Absolute Surface Step Energies: Accurate Theoretical Methods

Letter pubs.acs.org/JPCL

Absolute Surface Step Energies: Accurate Theoretical Methods Applied to Ceria Nanoislands Sergey M. Kozlov,† Francesc Viñes,† Niklas Nilius,‡ Shamil Shaikhutdinov,‡ and Konstantin M. Neyman*,†,§ †

Departament de Química Física and Institut de Química Teòrica i Computacional (IQTCUB), Universitat de Barcelona, C/Martí i Franquès 1, 08028, Barcelona, Spain ‡ Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, 14195 Berlin, Germany § Institució Catalana de Recerca i Estudis Avançats (ICREA), 08010 Barcelona, Spain ABSTRACT: Steps are ubiquitous structural features present on surfaces. They often exhibit unique physical and chemical properties, which in many cases are of even greater practical importance than the respective properties of surface terraces. Herein, we present two new methods to derive absolute values of surface step energies from electronic structure calculations, the efficiency and accuracy of which are superior to the methods previously employed. Application of these advanced methods allowed us to successfully explain the multitude of nanoisland shapes and orientations observed to form on CeO2(111) films via scanning tunneling microscopy studies. These methods are readily applicable to various systems and provide insights into the abundance and form of two-dimensional nanostructures, which are controlled by step energies. SECTION: Surfaces, Interfaces, Porous Materials, and Catalysis

N

anoislands are commonly observed nanostructures.1−5 Their shape, orientation,6−8 and thereby governed properties4 are determined by step energies, similar to nanoparticles, whose shape9 and other properties10−13 are often controlled by the surface energies of the exposed nanofacets. Specifically, steps, which limit the growth of nanoislands, are known to have different electronic features,4,14 adsorptive properties,15 and chemical activity16−18 compared to surface terraces. They also serve as nucleation centers19 and, thus, play a crucial role in catalysis,20 semiconductor technologies,3 and nanotechnology.14 Notably, structural and other properties of steps observed on vicinal or rough surfaces may be very similar to those present at the perimeters of nanoislands. Properties of steps have been intensely studied. Nevertheless, the pivotal issues of step energy and stability have been scarcely addressed theoretically (with the pertinent studies basically limited to refs 6, 21−26) or experimentally.7,8,27 This is partially due to the lack of accurate, reliable, and, at the same time, efficient methods to derive step energies from electronic structure calculations. The existing methods either provide step energies in a rather frugal way,23−25 which may produce inaccurate results in practice, or derive them from surface energies of vicinal surfaces,25,26 which requires numerous calculations. Here, we propose two novel methods of step energy calculation (applicable, in principle, to any material), hitherto never communicated, which are not only much less computationally demanding but also more accurate than the existing approaches. We have applied these methods to rationalize, in terms of step energies derived from density functional (DFT) calculations, various shapes and orientations of nanoislands formed on CeO2(111) films, as observed in our scanning tunneling microscopy (STM) studies. Ceria surfaces have been extensively characterized1,4,5 due to the importance of ceria in catalysis28−30 © 2012 American Chemical Society

and energy-related technologies.31−33 Atomic resolution studies have shown that steps on the CeO2(111) surface are, in general, stoichiometric and have the arrangements of atoms similar to that of the bulk.1,4,5 As mentioned above, ceria steps have different electronic4 and chemical properties17,18 compared to CeO2(111) terraces and serve as nucleation centers for metal atoms34 and nanoparticles.20 The methods that we have developed pave the way toward the understanding and subsequent control of the abundance, shape, and other related properties of nanoislands on various materials, eventually enabling their potential use in nanotechnology applications. Indeed, these methods enable calculations of step energies in a variety of experimental conditions, which will help to tune the preparation procedures to preferentially form a certain step type. In the absence of steps, the total energy, E, of a perfectly flat slab, an infinite 2D crystal limited in the third dimension by two planes, is usually expressed as35−37

E = εN + 2γA

(1)

Here, the first term corresponds to the bulk energy produced by the overall number N of (smallest) stoichiometric units forming the slab with energy ε per unit. The second one represents the destabilizing contribution from the two surfaces of the slab, which are assumed to have the same atomic arrangement, each with area A and specific surface energy γ. If steps (line defects) and point defects are present on the surface (Figure 1), the total energy includes two more terms E = εN + 2γ0A 0 +

∑ (βi + γ0Δi)Li + ∑ ξj

(2)

Received: May 29, 2012 Accepted: July 6, 2012 Published: July 6, 2012 1956

dx.doi.org/10.1021/jz3006942 | J. Phys. Chem. Lett. 2012, 3, 1956−1961

The Journal of Physical Chemistry Letters

Letter

Figure 1. Illustration of terms in eq 2.

where the second term is the contribution from two terraces of a low-index surface, each with area A0 and specific surface energy γ0. The third term is the sum over all steps with length Li, width projected on the terrace plane Δi (if a step is normal to the surface, Δi = 0), and step energy βi per unit length. The last term represents the energy contributions from corners between steps and other point defects, which will not be discussed further due to the absence of these features in the models addressed in this work. Equation 2 assumes that (i) steps and the two slab surfaces are well separated and do not interact with each other and (ii) the arrangement of atoms on the surface is not changed upon crossing the step. Equation 2, however, is seldom used22,24 because it yields limited accuracy in practical calculations. Such inaccuracies emerge if one derives ε and γ0 values from calculations of systems different from a rough slab (i.e., containing terraces and steps) and then applies eq 2.36,37 For example, let us consider a case where the calculated energy of the smallest stoichiometric unit in the bulk, εb, is different from the actual ε in the rough slab by only 0.007 eV (this value and those presented further in this paragraph are based on the typical calculations performed in the present work). A variety of technical reasons, such as different k-points sets, the number of plane waves in the basis sets, and the geometry optimization details in the calculations of the two systems, may produce this subtle energy difference. If one now uses εb instead of ε in eq 2, this difference is multiplied by N = 50 for the 0.382 nm wide cell with two steps typically used in the present work, thus introducing a 0.46 eV/ nm error in the derived step energy. Due to similar reasons, the difference between terrace surface energies calculated in a rough slab supercell and the smallest terrace (1 × 1) unit cell may be up to 0.15 eV/nm2. This value multiplied by a surface area of 1.22 nm2 (on each side of the slab for the typical 0.382 nm wide unit cells) introduces an additional error of 0.48 eV/nm to the step energy. Thus, the overall uncertainty is ∼1 eV/nm, which makes the direct approach inapplicable to many practical problems. A way to circumvent this drawback, and decisively reduce the related inaccuracy, is to perform only calculations of rough slabs with the slightly varying structure and dimensions of the unit cells. To do so, let us consider a periodic slab exposing only one type of steps, repeated with periodicity L (Figure 2a). Following eq 2, the energy per primitive unit cell of the slab is E = (εt + 2γ0wl)L0 /l + 2(β + γ0Δ)w

Figure 2. The geometric parameters used in eqs 3−5 for (a) a slab terminated with vicinal surfaces and (b) a stripe on a substrate.

which is equivalent to the known formula22,25,26 for the energy of a vicinal surface γ = [γ0(L0 + Δ) + β ]/L = γ0 cos θ + β /L

(4)

where θ is the misalignment angle between the vicinal and lowindex surfaces. Using this formula, one can derive β as the slope of the linear regression of γ/cos θ as a function of (L cos θ)−1. In the following, this method will be referred to as vicinal-γ. In practice, however, derivation of the slope energy from ab initio calculations using this approach requires linear regression of γ/cos θ versus (L cos θ)−1, with each point of the data set calculated, in turn, by linear regression of E versus t. This makes the established approach computationally expensive or even prohibitive for accurate calculations. By accuracy, we mean 95% confidence intervals for coefficients obtained in linear regression, and thus, the term accuracy used here is unrelated to the intrinsic accuracy of the electronic structure calculation. To advance this approach, we pose an alternative, the socalled vicinal-E method, where one can directly calculate the value of β + γ0Δ as the intercept of the linear regression of E(2w)−1 as a function of L0/l via eq 3. In this way, the method avoids calculation of the surface energy of a vicinal surface and solves the issue of computational efficiency. Note, that γ0Δ has to be subtracted from the obtained value for the steps with nonzero width. Because the monolayer step width is typically ∼0.1 nm, the uncertainty introduced by this correction is only ∼0.01 eV/nm due to the ∼0.1 eV/nm2 accuracy of independent calculations of the terrace surface energy. To cross-check the validity of the approaches employing vicinal surfaces, we also consider rough slab models in the form of periodic, one-layer high, linear stripes on a flat substrate (Figure 2b). Using eq 2, the energy of the slab’s primitive unit cell containing two slopes can be expressed as E = εtĹ /l + εĹ/l + 2γ0wL + w ∑ βi

(3)

(5)

Here, t ́ is the thickness of the substrate in stoichiometric layers, L (Ĺ) is the length of the unit cell (stripe) in the direction perpendicular to the step. From this expression, one can derive the sum of the step energies, ∑ βi, as the intercept of a linear regression of E − Esub as a function of Ĺ, where Esub is the energy of the unit cell of the substrate without the stripe. This method, further referred to as stripe-E, does not require calculation of any surface energy, but only provides ∑ βi, not each βi separately. Note that this method can be generalized to the case where different terraces have different terminations or even when the stripe and the substrate have distinct nature, that is, composed of different materials.

Here, t is the number of stoichiometric layers composing the slab (i.e., its thickness), w is the width of the primitive unit cell (one stoichiometric unit) in the direction along the step, L0 is the length of the low-index surface terrace, and l is the distance between adjacent rows of stoichiometric units in the direction normal to the step. Thus, L0/l is the number of stoichiometric units composing the surface of the low-index terrace. Calculating the surface energy defined in eq 1 as the intercept of the linear regression of E as a function of t in eq 3,35−37 one gets γA = γ0wL0 + (β + γ0Δ)w 1957



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Table 1. Step Energies β, in eV/nm, Calculated with the Vicinal-γ and Vicinal-E Methodsa

In our calculations of the energies of steps on ceria we used slabs featuring vicinal surfaces with steps like those sketched in Figure 3 (more details on the step structures can be found in

step type type type type type

I II II* III

vicinal-γ 1.39 2.12 4.03 2.01

± ± ± ±

vicinal-E

0.27 0.11 0.15 0.12

1.50 2.06 4.20 2.04

± ± ± ±

0.08 (1.30) 0.06 (1.72) 0.09 (4.04) 0.20 (1.97)

a

Estimated step energies at a temperature of 1000 K are given in parentheses.

The vicinal-γ method operates with surface energies and thus requires calculations of slabs with varying thicknesses. In the present work, slabs of 3, 4, and 5 O−Ce−O trilayers were considered, which are sufficiently thick for reliable calculations of surface energies.21,39 The latter were calculated for four vicinal surfaces with different terrace lengths. Thus, each step energy obtained by the vicinal-γ method required 12 electronic structure calculations. To use the vicinal-E method, the calculation of slabs with the same thickness is sufficient. Thus, only four electronic structure calculations were performed to obtain each step energy value. The difference between step energies calculated for three-, four-, and five-layer slabs is less than 0.07 eV/nm, and only the values for the latter slabs are presented in Table 1. In this case, the uncertainties are estimated as a sum of 95% confidence intervals for β + γ0Δ, with uncertainty introduced by subtracting the γ0Δ term (∼0.01 eV/nm). The comparison of step energies calculated by the conventional, vicinal-γ method and those obtained with our vicinal-E method shows that the difference between them is within the estimated error. However, the accuracy is higher for the vicinalE method (0.1 eV/nm on average) than that for the vicinal-γ method (0.16 eV/nm on average), despite the three times fewer calculations required for the former. This ratio would grow if a higher accuracy was required because using the vicinal-γ method one would need to consider slabs with a broader range of thicknesses to calculate more accurate surface energies. When step energies are calculated with the stripe-E method, only the sum of the energies of steps on both sides of the stripe is gained. Thus, sums of step energies calculated with the different methods are provided in Table 2. Stripe-E step energies

Figure 3. Atomic structures of considered models of ceria. Side views of (a) the CeO2 slab terminated by a vicinal surface, featuring type I steps and (b) the CeO2 stripe with type I and II steps. Ce atoms are displayed as gray spheres and O atoms are shown as red spheres. Top views of stoichiometric (c) type I, (d) type II, (e) type II*, and (f) type III steps with the height of one O−Ce−O trilayer on CeO2(111) terraces. O atoms located in the first, third, fourth, and sixth atomic layers are displayed as big dark, big bright, small dark, and small bright red spheres, respectively. Ce atoms in the second and fifth atomic layers are displayed as big and small gray spheres, respectively. Crystal directions on the surface are also shown.

refs 4 and 38). The slabs consisted of t = 3, 4, and 5 O−Ce−O trilayers and were completely relaxed during geometry optimization. Terrace length, L0, ranged from 7l to 10l for slabs with type I, II, and II* steps and 9l and 12l for slabs with type III steps. This yielded a distance between steps of L > 2.2 nm in the first case and L > 1.7 nm in the second case. Stoichiometric CeO2 stripes on the CeO2(111) substrate had a height of one O−Ce−O trilayer and width, Ĺ, of up to 13l. The substrate was modeled by a two-trilayers-thick slab. A rectangular 24l × w (7.935 × 0.382 nm2) supercell was used for calculations of ⟨110⟩ stripes, while ⟨211⟩ stripes were modeled in a 24l × w supercell with lattice vectors of 5.291 and 0.661 nm along the [211̅ ]̅ and [112]̅ directions, respectively. Cell dimensions did not vary with the stripe width. During geometry optimization, the stripe and the topmost trilayer of the substrate were relaxed, while the bottom O−Ce−O trilayer was fixed. The surface energy of CeO2(111) was calculated to be γ0 = 4.00 ± 0.11 eV/ nm2. Step widths, Δ, were 0.11 nm for type I steps, −0.11 nm for type II and type II* steps, and 0 for type III steps. Step energies on the aforementioned models were calculated using all three methods. First, we discuss results from the vicinal-γ and vicinal-E methods, both of which derive step energies from the energies of slabs terminated by vicinal surfaces (Table 1).

Table 2. Energies β, in eV/nm, of Step Pairs Appearing on Stripes on the CeO2(111) Surface step types

stripe-E

vicinal-γ

vicinal-E

I + II I + II* III + III

3.65 ± 0.20 6.01 ± 0.04 4.28 ± 0.07

3.51 ± 0.38 5.42 ± 0.42 4.03 ± 0.24

3.56 ± 0.15 5.70 ± 0.17 4.08 ± 0.39

are, in general, somewhat larger than the vicinal-γ and vicinal-E values, but only in the type I + II* case does the difference somewhat exceed the estimated accuracy of the former methods. The stripe-E method also yields tighter confidence intervals than the other methods when a comparable number of calculations is used in the linear interpolation. Stripes of various thickness were calculated in each case, but total energies of only four systems were used in every case. The reason for this is that inclusion of stripes with widths less than 1.5 nm made confidence intervals broader. Note that inclusion of a few more structures easily improves the accuracy of the calculations by an order of magnitude, for example, the step energy of I + II is calculated to be 3.71 ± 0.05 using six systems. The employed 1958



Letter

Figure 4. STM images (200 nm × 200 nm) of thin CeO2(111) films grown on Ru(0001) and annealed in 10−6 mbar of O2 at 980 (a,b) and ∼1100 K (c). The films expose atomically flat terraces and single-layer islands and pits. The characteristic island shapes are detailed in the insets (30 nm × 30 nm). While the steps of type I and II dominate film (a), the steps are preferentially of type III in image (c). Image (b) illustrates the intermediate state with several step orientations. More roundish islands, observed in (b), are indicative for a gradual transition of the preferred step orientation.

which have been described for the first time within this work. All three methods were found to yield similar values of step energies, but the new methods require significantly fewer calculations to produce results with even higher accuracy. We have applied the three methods to the analysis of monolayer step energies on CeO2(111) films, which determine the shape and orientation of nanoislands detected and characterized by STM. The computationally derived stabilities for various conceivable steps are in very good agreement with STM data. As such, we have acquired in-depth knowledge of the structure−stability relations for islands abundantly formed on the CeO2(111) surface, nanoobjects highly important for understanding ceria reactivity at the nanoscale. This study paves the way for investigations of step stabilities on various surfaces with different compositions. Such studies are presently scarce due to the high cost of sufficiently accurate step energy calculations when using conventional methodologies.

eqs 2−5 assume that adjacent steps do not interact with each other, which holds only at sufficiently large (1.5 nm) step separation. For slabs terminated by vicinal surfaces, the distances between adjacent steps exceed 1.7 nm, thereby fulfilling this condition. The differences in the step energies manifest themselves via the preferential formation of step edges with a certain orientation. This is illustrated by STM images (Figure 4) of the CeO2(111) thin films grown on Ru(0001), which expose atomically flat nanoislands (and pits), measuring a single O−Ce−O trilayer in height (depth). Atomically resolved STM images, in combination with tunneling conductance spectroscopy, have identified type I, II, and III steps4 but no type II* steps, in agreement with the high calculated formation energy of the latter. Figure 4a shows a film with an abundance of nanoislands in truncated triangular shapes. The islands are bordered by three long, type I steps and three short, type II steps (see inset in Figure 4a). The ratio between the lengths of type I and type II steps LI/LII = 2 ± 0.3 was measured on a sample prepared by oxidation at ∼1000 K, which clearly indicates that the type I steps are thermodynamically more stable than type II steps, in agreement with the calculated data presented above. Applying the Wulff construction scheme9 to the nanoisland shapes,7,8 one will get βi/βj = Ri/Rj, where R is the (shortest) distance from the nanoisland’s center of mass to the step. For an island of truncated hexagonal shape bordered by type I and type II steps, this ratio transforms to β1 β2

=

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EXPERIMENTAL AND COMPUTATIONAL DETAILS Periodic DFT calculations were carried out using the VASP package43 and employing the Perdew−Wang (PW91)44 exchange− correlation functional of generalized gradient approximation type, augmented with the Hubbard self-interaction correction45 (GGA+U) scheme, with Ueff = 4 eV.46 Other computational parameters can be found elsewhere.4 Test calculations done without the self-interaction correction yielded step energies ∼0.2 eV/nm different from those obtained with it. To estimate the contribution of harmonic vibrations to the Gibbs energy, ΔG, at finite temperature, the vibrational frequencies of CeO2 units located on the steps and on the terrace were calculated with a finite differences technique. The ΔG of the step energy was then calculated as the difference between ΔG of the CeO2 unit on the step and that on the terrace. Thin ceria films of ∼3 nm thickness were prepared in two different experimental setups using virtually identical preparation protocols,42 albeit the precise recipe had to be adapted for each setup.4 Briefly, the clean Ru(0001) surface was oxidized in 10−6 mbar of O2 at 600 K. Then Ce was deposited in 10−6 mbar of O2 at ∼100 K in amounts approximately equivalent to form one monolayer of CeO 2 (111) on Ru(0001). Subsequently, the temperature was slowly increased and kept at 700 K during deposition of the next several layers. After the deposition, the sample was annealed at ∼1000 K at the same oxygen pressure.

2 + L1/L 2 = 0.8 1 + 2L1/L 2

which is in excellent agreement with the ratio of 0.76 estimated at 1000 K (see Table 1). Analysis of numerous films prepared in our laboratories suggests that higher oxidation temperatures are necessary to produce type III structures, which is in good agreement with the higher formation energy of type III + III step pairs, as compared to type I + II pairs (see Table 2). Also, the precise experimental conditions (e.g., oxidation temperature and oxygen pressure, presence of adsorbates, the metal support in the case of the thin films, etc.) may affect the size, orientation, and shape distribution of nanoislands.4,15,40−42 Good agreement between our experimental and theoretical results suggests that the step structures observed here are thermodynamically stable and not significantly affected by kinetic limitations. In summary, we critically compared the conventional method of calculating step energies through surface energies of slabs terminated by vicinal surfaces with two alternative methods,

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 1959



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Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support by the Spanish MICINN (FIS2008-02238) and the Generalitat de Catalunya (2009SGR1041) and the access to computational resources of the Red Española de Supercomputación. S.M.K is also grateful to the Spanish Ministerio de Economia y Competitividad for the predoctoral FPU Grant AP2009-3379. F.V. thanks the Spanish MICINN for the postdoctoral grant under the program Juan de la Cierva JCI-2010-06372. We thank M. Baron, J.-F. Jerratsch, and X. Shao for STM data, as well as W. Kaden for his comments to the text. This work is part of the COST Action CM1104 “Reducible oxide chemistry, structure and functions”.

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Absolute Surface Step Energies: Accurate Theoretical Methods

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