Low-energy structures of clusters supported on metal fcc(110) surfaces
© Zhang et al; licensee Springer. 2011
Received: 16 June 2011
Accepted: 15 December 2011
Published: 15 December 2011
The low-energy structures (LESs) of adatom clusters on a series of metal face-centered 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 nearest-neighbor (NNN) adatom-adatom interaction are distinguished, although the NNN atomic interaction is much weaker than the nearest-neighbor 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.
In the next-generation microelectronics and ultra-high-density recording, the fully monodispersed nanostructures are believed to be one of the most promising materials . 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 [2–10]. However, such studies mainly focus on the lowest-energy structures. For the structures with energy close to the lowest one, which are named low-energy structures here, investigations and discussions are far from enough. At the usual experimental temperature, besides the lowest-energy structure, the low-energy ones also appear frequently owing to thermal effect and usually play significant role in many surface thermodynamic processes . In earlier publications, the low-energy 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 . 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).
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 first-principle 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 embedded-atom method (EAM) potential given by Oh and Johnson  and the potential developed by Rosato, Guillopé, and Legrand (RGL) on the basis of the second-moment approximation to the tight-binding model [15, 16], respectively. While, for Pt and Ag, the atomic interactions are all modeled by the surface-embedded 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 semi-infinite 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
Equation 5 shows that the energy difference of two structures results from the different nearest-neighbor and effective next nearest-neighbor adatom-adatom interactions.
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 lowest-energy structure should have the smallest structure factor Φ.
On Ni(110), Cu(110), and Ag(110) surfaces, different from the case on Pt(110), the calculation shows that the bond energy is positive. Then, , which means, according to Equations 7 and 8, the structures with low energy on Ni(110), Cu(110), and Ag(110) surfaces should have proper numbers of rows and lines to ensure low structure factor Φ. For example, n = 15 on Cu(110) as shown in Figure 1a, the proper values include r = 2, l = 8; r = 1, l = 15; r = 3, l = 6, etc., because the structures with these values have low energy and all of them are included in the LES group. If the structures with the same row are classified as one structure type, then the LES group on Cu(110), also on Ni(110) and Ag(110) surfaces, contains several types of structures. When the cluster size increases, it is easy imaginable that the structure types will change for keeping the proper values of r and l. It is indeed true as shown in our GA optimization results and closely related to the type change of the lowest-energy structure, the details of which are described later.
The energies of NN and effective NNN bonds and their ratio on metal surfaces
Corresponding to the type change of the lowest-energy structure, the low-energy 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 two-row island appears in the LES group. Our GA optimization shows that when the cluster size n increases, the energy of two-row island is increasingly lower than that of the linear chain, and at n = 12, as mentioned above, the two-row island becomes the lowest-energy structure of the cluster. When the size increases further, the linear chain gradually disappears from the LES group, meanwhile the three-row island appears. The two-row 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 two-row island is increasingly higher than that of three-row island. At n = 35, as mentioned above, the three-row island becomes the lowest-energy structure. When cluster size increases further, the two-row islands are gradually excluded from the LES group, meanwhile the four-row island appears in the LES group. At that time, the three-row 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 low-energy 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 . For example, on Ag(110) surface, the speed of the replacement with the cluster size increasing is much slower than that on Cu(110) like Figures 4 and 5 for the change of the lowest-energy structure.
where a is distance between two nearest neighbor atoms. Note that we have used for large clusters and assumed that each NN bond has the same length in Equation 14. Therefore, the equilibrium shape of large cluster only relates with ξ, i.e., the ratio of NN and NNN bond energies. If the cluster has large ξ, the aspect ratio A is small, and then the equilibrium shape is long in  direction and narrow in  direction. If the ξ is small, then the equilibrium shape with large aspect ratio A appears short and wide. For clusters on Ag(110), as shown in Table 1, our calculation shows that A is small, only 0.038. Such aspect ratio suggests the equilibrium shape of large clusters on Ag(110) is strip-like in  direction, and it is consistent with the experimental observation in general .
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 nearest-neighbor adatom-adatom 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. . 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 two-body interaction.
Groups of low-energy 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 tight-binding potentials. In order to explain or understand the low-energy structures, we give a model based on the simplified atom-atom interactions. The result shows that the difference of the low-energy structure on different surface is due to the effective NNN adatom-adatom 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 strip-like in  direction. The surface reconstruction is related to the NNN atomic interaction. On Pt(110) surface, the surface is likely to reconstruct naturally at room temperature because of the repulsive NNN atomic interaction. On other surfaces, e.g., Cu(110), however, owing to the attractive NNN atomic interaction, the natural surface reconstruction is unlikely to occur. These results are basically in accordance with the experimental observations.
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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