Atomistic study of metal clusters supported on oxide surface

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Surface Science 602 (2008) 46–53 www.elsevier.com/locate/susc

Atomistic study of metal clusters supported on oxide surface Y. Long

a,b,*

, N.X. Chen

a,c

a

Department of Physics, Tsinghua University, Beijing 100084, China Center for Advanced Study, Tsinghua University, Beijing 100084, China Institute for Applied Physics, University of Science and Technology, Beijing 100083, China b

c

Received 28 June 2007; accepted for publication 24 September 2007 Available online 1 October 2007

Abstract Metal clusters on oxide surface are a widely studied topic in surface science and technology. In this work, we use the ab initio based ˚ to 90 A ˚ . The clusters have a basic polyhedron pair potentials to study the shape evolution of these clusters with their width from 3 A shape covered by (0 0 1) and (1 1 1) faces, with four undetermined parameters. The main purpose of this work is to determine the structure parameters numerically. Here, we use a combination of energy minimization calculation, least square method and Lagrange multiplier method, and go through a series of metals including Ag, Al, Au, Pd and Rh. As a result, we find that these clusters have a truncated octahedron structure on MgO(0 0 1) surface, with a square contact face for Ag, Al and Au, and an octagon one for Pd and Rh. Also, we see that misfit dislocation appears when the cluster becomes large, first at the edge, then inside the contact area. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Metal cluster on oxide surface; Lagrange multiplier method

1. Introduction Metal clusters supported on oxide surface constitute a system with special properties in electricity, optics, magnetics, etc. They play an important role in various applications including catalysts, optoelectronics, magnetic devices, metal–ceramic sensors, etc. [1,2]. It is of fundamental importance in surface science to study the cluster structure atomistically. In the past decade, several experimental efforts have been put forward to treat this problem, by Barbier, Renaud, Graoui, Giorgio, Henry, etc. [3–10]. They mainly pay attention to the transition metal clusters such as Ag, Au, Pd and Ni, by the use of X-ray photoelectron spectroscopy (XPS), grazing incidence X-ray diffraction (GIXD), grazing incidence small angle X-ray scattering (GISAXS), high-resolution transmission electron microscopy (HRTEM), etc. The *

Corresponding author. Address: Department of Physics, Tsinghua University, Beijing 100084, China. E-mail address: [email protected] (Y. Long). 0039-6028/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2007.09.043

cluster structure is experimentally determined as a polyhedron shape covered by (0 0 1) and (1 1 1) faces, see Fig. 1 of the truncated octahedron case. This kind of cluster structure contains four parameters: l1, l2, h1 and h2, where l1 denotes the width of the cluster, it is used as the ‘cluster size’ in the following work, l2 denotes the side width, h1 denotes the lower half height, and h2 denotes the upper half height. Note that for different parameter combinations, the cluster can be in different shape. For example, if h1 = 0, it is half octahedron structure, if h2 = 0, it is plain stage structure, etc. The construction rule presented in Fig. 1 for metal clusters on MgO is the start point of our theoretical work. Based on the various parameter combinations, we can search through not only the original truncated octahedron structure, but also the half octahedron, plain stage, pyramid ones, etc. In summary, this theoretical work is built on some experimental results, and goes deep into a numerical determination of the structure parameters l1, l2, h1 and h2. However, a critical problem should be solved before we go into the numerical work. Note that the cluster contains several

Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

Fig. 1. The polyhedron structure of metal cluster on MgO, covered by (0 0 1) and (1 1 1) faces, with four structure parameters.

˚ . It is a timethousands of atoms when its size reaches 90 A consuming work to search through all the possible parameter combinations just by an energy minimization program without any tips (there are several hundreds of possible combinations need to be considered in this case). So we use a combination of energy minimization calculation, least square method and Lagrange multiplier method to treat this problem, reduce the calculation effort by using an approximate energy expression. In this method, energy minimization calculation is an important component. It is based on a series of model potentials for the target system. Here we use the pair potentials extracted from ab initio cohesive and adhesive energies by a Chen–Mo¨bius inversion method [11–13]. This pair-wise model seems a very simple approximation, but it is quite practical and can give a reasonable description of the structure and energy properties for metal/oxide interface, see our previous work Ref. [12] for its application. The following work consists of five parts. First, in Section 2, we present the atom–atom, atom–ion and ion–ion potentials used in this work, which are extracted from ab initio cohesive and adhesive energies. Second, the method to get the lowest energy structure for metal clusters on MgO(0 0 1) surface is introduced in Section 3. It is a combination of energy minimization, least square method and Lagrange multiplier method. Next, Section 4 shows the calculated results for a series of metals, including Ag, Al, Au, Pd and Rh. The structure parameters l1, l2, h1 and h2 are numerically determined, and misfit dislocation is found when the cluster becomes large. At last, Sections 5 and 6 are the discussion and conclusion.

47

from ab initio cohesive and adhesive energies by the Chen–Mo¨bius inversion method in our previous work [11–14], so we just present some sample potential curves here (see Fig. 2 of UAu–Au, UAu–O and UAu–Mg as the examples). In particular, the potentials UM–M for metal side are derived from the ab initio results of bulk materials [13]. This is because we consider the large clusters of the tens of angstroms in this work, whose atomic arrangements are just like bulk metals. Second, the ion–ion interactions have a complex constitution [15], with both the short-range and long-range parts (U = USR + ULR). Among them, the short-range interaction is the same with the atom–atom and atom–ion interactions, as shown in Fig. 2, and the long-range part is Coulomb interaction, determined by an effective charge Qeff of 2e in this case: LR LR ULR Mg–Mg ¼ UO–O ¼ UMg–O ¼

Q2eff 4p0 r

ð1Þ

Now, we pay attention to the validity of these pair potentials in studying metal clusters on oxide surface. Among the three kinds of interactions, the atom–atom and atom–ion ones play a more important role in determining the cluster structure, while the ion–ion ones are just to hold the substrate. So we mainly pay attention to the former two. First, for checking the atom–atom interactions, we consider the surface energy of pure metals, as shown in Table 1. For convenience, the theoretical values are extracted from the least square calculation result in the next section (see Table 2), and the experimental values are from

Fig. 2. Some potential curves of the atom–atom, atom–ion and ion–ion interactions.

2. Potential model The metal–oxide system contains three kinds of interactions: the atom–atom potentials UM–M (M = Ag, Al, Au, Pd and Rh) inside the metal clusters, the ion–ion potentials UO–O, UO–Mg and UMg–Mg inside oxide, and the atom–ion potentials UM–O and UM–Mg across the interface. First, the atom–atom and atom–ion interactions are short-range potentials. They have already been extracted

Table 1 Surface energies for Ag, Al, Au, Pd and Rh, both from theoretical calculation and experiments (J/m2)

This work Expt. [16]

Ag

Al

Au

Pd

Rh

1.58 1.25

1.82 1.20

1.93 1.55

2.01 2.10

2.68 2.75

The theoretical values are extracted from Table 2 by changing the unit ˚ 2 to J/m2 with the relation of 1 eV/A ˚ 2 = 16.02 J/m2. from eV/A

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Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

Table 2 The energy parameters for Ag, Al, Au, Pd and Rh clusters ˚ 2) ead (eV/A ˚ 2) e111 (eV/A ˚ 2) e001 (eV/A ˚) k1 (eV/A ˚) k2 (eV/A ˚) l1 (eV/A ˚) l2 (eV/A em (eV) C (eV)

Ag

Al

Au

Pd

Rh

0.1198 0.0956 0.1004 2.1979 0.8548 0.5495 0.1030 2.9234 1.8983

0.1124 0.1157 0.1059 3.5913 0.8662 0.9973 0.1353 3.0523 0.0623

0.1178 0.1115 0.1211 2.8276 1.1344 0.5506 0.1343 3.5972 3.0501

0.1417 0.1278 0.1327 2.0266 0.9491 1.0969 0.1913 4.1291 1.0441

0.1656 0.1784 0.1813 2.8381 1.2137 1.7097 0.2295 5.7118 0.0609

Ref. [16]. As a result, we see that the theoretical surface energies are in agreement with the experimental ones. So the atom–atom potentials are credible in describing the energy properties of metal clusters. Also, we calculate the surface relaxation made by these potentials, as shown in Fig. 3. It demonstrates that for the (0 0 1) surface, the layer distance di decreases when it is away from the surface. Unfortunately, this outward relaxation is different from the correct surface relaxation—which is inward due to Ref. [17]. But we also think that this pair potential approach is practical and sometimes reasonable for the present work, because of its success in predicting the surface energies. Second, for the atom–ion interactions, we consider the adhesive energy curves for several interface structures.

Here we also need to invoke some results in the latter part of this work, as shown in Fig. 10. When the metal cluster is supported on MgO(0 0 1) surface, the metal atom is on the O site inside the contact face, while it is on Mg site at the edge of the contact face or in the dislocation core. As a result, we check the atom–ion potentials at three interface structures, of metal on O site, Mg site and hollow site (H site), respectively, as shown in Fig. 4. Note that, the H site structure is the transition part between the O site and Mg site ones. For the checking, we use the atom–ion potentials to recalculate the ab initio adhesive energies for the three interface models. Here, the first-principles calculation is performed by CASTEP code [18,19] with generalized gradient approximation (GGA). About the parameter setting, plane-wave cut-off energy is 340 eV, and k-point spacing ˚ 1, generated by Monkhorst–Pack scheme is 0.05 A [20,21]. Fig. 5 shows the resultant adhesive energies for the O site, H site and Mg site structures both by ab initio method and pair potentials. It demonstrates that the adhesive energies obtained by the two ways are fitted good, so these potentials are credible in describing the interface structure between metal clusters and MgO(0 0 1) surface. 3. Methodology In this section, we answer two questions: how to get the cluster energy, and how to determine the structure param-

Fig. 3. (a) The (0 0 1) surface of fcc metal. (b) The distances between metal layers near the surface.

Fig. 4. The O site, H site and Mg site structures of metal/MgO interface. (a) The vertical view, (b) the O site structure, (c) the H site structure and (d) the Mg site structure.

Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

49

Fig. 5. The adhesive energies of O site, H site and Mg site structures. Scatter symbols indicate ab initio results, and lines indicate pair potential results. (a) Adhesive energies for O site structures, (b) for H site structures and (c) for Mg site structures.

eters. Usually, the cluster energy is obtained by an energy minimization program, but it is not enough in this work, for we need to search through a large amount of possible structures to find out the lowest energy states for a certain cluster (this count maybe several hundreds for the cluster ˚ large). So, we use an approximate energy size being 90 A expression to reduce the computational time, based on the energy minimization calculation and least square method. For metal cluster on MgO, the cluster energy contains several parts, including the lattice energy, surface energy, adsorption energy, edge energy, etc. In particular, for surface energy, we need to consider two cases, the (0 0 1) and (1 1 1) faces, as shown in Fig. 1. A reasonable energy expression should contain all these contributions e ¼ ead scon þ e111 s111 þ e001 s001 þ k1 l1 þ k2 l2 þ l1 h1 þ l2 h 2 þ e m n þ C

ð2Þ

where e denotes the cluster energy, scon denotes the contact face area, s111 and s001 denote the (1 1 1) and (0 0 1) face areas, ead, e111 and e001 denote the corresponding adsorption energy and surface energies, n denotes the number of atoms in cluster, and em denotes the lattice energy. Furthermore, the energy parts of edges and apexes on the cluster are also considered in this formula, where l1, l2, h1 and h2 are proportional to the edge lengths, and k1, k2, l1 and l2 denote the corresponding edge energies, accompanied with C denotes the apex energy. To get the parameters in Eq. (2), the least square method is introduced. For this purpose, we define two vectors e ¼ ½ ead s ¼ ½ scon

e111 s111

e001 s001

k1

k2

l1

l2

em

l1

l2

h1

h2

n

C 1

ð3Þ

where e denotes the set of energy parameters we want, and s denotes the set of corresponding contact face area, surface areas, edge lengths, etc. Based on the abbreviations, the energy formula is rewritten as e ¼ esT

ð4Þ T

where s denotes the transposition of s.

For determining e, we need to calculate a series of sample models by an energy minimization program resulting in the equations ei ¼ esTi ;

i ¼ 1; 2; . . . ; n

ð5Þ

where n is the number of sample models we have considered. Using the least square method [22], e is obtained by e ¼ ES T ðSS T Þ

1

ð6Þ ½sT1 ; sT2 ; . . . ; sTn .

in which E = [e1, e2, . . ., en], and S ¼ As a result, we get an approximate energy expression for metal cluster on MgO, and use it to determine the cluster structures in the following work. By the way, for calculating the sample models, we use the MINIMIZER module of Cerius2 software [18]. The resultant energy parameters are listed in Table 2. From the table, we see that k1 and l1 are positive, and k2 is negative, accompanied with l2 which is very small. It is worth to give a brief discussion about this fact. In Eq. (2), the four terms k1l1 + k2l2 + l1h1 + l2h2 denote the edge energy part of the cluster. For an approximate evaluation, we calculate the total edge length of the cluster in Fig. 1. Ignore p the ffiffiffi complex pffiffi deduction process, it is equal to 12l1  6 2l2 þ ð16 ð2Þ  8Þh1 . Turn back to the edge energy, it is approximately proportional to this total edge length, so pffiffiffi pffiffiffi k1 l1 þ k2 l2 þ l1 h1 þ l2 h2  e0 ð12l1  6 2l2 þ ð16 2  8Þh1 Þ ð7Þ where e0 denotes the factor of proportionality. Note that at the edge of a metal cluster, some metal–metal bonds are broken, so that e0 > 0. From Eq. (7), it is easy to see that k1 and l1 are positive, and k2 is negative, accompanied with l2, which must be very small. Now, we turn back to the cluster energy formula (see Eq. (2)), talk about its advantage and shortage. First of all, the energy formula can greatly reduce the effort in energy minimization calculation, so we can study the metal ˚ . But we must note that some clusters as large as 90 A features have been ignored in this approximation. For

50

Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

ing work Fig. 9). So we consider a conditional extremum problem minfeðl1 ; l2 ; h1 ; h2 Þg

at nðl1 ; l2 ; h1 ; h2 Þ ¼ constant

ð8Þ

where e(l1, l2, h1, h2) is the energy expression in Eq. (2), and n(l1, l2, h1, h2) is the function of atom number in cluster. Unfortunately, n(l1, l2, h1, h2) has a complex functional form. It is a difficult problem to find out all the parameter combinations for a certain atom number n. So we change Eq. (8) into another form. By using the Lagrange multiplier method [26], Eq. (8) is rewritten as

Fig. 6. A process to get the lowest energy structure of metal cluster on MgO.

minfeðl1 ; l2 ; h1 ; h2 Þ  g  nðl1 ; l2 ; h1 ; h2 Þg

example, the distortion of charge distribution when the cluster is very small, the many-body effect on the metal/ oxide interface, etc. As a result, our method is suitable for large clusters but not the small clusters. For the latter ones, first-principles calculation is preferred [23–25]. Next, we pay attention to another question, how to determine the cluster structure energetically? Factually, we need to optimize the structure parameters under the condition of atom number n, which is fixed. This is because the cluster structure cannot be obtained just by comparing its energy or per-atom energy without any constraint— both of them decrease with the cluster size (see the follow-

ð9Þ

where g is the Lagrange multiplier, it changes the conditional extremum problem into an extreme value problem. In this work, g can be predetermined, and a set of l1, l2, h1 and h2 is obtained for each g by an exhaust algorithm using the MATLAB software [27]. Lagrange multiplier theory [26] has proved that the resultant parameter combination of l1, l2, h1 and h2 achieves the lowest energy state when the atom number is n(l1, l2, h1, h2). As a result, the process to get the lowest energy cluster structure consists of three steps, first deriving the approximate energy expression, second exhausting the structure

Table 3 ˚) The structure parameters l2, h1 and h2 as the function of cluster size l1, for Ag, Al, Au, Pd and Rh clusters (A Ag

Al

Au

Pd

Rh

l1

l2

h1

h2

l2

h1

h2

l2

h1

h2

l2

h1

h2

l2

h1

h2

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 55 58 61 64 67 70 73 76 79 82 85 88 91

0 4 4 4 8 8 12 12 17 17 17 21 21 25 25 25 30 30 34 34 34 38 38 43 43 43 47 47 51 51

0 2 2 4 4 4 6 8 8 10 10 12 12 15 15 17 19 19 21 21 23 23 23 25 25 28 30 30 32 32

2 4 4 6 6 8 10 12 12 12 15 17 17 19 21 21 23 23 25 25 28 30 30 32 32 34 36 38 38 41

0 0 4 4 8 8 12 12 12 17 17 21 21 25 25 30 30 30 34 34 38 38 43 43 43 47 47 51 51 51

0 2 2 2 4 4 6 6 8 10 10 12 12 15 15 17 17 19 21 21 23 23 25 25 28 28 30 32 32 34

2 4 6 8 10 12 10 10 12 12 15 15 17 19 19 21 21 21 23 25 25 25 28 30 30 32 32 34 34 36

0 4 4 8 8 8 8 12 12 17 17 17 21 21 25 25 25 30 30 30 34 34 38 38 38 43 43 47 47 47

0 2 2 4 4 6 8 8 8 10 12 12 15 15 17 19 19 21 23 23 25 25 28 28 30 30 32 34 34 36

2 2 4 6 8 8 10 10 12 15 15 17 19 19 21 21 23 25 25 28 28 30 32 32 34 36 36 38 41 41

0 4 4 8 8 8 12 12 17 17 21 21 21 25 25 30 30 34 34 34 38 38 43 43 47 47 51 51 56 56

0 0 0 2 2 2 4 4 6 6 8 8 8 10 10 12 12 15 15 17 17 17 19 19 21 21 21 23 23 23

2 4 4 6 6 8 8 10 12 12 15 15 17 19 19 21 21 23 25 25 28 28 30 30 32 34 36 36 38 38

0 – 4 8 8 12 12 17 17 17 21 21 25 25 30 30 34 34 34 38 38 43 43 47 47 51 51 56 56 51

0 – 0 0 2 2 2 4 4 6 6 6 6 6 8 8 10 10 12 12 12 15 15 17 15 17 17 19 19 25

2 – 4 6 6 8 8 10 12 12 15 15 17 19 19 21 21 23 23 25 28 28 30 32 32 34 34 36 36 41

Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

parameter combinations, and third relaxing the cluster again, as shown in Fig. 6. Note that at the last step (when the structure parameters have been obtained), we use the energy minimization program [18] again to get the relaxed cluster structures for advanced study. 4. Result Now, we are in the position to calculate the structure parameters l1, l2, h1, and h2 for metal clusters on MgO(0 0 1) surface by using the above ideas. Factually, these parameters are obtained as the functions of Lagrange multiplier g (see Eq. (9)). But here we are interested in the evolution of cluster structure against its size l1, so we rewrite the rest parameters as the functions of l1, see Table 3. For an intuitive view, Fig. 7 shows some sample models of the clusters on MgO due to the above parameters, for ˚ . As we have mentioned in the last parawhich l1 is 45 A graph of Section 3, these models have been relaxed by using the Cerius2 software [18]. Some basic structure properties can be extracted from Fig. 7. First, the metal clusters have a truncated octahedron structure, which is proved to be energetically preferable than the others, such as the half octahedron, plain stage, pyramid ones, etc. Comparing the cluster structures between different metals, they have different kinds of contact faces on MgO(0 0 1) surface. For Ag, Al and Au, this contact face is square, but for Pd and Rh, it is octagon. From Fig. 1 we see that the geometrical structure of the

51

contact face is related to the value of h1  12 l2 , which is square for h1  12 l2 > 0, and octagon for h1  12 l2 < 0. Fig. 8 shows the values of h1  12 l2 for the metal clusters considered in this work. It is greater than 0 for Ag, Al and Au, but less than 0 for Pd and Rh. Next, we calculate the interfacial distance for metal clusters on MgO, as shown in Table 4. It is defined as the vertical distance between the first monolayer of metal side and the last monolayer of MgO side. From the table, we see that the theoretical values are in agreement with the available experiments. It shows a successful prediction of the interfacial distances by our potential model and method.

˚ to 90 A ˚. Fig. 8. The values of h1  12 l2 for metal clusters being from 3 A The lines are just to guide eyes.

˚ . (a) Ag cluster, (b) Al cluster, (c) Au cluster, (d) Pd cluster Fig. 7. The lowest energy structures of metal clusters on MgO(0 0 1) surface, for which l1 = 45 A and (e) Rh cluster.

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Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

Table 4 The theoretical and experimental interfacial distances for M/MgO(0 0 1) interface (M = Ag, Al, Au, Pd, Rh)

This work Expt.

Ag/MgO

Al/MgO

Au/MgO

Pd/MgO

Rh/MgO

2.60 2.52 [28]

2.35 –

2.65 2.08 [3]

2.25 2.22 [29,30]

2.10 –

And then, based on the structure parameters, we calculate the energy per atom (hei) for these clusters, as shown in Fig. 9. It demonstrates that hei decreases with l1, so the cluster tend to grow up if there are extra metal atoms. By the way, there is no minimum point at the hei curves. It means that for determining the lowest energy structure, we need to consider the constraint condition n(l1, l2, h1, h2) = constant, as mentioned in the third last paragraph of Section 3. At last, the misfit dislocation in metal clusters is taken into account. It is caused by the misfit between the metal a am and MgO lattices, which is equal to 2 aMgO , as shown in MgO þam Table 5 (where am and aMgO denote the lattice constants of metal and MgO sides, respectively). Due to the calculation, the misfit dislocation appears when the cluster is large enough, and relaxes the strain energy caused by the lattice misfit. Fig. 10 shows Rh clusters as the examples. It demonstrates that misfit dislocation appears when the cluster is ˚ large (Fig. 10b), with metal on Mg site at the disloca90 A tion core, and on O site at the other part of the contact face. But the dislocation is not so viewable when the cluster ˚ for example, see Fig. 10a). It is at the is small (l1 = 45 A edge of the contact face, where the metal atom is also on the Mg site. Factually, for Rh clusters, the dislocation moves inside ˚ . About the contact face when the cluster size l1 > 84 A ˚ , (b) Fig. 10. The cross section of Rh clusters on MgO. (a) l1 = 45 A ˚. l1 = 90 A

Ag, Al, Au and Pd clusters, they have smaller misfit than Rh (see Table 5). As a result, this separate line is greater ˚ for them, which is out of the range considered than 90 A in this work. Here, we just see the dislocation at the edge of the contact face for these four metals. But according to the case of Rh, we believe that the misfit dislocations also can move inside the contact face when the clusters ˚ cases is are large enough. The investigation of l1 > 90 A our future work. ˚ to 45 A ˚ . The lines Fig. 9. The energy per atom for metal clusters from 3 A are just to guide eyes.

Table 5 The misfit for (Ag, Al, Au, Pd, Rh)/MgO interfaces Ag/MgO (%)

Al/MgO (%)

Au/MgO (%)

Pd/MgO (%)

Rh/MgO (%)

3.1

4.0

3.4

8.1

10.4

5. Discussion Now, we are in the position to compare our results with others’ theoretical and experimental works. First, we pay attention to the theoretical works of Vervisch, Mottet, Goniakowski et al. [31–36]. They mainly studied Pd clusters on MgO(0 0 1) surface, including the structural, mechanical, energetic, thermodynamic properties, etc. Especially, they

Y. Long, N.X. Chen / Surface Science 602 (2008) 46–53

have shown that Pd cluster has an octagon contact face on MgO [31], which is also repeated in our work by another way—a combination of energy minimization, least square method and Lagrange multiplier method. This agreement can be treated as a basic checking for the present work. Second, we pay attention to the available experimental results of metal clusters on MgO. These works are mainly focused on Au and Pd. For Au clusters, the truncated octahedron and half octahedron structures are found in Pauwels’ work [3], and the pyramid and half octahedron structures that grow on MgO surface step are reported in Kizuka’s work [9]. For Pd clusters, Graoui, Giorgio and Henry’s work [5] shows an evolution of the cluster structure when it is growing up. It is in a pyramid structure ˚ , and in a half octahedron when the cluster is less than 30 A ˚ structure between 40 and 50 A, results in a truncated octa˚ . Among these cluster hedron structure larger than 100 A structures for Au and Pd, the truncated octahedron case is reproduced in this work. It shows that the theoretical results are supported by experiments, and also gives some confidence to the prediction of Ag, Al and Rh clusters. However, the other cases (the pyramid and half octahedron ones) are not obtained in this theoretical work. This is the limitation of the present work, for the complex experimental conditions cannot be exactly reproduced by a simple energy minimization calculation. Comparing the theoretical and experimental works, the former can give an exact atomic description of metal clusters on MgO, including the values of the structure parameters. However, the latter cannot provide such detailed information, for the limitation of equipments and the complex surface condition in experiments. Furthermore, the theoretical work usually cannot give a perfect reproduction of the experimental results, as shown in this work, but it also can show some valuable hints. 6. Conclusion In the present work, we use a combination of energy minimization calculation, least square method and Lagrange multiplier method to study the structure properties of metal clusters on MgO(0 0 1) surface. The main part of this work is to determine the cluster structure numerically ˚ to 90 A ˚ . Based on the structure for the size from 3 A parameters, we find that Ag, Al, Au, Pd and Rh clusters have a truncated octahedron structure on MgO. Especially, the former three have a square contact face on MgO, while the latter two have an octagon contact face. Considering the experiments, this truncated octahedron structure has been reported for Au and Pd clusters. It shows that the theoretical work can give some valuable hints to experimental understanding. Also, we find that the misfit dislocation appears in these clusters, with metal atom on Mg site at the dislocation core, and on O site for the other part of the contact face. The dislocation is at the edge of the contact face for small clusters, and moves inside it when the cluster becomes large.

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