J. Semicond. > 2016, Volume 37 > Issue 12 > 122003
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SEMICONDUCTOR PHYSICS

The slabs for the rutile TiO2(110) surface

Xuechao Li, Jianhao Shi and Rundong Wan

+ Author Affiliations

 Corresponding author: Wan Rundong, Email:rdwan@kmust.edu.cn

DOI: 10.1088/1674-4926/37/12/122003

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Abstract: Traditionally, we use a slab to mimic a surface and we constrain the slab to have the bulk-terminated 2D lattice constants. Here we propose a different model in which we impose no constraints, allowing all coordinates including the 2D slab itself to relax. We perform DFT calculations on both models. We find that the unconstrained slabs yield better agreement with experimental results and they have lower total energies. The optimized 2D lattice constants of the unconstrained slabs eventually converge to the attached bulk value. The total energy difference reveals that, with odd number trilayers, the unconstrained slab is much closer to the corresponding constrained slab. The surface energies both converge to the individual values with the number of atomic layers.

Key words: rutile TiO2, slab, DFT, LDA/6-31G

TiO2 surfaces are the most studied surfaces, because of their widespread applications [1, 2]. A slab is one of the most popular theoretical models to study such surfaces. We build a slab by repeating the bulk unit cell, and then we cut the repeated cell along a specified direction to mimic a surface with certain thickness. On such a slab, we can relax the structure to find the minimum total energy state, i.e. the ground state. Such an optimized slab might describe the surface relaxation, which reflects the atomic displacements from the bulk-terminated structure. A surface is the interface between the bulk and the vacuum. The common belief is that such surfaces would maintain the 2D bulk-terminated lattice constants, because the surfaces are attached to the bulk. Therefore, in theoretical studies, people constrain a 2D slab with the bulk-terminated lattice constants, only letting the internal coordinates relax. However, few people focus on the investigation of unconstrained slabs. Here we propose a different model in which we impose no constraints, allowing all coordinates including the 2D slab lattice constants to relax. Provided the slab thickness is sufficient, eventually the lattice constants of the unconstrained slabs should converge to the bulk values. The constraint would increase the total energy of the slab, meaning the constrained slab would have higher total energy than that of the unconstrained slab. Consequently, the relaxed slabs of two models would have different structures, which might be negligible, depending upon the thickness of a slab. Some previous works have reported that the computed atomic relaxations or displacements are much more dependent on the thickness of the slab considered [3-5]. In this work, if a slab were thick enough, the relaxed slab from these two different models would be similar. When we have a medium thick slab, there should be noticeable difference. Furthermore, whether a true surface layer strictly maintains the bulk terminated 2D lattice constants remains an open question, because there are no previous theoretical or experimental efforts in this respect.

To test the hypothesis, we choose the rutile (110) surface, because there are verified experimental and theoretical results [6-15], which we could use for comparisons. Most other surfaces do not have such advantages. To the best of our knowledge, all previous DFT calculations on this surface constrain the slab to have the bulk-terminated lattice constants [16-26]. We calculate the structures with both models: the constrained 2D lattice constants model and the fully relaxed one. With regard to the exchange-correlation functional, we choose local density approximation (LDA) [27, 28]. Though LDA is a low level functional, it performs surprisingly well for most extended systems, comparable to a high-level functional calculation such as a hybrid functional one. Another disadvantage of the LDA is that it underestimates the bandgap. We could solve this issue by performing band structure calculation with a hybrid functional calculation on the LDA optimized structure. Nevertheless, we focus on the geometrical structure here, avoiding the band structure calculation. The bonus of the LDA is its implementation in any DFT packages. To describe the electronic wavefunctions, we choose 6-31G basis sets [29, 30]. We performed LDA/6-31G approach on both the unconstrained and constrained slabs from three to eleven trilayers thick, as mentioned before.

For thick slabs such as 10-trilayer or 11-trilayer, the computed displacements from two models are similar. Nevertheless, for the medium-thickness slabs, the unconstrained slabs yield systematically better agreement with the experiments than the corresponding constrained ones, with regard to the displacements. This implies that un-constrained slabs converge faster with regard to the displacements. Consequently, we need the thinner un-constrained slabs to mimic a surface. This might have potential importance in the surface adsorption or doping study [31-37], which generally needs a few times larger unit cell, leading to much larger computation resources. The un-constrained slabs also have lower total energies than the corresponding constrained ones. From this perspective, we think the un-constrained slab is a better model for a surface, because the un-constrained slabs have both lower energies and better displacements. Furthermore, the lattice constants of the unconstrained slabs will increasingly converge to the bulk values with the increase of the slab thickness. The problem is that we are not able to perform the study on very thick slabs, due to the high time consumption and the demand of resources of the Gaussian type of orbital. By comparing the total energy difference between the constrained slab and the corresponding unconstrained slab, we find that the difference is an order higher for the slabs of even number trilayers. For the thick slabs of odd number trilayers, the constrained slab is acceptable. Although the geometric structures of rutile TiO2 surfaces have been investigated in depth, this is the first work considering both the constrained and unconstrained slabs, using the LDA/6-31G approach. We think these findings would provide different perspectives on the TiO2 surface studies.

We first optimize the rutile bulk geometric structure in Gaussian 09 [38], using the crystallography information file exported from the Materials Studio internal database [39], putting the optimized structure back in Materials Studio Visualizer [39]. We build the slabs for three to eleven trilayers by cutting the repeated bulk perpendicular to the [110] direction. We show such a slab in Figure 1, where we can generalize the labeling to other atoms.

Figure  1.  (Color online) Atomic label of the 3-trilayer slab. The larger and smaller balls are titanium and oxygen atoms, respectively. Symmetric atoms are marked with an asterisk (*).
DownLoad: Full-Size Img PowerPoint

On these slabs, we optimize the structures with two different methods: in the first method, we constrain the slabs so that they have the bulk-terminated 2D lattice constants, whereas we relax the other internal coordinates; in the second method, we relax all coordinates, including the lattice constants of the slabs themselves, if we consider a slab as a unit cell. We use a 2D unit cell instead of 3D supercell to describe a slab, while most other researchers use a 3D supercell. This is one advantage of the Gaussian type of orbital to mimic a surface, because a 3D supercell will consume much bigger resources. If we use the unconstrained slab model, the slab could change its 2D shape, meaning we might obtain a monoclinic system, whereas the constrained slab maintains its tetragonal shape. We perform all these geometrical optimizations in Gaussian 09.

We use the LDA exchange-correlation functional and the 6-31G basis sets in the calculations. For the basis set of the titanium atoms, we remove the functions with exponential less than 0.12 to speed up calculation without losing the accuracy [40]. For consistency, we use an ultrafine grid to obtain the total energies in all calculations. For the bulk optimization, we use 3D periodic boundary conditions, with the default k-point mesh 10 × 10 × 16 and corresponding 804 k-points, because the Gaussian 09 has quite different performance profiles from other packages. Using more k-points helps the convergence of the periodic wavefunction at a much less expensive cost. For the 2D slabs, we use 2D periodic boundary conditions, with the default k-point mesh 30 × 14 and corresponding 212 k-points. For all geometry optimizations, we impose the following four convergence criteria simultaneously: the maximum component of force reaches 0.00045 au; the root-mean-square of the forces reaches 0.0003 au; the next step displacement reaches 0.0018 au; and the root-mean-square of the next step displacement reaches 0.0012 au.

The optimized rutile lattice parameters are 4.510 (4.593), 2.918 (2.958) and 0.3032 (0.3048) Å for a, c, and the internal coordinate u, respectively, where the values in the brackets are the corresponding experimental values [21]. Although there are differences, these differences are small, causing negligible effect. We also notice that the 6-31G basis sets are suitable for the molecular system studies in quantum chemistry, not for solids. Therefore, the agreement might be more satisfactory for a surface study, because a 2D slab is some kind of middle point between the 0D molecules and 3D solids. Consequently, the LDA/6-31G approach should yield better results for the slabs.

We show some displacements for both the unconstrained and constrained slabs of different thickness in Table 1. A displacement is the difference between a relaxed atomic position and the corresponding bulk-terminated position, and we can decompose the displacement along three directions shown in Figure 1. The positive values indicate outward displacements whereas the negative values indicate inward displacements. For the large thickness slabs, such as 11-trilayer and 10-trilayer slabs, the displacements are in good agreement with the experimental ones, especially true for the unconstrained slabs. In general, the displacements of an unconstrained slab are better than the displacements of the corresponding constrained slab, if we compare them with the experimental ones, regardless of the thickness of a slab. Comparing the displacements of the unconstrained 7-trilayer slab and the unconstrained 11-trilayer slab, we find that the outward displacements increase and the inward displacements of the Ti2, Ti3, and Ti6 also increase. We attribute this to the fact that Ti2 is the twofold atom on the outmost layer, which would cause naturally inward displacement. Because the Ti3 and Ti6 atoms are right below the Ti2 atoms, the inward displacements would propagate. With the increase of thickness, the attraction from the middle atoms becomes bigger, resulting in the bigger displacements of the Ti3 and Ti6 atoms. Comparing the displacements of the unconstrained 6-trilayer slab with the unconstrained 10-trilayer slab, the displacements show a mixed pattern. With the increasing thickness, the outward displacements of the O1, Ti1, O3, O5, Ti4, and Ti5 atoms decrease and the outward displacements of the O2, O4, and O9 atoms increase, whereas the inward displacements of Ti2, Ti3, and Ti6 atoms decrease. This shows that the slab of the even number trilayers and that of odd ones have opposite trends. The displacements along the surface direction show little change. For the constrained slabs, these observations are also true.

Table  1.  Displacement comparisons between the different methods in Å, where UC denotes the unconstrained slabs and C denotes the constrained slabs, respectively. The displacements are along the [110] direction, except the lateral ones with the perpendicular signs in the [110] direction. LEED-IV and SXRD mean the atomic displacements obtained from the analysis of low-energy electron diffraction data and surface X-ray diffraction data in Reference [15].
DownLoad: CSV  | Show Table

To further illustrate this point, we show some of the displacement patterns for both the unconstrained slabs and the constrained slabs, ranging from 5-trilayer to 11-trilayer thick in Figure 2. For both the constrained and unconstrained slabs, we observe the well-known oscillating pattern, which indicates that the slabs of even number trilayers and those of odd ones converge differently, but they converge approximately to the same values. This verifies that at large thickness, the agreement with the experimental results is true. This is also the reason why we exclude results of the 3-trilayer slab and the 4-trilayer slab, although most displacements of the 4-trilayer are in excellent agreement with the experimental ones. Nevertheless, the displacements of the Ti3, Ti4 and O6 atoms are -0.17, 0.25 and -0.06 Å, respectively. These values deviate greatly from the converged value.

Figure  2.  (Color online) Displacement patterns for both the unconstrained and constrained slabs.
DownLoad: Full-Size Img PowerPoint

We show the optimized 2D lattice constants of the optimized unconstrained slabs in Figure 3. They also display oscillating behavior. The deviations from the bulk-terminated 2D constants are small, because all of them are less than 1%. But the slab structure should eventually converge to the bulk constants. At this stage, because we have used a Gaussian type of orbital in the analysis, which is very time consuming and demanding of resources, we are not able to perform thick slab study. Eventually the lattice constants should converge to the bulk values. How thick it might be remains unsolvable to us. Using plane wave pseudo-potential approach might be helpful.

Figure  3.  (Color online) Optimized lattice constants of the unconstrained slabs.
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The surface energies change as a function of the thickness of both the unconstrained slabs and the constrained ones, as shown in Figure 4. We define the surface energy as the difference between the energy of the relaxed slab and the original bulk-terminated slab. They are both converging to individual values, and the constrained slab has lower converged value. Again, for the slabs of even number trilayers, the surface energies converge from above, and for the ones of odd number trilayers, the surface energies converge from below to the surface energy of infinite layers. This trend strongly depends on the number of trilayers in the slabs, that is, the presence or absence of the symmetry for the odd or even number trilayers. Furthermore, both the unconstrained and constrained slabs show a similar trend. This point is in excellent agreement with the oscillation behavior of the calculated surface energy obtained in previous publications [41-43].

Figure  4.  The surface energies of the unconstrained and constrained slabs.
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The total energy difference between the constrained slab and the corresponding unconstrained slab is more striking, as shown in Figure 5. The ground state of the 4-trilayer constrained slab is unstable relative to that of the unconstrained one, because the total energy difference is much larger. This also occurs in other constrained slabs of even number trilayers, becoming less striking for thick slabs. In this regard, the slabs of constrained odd number trilayers are more stable. Consequently, the displacements of the constrained slabs and the unconstrained ones are closer.

Figure  5.  Total energy difference between the constrained and unconstrained slabs.
DownLoad: Full-Size Img PowerPoint

The unconstrained slabs are slightly better in describing the displacements of the rutile (110) surface. This might be attributed to the fact that the unconstrained slab has lower total energy, thus the ground state is closer to the true surface. Although a constrained slab maintains the bulk-terminated 2D lattice constants, the corresponding unconstrained slab has very close lattice constants. Furthermore, we do not know whether the rutile (110) surface top layers maintain the same 2D lattice constants as those of the deep layers. There might be slight distortions, which is hard to determine experimentally. Our unconstrained slab calculations might suggest the existence of such distortions, but this hypothesis needs further theoretical or experimental verifications.

Nevertheless, the constrained slabs of odd number trilayers deviate less from the corresponding unconstrained slabs, which can be seen from the total energy difference. This is good news, because many DFT works use slabs of constrained odd number trilayers. Consequently, there are more symmetric operations, resulting in lesser computation resources. To describe the rutile (110) surface, we think that a slab must have minimum 7-trilayer to yield the most experimentally comparable displacements.

Using different exchange-correlation functionals and/or different basis sets might yield different results, which is beyond current study. As a general trend, using LDA should be more accurate than Hartree-Fock method and less accurate than using generalized gradient approximation, possibly even further less accurate than using hybrid functionals, which is very resources consumptive and difficult to implement in most codes. The 6-31G basis sets have shown their validity in extensive quantum chemistry calculations. Therefore, we believe that the LDA/6-31G approach should yield decent results. Nevertheless, although the results of using the more sophisticated approximations might be slightly different, the conclusions of the two kinds of slabs would not be compromised.

We have performed LDA/6-31G calculations on both the unconstrained and constrained slabs, ranging from three to eleven trilayers thick. The unconstrained slab shows lower total energy than the constrained one, which suggests its ground state is much closer to the true surface. Moreover, the unconstrained slab shows very close lattice constants to the corresponding constrained ones. Therefore, the unconstrained slabs are better in terms of the displacements. The total energy difference of the odd number trilayers is much smaller, thus suggesting the constrained slab of odd number trilayers is close to the corresponding unconstrained slab. The surface energies of the two different kinds of slabs both converge to the individual values. The constrained slab has lower converged value than the unconstrained one. Based on these results, to better describe the rutile (110) surface, we should consider at least a 7-trilayer thick slab in a surface study.

We thank D. J. Fox for the constructive discussion.



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Fig. 1.  (Color online) Atomic label of the 3-trilayer slab. The larger and smaller balls are titanium and oxygen atoms, respectively. Symmetric atoms are marked with an asterisk (*).

DownLoad: Full-Size Img PowerPoint
Fig. 2.  (Color online) Displacement patterns for both the unconstrained and constrained slabs.

DownLoad: Full-Size Img PowerPoint
Fig. 3.  (Color online) Optimized lattice constants of the unconstrained slabs.

DownLoad: Full-Size Img PowerPoint
Fig. 4.  The surface energies of the unconstrained and constrained slabs.

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Fig. 5.  Total energy difference between the constrained and unconstrained slabs.

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Table 1.   Displacement comparisons between the different methods in Å, where UC denotes the unconstrained slabs and C denotes the constrained slabs, respectively. The displacements are along the [110] direction, except the lateral ones with the perpendicular signs in the [110] direction. LEED-IV and SXRD mean the atomic displacements obtained from the analysis of low-energy electron diffraction data and surface X-ray diffraction data in Reference [15].
[1]
Diebold U. Structure and properties of TiO2 surfaces: a brief review. Appl Phys A, 2003, 76(5): 681 doi: 10.1007/s00339-002-2004-5
[2]
Pang C L, Lindsay R, Thornton G. Chemical reactions on rutile TiO2 (110). Chem Soc Rev, 2008, 37(10): 2328 doi: 10.1039/b719085a
[3]
Harrison N M, Wang X G, Muscat J, et al. The influence of soft vibrational modes on our understanding of oxide surface structure. Faraday Discussions, 1999, 114: 305 doi: 10.1039/a906386b
[4]
Zhang Y, Lin W, Li Y, et al. A theoretical study on the electronic structures of TiO2: effect of Hartree-Fock exchange. J Phys Chem B, 2005, 109(41): 19270 doi: 10.1021/jp0523625
[5]
Labat F, Baranek P, Adamo C. Structural and electronic properties of selected rutile and anatase TiO2 surfaces: an ab initio investigation. J Chem Theory Comput, 2008, 4(2): 341 doi: 10.1021/ct700221w
[6]
Charlton G, Howes P B, Nicklin C L, et al. Relaxation of TiO2 (110)-(1×1) using surface X-ray diffraction. Physl Rev Lett, 1997, 78(3): 495 doi: 10.1103/PhysRevLett.78.495
[7]
Hebenstreit E L D, Hebenstreit W, Diebold U. Structures of sulfur on TiO2 (110) determined by scanning tunneling microscopy, X-ray photoelectron spectroscopy and low-energy electron diffraction. Surf Sci, 2001, 470(3): 347 doi: 10.1016/S0039-6028(00)00849-9
[8]
Bennett R A, Pang C L, Perkins N, et al. Surface structures in the SMSI state; Pd on (1×2) reconstructed TiO2 (110). J Phys Chem B, 2002, 106(18): 4688 doi: 10.1021/jp0138328
[9]
Diebold U. Structure and properties of TiO2 surfaces: a brief review. Appl Phys A, 2003, 76(5): 681 doi: 10.1007/s00339-002-2004-5
[10]
Pillay D, Hwang G S. Structure of small Aun, Agn, and Cun clusters (n=2-4) on rutile TiO2 (110): a density functional theory study. Journal of Molecular Structure: THEOCHEM, 2006, 771(1): 129
[11]
Perron H, Domain C, Roques J, et al. Optimisation of accurate rutile TiO2 (110), (100), (101) and (001) surface models from periodic DFT calculations. Theoretical Chemistry Accounts, 2007, 117(4): 565 doi: 10.1007/s00214-006-0189-y
[12]
Busayaporn W. TiO2 (110) surface structure. University of Manchester, 2010
[13]
Lindsay R, Wander A, Ernst A, et al. Revisiting the surface structure of TiO2 (110): a quantitative low-energy electron diffraction study. Phys Rev Lett, 2005, 94(24): 246102 doi: 10.1103/PhysRevLett.94.246102
[14]
Cabailh G, Torrelles X, Lindsay R, et al. Geometric structure of TiO2 (110) (1×1): achieving experimental consensus. Phys Rev B, 2007, 75(24): 241403 doi: 10.1103/PhysRevB.75.241403
[15]
Busayaporn W, Torrelles X, Wander A, et al. Geometric structure of TiO2 (110) (1×1): confirming experimental conclusions. Phys Rev B, 2010, 81(15): 153404 doi: 10.1103/PhysRevB.81.153404
[16]
Ramamoorthy M, Vanderbilt D, King-Smith R D. First-principles calculations of the energetics of stoichiometric TiO2 surfaces. Phys Rev B, 1994, 49(23): 16721 doi: 10.1103/PhysRevB.49.16721
[17]
Bates S P, Kresse G, Gillan M J. A systematic study of the surface energetics and structure of TiO2 (110) by first-principles calculations. Surf Sci, 1997, 385(2): 386
[18]
Elliott S D, Bates S P. A first principles survey of stoichiometric (1×2) reconstructions on the rutile surface. Surf Sci, 2001, 495(3): 211 doi: 10.1016/S0039-6028(01)01328-0
[19]
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    Xuechao Li, Jianhao Shi, Rundong Wan. The slabs for the rutile TiO2(110) surface[J]. Journal of Semiconductors, 2016, 37(12): 122003. doi: 10.1088/1674-4926/37/12/122003
    X C Li, J H Shi, R D Wan. The slabs for the rutile TiO2(110) surface[J]. J. Semicond., 2016, 37(12): 122003. doi: 10.1088/1674-4926/37/12/122003.
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    Received: 19 January 2016 Revised: 19 July 2016 Online: Published: 01 December 2016

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      Article Navigation >  Journal of Semiconductors  > 2016  >  37(12): 122003
      Xuechao Li, Jianhao Shi, Rundong Wan. The slabs for the rutile TiO2(110) surface[J]. Journal of Semiconductors, 2016, 37(12): 122003. doi: 10.1088/1674-4926/37/12/122003 ****X C Li, J H Shi, R D Wan. The slabs for the rutile TiO2(110) surface[J]. J. Semicond., 2016, 37(12): 122003. doi: 10.1088/1674-4926/37/12/122003.
      Citation:
      Xuechao Li, Jianhao Shi, Rundong Wan. The slabs for the rutile TiO2(110) surface[J]. Journal of Semiconductors, 2016, 37(12): 122003. doi: 10.1088/1674-4926/37/12/122003 ****
      X C Li, J H Shi, R D Wan. The slabs for the rutile TiO2(110) surface[J]. J. Semicond., 2016, 37(12): 122003. doi: 10.1088/1674-4926/37/12/122003.
      shu

      The slabs for the rutile TiO2(110) surface

      DOI: 10.1088/1674-4926/37/12/122003

      • Department of Materials Physics and Chemistry, Kunming University of Science and Technology, Kunming 650093, China

      Funds:

      the National Natural Science Foundation of China 50974067

      Project supported in part by the Major Program of National Science Foundation of China 51090385

      Project supported in part by the Major Program of National Science Foundation of China (No. 51090385) and the National Natural Science Foundation of China (No. 50974067)

      More Information
      • Corresponding author: Wan Rundong, Email:rdwan@kmust.edu.cn
      • Received Date: 2016-01-19
      • Revised Date: 2016-07-19
      • Published Date: 2016-12-01

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