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Theoretical study on the composition dependence of Ga8−nAsn (n=0–8) clusters
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Physica B 403 (2008) 159–164 www.elsevier.com/locate/physb
Theoretical study on the composition dependence of Ga8nAsn (n ¼ 0–8) clusters Jian Yu, Bao-xing Li, Qiao-yan Chu, Shi-chang Zhan Department of Physics, Microfluidic Chip Institute, and Key Laboratory of Organosilicon Chemistry and Material Technology of Ministry of Education, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China Received 16 March 2007; received in revised form 13 August 2007; accepted 15 August 2007
Abstract Using full-potential linear-muffin-tin-orbital molecular-dynamics (FP-LMTO-MD) method, we have investigated the dependence of GaAs clusters with eight atoms on composition. It is found that the ground state structures for Ga-rich and As-rich clusters are cube structures. As the ratio between gallium atoms and arsenic atoms is close to one, structural distortion become increasingly severe, or even the clusters adopt other geometrical configurations as their ground state structures. The energy gap Eg between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), and the vertical electron affinity show a certain degree of even/odd alternation with cluster composition. Among nine Ga8nAsn (n ¼ 0–8) clusters, only a few of clusters have different energy orders between the ionic and neutral isomers with large binding energy. Some ionic structures would change into other configurations due to severe structural distortion. r 2007 Elsevier B.V. All rights reserved. PACS: 36.40.c; 71.15.m; 61.46.w Keywords: GaAs cluster; Binding energy; Stability
1. Introduction The study of clusters is important in understanding the nature of transitional forms between atoms and bulk. In recent years, a number of theoretical and experimental studies of the structures and properties of small clusters have been carried out [1,2]. In particular, mixed GaAs clusters are the focus of many studies [3] because of their importance of technology in constructing fast microelectric device. However, the investigation on the GaAs clusters is difficult due to the computational difficulties associated with the structural as well as permutational variations resulting from the presence of more than one element. A few studies of small GaAs clusters have been reported recently. Experimentally, Liu and co-workers performed the photofragmentation studies of GaxAsy cluster ions with x+y constant [4–6]. Laser vaporization followed by Corresponding author.
E-mail address: [email protected] (B.-x. Li). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.091
supersonic expansion was used to produce the mixed GaAs clusters and their positive and negative ions. For neutral GaxAsy clusters, the electron affinities (EA) exhibit an alternating behavior between even and odd clusters. The GaAs clusters with an odd total number of atoms have higher EAs than their neighboring even ones. For ionic GaxAsy clusters, there is the same even/odd alternation in the fragmentation ion products: odd daughters are much more abundant than the even daughters. Liao et al. [7] studied Ga3As2 and Ga2As3 clusters using the complete active space multiconfiguration self-consistent-field (CASSCF) followed by multireference singles+doubles configuration interaction (MRSDCI) computations, which included up to 1.9 million configurations [7]. They computed the properties of the electronic states in distorted and undistorted trigonal bipyramid, and edgecapped tetrahedron structures of the Ga3As2 cluster, and found two nearly degenerate structures and several excited states. But, the Ga2As3 cluster exhibits undistorted trigonal bipyramidal states.
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Lou et al. [8] calculated the electronic and geometrical structures of GaxAsy (x+y ¼ 3–10) clusters using the local-spin-density (LSD) method. They found that all evennumbered GaxAsy (x+y ¼ even) clusters have closed shell electronic structures. The ionization potential and EA are distinctively different from the neighboring odd-numbered clusters, displaying an even/odd alternating pattern as a function of cluster size. Piquini et al. [9] examined the structural and electronic properties of GanAsm (n+mp8) clusters and the corresponding positive and negative ions using the Hartree– Fock method followed by second-order perturbation theory. An alternating behavior in the ionization potential beyond three-atom clusters was verified. They observed a structural pattern where the embryonic forms have the geometric and electronic structures based on highly symmetrical configurations. Kwong et al. [10] found that the structures of Ga10nAsn (n ¼ 0–10) clusters strongly depend on the composition [10]. They calculated the binding energy of each cluster using ab initio Hartree–Fock method and found that Asrich clusters are more stable than Ga-rich clusters generally. They also found that the dipole moment, energy gap between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), vertical ionization energy, and vertical EA show an even/odd alteration with the number of gallium atoms (or of arsenic atoms) in the clusters. The clusters with odd number of gallium atoms generally have higher dipole moments than those clusters with even number of gallium atoms, while the latter generally has larger HOMO– LUMO gaps and higher vertical ionization energy than the former. But the vertical EA shows only a certain degree of even/odd alternation with cluster composition. Al-Laham and Raghavachari [11] explored the electronic structures and stabilities of GanAsn clusters (n ¼ 1–3) by means of accurate quantum-chemical calculations using ab initio effective core potentials (ECP). They found that Ga3As3 cluster has a capped trigonal bipyramid ground state structure, similar to those of Si6 and Al3P3. They also found several other local minima including planar and prismatic structures. The electronegativity difference between the constituents of a mixed cluster is a major factor in determining its bonding and geometrical properties. Yi [12] investigated the atomic and electronic structures of small GaAs clusters containing even numbers of atoms by using first-principles pseudopotential calculations. The most stable structures of the GaAs clusters significantly depend on its size and composition. The structures become substantially distorted as the cluster size increases. As-rich clusters have larger binding energies than Ga-rich ones. A planar rhombic structure, an octahedron, a cube, a cube with a capped Ga–As pair and a cube with two-capped Ga–As pairs are energetically favored for the neutral GanAsn (n ¼ 2–6) clusters, respectively. Recently, Zhao and co-workers used full-potential linear-muffin-tin-orbital molecular-dynamics (FP-LMTO-
MD) calculations to study the structure stabilities of Ga4As4, Ga5As5, Ga6As6, and Ga8As8 clusters [13–16]. They found that the ground state structure of the Ga4As4 cluster is an edge-capped pentagonal bipyramid. For the Ga5As5 cluster, they found that the energy of a two-capped cube is lower than the others. Similarly, they obtained a distorted bi-capped pentaprism, which is the most stable structure of the Ga6As6 cluster. For the Ga8As8 cluster, they found that the distorted cage-like structure with an interior arsenic atom is the most stable. They found that the Ga8As8 cluster is easy to form the distorted cage with an arsenic atom inside, which is very different from the structures of the Ga4As4 and Ga5As5 clusters. They also found that the Ga5As5, Ga6As6, and Ga8As8 present semiconductor-like properties. Yang et al. [17] investigated the electronic and geometric structures of GanAsn (n ¼ 4–6) cluster ions by using the same method. They also investigated the evolution of some Ga4As4, Ga5As5, and Ga6As6 clusters as a function of charging. Gallium atoms are more easily on capped atomic positions than arsenic atoms in the negative GanAsn clusters. Ghosh et al. [18] performed theoretical investigation on size-dependent structural, electronic, and optical properties of stoichiometric GanAsn clusters with n up to 100 by using a simplified linear combination of atomic orbital-densityfunctional theory-local density approximation-tight-binding (LCAO-DFT-LDA-TB) method. In optimized structures, gallium atoms move inwards while arsenic atoms move outwards. Both the HOMO–LUMO gap and the total energy per atom at least for small clusters are nonmonotonic function of the cluster size. There is a correlation between band gap and the stability of the clusters. Even though the reports above have been found, the investigations on the GaAs clusters are far from adequate. In order to understand the growing process of the clusters, and their structural evolution properties as the composition ratio between arsenic atoms and gallium atoms changes, we think it is essential to study the stability and structural properties of the GaxAsy(x6¼y) clusters. In this article, we mainly study the stability of Ga8n Asn (n ¼ 0–8) clusters and find out their structural characteristics as the composition ratio between two elements using FP-LMTO-MD method. 2. Method The FP-LMTO-MD method is a self-consistent implementation of the Kohn–Sham equations in the localdensity approximation [19–22]. In this method, space is divided into two parts: non-overlapping muffin-tin (MT) spheres centered at the nuclei and the remaining interstitial region. LMTOs are augmented Hankel function, and are augmented only inside the MT spheres rather than in the interstitial region [23–26]. Self-consistent calculations are carried out with convergence criteria of 105 a.u. on the
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total energy and 103 a.u. on the force. Refs. [19–22] describe in detail how the molecular dynamics method can be performed. For each specific cluster size, we performed an exhaustive search for minimum energy structures by using the FPLMTO-MD method. The accuracy of the FP-LMTO-MD method for investigating the cluster structures has been confirmed by previous studies (for example, see Refs. [27,28]) on small Sin and GanNn clusters, etc. In order to perform the systematic search for the equilibrium structures of the mixed GaAs clusters, we used the FP-LMTOMD method to calculate the structures and energies in a global wide search. Large numbers of initial geometric configurations as seeds were relaxed until the local minimum of the total energy was found. To make sure the obtained lowest energy structures are real local minima, we have applied normal-mode vibrational analyses. In view of the large number of different seed structures used, we feel confident that the equilibrium structures presented are accurate. However, we cannot exclude the possibility of more stable structures. 3. Structures and discussions Using the FP-LMTO-MD method, we have studied the stable structures of Ga8nAsn (n ¼ 0–8) clusters systematically. A number of stable structures have been obtained. But only two structures with high stability are presented in Fig. 1. When n ¼ 0, it is a pure Ga8 cluster. Its two structures with large binding energies are shown as A1 and A2 in Fig. 1. The ground state structure A1 is a cube structure. The result agrees with that obtained by Song and Cao [29]. Gong and Tosatti obtained the same structure using an ab initio molecular dynamics method and simulated annealing technique [30]. The second most stable structure A2, which is a bi-capped trigonal prism, is found to be 0.24 eV less stable than A1. The Ga7As1 cluster doped with one arsenic atom also accepts a cube B1 as its most stable structure. But B2 is obviously different from A2. It is 0.21 eV less stable than B1. When arsenic atoms are up to 2 or 3, the structures C1 and D1 corresponding to the lowest total energies still keep basic cube configurations. But their structural distortions are significant compared with A1 or B1. Meta-stable structure C2 is different from C1. The energy difference between them is up to 0.66 eV. Lou et al. [8] obtained an octahedron with two edge-capping atoms (see D4 in Fig. 1) as the ground state structure of Ga5As3 cluster using LSD method. Its symmetry is Cs. We re-optimize the structure using the FP-LMTO-MD method. It is found that the structure is unstable. It undergoes further structural distortion into more stable configuration D3 in Fig. 1. D3 is still not as stable as D1 in Fig. 1. Another stable structure D2 lies between structures D1 and D3. Its geometrical configuration is similar to D4. But the
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positions of arsenic and gallium atoms are different. D2 lies 0.43 eV above D1. Zhao et al. [13] had investigated Ga4As4 cluster in detail by using the same method. Its most stable structure is a capped pentagonal bipyramid shown as E1 in Fig. 1, which is obviously different from the cube structure. The second most stable structure E2 is a bi-capped octahedron. Our results are in agreement with those they obtained. F1 in Fig. 1 is the ground state structure for Ga3As5 cluster. Lou et al. [8] obtained the structure. Our result is in agreement with theirs. The second most stable structure F2 is a new distorted tetra-capped tetrahedron. F2 is found to be 0.12 eV less stable than F1. The lowest energy structure of Ga2As6 cluster is also a distorted cube shown as G1 in Fig. 1. Although it has a similar geometrical configuration to C1, it cannot be obtained by exchanging gallium atoms with arsenic atoms in C1. Its second most stable structure G2 is a different structure. As the number of arsenic atoms is up to seven, the most stable structure we obtained can be also viewed as a distorted cube (see H1 in Fig. 1). But Yi obtained H3 as its ground state structure using first-principle pseudopotential calculations [12]. We re-optimize the structure H3 using the FP-LMTO-MD method. It is found that it transfers into H2 in Fig. 1. H2 is 2.48 eV less stable than H1. For pure As8 cluster, I1 has maximum binding energy. It is a cube structure similar to A1. This result agrees with that obtained by Baruah et al. [31]. The second most stable structure I2 is a bi-capped trigonal prism, which is 1.02 eV less stable than H1. But Yi suggested that I3 is the most stable [12]. We re-optimize the structure I3. It is found that I3 is 4.04 eV less stable than I2. Analyzing the structures of the Ga8nAs8 (n ¼ 0–8) clusters studied, we have seen that all the ground state structures have a cube structure except for Ga5As3 and Ga4As4 clusters. As the composition ratio between two elements trends to one, the structural distortion becomes increasingly severe. For the Ga5As3 and Ga4As4 clusters, severe structural distortion results in different ground state structures. But for the rich-Ga or rich-As clusters, the cube structure shows high stability. This finding is somewhat different from the conclusion drawn by Kwong et al. [10]. They found that the structures of the GaAs clusters with 10 atoms are quite different from each other, indicating that the clusters depended strongly on composition. But from Fig. 1, it is seen that only the meta-stable structures are different from each other, displaying their composition dependence. Similarly, we have calculated the energy gaps between the HOMO and the LUMO. It is found that the dependence of the gaps on composition is similar to that obtained by Kwong et al. [10]. Kwong’s results show that the HOMO–LUMO gaps of the Ga10nAsn (n ¼ 0–10) clusters depend strongly on cluster composition. The clusters with even number of Ga atoms generally have larger HOMO–LUMO gaps than those clusters with odd
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162
o
o
A1
A2
B1
B2 o
o
o
o
o
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o
o o
o
C1
C2
D1
D2
o o
o
o o
o
o
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o
o o
o
o
o D3
D4
o
o
o
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o
o o
E2
E1
o
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o o
o
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o F1
F2
o o
o
o o
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o o
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o o
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o o
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H1
o o
o o o
G2
G1
o H3
o
o o I1
o o
o o
o
o
o
o
o
o o I2
o
o
o
o o I3
Fig. 1. The stable structures of Ga8nAs8 (n ¼ 0–8) clusters with large binding energies. Black sphere and white spheres with O in it refer to Ga atom and As atom, respectively.
number of Ga atoms. Our computational results for the Ga8nAsn (n ¼ 0–8) clusters are presented in Table 1. The data also suggest that the gaps of the Ga8nAsn (n ¼ 0–8) clusters show a certain degree of even/odd alternation with
cluster composition with the exception at n ¼ 6, which is similar to that of the Ga10nAsn (n ¼ 0–10) clusters with 10 atoms. Except for 0.67 eV of structure B1, other gaps are in the range of 1.0–1.7 eV. The values are smaller than those
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of the Ga10nAsn (n ¼ 0–10) clusters. But all the lowest energy clusters with eight atoms are singlet state like the clusters with 10 atoms because they have even number of valence electrons and furthermore all the electrons are paired together in their respective molecular orbitals. On the other hand, we have also calculated the vertical EA for the Ga8nAsn (n ¼ 0–8) clusters, which is shown as in Table 1. The vertical EA is the amount of energy released when the cluster obtains an electron from its neutral state, assuming that the anionic geometry remains unchanged. Examination of Table 1 shows that all the clusters have positive vertical EA, suggesting that the clusters have a tendency to gain an electron under normal conditions, as it is energetically favorable to do so. This is in agreement with the results obtained by Kwong et al. [10]. The vertical EA also shows an even/odd alternation with cluster composition. In general, the clusters with odd number of Ga atoms have larger vertical EA than those clusters with oven number of Ga atoms. A comparison between the GaAs clusters with 10 and 8 atoms reveals that both the HOMO–LUMO gap and vertical EA show the same even/odd alteration with the number of Gallium atoms (or of arsenic atoms) in the clusters. This suggests that the physical properties depend on cluster composition. But the dependence of the geometrical structures on composition is different for the clusters with different atomic number. Kwong et al. found that the structures of the Ga10nAsn (n ¼ 0–10) clusters depend strongly on composition. But, our calculated results show that most of the Ga8nAsn (n ¼ 0–8) clusters adopt a cube as their ground state structures. Different ground state structures for pure Ga10 and As10 clusters result in the dependence of the cluster structures with 10 atoms on composition. The clusters will undergo some Table 1 The energy gap Eg (in eV) between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), the vertical electron affinity (EA, in eV), and the binding energy Eb (in eV) for the lowest energy structures Structure A1 Eg (eV) EA (eV) Eb (eV)
B1
C1
D1
E1
F1
G1
H1
I1
1.16 0.67 1.48 1.41 1.42 1.53 1.39 1.46 1.64 1.96 2.48 1.95 3.22 0.52 3.08 4.74 5.91 5.42 5.48 8.70 11.09 12.20 13.51 13.48 13.47 14.83 15.82
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intermediate structures in the process transferring into the As10 cluster from the Ga10 cluster. Therefore, it is expected that the structures of the Ga10nAsn (n ¼ 0–10) clusters are quite different from each other. But the pure Ga8 and As8 clusters have the similar geometrical configurations. The similarity easily makes the As-rich and Ga-rich clusters adopt similar structures as their ground state structures. Therefore, the dependence of the ground state structures on cluster composition is different for the mixed GaAs clusters with different atomic number. We have performed calculations on the ionic structures corresponding to Fig. 1. It is found that the energy orders of a few of structures reverse. For example, the positive ionic structure A2+ is 1.79 eV more stable than A1+ (see Fig. 2). But, for neutral cluster, the latter has higher stability than the former. In addition, on charging the structures in Fig. 1, they would undergo significant struc tural distortion. For example, anionic D1- and D2structures in Fig. 2 are completely different from their corresponding neutral structures D1 and D2. The total binding energies of the GaAs clusters with eight atoms are also presented in Table 1. The energies increase as the number of Ga atoms decreases. This indicates that the As-rich clusters are more stable than the Ga-rich clusters. This is in consistent with other theoretical studies on the GaAs clusters. We have also calculated their second energy difference D2E(Ga8nAsn). D2E(Ga8nAsn) is defined as E(Ga8n1Asn+1)+ E(Ga8n+1Asn1)2E(Ga8nAsn). The values are presented in Table 2. The clusters with positive values of D2E are more stable than their nearest neighbors. It is clearly seen from the data in Table 2 that the B1, C1, E1, and H1 structures exhibit high stability. It is worth noting that the E1 structure is substantially stable compared with its neighbors. This structure is similar to the ground state structure for Ge8 cluster [32]. But its symmetry becomes C1h from C2v of the Ge8 cluster because of the Ga4As4 cluster containing two different elements. In the mixed GaAs clusters, the charge transfers from the Ga atom sites to the neighboring As atom sites. The edge-capping gallium atom in the E1 structure tilts up to the side of the top arsenic atom because of electrostatic attraction. In addition, two top atoms in the structure are As atom and Ga atom, respectively. The distance between the two atoms should be smaller than the corresponding distances of pure
o o o
o
o o
A1+
A2+
D1-
D2-
Fig. 2. A1+ and A2+ are the positive ionic structures corresponding to neutral structures A1 and A2. D1- and D2- are the distorted negative ionic structures corresponding to neutral structures D1 and D2. Black sphere and white spheres with O in it refer to Ga atom and As atom, respectively.
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Table 2 The second difference D2E(Ga8nAsn) (in eV) for Ga8nAsn (n ¼ 0-8) clusters Structure
B1
C1
D1
E1
F1
G1
H1
D2E(Ga8nAsn)
0.84
1.27
0.20
1.35
0.02
1.36
0.36
Ga8 and As8 clusters since they are composed of the same atoms. In order to verify this, we have investigated the two edge-capped pentagonal bi-pyramids of the Ga8 and As8 clusters corresponding to the E1 structure. Our calculated results suggest that the edge-capped pentagonal bipyramid for the Ga8 cluster is stable, but the structure for the As8 clusters is not stable. The distance between the two top atoms of the Ga8 cluster is 3.61 A˚, which is larger than 3.53 A˚ of the E1 structure. It is the result that is expected. Detailed examinations on the structures in Fig. 1 show that the Ga atoms are easily on the capping atom positions. The situation is significant in the structures containing about the same number of Ga and As atoms. For example, both of D2 and E1 structures have an edge-capping Ga atom with two bonds. Even though D4 structure has also an edge-capping As atom, it is not stable. In addition, the Ga atoms are more easily on the face-capping atom positions with three bonds than the As atoms (see D1, E2, F1, and F2 in Fig. 1). This is related to their bond characteristics. The electronic configurations of arsenic atom and gallium atom are 4s24p3 and 4s24p1, respectively. Arsenic atom can adopt sp, sp2, sp3 or sp3d1 hybrid, whereas gallium atom usually has only sp or sp2 hybrid. In addition, both of them can form s bonds, or p bonds with neighbor atoms. Therefore, in the mixed clusters, the gallium atoms are easily on the capping atom positions, which need fewer bonds.
4. Conclusions Using FP-LMTO-MD method, we have investigated the stable structures of the Ga8nAs8 (n ¼ 0–8) clusters. It is found that most of the ground state structures for the Ga8nAsn (n ¼ 0–8) clusters are cube structures. When the number of Ga atoms is close to that of As atoms, the mixed clusters would undergo severe structural distortion, or accept other geometrical configurations as their ground state structures. The HOMO–LUMO gap and the vertical EA show an even/odd alteration with the number of gallium atoms. On charging the structures, a few of the energy orders probably reverse. Some structures would change into other configurations due to severe structural distortion. Almost all the clusters have a gap with larger than 1 eV.
Acknowledgments A Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant no. 200320, the National Natural Science Foundation under Grant no. 10674039, and the Natural Science Foundation of Zhejiang Province (Grant no. R405097) supported this work. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at doi:10.1016/j.physb.2007. 08.091.
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Physica B 403 (2008) 159–164 www.elsevier.com/locate/physb
Theoretical study on the composition dependence of Ga8nAsn (n ¼ 0–8) clusters Jian Yu, Bao-xing Li, Qiao-yan Chu, Shi-chang Zhan Department of Physics, Microfluidic Chip Institute, and Key Laboratory of Organosilicon Chemistry and Material Technology of Ministry of Education, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China Received 16 March 2007; received in revised form 13 August 2007; accepted 15 August 2007
Abstract Using full-potential linear-muffin-tin-orbital molecular-dynamics (FP-LMTO-MD) method, we have investigated the dependence of GaAs clusters with eight atoms on composition. It is found that the ground state structures for Ga-rich and As-rich clusters are cube structures. As the ratio between gallium atoms and arsenic atoms is close to one, structural distortion become increasingly severe, or even the clusters adopt other geometrical configurations as their ground state structures. The energy gap Eg between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), and the vertical electron affinity show a certain degree of even/odd alternation with cluster composition. Among nine Ga8nAsn (n ¼ 0–8) clusters, only a few of clusters have different energy orders between the ionic and neutral isomers with large binding energy. Some ionic structures would change into other configurations due to severe structural distortion. r 2007 Elsevier B.V. All rights reserved. PACS: 36.40.c; 71.15.m; 61.46.w Keywords: GaAs cluster; Binding energy; Stability
1. Introduction The study of clusters is important in understanding the nature of transitional forms between atoms and bulk. In recent years, a number of theoretical and experimental studies of the structures and properties of small clusters have been carried out [1,2]. In particular, mixed GaAs clusters are the focus of many studies [3] because of their importance of technology in constructing fast microelectric device. However, the investigation on the GaAs clusters is difficult due to the computational difficulties associated with the structural as well as permutational variations resulting from the presence of more than one element. A few studies of small GaAs clusters have been reported recently. Experimentally, Liu and co-workers performed the photofragmentation studies of GaxAsy cluster ions with x+y constant [4–6]. Laser vaporization followed by Corresponding author.
E-mail address: [email protected] (B.-x. Li). 0921-4526/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2007.08.091
supersonic expansion was used to produce the mixed GaAs clusters and their positive and negative ions. For neutral GaxAsy clusters, the electron affinities (EA) exhibit an alternating behavior between even and odd clusters. The GaAs clusters with an odd total number of atoms have higher EAs than their neighboring even ones. For ionic GaxAsy clusters, there is the same even/odd alternation in the fragmentation ion products: odd daughters are much more abundant than the even daughters. Liao et al. [7] studied Ga3As2 and Ga2As3 clusters using the complete active space multiconfiguration self-consistent-field (CASSCF) followed by multireference singles+doubles configuration interaction (MRSDCI) computations, which included up to 1.9 million configurations [7]. They computed the properties of the electronic states in distorted and undistorted trigonal bipyramid, and edgecapped tetrahedron structures of the Ga3As2 cluster, and found two nearly degenerate structures and several excited states. But, the Ga2As3 cluster exhibits undistorted trigonal bipyramidal states.
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Lou et al. [8] calculated the electronic and geometrical structures of GaxAsy (x+y ¼ 3–10) clusters using the local-spin-density (LSD) method. They found that all evennumbered GaxAsy (x+y ¼ even) clusters have closed shell electronic structures. The ionization potential and EA are distinctively different from the neighboring odd-numbered clusters, displaying an even/odd alternating pattern as a function of cluster size. Piquini et al. [9] examined the structural and electronic properties of GanAsm (n+mp8) clusters and the corresponding positive and negative ions using the Hartree– Fock method followed by second-order perturbation theory. An alternating behavior in the ionization potential beyond three-atom clusters was verified. They observed a structural pattern where the embryonic forms have the geometric and electronic structures based on highly symmetrical configurations. Kwong et al. [10] found that the structures of Ga10nAsn (n ¼ 0–10) clusters strongly depend on the composition [10]. They calculated the binding energy of each cluster using ab initio Hartree–Fock method and found that Asrich clusters are more stable than Ga-rich clusters generally. They also found that the dipole moment, energy gap between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), vertical ionization energy, and vertical EA show an even/odd alteration with the number of gallium atoms (or of arsenic atoms) in the clusters. The clusters with odd number of gallium atoms generally have higher dipole moments than those clusters with even number of gallium atoms, while the latter generally has larger HOMO– LUMO gaps and higher vertical ionization energy than the former. But the vertical EA shows only a certain degree of even/odd alternation with cluster composition. Al-Laham and Raghavachari [11] explored the electronic structures and stabilities of GanAsn clusters (n ¼ 1–3) by means of accurate quantum-chemical calculations using ab initio effective core potentials (ECP). They found that Ga3As3 cluster has a capped trigonal bipyramid ground state structure, similar to those of Si6 and Al3P3. They also found several other local minima including planar and prismatic structures. The electronegativity difference between the constituents of a mixed cluster is a major factor in determining its bonding and geometrical properties. Yi [12] investigated the atomic and electronic structures of small GaAs clusters containing even numbers of atoms by using first-principles pseudopotential calculations. The most stable structures of the GaAs clusters significantly depend on its size and composition. The structures become substantially distorted as the cluster size increases. As-rich clusters have larger binding energies than Ga-rich ones. A planar rhombic structure, an octahedron, a cube, a cube with a capped Ga–As pair and a cube with two-capped Ga–As pairs are energetically favored for the neutral GanAsn (n ¼ 2–6) clusters, respectively. Recently, Zhao and co-workers used full-potential linear-muffin-tin-orbital molecular-dynamics (FP-LMTO-
MD) calculations to study the structure stabilities of Ga4As4, Ga5As5, Ga6As6, and Ga8As8 clusters [13–16]. They found that the ground state structure of the Ga4As4 cluster is an edge-capped pentagonal bipyramid. For the Ga5As5 cluster, they found that the energy of a two-capped cube is lower than the others. Similarly, they obtained a distorted bi-capped pentaprism, which is the most stable structure of the Ga6As6 cluster. For the Ga8As8 cluster, they found that the distorted cage-like structure with an interior arsenic atom is the most stable. They found that the Ga8As8 cluster is easy to form the distorted cage with an arsenic atom inside, which is very different from the structures of the Ga4As4 and Ga5As5 clusters. They also found that the Ga5As5, Ga6As6, and Ga8As8 present semiconductor-like properties. Yang et al. [17] investigated the electronic and geometric structures of GanAsn (n ¼ 4–6) cluster ions by using the same method. They also investigated the evolution of some Ga4As4, Ga5As5, and Ga6As6 clusters as a function of charging. Gallium atoms are more easily on capped atomic positions than arsenic atoms in the negative GanAsn clusters. Ghosh et al. [18] performed theoretical investigation on size-dependent structural, electronic, and optical properties of stoichiometric GanAsn clusters with n up to 100 by using a simplified linear combination of atomic orbital-densityfunctional theory-local density approximation-tight-binding (LCAO-DFT-LDA-TB) method. In optimized structures, gallium atoms move inwards while arsenic atoms move outwards. Both the HOMO–LUMO gap and the total energy per atom at least for small clusters are nonmonotonic function of the cluster size. There is a correlation between band gap and the stability of the clusters. Even though the reports above have been found, the investigations on the GaAs clusters are far from adequate. In order to understand the growing process of the clusters, and their structural evolution properties as the composition ratio between arsenic atoms and gallium atoms changes, we think it is essential to study the stability and structural properties of the GaxAsy(x6¼y) clusters. In this article, we mainly study the stability of Ga8n Asn (n ¼ 0–8) clusters and find out their structural characteristics as the composition ratio between two elements using FP-LMTO-MD method. 2. Method The FP-LMTO-MD method is a self-consistent implementation of the Kohn–Sham equations in the localdensity approximation [19–22]. In this method, space is divided into two parts: non-overlapping muffin-tin (MT) spheres centered at the nuclei and the remaining interstitial region. LMTOs are augmented Hankel function, and are augmented only inside the MT spheres rather than in the interstitial region [23–26]. Self-consistent calculations are carried out with convergence criteria of 105 a.u. on the
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total energy and 103 a.u. on the force. Refs. [19–22] describe in detail how the molecular dynamics method can be performed. For each specific cluster size, we performed an exhaustive search for minimum energy structures by using the FPLMTO-MD method. The accuracy of the FP-LMTO-MD method for investigating the cluster structures has been confirmed by previous studies (for example, see Refs. [27,28]) on small Sin and GanNn clusters, etc. In order to perform the systematic search for the equilibrium structures of the mixed GaAs clusters, we used the FP-LMTOMD method to calculate the structures and energies in a global wide search. Large numbers of initial geometric configurations as seeds were relaxed until the local minimum of the total energy was found. To make sure the obtained lowest energy structures are real local minima, we have applied normal-mode vibrational analyses. In view of the large number of different seed structures used, we feel confident that the equilibrium structures presented are accurate. However, we cannot exclude the possibility of more stable structures. 3. Structures and discussions Using the FP-LMTO-MD method, we have studied the stable structures of Ga8nAsn (n ¼ 0–8) clusters systematically. A number of stable structures have been obtained. But only two structures with high stability are presented in Fig. 1. When n ¼ 0, it is a pure Ga8 cluster. Its two structures with large binding energies are shown as A1 and A2 in Fig. 1. The ground state structure A1 is a cube structure. The result agrees with that obtained by Song and Cao [29]. Gong and Tosatti obtained the same structure using an ab initio molecular dynamics method and simulated annealing technique [30]. The second most stable structure A2, which is a bi-capped trigonal prism, is found to be 0.24 eV less stable than A1. The Ga7As1 cluster doped with one arsenic atom also accepts a cube B1 as its most stable structure. But B2 is obviously different from A2. It is 0.21 eV less stable than B1. When arsenic atoms are up to 2 or 3, the structures C1 and D1 corresponding to the lowest total energies still keep basic cube configurations. But their structural distortions are significant compared with A1 or B1. Meta-stable structure C2 is different from C1. The energy difference between them is up to 0.66 eV. Lou et al. [8] obtained an octahedron with two edge-capping atoms (see D4 in Fig. 1) as the ground state structure of Ga5As3 cluster using LSD method. Its symmetry is Cs. We re-optimize the structure using the FP-LMTO-MD method. It is found that the structure is unstable. It undergoes further structural distortion into more stable configuration D3 in Fig. 1. D3 is still not as stable as D1 in Fig. 1. Another stable structure D2 lies between structures D1 and D3. Its geometrical configuration is similar to D4. But the
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positions of arsenic and gallium atoms are different. D2 lies 0.43 eV above D1. Zhao et al. [13] had investigated Ga4As4 cluster in detail by using the same method. Its most stable structure is a capped pentagonal bipyramid shown as E1 in Fig. 1, which is obviously different from the cube structure. The second most stable structure E2 is a bi-capped octahedron. Our results are in agreement with those they obtained. F1 in Fig. 1 is the ground state structure for Ga3As5 cluster. Lou et al. [8] obtained the structure. Our result is in agreement with theirs. The second most stable structure F2 is a new distorted tetra-capped tetrahedron. F2 is found to be 0.12 eV less stable than F1. The lowest energy structure of Ga2As6 cluster is also a distorted cube shown as G1 in Fig. 1. Although it has a similar geometrical configuration to C1, it cannot be obtained by exchanging gallium atoms with arsenic atoms in C1. Its second most stable structure G2 is a different structure. As the number of arsenic atoms is up to seven, the most stable structure we obtained can be also viewed as a distorted cube (see H1 in Fig. 1). But Yi obtained H3 as its ground state structure using first-principle pseudopotential calculations [12]. We re-optimize the structure H3 using the FP-LMTO-MD method. It is found that it transfers into H2 in Fig. 1. H2 is 2.48 eV less stable than H1. For pure As8 cluster, I1 has maximum binding energy. It is a cube structure similar to A1. This result agrees with that obtained by Baruah et al. [31]. The second most stable structure I2 is a bi-capped trigonal prism, which is 1.02 eV less stable than H1. But Yi suggested that I3 is the most stable [12]. We re-optimize the structure I3. It is found that I3 is 4.04 eV less stable than I2. Analyzing the structures of the Ga8nAs8 (n ¼ 0–8) clusters studied, we have seen that all the ground state structures have a cube structure except for Ga5As3 and Ga4As4 clusters. As the composition ratio between two elements trends to one, the structural distortion becomes increasingly severe. For the Ga5As3 and Ga4As4 clusters, severe structural distortion results in different ground state structures. But for the rich-Ga or rich-As clusters, the cube structure shows high stability. This finding is somewhat different from the conclusion drawn by Kwong et al. [10]. They found that the structures of the GaAs clusters with 10 atoms are quite different from each other, indicating that the clusters depended strongly on composition. But from Fig. 1, it is seen that only the meta-stable structures are different from each other, displaying their composition dependence. Similarly, we have calculated the energy gaps between the HOMO and the LUMO. It is found that the dependence of the gaps on composition is similar to that obtained by Kwong et al. [10]. Kwong’s results show that the HOMO–LUMO gaps of the Ga10nAsn (n ¼ 0–10) clusters depend strongly on cluster composition. The clusters with even number of Ga atoms generally have larger HOMO–LUMO gaps than those clusters with odd
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162
o
o
A1
A2
B1
B2 o
o
o
o
o
o
o
o o
o
C1
C2
D1
D2
o o
o
o o
o
o
o
o
o o
o
o
o D3
D4
o
o
o
o
o
o o
E2
E1
o
o
o
o
o o
o
o
o
o
o
o
o
o
o F1
F2
o o
o
o o
o
o o
o
o
o
o
o o
o
o o
o
o o
H2
H1
o o
o o o
G2
G1
o H3
o
o o I1
o o
o o
o
o
o
o
o
o o I2
o
o
o
o o I3
Fig. 1. The stable structures of Ga8nAs8 (n ¼ 0–8) clusters with large binding energies. Black sphere and white spheres with O in it refer to Ga atom and As atom, respectively.
number of Ga atoms. Our computational results for the Ga8nAsn (n ¼ 0–8) clusters are presented in Table 1. The data also suggest that the gaps of the Ga8nAsn (n ¼ 0–8) clusters show a certain degree of even/odd alternation with
cluster composition with the exception at n ¼ 6, which is similar to that of the Ga10nAsn (n ¼ 0–10) clusters with 10 atoms. Except for 0.67 eV of structure B1, other gaps are in the range of 1.0–1.7 eV. The values are smaller than those
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of the Ga10nAsn (n ¼ 0–10) clusters. But all the lowest energy clusters with eight atoms are singlet state like the clusters with 10 atoms because they have even number of valence electrons and furthermore all the electrons are paired together in their respective molecular orbitals. On the other hand, we have also calculated the vertical EA for the Ga8nAsn (n ¼ 0–8) clusters, which is shown as in Table 1. The vertical EA is the amount of energy released when the cluster obtains an electron from its neutral state, assuming that the anionic geometry remains unchanged. Examination of Table 1 shows that all the clusters have positive vertical EA, suggesting that the clusters have a tendency to gain an electron under normal conditions, as it is energetically favorable to do so. This is in agreement with the results obtained by Kwong et al. [10]. The vertical EA also shows an even/odd alternation with cluster composition. In general, the clusters with odd number of Ga atoms have larger vertical EA than those clusters with oven number of Ga atoms. A comparison between the GaAs clusters with 10 and 8 atoms reveals that both the HOMO–LUMO gap and vertical EA show the same even/odd alteration with the number of Gallium atoms (or of arsenic atoms) in the clusters. This suggests that the physical properties depend on cluster composition. But the dependence of the geometrical structures on composition is different for the clusters with different atomic number. Kwong et al. found that the structures of the Ga10nAsn (n ¼ 0–10) clusters depend strongly on composition. But, our calculated results show that most of the Ga8nAsn (n ¼ 0–8) clusters adopt a cube as their ground state structures. Different ground state structures for pure Ga10 and As10 clusters result in the dependence of the cluster structures with 10 atoms on composition. The clusters will undergo some Table 1 The energy gap Eg (in eV) between the highest-occupied molecular orbital (HOMO) and the lowest-unoccupied molecular orbital (LUMO), the vertical electron affinity (EA, in eV), and the binding energy Eb (in eV) for the lowest energy structures Structure A1 Eg (eV) EA (eV) Eb (eV)
B1
C1
D1
E1
F1
G1
H1
I1
1.16 0.67 1.48 1.41 1.42 1.53 1.39 1.46 1.64 1.96 2.48 1.95 3.22 0.52 3.08 4.74 5.91 5.42 5.48 8.70 11.09 12.20 13.51 13.48 13.47 14.83 15.82
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intermediate structures in the process transferring into the As10 cluster from the Ga10 cluster. Therefore, it is expected that the structures of the Ga10nAsn (n ¼ 0–10) clusters are quite different from each other. But the pure Ga8 and As8 clusters have the similar geometrical configurations. The similarity easily makes the As-rich and Ga-rich clusters adopt similar structures as their ground state structures. Therefore, the dependence of the ground state structures on cluster composition is different for the mixed GaAs clusters with different atomic number. We have performed calculations on the ionic structures corresponding to Fig. 1. It is found that the energy orders of a few of structures reverse. For example, the positive ionic structure A2+ is 1.79 eV more stable than A1+ (see Fig. 2). But, for neutral cluster, the latter has higher stability than the former. In addition, on charging the structures in Fig. 1, they would undergo significant struc tural distortion. For example, anionic D1- and D2structures in Fig. 2 are completely different from their corresponding neutral structures D1 and D2. The total binding energies of the GaAs clusters with eight atoms are also presented in Table 1. The energies increase as the number of Ga atoms decreases. This indicates that the As-rich clusters are more stable than the Ga-rich clusters. This is in consistent with other theoretical studies on the GaAs clusters. We have also calculated their second energy difference D2E(Ga8nAsn). D2E(Ga8nAsn) is defined as E(Ga8n1Asn+1)+ E(Ga8n+1Asn1)2E(Ga8nAsn). The values are presented in Table 2. The clusters with positive values of D2E are more stable than their nearest neighbors. It is clearly seen from the data in Table 2 that the B1, C1, E1, and H1 structures exhibit high stability. It is worth noting that the E1 structure is substantially stable compared with its neighbors. This structure is similar to the ground state structure for Ge8 cluster [32]. But its symmetry becomes C1h from C2v of the Ge8 cluster because of the Ga4As4 cluster containing two different elements. In the mixed GaAs clusters, the charge transfers from the Ga atom sites to the neighboring As atom sites. The edge-capping gallium atom in the E1 structure tilts up to the side of the top arsenic atom because of electrostatic attraction. In addition, two top atoms in the structure are As atom and Ga atom, respectively. The distance between the two atoms should be smaller than the corresponding distances of pure
o o o
o
o o
A1+
A2+
D1-
D2-
Fig. 2. A1+ and A2+ are the positive ionic structures corresponding to neutral structures A1 and A2. D1- and D2- are the distorted negative ionic structures corresponding to neutral structures D1 and D2. Black sphere and white spheres with O in it refer to Ga atom and As atom, respectively.
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Table 2 The second difference D2E(Ga8nAsn) (in eV) for Ga8nAsn (n ¼ 0-8) clusters Structure
B1
C1
D1
E1
F1
G1
H1
D2E(Ga8nAsn)
0.84
1.27
0.20
1.35
0.02
1.36
0.36
Ga8 and As8 clusters since they are composed of the same atoms. In order to verify this, we have investigated the two edge-capped pentagonal bi-pyramids of the Ga8 and As8 clusters corresponding to the E1 structure. Our calculated results suggest that the edge-capped pentagonal bipyramid for the Ga8 cluster is stable, but the structure for the As8 clusters is not stable. The distance between the two top atoms of the Ga8 cluster is 3.61 A˚, which is larger than 3.53 A˚ of the E1 structure. It is the result that is expected. Detailed examinations on the structures in Fig. 1 show that the Ga atoms are easily on the capping atom positions. The situation is significant in the structures containing about the same number of Ga and As atoms. For example, both of D2 and E1 structures have an edge-capping Ga atom with two bonds. Even though D4 structure has also an edge-capping As atom, it is not stable. In addition, the Ga atoms are more easily on the face-capping atom positions with three bonds than the As atoms (see D1, E2, F1, and F2 in Fig. 1). This is related to their bond characteristics. The electronic configurations of arsenic atom and gallium atom are 4s24p3 and 4s24p1, respectively. Arsenic atom can adopt sp, sp2, sp3 or sp3d1 hybrid, whereas gallium atom usually has only sp or sp2 hybrid. In addition, both of them can form s bonds, or p bonds with neighbor atoms. Therefore, in the mixed clusters, the gallium atoms are easily on the capping atom positions, which need fewer bonds.
4. Conclusions Using FP-LMTO-MD method, we have investigated the stable structures of the Ga8nAs8 (n ¼ 0–8) clusters. It is found that most of the ground state structures for the Ga8nAsn (n ¼ 0–8) clusters are cube structures. When the number of Ga atoms is close to that of As atoms, the mixed clusters would undergo severe structural distortion, or accept other geometrical configurations as their ground state structures. The HOMO–LUMO gap and the vertical EA show an even/odd alteration with the number of gallium atoms. On charging the structures, a few of the energy orders probably reverse. Some structures would change into other configurations due to severe structural distortion. Almost all the clusters have a gap with larger than 1 eV.
Acknowledgments A Foundation for the Author of National Excellent Doctoral Dissertation of PR China under Grant no. 200320, the National Natural Science Foundation under Grant no. 10674039, and the Natural Science Foundation of Zhejiang Province (Grant no. R405097) supported this work. Appendix A. Supporting information Supplementary data associated with this article can be found in the online version at doi:10.1016/j.physb.2007. 08.091.
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