ARTICLE pubs.acs.org/JPCC
Distribution of Ti3þ Surface Sites in Reduced TiO2 N. Aaron Deskins,*,† Roger Rousseau,‡ and Michel Dupuis‡ † ‡
Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, Massachusetts 01609, United States Chemical and Material Sciences Division, Pacific Northwest National Laboratory, Richland, Washington 99352, United States
bS Supporting Information ABSTRACT: We describe a DFT þ U study of the (110) rutile surface with oxygen vacancies (Ov's). Oxygen vacancies leave behind two excess unpaired electrons per Ov, leading formally to the formation of two Ti3þ ions. We investigate the location of the Ti3þ ions within the first three surface layers. In total, we obtained 49 unique solutions of possible Ti3þ pairs, to examine the stability of all Ti types (e.g., five-coordinated surface Ti, six-coordinated surface Ti, subsurface sites, etc.). Our results show that subsurface sites are preferred but that many configurations are close in energy, within up to 0.30.4 eV of each other. In contrast to findings in previous work, we show that sites directly adjacent to the Ov's are unstable. Analysis of our results shows that the two Ti3þ ions within a pair behave independently of each other, as there are little electronic interactions between the excess electrons associated with these sites. We also examined the migration of Ti3þ sites from the surface into the bulk and find the surface locations to be preferred by ∼0.5 eV relative to the bulk. Our systematic results provide a comprehensive picture of excess electrons that indicates that they are not trapped or localized at specific sites but are distributed across several sites due to nearly degenerate Ti3þ states.
I. INTRODUCTION Metal oxides are used in a variety of fields, such as catalysis, gas sensing, and electronics. TiO2 is a prototypical reducible metal oxide and has been the object of many studies, discussed in several review articles.16 It is photoactive and thus used for catalytic organic photodegradation,1 and it is also well researched for water-splitting.7 It is also used as a support material, such as for Au8 or vanadium oxides.9 TiO2 can also be induced to become superhydrophilic under UV light.3 Oxygen vacancies (Ov's) are common in reducible metal oxides and can significantly affect their properties. Surface Ov's in TiO2 can serve as reactive sites (such as with H2O10,11) and can also affect binding of metal particles.1215 Ov's lead to the creation of unpaired electrons, or Ti3þ centers, (O2 þ 2Ti4þ f Ov þ 1/2O2 þ 2Ti3þ), and these Ti3þ ions may strongly influence surface chemistry through charge-transfer processes.1620 Thus, fundamental knowledge of the characteristics of Ov's is of utmost importance for understanding TiO2 and metal oxides in general. In this paper, we address several key aspects of Ov's in TiO2, such as preferred location of Ti3þ sites, degree of interaction between Ti3þ sites, and energetic location of Ov defect states. This work was performed using density functional theory (DFT) within the framework of the DFT þ U correction for electron localization of the Ti 3d orbitals in order to model atomic-level details of Ov's on rutile TiO2(110) terminated surfaces. Experimental results indicate that defect states in TiO2 due to Ov's are formed within the band gap (measured to have a width of 3 eV for rutile21) ∼1 eV below the conduction band.22 The most stable surface of rutile is the (110) surface, which is dominated by O atoms bridging two Ti atoms (Ob), and five-coordinated Ti (Ti5c); see Figure 1. Scanning tunneling microscopy (STM) r 2011 American Chemical Society
images of the (110) surface typically show Ov’s as bright spots along dark Ob rows when imaging unoccupied states.23 Images of occupied states, however, show that electrons may be spread across many atomic sites.24 Resonant photoelectron diffraction studies also indicate that electrons are spread across many Ti sites, including subsurface sites.25,26 There is much experimental research19,20,27 examining the exact nature of Ov's and their role in surface chemistry, but theoretical methods have advanced to the point that they can provide information that may be challenging to obtain experimentally. Recent reviews28,29 discuss the difficulty of modeling Ov’s in TiO2. DFT methodologies using the local spin density (LSD) approximation and in the generalized gradient approximation (GGA) lead to delocalization of unpaired electrons across several Ti sites with no appreciable formation of distinct Ti3þ centers. Two common methods to overcome this problem are hybrid exchange-correlation functionals and DFT þ U.30 Hybrid functionals mix in a fraction of HartreeFock exchange into the exchange-correlation functional to mitigate delocalization. An early work by Di Valentin et al.31 modeled the rutile (110) surface with both surface hydroxyls and an Ov using the B3LYP32,33 hybrid functional. They found the defect states to be near 1 eV below the conduction band, in agreement with experiment, and showed that it was possible to localize electrons to form Ti3þ sites using a hybrid functional. Their calculated solution consisted of one electron localized at a five-coordinated surface site and the other electron localized adjacent to the Ov at Received: January 5, 2011 Revised: March 1, 2011 Published: March 25, 2011 7562
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Figure 1. Schematic diagrams of the (110) reduced surface and possible locations of Ti3þ considered in this work. Individual Ti atoms (gray spheres) are indicated by numbers (a). The different types of Ti atoms, according to coordination number and position within the slab, are also indicated (b).
a six-coordinated surface site. Later work34 examined Ov’s in bulk anatase and found several solutions close in energy, some with localized electrons, while others with delocalized electrons. This later study also has shown that the percentage of HartreeFock exchange mixing can profoundly affect the solution, with too much HartreeFock exchange leading to unreasonable results. The degree of mixing can thus act as an ad hoc parameter to control the simulation result. Hybrid functionals have also been used to construct theoretical STM images with reasonable results.35,36 Although hybrid functionals can correctly lead to localized Ti3þ, they suffer from being computationally intensive and thus have seen limited applicability in solid-state calculations. No doubt larger computers and better algorithms may lead to the increased use of hybrid functionals to model periodic systems, although the search for the “best” functional to describe TiO2 and other oxides remains elusive. DFT þ U uses a different approach than hybrid functionals, where a correcting function is added to the Hamiltonian to ensure localization, with the U parameter controlling the strength of the function. DFT þ U adds little computational overhead to DFT so that it has been used extensively in modeling TiO2.16,34,3749 On the basis of its ability to localize electrons and its computational tractability, DFT þ U is a suitable method for our current study. Furthermore, Di Valentin et al.44 and Finazzi et al.34 modeled Ti3þ using B3LYP and DFT þ U and found both methods to give similar results. We have used DFT þ U to successfully model surfaces with a single hydroxyl (HOb)39 as well as transport of electrons and holes in bulk phases of TiO2.37,38 A recent CarParrinello molecular dynamics study48 used DFT þ U to model electron localization near Ov’s. They observed electron transfer during the short picosecond time scale of these trajectories, supporting our previous work where we predicted low barriers for adiabatic electron transfer in TiO2.37 They also observed subsurface sites to be
preferred, rather than surface sites. However, one difficulty of such calculations is that short time molecular dynamics simulations will only sample the most stable configurations. These calculations are also cumbersome, so only a handful of solutions were obtained, likely due to the short duration of the trajectories. A large focus in the literature has been placed on correctly simulating the location of defect states within the band gap or in obtaining theoretical STM images. Determining the physical location of excess electrons is also important, particularly when electron transfer between a surface and adsorbates is involved. We previously modeled39 the (110) surface with surface hydroxyls (HOb). In that work, we considered a surface with one HOb, that is, one Ti3þ ion per super cell, where we carefully controlled the location of Ti3þ in order to determine the relative stability of excess electrons at different Ti sites. We extend this approach in the current work by modeling the (110) surface with one Ov, that is, two Ti3þ per supercell, and taking a systematic approach to sampling a large number of possible Ti3þ locations, including both high- and low-energy solutions. This allows us to sample many possible Ti3þ types in the near-surface region and make definitive statements about Ti3þ stability, rather than sampling only much more restricted subsets of possible Ti3þ configurations, as has been done in the past. This approach also allows us to assess the magnitude of polaronpolaron interactions between two Ti3þ centers as well as Ti3þOv interactions. Our results indicate that the electrons are distributed across many sites due to energetic degeneracy, but that subsurface sites are preferred, a finding consistent with electron diffraction data.25 We also show a thermodynamic driving force for electrons to move toward the surface, rather than the bulk. Our article proceeds as follows. In section II, we discuss our methodology, whereas in section III, we present our results and discussion, and finally give our conclusions in section IV. 7563
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Table 1. Comparison of Selected Triplet and Open-Shell Singlet Energies with Electrons Localized at Various Ti Sitesa Ti sites
a
EtripletEo.s.s. (eV)
triplet orbital types
singlet orbital types
Ti3þTi3þ distance (Å)
12
0.01
dxz/dxy
dxz/dxy
3.0
34
0.01
dyz/dyz
dx2y2/dxz
6.6
13
0.06
dxz/dx2y2
dxy/dxz
3.6
311
0.00
dx2y2/dx2y2
dx2y2/dx2y2
8.9
360
0.07
dxz/dz2
dxz/dxy
3.5
390
0.12
dx2y2/dz2
dx2y2/dxz
8.6
Energies were calculated with U = 4.1 eV.
II. METHODOLOGY As mentioned, we previously modeled Ti3þ due to HOb as well as charged polarons39 and employ a similar method in the current work. All calculations were performed using the DFT code Quickstep,50,51 part of the CP2K package. Quickstep uses the Gaussian and plane-wave (GPW) method52 such that the valence electrons are represented by Gaussian functions and an additional auxiliary plane-wave basis used for the efficient solution of the electrostatic interactions. A double-ζ valence polarized basis set was used for the valence electrons, and the plane waves were expanded up to a cutoff of 300 Ry. The Brillouin zone was sampled with a reciprocal space mesh consisting of only the Γ-point. Core electrons were modeled by norm-conserving pseudopotentials.53,54 We employed the DFT þ U method30 to ensure electron localization. Three different U values were considered: 3.3, 4.1, and 5.4 eV, just as in our previous HOb study. As we discuss below, the main effect of different U values is to change the location of the defect state within the gap, while the relative energies among possible sites of electron localization remain essentially unchanged when different U values are used. The majority of the work will thus focus on results using a U value of 4.1 eV. The (110) rutile surface was modeled using the slab method in a three-dimensional simulation cell. Because DFT slab calculations can be quite computationally intensive, surface cells have often been limited to (2 2) or smaller. This leads to Ov surface fractions of 0.25 or larger. Such large Ov coverages may not be representative of experimental conditions where, for instance, coverages around 0.080.15 occur under ultra-high-vacuum (UHV) conditions.18,55 In the current work, we used a (4 2) surface cell that initially had eight Ob atoms. One Ob atom was removed, giving an Ov coverage of 0.125. In between periodic slabs was a vacuum region ∼15 Å wide. The slab was five OTiO layers thick (15 atomic layers), and the bottom OTi atomic layers were frozen while the remaining layers were allowed to relax during simulation. A few select slab calculations were run with nine OTiO layers (fully relaxed) in order to test the stability of the Ti3þ as a function of depth. We localized electrons at Ti atoms within the top three layers. For a (4 2) surface cell, this leads to 16 possible Ti sites per layer, or 48 unique Ti positions. With two Ti3þ, the number of possible combinations between 48 sites is 1128. Calculating such a large number of solutions is intractable, so we simplified the problem by classifying the different Ti according to their local geometry. Ti5c are classified as Type I, surface six-coordinated Ti are classified as Type II, whereas the subsurface sites are classified as Types IIIVI. The six different unique Ti types are shown in Figure 1. We selectively chose combinations of Ti3þ that represent a broad range of Ti3þ pairs, and, in total, 49 unique pairs where modeled.
Our procedure for Ti3þ localization was as follows. An Ov was created by removing one of the surface Ob, giving two excess electrons. We then modeled the reduced TiO2 surface with vanadium Vþ nuclei (or pseudopotentials) replaced at two select Ti sites and with a large U value. The larger positive charge of V and large U values favored electron localization at those sites and also created a polaronic distortion in the structure, necessary for Ti3þ stabilization.31,37 These V-substituted geometries and wave functions were then used to start a simulation with the correct reduced TiO2 surface (only Ti and O), and at reasonable U values. In this manner, we could control the final location of the Ti3þ. We also comment on the choice of spin multiplicity. With two excess electrons, there are three different possible electronic states: open-shell triplet spin, open-shell singlet spin, and closedshell singlet spin. The triplet state corresponds to a solution with two unpaired electrons of parallel spins in two different orbital states, the open-shell singlet corresponds to a solution with two unpaired electrons of opposite spins in two different orbital states, whereas the closed-shell singlet corresponds to a solution with two paired electrons of opposite spin in the same orbital state. For large spatial separation between unpaired electrons, we expect the triplet spin state and the open-shell singlet state to be very close in energy, with the open-singlet state slightly more stable. Unpaired electrons at large spatial separation (such as at Ti3þ sites in TiO2) that are forced to be spin-paired into a single orbital state give rise to an energetically unfavorable electronic state. This scenario is reminiscent of a H2 molecule at large separation. In the present case, the energy splitting between the triplet and the open-shell singlet states are mitigated by the 3d atomic states of the unpaired electrons in TiO2 and the superexchange contribution from the oxygen atoms bridging the Ti sites. We note that Di Valentin et al.31 found the closed-shell state to be unstable (by 0.6 eV) compared with the triplet state. We had difficulty controlling the localization of electrons for closedshell calculations, indicative of their relative instability, and, therefore, did not consider closed-shell states any further. Rather, we have focused on triplet and open-shell singlet solutions. In Table 1, we show selected configurations for which we calculated both triplet and open-shell singlet states. The energy differences between solutions localized on the same Ti3þ sites, but with different multiplicities, are very small, ranging from 0.01 to 0.12 eV. Essentially, the triplet and open-shell solutions are nearly degenerate. Table 1 also shows that different d orbitals may be populated for triplet and open-shell singlet wave functions but that the energies of such solutions are close. The distance between Ti3þ is seen to have little effect on triplet/openshell singlet splitting, indicating that the Ti3þ states interactions are minimal and essentially are independent of their relative 7564
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a
0
70 100
400
0
0
60 70
60
0
800
80
60
10
7565
140
0
IV
VI
IV
V
I
V IV
V
I
V
IV
VI
I
V I
III
III
V
III
III
V
III III
III
III
Ti Type 2
0.51
0.48
0.45
0.45
0.44
0.37 0.38
0.35
0.35
0.34
0.33
0.31
0.30
0.27 0.28
0.23
0.23
0.22
0.19
0.18
0.17
0.06 0.15
0.02
0.00
stabilization energy (eV)
10.8
7.4
5.9
8.9
3.0
7.6 7.4
6.5
5.9
4.8
4.5
4.6
8.9
7.2 6.6
3.5
5.5
3.6
8.6
5.9
7.5
3.0 7.2
6.6
7.2
Ti3þ Ti3þ distance (Å)
A U value of 4.1 eV was used. Energies are relative to the most stable configuration (60 90 ).
III
I
100
3
11
IV
V
40
4
I
IV I
10
3
8
00
3
00
00
0
3 20
I
300
3
13 8
I
9
7
IV
300
10
I
10
III
I
I I
I
IV
III
I
III
III
III III
III
III
Ti Type 1
3
1
00
6
11
0
300 4
8 3
3
6
3
0
90
8
3
6
10
90
60
6
Ti Site 2
Ti Site 1
13
1
5
13
13 13
1
1
100
10
3
40
100
1
1
300
8
20
3
00
13
100 100
1
80
Ti Site 1
15
2
II
II
II
II
300 13
II II
100 20
II
II 100
VI 300
IV
I
IV
II VI
II
V
I
IV
V
II
VI VI
II
III
Ti Type 1
200
20
13
1300
10 600
3
1300
1300
100
800
60
1400 300
70
1300
Ti Site 2
II
II
II
V
VI IV
VI
V
VI
IV
II
VI
IV VI
I
VI
VI
VI
V
III
VI V
III
VI
Ti Type 2
Table 2. Relative Stabilization Energies for Open-Shell Singlet Calculations with Electrons Localized at Various Ti Sitesa
1.76
1.55
1.54
1.30
1.17 1.20
1.15
1.07
1.01
1.01
1.00
0.99
0.96 0.96
0.87
0.85
0.76
0.74
0.69
0.69
0.64 0.67
0.63
0.52
stabilization energy (eV)
5.9
3.0
6.6
8.8
9.9 10.5
6.8
7.8
3.0
3.0
5.5
12.7
3.7 5.9
3.6
5.5
7.4
5.6
3.0
5.7
6.6 3.6
5.7
10.0
Ti3þ Ti3þ distance (Å)
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Figure 2. Location of the Ti3þ pair for the top five solutions (ae). The inset of (a) shows the spin density plot of the most stable open-shell singlet solution (60 90 ). The green contour indicates R spin, whereas the blue contour indicates β spin. dxz and dz2 orbitals are shown for the unpaired electrons.
locations in the slab. Convergence of open-shell singlet wave functions to desired solutions proved easier than triplet wave functions, so we, therefore, focused our efforts on modeling Ti3þ pairs as open-shell singlet. The remainder of our work discusses such results.
III. RESULTS AND DISCUSSION A. Location of Most Stable Ti3þ Sites. We calculated the
stability of several Ti3þ pairs and summarize these results in Table 2. The stabilization energies were calculated such that the most stable state is defined as 0.0 eV, whereas less stable states have positive energies. The stabilization energy of state i is defined according to the following: Estabilization i ¼ Ei Emost stable structure
ð1Þ
All atom sites are referred to by the labeling scheme shown in Figure 1. For some Ti3þTi3þ pairs, we found solutions that occupied the same Ti sites as those presented in Table 2, but were less stable. These less-stable solutions converged to different (less-stable) d orbitals configurations, and we do not discuss them further. A similar observation was found in our
HOb work.39 We find the 60 90 Ti3þ pair to be the most stable solution of the 49 combinations we have considered, and this solution has a zero stabilization energy; that is, it represents our zero of energy. The spin density of this solution is shown in Figure 2, indicating where the electrons are localized. Figure 2 also shows the top five most stable solutions that we calculated. More on these solutions will be said later. Nevertheless, several solutions are close in energy; the top 10 calculated solutions are all within 0.23 eV. Our top 25 calculated solutions are within 0.51 eV. Clearly, we have not exhaustively simulated every possible Ti3þ pair, but these results categorically indicate that a large number of solutions are close in energy. The Ti3þ ions are, therefore, likely to occupy several sites during finite time, rather than being confined to one particular location, in agreement with recent CarParrinello molecular dynamics simulations.48 This agrees with our previous work involving a surface HOb where we also showed several Ti3þ solutions to be close in energy. Our results point to the model of unpaired electrons occupying several sites in a Boltzmann-like distribution. The barriers for diffusion of excess electrons from one Ti site to another Ti site have been calculated to be low, typically less than 0.3 eV,37,38 some as low as 0.09 eV, indicating that kinetic 7566
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barriers preventing Ti3þ dispersion are negligible at ambient temperature. We further analyzed our results by comparing the stabilization energies of one Ti type versus another Ti type in Figure 3a. This
allows us to make general conclusions and correlations about the Ti3þ site preference in surfaces with Ov’s. The stabilization energies shown are averages of our data for each type pair and indicate patterns of stability and instability. The most noticeable feature is that pairs of Type III Ti’s are very stable, as indicated by the dark row/column for this type. Type II Ti’s are the least stable, as indicated by the light row/column. There are a large number of Ti3þ pairs that have stabilization energies less than 0.4 eV, such as Type I/I, Type I/III, Type I/IV, Type I/V, Type III/ III, Type III/IV, Type III/V, or Type IV/V. Thus, a large number of stable Ti3þ pairs are possible. The most unstable pairs (>0.9 eV) are Type I/II, Type II/II, Type II/IV, Type II/V, and Type II/VI. We note that the solution calculated by Di Valentin et al.31 is Type I/II, much less stable (∼0.9 eV) than the Type III/III solution proposed here. Morgan and Watson40 predicted a Type II/II solution, which we find to be unstable by >1.5 eV. More will be said comparing our work to others in section III.E below. We also examined the geometric changes near Ti3þ ions. We found that localization of electrons to create Ti3þ sites leads to polaronic distortions. As shown in Table 3, the TiO bonds at the Ti3þ sites for the most stable solutions increase in order to accommodate the excess electronic density. This is entirely consistent with previous work,31,34,37 and this distortion is, in fact, necessary for polaron localization. B. Identifying the Contributions to the Energy of Ti3þ States. To understand the relative energetics of these solutions, we need to consider that the relative stabilization energy of a given state arises from several factors: the energy of the polaron at a site of a given type, the repulsion between two polarons, the spinspin coupling between polaronic sites, and the interaction of the polaron with the Ov. In what follows, we deconstruct our relative stabilization energies in order to access the magnitude of these factors. To this end, we have further analyzed our results to determine the magnitude of any Ti3þTi3þ interactions by extracting from our results an approximate average energy for Ti3þ to exist at a site of a given type (IVI). This was accomplished by fitting our results to the following equation using a least-squares regression algorithm. EðOv Þ ¼ δI EI þ δII EII þ δIII EIII þ δIV EIV þ δV EV þ δVI EVI ð2Þ
Figure 3. Correlation between the Ti types and the relative stabilization energy. The type of Ti3þ for the first unpaired electron is indicated by the x axis, whereas the type of Ti3þ for the second unpaired electron is indicated by the y axis. The shading of the boxes corresponds to the stabilization energies of the combination of unpaired electrons. (a) Stabilization energies taken from averaging the actual DFT þ U data (see Table 2). (b) Stabilization energies taken from the fitting of eq 2 and summing the individual Ei energies (see Table 4).
In eq 2, E(Ov) is the total stabilization energy of an Ov solution (2 electron pair), EI is the energy of an electron on a type I Ti, EII is the energy of an electron on a type II Ti, and so forth. δI is the occupancy of an electron of Type I, which can be either 0 (no Ti3þ of this type) or 1 (1 electron on a Ti3þ of this type) or 2 (2 electrons on Ti3þ’s of this type). Note, Σδi = 2 for an Ov because we have two unpaired electrons. For example, the stabilization
Table 3. Lattice Distortions for the Five Most Stable Solutionsa average equatorial bond distortion
average equatorial bond distortion
average axial bond distortion
average axial bond distortion
maximum bond
minimum bond
2
Site 1 (Å)
Site 2 (Å)
Site 1 (Å)
Site 2 (Å)
distortion (Å)
distortion (Å)
Ti Ti Site Site1
a
60
90
0.11
0.04
0.01
0.08
0.14
0.01
60
100
0.10
0.07
0.01
0.06
0.14
0.01
60
70
0.07
0.09
0.08
0.04
0.14
0.01
70
100
0.08
0.08
0.06
0.06
0.11
0.05
60
400
0.08
0.07
0.07
0.08
0.12
0.03
Numbers reflect changes in TiO bond distances around the Ti3þ site relative to the stoichiometric surface (no Ti3þ present). 7567
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energy of the 13 Ti3þ pair would be EI þ EII (with δI = δII = 1), whereas the energy of the 360 pair would be EI þ EIII. This model completely ignores cross-terms, or interactions between Ti3þ, and assumes the stabilization of a Ti3þ to be independent of its pair partner Ti3þ. This assumption is validated by the present data, as discussed below. The values of EI through EVI were found by fitting our data to the indicated equation. That is, E(Ov) is the dependent variable, δi are independent variables, and EI through EVI are the fitted values. Table 4 gives our fitted values of the one-electron stabilization energies for the present Ov case, as well as stabilization energies for a hydroxylated surface (one excess electron) and a surface with a net 1 charge (also one excess electron), both taken from our previous work.39 The agreement in stabilization energies between the three surfaces is quite remarkable, indicating that Ti3þ ions in all surfaces are of similar character. The Type III (first subsurface layer) Ti3þ sites are the most energetically preferred in all surfaces, and Type I are the second-most preferred location. This result is suggestive of the interpretation that the energy to form a polaron pair at sites i and j is largely dominated by the sum of the single-site energies for site i and j. Furthermore, this also suggests that other factors, such as polaronpolaron interactions, have a significantly small contribution to the overall stability of a Ti3þ pair. To examine this finding, we calculated the deviation of our calculated stabilization energies obtained from the DFT simulations (in Table 2) and the stabilization energies found by eq 2. The maximum absolute deviation is 0.38 eV (the 10 20 pair), the minimum absolute deviation is 0.01 eV (the 311 and 300 300 pairs), and the average absolute deviation is 0.11 eV. Table 5 gives a summary of all the deviations for the Ti3þ pair types. Most of the deviations are small (0.9 eV). Several Ti sites are close in energy, indicative that occupation of several sites is possible during finite time and at finite temperature. This occupation may indeed appear as a “delocalization” to experimental techniques, such as resonance photoelectron diffraction. As it relates to modeling TiO2, our results show that many local solutions (different Ti locations) are possible, indicating the need to properly ensure that the “correct” final wave function is obtained rather than a particular local minimum. We also found Ti3þTi3þ interactions to be small. Finally, we have shown that there is a strong thermodynamic drive (∼0.5 eV) for electrons to migrate toward the near-surface region (first subsurface layer). This migration may be one reason why TiO2 is so photoactive, because electron-transfer reactions can take place in this surface region. 7571
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’ ASSOCIATED CONTENT
bS
Supporting Information. A description of how our DFT þ U algorithm affects our calculated band gaps is found in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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dx.doi.org/10.1021/jp2001139 |J. Phys. Chem. C 2011, 115, 7562–7572
