In the past few decades, the binary clusters have aroused considerable interest [1–6]. Silicon is a semiconductor element of great importance for applications in microelectronic technology and material science, and silicon clusters have been extensively investigated both experimentally and theoretically [7–13]. However, silicon favours sp3 hybridisation rather than sp2 hybridisation, which leads to rather asymmetric and reactive structures for pure silicon clusters and makes the formation of cage-like geometries unstable . So, stabilising these clusters is urgent need to be solved before any applications of them. In recent years, a great deal of studies shown that transition metal (TM)-doped Sin clusters can enhance the chemical stabilities and change the geometries and electronic properties of silicon clusters. On the experimental aspect, using an ion trap, Hiura et al.  studied the formation of a series of metal-doped silicon clusters M+@Sin (M=Hf, Ta, Re, Ir, and W; n=9, 11, 13, 14). Oharaa et al.  reported experimental evidence for the geometric and electronic structures of metal-encapsulating Si cage cluster ions TMSin–(TM=Ti, Hf, Mo, and W; n=8–18) and found that the electron affinity of TMSin shows local minima around n=15–16. Koyasu et al.  investigated the electronic properties of MSin (M=Sc, Ti, V, Y, Zr, Nb, Lu, Tb, Ho, Hf, and Ta; n=6–20) clusters using the anion photoelectron spectroscopy at 213 nm. The results indicated that ScSi16–, TiSi16, VSi16+, NbSi16+, and TaSi16+ are generated in large quantities due to both electronic and geometric closings. Kong et al.  explored the structural evolution and electronic properties of anionic silver-doped silicon clusters, AgSin– (n=3–12), using anion photoelectron spectroscopy and found that the structures of AgSin– clusters are dominated by exohedral structures with the Ag atom occupying the low coordinated sites.
Theoretically, Lu and Nagase  performed a theoretical study on the structure of TMSin (TM=W, Zr, Os, Pt, Co, etc.) clusters, the results of which have shown that the formation of the endoheral structure strongly depends on the size of the Sin cluster. Ma et al.  studied the structure and energetics of CoSin (n=1–13) clusters and found that the doped Co atom enhances the stability of the Sin clusters for n=7–13. Guo et al.  investigated the geometries, stabilities, and electronic properties of TiSin (n=1–6) cluster and found that the Ti atom encapsulated into Si cages when n=12. Recently, Lu et al.  computed the geometries and electronic properties of ScSin0,−1 (n=1–6) clusters. The calculated results shown that global minima of neutral ScSin and their anions are “substitutional structure”, which is derived from Sin+1 by replacing a Si atom with a Sc atom.
For zirconium–silicon clusters, Jackson and Nellermoe  computed the endohedral binding energy of Zr@Si20 cluster using the local-density approximation method. Kumar and Kawazoe [24, 25] investigated the structure, stabilities, and electronic properties of fullerene-like Zr@Si16 clusters and the magic behaviour of ZrSi15 and ZrSi16 clusters. Sun et al.  studied the structures, stabilities, vibrational spectra, and charge distributions of Zr@Si20. Wang and Han  reported equilibrium geometries, growth-pattern mechanisms, stabilities, and polarisabilities of ZrSin (n=1–16) clusters. However, to the best of our knowledge, few systematic theoretical studies on the structures and stabilities of double zirconium-doped silicon clusters have been performed. Only Wang et al.  explored the geometries, relative stabilities, electronic properties, and ionisation potentials of medium-sized Zr2Sin (n=16–24) clusters, and they found that the dominant geometry gradually varies from the fullerene-like structure to the body-centred polyhedral structure. In order to provide a detailed theoretical understanding and interpretation of small-sized Zr2Sin clusters, we performed a systematical density functional theory (DFT) investigation on the geometries, relative stabilities, electronic properties, and polarisabilities of the small-sized Zr2-doped Sin (n=1–11) clusters. It is expected that our theoretical study will be useful not only for deeply understanding the influence of Zr on different properties of Si but can also provide powerful guidelines for novel silicon-based material researches.
2 Computational Methods
In this work, the cluster structure prediction is based on the CALYPSO method [29–31], which has been successful in correctly predicting structures for various systems. A local version of particle swarm optimisation (PSO) algorithm is implemented to utilise a fine exploration of potential energy surface for a given non-periodic system. The PSO algorithm is introduced by Kennedy and Eberhart in 1990 [32, 33]. As a stochastic global optimisation algorithm, PSO is inspired by the social behaviour of birds flocking or fish schooling and designed to solve problems related to multidimensional optimisation . The lowest-energy candidate structures of the global minimum for each size are further to perform geometric optimisation using the hybrid DFT method B3LYP [34, 35], as implemented in the Gaussian 09 package . The effect of the spin multiplicities (singlet, triplet, and quintet) is also taken into account in the optimisation procedure. Finally, vibrational frequency calculations are performed at the same level theory to assure the nature of the stationary points. Because Zr is a TM element, the full electron calculations are rather time-consuming, so the effective core potential (ECP) including relativistic effects is considered to describe the 4s24p64d25s2 outermost valence electrons of Zr atom. Thus, the basis set labelled GENECP (6-311+G(d) for Si atoms and Lanl2dz for Zr atoms) is chosen.
In order to test the reliability of the B3LYP scheme for the description of the Zr2-doped Sin clusters, the bond length r (Å), vibrational frequency ω (cm−1), electron affinity EA (eV), and dissociation energy D (eV) of Si2, Zr2, and ZrSi dimers are calculated. The calculated results as well as the experimental values [37–40] are listed in Table 1. From this table, one can find that the calculated results based on the B3LYP method agree much better with the experimental data than the others. In addition, the bond length and frequency for the ZrSi dimer are fitting well with the theoretical results of Gunaratne and Hazra (r=2.57 Å and ω=349 cm−1) . So, we can use the chosen computational method to describe small-sized Zr2Sin (n=1–11) clusters in the present work.
3 Results and Discussion
3.1 Pure Silicon Cluster Sin (n=3–13)
For the sake of contrast, the geometries of pure Sin (n=3–13) clusters are also optimised using the B3LYP/6-311+G(d) method. The ground-state structures of Sin clusters for each size are shown in Figure 1.
3.2 Zirconium–Silicon Clusters Zr2Sin (n=1–11)
For Zr2Sin (n=1–11) clusters, the ground-state and some low-lying isomers are plotted in Figures 2 and 3. These isomers are designated na, nb, nc, and nd according to their energies from low to high. The corresponding symmetries and electronic states are also indicated in Figures 2 and 3.
For Zr2Si clusters, the ground-state isomer 1a is a singlet with C2v symmetry. The Zr–Si–Zr apex angle is 63.98°, in which the two Zr atoms are located on the two sides. The chain isomer Zr–Si–Zr 1b (D∞h) with Si atom in the middle is 1.47 eV higher in energy than that of 1a isomer. When n=2, the pyramid isomer 2a with Cs symmetry is proved to be the lowest-energy structure. The isomer 2b is a C2v symmetry butterfly structure with the Si–Si bond for the “body” of the insect plus four Zr–Si bonds at the edges of the “wings”. The isomer 2c is a planar rhombus structure. Isomers 2b and 2c are 0.45 and 0.52 eV higher in energy than that of 2a isomer, respectively. At n=3, the lowest-energy isomer 3a and two low-lying isomers 3b and 3c are obtained. The isomer 3a with C2v symmetry is a pyramid structure, which can be viewed as a substituted version of the Si5 cluster. The geometry of isomer 3b is similar to 3a, the difference between them is the site of Zr atom, and the energy of former is higher than the latter by 0.06 eV. The isomer 3c is a trigonal biyramid, and its energy is higher than that of isomer 3a by 0.87 eV. In the case of Zr2Si4 clusters, the tetragonal bipyramid isomer 4a is the lowest-energy structure, which can be viewed as one Zr atom capping the bottom of the pyramid structure 3a. Interestingly, the structure of 4a is in line with the previous results of Cr2Si4 and Mn2Si4– clusters obtained by Robles and Khanna . The 3D structure 4b with D2h symmetry, which originated from the metastable isomer Si6  by replacing two Si atoms with two Zr atoms, is the second isomer. Another two isomers 4c and 4d are higher in total energy than that of isomer 4a by 0.63 and 0.91 eV, respectively. Among the isomers of Zr2Si5, the optimised results show that the pentagonal bipyramid structure 5a with C2v symmetry is the most stable isomer. When the 4a isomer is capped with one Zr atom, the second isomer 5b is obtained. After one atom is capped on the low-lying isomer 4d, the boat-shape isomers 5c and 5d are generated. The isomers 5b, 5c, and 5d are weaker than 5a by 0.31, 0.52, and 0.63 eV, respectively. As for Zr2Si6 clusters, the ground-state isomer 6a and other three low-lying isomers 6b, 6c, and 6d are obtained within an energy range of 0.55 eV. The isomers 6a and 6b are two structures derived from the lowest-energy isomer 5a, in which the eighth Zr or Si atom is capped on the pentagonal bipyramid structure. The same-site preference is found for the ground-state structures of As2Si6  clusters. The isomer 6c is a substituted geometry, in which the Zr atoms replace Si atoms of Si8 structure. The isomer 6d is a hexagonal bipyramidstructure, which is similar to previous TM2Si6 (TM=Cr, Mn, Ni, Co, and Fe) [41, 43]. The energies for 6b, 6c, and 6d are higher than that of 6a by 0.02, 0.43, and 0.55 eV, respectively. With regard to the Zr2Si7 clusters, the isomer 7a with C2v symmetry is found to be the ground-state structure among all the isomers. When one Si atom is capped on the triangular face of 6d, the 7b isomer is generated. The 7c isomer is obtained by capping one Zr atom on the top of hexahedron Si8 cluster. The 7d isomer is identified as two atoms capping on the face of a Si-centred tetragonal bipyramid Si6. The energy of 7a is lower than those of 7b, 7c, and 7d isomers by 0.13, 0.77, and 0.91 eV, respectively. For n=8, five kinds of structures can be verified to be the minima. The isomer 8a is the most stable structure, which has C2v symmetry and 3B2 state. The second isomer 8b is similar to the geometry of 8a. However, the energy of 8b is higher than that of 8a by 0.04 eV. The other isomers 8c and 8d can be obtained by replacing different Si atoms in the most stable Si10 cluster, and their energies are higher than that of 8a isomer by 0.35 and 0.54 eV, respectively. For Zr2Si9 clusters, the lowest-energy isomers 9a is generated by capping one Si atom on the top of the pentagonal prism Si10 cluster. This configuration is similar to the ground-state structure of Mg2Si9 . The isomers 9b and 9c are obtained by capping the 8d and 8c structures of Zr2Si8 cluster with one Si atom. The isomer 9d is identified as four atoms capping on the face of a Si-centred tetragonal bipyramid Si6. The energy of 9b, 9c, and 9d are 0.42, 0.52, and 0.91 eV higher in energy than that of the ground-state structure, respectively. For n=10, the calculated results show that the isomer 10a is the most stable structure. The isomer 10b is a bicapped pentagonal antiprism structure with two Zr atoms on the vertex, which is 0.56 eV higher in energy than 10a. The other isomers 10c and 10d have weaker stability than 10a because their energies are higher than that of 10a isomer by 0.77 and 0.93 eV, respectively. When the number of silicon atoms is increased to 11, the calculated results show that the 3D isomer 11a is the ground-state structure, which has a similar structure to the second isomer of Mo2Si11 cluster . It should be pointed out that the isomer 11a is the first structure in which one Zr atom is encapsulated into the Si cage. When one Zr atom is encapsulated into the Si frame, the hexagonal prism isomer 11b is obtained. After two Si atoms of most stable Si13 cluster are replaced by Zr atoms, the isomer 11c is generated. The low-lying isomers 11b, 11c, and 11d lie above the lowest-energy isomer 11a by 0.53, 0.77, and 0.91 eV, respectively.
From the above discussion on the structures of Zr2Sin clusters, one can find that the lowest-energy geometries favour the 3D structures from n=2 onwards. The Zr2Sin−1 structure capped with one Zr or Si atom, and two Zr atoms substituted Sin cluster are the two dominant growth behaviours for different-sized Zr2Sin clusters. Starting at n=11, one Zr atom falls into the interior site of Si framework, which is similar to the other TM Zr2-, Mo2-, and Pd2-doped silicon clusters, while one TM atom encapsulated Si cage at n=16, 9, and 10, respectively [28, 45, 46].
3.3 Relative Stabilities
Using the computational method described in Section 2, a lot of physical parameters, such as the averaged binding energies Eb(n), fragmentation energy Ef(n), second-order energy difference Δ2E(n), and the highest occupied-lowest unoccupied molecular orbital (HOMO-LUMO) energy gap Egap of ground-state Zr2Sin (n=1–11) clusters are obtained and plotted in Figure 4. For comparison, the Eb(n) and Egap of pure Sin+2 clusters are also calculated. The Eb(n), Ef(n), and Δ2E(n) of Zr2Sin clusters can be defined as the following formulas :
where Et denotes the total energy of the lowest-energy Si atom, Zr atom, Zr2Sin cluster, Zr2Sin−1 cluster and Zr2Sin+1 cluster, respectively.
The Eb(n), Ef(n), and Δ2E(n) values of ground-state Zr2Sin clusters against the corresponding number of Si atoms are depicted in Figure 4a–c. As seen from Figure 4a, the curves of Eb(n) for Zr2Sin and Sin+2 clusters show the same gradual growing tendency with cluster size increasing, which implies that these clusters can continue to gain energy with cluster size. When n=4, 7, and 10, three peaks are found for Zr2Sin clusters, which implies that the clusters of Zr2Si4, Zr2Si7, and Zr2Si10 are more stable than their neighbouring clusters. In addition, the Eb(n) values of Zr2Sin clusters are obviously smaller than that of corresponding pure Sin+2 cluster, which indicates that the Zr atoms can reduce the chemical stabilities of silicon clusters with small size. The similar results are also found for Cu2Sin (n=1–8) and Ca2Sin (n=1–11) clusters [48, 49]. The fragmentation energy Ef(n) and second-order energy difference Δ2E(n) for Zr2Sin clusters are shown in Figure 4b. Three visible peaks of Ef(n) for the lowest-energy Zr2Sin clusters are found at n=4, 7, and 9, reflects that the Zr2Si4, Zr2Si7, and Zr2Si9 clusters have stronger relative stabilities compared to the corresponding neighbours. For Δ2E(n), maxima are found at n=4, 7, and 10, suggests that these clusters possess higher stabilities, which is consistent with the trend of Eb(n) shown in Figure 4a. It is worthwhile to note the Δ2E(n) value of Zr2Si7 is the largest among the observed local peaks, so Zr2Si7 clusters are considerably stable.
3.4 HOMO-LUMO Gaps and Charge Transfer
The HOMO-LUMO energy gap Egap is a sensitive indicator of the relative stabilities, which implies the ability electrons to jump from occupied orbital to an unoccupied orbital . Generally speaking, a smaller gap signifies a stronger chemical activities, while a larger gap associates with a higher stabilities. The Egap for the most stable Zr2Sin and Sin+2 clusters against the cluster size are plotted in Figure 4c. From this figure, one can see that the Zr2Sin clusters with n=4, 7, and 10 show relatively larger Egap, implies that Zr2Si4, Zr2Si7, and Zr2Si10 clusters have stronger relative stabilities than others, which is in line with the previous analysis of Ef(n) and Δ2E(n) displayed in Figure 4a and b. So, it can be concluded that Zr2Si4 and Zr2Si7 are the magic clusters. Meanwhile, one can also see that the Egap values of Zr2Sin clusters are significantly less than that of corresponding pure silicon clusters, which reflects that the doping of Zr atoms can reduce the chemical stabilities of Sin+2 clusters. In addition, one can see that the change of Egap has no odd–even oscillating character as those of Zr2Sin (n=16–24) .
In order to explore the charge transfer mechanisms of Zr2Sin (n=1–11) clusters, the natural population analysis (NPA) is carried out. The results are summarised in Table 2. From this table, one can see that the Zr atoms possess positive charges for n=1–6 and negative charges for n=7–11, implies that the charges in Zr2Si1–6 clusters transfer from Zr atoms to Si atoms, while in Zr2Si7–11 clusters charges transfer from Si atoms to Zr atoms, which is in line with the NPA analysis of Be2Sin clusters that we calculated before . Thus, the direction of charges transfer depends on the size of Zr2Sin clusters. Moreover, one can also find that the charges are equal for each Zr atom in Zr2Si1,3–5,7,8 clusters, whereas they are different in others, which may be caused by the reason that there are equal numbers of Zr–Si bonds in Zr2Si1,3–5,7,8 clusters; that is to say, the charge distribution depends on the symmetry of the cluster. This result is similar to that of Cu2Sin  but differs from the Zr2Sin (n=16–24) that charges are different for each Zr atom . It is to be observed that the encapsulated Zr atom in Zr2Si11 receives more charges from its surroundings than the surface-capped Zr atom does and that the encapsulated Zr atom has a tendency to interact with more silicon atoms with unequivalent bond lengths, which is coincidence with that for Mo2Sin and Pd2Sin clusters obtained by Han and co-workers [45, 46]. Also, one can find that the charge properties are different for different cluster size. This finding may be result from cluster structures change with increasing cluster size that lead to chemical bonds change in cluster.
To gain further information about the internal charge transfer of Zr atoms, the natural electron configurations (NEC) of Zr atoms in the ground-state Zr2Sin clusters are calculated (results are summarised in Table 2). For free Zr atom, the configuration of valence electrons is 4s24p64d25s2. With regard to the impurities, the NEC values show that the 5s orbital lose electron 1.12–1.71, while the 4d, 5p, and 5d orbitals receive 0.87–2.27, 0.13–0.96, and 0.01–0.08 electrons for Zr2Sin clusters, respectively. So, one can conclude: (1) when n=1–6, the charges in the ground-state Zr2Sin clusters transfer from the 5s orbital of Zr atoms to the Si atoms and the 4d, 5p, and 5d orbitals of Zr atoms. Conversely, the charges in the corresponding Zr2Sin clusters transfer from the 5s orbital of Zr atoms and Sin frame to the 4d, 5p, and 5d orbitals of Zr atoms for n=7–11; (2) the electronic charge distributions of Zr2Sin clusters are primarily governed by s-, p-, and d-orbital interactions.
3.5 Chemical Hardness and Chemical Potential
As we know, the maximum hardness principle (MHP) proposed by Pearson  can be used to characterise the relative stabilities of a system. So, in order to further study the chemical stabilities of the most stable Zr2Sin clusters, the chemical hardness (η) and chemical potential (μ) are calculated and the results are listed in Table 3. Considering the influence of impurity Zr atoms on the pure silicon clusters, the experiment VIP values of pure Sin clusters are also summarised in Table 3. According to the finite-difference approximation and the famous Koopmans theorem , the η and μ are defined as follows :
From Table 3, one can see that the VIP of Zr2Si possesses the smallest value, implies that the Zr2Si cluster is very easily ionised than other clusters. The VIPs for Zr2Si4 and Zr2Si7 are larger than their neighbours, which is in conformity with the previous results discussed in Egap and Eb(n). Compared with the pure Sin clusters, one can see that the VIP values of Sin+2 clusters are bigger than those of Zr2Sin clusters obviously, which reflects that it is more easy for Zr2Sin clusters to gain electron than Sin+2 clusters. This result indicates that the doped Zr atoms can reduce the chemical stabilities of silicon clusters. In addition, the relationships of η and μ versus n are shown in Figure 5. One can see that the η for the doped system shows a oscillating behaviour as the cluster size increases. Three visible peaks occur at n=4, 7, and 10 in the curves are found, which implies that Zr2Si4, Zr2Si7, and Zr2Si10 clusters keep higher stability compared with their adjacent clusters. The calculated results are in accord with the Koopmans theorem, in which a soft molecule has a small HOMO-LUMO gap and a hard molecule has a large HOMO-LUMO gap . For chemical potential μ, three prominent minimums for the most stable Zr2Sin clusters are found at n=4, 7, and 10. The local minimum in the chemical potential at n=4 suggests that this cluster size is more stable than its neighbouring cluster sizes.
3.6 Infrared and Raman Spectra
In order to further explore the structure of a cluster, it is necessary to study the infrared (IR) and Raman spectra of clusters. The IR and Raman spectrum of magic numbers Zr2Si4 and Zr2Si7 are plotted in Figure 6 and the Supporting Information (Fig. S1). From Figure 6a, one can see that there are three obvious peaks in the IR spectra. The strongest intense IR frequency at 362.2 cm−1 results from the stretching vibration of Zr5–Zr6 bond; the double-degenerate intense IR frequency at 185.9 cm−1 is assigned to the in-plane wagging vibration of Si1–Si4 bond; and the triply degenerate peak at 313.9 cm−1 corresponds to the in-plane wagging vibration of Si2–Si3 bond. As shown in Figure 6b, there are three strong vibration peaks in the Raman spectra. The maximum Raman activity at 395 cm−1 is assigned to the breath vibration of Zr2Si4 cluster. The other two sharp peaks in the Raman spectra at frequencies 222.7 and 291.8 cm−1 result from the breath vibration of Zr2Si4 cluster and wagging vibration of Si1–Si4 bond, respectively. The IR spectra of Zr2Si7 is shown in Figure 6c. The highest intense IR peak at 294.7 cm−1 corresponds to the stretching vibration of Zr8–Zr9 bond. The second peak at 423.5 cm−1 is due to the stretchingvibration of Si6–Si3, Si6–Si4, Si7–Si3, and Si7–Si4 bonds. The third peak at 423.5 cm−1 comes from the wagging vibration of Si3–Si4 bond. In Figure 6d, the stretching vibration of Si atom gives rise to the strongest Raman peaks at 370.1 cm−1. The second strong peak at 351.7 cm−1 is assigned to wagging vibration of Si2–Si3 and Si5–Si4 bonds.
In this work, a systematic theoretical investigation of the equilibrium geometries, growth patterns, relative stabilities, and electronic properties of Zr2-doped Sin (n=1–11) clusters has been performed by using DFT at the B3LYP/GENECP level. All the calculated results are summarised as follows:
The optimised results show that the ground-state Zr2Sin clusters adopt the 3D structures for n=2–11. The Zr2Sin−1 structure capped with one Zr or Si atom and two Zr atoms substituted Sin cluster are the dominant growth patterns for Zr2Sin clusters of various sizes (n=1–11).
According to the calculated results of averaged binding energies, fragmentation energy, second-order energy difference, and the HOMO-LUMO energy gaps, it is found that the impurity Zr atoms in the Zr2Sin clusters reduce the chemical stabilities of silicon host. Zr2Si4 and Zr2Si7 clusters show the strong stability due to their local peak of all the curves (Fig. 3b and c).
The natural population and natural electronic configuration analysis indicates that the charges in Zr2Si1–6 clusters transfer from Zr atoms to the Si atoms, whereas in Zr2Si7–11 clusters charges transfer from the Si atoms to the Zr atoms.
The Zr2Si4 cluster has the largest chemical hardness and the smallest chemical potential.
This work was supported by the National Natural Science Foundation of China (Nos. 11274235 and 11304167), Postdoctoral Science Foundation of China (Nos. 20110491317 and 2014T70280), Open Project of State Key Laboratory of Superhard Materials (No. 201405), Program for Science & Technology Innovation Talents in Universities of Henan Province (No. 15HASTIT020), and Natural Science Foundation of He’nan Educational Committee (No. 2011A140006).
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Published Online: 2015-08-01
Published in Print: 2015-10-01