The Scientific World Journal

Volume 2013 (2013), Article ID 468327, 9 pages

http://dx.doi.org/10.1155/2013/468327

Research Article

First-Principles Study on Stability and Magnetism of (, ) Clusters

Xiao Zhang, Bao-Xing Li, Zhi-wei Ma, and Jiao-jiao Gu

Department of Physics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

Received 23 March 2013; Accepted 27 April 2013

Academic Editors: A. Hermann, A. Ovchinnikov, and A. Savchuk

Copyright © 2013 Xiao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The investigation on the structures, stabilities, and magnetism of (, ) clusters has been made by using first principles. We found some new ground-state structures which had not been found before. These mixed species prefer to adopt three-dimensional (3D) structures starting from four atoms. All the ground-state structures for the Ni-Al clusters are different from those of the corresponding pure Al clusters with the same number of atoms except for three atoms. The Mulliken population analysis shows that some charges transfer from the Al atoms to the Ni atoms. ( = odd number) cations, Ni₂Al₆ neutral, Ni₂Al₁ and Ni₃Al cations and anions, and Ni₃Al₅ anion have the magnetic moments of 2 . The magnetic moments of NiAl₄ and NiAl₆ cluster neutrals and cations are 2  and 3 , respectively. All the other cluster neutrals and ions do not have any nontrivial magnetic moments. The 3d electrons in Ni atoms are mainly responsible for the magnetism of the mixed Ni-Al clusters.

1. Introduction

Aluminum is one of the most popular metals used in quantum-effect electronic devices. It attracts scientists’ great attention to its nanoscale clusters. Cohen et al. have explained the stability of small clusters by using jellium model because of the valence electrons of -like free electron [1, 2]. Khanna et al. have discovered that aluminum clusters have chemical properties similar to single atoms of metallic and nonmetallic elements when they react with iodine [3, 4]. For instance, cluster behaves like a single iodine atom, while cluster is similar to an alkaline earth atom. Fournier have obtained the ground-state structures of and by performing “Tabu Search” (TS) global optimizations directly on the BPW91/LANL2DZ potential energy surface [5]. They have found that all clusters have the lowest spin state as their ground state except Al₄ (triplet), (quartet), (triplet), and maybe (singlet and triplet are degenerate). Furthermore, and had special stability and larger HOMO-LUMO gaps.

The studies of the Al clusters doped with impurity atoms have been also reported. Ren and Li have investigated the structures, stabilities, and magnetism of zinc-doped ( clusters in detail by using first-principles density functional theory [6]. Ding and Li have done the calculation on by using the first-principles methods [7]. Li and Wang have performed first-principles calculations on the ground states of both neutral and anionic clusters [8].

Nickel is a transition metal. Aluminum clusters doped Ni atoms have many novel properties and chemical properties. neutral clusters have been investigated by Wen et al. using the density functional theory and generalized gradient approximation (GGA) with the exchange-correlation potential (BPW91) [9]. It is found that the atomization energies per atom have the same trend as the binding energies per atom for clusters. It is shown that is the relatively most stable structure in the series. Besides that, average magnetic moment decreases when alloyed with Al atoms than that in pure Ni clusters. Deshpande et al. have reported the magnetic properties of small clusters with calculated in the framework of density functional theory [10]. The overall magnetic moment of the cluster decreases with the sequential substitution of by Al atoms. This could be attributed to the antiferromagnetic alignment of small individual atomic moments arising due to the hybridization of Al-p and Ni-3d orbitals. Calleja et al. have reported ab initio molecular dynamics simulations of and clusters using a fully self-consistent density-functional method [11]. Recently, Sun et al. have performed the study on the growth behavior and electronic properties of clusters in the framework of density-functional theory (DFT) theoretically [12]. In the ground-state structures of clusters, the equilibrium site of Ni atom gradually moves from convex and surface to interior site as the number of Al atom varying from 2 to 14. Ferrando et al. have performed very thorough review on alloyed clusters [13]. Bailey et al. have presented their research results about nickel and aluminum clusters and nickel-aluminum nanoalloy clusters with up to 55 atoms [14]. The effect of doping atoms into pure clusters and vice versa has been investigated.

Although the investigations about , , and clusters mentioned above have been performed, our knowledge about the properties of the Ni-Al clusters with different atomic ratio is still limited. In this paper, our main goal is to explore the effect of the Ni atoms on the geometric, electronic, and magnetic properties of the pure clusters.

The paper is organized as follows: a brief account of the computational methodology is given in Section 2, followed by a detailed presentation and discussion of the structures of different size clusters in Section 3. A summary of the findings and conclusions is given in Section 4.

2. Computational Method

We optimized the Ni-Al cluster structures by using first-principles density functional theory. Our calculations were performed with the generalized gradient approximation (GGA) by means of the Becke-Perdew functional, which uses Becke’s [15] gradient correction to the local expression for the exchange energy and Perdew’s [16] gradient correction to the local expression of the correlation energy, as implemented in the Amsterdam Density Functional (ADF) codes [17]. The choice of such a gradient-corrected exchange-energy functional gives us satisfactory results, which are in good agreement with the experimental results. For example, the calculated vertical electron affinity and vertical ionization potential for cluster are 2.16 eV and 6.61 eV, respectively. The available experimental values are 2.25 eV and 6.45 eV, respectively [18, 19]. In the ADF program, molecular orbitals (MOs) were expanded using a large, uncontracted set of Slater-type orbitals (STOs): TZ2P. The TZ2P basis is an all-electron basis of triple-ζ quality, augmented by two sets of polarization functions. The frozen-core approximation for the inner-core electrons was used. The orbitals up to 3p  for nickel and 2p for aluminum were kept frozen. An auxiliary set of and STOs was used to fit the molecular density and to represent the Coulomb and exchange potentials accurately in each self-consistent field (SCF) cycle. The self-consistent field was converged to a value of in Hartree.

Frequency analyses at the stable structures are carried out at the same theoretical level to clarify if they are true minima or transition states on the potential energy surfaces of specific clusters. All of the most stable clusters in the paper are characterized as energy minima without imaginary frequencies.

3. Results and Discussions

Searching for the ground-state structures of the atomic clusters, especially the mixed clusters, is a difficult task. The following approach is applied in an attempt to find the global energy minimum structures. Firstly, about more than ten thousand initial geometrical configurations are produced by random selections of atomic positions within a three-dimensional box, or a cage, or a ball in real space. The distance between two nearest atoms is properly chosen to avoid overlapping or loosely packing. Next, the Amsterdam Density Functional (ADF) package is used for geometrical optimizations with their spins being not unrestricted. After the structural optimization on the initial configurations is performed, it is found that some structures are stable, but others are not convergent. Sometimes, several initial structures can gave out the same final structure after the first-principles optimizations. In the end, the structures with the largest binding energies calculated using generalized gradient approximation (GGA) are considered to be the ground-state geometries. The binding energy (BE) for the cluster is calculated according to the following atomization reaction: . It is defined by the following: , where is the total energy of the cluster. and are the energy of and atoms, respectively.

All the lowest energy structures of Al₃, Al₄, and Al₅ clusters are planar structures. They are an equilateral triangle, a planar quadrangle, a trapezoid-like, respectively. The ground-state structures for clusters are three-dimensional. The Al₆ and Al₇ clusters have a distorted tetragonal bipyramid () and an antitrigonal prism with a capping atom (), respectively. But, Fournier found that the ground-state structure for the Al₆ cluster was a distorted trigonal prism by using the BPW91LANL2DZ method [5]. For the Al₇ cluster, its ground-state structure can be obtained by putting an atom on one triangular face of the prism for the Al₆ cluster. We have also performed calculations on the two structures found by Fournier with the ADF program. It is found that both of them are metastable. Our results are 0.15 and 0.36 eV more stable than his, respectively. For the Al₈ cluster, our result is a transcapped octahedron with symmetry. But, in Fournier’s report, the ground-state structure is a distorted transcapped octahedron, or an antitrigonal prism with a face-capping atom and an edge-capping atom. We have performed structural optimization on his structure. It is found that his structure finally changes into the structure. The Al₉ cluster has a ground-state structure. The most stable structure of pure Al₁₀ cluster can be obtained from the Al₉ ground-state isomer by putting a capping atom. All of the structures obtained by the ADF program are in excellent agreement with those reported by Chuang et al. using a genetic algorithm coupled with a tight-binding interatomic potential [20]. They are shown in Figure 1.

Figure 1: The ground-state structures of the clusters.

The ground-state structures of clusters are presented in Figure 2. The ionic structures similar to the neutral structures are not repeated in Figure 2. But the structure of cluster cation is presented in Figure 2 because it has a severe distorted ground-state structure compared with its neutral and anion. Their symmetries are listed in Tables 1, 2, and 3. The and clusters and their ions have one three-dimensional structure, which are different from the planar structures of the Al₄ and Al₅ clusters. All the most stable structures for the cluster neutrals, anions and cations are three-dimensional structures, but they are obviously different from the structures of the corresponding clusters with the same number of atoms. Positive cluster ion with symmetry has a severe distorted ground-state structure compared with its neutral and anion (see the structure in Figure 2). Mulliken population analysis shows that nonuniform charge loss of the atoms in the structure results in its severe structure distortion.

Table 1: The symmetries, the total magnetic moments (, in ), and the binding energies (BE, in eV) for () clusters and their ions. The energy gaps (, in eV), the vertical electron affinities (EA, in eV), and vertical ionization potentials (IP, in eV) of neutral () clusters.

Clusters
Symmetries
(in )
BE (in eV)
(in eV)
EA (in eV)
IP (in eV)

Anion
Neutral
Cation
Anion
Neutral
Cation

NiAl

0
1
2
4.82
3.53
3.47
0.25
1.29
7.00

NiAl₂
C2v
1
0
1
8.18
6.94
0.15
1.06
1.24
6.79

NiAl₃
C3v
0
1
2
11.31
9.92
2.67
0.60
1.39
7.25

NiAl₄
C4v
1
2
3
14.10
12.21
5.17
1.18
1.89
7.04

NiAl₅
Cs
0
1
2
16.62
14.59
8.38
0.52
2.03
6.21

NiAl₆
C2v
1
2
3
20.10
17.64
11.25
0.26
2.46
6.39

NiAl₇
Cs
0
1
2
23.16
21.06
14.50
0.53
2.10
6.56

NiAl₈
Cs
1
0
1
25.80
23.56
17.47
0.73
2.24
6.09

NiAl₉
Cs
0
1
2
28.81
26.39
19.99
0.69
2.42
6.60

Table 2: The symmetries, the total magnetic moments (, in ), and the binding energies (BE, in eV) for Ni₂ () clusters and their ions. The energy gaps (, in eV), the electron affinities (EA, in eV), and ionization potentials (IP, in eV) of neutral Ni₂ () clusters.

Clusters
Symmetries
(in )
BE (in eV)
(in eV)
EA (in eV)
IP (in eV)

Anion
Neutral
Cation
Anion
Neutral
Cation

Ni₂Al
C2v
2
1
2
8.70
7.19

0.51
1.51
6.87

Ni₂Al₂
C2v
1
0
1
12.79
10.90
4.14
0.66
1.89
6.76

Ni₂Al₃
C2v
0
1
0
15.87
14.48
8.60
0.54
1.39
5.88

Ni₂Al₄
Cs
1
0
1
18.41
16.84
10.76
0.86
1.57
6.08

Ni₂Al₅a
C₁
0
1
0
21.58
19.61
13.73
0.41
1.97
5.88

Ni₂Al₆
C2v
1
2
1
25.40
23.05
16.69
0.33
2.35
6.36

Ni₂Al₇
C2v
0
1
0
28.14
26.07
20.27
0.43
2.07
5.80

Ni₂Al₈
C₁
1
0
1
30.65
28.50
22.50
0.65
2.15
6.00

Ni₂Al₉
C2v
0
1
0
33.74
31.37
25.39
0.44
2.37
5.98

Table 3: The symmetries, the total magnetic moments (, in ), and the binding energies (BE, in eV) for Ni₃ () clusters and their ions. The energy gaps (, in eV) and the electron affinities (EA).

Clusters
Symmetries
(in )
BE (in eV)
(in eV)
EA (in eV)
IP (in eV)

Anion
Neutral
Cation
Anion
Neutral
Cation

Ni₃Al
C2v
2
0
2
12.35
10.89
4.32
0.47
1.46
6.57

Ni₃Al₂
C2v
1
0
1
16.67
14.99
8.45
0.62
1.68
6.55

Ni₃Al₃
Cs
0
1
0
19.83
18.32
12.42
0.50
1.51
5.90

Ni₃Al₄
C2v
1
0
1
23.26
21.71
15.10
1.27
1.55
6.61

Ni₃Al₅
C2v
2
1
0
26.44
24.53
18.91
0.22
1.91
5.62

Ni₃Al₆
C2v
1
0
1
30.71
28.39
22.04
0.41
2.32
6.35

Ni₃Al₇
Cs
0
1
0
33.30
31.07
25.15
0.29
2.23
5.92

Ni₃Al₈a
C₁
1
0
1
35.62
33.43
27.71
0.48
2.19
5.72

Ni₃Al₉a
C₁
0
1
0
38.71
36.22
30.32
0.29
2.49
5.90

Figure 2: The ground-state structures of the clusters. The black ball refers to atom in the clusters.

The cluster’s magnetism is usually described by its magnetic moment. We define magnetic moment (in Bohr magnetron unit ) as the difference between the electronic occupation number of spin-up and spin-down. We think the cluster is magnetic if its structure with nonzero spin is more stable than that with zero spin. The investigation on the magnetic moments for the clusters shows that anions, neutrals, and cations for the and clusters have the magnetic moments of 1 , 0, 1 , respectively. Negative, neutral, and positive and clusters have the magnetic moments of 1 , 2 , and 3 , respectively. The clusters with odd-aluminum atoms have the magnetic moments of 1 . Their negative ions have no magnetic moment, but the positive ions show the magnetic moment of 2 .

The most stable structures of clusters are presented in Figure 3. It is found from observing Figure 3 that the geometries are also different from those in Figures 1 and 2. Obviously, the second impurity atom further changes the structures of host clusters. The lowest energy structure of cluster which is obtained by a random way has a symmetry. The second most stable structure which produced by substitution from has a Cs symmetry and lies 0.07 eV above structure. All the ground state structures of the , ,  and   clusters have the same symmetry. The cluster can be obtained by substitution, by a random way, or by putting a capping atom on the cluster. The cluster has a low symmetry.

Figure 3: The ground-state structures of the clusters. The black ball refers to Ni atoms in the clusters.

By performing calculation on the magnetic moments for the clusters, we obtain the following results. All the clusters have the total magnetic moment of 1.0  due to one unpaired electron. This suggests that the two impurity atoms do not change the magnetism of the clusters. On the other hand, the , , and clusters with even number of electrons have no magnetic moment because all the electrons are paired together in their respective molecular orbitals. But, unexpectedly, the cluster shows magnetic moment of 2 . Each Ni atom in the cluster has the local magnetic moment 0.40  and the total charge transferring from the Al atoms to them is −0.74 e. However, the average magnetic moment of each Al atom is 0.20 , which is only half of that of the atoms. The Mulliken population analyses indicate that the local magnetic moment 0.40  of atoms is mainly from the d orbital. On the other hand, if we add up the magnetic moments for the atoms and the atoms, respectively, the total magnetic moments for atoms are 1.2 , which is larger than 0.8  for atoms. Hence, atoms contribute more to total spin than Ni atoms.

The and cluster ions are expected to produce zero magnetic moment. But our calculations show that the ions have the magnetic moment of 2 . It is found that the and clusters ions have no magnetic moment. The magnetic moment for the and cluster ions is 1 . Furthermore, it is found that the electron added or removed in the neutral clusters does not change the basic geometrical structures.

The lowest energy structures of clusters are presented in Figure 4. All the structures are three-dimensional. They differ from the geometries of the and clusters in Figures 2 and 3. The clusters have or symmetry. The ground-state structure of the cluster is degenerate. The structure lies only 0.03 eV below the structure. The structure can be obtained by putting two capping atoms in the structure. The structure is 0.26 eV more stable than the structure.

Figure 4: The ground-state structures of the clusters. The black ball refers to Ni atoms in the clusters.

The clusters have the magnetic moment of 1.0  due to one unpaired single electron. The magnetic moment is trivial. For the clusters show no magnetic moment because the molecular orbitals are doubly occupied. For their ions, some clusters show magnetism. The and cluster ions are expected to produce zero magnetic moment. But our calculations indicate that they have the magnetic moment of 2 . All the , , and clusters ions including the cation have no magnetic moments. The magnetic moment for the , , and clusters ions is 1 , which is trivial.

To investigate the bonding character of the mixed - clusters in relation to molecular orbitals, we have also examined the density of states in terms of the contributions of the different orbital components (s, p, d) and the frontier orbital (HOMO and LUMO) states. The Mulliken population analyses indicate that some of charge transfer from the Al atoms to the atoms. Here, we take the cluster with the magnetic moment of 2  as an example to illustrate their bonding properties. The orbitals of HOMO and LUMO states of the cluster are shown in Figure 5. For the orbitals of the HOMO state, the cluster binds in the term of and orbital. Likewise, the binging characteristic varies from orbital to orbital and feature between and atoms for the LUMO state.

(a)

(b)

Figure 5: The HOMO and LUMO orbitals of the cluster.

In order to better understand the relative stability of the Ni-Al clusters, we have also investigated the second difference of cluster energies, the energy gaps, and their positive and negative ion clusters. The second difference of cluster energies , which is a sensitive quantity that reflects the relative stability of the mixed clusters. Maximum indicates that the cluster is more stable than its neighboring clusters. The maximums for the and  clusters are found at (see Figure 6). The conclusion is a good agreement with that obtained by Sun et al. [12]. For the clusters, the maximums are at and. It is found from observing Figure 6 that the minimum for the mixed clusters is at . This implies that the clusters are less stable than their neighboring clusters.

(a)

(b)

(c)

(d)

Figure 6: The second difference of the cluster energies, the energy gaps between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), the electron affinities (EA), and the ionization potentials (IP) for clusters.

The maximums of for the pure clusters are also at and. The stability probably can be explained by jellium model [2, 21]. In this model, the nuclei together with the innermost electrons form a positive charged background, whereupon the valence electrons coming from individual atoms are then subjected to this potential. The clusters containing 8, 20, 40, 58… valence electrons correspond to filled electronic shells and exhibit enhanced stability. But the stability is associated not only with closed electronic shells but also, to a lesser extent, with numbers of electrons just above and below the shell closing numbers [5]. The simultaneous stability of the neutral, anion, and cation of a given size can happen only for clusters of elements having more than one itinerant electron per atom. Therefore, the higher stability of and clusters is easily understood. It is found from observing Figure 6 that the mixed and clusters also have high stability. This indicates that the stability of the nickel-doped clusters is determined by the host clusters. But it should be noted that more heteroatoms would change their relative stability, as we have seen in the clusters. Al₅ cluster is less stable compared to other clusters. This is the reason that the clusters are less stable than their neighboring clusters.

The energy gaps as an important function of the cluster size present a characteristic quantity of metal clusters’ chemical activity. Here we define as the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). For the pure clusters, it is found that the Al₂ cluster has a large gap of 3.05 eV. Other gaps change within range from 0.32 to 0.84 eV. For the smaller mixed clusters, their energy gaps change drastically. But they show relatively small fluctuation as the atomic number increases. It is worth noting that the curves display an even/odd alternating pattern as a function of cluster size for the clusters. Even though the and clusters with larger than five atoms do not present the alternating pattern, their change trend is similar as the atomic number increases.

Finally, we have also investigated the positive and negative ion clusters. The electron affinities are the amount of energy released when the cluster obtains an electron from its neutral state. In our research, all the vertical electron affinities (EA) of the clusters are positive, suggesting that the clusters have a tendency to gain an electron, as it is energetically favorable to do so. The vertical electron affinities (EA) change with atomic number, shown as Figure 6. The graph of vertical electron affinities versus atomic number shows that the mixed clusters with one, two, and three nickel atoms have a similar change trend. For the positive ion clusters, similar energies called vertical ionization potentials are calculated. The graph of vertical ionization potentials versus atomic number displays different changes. For the ) clusters with two nickel atoms, the vertical ionization potentials (IP) show an even/odd alteration with the number of aluminum atoms. For the clusters with three nickel atoms, a similar change trend is observed. But, for the mixed clusters including one nickel atom, the property does not exist.

4. Conclusions

We have optimized the geometric structures of the mixed clusters by using first-principles density functional theory. Their ground-state structures are obtained. Most of them are different from those of the host Al clusters. The impurity Ni atoms result in the local structural distortion of the mixed clusters in comparison with the corresponding pure Al clusters. At the same time, we calculated magnetic moment of the mixed cluster neutrals and ions. Some of them present nontrivial magnetic moments. The positive ions, the neutral, theand positive and negative ions, and the negative ion possess the magnetic moments of 2 . and cluster neutrals and positive ions have the magnetic moments of 2  and 3 , respectively. But all the other cluster neutrals and ions do not have any nontrivial magnetic moments. Some of charge transfer from the atoms to the atoms. The local magnetic moment of atoms is mainly from the d orbital. Both the structural effect and chemical bonding are responsible for the magnetism of the mixed clusters.

Acknowledgments

The Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant No. 200320), the Natural Science Foundation of Zhejiang Province (Grant no. Y6100098), and the National Natural Science Foundation of China (Grant no. 11074062) supported this work.

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