Energetics of Rutile TiO2 Vicinal Surfaces with ... - ACS Publications

In this approach, EDM provides the energy for each atom so that we can determine the stability of different step configurations correctly. Even though...
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Energetics of Rutile TiO2 Vicinal Surfaces with ⟨001⟩ Steps from the Energy Density Method Bora Lee and Dallas R. Trinkle* Department of Materials Science and Engineering, University of Illinois at Urbana−Champaign, Urbana, Illinois 61801, United States ABSTRACT: Rutile TiO2 vicinal surfaces with ⟨001⟩ steps are investigated using the energy density method (EDM) based on density functional theory. In this approach, EDM provides the energy for each atom so that we can determine the stability of different step configurations correctly. Even though the energy variation due to the step is localized around the step edge, the step−step interaction is long ranged. The finite-size effect in the step−step interaction is identified using EDM. The oxygen vacancy at the step edge explains the atomic structure of the step observed by STM.

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TiO2 vicinal surfaces.6,7,19 The ⟨110⟩ step has the highest step energy even after it reconstructs,7 and the ⟨001⟩ step has only two stable structures that are ⟨001⟩O and ⟨001⟩Ti steps.19 Recently, Hardcastle et al.6 and Stausholm-Møller et al.7 computed the isolated step energies and the step−step interactions of the ⟨001⟩ step, and they showed that the ⟨001⟩O and ⟨001⟩Ti steps have similar step energies, and the step−step interactions for both steps are negligible. Complicating all of these step energy calculations is “odd−even” oscillations in the TiO2 (110) surface energy with the number of layers,23 which must be addressed carefully. The energy density method (EDM) based on DFT provides spatial energy distributions that can be integrated over appropriate volumes to give the energies of sets of atoms.24 The EDM approach removes ambiguities in some surface energy calculations, such as the GaAs (111) surface, which has different types of surfaces on the top and bottom of the slab geometry.25 In addition, the minimum slab thickness for finitesize convergence is obtained within a single calculation by comparing the EDM energies of the atomic layers to the energy of the bulk system. Similarly, we define defect formation energies from the energy density changes in the defect region. Defect energies obtained this way using EDM compare well with conventional total-energy DFT calculations.26,27 However, while EDM has been very useful for point and planar defects,28−31 there are not yet any studies that explore what can be learned for line defects with EDM. In this work, we apply the newly reformulated EDM24,26,27 to evaluate the isolated step energies and step−step interactions of rutile TiO2 vicinal surfaces with ⟨001⟩ steps. We calculate energies and geometries for two kinds of ⟨001⟩ steps: ⟨001⟩O and ⟨001⟩Ti steps. We choose these simpler step geometries without reconstruction to focus our study on the application of EDM to step energies. We find the TiO2(110)-p(1 × 1) surface is divided into two different configurations with different

INTRODUCTION The geometries and energetics of steps on surfaces play a crucial role in a wide range of surface phenomena. Step edges influence surface morphologies and have been observed to cause surface reconstructions1−3 and affect the shapes of islands on the surfaces.4−7 Surface steps can be active sites for catalysis8,9 and are preferential sites for the adsorption of adatoms and molecules10 and nucleation for metal nanoclusters.11,12 Steps on metal and semiconductor surfaces have been investigated extensively in experiment and theory,13,14 but structurally and electronically complex oxide surfaces have received less attention. Recent studies have computed the steps on MgO,15 CeO2,16 SrTiO3,17 and TiO2 surfaces3,6−9,18,19 due to their importance in catalytic applications. In particular, rutile TiO2 step structures and energies have been the subject of both experimental and computational studies to elucidate their role in the high catalytic activity of TiO2. Scanning tunneling microscopy (STM) studies observed three different types of monatomic height steps on the TiO2 (110) surfaces whose step edges are running along either [001], [111], or [110] directions.4,10,20,21 The ⟨001⟩ and ⟨111⟩ steps form during annealing on sputtered surfaces, and ⟨111⟩ steps occurring at low annealing temperatures and ⟨001⟩ steps become favorable at high annealing temperatures.4,20 The ⟨110⟩ steps are metastable and are formed by ion beam injection.10 The reconstructed ⟨111⟩ step edges are reported with strand structures and oxygen vacancies.3,8,9,22 Diebold et al. characterized the atomic structure of ⟨001⟩ steps using STM.21 They observed two different types of ⟨001⟩ steps: one which is terminated with bridging oxygen atoms ⟨001⟩O and the other is a reconstructed ⟨001⟩ step with exposed 5-fold titanium atoms ⟨001⟩Ti. They reported specific (1 × 4) reconstruction on the ⟨001⟩Ti step; however, other experimental studies show that the ⟨001⟩ step edges are rough and undercoordinated.3,22 Density functional theory (DFT) calculates the energies of specific atomic configurations and determines their relative stability. Previous DFT studies found that the ⟨111⟩ step with reconstructed step edge is the most stable step configuration on © XXXX American Chemical Society

Received: April 15, 2015 Revised: July 20, 2015

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DOI: 10.1021/acs.jpcc.5b03623 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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as shown in Figure 1. Considering the bridge or flat configurations on TiO2(110)-p(1 × 1) separately is not possible with conventional total-energy DFT calculations, but it is essential for calculating the step energy. If the rutile TiO2 vicinal surface slab geometry has the same step type on the top and bottom surfaces, then the varying thickness complicates the step energy calculation due to energy oscillations. Vicinal surface slabs with uniform thickness have different step types on the top and bottom surfaces, while vicinal surface slabs with varying thickness can have the same step type on both surfaces. Conventional total-energy DFT calculations of TiO2 vicinal surfaces require identical step types on each surface, however the accompanied varying thickness causes a problem with defining a proper reference energy for step energy calculations. The step energy is the difference between the total energy of the vicinal surface supercell and total energy of the TiO2(110) flat surface supercell. Rutile TiO2(110) surface calculations show oscillations in the surface energy as the slab thickness changes from odd to even numbers of layers due to electrostatic interactions, and these so-called odd−even oscillations persist for at least 10 layers.23 The vicinal surface energy also has odd−even oscillations, and these oscillations can be eliminated by calculating the energy difference between the vicinal and the (110) surface. However, these oscillations are not removed completely because the vicinal surface geometry with varying thickness is a mixture of even and odd layers, whereas the flat surface reference structure has only even or odd layers. To overcome the intrinsic limitations of conventional totalenergy DFT calculations for rutile TiO2 vicinal surfaces, we instead use EDM which gives the energy of each atom in the supercell. The energies of the bridge and flat configurations are provided by EDM separately. We choose uniform thickness vicinal surface slab geometries with different step types on each surface. The atoms in the bottom four layers are fixed at bulk TiO2 positions, while the rest of them are relaxed. Even though the calculated surface energy and step energy depends on the number of fixed layers,32 we focus on the relaxed atoms in the supercell and eliminate unwanted mixing of the odd−even oscillations with EDM. We perform DFT calculations33 on the TiO2 surfaces with the Vienna ab initio simulation package (VASP).34 We use the projector augmented wave (PAW) method, 35 and the exchange-correlation energy is treated in the Perdew−Burke− Ernzerhof36 version of the generalized gradient approximation functional (PBE-GGA). Electronic configurations of [Ne]3s23p64s23d2 and [He]2s22p4 are used for Ti and O atoms, respectively; this requires a plane-wave energy cutoff of 900 eV. We optimize all the geometries until the Hellman−Feynman forces are less than 0.02 eV/Å. Brillouin zone integration is performed by summing over a 4 × 4 × 6 Monkhorst−Pack37 mesh in bulk TiO2. The calculated lattice constants for the rutile phase of TiO2 are a = 4.649 Å, c = 2.970 Å, and u = 0.305. We calculate surface properties for the TiO2(110)-p(1 × 1) flat surface and five vicinal surfaces with Miller indices (n + 1, n, 0), where n = 1, ..., 5. The k-point densities for each slab are 6 × 4 × 1, 6 × 2 × 1, and 6 × 1 × 1 Monkhorst−Pack37 meshes for (110), (210) and (320) and (430), (540), (650), respectively. EDM integrates over Bader and charge-neutral volumes to determine atomic energies from a single DFT calculation.38 It requires a denser grid for the energy density than determined by the energy cutoff. The grid density for EDM calculations of the flat (110) surface is 40 × 90 × 600; the number of grid

contributions to the surface energy, and this energy partitioning is an important term in the step energy. The energy change due to the step is localized near the step edge for both ⟨001⟩O and ⟨001⟩Ti steps, while ⟨001⟩Ti step energy is higher than ⟨001⟩O step energy. A repulsive step−step interaction is observed, and the interaction changes with the distance between the steps. We calculate layer-by-layer stress distributions, and find a change in the stresses which reflects the changes in the step−step interaction. The formation energies of the oxygen vacancies on the vicinal surface are calculated and we explain the STM images.3,21

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METHODOLOGY Figure 1 shows two different step geometries of a ⟨001⟩ step on a TiO2 vicinal surface. Periodic boundary conditions produce

Figure 1. Schematic and atomic structures of TiO2 vicinal surfaces with a monatomic-height ⟨001⟩ step. The vicinal surface geometry is defined by the surface area Av, the miscut angle θ, and the length of the step t. The ⟨001⟩ step is defined by the terrace of the (110) surface and the step edge along the [001] direction in the oblique cell. The Miller index of the vicinal surface is (n + 1, n, 0), and the terrace width d between the steps is (n + 1/2) × (unit length along the [110] direction of the (110) surface). The side views of the relaxed structures of the two different types of (210) surfaces are shown, and steps on the top surface are considered. We show two ways to construct the step geometry: ⟨001⟩O and ⟨001⟩Ti. The TiO2(110)-p(1 × 1) surface contains two configurations: the bridge configuration consists of eight TiO2 formula units having a red bridging oxygen atom (Obr) termination and the f lat configuration consists of eight TiO2 formula units having a gray 5-fold coordinated titanium atom (Ti5c) termination. The terrace of the vicinal surface has a total of (2n + 1) bridge and flat configurations. The ⟨001⟩O step has (n + 1) bridge and n flat configurations, while the ⟨001⟩Ti step has n bridge and (n + 1) flat configurations. The bridge configuration is exposed at the ⟨001⟩O step edge, while the flat configuration is located at the ⟨001⟩Ti step edge.

steps with a finite width of flat terrace between them. The repeating steps generate step−step interactions that must be quantified to compute an isolated step energy. The vicinal surface is more computationally efficient than a stepped flat surface that has steps that face each other, because it has a wider terrace width for the same number of atoms. The steps on the vicinal surface result in the terrace no longer having the full periodicity of the TiO2(110)-p(1 × 1) surface. Every step introduces an extra half configuration of TiO2(110)-p(1 × 1) surface, either the bridge configuration or the flat configuration, B

DOI: 10.1021/acs.jpcc.5b03623 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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FLAT AND VICINAL SURFACES We compute energy differences for each TiO2 formula unit ΔETiO2 relative to either bulk or a flat surface reference using EDM. For the surface energy, ΔETiO2 is the energy of the surface structure referenced to the bulk. For the ⟨001⟩ step energy, ΔETiO2 is the energy of the TiO2 unit in the vicinal surface referenced to its corresponding TiO2 unit in the (110) surface. We define an energy ΔE for a region of interest as the summation of ΔETiO2 over TiO2 units, ΔE =

∑

Figure 2. Geometry and energy of the TiO2(110)-p(1 × 1) surface. The energy of TiO2 units with respect to the bulk value is given for the bridge configuration (orange), the flat configuration (light blue), and the sum of both (gray). The contribution to the surface energy is different between the bridge configuration and the flat configuration and determines how the step energies are defined.

ΔE TiO2

TiO2 units

(1)

For example, summing over TiO2 units in a given surface layer gives the energy of that layer (we call it ΔElayer). The surface energy is obtained by adding up ΔETiO2 layer by layer and referenced to bulk until we find the layer for which ΔElayer is almost zero. Similary, the step energy is calculated by summing up ΔETiO2 layer by layer and referenced to the flat surface, until we find the layer for which ΔElayer is negligible. Surface Reference for Step Energy. In general, one computes the isolated step energy from the difference of a vicinal surface energy and a reference surface energy, with careful size convergence to step−step interactions. The step energy βv of each vicinal surface is calculated from the difference between the energy of the vicinal surface and energy of the terrace, βv =

1 [A v γv − (A v cos θ )γ110] t

contributions of the bridge and flat configuration computed with EDM are necessary quantities to define the step energy. Reconciling the step energy from the energy density in eq 1 and the standard definition of eq 2 requires considering the bridge and flat configurations separately on the terrace of the vicinal surface. The energy of the terrace (Av cos θ)γ110 is expressed in terms of the number of bridge and flat configurations and their energies. As shown in Figure 1, the ⟨001⟩O and ⟨001⟩Ti steps have different numbers of bridge and flat configurations. Since the previous total-energy DFT calculations6,7 cannot extract the energy of bridge and flat configurations separately, those authors use half of γ110 to calculate the step energy. Therefore, the energy of the terrace for both ⟨001⟩O and ⟨001⟩Ti steps are the same as

(2)

(A v cos θ )γ110 = n(ΔE bridge + ΔEflat)

where γv is the vicinal surface energy, Av is the vicinal surface area, γ110 is the (110) surface energy, θ is the miscut angle, Av cos θ is the terrace area, and t is the length of the step. The step energy βv contains the step−step interaction when the terrace width d = Av cos θ/t is finite. Marchenko and Parshin39 derived step−step interactions from the strain fields generated by local atomic relaxations due to the step, and it is proportional to the inverse of the square of terrace width d, q βv = β0 + 2 (3) d

1 (ΔE bridge + ΔEflat) (4) 2 where ΔEbridge or ΔEflat is calculated from eq 1 by summing ΔETiO2 over TiO2 units in the top five layers of the bridge or the flat configurations, and n is the number of pairs of bridge and flat configurations in the vicinal surface. We emphasize that total-energy DFT calculations cannot determine ΔEbridge and ΔEflat separately, but only their sum. On the other hand, EDM calculates the individual contributions from the bridge and flat configurations and uses these as references for the step energies. This leads to a EDM-consistent definition of the terrace energy as +

where β0 is the isolated step energy and q is the interaction coefficient. Linear fitting of βv versus 1/d2 determines β0 and q. Figure 2 shows the difference in energy between the bridge and the flat configurations of TiO2(110)-p(1 × 1). The broken bonds of the Obr atom in the bridge configuration cause the energy difference to be higher than the flat configuration at the surface. The negligible energy difference of the sixth layer with respect to the bulk indicates that the sixth layer is bulk-like. Consequently, surface energies are computed by summing over TiO2 units from the first to the fifth layers. The energy differences of the bridge and flat configurations are ΔEbridge = 384.9 meV and ΔEflat = 115.4 meV. The (110) surface energy is γ110 = 25.6 meV/Å2, with a flat contribution of 5.9 meV/Å2 and a bridge contribution of 19.7 meV/Å2. The different energy

* (A v cos θ )γ110 ⎧ ⎪ n(ΔE bridge + ΔEflat ) + ΔE bridge for ⟨001⟩O =⎨ ⎪ ⎩ n(ΔE bridge + ΔEflat) + ΔEflat for ⟨001⟩Ti

(5)

The surface energy γ*110 computed from a total of (2n + 1) bridge and flat configurations of the vicinal surface is different from the surface energy γ110 computed from equal numbers of bridge and flat configurations. Table 1 summarizes the γ110 * surface energies and shows that γ*110 converges to the flat C

DOI: 10.1021/acs.jpcc.5b03623 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Table 1. Surface and Step Energies of the ⟨001⟩O and the ⟨001⟩Ti for Five Vicinal Surface Configurationsa ⟨001⟩O

⟨001⟩Ti

(210) (320) (430) (540) (650) (210) (320) (430) (540) (650)

d (Å)

γv (meV/Å2)

γ110 * (meV/Å2)

βv(γ110) (meV/Å)

βv(γ110 * ) (meV/Å)

9.86 16.44 23.01 29.59 36.16 9.86 16.44 23.01 29.59 36.16

33.5 30.7 29.5 28.5 28.0 34.0 30.9 29.5 28.5 27.9

29.5 28.3 27.4 27.0 26.7 20.7 22.8 23.7 24.1 24.3

100 97 98 93 92 106 101 98 94 89

57 50 50 45 43 149 144 141 136 131

a d is the terrace width separated by steps. The γv is the surface energy of the vicinal surface, and its integration error is less than ±0.1 meV/Å2. The step energy referenced to γ110 and γ*110 is βv(γ110) and βv(γ*110), respectively. Its integration error is less than ±3 meV/Å. The large difference in * ) between the ⟨001⟩O and the ⟨001⟩Ti configurations results from the differences in γ110 * , and γ110 * is converged to the γ110 = 25.6 meV/Å as d βv(γ110 increases.

significantly between ⟨001⟩O and ⟨001⟩Ti steps. The higher value of ΔETiO2 for the Ti5c unit on the ⟨001⟩Ti step edge is due to an additional dangling bond from an oxygen atom, which is known to be less stable from Obr on the (110) surface. However, the Obr unit on the ⟨001⟩O step edge is more stable than the Obr unit on the flat (110) surface due to atomic relaxation at the step edge. The energy ΔElayer shows that the energy of the sixth layer is close to the (110) surface for both step structures. The other vicinal surfaces show similar energy spatial distributions as the (540) surface, and their step energies are summarized in Table 1. Table 1 shows the differences in step energies using two * . We calculate step different reference energies, γ110 and γ110 energies βv(γ110) using eqs 2 and 4. The step energies βv(γ110) are similar for the ⟨001⟩O and ⟨001⟩Ti steps, and this agrees with previous step energy calculations.6,7 The step energies * ) are calculated using eqs 2 and 5. The step energy of the βv(γ110 ⟨001⟩Ti step is three times that of the ⟨001⟩O step. This indicates that the ⟨001⟩O step has lower formation energy than the ⟨001⟩Ti step. The difference between eq 4 and eq 2 is 1/2(ΔEbridge − ΔEflat) = −45 meV/Å for the ⟨001⟩O step and 1/2(−ΔEbridge + ΔEflat) = 45 meV/Å for the ⟨001⟩Ti step, which is nearly identical to the difference between βv(γ*110) and βv(γ110). Thus, the EDM method shows a difference in stability due to surface energy contributions that cannot be deduced from total energy calculations. Step−Step Interaction. Figure 4 shows two different isolated step energies for short-separated steps and longseparated steps. The value of βv decreases as d increases, but the slope of βv versus 1/d2 has two values depending on the step separation. The change of βv from (210) to (430) (shortseparated steps) is small, while βv dramatically decreases from (430) to (650) (long-separated steps). This means that there are different step−step interactions between short-separated steps and long-separated steps. Surfaces with short-separated steps and surfaces with long-separated steps both satisfy eq 3, but with different values for β0 and q. The positive slopes indicate that the step−step interactions are repulsive, but the interaction coefficient qL is larger than qS for both ⟨001⟩O and ⟨001⟩Ti steps. The weak interaction of short-separated steps agrees with previous calculations.6,7 The different interactions result in different isolated step energies. Experiments find that the terrace width is over 10 nm and enlarges under annealing.21 Therefore, the isolated step energy from the long-separated steps is expected to be a realistic estimate of the step energy.

surface energy γ110 = 25.6 meV/Å as d increases. As n approaches infinity, γ*110 equals to γ110 for both ⟨001⟩O and ⟨001⟩Ti steps. Step Energy. Figure 3 shows the localization of the energy change ΔETiO2 in ⟨001⟩O and ⟨001⟩Ti steps. The energy change ΔETiO2 of surface layers alternates between low energy for Obr units and high energy for Ti5c units. The energy change ΔETiO2 has large variations around the step edge, with values that differ

Figure 3. (Left) Relaxed geometry and energy map of the (540) surface for ⟨001⟩O and ⟨001⟩Ti steps. The alternating solid and dashed lines are guide lines for each layer. The atoms above the thicker dashed line are relaxed. The color blocking behind each TiO2 unit shows the energy relative to the (110) surface: red is higher, blue is lower. Large energy variations are localized around the step within one (110) surface unit cell. (Right) The energy change of each layer ΔElayer. The step energy is sum of ΔElayer from the surface to the fifth layer: the ⟨001⟩O step energy is 45 meV/Å and the ⟨001⟩Ti step energy is 136 meV/Å. The ⟨001⟩O step has one more bridge configuration, which contributes to a lower step energy than the ⟨001⟩Ti step, which has one more flat configuration. D

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Figure 5. Geometry and stress distribution of each layer for ⟨001⟩O and ⟨001⟩Ti. The five vicinal surfaces from (210) with d = 9.86 Å to (650) with d = 36.16 Å are presented as a single geometry with a break. Faded atoms are periodic images of the step supercell, and the atoms above the dashed line are relaxed. The layer-by-layer stress is obtained by the energy derivative of strain, and the strain is applied along the [110] direction. The layer-by-layer stress is lower for shortseparated steps (blue), while that is higher for long-separated steps (red). The (430) surface (purple) shows a large stress distribution of each layer and it indicates (430) surface belongs to the long-separated step.

Figure 4. Step energies for ⟨001⟩O and ⟨001⟩Ti as a function of 1/d2 and linear fits. The βv-intercept is the isolated step energy, and the slope is the step−step interaction coefficient. The error bar of step energy for each surface is the integration error. The (430) surface is a common structure to both short-separated and long-separated steps. The interaction changes from short-separated steps (blue line) with small interaction coefficient qS to long-separated steps (red line) with large interaction coefficient qL. The different interactions result in different isolated step energies: the isolated step energies from shortseparated steps (βS0) are overestimated.

surface is a long-separated step and has the minimum dimensions necessary for long-separated step behavior. Step-Edge Reconstruction. We calculate the oxygen vacancy formation energy to investigate the stability of oxygen vacancies on the TiO2 vicinal surfaces. In experiments, oxygen vacancies are easily produced during the preparation of samples by sputtering and annealing processes. We use the oxygen vacancy formation energy definition of Janotti et al.,40 1 E f (VO) = E(Ti nO2n − 1) − E(Ti nO2n) + E(O2 ) + μO 2

The isolated step energies of ⟨001⟩O and ⟨001⟩Ti steps are 38 ± 1 and 125 ± 3 meV/Å2, respectively. We consider the layer-by-layer stress to quantify changes in elastic interactions: the stress is the derivative of a layer’s EDMcomputed energy with respect to strain. The layer-by-layer stress σ[110] produced by strain along the [110] direction between steps is

σ[1 1̅ 0] =

1 ∂ΔE layer A v ∂ϵ[1 1̅ 0]

(7)

where E(TinO2n−1) is the energy of the defective structure, E(TinO2n) is the energy of the perfect structure, and E(O2) is the energy of an oxygen molecule. The O2 binding energy calculated by DFT depends on which exchange-correlation functional are used, and there is an alternative definition using H2O.41 However, we choose the O2 binding energy, because we are interested in the relative stability of oxygen vacancies. The chemical potential of oxygen μO is zero in the oxygen rich limit, while it is estimated to be −4.14 eV in the oxygen poor limit from the enthalpy difference between Ti2O3 and TiO2 bulk crystals. Figure 6 shows the favorable oxygen vacancy formation energies at ⟨001⟩Ti step edges and unfavorable formation energies at ⟨001⟩O step edges, compared with the terrace and in the center of the slab. The Ef(VO) at the step edge of the ⟨001⟩O step is higher than both the Ef(VO) on the terrace and in the center of the slab. The most stable oxygen vacancy site on the ⟨001⟩O terrace is the closest bridging oxygen atom to the step edge. On the other hand, the oxygen vacancy at the step edge of the ⟨001⟩Ti step is the most favorable site. The closest

(6)

where ΔElayer is the energy of each surface layer referenced to bulk, ϵ[110] is applied strain, and Av is the vicinal surface area. We apply 1% compressive strain along the [110] direction. Figure 5 shows the changes of layer-by-layer stress along the [110]-direction of five vicinal surfaces which correlate with the different step−step interactions. The stress distribution of short-separated steps (excluding the (430) surface) shows small changes through the slab. This means that the step−step interaction based on the elastic field along the [110]-direction is not significant for short-separated steps. The long-separated steps show a large surface stress distribution indicating that the long-separated steps have a large interaction between the steps along the [110]-direction. The result is consistent with their interaction coefficient. We use the (430) surface to fit the step− step interactions for both short-separated steps and longseparated steps, but the stress distribution shows that the (430) E

DOI: 10.1021/acs.jpcc.5b03623 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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step edge is easily formed on the ⟨001⟩O step, and the relaxed structure is consistent with bright spot from STM image. This implies that the step with oxygen vacancy near to the step edge is the ⟨001⟩O step.

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CONCLUSION We use the energy density method to investigate the stability of rutile TiO2 vicinal surfaces with the ⟨001⟩ steps. Energy partitioning by EDM is essential to correctly calculate the isolated step energy of different step configurations. In the atomic scale simulation, the terrace configuration on the vicinal surface can be different from the flat surface periodicity and its contributions are not negligible. The different terrace configurations on the vicinal surface complicate the step energy calculation in TiO2 and similar behavior is expected for other vicinal surfaces. Therefore, the spatial partitioning of energy through EDM correctly defines the reference energy and it results in accurate step energies. The step−step interactions change with respect to the distance between steps, and the stress analysis shows elastic interactions along the direction between steps is artificially small when the steps are too close. The different interaction leads to an incorrect isolated step energy. Therefore, it is important to determine the stress distribution in the step to understand the step−step interactions for any step calculation and predict experimentally relevant step energies. These results provide an insight into accurate modeling of all steps on surfaces or interfaces.

Figure 6. Formation energies for an oxygen vacancy on the terrace, at the step edge, and in the middle of the slab for ⟨001⟩O and ⟨001⟩Ti steps. The removal of an oxygen atom creates a vacancy row due to periodic boundary condition. Reported values are the oxygen formation energy corresponds to the oxygen rich limit, and the values in parentheses are estimated for the oxygen poor limit. The oxygen vacancy on the terrace nearest to the step edge is more stable than at the step edge for the ⟨001⟩O step, while the oxygen vacancy at the step edge is the most stable for the ⟨001⟩Ti. The most stable oxygen vacancy rows on the ⟨001⟩O and the ⟨001⟩Ti steps are enclosed in a dashed circle, and their relaxed structures are presented on the right.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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bridging oxygen atom to the step edge of ⟨001⟩Ti step has lower Ef(VO) than the other vacancy sites on the terrace. The oxygen vacancy formation energy at the center of the slab (our bulk reference) is higher by 0.90−1.22 eV for ⟨001⟩Ti step and 0.54−0.71 eV for ⟨001⟩O step than on the (110) terrace area. Therefore, the oxygen vacancy is created most easily at the ⟨001⟩Ti step edge, and the next most favorable sites are oxygen atoms closest to the step edge for both step structures. The most stable relaxed structures of the ⟨001⟩O and ⟨001⟩Ti steps with an oxygen vacancy are shown in Figure 6 on the right side. The ⟨001⟩Ti step with an oxygen vacancy at the step edge shows significant atomic relaxations around the step edge. This large distortion leads to the lower oxygen vacancy formation energy of the ⟨001⟩Ti step. Even though the ⟨001⟩Ti step has higher step energy than the ⟨001⟩O step, it can be placed on TiO2 vicinal surface due to low Ef(VO) and accompanied reconstruction. In addition, this relaxed structure could explain a specific (1 × 4) reconstruction of the ⟨001⟩Ti step.21 Diebold et al. observed that the ⟨001⟩Ti step edges with a (1 × 4) reconstructed bright row, which indicate Ti5c atoms, and the distance between them is the same with a length of four TiO2 units along the [001] direction.21 In the relaxed structure, the oxygen atom replaces the original Ti5c site and is exposed on the surface, and this implies the dark area between bright spots of the reconstructed ⟨001⟩Ti step. Therefore, three oxygen vacancies in every four TiO2 units result in the (1 × 4) reconstruction of ⟨001⟩Ti step edge. Bechstein et al. found that oxygen vacancies are formed close to the step edge of the ⟨001⟩ step.3 The Ef(VO) shows that the oxygen vacancy near to the

ACKNOWLEDGMENTS This research was supported by NSF/DMR Grant 14-10596. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant Number ACI-1053575.42

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DOI: 10.1021/acs.jpcc.5b03623 J. Phys. Chem. C XXXX, XXX, XXX−XXX