- Email: [email protected]
Geometries, stabilities and electronic properties of small Nb-doped gallium clusters: A density functional theory study
Physica B 406 (2011) 3544–3550
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Geometries, stabilities and electronic properties of small Nb-doped gallium clusters: A density functional theory study Shi Shun-Ping a, Cao Yi-Ping a,n, Zhai Ai-Ping a, Li Yang a, Jin Xing-Xing b a b
Department of Opto-electronics, Sichuan University, Chengdu 610065, PR China Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 May 2011 Accepted 8 June 2011 Available online 15 June 2011
The host Gan þ 1 and doped GanNb (n ¼ 1–9) clusters with several spin configurations have been systematically investigated by a relativistic density functional theory (DFT) with the generalized gradient approximation. The optimized equilibrium geometries tend to prefer the close-packed configurations for small Nb-doped gallium clusters up to n ¼9. The average binding energies per atom (Eb/atom), second-order differences of total energies (D2E), fragmentation energies (Ef) and HOMO–LUMO gaps of Gan þ 1 and GanNb (n ¼1–9) clusters are studied. The results indicate the doping of Nb atom in gallium clusters improves the chemical activities. In particular, the clusters with sizes of Ga4Nb and Ga7Nb are found to be more stable with respect to their respective neighbors. Our calculated vertical ionization potentials (VIPs) exhibit an obvious oscillating behavior with the cluster size increasing, except for Ga3 and Ga4Nb, suggesting the Ga3, Ga5, Ga7, GaNb, Ga3Nb, Ga6Nb and Ga8Nb clusters corresponding to the high VIPs. In the case of vertical electron affinities (VEAs) and chemical hardness Z, VEAs are slightly increasing whereas chemical hardness Z decreasing as GanNb cluster size increases. Besides, the doping of Nb atom also brings the decrease as the cluster sizes increases for atomic spin magnetic moments (mb). & 2011 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Gan þ 1 cluster GanNb clusters
1. Introduction Clusters are particularly interesting research fields because the properties of materials can be designed by exploring the enormous variability in the size, shape and composition of clusters, which are distinct from those of individual atoms and molecules or solid material. There are serial interests in transition-metal (TM) clusters, owing to the potential technological applications in photonic devices [1], modeling of compound semiconductors [2] and photoelectron spectroscopy [3]. Therefore, a great number of experimental and theoretical works about pure and doped TM clusters have been performed [4–26]. This is motivated by the fact that a TM atom doped in a small cluster of another metal can strongly change the properties of the host cluster [11–26]. Gallium is a semiconducting element of great importance in microelectronics industry. The properties of cluster could lead to an entirely new range of gallium-based applications in optoelectronic and other devices. For TM gallium clusters, there are enough investigations performed in both theoretical and experimental aspects [11–23,27–36]. For Ga2 dimer, Himmel and
n
Corresponding author. E-mail address: [email protected] (C. Yi-Ping).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.017
Gaertner [27] got the dissociation energy of De ¼145 kJ mol 1 and the force constant of f¼64.8 70.3 N m 1 with the aid of resonance Raman and UV/vis spectroscopy; they also experimentally observed that the wavenumber is 175.4 cm 1. The bond ˚ the frequency of 191 cm 1 and the dissociation length of 2.697 A, energy of 1.13 eV were also reported by Roos et al. [29]. Some theoretical calculations, which are mainly concerned with small gallium clusters [4–10], have been performed. Jones calculated the structures of Gan (n¼10) [4] clusters using the DFT. With increase in cluster size he found transitions from planar to non-planar structures at n¼5, and to states with minimum spin degeneracy at n ¼6, which are similar to our results about pure gallium cluster. Song and Cao [8] investigated the geometrical and electronic structures of Gan (n ¼2–26) clusters based on the generalized gradient approximation for the exchange correlation potential to the DFT. With increasing cluster size, the gallium clusters tend to adopt compact structures in their calculations. Doped gallium clusters [11–23] have been performed to focus a few experimental and theoretical studies. Song and Cao [11] found that the structures, binding energies and HOMO–LUMO gaps of GaxNy (x þyr8) clusters strongly depend on their size and composition. The lowest-energy geometries and electronic-structure properties of GanN (n ¼1–19) clusters have been investigated by Song et al. [15], who indicated that Ga3N, Ga7N and Ga15N
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
exhibit particularly higher stability and the N in GanN clusters is less ionic than that in bulk GaN. Clusters ions of gallium [36] were produced by the sputtering of pure metal targets; the stability of Ganþ clusters were obtained in their experimental, which contains a sufficient number of electrons to fill a shell in the spherical jellium model of cluster electronic structure. Although many studies have been taken on pure gallium clusters and doped gallium clusters, to our knowledge, surely systematic and theoretical investigations on niobium-doped gallium clusters have not been reported so far. The insights of geometrical and electronic structures of small Gan þ 1 and GanNb clusters are essential for further grasp of catalytic reactions. In order to reveal the electron properties of the GanNb (n ¼1–9) clusters, in this paper, we optimize all the possible geometrical structures of GanNb (n¼1–9) clusters by employing DFT approach to find the structural evolution and stability, and combined with pure gallium clusters for comparison. The binding energies (Eb), fragmentation energies (Ef), second-order energies difference (D2E) and electronic properties of Gan þ 1 and GanNb (n ¼1–9) clusters are carried out using DFT with basis sets LANL2DZ(f). We also perform the analysis on HOMO–LUMO gaps to gain insights into the electronic structures for the lowest-energy geometries of Gan þ 1 and GanNb clusters. The VIP and VEA are calculated, and the corresponding chemical hardness Z is also calculated, which reflects the electronic properties. Additionally, the magnetic moments of Gan þ 1 and GanNb clusters are predicted.
2. Computational details All simulations are carried out using the Gaussian 03W package [37], which can perform DFT calculations of atoms, molecules and clusters. The B3LYP method, which incorporates the combination of Becke’s hybrid exchange functional B3 [38] with the Lee et al. (LYP) [39] non-local functional is selected by us. In addition, valence basis sets LANL2DZ(f) [40–42] is chosen to Ga and Nb atoms, in which the core electrons are frozen by relativistic effective core potential (RECP) and the valence electrons are treated by a double-zeta basis set. The B3LYP was successfully used for Fe2, Co2, Ni2, Ru2, Rh2, Pd2, Os2, Ir2 and Pt2 dimers [43] and for GanAl (n¼1–15) clusters [23]. In order to estimate the accuracy of our chosen scheme (B3LYP/LANL2DZ) about representing the GanNb clusters, we first fulfill the test calculations on Ga2 and Nb2 dimers. The calculated bond lengths (Re), harmonic vibrational frequencies (oe) and dissociation energies (De) together with available experimental data are summarized in Table 1. The Ga2 dimer has a ground state with ˚ oe ¼143.3 cm 1 and De ¼1.17 eV, which are in Re ¼2.862 A, excellent agreement with the other theoretical calculations [28,29] and in reasonable agreement with the scarce experimental data (oe ¼165 cm 1, De ¼1.4 eV) [32]. Moreover, our results fit well with the previous investigation of Ref. [31] based on ab initio functional. The results given with 20s17p11d5f2g basis sets from Ref. [29] are also listed in Table 1, from which we can see that 20s17p11d5f2g basis sets generally present smaller dissociation energy with respect to experiment. To our knowledge, there is no experimental bond length for the Ga2 dimer. As for Nb2 dimer, the experimental bond length, vibrational frequency and dissociation energy are 2.08 A˚ [44], 424.9 cm 1 [45] and 5.22 eV [45], respectively. The other theoretical bond length, vibrational frequency ˚ 472 cm 1, and 5.8 eV [46], and dissociation energy are 2.08 A, ˚ 471.4 cm 1 respectively. Our calculation finds values of 2.13 A, and 4.54 eV, respectively, which is in closer agreement with the experimental and theoretical values. Therefore, the B3LYP/ LANL2DZ(f) scheme is reliable and accurate enough for describing the systems involving Ga and Nb atoms.
3545
Table 1 Computed bond lengths (Re), vibrational frequencies (oe) and dissociation energies (De) of dimers (Ga2, Nb2 and GaNb) and available experimental and previous theoretical data.
Ga2 This work Theory Experiment Nb2 This work Theory Experiment GaNb This work
Multiplicity (M)
Bond lengths ˚ (Re, A)
Frequencies (oe, cm 1)
Dissociation energies (De, eV)
3 3 3
2.862 2.746a –
143.3 162a 165c
1.17 1.13b 1.4c
3 3 3
2.129 2.08d 2.08e
471.4 472d 424.9f
4.54 5.8d 5.22f
7
2.622
206.3
1.67
a
Ref. [28]. Ref. [29]. Ref. [32]. d Ref. [44]. e Ref. [45]. f Ref. [47]. b c
1-a0
2-a0
3-a0
4-a0
5-a 0
6-a 0
7-a0
8-a 0
9-a 0
Fig. 1. Lowest energy structures of gallium clusters for each size.
3. Results and discussions 3.1. Lowest-energy structures The calculated lowest-energy structures of the Gan þ 1 and lowlying isomers of the GanNb clusters are shown in Figs. 1 and 2. The isomers are designated by n-a, n-b, n-c and so on, where n is the Ga atom number. For these structures, the point group symmetry, spin multiplicity, geometry property, relative energy DE (relative to the lowest-energy structure) and averaged binding energy Eb are listed in Table 2. The value of Eb is defined for Gan þ 1 and GanNb clusters as Eb ½Gan þ 1 ¼ ððn þ1ÞE½GaEðGan þ 1 ÞÞ=ðn þ 1Þ, Eb ½Gan Nb ¼ ðnE½Ga þ E½NbE½Gan NbÞ=ðn þ1Þ,
ð1Þ
where E is the total energy of relevant system. For the GaNb dimer with CNV symmetry, the corresponding septet state with ˚ oe ¼206.26 cm 1 and De ¼1.67 eV, the binding energy R¼2.62 A, is obtained as 0.83 eV. The most stable structure of the Ga3 cluster is an equilateral triangle structure with bond length of 2.67 A˚ and D3h symmetry.
3546
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
1-a
3-a
2-a
3-b
2-b
3-c
2-c
3-d
2-d
4-a
4-b
4-c
4-d
4-e
4-f
4-g
5-a
5-b
5-c
5-d
6-a
6-b
6-c
6-d
6-e
7-a
7-f
8-d
9-d
7-b
7-g
8-e
9-e
7-c
8-a
9-a
7-d
8-b
9-b
Table 2 Point group symmetries (PG), spin multiplicity, bond lengths, vibrational frequencies, relative energies and average binding energies of Gan þ 1 and GanNb (1–9) clusters. R and R1 denote the shortest Ga–Nb and Ga–Ga bond lengths, respectively; Freq denotes the lowest vibrational frequency of the Gan þ 1 and GanNb equilibrium geometry. Cluster
Isomer
PG
Spin
˚ R (A)
˚ R1 (A)
Freq (cm 1)
DE (eV) Eb (eV)
Ga2
1-a0
Dh
3
–
2.86
143.13
0.00
0.58
GaNb
1-a
CNV
7
2.62
–
206.25
0.00
0.83
Ga3
2-a0
D3h
2
–
2.67
86.31
0.00
0.79
Ga2Nb
2-a 2-b 2-c 2-d
CNV DNH C2V C2V
6 6 4 6
2.60 2.78 2.74 2.58
2.81 2.78 2.88 3.18
245.20 173.30 93.80 86.50
0.8307 1.0276 0.4139 0.00
0.93 0.87 1.07 1.21
Ga4
3-a0
D4h
3
–
2.74
49.29
0.00
1.09
Ga3Nb
3-a 3-b 3-c 3-d
C1 C1 CS CS
5 5 3 3
2.68 2.53 2.58 2.61
2.97 2.99 2.90 2.65
21.30 53.92 54.74 22.67
0.0861 0.00 0.2088 0.4153
1.43 1.45 1.40 1.34
Ga5
4-a0
Cs
2
–
2.82
22.10
0.00
1.06
Ga4Nb
4-a 4-b 4-c 4-d 4-e 4-f 4-g
C3V C3V CS CS C2V C1 C2V
4 6 2 4 4 4 2
2.70 2.70 2.50 2.66 2.66 2.66 2.50
2.89 2.97 3.03 2.87 2.87 2.87 2.50
29.32 12.38 48.06 57.31 57.31 57.46 48.01
0.6063 1.6139 0.2944 0.0006 0.0005 0.00 0.2944
1.52 1.32 1.58 1.64 1.63 1.65 1.57
Ga6
5-a0
C2
1
–
2.66
35.51
0.00
1.33
Ga5Nb
5-a 5-b 5-c 5-d
C1 C1 C1 C2V
1 7 5 1
2.60 2.58 2.74 2.52
2.68 2.70 2.69 2.83
23.80 39.05 36.21 31.99
0.4989 0.6202 0.00 0.0484
1.48 1.56 1.46 1.55
Ga7
6-a0
C1
2
–
2.61
20.72
0.00
1.44
Ga6Nb
6-a 6-b 6-c 6-d 6-e
CS C1 C1 C1 C2
4 6 4 4 2
2.58 2.76 2.60 2.67 2.58
2.62 2.69 2.65 2.76 2.67
34.73 17.81 23.88 10.72 36.84
0.2029 0.7027 0.1916 0.6565 0.00
1.62 1.54 1.52 1.55 1.65
7-e
8-c
9-c
9-f
Ga8
7-a0
D2h
1
-
2.62
30.93
0.00
1.49
Ga7Nb
7-a 7-b 7-c 7-d 7-e 7-f 7-g
C1 C1 C3 C1 C1 C1 C1
1 1 1 1 5 1 5
2.61 2.58 2.60 2.55 2.67 2.53 2.54
3.21 2.77 2.79 2.65 2.63 2.58 2.70
14.91 32.01 18.89 31.41 22.90 9.60 21.94
0.00 0.4547 0.2831 0.4102 0.7836 0.6074 1.1589
1.72 1.66 1.68 1.67 1.62 1.64 1.57
Fig. 2. Lowest energy structures and low-lying isomers for GanNb (n¼ 1–9) clusters.
The linear and triangular configurations are selected as the initial structures to optimize for Ga2Nb clusters. The possible geometries of the Ga2Nb clusters maintaining CNV, DNH and C2V symmetries are considered. The C2V structure is lower in energy than the linear CNV and DNH isomers, which indicates that the linear isomers are less stable than the triangle structure, and the most stable structure of Ga2Nb clusters is 2-d, which is an isosceles triangular structure with the Ga–Nb bond length, the Ga–Ga bond ˚ 3.178 A˚ and 1.21 eV, length and the binding energy are 2.575 A, respectively. The binding energies of 0.28, 0.34 and 0.14 eV are more stable than the other partners. Ga4 is a square with D4h symmetry, corresponding to electronic state is 3Au’’. Our result is excellent with the previous theory calculation of Ref. [23]. We obtained four isomers in geometry optimization of Ga3Nb clusters, and the first three-dimensional (3D) structure occurs at n ¼3. But the planar structures of Ga3Nb clusters are proved not to be equilibrium geometries. A Nb atom is capped the top of the Ga3 triangle, which result in a tetrahedral configuration; the most stable structure (3-b) of Ga3Nb clusters is a spin quintuplet state with the binding energy of 1.45 eV, which is 0.36 eV lower in stability with respective to host Ga4 clusters; the quintuplet Ga3Nb clusters with the Ga–Nb bond length of
Ga9
8-a0
C1
2
–
2.58
18.12
0.00
1.47
Ga8Nb
8-a 8-b 8-c 8-d 8-e
C1 C1 C1 C1 C1
6 2 2 6 2
2.64 2.52 2.61 2.70 2.61
2.67 2.63 2.66 6.62 2.62
35.01 12.66 30.59 26.30 21.72
0.7389 0.3070 0.00 0.6819 0.2152
1.62 1.67 1.70 1.63 1.68
Ga10
9-a0
C2
1
–
2.56
37.38
0.00
1.52
Ga9Nb
9-a 9-b 9-c 9-d 9-e 9-f
C1 C1 C1 C1 Cs Cs
1 5 3 1 5 1
2.54 2.55 2.60 2.62 2.56 2.63
2.57 2.66 2.66 2.72 2.67 2.58
19.97 14.67 29.76 12.46 9.87 13.63
0.5042 0.0801 0.00 0.1990 0.0800 0.2163
1.67 1.71 1.72 1.70 1.71 1.69
˚ corresponding to 2.53 A˚ and the Ga–Ga bond length of 2.53 A, electronic state, is 5A. In addition, other three isomers, 3-a, 3-c and 3-d, which have relative energies of 0.0861, 0.2088 and 0.4153 eV, respectively, are obtained in our optimization. It is less stable than the 3-b.
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
to the ground-state structure 9-c constructed on the basis of the 8-c. The 9-a, 9-d and 9-f isomers correspond to singlet spin state, and for 9-b and 9-e isomers, the quintet state is favored. The lowest-energy isomer (9-c) with binding energy of 1.72 eV favors triplet spin multiplicity. The similar isomers 9-b and 9-e have identical spin multiplicity and structures, which are nine gallium atoms encompassing niobium atom, but their relative energy is different. From the above discussion, it is remarkable that the lowest energy configurations of gallium and niobium-doped gallium clusters favor 3D structures and prefer the low spin multiplicity. The results of the present calculation show interesting patterns. It is found that the lowest-energy structures of Nb substituted Gan þ 1 cluster are different for the lowest-energy structures of pure Gan þ 1 cluster, which are dominant growth patterns. This may be due to the fact that the Nb has a closed-shell electronic configuration 5s, and the single 4p outermost valence electron corresponds to the Ga atom. The smaller the clusters are, the more obvious the structural distortion is.
3.2. Stabilities The understanding of the variation in the formation energy properties of clusters is important for catalysis and can provide a good way to show the relative local stability of small clusters [47]. So the properties of Ga1–10 and Ga1–9Nb clusters will be reflected by the average binding energies (Eb), second-order differences of total energies (D2E) and fragmentation energies (Ef). We will focus on the changes of stabilities after doping the Nb atom in gallium clusters in this section. We plot the average binding energies per atom (Eb) of the lowest-energy structures of GanNb clusters as function of cluster size in Fig. 3. The corresponding binding energy of Gan þ 1 cluster are also plotted. From Fig. 3 one can see that the atomic average binding energies of Gan þ 1 clusters have an increasing tendency with the clusters size growing, and the mutative behaviors of GanNb clusters are close to that of bare gallium clusters. But the average binding energies of the most stable GanNb cluster are higher than those of the pure Gan þ 1 cluster. It indicates that the doped Nb atom in the Gan clusters contributes to increase the stabilities of the gallium framework. Especially, for n ¼1–4, the Eb increases rapidly from 0.83 eV for GaNb to 1.65 eV for Ga4Nb, which corresponds to the structure transition from two to three dimension. The Eb increases gradually in the range n¼ 5–9, in which the rate of increase becomes weak. The results are similar to N-doped gallium clusters [15]. It is also found that the binding energy rise monotonically with cluster size and contains two minor bumps at n¼5 and 8, which indicate that the Ga5Nb and Ga8Nb clusters have weaker stabilities than its respective neighbors. 1.8 Binding Enenrgy per atom Eb (eV)
The structure of Ga5 we obtained is considered as a buckled structure with CS symmetry, which is excellent with Ref. [8]. A variety of possible structures are considered initially for the Ga4Nb clusters. The computational results reveal that the well-defined 4-f isomer has higher total bonding energy and cannot be competitive with the other isomers optimized in stability. The structures ((4-d), (4-e) and (4-f)) with CS, C2V and C1 symmetry are similar to the Ga5 [6] configuration, but the Ga–Nb bond length, Ga–Ga bond length and binding energy are little different. The structures with the four-coordinated Nb atoms located on the Ga–Ga–Ga triangle ring, which are 0.13 eV, 0.33 eV, 0.07 eV and 0.08 eV of 4-a, 4-b, 4-c and 4-g are less stable in binding energy to the 4-f one. The lowest-energy structure for Ga6 is a prism structure (C2) with binding energy of 1.33 eV as the ground state. The structures obtained for Ga5Nb clusters have C1 and C2V symmetries. One of these structures with C1 (5-b) symmetry is the most stable. The Ga–Nb bond length and Ga–Ga bond length in it are R ¼2.58 and ˚ respectively, from our calculations. The corresponding R1 ¼2.70 A, binding energy and electronic state are 1.56 eV and 7A. The (5-a, 5-c) isomers with the same symmetry of C1 hold different spin multiplicities and binding energies to 5-b geometry. The structure of 5-d isomer with C2V symmetry and spin multiplicity (PG¼1) is similar to the Ga6 configuration [6]. All Ga5Nb isomers, whose binding energies are in the range 1.46–1.56 eV, are less stable than the host Ga6 clusters. In the case of n¼7, the pure Ga7 adopted the distorted capped trigonal prism with binding energy 1.44 eV. As shown in 6-a, 6-b, 6-c, 6-d and 6-e in Fig. 2, the most stable structure of Ga6Nb is pentagonal bipyramid (PBP) configuration with C2 symmetry as ground-state structure. It can be viewed as the trigonal prism of Ga6 with one open edge capped by an additional Nb atom. The binding energy of the most stable (6-e) isomer is 0.21 eV, which is higher than the C1 ground state of Ga7. In addition, the stable 6-b structure is yielded when one Nb is directly capped on the boatlike Ga6 cluster. As mentioned above, the other isomers can be optimized to be a minimum, and the corresponding bond lengths, frequencies and binding energies are listed in Table 2. The lowest-energy structure of Ga8 cluster corresponds to one capped octahedron geometry with D2h symmetry. The bonding energy of the ground-state structure is 1.49 eV. Several isomers exist in the Ga7Nb clusters are shown in Fig. 2, in which the Nb atom is located with the largest number of nearest neighbored Ga ˚ and Ga–Ga bond length atoms. Ga–Nb bond length (2.605 A) ˚ of the most stable structure (7-a) for Ga7Nb clusters are (3.205 A) optimized in the present work. This structure has binding energy as high as 1.72 eV. The results confirm our above-mentioned prediction that the Nb-centered structure is more stable than the Ga-capped structure. Other isomers, 7-b, 7-c, 7-d, 7-e, 7-f and 7-g, are 0.4547, 0.2831, 0.4102, 0.7836, 0.6074 and 1.1589 eV, respectively, of the relative energies are less stable than 7-a. For Ga9 cluster, the lowest-energy structure is a pentagonal arrangement of atoms with C1 symmetry. A niobium-doped 8-atom gallium cluster with 2.60 A˚ of the Ga–Ga bond length and 2.66 A˚ of the Ga–Nb length bond as the ground state, which is the most stable structure, are shown in Fig. 2 (8-c). For the case of Ga8Nb, the edge capped cube (8-c) with doublet spin state is 1.70 eV of binding energy, which is more stable than the face capped one with doublet state 8-b and 8-e isomers. Isomers (8-a, 8-d) have the sextet state. The binding energies of 8-a, 8-b, 8-d and 8-e are 1.62, 1.67, 1.63 and 1.68 eV, which are lower than 0.08, 0.03, 0.07 and 0.02 eV of the most stable structure (8-c), respectively. The ground-state structure obtained for Ga10 has C2 symmetry and it can be built from Ga9 wedges. Our structure for Ga10 is in agreement with predicted by Bin [8]. As for the Ga9Nb cluster, the 9-a, 9-b, 9-c, 9-d, 9-e and 9-f isomers are obtained by one Ga atom on different sites of Ga8Nb cluster. Only 0.02 eV more stable relative
3547
1.6 1.4 1.2 Gan+1 GanNb
1.0 0.8 0.6 0
2
4 6 Cluster size n
8
10
Fig. 3. Binding energies per atom (Eb/atom) of the Gan þ 1 and GanNb clusters with n¼1–9.
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
The second-order differences of total energies (D2E) for the lowest-energy structures of Gan þ 1 and GanNb clusters with sizes of n ¼1–9 are also evaluated and plotted as a function of cluster size in Fig. 4. The D2E is often compared directly with the relative abundances determined in mass spectroscopy experiments. The values of D2E are defined for Gan þ 1 and GanNb clusters:
D2 E½Gan þ 1 ¼ E½Gan þ 2 þ E½Gan 2E½Gan þ 1 , D2 E½Gan Nb ¼ E½Gan þ 1 Nb þ E½Gan1 Nb þ 2E½Gan Nb,
ð2Þ
where E is the total energy of relevant system. As shown in Fig. 4, the Gan þ 1 clusters stabilities exhibit pronounced odd–even alternations. However, it is clearly found that the doping impurity Nb atom makes the stable pattern of the host cluster differently from n ¼1 to n¼6, the changes of these values are very small. The odd– even alternations are close to pure Gan þ 1 clusters when n ¼6, 7, and 8. In additional, Ga7Nb cluster with positive D2E value has stronger stabilities than its respective neighbors, it is found that the clusters with n ¼7 are particularly stable. This conclusion is in excellent agreement with pure Gan þ 1 cluster [8]. In Al-doped gallium clusters, n ¼7 and 13 are also the highest of the secondorder differences of total energies (D2E) [23]. To further confirm the relative stabilities of the Ga1–10 and Ga1–9Nb clusters, we evaluate the fragmentation energies, which are plotted in Fig. 5. Fragmentation energy (Ef) expresses the thermodynamic stability of cluster, and the greater value indicates the more stable cluster. The values of fragmentation energies are estimated with equations: Ef ðGan þ 1 Þ ¼ EðGan Þ þEðGaÞEðGan þ 1 Þ, Ef ðGan NbÞ ¼ EðGan1 NbÞ þEðGaÞEðGan NbÞ,
ð3Þ
where E is the total energy of relevant system. From Fig. 5, one can see that the lower fragmentation channel corresponds to the 1.0
Δ2E (eV)
0.5 0.0 -0.5 Gan+1 GanNb
-1.0 -1.5
Gan+1 GanNb
2.5 2.0 1.5 1.0 0.5 0.0 0
2
4 6 Cluster size n
8
10
Fig. 6. HOMO-LUMO gaps for the studied Gan þ 1 and GanNb clusters with n¼1–9.
loss of the Ga atom. The Ga4Nb and Ga7Nb clusters with high fragmentation energies suggest their higher stabilities again. This conclusion is consistent with the results from second-order differences energies mentioned above. But for n ¼5 and 8, the values of Ef ¼1.18 and 1.59 eV, which show that removing a niobium or gallium atom from the cluster needs less energy, namely the thermodynamic stability of Ga5Nb and Ga8Nb clusters are poor. The gaps between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are generally considered as a useful quantity to evaluate the stability of one system, which is a useful parameter to study the metal– metal transition for clusters [48]. Fig. 6 shows us the HOMO–LUMO gaps for the most stable geometries of the studied Gan þ 1 and GanNb clusters, from which one can see that HOMO–LUMO gaps of Gan þ 1 and GanNb clusters exhibit similar behavior, that is, the HOMO–LUMO gaps increase in principle with the increase in cluster size from n¼1 to n¼3, but the HOMO–LUMO gaps decrease with the increase in cluster size from n¼3 to n¼9. The GanNb (n¼5–9) clusters have lower HOMO–LOMO energy gaps than other clusters (n¼1–4), implying that they have weaker chemical reactivity. No clear odd–even oscillations are found for the HOMO–LUMO gaps of Gan þ 1 and GanNb clusters. This phenomenon is also presented in other doped clusters, like GanN systems [11]. 3.3. Electronic properties
-2.0 0
1
2
3
4 5 6 Cluster size n
7
8
9
Fig. 4. Second-order difference of total energies of Gan þ 1 and GanNb clusters with n¼ 1–9.
2.8 Fragmentation energy Ef (eV)
3.0 HOMO-LUMO gap(eV)
3548
2.6
VIP½Gan þ 1 ¼ E½Ganþþ 1 E½Gan þ 1 ,
Gan+1 GanNb
2.4
In cluster science, ionization potential and electron affinity are used as important properties to study the change in electronic structure of the cluster size. Based on the most stable configurations, the vertical ionization potential (VIP) and electron affinity (VEA) are calculated for the lowest-energy structures. The VIP and VEA are evaluated with þ
VIP½Gan Nb ¼ E½Gan Nb E½Gan Nb,
2.2
VEA½Gan þ 1 ¼ E½Gan þ 1 E½Gan þ 1 ,
2.0
VEA½Gan Nb ¼ E½Gan NbE½Gan Nb ,
ð4Þ
1.8 1.6 1.4 1.2 1.0 0.8 0
2
6 4 Cluster size n
8
10
Fig. 5. Fragmentation energies (Ef) of ground stable structures of the Gan þ 1 and GanNb clusters with n¼ 1–9.
where E is the total energyof corresponding systems based on the ground-state structures of Gan þ 1 and GanNb clusters. The calculations þ of single point energies for charged systems (Ganþþ 1 , Ga n þ 1 , GanNb and GanNb ) are also performed at B3LYP/LANL2DZ(f) level of theory and considered the low spin multiplicities. With the VIP and VEA, the chemical hardness Z, which is established as an electronic quantity that in many cases may be used to characterize the relative stability of molecules and aggregate through the principle of maximum hardness proposed by pearson [49], can be easily obtained with Z ¼VIP–VEP under the finite difference approximation [50].
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
Magnetic moment per atom (µb)
7.0 Gan+1 GanNb
6.8
VIP (eV)
6.6 6.4 6.2 6.0 5.8 Gan+1 GanNb
VEA (eV)
2.0
Gan+1 GanNb
2.5 2.0 1.5 1.0 0.5 0.0 2
4 6 Cluster size n
8
10
Fig. 8. Magnetic moments (m/atom) vary with cluster sizes.
1.8 1.6
and GanNb clusters are plotted as a function of cluster sizes in Fig. 8. In Ganþ 1 system, from Ga2 to Ga5, the spin magnetic moments decrease. Ga6, Ga8 and Ga10 clusters exhibit nonmagnetic ground state. As for GanNb clusters, the GaNb dimer has the highest atomic magnetic moment of 3.0mb/atom in all studied sizes, and the atomic magnetic moments (mb/atom) decrease as the cluster sizes increase. There is a sharp decrease from n¼1 (3.0mb/atom) to n¼4 (0.6mb/atom) in the curve of magnetic moments. Staring from n ¼7, the moments gradually increase as the cluster sizes increase. When the number of Ga atoms is 7, the cluster is a closed-shell system, whose a and b spin orbitals degenerate, so the corresponding magnetic moment is zero.
1.4 1.2 1.0
Hardness (eV)
3.0
0
2.2
3549
0.8 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6
Gan+1 GanNb
4. Conclusions
0
2
4 6 Cluster size n
8
10
Fig. 7. Vertical ionization potentials (VIP), vertical electron affinities (VEA) and chemical hardness (Z) of the Gan þ 1 and GanNb clusters with n ¼1–9.
These values of VIP, VEA and Z are plotted in Fig. 7, from which one can see that VIPs show an obvious oscillating behavior with the cluster size increasing, except for Ga3 in Ganþ 1 cluster and Ga4Nb in GanNb cluster. However, it is clearly found that the doping impurity atom makes the stable pattern of the host cluster contrary. This suggests that the clusters with a closed electronic shell are electronically stabilized and show high VIPs, and the maxima are found at GaNb configurations from GanNb cluster. The VEAs of the most stable structures of Ganþ 1 and GanNb (n¼1–9) clusters are also investigated as shown in Fig. 7. We can see that the values of VEAs increase with the cluster size increasing. In addition Ga4Nb and Ga7Nb have lower values than their respective neighbors, indicating the lower stabilities of neighbors cluster. In the previous study, Zhao et al. [7] also found that the values of VEAs increase with the cluster size increasing using B3LYP method for GanN clusters. As the cluster size increases, the chemical hardness Z of the Gan þ 1 cluster first drops rapidly in small size range, i.e. n¼ 2–4. There is a noticeable rise of chemical hardness Z from Ga4 to Ga5. From n¼4, the chemical hardness Z gradually increases with cluster size with weak even–odd oscillations. But as for chemical hardness Z of GanNb clusters, the values decrease as cluster sizes increase, except for Ga6Nb. 3.4. Magnetisms For the ground-state structures of Gan þ 1 and GanNb clusters (n¼ 1–9), the atomic spin magnetic moments of the studied Gan þ 1
In this paper, the geometric structures, stabilities and electronic properties of Gan þ 1 and GanNb (n ¼1–9) clusters are investigated by DFT with the generalized gradient approximation. By optimizing abundant possible geometries with different spin multiplicities, low-lying geometries are obtained. The relative stabilities of small gallium and Nb-doped gallium clusters are determined by estimating the average binding energies per atom (Eb/atom), second-order differences of total energies (D2E), fragmentation energies (Ef), HOMO–LOMO gaps, vertical ionization potential (VIP) and electron affinity (VEA). The atomic spin magnetic moment (mb) is also analyzed. All results are summarized as follows: (i) A lot of initial configurations are optimized to obtain the most stable structures for Gan þ 1 and GanNb (n¼1–9) clusters. The optimized geometries reveal that the lowest energy configurations of Nb-doped gallium clusters structures are different from pure gallium clusters for n ¼1–9. (ii) The binding energy, second-order difference of energy, fragmentation energy and HOMO–LUMO energy gap are studied as a function of cluster size for each most stable ground state Gan þ 1 and GanNb (n¼1–9) clusters. The calculated results the mutative behaviors of GanNb are close to that of bare gallium clusters, but the average binding energies of the most stable GanNb clusters are higher than those of the pure Gan þ 1 clusters. The D2E of Gan þ 1 clusters stabilities exhibit pronounced odd–even alternations. The Ga4Nb and Ga7Nb clusters with positive D2E values have stronger stabilities than their respective neighbors. The fragmentation energies show an obvious oscillating behavior with the cluster size increasing, except for Ga3Nb and Ga6Nb. In addition, the HOMO–LUMO gaps of Gan þ 1 and GanNb clusters exhibit similar behavior, and the
3550
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
HOMO–LUMO gaps of GanNb (n ¼1–9) are in the range1.6 2.9 eV, indicating the semiconductor-like behaviors of small GanNb clusters with sizes nr9. (iii) VIPs show an obvious oscillating behavior with the cluster size increasing, except for Ga3 and Ga4Nb. But the doping impurity atom makes the stable pattern of the host cluster contrary. The values of VEAs increase with the cluster size increase. The chemical hardness Z with weak even–odd oscillations for GanNb cluster whereas the values decrease as cluster sizes increase for GanNb cluster. Ga6, Ga8 and Ga10 clusters exhibit nonmagnetic ground state and the doping of Nb atom brings the reduction as the cluster sizes increases, in which there is a sharp decrease from n ¼1 (3.0mb/atom) to n ¼4 (0.6mb/atom) in the curve of magnetic moments.
Acknowledgment This work was supported by National Science and Technology Major Project of China under Grant no. 2009ZX02204-008. We are very grateful to Yanfang Li for many helpful discussions. Our thanks are also to Dr Jiguang Du for practical help. References [1] A. Podhorodecki, M. Banski, J. Misiewicz, C. Lecerf, P. Marie, J. Cardin, X. Portier, J. Appl. Phys. 108 (2010) 063535. [2] A. Karsten, N. Kai, N. Janne, K. Antti, Phys. Rev. B 66 (2002) 0305205. [3] C.J. Zhang, W.H. Jia, Chem. Phys. 327 (2006) 144. [4] R.O. Jones, J. Chem. Phys. 99 (1993) 1194. [5] N. Gaston, A.J. Parker, Chem. Phys. Lett. 11 (2010) 006. [6] X.G. Gong, Phys. Lett. A 166 (1992) 369. [7] Y. Zhao, W.G. Xu, Q.S. Li, J. Phys. Chem. A 108 (2004) 7448. [8] B. Song, P.L. Cao, J. Chem. Phys. 123 (2005) 144312. [9] S. Krishnamurty, K. Joshi, S. Zorriasatein, D.G. Kanhere, J. Chem. Phys. 127 (2007) 054308. ¨ [10] A. Schnepf, H. Schnockel, Angew. Chem. Int. Ed. 41 (2002) 3532. [11] B. Song, P.L. Cao, Phys. Lett. A 328 (2004) 364. [12] B. Song, P.L. Cao, B.X. Li, Phys. Lett. A 315 (2003) 308. [13] A.K. Kandalam, M.A. Blanco, R. Pandey, J. Phys. Chem. B 106 (2002) 1945. [14] J.J. Zhao, B.L. Wang, X.L. Zhou, X.S. Chen, W. Lu, Chem. Phys. Lett. 422 (2006) 170. [15] B. Song, C.H. Yao, P.L. Cao, Phys. Rev. B 74 (2006) 035306. [16] J.J. BelBruno, Heterocyl. Chem. 11 (2000) 281. [17] A. Costales, A.K. Kandalam, R. Pandey, J. Phys. Chem. B 107 (2003) 4508. [18] P. Karamanis, C. Pouchan, Phys. Rev. A 77 (2008) 013201. [19] M.A. Al-Laham, K. Raghavachari, Chem. Phys. Lett. 187 (1991) 13.
[20] P. Chandrachud, K. Joshi, D.G. Kanhere, Phys. Rev. B 76 (2007) 235423. [21] E.L. Li, X.M. Luo, W. Shi, X.W. Wang, J. Mol. Struct. 723 (2005) 79. [22] J.J. BelBruno, E. Sanville, A. Burnin, A.K. Muhangi, A. Malyutin, Chem. Phys. Lett. 478 (2009) 132. [23] L. Guo, Comp. Mater. Sci. 45 (2009) 951. [24] J.G. Du, X.Y. Sun, J. Chen, G. Jiang, J. Phys. B: At. Mol. Phys. 43 (2010) 205103. [25] J.X. Yang, C.F. Wei, J.J. Guo, Physica B 405 (2010) 4892. [26] Y.F. Li, X.Y. Kuang, S.J. Wang, Y.R. Zhao, J. Phys. Chem. A 114 (2010) 11691. [27] H.J. Himmel, B. Gaertner, Chem. Eur. J. 10 (2004) 5936. [28] K.K. Das, J. Phys., B. At., Mol. Opt. Phys. 30 (1997) 803. ˚ Malmqvist, V. Veryazov, P.O. Widmark, J. Phys. Chem. [29] B.O. Roos, R. Lindh, P.A. A 108 (2004) 2851. ¨ ¨ [30] O. Zuger, U. Durig, Phys. Rev. B 46 (1992) 7319. [31] S.A. Hameed, JKAU: Sci. 21 (2009) 227. [32] K. Huber, G. Herzberg, Constants of Diatomic Molecule, Van Nostrand Reinhold, New York, 1979. [33] K. Balasubramanian, J. Phys. Chem. 94 (1990) 7764. [34] G.A. Breaux, R.C. Benirschke, T. Sugai, B.S. Kinnear, M.F. Jarrold, Phys. Rev. Lett. 97 (2003) 215508. ¨ W. Eberhardt, J. Chem. Phys. 100 (1994) 995. [35] C.Y. Cha, G. Gantefor, [36] F.L. King, M.M. Ross, Chem. Phys. Lett. 164 (1989) 131. [37] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery Jr, T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, J.A. Pople, Gaussian 03, Revision B 02, Gaussian, Inc., Pittsburgh PA, 2003. [38] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [39] L. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [40] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [41] W.R. Wadt, P.J. Hay, J. Chem. Phys. 82 (1985) 284. [42] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [43] J.G. Du, X.Y. Sun, H.Y. Wang, Int. J. Quant. Chem. 108 (2008) 1505. [44] L. Goodwin, D.R. Salahub, Phys. Rev. A 47 (1993) 774. [45] D.A. Hales, L. Lian, P.B. Armentrout, Int. J. Mass Spectrom. Ion Process. 102 (1990) 269. [46] A.M. James, P. Kowalczyk, R. Fournier, B. Simard, J. Chem. Phys. 99 (1993) 8504. [47] D.B. Zhang, J. Shen, J. Chem. Phys. 120 (2004) 5104. [48] M. Salazar-Villanueva, P.H. Herna´ndez Tejeda, U. Pal, J.F. Rivas-Silva, J.I. Rodrı´guez Mora, J.A. Ascencio, J. Phys. Chem. A 110 (2006) 10274. [49] R.G. Pearson, Chemical Hardness: Applications from Molecules to Solids, Wiley-VCH, Weinheim, 1997. [50] P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Geometries, stabilities and electronic properties of small Nb-doped gallium clusters: A density functional theory study Shi Shun-Ping a, Cao Yi-Ping a,n, Zhai Ai-Ping a, Li Yang a, Jin Xing-Xing b a b
Department of Opto-electronics, Sichuan University, Chengdu 610065, PR China Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, PR China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 9 May 2011 Accepted 8 June 2011 Available online 15 June 2011
The host Gan þ 1 and doped GanNb (n ¼ 1–9) clusters with several spin configurations have been systematically investigated by a relativistic density functional theory (DFT) with the generalized gradient approximation. The optimized equilibrium geometries tend to prefer the close-packed configurations for small Nb-doped gallium clusters up to n ¼9. The average binding energies per atom (Eb/atom), second-order differences of total energies (D2E), fragmentation energies (Ef) and HOMO–LUMO gaps of Gan þ 1 and GanNb (n ¼1–9) clusters are studied. The results indicate the doping of Nb atom in gallium clusters improves the chemical activities. In particular, the clusters with sizes of Ga4Nb and Ga7Nb are found to be more stable with respect to their respective neighbors. Our calculated vertical ionization potentials (VIPs) exhibit an obvious oscillating behavior with the cluster size increasing, except for Ga3 and Ga4Nb, suggesting the Ga3, Ga5, Ga7, GaNb, Ga3Nb, Ga6Nb and Ga8Nb clusters corresponding to the high VIPs. In the case of vertical electron affinities (VEAs) and chemical hardness Z, VEAs are slightly increasing whereas chemical hardness Z decreasing as GanNb cluster size increases. Besides, the doping of Nb atom also brings the decrease as the cluster sizes increases for atomic spin magnetic moments (mb). & 2011 Elsevier B.V. All rights reserved.
Keywords: Density functional theory Gan þ 1 cluster GanNb clusters
1. Introduction Clusters are particularly interesting research fields because the properties of materials can be designed by exploring the enormous variability in the size, shape and composition of clusters, which are distinct from those of individual atoms and molecules or solid material. There are serial interests in transition-metal (TM) clusters, owing to the potential technological applications in photonic devices [1], modeling of compound semiconductors [2] and photoelectron spectroscopy [3]. Therefore, a great number of experimental and theoretical works about pure and doped TM clusters have been performed [4–26]. This is motivated by the fact that a TM atom doped in a small cluster of another metal can strongly change the properties of the host cluster [11–26]. Gallium is a semiconducting element of great importance in microelectronics industry. The properties of cluster could lead to an entirely new range of gallium-based applications in optoelectronic and other devices. For TM gallium clusters, there are enough investigations performed in both theoretical and experimental aspects [11–23,27–36]. For Ga2 dimer, Himmel and
n
Corresponding author. E-mail address: [email protected] (C. Yi-Ping).
0921-4526/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2011.06.017
Gaertner [27] got the dissociation energy of De ¼145 kJ mol 1 and the force constant of f¼64.8 70.3 N m 1 with the aid of resonance Raman and UV/vis spectroscopy; they also experimentally observed that the wavenumber is 175.4 cm 1. The bond ˚ the frequency of 191 cm 1 and the dissociation length of 2.697 A, energy of 1.13 eV were also reported by Roos et al. [29]. Some theoretical calculations, which are mainly concerned with small gallium clusters [4–10], have been performed. Jones calculated the structures of Gan (n¼10) [4] clusters using the DFT. With increase in cluster size he found transitions from planar to non-planar structures at n¼5, and to states with minimum spin degeneracy at n ¼6, which are similar to our results about pure gallium cluster. Song and Cao [8] investigated the geometrical and electronic structures of Gan (n ¼2–26) clusters based on the generalized gradient approximation for the exchange correlation potential to the DFT. With increasing cluster size, the gallium clusters tend to adopt compact structures in their calculations. Doped gallium clusters [11–23] have been performed to focus a few experimental and theoretical studies. Song and Cao [11] found that the structures, binding energies and HOMO–LUMO gaps of GaxNy (x þyr8) clusters strongly depend on their size and composition. The lowest-energy geometries and electronic-structure properties of GanN (n ¼1–19) clusters have been investigated by Song et al. [15], who indicated that Ga3N, Ga7N and Ga15N
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
exhibit particularly higher stability and the N in GanN clusters is less ionic than that in bulk GaN. Clusters ions of gallium [36] were produced by the sputtering of pure metal targets; the stability of Ganþ clusters were obtained in their experimental, which contains a sufficient number of electrons to fill a shell in the spherical jellium model of cluster electronic structure. Although many studies have been taken on pure gallium clusters and doped gallium clusters, to our knowledge, surely systematic and theoretical investigations on niobium-doped gallium clusters have not been reported so far. The insights of geometrical and electronic structures of small Gan þ 1 and GanNb clusters are essential for further grasp of catalytic reactions. In order to reveal the electron properties of the GanNb (n ¼1–9) clusters, in this paper, we optimize all the possible geometrical structures of GanNb (n¼1–9) clusters by employing DFT approach to find the structural evolution and stability, and combined with pure gallium clusters for comparison. The binding energies (Eb), fragmentation energies (Ef), second-order energies difference (D2E) and electronic properties of Gan þ 1 and GanNb (n ¼1–9) clusters are carried out using DFT with basis sets LANL2DZ(f). We also perform the analysis on HOMO–LUMO gaps to gain insights into the electronic structures for the lowest-energy geometries of Gan þ 1 and GanNb clusters. The VIP and VEA are calculated, and the corresponding chemical hardness Z is also calculated, which reflects the electronic properties. Additionally, the magnetic moments of Gan þ 1 and GanNb clusters are predicted.
2. Computational details All simulations are carried out using the Gaussian 03W package [37], which can perform DFT calculations of atoms, molecules and clusters. The B3LYP method, which incorporates the combination of Becke’s hybrid exchange functional B3 [38] with the Lee et al. (LYP) [39] non-local functional is selected by us. In addition, valence basis sets LANL2DZ(f) [40–42] is chosen to Ga and Nb atoms, in which the core electrons are frozen by relativistic effective core potential (RECP) and the valence electrons are treated by a double-zeta basis set. The B3LYP was successfully used for Fe2, Co2, Ni2, Ru2, Rh2, Pd2, Os2, Ir2 and Pt2 dimers [43] and for GanAl (n¼1–15) clusters [23]. In order to estimate the accuracy of our chosen scheme (B3LYP/LANL2DZ) about representing the GanNb clusters, we first fulfill the test calculations on Ga2 and Nb2 dimers. The calculated bond lengths (Re), harmonic vibrational frequencies (oe) and dissociation energies (De) together with available experimental data are summarized in Table 1. The Ga2 dimer has a ground state with ˚ oe ¼143.3 cm 1 and De ¼1.17 eV, which are in Re ¼2.862 A, excellent agreement with the other theoretical calculations [28,29] and in reasonable agreement with the scarce experimental data (oe ¼165 cm 1, De ¼1.4 eV) [32]. Moreover, our results fit well with the previous investigation of Ref. [31] based on ab initio functional. The results given with 20s17p11d5f2g basis sets from Ref. [29] are also listed in Table 1, from which we can see that 20s17p11d5f2g basis sets generally present smaller dissociation energy with respect to experiment. To our knowledge, there is no experimental bond length for the Ga2 dimer. As for Nb2 dimer, the experimental bond length, vibrational frequency and dissociation energy are 2.08 A˚ [44], 424.9 cm 1 [45] and 5.22 eV [45], respectively. The other theoretical bond length, vibrational frequency ˚ 472 cm 1, and 5.8 eV [46], and dissociation energy are 2.08 A, ˚ 471.4 cm 1 respectively. Our calculation finds values of 2.13 A, and 4.54 eV, respectively, which is in closer agreement with the experimental and theoretical values. Therefore, the B3LYP/ LANL2DZ(f) scheme is reliable and accurate enough for describing the systems involving Ga and Nb atoms.
3545
Table 1 Computed bond lengths (Re), vibrational frequencies (oe) and dissociation energies (De) of dimers (Ga2, Nb2 and GaNb) and available experimental and previous theoretical data.
Ga2 This work Theory Experiment Nb2 This work Theory Experiment GaNb This work
Multiplicity (M)
Bond lengths ˚ (Re, A)
Frequencies (oe, cm 1)
Dissociation energies (De, eV)
3 3 3
2.862 2.746a –
143.3 162a 165c
1.17 1.13b 1.4c
3 3 3
2.129 2.08d 2.08e
471.4 472d 424.9f
4.54 5.8d 5.22f
7
2.622
206.3
1.67
a
Ref. [28]. Ref. [29]. Ref. [32]. d Ref. [44]. e Ref. [45]. f Ref. [47]. b c
1-a0
2-a0
3-a0
4-a0
5-a 0
6-a 0
7-a0
8-a 0
9-a 0
Fig. 1. Lowest energy structures of gallium clusters for each size.
3. Results and discussions 3.1. Lowest-energy structures The calculated lowest-energy structures of the Gan þ 1 and lowlying isomers of the GanNb clusters are shown in Figs. 1 and 2. The isomers are designated by n-a, n-b, n-c and so on, where n is the Ga atom number. For these structures, the point group symmetry, spin multiplicity, geometry property, relative energy DE (relative to the lowest-energy structure) and averaged binding energy Eb are listed in Table 2. The value of Eb is defined for Gan þ 1 and GanNb clusters as Eb ½Gan þ 1 ¼ ððn þ1ÞE½GaEðGan þ 1 ÞÞ=ðn þ 1Þ, Eb ½Gan Nb ¼ ðnE½Ga þ E½NbE½Gan NbÞ=ðn þ1Þ,
ð1Þ
where E is the total energy of relevant system. For the GaNb dimer with CNV symmetry, the corresponding septet state with ˚ oe ¼206.26 cm 1 and De ¼1.67 eV, the binding energy R¼2.62 A, is obtained as 0.83 eV. The most stable structure of the Ga3 cluster is an equilateral triangle structure with bond length of 2.67 A˚ and D3h symmetry.
3546
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
1-a
3-a
2-a
3-b
2-b
3-c
2-c
3-d
2-d
4-a
4-b
4-c
4-d
4-e
4-f
4-g
5-a
5-b
5-c
5-d
6-a
6-b
6-c
6-d
6-e
7-a
7-f
8-d
9-d
7-b
7-g
8-e
9-e
7-c
8-a
9-a
7-d
8-b
9-b
Table 2 Point group symmetries (PG), spin multiplicity, bond lengths, vibrational frequencies, relative energies and average binding energies of Gan þ 1 and GanNb (1–9) clusters. R and R1 denote the shortest Ga–Nb and Ga–Ga bond lengths, respectively; Freq denotes the lowest vibrational frequency of the Gan þ 1 and GanNb equilibrium geometry. Cluster
Isomer
PG
Spin
˚ R (A)
˚ R1 (A)
Freq (cm 1)
DE (eV) Eb (eV)
Ga2
1-a0
Dh
3
–
2.86
143.13
0.00
0.58
GaNb
1-a
CNV
7
2.62
–
206.25
0.00
0.83
Ga3
2-a0
D3h
2
–
2.67
86.31
0.00
0.79
Ga2Nb
2-a 2-b 2-c 2-d
CNV DNH C2V C2V
6 6 4 6
2.60 2.78 2.74 2.58
2.81 2.78 2.88 3.18
245.20 173.30 93.80 86.50
0.8307 1.0276 0.4139 0.00
0.93 0.87 1.07 1.21
Ga4
3-a0
D4h
3
–
2.74
49.29
0.00
1.09
Ga3Nb
3-a 3-b 3-c 3-d
C1 C1 CS CS
5 5 3 3
2.68 2.53 2.58 2.61
2.97 2.99 2.90 2.65
21.30 53.92 54.74 22.67
0.0861 0.00 0.2088 0.4153
1.43 1.45 1.40 1.34
Ga5
4-a0
Cs
2
–
2.82
22.10
0.00
1.06
Ga4Nb
4-a 4-b 4-c 4-d 4-e 4-f 4-g
C3V C3V CS CS C2V C1 C2V
4 6 2 4 4 4 2
2.70 2.70 2.50 2.66 2.66 2.66 2.50
2.89 2.97 3.03 2.87 2.87 2.87 2.50
29.32 12.38 48.06 57.31 57.31 57.46 48.01
0.6063 1.6139 0.2944 0.0006 0.0005 0.00 0.2944
1.52 1.32 1.58 1.64 1.63 1.65 1.57
Ga6
5-a0
C2
1
–
2.66
35.51
0.00
1.33
Ga5Nb
5-a 5-b 5-c 5-d
C1 C1 C1 C2V
1 7 5 1
2.60 2.58 2.74 2.52
2.68 2.70 2.69 2.83
23.80 39.05 36.21 31.99
0.4989 0.6202 0.00 0.0484
1.48 1.56 1.46 1.55
Ga7
6-a0
C1
2
–
2.61
20.72
0.00
1.44
Ga6Nb
6-a 6-b 6-c 6-d 6-e
CS C1 C1 C1 C2
4 6 4 4 2
2.58 2.76 2.60 2.67 2.58
2.62 2.69 2.65 2.76 2.67
34.73 17.81 23.88 10.72 36.84
0.2029 0.7027 0.1916 0.6565 0.00
1.62 1.54 1.52 1.55 1.65
7-e
8-c
9-c
9-f
Ga8
7-a0
D2h
1
-
2.62
30.93
0.00
1.49
Ga7Nb
7-a 7-b 7-c 7-d 7-e 7-f 7-g
C1 C1 C3 C1 C1 C1 C1
1 1 1 1 5 1 5
2.61 2.58 2.60 2.55 2.67 2.53 2.54
3.21 2.77 2.79 2.65 2.63 2.58 2.70
14.91 32.01 18.89 31.41 22.90 9.60 21.94
0.00 0.4547 0.2831 0.4102 0.7836 0.6074 1.1589
1.72 1.66 1.68 1.67 1.62 1.64 1.57
Fig. 2. Lowest energy structures and low-lying isomers for GanNb (n¼ 1–9) clusters.
The linear and triangular configurations are selected as the initial structures to optimize for Ga2Nb clusters. The possible geometries of the Ga2Nb clusters maintaining CNV, DNH and C2V symmetries are considered. The C2V structure is lower in energy than the linear CNV and DNH isomers, which indicates that the linear isomers are less stable than the triangle structure, and the most stable structure of Ga2Nb clusters is 2-d, which is an isosceles triangular structure with the Ga–Nb bond length, the Ga–Ga bond ˚ 3.178 A˚ and 1.21 eV, length and the binding energy are 2.575 A, respectively. The binding energies of 0.28, 0.34 and 0.14 eV are more stable than the other partners. Ga4 is a square with D4h symmetry, corresponding to electronic state is 3Au’’. Our result is excellent with the previous theory calculation of Ref. [23]. We obtained four isomers in geometry optimization of Ga3Nb clusters, and the first three-dimensional (3D) structure occurs at n ¼3. But the planar structures of Ga3Nb clusters are proved not to be equilibrium geometries. A Nb atom is capped the top of the Ga3 triangle, which result in a tetrahedral configuration; the most stable structure (3-b) of Ga3Nb clusters is a spin quintuplet state with the binding energy of 1.45 eV, which is 0.36 eV lower in stability with respective to host Ga4 clusters; the quintuplet Ga3Nb clusters with the Ga–Nb bond length of
Ga9
8-a0
C1
2
–
2.58
18.12
0.00
1.47
Ga8Nb
8-a 8-b 8-c 8-d 8-e
C1 C1 C1 C1 C1
6 2 2 6 2
2.64 2.52 2.61 2.70 2.61
2.67 2.63 2.66 6.62 2.62
35.01 12.66 30.59 26.30 21.72
0.7389 0.3070 0.00 0.6819 0.2152
1.62 1.67 1.70 1.63 1.68
Ga10
9-a0
C2
1
–
2.56
37.38
0.00
1.52
Ga9Nb
9-a 9-b 9-c 9-d 9-e 9-f
C1 C1 C1 C1 Cs Cs
1 5 3 1 5 1
2.54 2.55 2.60 2.62 2.56 2.63
2.57 2.66 2.66 2.72 2.67 2.58
19.97 14.67 29.76 12.46 9.87 13.63
0.5042 0.0801 0.00 0.1990 0.0800 0.2163
1.67 1.71 1.72 1.70 1.71 1.69
˚ corresponding to 2.53 A˚ and the Ga–Ga bond length of 2.53 A, electronic state, is 5A. In addition, other three isomers, 3-a, 3-c and 3-d, which have relative energies of 0.0861, 0.2088 and 0.4153 eV, respectively, are obtained in our optimization. It is less stable than the 3-b.
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
to the ground-state structure 9-c constructed on the basis of the 8-c. The 9-a, 9-d and 9-f isomers correspond to singlet spin state, and for 9-b and 9-e isomers, the quintet state is favored. The lowest-energy isomer (9-c) with binding energy of 1.72 eV favors triplet spin multiplicity. The similar isomers 9-b and 9-e have identical spin multiplicity and structures, which are nine gallium atoms encompassing niobium atom, but their relative energy is different. From the above discussion, it is remarkable that the lowest energy configurations of gallium and niobium-doped gallium clusters favor 3D structures and prefer the low spin multiplicity. The results of the present calculation show interesting patterns. It is found that the lowest-energy structures of Nb substituted Gan þ 1 cluster are different for the lowest-energy structures of pure Gan þ 1 cluster, which are dominant growth patterns. This may be due to the fact that the Nb has a closed-shell electronic configuration 5s, and the single 4p outermost valence electron corresponds to the Ga atom. The smaller the clusters are, the more obvious the structural distortion is.
3.2. Stabilities The understanding of the variation in the formation energy properties of clusters is important for catalysis and can provide a good way to show the relative local stability of small clusters [47]. So the properties of Ga1–10 and Ga1–9Nb clusters will be reflected by the average binding energies (Eb), second-order differences of total energies (D2E) and fragmentation energies (Ef). We will focus on the changes of stabilities after doping the Nb atom in gallium clusters in this section. We plot the average binding energies per atom (Eb) of the lowest-energy structures of GanNb clusters as function of cluster size in Fig. 3. The corresponding binding energy of Gan þ 1 cluster are also plotted. From Fig. 3 one can see that the atomic average binding energies of Gan þ 1 clusters have an increasing tendency with the clusters size growing, and the mutative behaviors of GanNb clusters are close to that of bare gallium clusters. But the average binding energies of the most stable GanNb cluster are higher than those of the pure Gan þ 1 cluster. It indicates that the doped Nb atom in the Gan clusters contributes to increase the stabilities of the gallium framework. Especially, for n ¼1–4, the Eb increases rapidly from 0.83 eV for GaNb to 1.65 eV for Ga4Nb, which corresponds to the structure transition from two to three dimension. The Eb increases gradually in the range n¼ 5–9, in which the rate of increase becomes weak. The results are similar to N-doped gallium clusters [15]. It is also found that the binding energy rise monotonically with cluster size and contains two minor bumps at n¼5 and 8, which indicate that the Ga5Nb and Ga8Nb clusters have weaker stabilities than its respective neighbors. 1.8 Binding Enenrgy per atom Eb (eV)
The structure of Ga5 we obtained is considered as a buckled structure with CS symmetry, which is excellent with Ref. [8]. A variety of possible structures are considered initially for the Ga4Nb clusters. The computational results reveal that the well-defined 4-f isomer has higher total bonding energy and cannot be competitive with the other isomers optimized in stability. The structures ((4-d), (4-e) and (4-f)) with CS, C2V and C1 symmetry are similar to the Ga5 [6] configuration, but the Ga–Nb bond length, Ga–Ga bond length and binding energy are little different. The structures with the four-coordinated Nb atoms located on the Ga–Ga–Ga triangle ring, which are 0.13 eV, 0.33 eV, 0.07 eV and 0.08 eV of 4-a, 4-b, 4-c and 4-g are less stable in binding energy to the 4-f one. The lowest-energy structure for Ga6 is a prism structure (C2) with binding energy of 1.33 eV as the ground state. The structures obtained for Ga5Nb clusters have C1 and C2V symmetries. One of these structures with C1 (5-b) symmetry is the most stable. The Ga–Nb bond length and Ga–Ga bond length in it are R ¼2.58 and ˚ respectively, from our calculations. The corresponding R1 ¼2.70 A, binding energy and electronic state are 1.56 eV and 7A. The (5-a, 5-c) isomers with the same symmetry of C1 hold different spin multiplicities and binding energies to 5-b geometry. The structure of 5-d isomer with C2V symmetry and spin multiplicity (PG¼1) is similar to the Ga6 configuration [6]. All Ga5Nb isomers, whose binding energies are in the range 1.46–1.56 eV, are less stable than the host Ga6 clusters. In the case of n¼7, the pure Ga7 adopted the distorted capped trigonal prism with binding energy 1.44 eV. As shown in 6-a, 6-b, 6-c, 6-d and 6-e in Fig. 2, the most stable structure of Ga6Nb is pentagonal bipyramid (PBP) configuration with C2 symmetry as ground-state structure. It can be viewed as the trigonal prism of Ga6 with one open edge capped by an additional Nb atom. The binding energy of the most stable (6-e) isomer is 0.21 eV, which is higher than the C1 ground state of Ga7. In addition, the stable 6-b structure is yielded when one Nb is directly capped on the boatlike Ga6 cluster. As mentioned above, the other isomers can be optimized to be a minimum, and the corresponding bond lengths, frequencies and binding energies are listed in Table 2. The lowest-energy structure of Ga8 cluster corresponds to one capped octahedron geometry with D2h symmetry. The bonding energy of the ground-state structure is 1.49 eV. Several isomers exist in the Ga7Nb clusters are shown in Fig. 2, in which the Nb atom is located with the largest number of nearest neighbored Ga ˚ and Ga–Ga bond length atoms. Ga–Nb bond length (2.605 A) ˚ of the most stable structure (7-a) for Ga7Nb clusters are (3.205 A) optimized in the present work. This structure has binding energy as high as 1.72 eV. The results confirm our above-mentioned prediction that the Nb-centered structure is more stable than the Ga-capped structure. Other isomers, 7-b, 7-c, 7-d, 7-e, 7-f and 7-g, are 0.4547, 0.2831, 0.4102, 0.7836, 0.6074 and 1.1589 eV, respectively, of the relative energies are less stable than 7-a. For Ga9 cluster, the lowest-energy structure is a pentagonal arrangement of atoms with C1 symmetry. A niobium-doped 8-atom gallium cluster with 2.60 A˚ of the Ga–Ga bond length and 2.66 A˚ of the Ga–Nb length bond as the ground state, which is the most stable structure, are shown in Fig. 2 (8-c). For the case of Ga8Nb, the edge capped cube (8-c) with doublet spin state is 1.70 eV of binding energy, which is more stable than the face capped one with doublet state 8-b and 8-e isomers. Isomers (8-a, 8-d) have the sextet state. The binding energies of 8-a, 8-b, 8-d and 8-e are 1.62, 1.67, 1.63 and 1.68 eV, which are lower than 0.08, 0.03, 0.07 and 0.02 eV of the most stable structure (8-c), respectively. The ground-state structure obtained for Ga10 has C2 symmetry and it can be built from Ga9 wedges. Our structure for Ga10 is in agreement with predicted by Bin [8]. As for the Ga9Nb cluster, the 9-a, 9-b, 9-c, 9-d, 9-e and 9-f isomers are obtained by one Ga atom on different sites of Ga8Nb cluster. Only 0.02 eV more stable relative
3547
1.6 1.4 1.2 Gan+1 GanNb
1.0 0.8 0.6 0
2
4 6 Cluster size n
8
10
Fig. 3. Binding energies per atom (Eb/atom) of the Gan þ 1 and GanNb clusters with n¼1–9.
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
The second-order differences of total energies (D2E) for the lowest-energy structures of Gan þ 1 and GanNb clusters with sizes of n ¼1–9 are also evaluated and plotted as a function of cluster size in Fig. 4. The D2E is often compared directly with the relative abundances determined in mass spectroscopy experiments. The values of D2E are defined for Gan þ 1 and GanNb clusters:
D2 E½Gan þ 1 ¼ E½Gan þ 2 þ E½Gan 2E½Gan þ 1 , D2 E½Gan Nb ¼ E½Gan þ 1 Nb þ E½Gan1 Nb þ 2E½Gan Nb,
ð2Þ
where E is the total energy of relevant system. As shown in Fig. 4, the Gan þ 1 clusters stabilities exhibit pronounced odd–even alternations. However, it is clearly found that the doping impurity Nb atom makes the stable pattern of the host cluster differently from n ¼1 to n¼6, the changes of these values are very small. The odd– even alternations are close to pure Gan þ 1 clusters when n ¼6, 7, and 8. In additional, Ga7Nb cluster with positive D2E value has stronger stabilities than its respective neighbors, it is found that the clusters with n ¼7 are particularly stable. This conclusion is in excellent agreement with pure Gan þ 1 cluster [8]. In Al-doped gallium clusters, n ¼7 and 13 are also the highest of the secondorder differences of total energies (D2E) [23]. To further confirm the relative stabilities of the Ga1–10 and Ga1–9Nb clusters, we evaluate the fragmentation energies, which are plotted in Fig. 5. Fragmentation energy (Ef) expresses the thermodynamic stability of cluster, and the greater value indicates the more stable cluster. The values of fragmentation energies are estimated with equations: Ef ðGan þ 1 Þ ¼ EðGan Þ þEðGaÞEðGan þ 1 Þ, Ef ðGan NbÞ ¼ EðGan1 NbÞ þEðGaÞEðGan NbÞ,
ð3Þ
where E is the total energy of relevant system. From Fig. 5, one can see that the lower fragmentation channel corresponds to the 1.0
Δ2E (eV)
0.5 0.0 -0.5 Gan+1 GanNb
-1.0 -1.5
Gan+1 GanNb
2.5 2.0 1.5 1.0 0.5 0.0 0
2
4 6 Cluster size n
8
10
Fig. 6. HOMO-LUMO gaps for the studied Gan þ 1 and GanNb clusters with n¼1–9.
loss of the Ga atom. The Ga4Nb and Ga7Nb clusters with high fragmentation energies suggest their higher stabilities again. This conclusion is consistent with the results from second-order differences energies mentioned above. But for n ¼5 and 8, the values of Ef ¼1.18 and 1.59 eV, which show that removing a niobium or gallium atom from the cluster needs less energy, namely the thermodynamic stability of Ga5Nb and Ga8Nb clusters are poor. The gaps between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are generally considered as a useful quantity to evaluate the stability of one system, which is a useful parameter to study the metal– metal transition for clusters [48]. Fig. 6 shows us the HOMO–LUMO gaps for the most stable geometries of the studied Gan þ 1 and GanNb clusters, from which one can see that HOMO–LUMO gaps of Gan þ 1 and GanNb clusters exhibit similar behavior, that is, the HOMO–LUMO gaps increase in principle with the increase in cluster size from n¼1 to n¼3, but the HOMO–LUMO gaps decrease with the increase in cluster size from n¼3 to n¼9. The GanNb (n¼5–9) clusters have lower HOMO–LOMO energy gaps than other clusters (n¼1–4), implying that they have weaker chemical reactivity. No clear odd–even oscillations are found for the HOMO–LUMO gaps of Gan þ 1 and GanNb clusters. This phenomenon is also presented in other doped clusters, like GanN systems [11]. 3.3. Electronic properties
-2.0 0
1
2
3
4 5 6 Cluster size n
7
8
9
Fig. 4. Second-order difference of total energies of Gan þ 1 and GanNb clusters with n¼ 1–9.
2.8 Fragmentation energy Ef (eV)
3.0 HOMO-LUMO gap(eV)
3548
2.6
VIP½Gan þ 1 ¼ E½Ganþþ 1 E½Gan þ 1 ,
Gan+1 GanNb
2.4
In cluster science, ionization potential and electron affinity are used as important properties to study the change in electronic structure of the cluster size. Based on the most stable configurations, the vertical ionization potential (VIP) and electron affinity (VEA) are calculated for the lowest-energy structures. The VIP and VEA are evaluated with þ
VIP½Gan Nb ¼ E½Gan Nb E½Gan Nb,
2.2
VEA½Gan þ 1 ¼ E½Gan þ 1 E½Gan þ 1 ,
2.0
VEA½Gan Nb ¼ E½Gan NbE½Gan Nb ,
ð4Þ
1.8 1.6 1.4 1.2 1.0 0.8 0
2
6 4 Cluster size n
8
10
Fig. 5. Fragmentation energies (Ef) of ground stable structures of the Gan þ 1 and GanNb clusters with n¼ 1–9.
where E is the total energyof corresponding systems based on the ground-state structures of Gan þ 1 and GanNb clusters. The calculations þ of single point energies for charged systems (Ganþþ 1 , Ga n þ 1 , GanNb and GanNb ) are also performed at B3LYP/LANL2DZ(f) level of theory and considered the low spin multiplicities. With the VIP and VEA, the chemical hardness Z, which is established as an electronic quantity that in many cases may be used to characterize the relative stability of molecules and aggregate through the principle of maximum hardness proposed by pearson [49], can be easily obtained with Z ¼VIP–VEP under the finite difference approximation [50].
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
Magnetic moment per atom (µb)
7.0 Gan+1 GanNb
6.8
VIP (eV)
6.6 6.4 6.2 6.0 5.8 Gan+1 GanNb
VEA (eV)
2.0
Gan+1 GanNb
2.5 2.0 1.5 1.0 0.5 0.0 2
4 6 Cluster size n
8
10
Fig. 8. Magnetic moments (m/atom) vary with cluster sizes.
1.8 1.6
and GanNb clusters are plotted as a function of cluster sizes in Fig. 8. In Ganþ 1 system, from Ga2 to Ga5, the spin magnetic moments decrease. Ga6, Ga8 and Ga10 clusters exhibit nonmagnetic ground state. As for GanNb clusters, the GaNb dimer has the highest atomic magnetic moment of 3.0mb/atom in all studied sizes, and the atomic magnetic moments (mb/atom) decrease as the cluster sizes increase. There is a sharp decrease from n¼1 (3.0mb/atom) to n¼4 (0.6mb/atom) in the curve of magnetic moments. Staring from n ¼7, the moments gradually increase as the cluster sizes increase. When the number of Ga atoms is 7, the cluster is a closed-shell system, whose a and b spin orbitals degenerate, so the corresponding magnetic moment is zero.
1.4 1.2 1.0
Hardness (eV)
3.0
0
2.2
3549
0.8 5.4 5.2 5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6
Gan+1 GanNb
4. Conclusions
0
2
4 6 Cluster size n
8
10
Fig. 7. Vertical ionization potentials (VIP), vertical electron affinities (VEA) and chemical hardness (Z) of the Gan þ 1 and GanNb clusters with n ¼1–9.
These values of VIP, VEA and Z are plotted in Fig. 7, from which one can see that VIPs show an obvious oscillating behavior with the cluster size increasing, except for Ga3 in Ganþ 1 cluster and Ga4Nb in GanNb cluster. However, it is clearly found that the doping impurity atom makes the stable pattern of the host cluster contrary. This suggests that the clusters with a closed electronic shell are electronically stabilized and show high VIPs, and the maxima are found at GaNb configurations from GanNb cluster. The VEAs of the most stable structures of Ganþ 1 and GanNb (n¼1–9) clusters are also investigated as shown in Fig. 7. We can see that the values of VEAs increase with the cluster size increasing. In addition Ga4Nb and Ga7Nb have lower values than their respective neighbors, indicating the lower stabilities of neighbors cluster. In the previous study, Zhao et al. [7] also found that the values of VEAs increase with the cluster size increasing using B3LYP method for GanN clusters. As the cluster size increases, the chemical hardness Z of the Gan þ 1 cluster first drops rapidly in small size range, i.e. n¼ 2–4. There is a noticeable rise of chemical hardness Z from Ga4 to Ga5. From n¼4, the chemical hardness Z gradually increases with cluster size with weak even–odd oscillations. But as for chemical hardness Z of GanNb clusters, the values decrease as cluster sizes increase, except for Ga6Nb. 3.4. Magnetisms For the ground-state structures of Gan þ 1 and GanNb clusters (n¼ 1–9), the atomic spin magnetic moments of the studied Gan þ 1
In this paper, the geometric structures, stabilities and electronic properties of Gan þ 1 and GanNb (n ¼1–9) clusters are investigated by DFT with the generalized gradient approximation. By optimizing abundant possible geometries with different spin multiplicities, low-lying geometries are obtained. The relative stabilities of small gallium and Nb-doped gallium clusters are determined by estimating the average binding energies per atom (Eb/atom), second-order differences of total energies (D2E), fragmentation energies (Ef), HOMO–LOMO gaps, vertical ionization potential (VIP) and electron affinity (VEA). The atomic spin magnetic moment (mb) is also analyzed. All results are summarized as follows: (i) A lot of initial configurations are optimized to obtain the most stable structures for Gan þ 1 and GanNb (n¼1–9) clusters. The optimized geometries reveal that the lowest energy configurations of Nb-doped gallium clusters structures are different from pure gallium clusters for n ¼1–9. (ii) The binding energy, second-order difference of energy, fragmentation energy and HOMO–LUMO energy gap are studied as a function of cluster size for each most stable ground state Gan þ 1 and GanNb (n¼1–9) clusters. The calculated results the mutative behaviors of GanNb are close to that of bare gallium clusters, but the average binding energies of the most stable GanNb clusters are higher than those of the pure Gan þ 1 clusters. The D2E of Gan þ 1 clusters stabilities exhibit pronounced odd–even alternations. The Ga4Nb and Ga7Nb clusters with positive D2E values have stronger stabilities than their respective neighbors. The fragmentation energies show an obvious oscillating behavior with the cluster size increasing, except for Ga3Nb and Ga6Nb. In addition, the HOMO–LUMO gaps of Gan þ 1 and GanNb clusters exhibit similar behavior, and the
3550
S. Shun-Ping et al. / Physica B 406 (2011) 3544–3550
HOMO–LUMO gaps of GanNb (n ¼1–9) are in the range1.6 2.9 eV, indicating the semiconductor-like behaviors of small GanNb clusters with sizes nr9. (iii) VIPs show an obvious oscillating behavior with the cluster size increasing, except for Ga3 and Ga4Nb. But the doping impurity atom makes the stable pattern of the host cluster contrary. The values of VEAs increase with the cluster size increase. The chemical hardness Z with weak even–odd oscillations for GanNb cluster whereas the values decrease as cluster sizes increase for GanNb cluster. Ga6, Ga8 and Ga10 clusters exhibit nonmagnetic ground state and the doping of Nb atom brings the reduction as the cluster sizes increases, in which there is a sharp decrease from n ¼1 (3.0mb/atom) to n ¼4 (0.6mb/atom) in the curve of magnetic moments.
Acknowledgment This work was supported by National Science and Technology Major Project of China under Grant no. 2009ZX02204-008. We are very grateful to Yanfang Li for many helpful discussions. Our thanks are also to Dr Jiguang Du for practical help. References [1] A. Podhorodecki, M. Banski, J. Misiewicz, C. Lecerf, P. Marie, J. Cardin, X. Portier, J. Appl. Phys. 108 (2010) 063535. [2] A. Karsten, N. Kai, N. Janne, K. Antti, Phys. Rev. B 66 (2002) 0305205. [3] C.J. Zhang, W.H. Jia, Chem. Phys. 327 (2006) 144. [4] R.O. Jones, J. Chem. Phys. 99 (1993) 1194. [5] N. Gaston, A.J. Parker, Chem. Phys. Lett. 11 (2010) 006. [6] X.G. Gong, Phys. Lett. A 166 (1992) 369. [7] Y. Zhao, W.G. Xu, Q.S. Li, J. Phys. Chem. A 108 (2004) 7448. [8] B. Song, P.L. Cao, J. Chem. Phys. 123 (2005) 144312. [9] S. Krishnamurty, K. Joshi, S. Zorriasatein, D.G. Kanhere, J. Chem. Phys. 127 (2007) 054308. ¨ [10] A. Schnepf, H. Schnockel, Angew. Chem. Int. Ed. 41 (2002) 3532. [11] B. Song, P.L. Cao, Phys. Lett. A 328 (2004) 364. [12] B. Song, P.L. Cao, B.X. Li, Phys. Lett. A 315 (2003) 308. [13] A.K. Kandalam, M.A. Blanco, R. Pandey, J. Phys. Chem. B 106 (2002) 1945. [14] J.J. Zhao, B.L. Wang, X.L. Zhou, X.S. Chen, W. Lu, Chem. Phys. Lett. 422 (2006) 170. [15] B. Song, C.H. Yao, P.L. Cao, Phys. Rev. B 74 (2006) 035306. [16] J.J. BelBruno, Heterocyl. Chem. 11 (2000) 281. [17] A. Costales, A.K. Kandalam, R. Pandey, J. Phys. Chem. B 107 (2003) 4508. [18] P. Karamanis, C. Pouchan, Phys. Rev. A 77 (2008) 013201. [19] M.A. Al-Laham, K. Raghavachari, Chem. Phys. Lett. 187 (1991) 13.
[20] P. Chandrachud, K. Joshi, D.G. Kanhere, Phys. Rev. B 76 (2007) 235423. [21] E.L. Li, X.M. Luo, W. Shi, X.W. Wang, J. Mol. Struct. 723 (2005) 79. [22] J.J. BelBruno, E. Sanville, A. Burnin, A.K. Muhangi, A. Malyutin, Chem. Phys. Lett. 478 (2009) 132. [23] L. Guo, Comp. Mater. Sci. 45 (2009) 951. [24] J.G. Du, X.Y. Sun, J. Chen, G. Jiang, J. Phys. B: At. Mol. Phys. 43 (2010) 205103. [25] J.X. Yang, C.F. Wei, J.J. Guo, Physica B 405 (2010) 4892. [26] Y.F. Li, X.Y. Kuang, S.J. Wang, Y.R. Zhao, J. Phys. Chem. A 114 (2010) 11691. [27] H.J. Himmel, B. Gaertner, Chem. Eur. J. 10 (2004) 5936. [28] K.K. Das, J. Phys., B. At., Mol. Opt. Phys. 30 (1997) 803. ˚ Malmqvist, V. Veryazov, P.O. Widmark, J. Phys. Chem. [29] B.O. Roos, R. Lindh, P.A. A 108 (2004) 2851. ¨ ¨ [30] O. Zuger, U. Durig, Phys. Rev. B 46 (1992) 7319. [31] S.A. Hameed, JKAU: Sci. 21 (2009) 227. [32] K. Huber, G. Herzberg, Constants of Diatomic Molecule, Van Nostrand Reinhold, New York, 1979. [33] K. Balasubramanian, J. Phys. Chem. 94 (1990) 7764. [34] G.A. Breaux, R.C. Benirschke, T. Sugai, B.S. Kinnear, M.F. Jarrold, Phys. Rev. Lett. 97 (2003) 215508. ¨ W. Eberhardt, J. Chem. Phys. 100 (1994) 995. [35] C.Y. Cha, G. Gantefor, [36] F.L. King, M.M. Ross, Chem. Phys. Lett. 164 (1989) 131. [37] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery Jr, T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, J.A. Pople, Gaussian 03, Revision B 02, Gaussian, Inc., Pittsburgh PA, 2003. [38] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [39] L. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [40] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 270. [41] W.R. Wadt, P.J. Hay, J. Chem. Phys. 82 (1985) 284. [42] P.J. Hay, W.R. Wadt, J. Chem. Phys. 82 (1985) 299. [43] J.G. Du, X.Y. Sun, H.Y. Wang, Int. J. Quant. Chem. 108 (2008) 1505. [44] L. Goodwin, D.R. Salahub, Phys. Rev. A 47 (1993) 774. [45] D.A. Hales, L. Lian, P.B. Armentrout, Int. J. Mass Spectrom. Ion Process. 102 (1990) 269. [46] A.M. James, P. Kowalczyk, R. Fournier, B. Simard, J. Chem. Phys. 99 (1993) 8504. [47] D.B. Zhang, J. Shen, J. Chem. Phys. 120 (2004) 5104. [48] M. Salazar-Villanueva, P.H. Herna´ndez Tejeda, U. Pal, J.F. Rivas-Silva, J.I. Rodrı´guez Mora, J.A. Ascencio, J. Phys. Chem. A 110 (2006) 10274. [49] R.G. Pearson, Chemical Hardness: Applications from Molecules to Solids, Wiley-VCH, Weinheim, 1997. [50] P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev. 103 (2003) 1793.