pubs.acs.org/Langmuir © 2010 American Chemical Society
Modeling Localized Photoinduced Electrons in Rutile-TiO2 Using Periodic DFTþU Methodology† Abdesslem Jedidi,‡,§ Alexis Markovits,‡ C. Minot,*,‡ Sarra Bouzriba,‡,§ and Manef Abderraba§ ‡
Universit e Pierre et Marie Curie-Paris6, UMR CNRS 7616 LCT (Laboratoire de Chimie Th eorique), Paris, F-75005, France, and §Unit e de Recherche de Physico-Chimie Mol eculaire, Institut Pr eparatoire des Etudes Scientifiques et Techniques de Tunis, Boite postale BP51, 2070 La Marsa, Tunisia Received April 6, 2010. Revised Manuscript Received June 7, 2010
We propose a theoretical model for photocatalytic processes on titanium dioxide, described by its most stable phase and surface, rutile-TiO2(110). The excitation induced by light promotes electrons from the valence band to the conduction band. In this context, one important requirement is having a correct value of the magnitude of the electronic gap. The use of GGAþU or LDAþU functional with an appropriate U value allows this. The U correction has little consequence on the adsorption strength itself on the TiO2(110) surface. For the ground state, it only yields a slight increase of the interaction strength of some test molecules; the surface basicity is somewhat enhanced. This is interpreted by the shift of TiO2 vacant levels. Photoexcitation is taken into account by imposing two unpaired electrons per cell of the same spin. The size of the cell therefore determines the number of excitations per surface area; the larger the cell, the smaller the electron-hole surface concentration and the smaller the energy for electronic excitation. For the excited state, careful attention must be focused on the localization of the excited electron and of the hole which are crucial for the determination of the lowest electronic states and for the surface reactivity. We found that the excited electron is localized on a pentacoordinated surface titanium atom while the hole is shared by two surface oxygen atoms not too far from it. The electronic levels associated to the reduced titanium atoms are low in energy; the projected density of states is superposed onto the valence band.
1. Introduction Among the properties of TiO2, the best known and the least studied by theoretician concerns its use as photocatalyst. It was first recognized in 1972 for the photoelectrolysis of water on TiO2 anodes.1 Under radiation, TiO2 is capable of oxidizing organic impurities in aqueous solution and decomposing water molecules into hydrogen and oxygen.2 Many studies were then carried out on TiO2 or SrTiO3(111) surfaces, for understanding how H2O could be an interesting source for H2 production.3-6 Somorjai and co-workers have contributed to this effort, demonstrating, for instance, that water dissociation over TiO2 surfaces was favored with reduced surface Ti3þ.7 Since then, many other applications have been presented,8-10 including photoelectrolysis,11 photocatalysis,12-15 and color photography.12 TiO2 is applied to the treatment of pollutants and the chemical conversion of solar † Part of the Molecular Surface Chemistry and Its Applications special issue. *To whom correspondence should be addressed. Telephone: þ33(0)144272505. Fax: þ33(0)144274117. E-mail:
[email protected].
(1) Fujishima, A.; Honda, K. Nature 1972, 238, 37. (2) Linsebigler, A.; Lu, G.; Yates, J. T., Jr. Chem. Rev. 1995, 95, 735–758. (3) Ferrer, S.; Somorjai, G. A. Surf. Sci. 1980, 97(1), L304–L308. (4) Ferrer, S.; Somorjai, G. A. Surf. Sci. 1980, 94(1), 41–56. (5) Wagner, F. T.; Ferrer, S.; Somorjai, G. A. Surf. Sci. 1980, 101(1-3), 462– 474. (6) Somorjai, G. A. J. Met. 1979, 31(12), 146–146. (7) Lo, J. W.; Chung, Y. W.; Somorjai, G. A. Surf. Sci. 1978, 71, 199–219. (8) Diebold, U. Surf. Sci. Rep. 2003, 48, 53–229. (9) Duncan, W. R.; Prezhdo, O. V. Annu. Rev. Phys. Chem. 2007, 58, 143–184. (10) Duncan, W. R.; Craig, C. F.; Prezhdo, O. V. J. Am. Chem. Soc. 2007, 129, 8528–8543. (11) Lewis, N. S. J. Electroanal. Chem. 2001, 508, 1–10. (12) Liu, D.; Hug, G.; Kamat, P. J. Phys. Chem. 1995, 99, 16768–16775. (13) Ohno, T.; Sarakawa, K.; Matsumara, M. New J. Chem. 2002, 26, 1167– 1170. (14) O’Regan, B.; Gr€atzel, M. Nature 1991, 353, 737–740. (15) Wachs, I. E.; Saleh, R. Y.; Chan, S.; Chersich, C. CHEMTECH 1985, 756– 761.
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energy;13 a wide variety of new designs are being investigated, including dye-sensitized semiconductor solar cells or Gr€atzel cells. Intensive research activity recently has been prompted by the key role of TiO2 in the injection process in a photochemical solar cell with high conversion efficiency.14 TiO2 is a wide band gap semiconductor (3.05 eV for rutile and 3.18 eV for anatase) and can only absorb about 5% of the sunlight in the ultraviolet light region, which substantially limits its practical application. TiO2 crystallizes in three different phases: rutile (the most stable one), anatase, and brookite. However, opinion is divided about the importance of the specific nature of the phases; differences in chemical behavior for different phases may be large. Somorjai suggested that the lower efficiency in the rutile phase originates in the very fast recombination of the electron-hole pair and the relatively low amount of reactants and hydroxides attached to the surface.16 According to Wachs et al., it could also originate from the preparation and impurities.15 H adsorption is structure insensitive, being similar when relaxations of the surfaces are included,17 whereas H2O adsorption, molecular or dissociative, depends on the surface structure.18 Anatase was first thought to be a better photocatalyst.19,20 The standard TiO2 powder21 contains anatase and rutile phases in a ratio of about 3:1. It has however been reported22 that the surface of anatase particles is (16) Somorjai, G. A. Chemistry in two dimensions: Surface; Cornell University Press: Ithaca, NY, 1981. (17) Bouzoubaa, A.; Markovits, A.; Calatayud, M.; Minot, C. Surf. Sci. 2005, 583, 107–117. (18) Di Valentin, C.; Tilocca, A.; Selloni, A.; Beck, T. J.; Klust, A.; Batzill, M.; Losovyj, Y.; Diebold, U. J. Am. Chem. Soc. 2005, 127, 9895–9903. (19) Gr€atzel, M. Comments Inorg. Chem. 1991, 12, 93–111. (20) Tang, H.; Levy, F.; Berger, H.; Schmid, P. E. Phys. Rev. B 1995, 52, 7771– 7774. (21) Ohno, T.; Sarukawa, K.; Tokieda, K.; Matsumura, M. J. Catal. 2001, 203, 82–86. (22) Bickley, R. I.; Gonzalezcarreno, T.; Lees, J. S.; Palmisano, L.; Tilley, R. D. J. Solid State Chem. 1991, 92, 178–190.
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Figure 1. The 110 surface of rutile with labeling of the fivefold coordinated Ti atoms and of the bridging atoms. Localization on the Ti atoms is always made on Ti0.
transformed to the rutile structure. Rutile particles with a small surface area are efficient for splitting water,22 which is an important reaction to convert light energy into chemical energy. Ohtani et al.23 reported the markedly high photocatalytic activity of brookite nanocrystallites as compared to that of rutile and anatase. Koelsch et al.24 deposited brookite as a thin film from a stable dispersion and proposed brookite as a good candidate for photovoltaic devices. The rutile TiO2(110) surface is one of the most important surface models for metal oxides. The perfect rutile TiO2(110) surface has been extensively studied (see, for instance, refs 2 and 25-30). The structure is made of alternating horizontal and vertical polymers31 which makes the surface at the same time reactive and stable.2,32,33 The reconstruction is weak, contrasting with other surfaces that form (110) facets upon heating,26,29,34 and is restricted to a small relaxation and rumpling of the bridging oxygen atoms.27,30 The TiO2 valence band (VB) is formed by oxygen orbitals while the conduction band (CB) originates from the d-orbitals of the titanium atoms. The gap between the valence and CBs of bulk TiO2 is 3.05 eV,25,28 while the first exciton energy is 3.57 eV. A gap is also found by calculations35,36 for regular slabs with a [110] orientation preserving the stoichiometry (2.06 and 1.49 eV for LDAþU and GGAþU, respectively). The adsorption on the (110) surface is then controlled by the electron counts that maintain this gap.35 In this paper, we investigate the nature of the excited state of a rutile TiO2(110) surface using a periodic density functional theory (DFT) approach, testing LDAþU and GGAþU methodologies, using plane waves as implemented in the VASP code. We show that the U correction allows a reasonable description of the excited state provided that attention is paid to localization. Such (23) Ohtani, B.; Handa, J.; Nishimoto., S.; Kagiya, T. Chem. Phys. Lett. 1985, 120, 292–294. (24) Koelsch, M.; Cassaignon, S.; Guillemoles, F. J.; Jolivet, J. R. Thin Solid Films 2002, 403, 312–319. (25) http://www.oxmat.co.uk/Crysdata/tio2.htm. (26) Barteau, M. A. Chem. Rev. 1996, 96, 1413–1430. (27) Charlton, G.; Howes, P. B.; Nicklin, C. L.; Steadman, P.; Taylor, J. S. G.; Muryn, C. A.; Harte, S. P.; Mercer, J.; McGrath, R.; Norman, D.; Turner, T. S.; Thornton, G. Phys. Rev. Lett. 1997, 78, 495–498. (28) Cronemeyer, D. C. Phys. Rev. 1952, 87, 876–886. (29) Henrich, V. E. Rep. Prog. Phys. 1985, 48, 1481–1541. (30) Ramamoorthy, M.; Vanderbilt, D.; King-Smith, R. D. Phys. Rev. B 1994, 49, 16721–16727. (31) Fahmi, A.; Minot, C. Surf. Sci. 1994, 304, 343–359. (32) Minot, C. Theoretical approaches of the reactivity at MgO(100) and TiO2(110) surfaces. In Progress in Theoretical Chemistry and Physics; Chaer Nascimento, M. A., Ed.; Kluwer: Dordrecht, 2001; Vol. 7, pp 241-249. (33) Ahdjoudj, J.; Markovits, A.; Minot, C. Catal. Today 1999, 50, 541–551. (34) Firment, L. E. Surf. Sci. 1982, 116, 205–216. (35) Calatayud, M.; Markovits, A.; Minot, C. THEOCHEM 2004, 709, 87–96. (36) Vogtenhuber, D.; Podloucky, R.; Neckel, A.; Steinemann, S. G.; Freeman, A. J. Phys. Rev. B 1994, 49(3), 2099–2103. (37) Shapovalov, V.; Stefanovich, E. V.; Truong, T. N. Surf. Sci. Lett. 2002, 498, L103–L108.
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a model should be useful to understand the early stage of photocatalytic processes. To our knowledge, the only similar approaches37,38 were done using ab initio embedded cluster methodology. The paper is organized as follows. We first present models and strategy (section 2) and methodology used (section 3). Then, in section 4, we present tests for the validity of the U correction for adsorption in the ground state, showing that the main results are preserved. Finally, in section 5, we discuss the results for the excited state, focusing on the electron-hole localization.
2. Models and Strategy In this paper, we investigate consequences of absorbing light on the (110) surface of rutile, clean or covered. We choose to describe the solid using periodicity, since this is more appropriate for a description of a solid, even though an accurate methodology for excited states such as TD-DFT is then lacking. For the rutile surface, the ground state is a low spin state (singlet). An excited state is obtained by imposing a high spin state that must promote an electron from the VB to the CB. Per unit cell, this corresponds to impose two electrons with the same spin. It also creates a Frenkel exciton (hole in the VB, electron in the CB). Hereafter, we call “excitation energy”, EE, the energy difference between the triplet electronic state and the fundamental singlet state. In order to describe reactivity after light absorption, we are more concerned by an appropriate description of the excited state after electronic and geometric relaxation than by excitation itself. Note that the photon energy required for the absorption of light should correspond to a vertical transition (which does not include the relaxation of the excited state); then it should be more directly related to the magnitude of the band gap. The transition should also lead to a singlet state (antiferromagnetic state) that is difficult to model. Nevertheless, the triplet state (ferromagnetic) should not be very different for energy and localization than the singlet. The amount of EE varies with the size of the unit cell considered. The convergence with the unit cell is therefore essential to check and will be discussed in section 4. The model consists of slabs of nine atomic layers (three Ti2O4 layers) of rutile oriented in the (110) direction (Figure 1). We have used the experimental cell parameters which yield 1.9485 and 1.9800 A˚ Ti-O bond lengths.39 Theoretical results obtained from DFT are close to them as discussed in ref 40 (1.932/1.962 A˚ for LDA; 1.961/2.002 A˚ for GGA-PW91). The U correction (38) Belelli, P. G.; Ferullo, R. M.; Branda, M. M.; Castellani, N. J. Appl. Surf. Sci. 2007, 254(1), 32–35. (39) Hyde, B. G.; Anderson, S. Inorganic Crystal Structures; John Willey & Sons: New York, 1989. (40) Labat, F.; Baranek, P.; Domain, C.; Minot, C.; Adamo, C. J. Chem. Phys. 2007, 126, 154703.
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Figure 2. Adsorption on rutile TiO2: (a) dissociated water, (b) ammonia, and (c) CO2 in parallel mode (top-view).
contributes to a slight increase (1.995/2.001 for LDAþU; 2.01/ 2.03 for GGA-PW91þU). To avoid spin delocalization on the two opposite surfaces of the slab, we have fixed the atoms of the back face at the bulk position and saturated them by adsorption of water molecules (OH on Ti5 and H on bridging O2, with the indices 5 or 2 indicating the atom coordination). The geometry of the three top layers was optimized. This model allows us to consider large unit cells (small amount of excitations) and should be sufficient to draw conclusions on trends. It allows spin localization on the unsaturated surface and improves the convergence of properties with the slab thickness. Note that the saturation of the dangling bonds from the back face contributes to enlarge the energy gap for the slab (from 1.77 to 2.06 eV using LDAþU).
3. Computational Details We performed spin polarized calculations based on density functional theory DFTþU in the local density approximation, LDA, or in the generalized gradient approximation, GGA (using the PW91 exchange-correlation functional), LDAþU or GGAþU, as implemented in the VASP code,41-44 which uses a plane wave basis set (with a kinetic energy cutoff at 400 eV). The electron-ion interactions were described by projector augmented wave (PAW) pseudopotentials.45 Relativistic effects were partially taken into account through the use of relativistic scalar pseudopotentials. The calculations were performed by sampling the Brillouin zone in a 6 12 1 Monkhorst-Pack set for the unit cell and adapted to the cell multiplicity (6 2 1 for the p(1 6) cell). The densities of states were computed with the same or denser grids. Some test calculations have been carried out for checking the energy convergence of supercells with respect to k-points. The energies were computed within the tetrahedron method with Bl€ochl correction. For the LDAþU, we first made calculations for the bulk rutile structure with different U. The gap increases nearly linearly with U and reaches 3 eV for U = 10 eV. Similarly, this value is obtained with U = 8 eV using GGAþU. The gap is indirect; the highest occupied molecular orbital (HOMO) is at the Γ point while the lowest unoccupied molecular orbital (LUMO) is at the (0.5, 0.5, 0.) k-point. Slab calculations were done imposing these values. As already mentioned, this leads to the presence of smaller gaps: 2.06 eV (LDAþU) and 1.49 eV (GGAþU), with the surface states reducing the HOMO-LUMO energy difference. (41) (42) (43) (44) (45)
Kresse, G.; Hafner, J. Phys. Rev. B 1993, 47, 558–561. Kresse, G.; Hafner, J. Phys. Rev. B 1994, 49, 14251–14269. Kresse, G.; Hafner, J. Mater. Sci. 1996, 6, 15–50. Kresse, G.; Furthm€uller, J. Phys. Rev. B 1996, 54, 11169–11186. Kresse, G.; Joubert, J. Phys. Rev. B 1999, 59, 1758–1775.
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The geometry optimizations were carried out until the forces remaining on the atoms were less than 0.01 eV/A˚, and a dipole correction has been considered to minimize dipole-dipole interaction between successive slabs. The calculations being periodic in three dimensions, a vacuum width of at least 10 A˚ ensures that the interaction between successive slabs is negligible. We report results obtained after imposing an initial localization of the spin. This has been performed using the MAGMOM option in VASP. Without this option, spin is much delocalized; corresponding energies are much higher and results are unphysical. The use LDAþU and GGAþU has the advantage of simplicity. The B3LYP46,47 methodology is superior but more difficult to run with codes using plane wave basis sets; it requires the use of atomic basis sets (or projection on such basis sets). We conducted tests using the CRYSTAL code48 that is also periodic, using the same slabs and atomic basis set.49
4. Tests for Adsorption on the Ground State Before studying adsorption for an excited state, we must verify that adsorption on the ground state is well described using LDAþU. In the comparison between reactivity before or after excitation, we want to be sure that the U parameter does not change the known results without light absorption. Besides, local electron excitation should not modify the binding of adsorbates present at sites that are not sites of excitation. We have chosen to adsorb molecules on a p(1 2) unit cell allowing pairs of surface sites, with one being available for a localized excitation and a possible specific photochemical process. Besides, considering a p(1 1) unit cell would correspond later to a too large excitation concentration. The results are shown in Table 3, and representative adsorption modes are shown in Figure 2. Adsorption energies are calculated with the formula: Eads=Esupersystem - (Esurf þ Eadsorbate). According to the convention, an exothermic process has positive adsorption energy. The main results expected for adsorption on the ground state of the (110) rutile surface are as follows: (i) The dominant acidic character of the clean surface; molecules adsorb with the formation of a dative bond to the Ti5 surface cations. The most basic molecules are more firmly bound to Ti from the (110) surface, NH3 more than H2O (without dissociation) and much more than CO2. (ii) GGA for the p(1 2) unit cell gives the water (46) Becke, A. D. J. Chem. Phys. 1993, 98, 5648–5652. (47) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1988, 37(2), 785–789. (48) Dovesi, R.; Saunders, V. R.; Roetti, C.; Orlando, R.; Zicovich-Wilson, C. M.; Pascale, F.; Civalleri, B.; Doll, K.; Harrison, N. M.; Bush, I. J.; D’Arco, P.; Llunell, M. CRYSTAL06 User’s Manual; University of Torino: Torino, 2006. (49) Markovits, A.; Fahmi, A.; Minot, C. THEOCHEM. 1996, 371, 219–235.
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Figure 3. Excitation energy (EE in eV) as a function of the cell multiplicity, n, for the p(1 n) TiO2 slab (three layers with saturation). Circles correspond to p(2 2) and p(2 3) cells; EEs are the same than for the p(1 n) cells of the same size within 8% or less.
dissociation. Dissociation is controversial and may depend on the coverage and the presence of defects. Experimental measurements have detected surface hydroxyl groups.50,51 Theoretical studies yield different results: water can be dissociated52-58 or not59-62 on TiO2 surfaces. The LDA and GGA equally reproduce these features, LDA with larger adsorption energies than GGA. A difference appears for the CO2 adsorption. LDA favors the mode parallel to the surface where two O atoms are bound to the surface Ti atoms. GGA with smaller values privileges the binding of a single O atom, leading to the perpendicular mode; in both cases, CO2 behaves as a weak base while the adsorption site for TiO2 is the acidic center. Introducing a U correction shifts the TiO2 vacant levels upward in energy; therefore, the balance between the role of Frontier orbitals is modified, making the influence of the HOMO more important. It follows that the basicity of the surface O atoms increases and that dissociation is easier. In agreement with this analysis, water dissociation is facilitated for the U correction (see Table 3). A second remark emerges from Table 3 that displays the heats of adsorption for test molecules. All the adsorptions are larger than those for U = 0. At first sight, this is unexpected for a molecular adsorption; indeed, if the LUMO levels (CB) are shifted toward upper energies, we should observe a decrease of the heats of adsorption following an increase in the energy difference of the frontier orbitals. The reason is to be found in the ionicity of the adsorption site. On the naked surface, unsaturated Ti atoms reinforce their binding to their neighbors by developing a partial covalent character for the Ti-O bonds which implies a certain amount of population on Ti. The U correction, effective only on the Ti orbitals, is then destabilizing. Under adsorption, the Ti cation has a larger coordination and is more (50) Henrich, V. E.; Dresselhaus, G.; Zeiger, H. J. Solid State Commun. 1977, 24, 623–626. (51) Pan, J.-M.; Maschoff, B. L.; Diebold, U.; Madey, T. E. J. Vac. Sci. Technol. 1992, A 10, 2470–2476. (52) Fahmi, A.; Minot, C. J. Organomet. Chem. 1994, 478, 67–73. (53) Ferris, K. F.; Wang, L.-Q. J. Vac. Sci. Technol. 1998, A 16, 956–960. (54) Goniakowski, J.; Bouette-Russo, S.; Noguera, C. Surf. Sci. 1993, 284, 315– 327. (55) Goniakowski, J.; Gillan, M. J. Surf. Sci. 1996, 350, 145–158. (56) Goniakowski, J.; Noguera, C. Surf. Sci. 1995, 330, 337–349. (57) Lindan, P. J. D.; Harrison, N. M.; Gillan, M. J. Phys. Rev. Lett. 1998, 80, 762–765. (58) Lindan, P. J. D.; Harrison, N. M.; Holender, J. M.; Gillan, M. J. Chem. Phys. Lett. 1996, 261, 246–252. (59) Brinkley, D.; Dietrich, M.; Engel, T.; Farrall, P.; Gantner, G.; Schafer, A.; Szuchmacher, A. Surf. Sci. 1998, 395, 292–306. (60) Casarin, M.; Maccato, C.; Vittadini, A. J. Phys. Chem. B 1998, 102, 10745– 10752. (61) Langel, W. Surf. Sci. 2002, 496, 141–150. (62) Stefanovich, E. V.; Truong, T. N. Chem. Phys. Lett. 1999, 299, 623–629.
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ionic. It is then less populated and the destabilization with U is lower. This explains an overall increase of the heats of adsorption with U that was unexpected to us.
5. Results and Discussion 5.1. Excitation Energies Using Periodic DFT and Convergence with the Size of the Unit Cell. We force an electron excitation by imposing a spin state. In a rigid band model, one electron must then occupy the lowest energy level of the CB. One electron promotion per primitive cell leads to a high concentration, since there are as many promoted electrons as there are equivalent sites. This leads to a ferromagnetic arrangement. The excitation should then correspond to the promotion of electrons occupying the complete highest band of the VB to a state where the entire lowest band of the CB is occupied. The average energy difference between excited and ground states is larger than the band gap. The lowest-energy promotion corresponds to the band gap and a convergence to this value is expected when the excitation concentration decreases. Since the lifetime of the excited state should be small, we should not expect a large excitation concentration. In a periodic model, we must therefore use a large enough cell. When the size of the unit cell increases by a factor n, the promotion affects a fraction 1/n of these bands (the upper part of the VB and the lower part of the CB). The energy difference is then smaller, and the excitation energy should decrease. In Figure 3, we display the excitation energies calculated for single excitation, increasing the cell size (n = 1, 6) along the [001] direction. Tests have also been performed for the p(2 2) and p(3 2) cells using a GGAþU functional. The excitation energies are the same as those for the p(1 4) and p(1 6) cells of equivalent multiplicity. The very first observation is that the results are much more sensitive to the U parameter than to the choice of the DFT functional. Indeed, EEs calculated with LDA and GGA are nearly the same while the introduction of U induces a decrease of ∼1 eV. Let us first note that, using LDAþU or GGAþU, the excitation values decrease with increasing n, that is, decrease when the concentration of excited electrons decreases. They converge on a limit, with the convergence being reached in practice already for n = 3. Note that the initial electron localization is very important for a correct description of the excited state. Otherwise, spin is distributed among several surface Ti ions and the results are not physical. The excitation energy considering localized states is ∼0.87 eV, a value clearly smaller than the band gap of the slab calculated for the ground state. This will be discussed in more detail in section 5.2 hereafter. In Figure 3, the dependence on n (size of the unit cell) should be correlated to the interaction of the spin on neighboring sites. DOI: 10.1021/la101359m
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Jedidi et al. Table 1. Main Contributions to the Spin Density for the Different p(1 n) Cells Calculated by Integration of the Spin Density
Figure 4. LDAþU spin density isosurface distribution for the photoactivated TiO2 surface (one excitation per primitive cell). (a) Top view of p(1 3). Spin is shown in yellow. (b) Side view of p(1 1). Saturation of the face below (red labels) allows localization on the face above. Electrons are on the fivefold coordinated Ti atoms, and holes are on the bridging oxygen atoms.
For the primitive cell, promotion occurs on all the Ti5 surface atoms and leads to a ferromagnetic spin arrangement. Compared with the p(1 6) unit cell, the single promotion for the primitive (1 1) cell corresponds to six successive excitations. The trend shows that successive excitations involve larger and larger EEs (the EEs are larger for small n values in Figure 3), reflecting that an increase in the number of excitations is not energetically favorable. The consideration of successive p(1 n) unit cells then involves the most uniform description for a given excitation concentration without two excitations occurring at neighboring places: the use of the p(1 2) cell, corresponding to three excitations per p(1 6) cell, represents the most favorable case where excitations do not take place at vicinal places. 5.2. Localization of Electron and Hole. Saturation of the back face of the slab helps to localize the electron and hole on the bare surface. The holes and excited electrons within triplet structures tend to be localized on the least coordinated atoms and therefore on the bare surface. These sites are less charged and do not require as much as the regular sites to be stabilized by neighboring counterions. This is a general rule,63 with some exceptions possibly occurring due to the electron-hole interaction.64,65 Upon photoexcitation, the fivefold coordinated Ti4þ ions are reduced to Ti3þ and the bridging oxygen O2- ions are oxidized to O-. Figure 4 shows the spin distribution on these atoms. Our result resembles those of Belelli et al.38 who found, according to M€ulliken population analysis, spin density on fivefold coordinated Ti and twofold bridging oxygen in the case of photoactivated TiO2(001) anatase. They are in contrast with the conclusion of Shapalov et al.37 These authors have used an embedded cluster to model the rutile TiO2(110) surface, and they have found for the triplet state that the unpaired electron is localized in two-sublayer titanium and the hole is on one-sublayer oxygen. They concluded from this localization that both the hole and the unpaired electron are unlikely to participate in the surface reactions. Doubling or more the unit cell allows distinguishing between several fivefold coordinated Ti or bridging oxygen atoms. (63) Nakamura, R.; Okamura, T.; Ohashi, N.; Imanishi, A.; Nakato, Y. J. Am. Chem. Soc. 2005, 127, 12975–12983. (64) Qu, Z.-w.; Kroes, G.-J. J. Phys. Chem. B 2006, 110(18), 8998–9007. (65) Qu, Z.-w.; Kroes, G.-J. J. Phys. Chem. C 2007, 111, 16808–16817.
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cell
double
triple
quadruple
quintuple
sextuple
Ti0 O1 þ O-1 O0
0.992 0.463 0.096
0.984 0.470 0.036
0.982 0.396 0.037
0.932 0.356 0.004
0.983 0.357 0.002
As mentioned in the previous section, we have forced the initial localization of the spin on individual atoms; this spin distribution was not always strictly maintained where it was initially imposed, and we have found some electrons switching place; however, in the final result, we always obtained a localized spin on very few number of atoms and this localization always led to a stabilization of the excited state. Localization for the multiple cells is always important on a single Ti atom (see Table 1), and we call hereafter Ti0 the reduced Ti atom. Then, localization on O atoms is always referred relative to Ti0 (see Figure 1), with the index indicating the distance to Ti0 (0 for the first neighbors, 1 for the second neighbors ...). The localization on O is less pronounced than that on Ti0; however, there is 1 order of magnitude between the O atoms bearing the largest spin density and the others (Table 1). For the double unit cell, localization on the bridging O atoms takes place on O1. Localization on O0 would result in sharing the distribution of these atoms in stripes, while localization on O1 makes two-dimensional nets. The localization splits the set of atoms (Ti5 and bridging O) into a set of atoms with spin and a set of atoms not affected by the excitation (neither reduced nor oxidized). The latter remain strongly charged and are better connected by nets. For larger multiple unit cells, the localization remains similar, always occurring on O1 (and O-1 equivalent to O1). We did not succeed to localize on a single oxygen, breaking the symmetry between O1 and O-1. Localization did not take place on a more distant O for the quadruple, quintuple, or sextuple unit cells. We present in Figures 4 and 5 the LDAþU spin distribution and the projected density of states (PDOS) curves for the triple cell in the triplet state, respectively. It is worth noticing that spin integration shows that, with pure LDA and GGA, we do not obtain spin localization. For instance then, every surface Ti6 bears some nonnegligible spin. We conclude that the use of U is necessary. Figure 5 shows the PDOS LDAþU curves on Ti0 and O1 for the p(3 1) cell in triplet electronic state with an energy range from ∼ -15 to ∼ -1 eV, with the Fermi level (at 8 eV) erasing the upper part of the VB (extending from ∼ -13 to -8 eV). The integrated curve of the total DOS (not shown) shows two steps; the first one, lower in energy, is due to a large Ti contribution of alpha spin; it corresponds to the electron that reduces the pentacoordinated Ti surface site. The unpaired electron on Ti corresponds to a broad PDOS peak overlapping the valence band mostly made of the 2p(O) orbitals. The other step, close to the Fermi energy, corresponds to the PDOS on O1 for the main part and corresponds to the unmatched electron of the hole. The hole itself is visible on the beta peak just above the Fermi level. The DOS analysis makes understandable why the EEs for LDAþU (GGAþU) are not larger than those for LDA (GGA), although the introduction of U enlarges the magnitude of the electronic band gap: the electron of reduction on Ti0 is not a state in the gap that poorly detaches from the VB; it is significantly stabilized below the top of the VB. On the contrary, the Fermi level separates the two spin components of the hole. Langmuir 2010, 26(21), 16232–16238
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Figure 5. LDAþU DOS for the triple cell. Projected density of states on Ti0 (full line) and O(1 (dotted line). Both of the unpaired electrons lie below the Fermi level. Our results are in contrast with a rigid band model where an unpaired electron is transferred from the valence band to the conduction band. Table 2. Relaxation Occurring at the Surface under Electron Excitationa cell
double
triple
quadruple
quintuple
sextuple
z(Ti1)-z(Ti0) 0.07 0.07 0.07 0.08 0.09 0.060 0.025 0.025 0.016 0.016 z(O1)-z(O0) a The vertical displacements (outward) are in angstroms (A˚).
Table 3. Adsorption Energy (eV) on the p(1 2) Unit Cell for the Ground State (eV)a CO2
H2O
NH3
parallel perpendicular molecular dissociated molecular LDA LDAþU (U = 10 eV) GGA GGAþU (U = 8 eV) a
0.51 0.64
0.31 0.59
1.16 1.49
1.33 1.99
1.47 1.83
0.13 0.16
0.23 0.15
0.71 0.97
0.84 1.39
0.97 1.26
A positive value corresponds to an exothermic process.
We have also performed two periodic calculations with localized basis sites described in ref 66 using the CRYSTAL code and Hartree-Fock and B3LYP46,47 methodology. The geometry of the atoms of the slab is frozen at their bulk position. The DOS obtained are similar to those of the LDAþU or GGAþU results. The EEs calculated using Hartree-Fock and B3LYP methodologies are 1.62 and 0.75 eV, respectively. These values are noticeably smaller than the gap calculated for the slab: 8.3 eV, overestimated by Hartree-Fock, and 1.9 eV using B3LYP that provides a value close to LDAþU. A rigid band model is therefore an oversimplified view of the excited slab, and the LDAþU and GGAþU methods correctly describe the surface after excitation. Let us add that we also investigated the DOS for the DFT calculations without U. A spin polarization occurs at the Fermi level given what is indicated on the output. However, a careful attention to the DOS does not give confidence to this result: the electron count where the spin polarization occurs is less than it should be. The exact electron count corresponding to a small increase shifts the Fermi level into the gap where no polarization occurs. Calculations therefore failed to give a correct description of an excited state. The excitation induces a modification of the geometry at the excitation sites. Vertical relaxations are displayed in Table 2. (66) Markovits, A.; Ahdjoudj, J.; Minot, C. Surf. Sci. 1996, 365, 649–661.
Langmuir 2010, 26(21), 16232–16238
The atoms bearing the spin (electron or hole) always move outward toward the surface. Several explanations are consistent with these relaxations. These atoms, that are always surface ions, have the smallest charges and thus can be less bound than the others, since large charges have to be balanced by interaction with neighboring ions of opposite charge. In terms of orbitals, Ti3þ should be hybridized to accommodate an electron, with the 4s contribution being stabilizing. As a result, the odd electron occupies a dangling bond emerging from the surface available to react with an appropriate radical species.
6. Conclusion We have proposed a model in order to study photocatalysis involving the use of a periodic slab representing the TiO2(110) surface of the rutile phase and the DFTþU methodology. We first carried out some tests on adsorption showing that U functionals do not affect the adsorption modes when adsorption is thermal. Only small differences appear; the dominant surface character is still acidic but it is reduced. Indeed, the vacant levels, mainly associated with surface Ti, are shifted and the role of frontier orbitals is changed: the influence of the HOMO is increased. From a quantitative point of view, all the interaction strengths are increased. The introduction of the U parameter, when calculating adsorption energies, destabilizes more the clean surface, that is, the reference, than the surface with an adsorbate. In order to model the excited state after excitation induced by light absorption, we imposed two electrons of the same spin per unit cell. At variance with DFT without U, the use of the LDAþU or GGAþU functional provides a reasonable description. It permits reproduction of the experimental electronic gap magnitude which is required in this approach. In a rigid band model, an electron is supposed to be promoted from the valence band (mainly located on surface oxygen), where a hole is created, to the conduction band (mainly with titanium character). However, the rigid band model then appears as oversimplified. The stabilization of the reduced Ti atom is strong, reducing the excitation energy to a value that is smaller than the band gap. Localization of the electron has to be carefully considered. The unpaired excited electron is on a single surface pentacoordinated titanium atom, and holes are on the two surface O atoms which are second neighbors to the titanium. DOI: 10.1021/la101359m
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The concentration of hole-electron pairs on the surface depends on the size of the unit cell. A small cell yields a high concentration, while a very large cell decreases it. The excitation energy decreases with increasing size of the cell and quickly reaches a limit.
16238 DOI: 10.1021/la101359m
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Acknowledgment. The authors thank Dr. Ilaria Ciofini, Prof. Carlo Adamo, and Prof. Michel Van Hove for fruitful discussions. Bilateral agreements between France and Tunisia are acknowledged for their support (PHC-Utique, CMCU n°09G1212).
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