Abstract
The lowenergy structures (LESs) of adatom clusters on a series of metal facecentered cubic (fcc) (110) surfaces are systematically studied by the genetic algorithm, and a simplified model based on the atomic interactions is developed to explain the LESs. Two different kinds of LES group mainly caused by the different next nearestneighbor (NNN) adatomadatom interaction are distinguished, although the NNN atomic interaction is much weaker than the nearestneighbor interaction. For a repulsive NNN atomic interaction, only the linear chain is included in the LES group. However, for an attractive one, type of structure in the LES group is various and replace gradually one by one with cluster size increasing. Based on our model, we also predict the shape feature of the large cluster which is found to be related closely to the ratio of NN and NNN bond energies, and discuss the surface reconstruction in the view of atomic interaction. The results are in accordance with the experimental observations.
PACS: 68.43.Hn; 68.43.Fg.
Keywords:
supported cluster; structure; shape; metal surfaceIntroduction
In the nextgeneration microelectronics and ultrahighdensity recording, the fully monodispersed nanostructures are believed to be one of the most promising materials [1]. In order to fabricate such nanostructures, knowledge of the morphology of nanoclusters on surfaces becomes enormously important. So far, numerous experimental observations and theoretical investigations into structures of clusters have been reported on transition and noble fcc metal surfaces, e.g., fcc(111), fcc(100), and fcc(110) surfaces [210]. However, such studies mainly focus on the lowestenergy structures. For the structures with energy close to the lowest one, which are named lowenergy structures here, investigations and discussions are far from enough. At the usual experimental temperature, besides the lowestenergy structure, the lowenergy ones also appear frequently owing to thermal effect and usually play significant role in many surface thermodynamic processes [11]. In earlier publications, the lowenergy structures of adatom cluster on fcc(111) have been systematically studied, and it has been shown how the atomic interactions determine the equilibrium structures and shapes of the supported clusters [10]. In order to get a global view on the morphology of supported homoepitaxial clusters, here we investigate further a series of metal homoepitaxial clusters on fcc(110) surfaces, whose structure characteristics are far different from those of fcc(111).
Calculation method
Four metal homoepitaxial systems are investigated: Ni, Cu, Pt, and Ag. The atomic interactions are described by semiempirical potentials. The semiempirical potential might not be as accurate as the firstprinciple method in describing atomic interaction, but it enables us to study systematically clusters in a large size range, which is quite expensive for the latter one. Considering of the shortcoming of the semiempirical method, here we focus on the relationship between the atomic interaction and the structure of cluster, which is not sensitive to the accuracy of potential. However, we still choose the potentials carefully that nicely describe the surface diffusion [12,13]. For Ni and Cu, the atomic interactions are described by the embeddedatom method (EAM) potential given by Oh and Johnson [14] and the potential developed by Rosato, Guillopé, and Legrand (RGL) on the basis of the secondmoment approximation to the tightbinding model [15,16], respectively. While, for Pt and Ag, the atomic interactions are all modeled by the surfaceembedded atom method (SEAM) potential given by Haftel and Rosen for the surface environment [17,18].
Clusters are put on a slab containing 12 atom layers in Z direction, in which three bottoms of them are fixed to simulate a semiinfinite slab, while the atom numbers in X and Y directions vary with the cluster size n. Periodic boundary conditions are applied in X and Y directions. The clusters with size n = 2 to 39 are studied. Structures are optimized according to their energy by the genetic algorithm (GA), which has been described in detail in our previous publications [7,8].
Results and discussion
In the present work, we investigate the structures whose energy differences with the
lowest one are smaller than 0.12 eV. These structures are defined as the lowenergy
structures (LESs). According to the Boltzmann distribution, the probability of finding
a structure whose energy is 0.12 eV higher than the lowest one is less than 1% at
room temperature. Under the definition of LES above, we see that the lowenergy structures
obtained by our genetic algorithm are all twodimensional on the surfaces studied
here, i.e., threedimensional structures are excluded from the LES group for their
higher energy. However, on the different surface, the structure features are different
as expected. On Ni(110), Ag(110), and Cu(110) surfaces, various types of structures
are included in the LES group. In Figure 1, for example, the lowenergy structures of cluster n = 15 obtained by our genetic algorithm on Cu(110) and Pt(110) are given. On Cu(110)
surface, as shown in Figure 1a, both the linear chain and twodimensional islands appear. The energy of linear chain
is lower than that of threerow islands and higher than those of short tworow islands.
While on Pt(110) surface, as shown in Figure 1b, there is only one structure type in the LES group, i.e., the linear chain along
the [
Figure 1. Lowenergy structures of cluster n = 15 (a) on Cu(110) and (b) on Pt(110) surface. From structures 1 to 7, the energy is increasingly higher. The nearestneighbor (NN), next nearestneighbor (NNN), and third nearestneighbor (TNN) bonds are indicated in (c).
In order to understand the results on Ni(110), Ag(110), Cu(110), and Pt(110) surfaces, and give a general relationship between the structure and the atomic interaction, we try to give a simplified or approximated model in the following for describing the energy of the system, which is based on only the twobody atomic interaction. We decompose the total internal energy E of the system into three parts:
where E_{aa}, E_{as}, and E_{slab }refer to the energies contributed by the adatomadatom interaction, the adatomsubstrate interaction, and the bare slab internal interaction, respectively. For E_{aa}, we consider the nearestneighbor (NN), next nearestneighbor (NNN), and third nearestneighbor (TNN) interactions, and then E_{aa }can be written as:
where C_{nn}, C_{nnn}, and C_{tnn }refer to the numbers of NN, NNN, and TNN bonds, respectively. E_{nn}, E_{nnn}, and E_{tnn }are the energies of NN, NNN, and TNN bonds, respectively. As shown in Figure 1c, one NNN bond generally corresponds to two TNN bonds, i.e., C_{tnn }≈ 2C_{nnn}. Therefore, the last two terms in the right side of Equation 2 can be written as
C_{nnn}E_{nnn}+C_{tnn}E_{tnn}=C_{nnn}(E_{tnn+}2E_{tnn}). For convenience, we set (E_{nnn }+ 2E_{tnn}) as
The values of E_{nn }and
Considering that the energy contributed by the bare slab internal interaction, i.e., E_{slab}, can be approximately viewed as invariant, we then get the energy difference ΔE between the two structures as following:
Equation 5 shows that the energy difference of two structures results from the different nearestneighbor and effective next nearestneighbor adatomadatom interactions.
By examining the structure feature of cluster on fcc(110) surface, one can see the numbers of NN and NNN bonds satisfy:
in which r and l are the numbers of rows and lines of the cluster, respectively. For example, structure 1 in Figure 1a, r = 2 and l = 8, then C_{nn }= 13 and C_{nnn }= 7. With Equations 4 and 6, total internal energy can be written as:
where
With this simplified model Equation 7, for different structures of a cluster, we can predict their energy sequence just by comparing the values of Φ, which can be easily obtained by counting the numbers of rows and lines. Note that the bond energy E_{nn }is always positive, the larger structure factor Φ then means the higher energy of the structure, and vice versa. In other words, the lowestenergy structure should have the smallest structure factor Φ.
On Pt(110) surface, our calculation shows that the bond energy
Figure 2. The relative energy distribution. For simplification only the lowest energies of liner chain (solid squares), broken chain (open squares), and tworow island (dots) on Pt(110) surface are given. The structures of cluster n = 15 as an example are shown in the right. The number in the bracket means the number of NN and NNN bonds, respectively.
On Ni(110), Cu(110), and Ag(110) surfaces, different from the case on Pt(110), the
calculation shows that the bond energy
In our previous work [8], we reported the type change of the lowestenergy structure at critical size
For example, on Cu(110) surface, our calculation gives the ratio of E_{nn }and
Figure 3. The lowestenergy structures with two and three rows for cluster n = 34, 35 and 36, respectively.
is still valid. If we continue to subtract one atom, i.e., n = 34, the Rrow structure with oneless line comparing with that of n = 36 appears, as shown in Figure 3, because such structure can keep the energy being lowest according to the model Equation
7. As a result, from Rrow structure to R + 1 one at n = 34, dl = 5 no longer satisfies Equation 9. Therefore, on Cu(110) surface, the critical size
for tworow type changing to threerow one should be
With Equations 9 and 10, we can finally fix the critical size, which satisfies:
Therefore, according to Equation 11, we can predict the type change of the lowestenergy
structure. Still take Cu(110) as an example (ξ = 5.26), from Equation 11, the change of the lowestenergy structure from one linear
chain to tworow island will occur at
Figure 4. The relative energy distribution. The energies of liner chain (solid squares), broken chain (open squares), tworow islands (dots), and threerow islands (open circles) are given on Cu(110) surface. Only the lowest energies are considered as before.
Table 1. The energies of NN and effective NNN bonds and their ratio on metal surfaces
Figure 5. The relative energy distribution. Same as Figure 4, but for Ag(110) surface.
Corresponding to the type change of the lowestenergy structure, the lowenergy structures
studied here show an interesting stepwise replacement in type with the cluster size
increasing. For example, on Cu(110), there is only linear chain in the LES group for
n ≤ 5. At n = 6, the tworow island appears in the LES group. Our GA optimization shows that when
the cluster size n increases, the energy of tworow island is increasingly lower than that of the linear
chain, and at n = 12, as mentioned above, the tworow island becomes the lowestenergy structure of
the cluster. When the size increases further, the linear chain gradually disappears
from the LES group, meanwhile the threerow island appears. The tworow island maintains
in the group. At n = 16, there is no linear chain in the LES group. When the cluster size becomes much
larger than 16, similar to the case of linear chain, the energy of tworow island
is increasingly higher than that of threerow island. At n = 35, as mentioned above, the threerow island becomes the lowestenergy structure.
When cluster size increases further, the tworow islands are gradually excluded from
the LES group, meanwhile the fourrow island appears in the LES group. At that time,
the threerow island maintains in the group. In one word, when the cluster size increases,
the structures with more rows replace the ones with fewer rows step by step. The stepwise
replacement of the lowenergy structures also appears on Ag(110) and Ni(110) surfaces,
the difference is only the speed of the replacement owing to the different ratio ξ and then
In terms of NN and NNN atomatom interactions, we give a simple model Equation 7 to describe the energy of the cluster adsorbed on fcc(110) surface. In the above, we see that the model explains well the distinguishing features of the lowenergy structures obtained by our GA optimization, including structure type varying with the surface species and cluster size, which suggest that the model is reasonable. The most important is that based on our model Equation 7, we can further explore the equilibrium shape of large islands on fcc(110) surfaces, which is difficult to be obtained directly by GA optimization owing to the heavy computation. For numbers of rows and lines in cluster, we have
where d is the number of atoms needed for the cluster to form a complete r × l rectangular island, and it satisfies 0 ≤ d < l. For example, the structure (7) in Figure 1a has d = 3. When the cluster size n increases, the value of d linearly oscillates from 0 to (l − 1). Considering that the problem we are interested in here is the general shape of the lowenergy islands in equilibrium state, we take the average value of d in Equation 12, i.e., d = l/2. Note that r and l need to have proper values to minimize the energy of cluster, i.e., minimize Φ Equation 8, and then with Equation 12 and d = l/2, we have:
If the right side of Equation 13 is not an integer, then the close one which minimizes Φ is taken as the value for r or l. Then, we obtain the aspect ratio A of the equilibrium island:
where a is distance between two nearest neighbor atoms. Note that we have used
From the distinguishing features of the structures in LES group and the simplified
atomic interaction model Equation 7, we can further discuss the surface reconstruction
qualitatively. On Pt(110), as mentioned above, there is only linear chain in the LES
group, the reason is that the effective next nearestneighbor atomic interaction in
cluster is repulsive. The islands are all excluded from the LES group. The result
suggests that the Pt adatoms on Pt(110) do not tend to form closepacked configuration
but prefer the loose one which is continuous in [
Figure 6. Two types of structures for large adatom cluster on fcc(110) surface. Loose (a) and compact (b) configurations of cluster n = 126.
For cluster on other surfaces, e.g., Cu(110) and Ag(110), different from the case on Pt(110), the compact configuration has much lower energy than the loose one because the effective next nearestneighbor adatomadatom interaction is attractive as mentioned above. Then, on Cu(110) and Ag(110) surfaces, the compact structure such as island (b) in Figure 6 has much higher frequency than structure (a). Therefore, contrary to Pt(110) surface, the Cu(110) and Ag(110) surfaces are unlikely to occur (1 × 2) reconstruction naturally, which are in good agreement with the observation of Zhang et al. [21]. These accordant results including the shape of large islands and the surface reconstruction reflect that our model Equation 7 really works although it is just based on the simplified twobody interaction.
Conclusion
Groups of lowenergy structures are obtained for clusters adsorbed on Ag(110), Ni(110),
Cu(110), and Pt(110) surfaces by the genetic algorithm based on the EAM, SEAM, and
tightbinding potentials. In order to explain or understand the lowenergy structures,
we give a model based on the simplified atomatom interactions. The result shows that
the difference of the lowenergy structure on different surface is due to the effective
NNN adatomadatom interaction although it is very weak comparing to the NN atomic
interaction. For a repulsive NNN atomic interaction, e.g., on Pt(110), there is only
one type of structure in the LES group, i.e., linear chain. For an attractive NNN
atomic interaction, e.g., on Ag(110), Ni(110), and Cu(110) surfaces, the structure
type in the LES group is various, and when the cluster size increases the structure
type with fewer rows will be gradually excluded from the LES group and replaced by
the new one with more rows. The speed of replacement with the cluster size is determined
by the ratio of the NN and NNN bond energies ξ. Based on our model, we also discuss the aspect ratio of the large island and the
surface reconstruction on fcc(110) in the view of atomic interaction. It is shown
that the aspect ratio is inversely proportional to ξ. On Ag(110) surface, for example, owing to large ξ, the equilibrium shape of the large island is striplike in [
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
PZ, LXM, HZS, and JHZ wrote the computer program together, and PZ also performed the simulations and other calculations. WXZ, XJN, and JZ corrected the program and developed the algorithm. PZ and JZ give the model and explained the results. All the authors participated in the revision and approval of the manuscript.
Acknowledgements
The calculations are performed at the National High Performance Computing Center of Fudan University and Shanghai Supercomputing Center. This work is supported by Chinese NSF (no. 11074042), Major State Basic Research Development Program of China (973 Program) (no. 2012CB934200), and Innovation Program of Shanghai Municipal Education Commission (no. 10ZZ02).
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