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A First Principles Study on Charge Dependent Diffusion of Point Defects in Rutile TiO2 Abu Md. Asaduzzaman*,† and Peter Kru¨ger‡ Department of Chemistry, UniVersity of Manitoba, Winnipeg R3T 2N2, Canada, and ICB, UMR 5209 UniVersite´ de BourgognesCNRS, 21078 Dijon, France ReceiVed: August 23, 2010; ReVised Manuscript ReceiVed: September 15, 2010
A first principles theoretical study on the diffusion mechanism of Ti interstitials and O vacancies in rutile TiO2 is reported. We find that the diffusion depends strongly on the defect charge. Weakly charged Ti ions diffuse preferentially through the open channels along the c axis with a barrier of ∼0.4 eV. Ti4+ ions, however, diffuse perpendicular to c by an interstitialcy mechanism with a barrier of ∼0.2 eV. Neutral oxygen vacancies diffuse along the c axis with a barrier of 0.65 eV. Introduction Defects in a transition metal oxide play an important role in crystalline transport, affect mechanical properties through interaction with dislocations and stacking-faults, modify surface chemistry, and alter electrical and optical properties. In the case of titanium dioxide, properties such as catalytic activity, occurrence of strong metal-support interactions, and oxide growth rate in the electrochemical cells are affected by point defects.1-3 TiO2 can hardly be oxidized but easily reduced to TiO2-x contains extra Ti and/or missing O atoms. Thus the main point defects are Ti interstitials (TiI) and O vacancies (VO).4,5 Which of the two defects dominates the ionic transport in bulk and at the surface has been the subject of many papers but has nonetheless remained controversial. At high temperatures (>1750 K) or strongly reducing conditions TiI seems to dominate, but at low T and less reducing conditions VO might be the major defect type in the bulk.4 For TiO2 surfaces prepared under ultrahigh vacuum, oxygen vacancies are abundant. However there is evidence that Ti interstitial diffusion plays a major role in bulk-assisted reoxidation of the rutile (110) surface6 and in dissociative oxygen adsorption.7 For a deeper understanding of these phenomena, the microscopic diffusion processes need to be clarified. For Ti interstitials, two main diffusion mechanism have been suggested: (i) parallel to the c axis through the “open channels” and (ii) perpendicular to the c axis through an interstitialcy process (see Figure 1). In the interstitialcy mechanism, the diffusing interstitial Ti atom pushes another Ti atom out of its lattice site into the next interstitial site, such that the “kicked-out” atom becomes the new interstitial atom. Huntington and Sullivan8 argued that (i) is the most likely process. Later, Hoshino et al.9 found experimental evidence that 44Ti diffuses faster perpendicular than parallel to c, which suggests that the interstitialcy mechanism (ii) is the dominant process. The diffusion mechanism of transition metal impurities in TiO2 is closely related to that of Ti interstitials and may give insights into the self-diffusion problem. For bulk TiO2, Sasaki et al.10 found that the diffusion path of transition metal impurities strongly depends on the charge of the defect. Divalent cations diffuse along the open channels (process i), whereas trivalent * To whom correspondence should be addressed, asaduzza@ cc.umanitoba.ca. † University of Manitoba. ‡ UMR 5209 Universite´ de BourgognesCNRS.
Figure 1. Diffusion path of an interstitial Ti (marked by 1 and 2) perpendicular to the c axis through the interstitialcy mechanism. The Ti defect atom at interstitial site 1 kicks out a neighboring Ti atom at lattice site K. The kicked-out atom moves to the next interstitial site 2 and becomes the defect. The diffusion in the “open channel” corresponds to the equivalent of 1 along the [001]. The blue and red balls correspond to oxygen and titanium atoms, respectively.
cations preferentially diffuse via the interstitialcy mechanism (ii). This rule was found to hold even for mixed valent impurities such as Mn and Fe, which can have both the +2 and the +3 charge state in TiO2. The diffusion of adsorbed transition metal atoms into the TiO2 substrate was also reported. For example, there is experimental evidence that adsorbed V atoms diffuse into TiO2(110) by kicking out a 6-fold surface Ti atom.11,12 We have confirmed this kick-out process for V and Mo adatoms in our previous theoretical studies13,14 and found diffusion barriers of the order of half an electronvolt. Recently Iddir et al.15 have reported a first principles study on the diffusion of interstitial Ti and oxygen vacancies, but diffusion barriers were only calculated for the fully charged ions, Ti4+ and VO2+. In view of the experimental evidence10 for a charge dependence of the metal impurity diffusion, it is important to study the defect diffusion also for other charge states. In this article we present a density functional study on the possible diffusion mechanism of interstitial Ti as a function
10.1021/jp107986a 2010 American Chemical Society Published on Web 11/02/2010
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Asaduzzaman and Kru¨ger TABLE 1: Diffusion Barriers as a Function of Charge State and Diffusion Directiona nominal defect type
Ti0
Ti2+
Ti4+
barrier height |c (eV) barrier height ⊥c (eV) Bader charge on interstitial Ti (e)
0.45 0.54 +2.25
0.35 0.46 +2.30
0.31 0.23 +2.47
a
Figure 2. Different oxygen vacancy (marked by V) diffusion paths. New vacancy positions are shown by arrows. The presentation is the same as that given in Figure 1.
of the defect charge. We find that the preferential diffusion path depends strongly on defect charge and we discuss the result in the light of the experimental literature. We have also studied oxygen vacancy diffusion assuming a neutral vacancy. For this defect we find a highly anisotropic diffusion with strong preference for the c direction. Computational Procedure All calculations were performed using the plane-wave code VASP16-18 within density functional theory and the generalized gradient approximation “Perdew-Wang 91” and ultrasoft pseudopotentials as provided in the package.19 The systems were modeled with a (2 × 4 × 2) supercell containing 192 atoms ((1 for TiI and VO). We performed a few tests with a 240atom 2 × 5 × 2 supercell and found negligible differences in the reported energies. The plane wave cut off energy was 400 eV, and only the Γ-points were used. All calculations were done without spin polarization since in previous similar studies13,14 we found that spin-polarization changes the structural energies only in a negligible way. Previously optimized structural parameters of bulk TiO2 have been used. Results and Discussion There is a natural interstitial site for cations in TiO2, namely, the center of the empty oxygen octahedra; see Figure 1. We found that this site is the most stable one for Ti impurities irrespective of their charge. When TiI is present, the surrounding atoms relax to accommodate the defect. A TiI is bonded octahedrally to six oxygen atoms and has two short bonds to Ti atoms. Upon relaxation these two Ti atoms move away from TiI by 0.33 Å from their bulk position. Among the six nearest neighbor O atoms, four move inward by 0.15 Å and the other two move outward by 0.21 Å from their bulk positions. For a VO defect we obtained the following atomic relaxations. The three nearest neighbor Ti atoms (at ∼2 Å from VO) move away from the vacancy by 0.33 Å and the four next nearest Ti atoms (at ∼3.5 Å) by 0.10 Å. The nearest O atom (II in Figure 2) moves toward the vacancy by 0.20 Å. These relaxations are in good agreement with those of Iddir et al.15 The diffusion mechanisms of TiI moving |c or ⊥c are shown in Figure 1. The calculated diffusion barriers are given in Table 1 for three different charge states. Using essentially the same computational method as ours (although with somewhat different numerical parameter settings, e.g., 300 eV cutoff energy, treating Ti 3p as valence electrons) Iddir et al.15 found the following
See Figure 1 for the diffusion paths.
barriers (in eV): 0.7 for Ti0 |c, 0.37 for Ti4+ |c, and 0.225 for Ti4+ ⊥c. They did not report values for the three other cases in Table 1. The barriers for Ti4+ agree well with our values. There is some disagreement with the Ti0 |c value, but the trend as a function of charge state for this diffusion path is the same in both calculations. Our calculated barriers |c for Ti4+, Ti2+, and Ti0 scale very nearly linearly with the ionic radius (see Figure S1 in the Supporting Information). These results, together with the fact that we have obtained a value for Ti4+ in agreement with Iddir et al.,15 make us very confident that our Ti0 value is also correct. From Table 1 we see that for both diffusion directions, the barrier height decreases with increasing ionic charge. This is expected since higher charged cations have a smaller ionic radius and can thus diffuse more easily. The decrease of barrier height is more pronounced for diffusion |c than for diffusion ⊥c. This leads to a reversal of the preferred diffusion path when increasing the charge from Ti2+ to Ti4+. Indeed, the lowest energy barrier is |c for Ti0 and Ti2+, but it is ⊥c for Ti4+. So, surprisingly, we find that the preferred diffusion direction is charge dependent and switches from |c for neutral or weakly charged Ti defects to ⊥c for the maximally charged cation Ti4+. Note that qualitatively the same charge dependence of the diffusion mechanism was found experimentally for transition metal impurities in TiO2 by Sasaki et al.10 While divalent cations diffuse through the open channels |c, trivalent cations mainly diffuse via the interstitialcy process ⊥c. Our results for the Ti4+ defect, which we found to diffuse preferentially ⊥c by the interstitialcy mechanism, agree with the experiments by Hoshino et al.9 These authors, who based their analysis on Ti4+ defects, found that the self-diffusion ⊥c is by a factor 1.2 to 1.6 larger than the diffusion |c in the temperature range of 1000-1500 °C. Moreover for Ti3+ defects, they have suggested that the ratio of diffusion coefficients D⊥c/D|c is reduced with respect to Ti4+, so the diffusion process |c becomes (relatively) more favorable for less charged defects. This is the same trend for the chargedependent Ti diffusion that we have found here. Ti interstitials in TiO2 are usually considered to be in the +4 or +3 charge state.9,10 This knowledge is based on the analysis of experimental data with models containing ions with integer charge states and free carriers. Let us note that these experimentally derived formal defect charges are not directly comparable with the Ti charge states of our first principles electronic structure calculations. In the calculations, only the total number of electrons in the supercell is fixed and has an integer value. The actual charge on the interstitial site has a different value, which is determined by the self-consistent solution of the electronic structure problem. In order to avoid confusion between nominal defect charges considered in the calculation and the formal ionic charges usually considered in the experimental literature, we shall introduce a small typographic distinction. We continue to write Tin+ for a Ti interstitial with nominal (supercell) charge +n, but we shall henceforth write Ti+n for a Ti cation with formal charge +n. We have calculated the actual charge of the Ti interstitial for the three different nominal defect types using the Bader analysis.
Diffusion in Rutile TiO2 The results are given in the last line of Table 1. It can be seen that the nominally neutral Ti0 interstitial is in reality a cation. The electrons that are missing on the defect are distributed over the supercell, mainly on other Ti sites, which are slightly reduced. These extra electrons on the other Ti atoms have a localized or a delocalized character as the case may be.20 The delocalized electrons can be identified with free n-type carriers of the diffusion models. When going from the nominal Ti0 to the Ti4+ defect calculation, the Bader charge on the Ti interstitial varies only by a fraction of an electron as seen from Table 1. The values are in the same range as those obtained for (lattice) Ti ions in bulk TiO2 (+2.53e) and Ti2O3 (+2.21e), where the formal cations are Ti+4 and Ti+3, respectively. From these bulk references, we may identify a Ti Bader charge of ∼2.5e with a formal Ti+4 cation and a Bader charge of ∼2.2e with a formal Ti+3 cation. It is seen that the Ti4+ calculation corresponds well to the formal Ti+4 cation, as expected, but the best model for a formal Ti+3 interstitial would be the Ti0 calculation. Then the question arises what nominal charge state should be chosen in the calculation. The best choice will in general depend on various thermodynamical parameters, in particular on the electronic chemical potential (Fermi energy), which in turn depends on oxygen pressure, temperature, and the concentrations of all defect types (impurities, Ti and O interstitials and vacancies). Let us consider the simplest case of bulk Ti1+xO2 with no other defects than Ti interstitials. The correct choice for this case is the Ti0 calculation because only insertion of Ti0 keeps the system charge neutral. Ti0 should still be the appropriate choice when all other defects are of n-type, which includes in particular oxygen vacancies. When p-type defects (Ti vacancies, O interstitials, and extrinsic p-dopants such as substitutional AlTi impurities) are abundant, however, the Ti4+ defect might be the most appropriate choice. It is seen from Table 1 that for both diffusion mechanisms, the Ti2+ barriers lie between those of Ti0 and Ti4+. By interpolating between the points, we can estimate that switching of the preferential diffusion process from |c to ⊥c occurs for a nominal defect type of Ti2.6+, which, as we have identified above, corresponds a formal ionic charge between Ti+3 and Ti+4. Summarizing these arguments, we may conclude that formal Ti+4 interstitials are well described by our Ti4+ calculation and diffuse preferentially ⊥c through the interstitialcy process. Formal Ti+3 interstitials, however, are best modeled with the Ti0 calculation and diffuse |c through the open channels. Next we consider the oxygen vacancies (VO) diffusion. In order to better understand the results, we first have a look at the orientation dependence of the O-Ti bonding in rutile. The Ti sublattice can be seen as an arrangement of dense [001] rows (Ti-Ti distance ∼3 Å). Each O atom is strongly bound (by two Ti-O bonds) to one Ti row and more weakly bound (by only one Ti-O bond) to another Ti row (see Figure 2). There are three inequivalent paths for VO diffusion, labeled I-III in Figure 2. The calculated barriers for a nominally neutral oxygen vacancy diffusing along these paths are 0.65 (I), 0.96 (II), and 1.71 eV (III). In path I along [001], the diffusing O atom stays all the way between both Ti rows it is attached to. It thus remains well bonded, and the barrier is consequently rather low. In path II the diffusing O atom crosses a Ti row. Only one of the three Ti-O bonds is broken in the process, and so a low barrier may be expected. However, by crossing the Ti row, the O atom has to pass between two Ti sites that are only 3 Å apart. So there is steric hindrance, which implies a shortening of the Ti-O bonds and/or substantial lattice
J. Phys. Chem. C, Vol. 114, No. 46, 2010 19651 distortions and thus repulsive interactions along the path. This may explain why path II has a higher barrier than path I. In path III finally, the diffusing O atom detaches from the Ti row to which it is strongly bound and two Ti-O bonds are completely broken. So we can expect a high barrier, and we indeed found that the barrier of path III is much higher than the other two. We note that the present results suggest a highly anisotropic VO diffusion. Macroscopic diffusion perpendicular to c necessarily involves path III whose barrier is 2.5 times higher than that of path I along c. Although path II is an atomic motion ⊥c, it cannot lead to (macroscopic) diffusion ⊥c, not even in combination with path I. This is because through path II the O vacancy can only jump from one side to the other of the same Ti row but can never detach from this row. Here we have considered only the nominally neutral oxygen vacancy VO0. Following the same line of arguments as for the Ti interstitial, we expect the neutral charge state to be the correct choice for the oxygen vacancy if only n-type defects (in particular VO and TiI) are present. Under strongly oxidizing conditions, however, the most appropriate nominal charge state will be single (VO+) or double charged positive (VO2+). Iddir et al.15 calculated the diffusion barriers of a VO2+ defect. They found the values 1.77, 0.69, and 1.1 eV for barriers I, II, and III, (called A, B, and C in ref 15), respectively. This is the same energy range as our values for VO0, but the relative stability of the paths is different. Combining the results of Iddir et al.15 with ours, we may conclude that also the VO diffusion mechanism is highly charge dependent. While a neutral vacancy diffuses essentially only parallel to the c axis using path I, a double charged vacancy diffuses more easily perpendicular to the c axis by a combination of processes II and III. We also note that the lowest possible diffusion barrier of VO (0.65 eV) is larger than the highest possible barrier of TiI (0.54 eV, see Table 1). This means that at comparable concentrations of Ti interstitials and O vacancies, the ionic transport is dominated by Ti diffusion, as suggested by Henderson6 from surface experiments. Conclusion We have carried out a first principles study on the diffusion mechanism of two main point defects in rutile TiO2, that is Ti interstitials and O vacancies. We find that Ti interstitials can diffuse either along the c axis through the open channels or via the interstitialcy mechanism perpendicular to c. The barriers for both processes are in the range of 0.2-0.5 eV and depend quite strongly on the charge of the interstitial defect. Weakly charged Ti ions diffuse preferentially through the open channels along c, but fully charged Ti4+ ions diffuse more easily perpendicular to c. This result nicely agrees with the experimentally observed charge dependent diffusion of 3d transition metal impurities in TiO2. We have clarified the relation between ionic charges used in first principles calculations with formal ionic charges considered in experiments. For the Ti interstitial we found good correspondence between the Ti0 (Ti4+) calculation and the formal Ti+3 (Ti+4) ion. For oxygen vacancies, the barriers of all three inequivalent diffusion paths have been calculated for nominally neutral vacancies. The diffusion path c is found to have the lowest barrier resulting in a highly anisotropic diffusion. Acknowledgment. We acknowledge access to HPC resources from DSI-CCUB (Universite´ de Bourgogne) where the calculations were performed.
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Supporting Information Available: The barrier heights |c of Ti4+, Ti2+, and Ti0 plotted as a function of their ionic radii (Figure S1). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Casillas, N.; Synder, S. R.; Smyrl, W. H.; White, H. S. J. Phys. Chem. 1991, 95, 7002. (2) Yu, N.; Halley, J. W. Phys. ReV. B 1995, 51, 4768. (3) Tauster, S. J.; Fung, S. C.; Garten, R. L. J. Am. Chem. Soc. 1978, 100, 170. (4) He, J.; Behera, R. K.; Finnis, M. W.; Li, X.; Dickey, E. C.; Phillpot, S. R.; Sinnot, S. B. Acta Mater. 2007, 55, 4325, and references therein. (5) Nowotny, M. K.; Bak, T.; Nowotny, J. J. Phys. Chem. B 2008, 110, 16292. (6) Henderson, M. A. Surf. Sci. 1999, 419, 174. (7) Wendt, S. Science 2008, 320, 1755. (8) Huntington, H. B.; Sullivan, G. A. Phys. ReV. Lett. 1965, 14, 177.
Asaduzzaman and Kru¨ger (9) Hoshino, K.; Peterson, N. L.; Willey, C. L. J. Phys. Chem. Solids 1985, 46, 1397. (10) Sasaki, J.; Peterson, N. L.; Hoshino, K. J. Phys. Chem. Solids 1985, 46, 1267. (11) Sambi, M.; Della Negra, M.; Granozzi, G. Thin Solid Films 2001, 400, 26. (12) Agnoli, S.; Castellarin-Cudia, C.; Sambi, M.; Surnev, S.; Ramsey, M. G.; Netzer, F. P. Surf. Sci. 2003, 546, 117. (13) Asaduzzaman, A. M.; Kru¨ger, P. Phy. ReV. B 2007, 76, 115412. (14) Asaduzzaman, A. M.; Kru¨ger, P. J. Phys. Chem. C 2008, 112, 4622. ¨ gu¨t, S.; Zapol, P.; Browning, N. D. Phys. ReV. B 2007, (15) Iddir, H.; O 75, 073203. (16) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (17) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (18) Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, 558. (19) Kresse, G.; Hafner, J. J. Phys.: Condens. Matter. 1994, 6, 8245. (20) The question of defect charge localization is very subtle and depends, from a theoretical point of view, strongly on the exchangecorrelation potential, see: Di Valentin, C.; Pacchioni, G.; Selloni, A. Phys. ReV. Lett. 2006, 97, 166803.
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