Slab models of rutile TiO2 (110) surface: DFT and DFT+U calculations

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The F2B slab model weakens the oscillations and calculations on F2B model quickly converge. However, the F2B model leads to artificial narrowness of band gap. Besides, when the number of layers increases, surface energy obtained from all three slab models approaches similar values. In particular, values of surface energy from DFT calculations converge to the experimental range for all three slab models.

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Nội dung Text: Slab models of rutile TiO2 (110) surface: DFT and DFT+U calculations

Cite this paper: Vietnam J. Chem., 2023, 61(5), 563-570 Research article DOI: 10.1002/vjch.202200153 Slab models of rutile TiO2 (110) surface: DFT and DFT+U calculations Tran Thi Thoa1, Trang Thuy Nguyen2, Hoang Van Hung1*, Nguyen Thi Minh Hue1* 1 Faculty of Chemistry and Center for Computational Science, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi 10000, Viet Nam 2 Key Laboratory for Multiscale Simulation of Complex Systems, University of Science, Vietnam National University – Hanoi, 19 Le Thanh Tong, Hoan Kiem, Hanoi 10000, Viet Nam Submitted August 18, 2022; Revised October 3, 2022; Accepted August 2, 2023 Abstract The dependence of rutile TiO2 (1 1 0) surface’s structural and electronic properties on the number of layers has been carefully investigated with DFT and DFT+Ud,p approaches using three slab models. We have found that oscillations of surface energy, electronic properties from DFT calculations on the three models are stronger than those from corresponding DFT+Ud,p calculations. The even-odd fluctuations were demonstrated to relate to over delocalization of electrons, especially electrons in Ti 3d orbitals. The Ud,p corrections in DFT+Ud,p enabled to restrict the oscillations. The F2B slab model weakens the oscillations and calculations on F2B model quickly converge. However, the F2B model leads to artificial narrowness of band gap. Besides, when the number of layers increases, surface energy obtained from all three slab models approaches similar values. In particular, values of surface energy from DFT calculations converge to the experimental range for all three slab models. Keywords. Rutile (1 1 0) surface, surface energy, atomic displacements, band gap, DFT, DFT+Ud,p. 1. INTRODUCTION calculations are the same, surface energy obtained A (1 1 0) surface is the thermodynamically most from different slab models converged to different stable crystal face of rutile TiO2 and gains a lot of values instead of the same or similar values. For attention of scientists.[1] Structural and electronic example, in the study of H. Perron et al., converged properties of the surface play crucial roles in variety surface energy received the value of 0.50 J/m2 for of applications such as photocatalysis, sensors, fully relaxing all atom (FR), whereas 0.60-0.70 J/m2 photovoltaics.[2-4] Although a wide number of was shown in the model which one or two internal calculations based on DFT with local density layers are fixed to bulk atomic positions (FIL).[16] The approximation (LDA) and generalized gradient discrepancy of 0.13 J/m2 in converged surface energy approximation (GGA) were published, most obtained was also seen between fixed bottom layer (F2B) and (1 1 0) surface energy has large deviation from FR models with 12 layers.[17] experimental observation. Experimental surface Furthermore, it is known that the DFT method energy of rutile (1 1 0) surface was recorded about underestimates band gap of rutile TiO2. This is 0.28-0.38 J/m2.[5] Difference in the experimental consequence of self-interaction errors (SIE). One of surface energy values of a solid derivates from not effective and economical methods to deal with the only mistakes and different performing conditions of limitation is using DFT+U method. Depend on each the contact angle measurements, kind of measuring specific purpose, U should be taken different values. liquids used in the contact angle but also objectively For example, for defective systems of TiO2 such as O different mathematical formulas in methods of vacancies or Ti interstitials, Ud = 3-6 eV is good for determining surface energy.[6,7] Meanwhile, describing defective states.[18,19] Meanwhile, for calculations with LDA reported large surface energy, composite of TiO2, Ud value could be choose to be 6 nearly 0.89 J/m2.[8,9] Results of surface energy from eV,[20] even 9 eV.[21] Our previous work demonstrated GGA are smaller but still far from experimental that the combination of Ud and Up corrections helps values, 0.74 J/m2 (PBE),[10] 0.73 J/m2 (GGA of simultaneously increase the band gap and decrease Perdew and Wang),[11] 0.57 J/m2 (PBE),[12,13] 0.47 discrepancy of lattice constants of rutile TiO2 bulk in J/m2 (PBE).[14,15] Besides, when all parameters in comparison with DFT approach.[22] We found that a 563 Wiley Online Library © 2023 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH
25728288, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/vjch.202200153 by Readcube (Labtiva Inc.), Wiley Online Library on [01/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Vietnam Journal of Chemistry Nguyen Thi Minh Hue et al. pair of Ud = 7 eV, Up = 10 eV was one of the best Optimization the rutile TiO2 bulk with the DFT pairs for calculations on the bulk. If the DFT+Ud,p is method obtained lattice constants, a = 4.644 Å, c = applied to rutile (1 1 0) surface, how does properties 2.966 Å, and u = 0.304 Å, which is in good agreement of the surface change in calculations for each of the with the experimental values with discrepancy from three slab models? 0.33-1.24%.[30] As the limitation of the DFT, the With the aim of shedding more light on surface calculated band gap of rutile bulk is 1.67 eV. The energy as well as other properties of rutile TiO2 result is consistent with the previous studies.[31-33] (1 1 0) surface, we have carried out calculations based Meanwhile, the corresponding DFT+ Ud,p calculations on DFT and DFT+Ud,p methods for the three different on the bulk result in band gap of 2.92 eV and lattice slab models. The first model corresponds to fully constants a = 4.671 Å, c = 3.068 Å, and u = 0.305 Å. relaxing all atoms (FR). The model in which atoms in From optimized bulk, rutile TiO2 (1 1 0) surface the two bottom layers fixed to their positions in was constructed in the form of a slab model with the relaxed bulk is the second model (F2B). The last is a bulk- terminated lattice constants. The slab contains model with one or two inner layers fixed to their layers (figure 1). Each layer is composed of three positions in the optimized bulk so that the layers on atomic planes: a plane of Ti2O2 in the middle and two both sides of slab are symmetrical through fixed atomic O planes above and below the Ti2O2 plane. layers (FIL). Our result showed that the surface energy for all the three models quickly goes to the similar values. In particular, the surface energy from DFT calculations reaches the experimental value for all three slab models with enough number of layers. Atomic displacements and electronic properties of rutile (1 1 0) surface have been also investigated in the work. We have found that even-odd oscillation of surface’s properties relates to artificial delocalization of electrons in DFT approach. The oscillation could be limited with U corrections. Figure 1: Rutile (1 1 0) surface (side view) 2. COMPUTATIONAL METHOD All the three slab models were examined in both the DFT and DFT+Ud,p methods. In the next section, All calculations were performed using the Vienna Ab convergence of surface energy with respect to the initio Simulation Package (VASP).[23] The core number of layers for each kind of slab models was electrons were treated due to the augmented-wave studied. The surface energy is defined as difference in (PAW) approach. Meanwhile, valence electrons of Ti the total energy between the slab (Eslab) and the bulk (3d24s2) and oxygen (2s22p4) were presented by plane phase (Ebulk) with an equal number of TiO2 units as waves with the cutoff energy of 450 eV.[24] The the slab, divided by the total exposed area. Perdew-Burke-Ernzerhof (PBE) functional within the Thus, for either the FR or FIL models, the generalized gradient approximation (GGA) was expression of surface energy is: applied for the exchange-correlation interaction [25-27]. For all calculations, the threshold of 10- 6 eV and 0.01 eV/Å were employed for energy and atomic force, E slab − NE bulk rel rel E =E = FR surf FIL surf respectively. In this paper, the DFT+ Ud,p was carried 2A out in formulation of Dudarev with J = 0 eV.[28] Pair rel rel where E slab , E bulk is the total energy of a relaxed slab of Ud = 7 eV and Up = 10 eV were utilized as previous and relaxed bulk unit cell. N is a ratio of the number work.[22] A The Brillouin zone was sampled using of atoms in the slab to the number of atoms in the 741 and 42421 G-centered Monkhorst-Pack bulk. A is the surface area of the slab. The factor of 2 grids for relaxation and density of state (DOS) indicates that each slab includes two surfaces. calculation, respectively.[29] The tetrahedron method For the F2B model, the surface energy was was used to calculate DOS, PDOS. A vacuum computed as the following: thickness of 12 Å is enough to eliminate interaction between periodic images. E no−rel − NE bulk E slab − E slabrel rel rel no− E surfB = slab F2 + 2A A 3. RESULTS AND DISCUSSION no− rel rel real Here E slab , E bulk , and E slab are energy of the unrelaxed slab, the relaxed bulk, and the relaxed slab 3.1. The surface energy © 2023 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.de 564
25728288, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/vjch.202200153 by Readcube (Labtiva Inc.), Wiley Online Library on [01/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Vietnam Journal of Chemistry Slab models of rutile TiO2 (110) surface… with the two bottom layers fixed, respectively. The results of surface energy were shown in table 1 and figure 2. Table 1: Surface energy of rutile (1 1 0) surface calculated for the three slab models based on DFT and DFT+Ud,p approaches Number DFT DFT+ Ud,p FR of layers Esurf (J/m2) F2 EsurfB (J/m2) FIL E surf (J/m2) FR Esurf (J/m2) F2 EsurfB (J/m2) FIL E surf (J/m2) 3 0.68 0.706 0.717 1.083 1.102 1.103 4 0.353 0.471 0.647 0.955 0.983 1.047 5 0.516 0.475 0.524 0.980 0.970 0.983 6 0.364 0.398 0.493 0.939 0.940 0.959 7 0.443 0.408 0.446 0.929 0.921 0.930 8 0.360 0.369 0.431 0.904 0.899 0.911 9 0.401 0.373 0.402 0.887 0.878 0.887 10 0.350 0.349 0.391 0.866 0.857 0.868 11 0.371 0.348 0.372 0.846 0.837 0.847 12 0.338 0.363 0.826 0.816 0.828 (a) (b) Figure 2: Surface energy versus the number of layers for the three slab models according to DFT approach (a) and DFT+Ud,p approach (b). The region formed by red lines is experimental result[5] In general, when the number of layers increases, be seen because of the two fixed bottom layers. The the change in surface energy of the three slab models F2B model does not possess any symmetry element decreases and surface energy converges to similar when the number of layers changes. On the contrary, values (figure 2). However, in DFT calculations, the the constraint on atoms in inner layers creates mirror three slab models follow in different directions to plane in FIL model, which avoids the even-odd reach convergence (figure 2a). To illustrate, the FR oscillation and causes surface energy of the FIL model indicates a strong oscillation in surface energy model equal or higher than that of other slab models. between the odd and even number of layers. Here Interestingly, surface energy of all the three slab slabs with odd number of layers always gives large models converges to the experimental range in DFT surface energy, whereas a smaller surface energy formalism. Here the convergence speed of the F2B belongs to a slab with even number of layers. The model is the largest, only requirement of 8 layers to result is attributed to presence of a mirror symmetry reach experimental result, followed by the FR model plane when the number of layers is odd. The with 10 layers. The surface energy of the FIL model appearance of the symmetry element makes converges to experimental range from 11 layers. In equivalent atoms on the two sides of the slab move particular, for the F2B model, when the number of symmetrically through the mirror plane in relaxation. layers is even, only need 4 layers the surface energy Meanwhile, there is no symmetrical plane in slabs could reach experimental values. with even number of layers. The oscillation of surface By comparison, for DFT+Ud,p calculations (figure energy fits to previous studies.[8,9,15,34] The oscillation 2b), the even-odd fluctuation of surface energy was of the F2B model is much weaker and seems not to only seen for FR slab model with small number of © 2023 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.de 565
25728288, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/vjch.202200153 by Readcube (Labtiva Inc.), Wiley Online Library on [01/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Vietnam Journal of Chemistry Nguyen Thi Minh Hue et al. layers (figure 2b). The oscillation completely sufficient number of layers in both the DFT and vanishes from FR slab model with more than 6 layers. DFT+Ud,p methods. Thus, comparison with DFT approach, DFT+Ud,p method limits the even-odd oscillation of surface 3.2. Atomic displacements energy. Besides, the difference in surface energy of three slab models with more than 6 layers seems The rutile TiO2 (1 1 0) surface consists of unsaturated insignificant. When the number of layers increases, atoms in the outermost plane - exposed atoms, five- the surface energy of all the slab models gradually fold coordinated titanium atoms (Ti5c) and two-fold decreases. For example, for slab with 4 layers, the FR, coordinated oxygen atoms (O2c or bridging oxygen). F2B and FIL models have surface energy of 0.955, Besides, the rutile (1 1 0) surface contains fully 0.983, and 1.047 J/m2, respectively. When the coordinated atoms as in the bulk, Ti6c and O3c number of layers is 12, all the three slab models gain (referred to as in-plane oxygen) atoms (figure 1). the surface energy of nearly 0.82 J/m2. The results After relaxation, the outermost atoms of (1 1 0) could be compared with 0.86 J/m2 (Ud = 4.2 eV, 4 surface move along z-direction outward or inward layers),[35] 0.95 J/m2 (Ud = 4.2 eV),[36] 0.83 J/m2 (Ud = from their bulk positions. Results of the atomic 4.2 eV, 5 layers).[37] Therefore, surface energy for all displacements for slab models with 5 layers were the three slab models reaches the similar values after summarized in table 2. Table 2: Atomic displacements (Å) away from the bulk of TiO2 (1 1 0) surface calculated for three slab models with five layers in DFT and DFT+ Ud,p approaches. A negative value indicated that the atom moves inwards the bulk along z-direction DFT DFT+U Atom FR F2B FIL FR F2B FIL SXRD[38] LEED[39] model model model model model model Ti6c 0.228 0.293 0.219 0.166 0.205 0.166 0.250.01 0.250.03 (0.013) (0.054) (0.015) (0.054) Ti5c -0.12 -0.097 -0.133 -0.076 -0.06 -0.076 -0.110.01 -0.190.03 (0.003) (0.013) (0.024) (0.04) (0.024) O3c 0.211 0.245 0.2 0.171 0.193 0.172 0.170.03 0.270.08 (0.019) (0.018) O2c 0.046 0.109 0.038 0.019 0.057 0.019 0.100.04 0.100.05 (0.014) (0.031) (0.012) (0.031) (0.031) Values in parentheses are discrepancy between calculated values and the closest value of experiment. For the three slab models, the z-movement of each are in agreement with the experimental results. outermost atoms has the same direction for both the However, displacement of Ti5c in the F2B model is DFT and DFT+Ud,p calculations. The Ti6c atoms on not good as the other slab models. the (1 1 0) surface relax outwards, whereas Ti5c atoms move inwards in z-direction. Meanwhile, both 3.3. Band gap O2c and O3c atoms move outwards in relaxation. The result is in good agreement with experimental results The dependence of band gap versus the number of of SXRD and LEED.[38,39] However, there is deviation layers have been calculated for three slab models with between theoretical and experimental result in atomic DFT and DFT+Ud,p approaches. Results were displacements, about 0.003-0.054 Å. The discrepancy indicated in figure 3. may be acribed to approximations in DFT, DFT+Ud,p For DFT calculations, the strong even-odd approaches. Results from the DFT calculations have oscillation of band gap is seen in the FR slab model smaller deviation than those of the DFT+Ud,p. For the (figure 3a). The slabs with even number of layers DFT results (table 2), the displacement of Ti5c in the have a larger band gap than the bulk, whereas band FR slab model is closer to the experimental results gap of slabs with odd number of layers is smaller than than that in the other two models. But displacement that of bulk. The oscillation is ascribed to of O2c in FR slab model is not as good as that in the hybridization of Ti 3d and O 2p orbitals among FIL slab model. For DFT+U calculations, layers.[34] For even-layered slabs, the hybridization displacements of Ti6c, O3c, O2c in the F2B model results in bilayers. Meanwhile, the Ti(3d)-O(2p) © 2023 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.de 566
25728288, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/vjch.202200153 by Readcube (Labtiva Inc.), Wiley Online Library on [01/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Vietnam Journal of Chemistry Slab models of rutile TiO2 (110) surface… hybridization expand throughout odd-layered slabs approach bulk’s band gap and reach about 1.60 eV for which have a mirror symmetry. The amplitude of the slab model with 11 layers, the F2B model quickly oscillation reduces when the number of layers converges to very small value of band gap, nearly increases. On the contrary, the FIL and F2B slab 1.22 eV. The narrowness of band gap could be models do not show strong even-odd oscillation. interpreted from PDOS results (figures 4 and 5). Furthermore, while the FR and FIL models slowly (a) (b) Figure 3: Dependence of band gap of (1 1 0) surface on the number of layers calculated from DFT (a), DFT+Ud,p (b) approaches. The red line shows the band gap of rutile bulk calculated from DFT method (a) (b) (c) Figure 4: Density of states of slab with 5 layers for FR (a), F2B (b), and FIL models (c). The black, red, and violet lines represent the total DOS, PDOS of Ti, O atoms, respectively. The Fermi level was fixed to zero From figure 4, it can see that for all three slab The shortening of band gap in F2B model were models, Ti 3d orbitals mainly contribute to the also seen in results from DFT+Ud,p method, 1.57 eV conduction band. Meanwhile, majority of the upper for F2B model and about 2.60 for FR and FIL models. valence band is composed of O2p orbitals. To further By the same way, it could be seen that shortening is analyse contribution of each layer in slab to valence attributed to fixing atoms in two bottom layers. and conduction bands, we have plotted PDOS in more Interestingly, from results of DFT+Ud,p detail (figure 5). Inspection of the PDOS calculations (figure 3b), the even-odd oscillation of demonstrated that the conduction band comes from band gap was only indicated in slab models with Ti3d orbitals in all layers (figures 5a,c,e). The small number of layers, about less 6 layers for FR contributions of layers to the conduction band are model, less 7 layers for FIL model, and less than 5 nearly similar. In the same way, the O2p orbitals from layers for F2B model. The band gap values of the all layers of the slab are responsible for the valence three slab models rapidly converged in DFT+U band (figures 5b,d,f). There are not large difference in calculations compared with DFT calculations. The contributions of O2p orbitals in layers to the valence result implies that the even-odd oscillation relates to band. Analysis of PDOS for other slabs gives the over delocalization of electrons from DFT same result. Therefore, artificial narrowness of band calculation. On the other hand, the FR slab model gap of F2B model derives from freezing two bottom shows the largest even-odd oscillation in both DFT layers. and DFT+Ud,p formalisms. © 2023 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.de 567
25728288, 2023, 5, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/vjch.202200153 by Readcube (Labtiva Inc.), Wiley Online Library on [01/05/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Vietnam Journal of Chemistry Nguyen Thi Minh Hue et al. (a) (b) (c) (d) (e) (f) Figure 5: PDOS of atoms in each layer: Ti atoms in FR model (a), FIL (b), F2B (c); and O atoms in FR (d), FIL (e), F2B (f) (a) (b) Figure 6: Dependence of electronic properties on the number layer for FR model based on DFT (a) and DFT+Ud,p (b) methods Deeper analysis of results from FR slab models surface energy in DFT+Ud,p calculations approaches realizes that the even-odd oscillation mainly derives to similar values for the three slab models with more from fluctuation of bottom conduction band (figure than 6 layers. Meanwhile, surface energy from DFT 6). As above analysis, the majority of conduction calculations achieves experimental range for all the band is composed of Ti3d orbitals (figure 4). Thus, three slab models: The F2B slab model only needs 8 the U correction restricts even-odd oscillation of layers to reach the experiment while FR, FIL slab structural and electronic properties with slab models require 10, 11 layers, respectively. Besides, thickness by virtue of decrease in delocalization of atomic displacements of outermost atoms in DFT and electrons, especially 3d electrons. DFT+Ud,p calculations for the three slab models are compared with experiments. Analysis PDOS 4. CONCLUSION indicated that the oscillation is mainly related to over delocalization of Ti3d orbitals. The convergence- We have studied structural and electronic properties speed of surface’s properties with respect to the of rutile TiO2 (1 1 0) using three slab models: slab number of layers is the highest for calculations using with all atoms relaxed (FR), slab with fixing atoms in the F2B model. However, band gap gained from the one or two internal layers (FIL), and slab with fixed F2B model is about 1.22 eV, which is much smaller atoms in two bottom layers (F2B). Calculations were than that of the bulk and other slab models. The performed with DFT and DFT+Ud,p methods. artificial shortening in band gap is a consequence of Hubbard parameters, Ud and Up, were employed for fixing two bottom layers in F2B model. Therefore, Ti 3d and 2p orbitals, respectively. The obtained depending on the specific purpose of study, we should © 2023 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.de 568
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