discrepancy between the two studies is not related to differences in the computational details, such as the choice of supercell, slab thickness, or sampling of the Brillouin zone. It is likely that Morgan and Watson missed the

Figure 3. Some of the lowest-energy structures corresponding to a partially reduced rutile TiO2(110) surface. Cyan spheres indicate the positions of the Ti3þ atoms. (a) Both electrons are located on two subsurface Ti atoms. (b) One electron is located on a 5c-Ti atom and the second is located on a subsurface Ti atom. (c) Both electrons are located on two different 5c-Ti atoms. (d) Lowest energy structure in which one of the electrons occupies the vacancy site (i.e., it is located on one of the two 6c-Ti* atoms). The second electron sits on a subsurface Ti atom. (e) One electron is located on a 6c-Ti* atom while the second is located on a nearby 5c-Ti atom. (f) Both electrons are located on the two 6c-Ti* atoms (i.e., they occupy the vacancy site). The relative energies (ΔE in electronvolts) indicate by how much the energy of a structure exceeds that of the structure shown in panel a. Calculations were performed at the Γ-point by use of the PBEþU (U = 4.2 eV) functional, a [3 2] supercell, and a slab composed of four stoichiometric layers. The structures are triplets (Ns = 2).

states of lower energies because they did not attempt to make polarons and find their preferred location. The lowest energy structure found by the B3LYP calculations of Di Valentin et al.67 belongs to the fifth category defined here: 4699



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Table 2. Bridging Oxygen Vacancy Formation Energy of the Rutile TiO2(110) Surfacea Evf (eV) supercell

NL

[3 2]

4

[3 2] [5 2] [3 2]

NKP

3a

3b

3c

3d

3e

3f

1b

3.26

3.27

3.36

3.58

3.70

4.36

4

5c

3.30

3.31

3.29

3.61

3.68

4.08

4 6

1b 1b

3.16 3.04

3.18 3.11d

3.20 3.20

3.54 3.47

3.58 3.62

4.03

a

See eq 1. The results were obtained by use of the PBEþU (U = 4.2 eV) functional and the PAW pseudopotentials. The vacancy concentrations are 17% and 10% for the [3 2] and [5 2] supercells, respectively. NL is the number of stoichiometric layers in the slab. Here we report only the results obtained when the positions of the atoms located in the bottom stoichiometric layer are fixed. For the 6-layer slab (18 atomic layers), the variations of Evf are smaller than 0.1 eV when more layers are fixed, which we consider negligible. NKP is the number of irreducible k-points used to sample the Brillouin zone. The structures are triplets (Ns = 2). b Calculation performed at the Γ-point only. c A 3 2 1 Monkhorst-Pack grid has been used. d The two electrons are not located on the same Ti atom as shown in Figure 3.

one UPE is on a subsurface Ti3þ atom and the other on the 6cTi* atom. In the present work this has higher energy than the structures in the first four categories. Moreover, the lowest energy structure in the fifth category, that we obtain, is lower by 0.1 eV than the structure obtained by Di Valentin et al.67 It is not clear whether this difference is caused by differences between B3LYP and PBEþU (U = 4.2 eV). Also, finding the lowest energy state was not a goal in the paper of Di Valentin et al.

4. DEPENDENCE OF VACANCY FORMATION ENERGY ON SUPERCELL SIZE, SLAB THICKNESS, AND BRILLOUIN ZONE SAMPLING The electronic properties and the vacancy formation energy of the rutile TiO2(110) surface depend on the slab thickness, the number of layers allowed to relax, and the size of the supercell.99-103 Variations as large as 0.84 eV (corresponding to a 30% deviation) are possible when these parameters are changed.99,100 To study these variations we reoptimized the structures shown in Figure 3 (as well as a few others) for different values of the parameters listed above. The results are presented in Table 2. A comparison of the first row with the second row shows that increasing the number of k-points from 1 to 5 does not change significantly the values of Evf obtained with the [3 2] supercell. Excluding the structure displayed in Figure 3f, which is very unstable compared to the other structures, the largest variation of Evf with the number of k-points is -0.07 eV. We consider this difference negligible. The effect of the size of the supercell on Evf can be seen by comparing the first and third rows of Table 2. Evf decreases as the size of the supercell increases (i.e., when going from [3 2] to [5 2]). This is expected because the vacancy-vacancy interaction is repulsive.99,100 Increasing the supercell size from [3 2] to [5 2] decreases the vacancy concentration from 17% to 10% and increases the distance between a vacancy and its periodic replicas, thus decreasing the vacancy-vacancy repulsion energy. Variations up to -0.16 eV are observed for the first five categories (see Figure 3a-e). However, the ordering in the stability of the structures representing each category is unchanged.

In the last row of Table 2 we report the results obtained by use of a [3 2] supercell and a slab composed of six stoichiometric layers, in which the positions of the atoms located in the bottom stoichiometric layer are fixed to their corresponding bulk positions. Again, there are some changes in the vacancy formation energies but no change in the ordering of the stability of various structures. The values of Evf obtained by fixing three and two stoichiometric layers (i.e., allowing only the atoms located in the top three or four stoichiometric layers to relax, instead of the top five layers) are not shown here. They differ from those presented in Table 2 by 0.12 eV at most. Comparison between the first and last rows of Table 2 shows that increasing the slab thickness from four to six stoichiometric layers decreases the oxygen vacancy formation energy. The largest variation (-0.22 eV) is observed when both electrons are located on subsurface Ti atoms (Figure 3a). It is important to mention that we did not search for the optimal positions of the Ti3þ atoms in the subsurface layers. We have restricted the positions of the UPEs to the second and third stoichiometric layers (relative to the surface layer), as we did for the slab composed of four stoichiometric layers. Consequently, the values of Evf reported in Table 2 in the column labeled 3a may not be the lowest possible energy with respect to the location of Ti3þ in the subsurface. However, this does not affect the conclusion that structures in which one or two UPEs are located on a 6c-Ti* atom have significantly higher energy than the others.

5. DEPENDENCE OF VACANCY FORMATION ENERGY ON U We repeated the calculations reported above by using values of U ranging from 1.0 to 6.0 eV in increments of 0.5 eV, a [3 2] supercell, and slabs composed of four stoichiometric layers. For U equal to 3.0 and 5.5 eV we reoptimized all the structures obtained using U = 4.2 eV, while for the other values of U we optimized only a few low-energy structures obtained for the six categories shown in Figure 3. Figure 4 shows the oxygen vacancy formation energy (see eq 1 for the definition of Evf) as a function of U. The values reported in that figure correspond to the lowest energy structure obtained for each of the six categories shown in Figure 3. The lowest-energy structure within each category might change with U and have different locations for the Ti3þ atoms than the locations given in Figure 3 (but the locations are of the same kind). Since the energies of various structures within a category are very close to each other, we do not show these structural changes. For example, consider the lowest energy structure in which both Ti3þ atoms are located in the subsurface. For U e 4.5 eV, the DFT calculations show that one Ti3þ atom is in the second layer and the other one is in the third layer as in Figure 3a. In contrast, starting from U = 5.0 eV, for the same category, the two Ti3þ atoms are located in the second layer (the picture of this structure is not shown). Nevertheless, the energies of all structures within a given category are very close to each other. For U e 2.0 eV, the system refuses to make stable polarons. No matter where we try to localize the electrons, the optimization procedure leads to the same structure; the polaronic distortions created by us in the structure, at the start of geometry optimization, disappear during the optimization. The two KS orbitals created when the vacancy is made are delocalized over subsurface Ti atoms (see Figure 2a,b for the case of U = 0 eV). 4700



Figure 4. Effect of the parameter U on the oxygen vacancy formation energy (Evf in electronvolts) associated with the removal of one protruding bridging oxygen atom on the clean stoichiometric rutile TiO2(110) surface. The energies are for triplets (Ns = 2) and were obtained by use of the PBEþU functional, the PAW pseudopotential, a [3 2] supercell, and a slab composed of four stoichiometric layers. The calculations were performed at the Γ-point only. For U g 2.5 eV, several values are reported. They correspond to the positions occupied by the two electrons associated with the creation of the defect. We use the following notation: (O, black) both electrons are delocalized over several subsurface Ti atoms; (0, red) one electron is localized on a subsurface Ti atom and the other one is delocalized over several subsurface Ti atoms; (4, green) one electron is localized on a 5c-Ti atom and the other one is delocalized over several subsurface Ti atoms; (b, blue) both electrons are localized on two different subsurface Ti atoms (see Figure 3a); (2, magenta) one electron is localized on a subsurface Ti atom while the second one is localized on a 5c-Ti atom (see Figure 3b); (9, red) both electrons are localized on two 5c-Ti atoms (see Figure 3c); (þ, orange) one electron is localized on a subsurface Ti atom and the other one is localized on a 6c-Ti* atom (see Figure 3d); (*, violet) one electron is localized on a 5c-Ti atom and the other one is localized on a 6c-Ti* atom (see Figure 3e); (, cyan) both electrons are localized on the two 6c-Ti atoms (see Figure 3f).

For U = 2.5 eV, the system has local energy minima for five types (i.e., categories) of states. In the state having the lowest energy (open black circles in Figure 4), both UPEs are delocalized over the subsurface atoms. U = 2.5 eV is the highest value of U for which such a state corresponds to a minimum of the total energy (i.e., no open black circle is present in Figure 4 for U > 2.5 eV). U higher than 2.5 eV will localize at least one electron. We describe next the other states in order of increasing energy. The open red square in Figure 4 represents a state in which one UPE is localized on a subsurface Ti atom and the other is delocalized on several subsurface Ti atoms. This type of state corresponds to a minimum on the total energy surface only for U = 2.5 eV (no other open red square is present in Figure 4). The open green triangle corresponds to a state in which one UPE is localized on a 5c-Ti atom on the surface and the other is delocalized over several subsurface Ti atoms. Such a state corresponds to a minimum on the potential surface only if U = 2.5 eV. Next are the states represented by a filled blue circle and a filled red square. The filled blue circle corresponds to a state in which both UPEs are localized on subsurface Ti atoms. In the state represented by the filled red square, both UPEs are localized on two 5c-Ti atoms in the surface layer. These two kinds of states are present for all U g 2.5 eV. U = 2.5 eV is the first value of U for which there are states in which both UPEs are localized, but these states have the highest energy. Also, U = 2.5 eV is the highest

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value of U for which total energy minima corresponding to at least one delocalized UPE are present. All minima found for U g 3.0 eV correspond to two localized UPEs and differ only through the location of the Ti3þ atoms formed by electron localization. The reader can follow the evolution of the states and their energy in Figure 4. We note several features. The states in which at least one UPE is located on a 6c-Ti* atom (i.e., orange þ, violet *, and cyan ) have the highest energy for all U for which they are present. In particular, the energy of the state in which two UPEs are located on the two 6c-Ti* sites is the highest at all U. This state is present only for 3.5 g U g 6 eV. Between U = 3.0 and 4.5 eV, the states represented by the filled blue circle (two UPEs localized on different subsurface Ti atoms) and the filled magenta triangle (one UPE on a 5c-Ti on the surface and the other on a subsurface Ti atom) are most stable. The energy difference between them is within the error of DFT. For U g 5.0 eV the state represented by a filled red square (both UPE localized on 5c-Ti atoms in the surface layer) has the lowest energy, but the filled magenta triangle and the filled blue circle are close to it. It is also interesting to note that for U e 2.5 eV the energy of vacancy formation grows slightly with U, but for U g 2.5 eV we observe a rapid decrease of the vacancy formation energy with U. Within the error of DFT we can say, however, that as U varies in this range, the nature of the low-energy states does not change (i.e., these states do not have an UPE on the 6c-Ti* sites).

6. DEPENDENCE OF AU ATOM ADSORPTION ENERGY ON UNPAIRED ELECTRON LOCATION So far we have shown that the formation of polarons on different Ti atoms affects substantially the energy of oxygen vacancy formation. It is natural to ask whether polaron formation may affect other quantities. In principle, any transformation that creates or removes unpaired electrons in a reducible oxide may be affected by polaron formation. As an example, we consider the adsorption of a Au atom on a reduced rutile TiO2(110) surface. Previous work has shown that a Au atom adsorbed on a 5c-Ti atom or at the vacancy site (to bind to the 6c-Ti* atoms) gains electron charge.69,77,104 Formally this means that when a Au atom is adsorbed on the reduced surface, one UPE binds to the Au and the other is left in the oxide. The binding energy of gold is likely to depend of the location of that UPE in the oxide. We used PBEþU (U = 4.2 eV) and a [4 2] supercell to study the position preferred by this electron by creating a polaron on a 5cTi site, or on a 6c-Ti site, or on a 6c-Ti* site, or on a Ti atom in the subsurface. The binding energy of the Au atom to the vacancy site varies by as much as 0.7 eV depending the position of the UPE (which can be any one of the states shown in Figure 5). The least-stable state is the one in which the UPE is located on a 6c-Ti (see Figure 5g). The amount of charge gained by Au is approximately 0.5e, regardless of the location of the Ti3þ atom. These observations apply also when Au1 is adsorbed on top of a 5c-Ti atom located far from the vacancy site (not shown here). These exploratory calculations suggest that whenever we deal with a system in which UPEs are involved, we need to examine where they are located. This has also been shown for the TiO2 surface on which one adsorbed a H atom.79 We suspect that this type of behavior may be encountered on all reducible oxides. 4701



Figure 5. Low-energy structures corresponding to the binding of a Au atom to a protruding bridging oxygen vacancy on the clean, partially reduced rutile TiO2(110). The relative energy (ΔE) indicates by how much the energy of a structure exceeds that of the lowest energy structure shown in panel a. De[Au] gives the energy required to desorb Au from the vacancy site. The energies are in electronvolts. q[Au] indicates the amount of charge (in e) gained by Au1 obtained from a Bader charge analysis.95-98 The calculations were performed at the Γpoint by use of the PBEþU (U = 4.2 eV) functional, the PAW pseudopotential, a [4 2] supercell, and a slab composed of four stoichiometric layers. The structures are doublets (Ns = 1).

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7. SUMMARY AND CONCLUSIONS When an oxygen vacancy is made, by removing a bridging oxygen atom from the surface, two of the electrons involved in the Ti-O bonds being broken are removed with the neutral oxygen atom and two are left behind in the neutral oxide. Here we studied what happens with the electrons left behind and how their fate affects the energy of oxygen vacancy formation and the chemistry of the surface. To answer this question we use DFT with a PBEþU functional. The results depend on the value of U and we perform calculations by varying U between 0.0 and 6.0 eV, in increments of 0.5 eV. An unpaired electron can become localized on a Ti atom by forming a polaron, in which the Ti atom is reduced to Ti3þ and the oxygen atoms near the newly formed Ti3þ atom move away from it. The polaron breaks the symmetry of the surface and it will not be formed if this symmetry breaking is prevented during the optimization of the structure. We find that for U e 2.0 eV the system does not form polarons even when we distort initially the positions of the oxygen atom. When the structure is optimized, the distorted oxygen atoms relax and the unpaired electrons become delocalized. There are three categories of states that are stable for U > 2.5 and whose energies differ from each other by very small amounts (see Figure 4). In these states the UPEs are localized in one of the following: either on two 5c-Ti atoms on the surface, or on one 5cTi surface and somewhere in the slab, or on two Ti atoms in the slab. Since the energies of these states are very close to each other, we expect all three kinds of states to be present on the reduced surface. Therefore, our prediction is that, if GGAþU with U > 2.5 eV is qualitatively correct, and if the temperature is not very low, the reduced oxide will have electrons localized on 5c-Ti atoms and on subsurface Ti. Probes that average over the properties of the sample, such as photoelectron diffraction,55 will conclude that the electrons are delocalized over the 5c-Ti and some of the Ti atoms in subsurface. In principle, local probes such as scanning tunneling microscopy (STM) should distinguish a state with the electron localized on a 5c-Ti atom from one in which the electron is subsurface. However, polaron mobility is fairly high,75 so the UPEs can migrate easily from one position to another. If the time to image a site is longer than the time it takes the polaron to jump from site to site, the STM will find an electron at any of the sites that the polaron can visit. This picture is in agreement with the ab initio molecular dynamics results of a recent paper by Marx and co-workers.105 We are grateful to one of the reviewers for bringing to our attention this article. This may explain why recent STM studies22 find that the UPEs are delocalized. To explore whether the polaron position affects surface chemistry, we examined the adsorption of a Au atom on a surface having an oxygen vacancy. The Au atom is electrophilic and takes electrons from the surface. So, formally, adsorbing Au “ties up” one of the unpaired electrons created when the vacancy was formed. There is still one unpaired electron left in the oxide, and for values of U greater than 2.5 eV, this electron is localized on a Ti atom. The binding energy of Au (to a reduced rutile surface) depends on the location of the Ti atom to which the electron is bound. The binding energy can change by as much 0.7 eV when the location of this electron changes. We have focused here on TiO2(110) and the connection between the magnitude of U, the energy of oxygen vacancy formation, the location of the polarons, and the manner in which polaron formation affects the adsorption of Au. However, we 4702


The Journal of Physical Chemistry C think that the issues raised here are more general. They are likely to be important whenever one examines a process in which unpaired electrons are created at the surface of a reducible oxide. A reducible oxide is needed because the whole issue appears because Ti in TiO2 can be reduced by the unpaired electrons from Ti4þ to Ti3þ. It is this reduction process that provides a home for the unpaired electrons and the polaron formation is merely a mechanism for stabilizing this home. We propose that one should examine polaron formation and electron localization whenever one makes an oxygen vacancy on a reducible oxide. In fact, one can speculate that this polaron formation and electron localization needs to be examined whenever one makes an anion vacancy at the surface of the salt of a reducible cation. Furthermore, there is no a priori reason why these phenomena are confined to making vacancies. Any other process that creates one (or several) unpaired electrons, such as adsorbing an alkali atom or a H atom (to make a hydroxyl), may be affected by the localization of this electron (i.e., whether it is localized or not, and if it is localized, which location has the lowest energy). The adsorption of Au on the reduced TiO2 surface is a good example of such a process. While some qualitative trends do not depend on the value of U, the energy of vacancy formation does. So far we do not have a reliable method for determining U and, in addition, there is no guarantee that DFTþU gives reliable results, regardless of how we chose U. We advocate that almost all chemistry problems require calculations of total energy difference and therefore the chemist should fit U to give correctly the energy of some reaction involving the oxide under examination. For example, one might fit the energy to convert TiO2 to Ti2O3 by using water as an oxidant (to avoid O2, a notoriously difficult molecule for DFT) or the reaction of the oxide with HX (X = halogen) to make water and the halide.106 However, while this procedure is chemically reasonable, there is no guarantee that it is any better than other choices. We are skeptical of attempts of determining the value of U by adjusting it to give the correct band gap or to match the result of some spectroscopic measurement of an one-electron excitation (i.e., photoelectron or electron energy loss spectra) with the energy of a Kohn-Sham orbital. The role of the KohnSham equation in DFT is to provide the correct electron density, not the one electron excitation spectra. Of course, one would like to compare the calculated energy of vacancy formation to the experiment and use this comparison to determine U. Unfortunately, we are not aware of any measurements of vacancy formation energy, on single crystals, in ultrahigh vacuum. Measurements on powder oxides are subject to many uncertainties. First, it is difficult to prepare a sample which is free of Ti interstitial, or dopants. Very low levels of such dopants can change substantially the energy of vacancy formation. Furthermore, we are interested in the energy of forming oxygen vacancies at the surface, and unless the surface area (per gram) is very large, it is difficult to distinguish the oxygen molecules desorbing from the surface layer from those desorbing from the bulk. The energy of vacancy formation depends on the crystal face, and measurements on crystalline powders having a variety of faces would be hard to interpret. Finally, the likely presence of hydroxyls or carbonates on the surface affects the energy of vacancy formation. In addition, it is difficult to imagine experiments that will determine reliably where the unpaired electrons are localized and how this localization is connected to the adsorption energy of a given species.

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’ AUTHOR INFORMATION Corresponding Author

*Phone 805-893-2256; fax 805-893-4120; e-mail metiu@chem. ucsb.edu.

’ ACKNOWLEDGMENT We thank Dr. Zhenpeng Hu for many valuable discussions. This research was supported by the University of California Lab Fees Program (09-LR-08-116809), the U.S. Department of Energy (DE-FG02-89ER140048), and the Air Force Office of Scientific Research (FA9550-09-1-0333). We made use of the computer facility of the California NanoSystems Institute, funded in part by the National Science Foundation (CHE-0321368). Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract DE-AC02-06CH11357. ’ REFERENCES (1) Pang, C. L.; Lindsay, R.; Thornton, G. Chem. Soc. Rev. 2008, 37, 2328. (2) Pacchioni, G. J. Chem. Phys. 2008, 128, No. 182505. (3) Cinquini, F.; Di Valentin, C.; Finazzi, E.; Giordano, L.; Pacchioni, G. Theor. Chem. Acc. 2007, 117, 827. (4) Thompson, T. L.; Yates, J. T., Jr. Top. Catal. 2005, 35, 197. (5) Ganduglia-Pirovano, M. V.; Hofmann, A.; Sauer, J. Surf. Sci. Rep. 2007, 62, 219. (6) Henderson, M. A. Surf. Sci. 1999, 419, 174. (7) Diebold, U. Surf. Sci. Rep. 2003, 48, 53. (8) Doornkamp, C.; Ponec, V. J. Mol. Catal. A: Chem. 2000, 162, 19. (9) Vannice, M. A. Catal. Today 2007, 123, 18. (10) Mars, P.; van Krevelen, D. W. Chem. Eng. Sci. 1954, 3 (Spec. Suppl.), 41. (11) Kim, D.; Hong, J.; Park, Y. R.; Kim, K. J. J. Phys.: Condens. Matter 2009, 21, 195405. (12) Han, G.; Hu, S.; Yan, S.; Mei, L. Phys. Status Solidi RRL 2010, 3, 148. (13) Aiura, Y.; Nishihara, Y.; Haruyama, Y.; Komeda, T.; Kodaira, S.; Sakisaka, Y.; Maruyama, T.; Kato, H. Physica B 1994, 194-196, 1215. (14) Mutombo, P.; Kiss, A. M.; Berk o, A.; Chab, V. Modelling Simul. Mater. Sci. Eng. 2008, 16, No. 025007. (15) Onda, K.; Li, B.; Petek, H. Phys. Rev. B 2004, 70, No. 045415. (16) Lindan, P. J. D.; Harrison, N. M.; Gillan, M. J.; White, J. A. Phys. Rev. B 1997, 55, 15919. izmar, E.; Potzger, K.; Krause, M.; Talut, G.; Helm, (17) Zhou, S.; C M.; Fassbender, J.; Zvyagin, S. A.; Wosnitza, J.; Schmidt, H. Phys. Rev. B 2009, 79, No. 113201. (18) Pabisiaka, T.; Kiejna, A. Solid State Commun. 2007, 144, 324. (19) Egdell, R. G.; Eriksen, S.; Flavell, W. R. Solid State Commun. 1986, 60, 835. (20) Chun-Ru, W.; Yin-Sheng, X. Surf. Sci. 1989, 219, L537. (21) Zhang, Z.; Ge, Q.; Li, S.-C.; Kay, B. D.; White, J. M.; Dohnalek, Z. Phys. Rev. Lett. 2007, 99, No. 126105. (22) Minato, T.; Sainoo, Y.; Kim, Y.; Kato, H. S.; Aika, K.; Kawai, M.; Zhao, J.; Petek, H.; Huang, T.; He, W.; Wang, B.; Wang, Z.; Zhao, Y.; Yang, J.; Hou, J. G. J. Chem. Phys. 2009, 130, No. 124502. (23) Zhang, Y.; Lin, W.; Li, Y.; Ding, K.; Li, J. J. Phys. Chem. B 2005, 109, 19270. (24) Morgan, B. J.; Watson, G. W. Surf. Sci. 2007, 601, 5034. (25) Calzado, C. J.; Hernandez, N. C.; Sanz, J. F. Phys. Rev. B 2008, 77, No. 045118. (26) Finazzi, E.; Di Valentin, C.; Pacchioni, G. J. Phys. Chem. C 2009, 113, 3382. (27) Morgan, B. J.; Watson, G. W. J. Phys. Chem. C 2010, 114, 2321. 4703


The Journal of Physical Chemistry C (28) Mattioli, G.; Filippone, F.; Alippi, P.; Bonapasta, A. A. Phys. Rev. B 2008, 78, No. 241201R. (29) Yang, K.; Dai, Y.; Huang, B.; Feng, Y. P. Phys. Rev. B 2010, 81, No. 033202. (30) Morgan, B. J.; Scanlon, D. O.; Watson, G. W. e-J. Surf. Sci. Nanotechnol. 2009, 7, 389. (31) Morgan, B. J.; Watson, G. W. J. Phys. Chem. C 2009, 113, 7322. (32) Nolan, M.; Elliott, S. D.; Mulley, J. S.; Bennett, R. A.; Basham, M.; Mulheran, P. Phys. Rev. B 2008, 77, No. 235424. (33) Bredow, T.; Pacchioni, G. Chem. Phys. Lett. 2002, 355, 417. (34) Sterrer, M.; Yulikov, M.; Risse, T.; Freund, H.-J.; Carrasco, J.; Illas, F.; Di Valentin, C.; Giordano, L.; Pacchioni, G. Angew. Chem., Int. Ed. 2006, 45, 2633. (35) Islam, M. M.; Bredow, T.; Gerson, A. Phys. Rev. B 2007, 76, No. 045217. (36) Cho, E.; Han, S.; Ahn, H.-S.; Lee, K.-R.; Kim, S. K.; Hwang, C. S. Phys. Rev. B 2006, 73, No. 193202. (37) Cui, X.; Wang, B.; Wang, Z.; Huang, T.; Zhao, Y.; Yang, J.; Hou, J. G. J. Chem. Phys. 2008, 129, No. 044703. € g€ut, S.; Zapol, P.; Browning, N. D. Phys. Rev. B 2007, (38) Iddir, H.; O 75, No. 073203. (39) Von Oertzen, G. U.; Gerson, A. R. Int. J. Quantum Chem. 2006, 106, 2054. (40) Kowalski, P. M.; Meyer, B.; Marx, D. Phys. Rev. B 2009, 79, No. 115410. (41) Shu, D.-J.; Ge, S.-T.; Wang, M.; Ming, N.-B. Phys. Rev. Lett. 2008, 101, No. 116102. (42) Jug, K.; Nair, N. N.; Bredow, T. Surf. Sci. 2005, 590, 9. (43) Yu, N.; Halley, J. W. Phys. Rev. B 1995, 51, 4768. (44) Lindan, P. J. D.; Muscat, J.; Bates, S.; Harrison, N. M.; Gillan, M. Faraday Discuss. 1997, 106, 135. (45) Schierbaum, K. D.; Xu, W.-X. Int. J. Quantum Chem. 1996, 57, 1121. (46) He, J.; Sinnott, S. B. J. Am. Ceram. Soc. 2005, 88, 737. (47) Hameeuw, K.; Cantele, G.; Ninno, D.; Trani, F.; Iadonisi, G. Phys. Status Solidi A 2006, 203, 2219. (48) Ramamoorthy, M.; King-Smith, R. D.; Vanderbilt, D. Phys. Rev. B 1994, 49, 7709. (49) Bouzoubaa, A.; Markovits, A.; Calatayud, M.; Minot, C. Surf. Sci. 2005, 583, 107. (50) Wang, S.-G.; Wen, X.-D.; Cao, D.-B.; Li, Y.-W.; Wang, J.; Jiao, H. Surf. Sci. 2005, 577, 69. (51) Finazzi, E.; Di Valentin, C.; Pacchioni, G.; Selloni, A. J. Chem. Phys. 2008, 129, No. 154113. (52) Di Valentin, C. J. Chem. Phys. 2007, 127, No. 154705. (53) Di Valentin, C.; Pacchioni, G.; Selloni, A. J. Phys. Chem. C 2009, 113, 20543. (54) Papageorgiou, A. C.; Beglitis, N. S.; Panga, C. L.; Teobaldic, G.; Cabailha, G.; Chena, Q.; Fisher, A. J.; Hofer, W. A.; Thornton, G. Proc. Natl. Acad. Sci. U.S.A. 2010, 107, 2391. (55) Kr€uger, P.; Bourgeois, S.; Domenichini, B.; Magnan, H.; Chandesris, D.; Fevre, P. L.; Flank, A. M.; Jupille, J.; Floreano, L.; Cossaro, A.; Verdini, A.; Morgante, A. Phys. Rev. Lett. 2008, 100, No. 055501. (56) Wendt, S.; Sprunger, P. T.; Lira, E.; Madsen, G. K. H.; Li, Z.; Hansen, J. Ø.; Matthiesen, J.; Blekinge-Rasmussen, A.; Lægsgaard, E.; Hammer, B.; Besenbacher, F. Science 2008, 32, 1755. (57) Henrich, V. E.; Dresselhaus, G.; Zeiger, H. J. Phys. Rev. Lett. 1976, 36, 1335. (58) Zhang, Z.; Jeng, S.-P.; Henrich, V. E. Phys. Rev. B 1991, 43, 12004. (59) Bechstein, R.; Gonzalez, C.; Sch€atte, J.; Jelínek, P.; Perez, R.; K€uhnle, A. Nanotechnology 2009, 20, No. 505703. (60) Yurtsever, A.; Sugimoto, Y.; Abe, M.; Morita, S. Nanotechnology 2010, 21, No. 165702. (61) Yoshida, K.; Kawai, T.; Nambara, T.; Tanemura, S.; Saitoh, K.; Tanaka, N. Nanotechnology 2006, 17, 3944. (62) Yim, C. M.; Pang, C. L.; Thornton, G. Phys. Rev. Lett. 2010, 104, No. 036806.

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(63) Teobaldi, G.; Hofer, W.; Bikondoa, O.; Pang, C. L.; Cabailh, G.; Thornton, G. Chem. Phys. Lett. 2007, 437, 73. (64) Enevoldsen, G. H.; Foster, A. S.; Christensen, M. C.; Lauritsen, J. V.; Besenbacher, F. Phys. Rev. B 2007, 76, No. 205415. (65) Bikondoa, O.; Pang, C. L.; Ithnin, R.; Muryn, C. A.; Onishi, H.; Thornton, G. Nat. Mater. 2006, 5, 189. (66) Stausholm-Møller, J.; Kristoffersen, H. H.; Hinnemann, B.; Madsen, G. K. H.; Hammer, B. J. Chem. Phys. 2010, 133, No. 144708. (67) Di Valentin, C.; Pacchioni, G.; Selloni, A. Phys. Rev. Lett. 2006, 97, No. 166803. (68) Errico, L. A.; Renteria, M.; Weissmann, M. Phys. Rev. B 2005, 72, No. 184425. (69) Chretien, S.; Metiu, H. J. Chem. Phys. 2007, 126, No. 104701. (70) Cococcioni, M.; de Gironcoli, S. Phys. Rev. B 2005, 71, No. 035105. (71) Fabris, S.; de Gironcoli, S.; Baroni, S.; Vicario, G.; Balducci, G. Phys. Rev. B 2005, 71, No. 041102R. (72) Scherlis, D. A.; Cococcioni, M.; Sit, P.; Marzari, N. J. Phys. Chem. B 2007, 111, 7384. (73) Bennett, R. A.; Mulley, J. S.; Basham, M.; Nolan, M.; Elliott, S. D.; Mulheran, P. A. Appl. Phys. A: Mater. Sci. Process. 2009, 96, 543. (74) Nolan, M.; Mulley, J. S.; Bennett, R. A. Phys. Chem. Chem. Phys. 2009, 11, 2156. (75) Deskins, A. N.; Dupuis, M. Phys. Rev. B 2007, 75, No. 195212. (76) Chretien, S.; Metiu, H. J. Chem. Phys. 2008, 129, No. 074705. (77) Chretien, S.; Metiu, H. J. Chem. Phys. 2007, 127, No. 244708. (78) Deskins, N. A.; Rousseau, R.; Dupuis, M. J. Phys. Chem. C 2010, 114, 5891. (79) Deskins, N. A.; Rousseau, R.; Dupuis, M. J. Phys. Chem. C 2009, 113, 14583. (80) Kresse, G.; Hafner, J. Phys. Rev. B 1993, 47, 558. (81) Kresse, G.; Hafner, J. Phys. Rev. B 1994, 49, 14251. (82) Kresse, G.; Furthm€uller, J. Comput. Mater. Sci. 1996, 6, 15. (83) Kresse, G.; Furthm€uller, J. Phys. Rev. B 1996, 54, 11169. (84) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (85) Bl€ochl, P. E. Phys. Rev. B 1994, 50, 17953. (86) Kresse, G.; Joubert, D. Phys. Rev. B 1999, 59, 1758. (87) Dudarev, S. L.; Botton, G. A.; Savrasov, S. Y.; Humphreys, C. J.; Sutton, A. P. Phys. Rev. B 1998, 57, 1505. (88) Rossella, F.; Galinetto, P.; Mozzati, M. C.; Malavasi, L.; Fernandez, Y. D.; Drera, G.; Sangaletti, L. J. Raman Spectrosc. 2009, 41, 558. (89) Zhao, Q.; Wu, P.; Li, B. L.; Lu, Z. M.; Jiang, E. Y. J. Appl. Phys. 2008, 104, No. 073911. (90) Galinetto, P.; Mozzati, M. C.; Vercesi, C.; Malavasi, L.; Fernandez, Y. A. D.; Rossella, F.; Drera, G.; Sangaletti, L. J. Phys.: Conf. Ser. 2010, 200, No. 072030. (91) Burdett, J. K.; Hughbanks, T.; Miller, G. J.; Richardson, J. W.; Smith, J. V. J. Am. Chem. Soc. 1987, 109, 3639. (92) Hill, R. J.; Howard, C. J. J. Appl. Crystallogr. 1987, 20, 467. (93) Monkhorst, H. J.; Pack, J. D. Phys. Rev. B 1976, 13, 5188. (94) Makov, G.; Payne, M. C. Phys. Rev. B 1995, 51, 4014. (95) Bader, R. F. W. Atoms in Molecules - A Quantum Theory; Oxford University Press: Oxford, U.K., 1990. (96) Henkelman, G.; Arnaldsson, A.; Jonsson, H. Comput. Mater. Sci. 2006, 36, 354. (97) Sanville, E.; Kenny, S. D.; Smith, R.; Henkelman, G. J. Comput. Chem. 2007, 28, 899. (98) Tang, W.; Sanville, E.; Henkelman, G. J. Phys.: Condens. Matter 2009, 21, No. 084204. (99) Vijay, A.; Mills, G.; Metiu, H. J. Chem. Phys. 2003, 118, 6536. (100) Rasmussen, M. D.; Molina, L. M.; Hammer, B. J. Chem. Phys. 2004, 120, 988. (101) Bredow, T.; Giordano, L.; Cinquini, F.; Pacchioni, G. Phys. Rev. B 2004, 70, No. 035419. (102) Hameeuw, K. J.; Cantele, G.; Ninno, D.; Trani, F.; Iadonisi, G. J. Chem. Phys. 2006, 124, No. 024708. 4704



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(103) Thompson, S. J.; Lewis, S. P. Phys. Rev. B 2006, 73, No. 073403. (104) W€orz, A. S.; Heiz, U.; Cinquini, F.; Pacchioni, G. J. Phys. Chem. B 2005, 109, 18418. (105) Kowalski, P. M.; Camellone, M. F.; Nair, N. N.; Meyer, B.; Marx, D. Phys. Rev. Lett. 2010, 105, No. 146405. (106) Li, B.; Metiu, H. J. Phys. Chem. C 2010, 114, 12234. (107) Poirier, J.-P.; Tarantola, A. Phys. Earth Planet. Int. 1998, 109, 1. (108) Manghnanian, M. H.; Fisher, E. S.; Brower, W., Jr. J. Phys. Chem. Solids 1972, 33, 2149.

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Electronic Structure of Partially Reduced Rutile TiO2 (110) Surface

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