Configuration evolution of Aln(n=3,4,6,13,19) clusters studied using linear synchronous transit method
PENG Ping(彭 平), LI Gui-fa(李贵发), YANG Feng (杨 峰), TIAN Ze-an(田泽安)
ZHENG Cai-xing(郑采星), HAN Shao-chang(韩绍昌)
School of Materials Science and Engineering, Hunan University, Changsha 410082, China
Received 10 April 2006; accepted 25 April 2006
Abstract: The evolution of configurations of Aln (n=3,4,6,13,19) clusters were investigated using linear synchronous transit (LST) method. The stable structures of Al3, Al4, Al6, Al13, Al19 clusters were confirmed to be triangle, rhombus, octahedron, icosahedron and double icosahedron, respectively. For Al6 and Al19 clusters there are metastable structures of parallelogram and octahedron, respectively, whereas in the Al3, Al4 and Al13 clusters, no metastable configuration are validated. A large energy gap and a low energy barrier between the parallelogram and the octahedron of the Al6 cluster indicate the transformation from its metastable configuration to stable octahedron to be rather easy. By contrast, a small energy gap and a high energy barrier between the stable and metastable structures of Al19 cluster mean its configuration evolution from the octahedron to the double icosahedron occurs hardly, therefore the metastable octahedron configuration of Al19 cluster can be extensively detected in experiments and simulations.
Key words: first-principles calculation; linear synchronous transit method; Al cluster; configuration evolution
1 Introduction
Recently Aln clusters attract many researchers’ attention because of its simple electronic structures and unique conductive characteristics, i.e., an obvious variation of its conductibility from an isolator to a conductor with the addition in atoms of Aln cluster. In the past two decades, a lot of theoretical and experimental researches[1-8] had been done. LIU et al[1] simulated the formation and evolution of Aln clusters during rapid cooling processes of liquid metal systems by a molecular dynamics (MD) method. The results demonstrate that only an icosahedral atom aggregations of (12 0 12 0) related to the 1551 bond-types and their combinations can easily grow and have inheritable characters, and no high symmetrical cuboctahedrons or decahedrons were detected. RAO et al[5] investigated the geometries, ionization energies and binding energies of Aln (n=2-15) clusters by means of Gaussian program based on density functional theory (DFT). The results show that the stable configurations of Al3, Al4, Al6 and Al13 are triangle, rhombus, octahedron, icosahedron, respectively, and their binding energy increases with the addition of Al atoms but are far lower than that of a Al crystal. Meanwhile, several experimental investigations were also carried out, in which the ionization energy[7] of Aln (n≤80) clusters and their binding energy[8] (n≤27) were measured by photoionization spectroscopes. However, up to now there is no experiment report on the configuration evolution of small Aln clusters. Besides, although the previous theoretical investigations[1-6] characterized the stable configurations of small clusters, it still lacks of a consideration on whether there are metastable structures in these clusters and how metastable structures evolve into stable configurations if there are. In order to understand the formation mechanism and configuration evolution characteristics of Aln clusters in the cooling process of liquid metal, a deep investigation of structure stabilities of Aln (n=3, 4, 6, 13, 19) cluster isomers was performed in this study by a first-principles pseudo-potential plane-wave method. A special consideration was focused on the evolution of these cluster isomers.
2 Calculation models and method
CASTEP (Cambridge serial total energy package) program[9] based on density functional theory (DFT) was adopted. A periodic boundary condition was employed, in which crystal wave functions were developed by plane wave basis sets. The model design is based on the method brought forward by WANG et al[10, 11], in which a supercell with crystal lattice constant of 200 nm was set up firstly, and then the cluster model was placed in the supercell, by which the interaction between cluster-cluster was omitted. On the basis of results calculated by RAO et al[5] and LLOYD et al[6], the original models of Aln clusters were designed, as shown in Fig.1, in which the initial Al-Al bond distance was set as dc =0.286 3 nm. During the optimization and total energy calculation of Aln clusters, the electronic exchange-correlation energy functions represented with the PBE type based on generalized gradient approximation (GGA) were used and the atomic potential functions were set as a norm- conserving form. 3s23p1 of Al were dealt with valence electrons, while the other orbital electrons were regarded as core electrons. The kinetic energy cutoff was set as 300 eV. Special K-point method in MONKHORST-pack scheme was used during the course of BRILLOUIN zone integral. FFT mesh was set as (24×24×24). All atomic positions in the supercells were relaxed under not fixing cluster symmetry according to the total energy and force using the BFGS scheme, based on the cell optimization criterion (RMS force of 0.5 eV/nm, RMS stress of 0.1 GPa, and RMS displacement of 0.02 nm). The calculation of total energy and electronic structure was followed by cell optimization with SCF tolerance of 2×10-5 eV under GGA-PBE potential.
3 Results and discussion
3.1 Stable structures
The structure stability of clusters can be evaluated by their binding energy (Eb) and HOMO- LUMO energy gap (ΔEH-L), i.e., low Eb or high ΔEH-L indicates a cluster combines firmly and is uneasy to be dissociated[5]. Herein the binding energy of Aln clusters was calculated according to the following Eqn.[5]:
(1)
where Etotal(Aln) is the total energy of Aln clusters, E(Al) is the energy of free atom Al, which is equal to 53.676 eV. The result is shown in Table 1. From Table 1 it can be seen that the binding energy Eb of Al3 triangle is lower than that in its line structure, which suggests that the stable structure of Al3 cluster is triangle. For Al4 cluster, a larger binding energy of tetrahedron than that of rhombus indicates its stable structure is still a planar type. As far as Al6 cluster, its stable structure should be octahedron [2, 3, 5] because the Eb of octahedron structure is lower than that of parallelogram. Similarly, the stable structures of both Al13 and Al19 clusters are icosahedrons rather than cuboctahedron or decahedron with high symmetries[5].
Fig.1 Calculation models of Aln clusters: (a1) Al3 (line); (a2) Al3 (triangle); (b1) Al4 (rhombus); (b2) Al4 (tetrahedron); (c1) Al6 (parallelogram); (c2) Al6 (octahedron); (d1) Al13 (icosahedron); (d2) Al13 (decahedron); (d3) Al13 (cuboctahedron); (e1) Al19 (octahedron); (e2) Al19 (double icosahedron)
Table 1 Binding energy Eb and HOMO-LUMO energy gap ΔEH-L of Aln (n=3, 4, 6, 13, 19) clusters
Fig.2 illustrates the electronic density of state (DOS) of Aln clusters. It can be seen that a gradual change of the DOS from discrete to quasi-continuous then to continuous spectra with the addition in atoms of Aln clusters, which indicates that the s-p hybridization of 3s and 3p electrons of Al atoms in Aln clusters is strengthened. Based on Fig.2, HOMO-LUMO energy gaps ΔEH-L of Aln clusters were also calculated, as shown in Table 1. In the three dimension clusters of Al6, Al13 and Al19, most of the configurations with lager energy gaps have lower binding energies, which indicates those configurations are their stable arrangements. However this correspondence of ΔEH-L with Eb is not very good for Al3 and Al4 clusters with a planar structure. The discrete characteristics of DOS spectra of these clusters should be responsible for this distinguishment.
Interestingly, by comparing the binding energy Eb and the HOMO-LUMO energy gap ΔEH-L as shown in
Fig. 2 DOS curves of Al3, Al4, Al6, Al13, Al19 clusters
Table 1, it is found that icosahedral structures, especially, the icosahedron of Al13 cluster with a typical closed-shell character[5], are more stable relative to other configurations among the above Aln clusters. Although an effect of temperature on the stability of Aln clusters is not considered in the present work, the thermodynamic tendency that Al atoms easily aggregate to icosahedral structures as well as their assemblages and preserve them unchanged during cooling should be undoubted. The reason may be that only icosahedron atom groups or their combinations and few octahedron as well as decahedron clusters with high symmetries were detected in MD simulation of the solidification of liquid metal Al[1].
3.2 Configuration evolution
Because the binding energies of different Aln cluster configurations are very close to each other, metastable structures of these Aln clusters should be considered except for their stable configurations. Herein an investigation on the transformation of different Aln cluster configurations was carried out by a linear
synchronous transit (LST) method[12], as shown in Fig.3. It can be seen that the transformations of triangle, rhombus and icosahedron from linear, tetrahedron and decahedron or cuboctahedron of Al3, Al4 and Al13 clusters, respectively, are all exothermic reactions and there are no energy barriers and transition states. These evolutions can be completed automatically. Hence for Al3, Al4 and Al13 clusters, there does not exist any metastable configurations except for their stable structures. However, for Al6 and Al19 clusters, a different evolution tendency from Al3, Al4 and Al13 clusters can be seen. The evolutions cannot take place automatically without outside driving forces. These are two transition states for the transformations to octahedron from parallelogram in Al6 cluster and from octahedron to double-icosahedron in Al19 cluster. In order to transfer into their stable octahedron and double-icosahedron configurations, Al6 and Al19 clusters must overcome energy barriers of ΔE2≈0.841 eV and 3.295 eV, respectively. Therefore, the parallelogram and the octahedron are the metastable structures of Al6 and Al19 clusters, respectively.
For Al6 cluster, a low energy barrier ΔE2 in the transformation from parallelogram to octahedron and a large energy difference (ΔE1=0.526 eV) between its metastable and stable configurations mean that this structure evolution is rather easy. Whereas, a high energy barrier ΔE2 in the evolution from the metastable to stable structures of Al19 cluster and a low energy difference (ΔE1=0.082 eV) between its double-
icosahedron and octahedron configurations indicate that this development will suffer more difficulty relative to that in Al6 cluster. The reason may be that an isomer of the Al19 double- icosahedron cluster, i.e., octahedron, and few decahedron and cuboctahedrons, an isomer of Al13 cluster icosahedron, were detected in annealing experiments made by LLOYD et al[6] and in MD simulation taken by YAO et al[13].
4 Conclusions
The stable structures of Al3, Al4, Al6, Al13, Al19 cluster are triangle, rhombus, octahedron, icosahedron, and double-icosahedron, respectively. In Al3, Al4 and Al13 clusters no metastable configuration structure are validated by the LST method. The transition states in the transformations from parallelogram to octahedron of Al6 cluster and from octahedron to double-icosahedron of Al19 cluster indicate the parallelogram and the octahedron are the metastable structures of Al6 and Al19 clusters, respectively. For Al6 cluster, a large energy difference within parallelogram and octahedron and its low transformation energy barrier imply the development from its metastable to stable structures is easy. By
Fig.3 Schematic diagrams on evolution of configuration structures of Aln (n=3, 4, 6, 13, 19) clusters
contraries, a low energy difference between double- icosahedron and octahedron of Al19 cluster and its high transformation energy barrier mean that its configuration evolution is rather difficult. Therefore the octahedron, i.e., a metastable structure in Al19 cluster, can be detected extensively in experiments and simulations.
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(Edited by CHEN Can-hua)
Foundation item: Project(104139) supported by the Ministry of Education of China; Project(03-Y3069) supported by the Hunan Province Natural Science Fund
Corresponding author: PENG Ping ; Tel: +86-731-8821610; Fax: +86-731-8821483; E-mail: ppeng@hnu.cn
