CHINESE JOURNAL OF PHYSICS VOL. 48, NO. 1 FEBRUARY University of the Philippines Los Banos, Laguna, Philippines 4031 (Received October 17, 2008)

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1 CHINESE JOURNAL OF PHYSICS VOL. 48, NO. 1 FEBRUARY 2010 Atomic and Electronic Structures of AuSi n (n = 1 16) Clusters: A First-Principles Study Feng-Chuan Chuang, 1 Chih-Chiang Hsu, 1 Yun-Yi Hsieh, 1 and Marvin A. Albao 1, 2 1 Department of Physics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan 2 Physics Division, Inst of Mathematical Sciences and Physics, University of the Philippines Los Banos, Laguna, Philippines 4031 (Received October 17, 2008) The structures of AuSi n (n = 1 16) clusters are investigated systematically using firstprinciples calculations. The lowest energy isomers exhibit a preference toward an exohedral rather than endohedral structure. Our studies suggest that AuSi n clusters with n = 5 and 10 are relatively stable isomers. We observed no significant alteration in the cluster s inner core structure for sizes n = 6, 7, 10, 11, 12, 14, and 15 in the presence of doping. Moreover, an analysis of the fragmentation energies is presented in detail. Our studies further indicate that doping of an Au atom significantly decreases the gaps between the highest occupied molecular orbital and the lowest unoccupied molecular orbital for n > 7. Additionally, we report on similar results obtained for CuSi n (n = 1 16) and AgSi n (n = 14, 15, and 16) and compare them with those of AuSi n clusters. Next, the low energy isomers for certain sizes of CuSi n (n = ) clusters are selected for further optimizations using the Gaussian 03 package. We found that for CuSi n (n = ), the endohedral isomers have lower energies than their exohedral counterparts, consistent with a recent study by Janssens et al. [Phy.Rev.lett.99,063401(2007)] in which a similar trend was observed. PACS numbers: Bc, c I. INTRODUCTION Metal-doped silicon clusters or cages have been the focus of extensive theoretical and experimental investigations [1 3]. Unlike bulk materials, a complete understanding of their properties has yet to be reached, despite significant headway gained in the field over the years. Among theoretical material scientists, the clusters novel properties in relation to the bulk present an opportunity to develop and refine existing theories of materials. For materials engineers, the challenge has been to explore potential applications in the microelectronics industry that take advantage of the clusters unique behavior. Not surprisingly, numerous theoretical studies [3 5] have been devoted to the determination of equilibrium geometries, electronic and bonding structures, as well as structural transitions of different sizes of both pure silicon and metal-doped silicon clusters. Compared with other metal-doped silicon cluster investigations [2 5], studies on Group I-B (Cu, Ag, and Au) metals have been mostly limited to CuSi n clusters [1, 6-12]. Following Beck s [1] observation of CuSi n clusters of various sizes, particularly n = 10, interest in copper-silicon clusters has remained high and sustained through the years, as evihttp://psroc.phys.ntu.edu.tw/cjp 82 c 2010 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 83 denced by a number of theoretical studies [7 10]. Recently, in a new attempt to fully explore Group IB metal doped silicon clusters, Duncan and his coworkers performed studies [13, 14] on the MSi n (M= Cu, Ag, and Au) clusters. Prior to this work, only copper-silicon clusters had been produced, and the production of silver and gold had so far been missing. In their report, Duncan s group explored and highlighted some of the most desirable properties of Group IB metal-silicon clusters [13, 14]. Their studies confirmed the similarities in the mass spectra of these systems [13]. Furthermore, they reported that for the case of a copper-silicon and silver-cluster [14], MSi n (n = 7 and 10) are the most abundant products. Photodissociation data, on the other hand, revealed that the most abundant fragment ions are Si + 7 and Si+ 10 for AgSi+ 7 and AgSi+ 10, respectively. For CuSi 7 the primary fragments detected are the Si + 7 from CuSi+ 7, and Si+ 6, CuSi+ 6, and Si+ 10 from CuSi+ 10. Together, these findings suggest that the clusters structure should have an external metal, and further reinforce the idea that both Ag-Si and Cu-Si bondings are weaker on average than Si-Si bonding. Notwithstanding the obvious similarities in the mass spectral data among the three metals, some differences exist in the photodissociation data in the form of available fragmentation channels for n = 10. Partly in response to the new findings by Jaeger et al. [14], we have performed a first-principles study on AgSi n (n = 1 13) in a previous work [15] and theoretically confirmed that the Ag atom is indeed exohedral. In a related experimental work, Janssen and co-workers reported that the physisorption of Ar is very sensitive to the position of the transition metal (TM) atom(s) on TM doped silicon clusters, and thus could serve as an indicator of whether the TM atom(s) is(are) exohedral or endohedral [16]. The same study revealed that for CuSi n clusters with n 12 the argon-complex formation is unlikely, thus suggesting a caged structure. Next, in analogy with AgSi n and CuSi n clusters, the most abundant gold-silicon product turns out to be AuSi 7, with AuSi 10, AuSi 11, and AuSi 12 somewhat less so [13], a phenomenon not replicated in the AgSi n and CuSi n cluster spectral data. However, aside from the published result in Ref. [13], no additional photodissociation data on AuSi n clusters has been reported to date. This seeming lack of data notwithstanding, it should be pointed out that the caged structure for AuSi n clusters has been included in numerous comparative studies in relation to other transition metals [17-20]. Nevertheless, whether the Au atom is exohedral or endohedral remains an open question. Thus, we believe that an extensive and definitive study on AuSi n clusters which aims to address unresolved issues, as well as one that highlights the similarities and differences among the Group IB metals, is still relevant and useful. In this paper, we first focus on AuSi n clusters and later extend the work by performing comparative studies between AuSi n and the other Group IB metal-silicon clusters, with the overall goal of making general observations regarding this family of metals. This paper is organized as follows: In Sect. II, we describe our computational methods. In Sect. III, we present the bulk of our work, which consists mostly of our efforts to determine the optimized structures for each metal cluster size, along with an analysis of the stability of these clusters. We end this article with Sect. IV highlighting the key findings and insights derived from this study.

3 84 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 TABLE I: Summary of the predicted lowest energy isomers of various metals, sizes, and structure types (endohedral or exohedral). The corresponding figure for the lowest isomer is indicated in the table. exohedral endohedral VASP Gaussian VASP Gaussian size, n Au Ag Cu Cu Au Ag Cu Cu Fig. Fig. Fig. Fig. Fig. Fig. Fig. Fig. 10 3(e) 3(e) 3(e) 3(f) 3(i) 3(j) 3(l) 3(j) 11 4(a) 4(b) 4(c) 4(c) 4(d) 4(d) 4(e) 4(e) 12 5(a) 5(b) 5(d) 5(c) 5(e) 5(e) 5(e) 5(g) 13 5(h) 5(i) 5(j) 5(i) 5(k) 5(k) 5(l) 5(l) 14 6(a) 6(a) 6(a) 6(a) 6(d) 6(d) 6(e) 6(e) 15 7(a) 7(a) 7(a) 7(a) 7(d) 7(e) 7(e) 7(d) 16 8(a) 8(b) 8(c) 8(c) 8(e) 8(e) 8(f) 8(e) II. COMPUTATIONAL DETAILS The calculations for both metal-silicon clusters and pure silicon clusters were done within the generalized gradient approximation (GGA), as parameterized by Perdew, Burke, and Ernzerhof (PBE) [21] for spin polarized density functional theory (DFT) [22] using projector-augmented-wave potentials (PAW) [23] as implemented in the Vienna Ab initio Simulation Package (VASP) [24]. The choice of the PBE functional and PAW pseudopotential is due to the inclusion of transition metals in the atomic cluster system. The kinetic energy cutoff is set to ev (18.03 Ry), ev (18.36 Ry), and ev (20.08 Ry), for MSi n, with M = Au, Ag, and Cu, respectively. The structural optimization was done with the conjugate gradient algorithm and with symmetry, until the forces on the atoms were less than ev/å. The length of the supercell was set to 15 Å. Every cluster is rotated such that the vector defined by the longest pair distance between any two atoms within the cluster lies along the diagonal direction of the simulation box. To optimize certain proposed models, a quasi-newtonian algorithm was used to relax the systems to their local minima. Next, initial test runs to ensure the validity of our methodology [15] were done, by verifying that the results on pure silicon clusters Si n (n = 2 16) are in agreement with the previous studies by Ho et al. [25], Liu et al. [26], and Lu et al. [27]. In order to classify the Au atom s structure (in relation to the Si cluster) as either endohedral or exohedral, we generated numerous initial structures for each Au-doped silicon cluster size, then allow each of these structures to reach their optimized geometry via a series of structural optimizations using first-principles total energy calculations. For this purpose, proposed candidates from the literature are taken as the initial configurations [2, 8, 15, 17, 18, 28-33] for the optimization procedures that follow. Two other techniques are used to generate more candidate structures. In one such technique, we replace one Si atom in a pure Si n+1 with an Au atom, resulting in a new structure for AuSi n. In the

4 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 85 FIG. 1: (Color online) Si dimer, Au dimers, and Au silicides, along with their bond lengths. The lowest energy isomers of neutral AuSi n (n = 2 5) are shown. The numbers under the structures are the relative energies per atom (in mev/atom) with respect to that of the lowest energy isomer of the same size. The Si atoms and Au atoms are shown in yellow (light gray) and grey (dark gray), respectively. other, we cap a pure Si n cluster with an Au atom at selected positions to create an AuSi n cluster. Further complementing these aforementioned techniques is an approach using a cluster growth model to generate hundreds of initial candidates for further optimizations. Together, these techniques yielded a total of over 150 (for sizes n = 9 and 10), 220 (for sizes n = 11 and 12) and 600 (for sizes n 13) initial structures. However, we caution the reader that our current approach may not be appropriate in cases where a silicon cluster is doped with two or more additional metal atoms. More complicated cases such as these may require more sophisticated approaches, such as a genetic algorithm [9, 34]. Here we should mention that in order to make the comparison among the lowest energy structures for the three metals valid, we deem it necessary to re-do the CuSi n (n = 1 16) clusters calculations instead of relying on previously known results for Cu [8]. This would ensure that the same level of accuracy is employed. Also, we extend our calculations to include results for the AgSi n clusters (n = 14, 15, and 16), which were not the focus of our prior work [15]. Lastly, the low energy isomers for certain sizes of CuSi n (n = 10 16) clusters are selected for further optimizations using the Gaussian 03 program [35]. As in

5 86 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 FIG. 2: (Color online) The lowest energy isomers of neutral AuSi 6, AuSi 7, and AuSi 8 clusters. the previous studies [8], we chose the G(d) basis set for both Cu and Si and applied the B3LYP method in order to account for the effect of inner core orbitals. III. RESULTS AND DISCUSSIONS III-1. Structures of AuSi n (n = 1 to 16) In this subsection, key results of our comprehensive search for the lowest energy structures of AuSi n (n = 1 16) clusters are first presented. Next, we will compare and contrast these with AgSi n and CuSi n clusters, thus enabling us to make general observations regarding the properties of the entire class of Group IB metals. Figs. 1 8 illustrate selected low energy isomers for each size. We would like to stress that the discussions in this subsection are centered on the results of our VASP (PBE/PAW) calculations. However, meaningful comparison with previous related studies on Cu [8] means that for CuSi n (n = 10 16) we need to use the Gaussian 03 package (B3LYP/6-311+G(d)), as this was the one employed in those studies. Starting with the smallest cluster AuSi, our calculations reveal that the bond lengths of the Si dimer, Au dimer, and AuSi dimer are 2.28, 2.52, and 2.25 Å, respectively. The bond length of AuSi turns out to be shorter than that of AgSi by 0.11 Å( 4.7%). This calculated AuSi bond length (2.25 Å) agrees with the values found in both an experiment (2.26 Å ) [6] and another theoretical study (2.25 Å) [7].

6 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 87 FIG. 3: (Color online) The lowest energy isomers of neutral AuSi 9 and AuSi 10. The numbers in the parentheses are for CuSi 9 and were obtained using the VASP package only, whereas the others are for CuSi 10 and were obtained using the VASP and Gaussian 03 packages, respectively. For AuSi 2, the optimized geometry shown in Fig. 1(a) is an isosceles triangle wherein the bond lengths are 2.43 Å for Au-Si and 2.33 Å for Si-Si. Here we remark that AuSi 2, AgSi 2, and CuSi 2 [15] all share the same lowest structure. Previously, we found that AgSi 3 shows a preference for a planar rhombus structure for its lowest energy isomer [15]. Our data on AuSi 3 and CuSi 3 indicate that this structure, as shown in Fig. 1(e), is common for

7 88 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 FIG. 4: (Color online) The lowest energy isomers of neutral AuSi 11. The numbers in the parentheses were obtained using the VASP and Gaussian 03 packages, respectively. all Group IB metals, though naturally Si-Si angles and metal-si bond lengths vary for each metal cluster [8, 15, 20]. Turning to the next bigger cluster, AuSi 4, our analysis indicates that a structure as shown in Fig. 1(j) wherein the Au atom is bonded to 3 Si atoms and collectively forms a distorted rectangular pyramidal structure is the most stable. For comparison, our analysis shows that CuSi 4 also shares this lowest structure (Fig. 1(j)), in agreement with a previous study [8], and that Fig. 1(k) is the lowest energy isomer for AgSi 4. As the Au-doped clusters get progressively larger, a trend toward increasing propensity for adsorption of an additional Au atom on the stable pure silicon clusters slowly emerges. A case in point is AuSi 5, where the lowest structure appears to be that of an Au atom adsorbed on a bipyramid (hexahedron) of Si 5, as illustrated in Fig. 1(o). We remark that other metals in this family, AgSi 5 and CuSi 5, similarly adopt this optimized geometry. For AuSi 6, one can create stable isomers such as those presented in Figs. 2(a), (b), and (c) by having one Au atom and one Si atom cap a bipyramid (hexahedron) of Si 5. Of these, the one shown in Fig. 2(a) is the lowest energy isomer for AuSi 6. For comparison, we included its lowest energy isomer counterparts for AgSi 6 and CuSi 6 in Figs. 2(b) and (c), respectively. The latter in particular shows an isomer which can be generated by substituting one Si in a pentagonal bipyramid of Si 7 with a metal atom. Other isomers of

8 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 89 FIG. 5: (Color online) The lowest energy isomers of neutral AuSi 12 and AuSi 13. interest are also shown for completeness. The low energy isomers for AuSi 7 are similarly presented in Figs. 2(f), (g), (h), (i), and (j). Of these, the one with the lowest energy (Fig. 2(f)) appears to be a pentagonal bipyramid of Si 7 in which an Au atom is attached to the side and slightly off the plane defined by the pentagon. In contrast, the corresponding metal atom in the lowest energy isomer counterparts for AgSi 7 and CuSi 7 is coplanar with the pentagon, as shown in Fig. 2(i).

9 90 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 FIG. 6: (Color online) The lowest energy isomers of neutral AuSi 14. Reminiscent of the AgSi n clusters behavior in our previous work, certain relatively large clusters increasingly show a tendency to form lowest structures by substituting an Si atom with a metal atom which is then subject to further relaxations. This trend slowly emerges starting with AuSi 8, and continues for bigger clusters. This new trend (substitution) is evident in Fig. 2(k) wherein the lowest energy isomer of AuSi 8 is such that one of the Si atoms in the lowest energy isomer of Si 9 is replaced by an Au atom. This same structure (Fig. 2(k)) is shared by CuSi 8, whereas AgSi 8 prefers a different structure which is displayed in Fig 2(l). To arrive at the latter, it is helpful to imagine a top Si atom of the lowest energy isomer for Si 9 (bicapped pentagonal bipyramid) being replaced by a metal atom, which then gradually moves away from this initial position. Shown in Fig. 2(m) is one Au atom capping the lowest energy isomer of Si 8 [26]. However, for AuSi 9, the lowest energy isomer illustrated in Fig. 3(a) is not structurally related to Si 9 and has not in fact been previously identified. On the other hand, the next lowest energy isomer shown in Fig. 3(b) is something which we have already seen in AgSi 9. Fig. 3(c) is the lowest energy structure for CuSi 9. Moving up to the next bigger cluster, we found that in fact all three Group IB metals adopt the same lowest energy isomer for n = 10. In addition, we also found that doping is generally favored outside for MSi 10, as illustrated in Fig. 3(e). These structures tend to retain the tricapped trigonal prism (TTP) of Si 9 with one silicon and one Group IB metal

10 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 91 FIG. 7: (Color online) The lowest energy isomers of neutral AuSi 15. atom capping the TTP. Having covered them previously in Ref. 15, however, it suffices to show two representative open structures from that study [15] in Figs. 3(g) and (h). We also examined the other known structural models of caged metal-doped Si 10 [8, 10, 15, 17, 31, 32], as shown in Figs. 3 (i), (j), (k), and (l). All these are indicative that the center-site structures are unfavorable for MSi 10 (M= Cu, Ag, and Au), as they are relatively high in energy compared to those in the adsorption and substitutional structures. Here we reiterate that all three Group IB metals adopt the same lowest energy isomer for n = 10 in Fig. 3(e). However, the earlier study [8] done for CuSi 10 indicates that Fig. 3(f) is the lowest energy isomer. Two CuSi 10 isomers as seen in Figs. 3(e) and (f) have the same TTP unit but with a different adsorption site for the Cu atom and are nearly degenerate in energy by mev per atom in our VASP calculations. Furthermore, a separate Gaussian 03 package calculation is in agreement with a previous study [8], and thus confirms that the isomer in Fig. 3(f) is indeed the most stable. In addition, the same calculations show that the isomers in Figs. 3(e) and (f) are degenerate and differ only by 0.48 mev per atom. Lastly, we verified that the caged and open structures as shown in Figs. 3(g) (l) are unfavorable for CuSi 10. We remark that for CuSi 10 the isomer in Fig. 3(h) relaxes to the structure in Fig. 3(i), whereas the isomers in Figs. 3(k) and (l) are reduced to the same configuration. Next, the six low energy isomers for AuSi 11 are presented in Fig. 4. Of these, the

11 92 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 FIG. 8: (Color online) The lowest energy isomers of neutral AuSi 16. lowest structure, shown in Fig. 4(a), is one wherein two silicon atoms and one gold atom cap the Si 9 cluster. A similar structure can be generated using the same procedure in which a pure Si cluster is capped with a metal atom and two Si atoms, but with the three atoms assuming different positions, resulting in slightly higher energies. This is illustrated in Fig. 4(b), which shows the lowest energy isomer for AgSi 11. Also, for CuSi 11, the most stable structure is shown in Fig. 4(c). For AuSi 12 and AgSi 12, the lowest structures favored vary, with a metal latching on to different Si atoms from the lowest energy isomer for pure Si 12 obtained from Ref. [26]. The isomers shown in Figs. 5(a), (b), and (d) are the lowest energy isomer for AuSi 12, AgSi 12, and CuSi 12, respectively. Figs. 5(e), (f), and (g) are the known Si 12 caged clusters formed by encapsulation of a metal atom [2, 18]. Just like in CuSi 10 and CuSi 11, we compare our result with the previous study [8]. For CuSi 12, we found that the energies of isomers as shown in Figs. 5(b), (c), and (e) are very close to the one in Fig. 5(d). As we can see, the result for CuSi 12 based on the VASP calculations shows that the energy is almost degenerate, making it impossible to make definitive conclusions. Therefore, we further optimized the CuSi 12 clusters using the Gaussian 03 package and confirmed that the caged isomer shown in Fig. 5(g) is indeed the lowest energy isomer. The isomers in Figs. 5(e) and (g) are energetically degenerate.

12 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 93 FIG. 9: (Color online) The relative energy per atom of the lowest endohedral isomer (in mev/atom) with respect to the lowest exohedral isomer as a function of the number of Si atoms in the clusters. Four low energy isomers for AuSi 13 are presented in Fig. 5, with the lowest energy isomer shown in Fig. 5(h). The next lowest isomer, shown in Fig. 5(i), is also the lowest energy isomer for AgSi 13, and is generated simply by capping the lowest energy isomer of Si 13 with a metal atom. The isomer shown in Fig. 5(j), on the other hand, is formed via a substitution of a metal atom with a Si atom in the lowest energy isomer of Si 14. Moreover, this isomer (Fig. 5(j)) has the lowest energy for CuSi 13. We further optimized the CuSi 13 clusters using the Gaussian 03 package and confirmed that the caged isomer, as shown in Fig. 5(l), is lowest in energy for CuSi 13. The trend in which the lowest energy isomer is a cluster of Si atoms capped with an Au atom continues for AuSi 14, as illustrated in Fig. 6(a). Its structure can be characterized as consisting of a TTP core plus 5 Si atoms and one Au atom on one side. The next two lowest energy isomers shown in Figs. 6(b) and (c) can be described as a pure Si 13 capped with a Au atom as well as Si atoms pointed to by the red arrows. For n = 14, 15, and 16, we also illustrate other low energy caged structures based on several ZrSi n structures from Ref. [30]. Here we remark that AuSi 14 and AgSi 14 share the same lowest structure using the VASP package. For CuSi 14, we note that the caged isomer in Fig. 6(e) is nearly energetically degenerate with Fig. 6(a) by 2.55 mev per atom. We further optimized CuSi 14 using the Gaussian 03 package. Our results confirmed that the caged isomer in Fig. 6(e) is

13 94 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 FIG. 10: (Color online) (a) Binding energy per atom and (b) second difference in binding energy (per atom) versus the number of silicon atoms in a cluster. (c) the HOMO-LUMO gap versus the number of silicon atoms in a cluster.

14 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 95 FIG. 11: (Color online) (a), (b), and (c) Fragmentation energy ( E n,m = E(MSi m ) + E(Si n m ) E(MSi n ), M = Au, Ag, and Cu) vs. the number of silicon atoms in a cluster. The legend explains the labeling of the smaller fragments.

15 96 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 FIG. 12: (Color online) The embedding energy of the listed endohedral isomers in Table I for MSi n (M= Cu, Ag, and Au; n = 10 16) are plotted as a function of the number of Si atoms in the clusters. indeed the lowest energy isomer. For AuSi 15 the two lowest energy isomers, as illustrated in Figs. 7(a) and (b), can be regarded as that of the lowest energy of a pure silicon cluster isomer (Si 15 from Ref. [25]) and an Au atom positioned at specific locations at its exterior. Here we note that AuSi 15, AgSi 15, and CuSi 15 all share the same lowest structure. Another low energy isomer is formed by joining two TTP units and capping the resulting structure with an Au atom as illustrated in Fig. 7(c). We are also able to report that calculations using the Gaussian 03 package likewise reveals that this endohedral isomer, as shown in Fig. 7(d), is the lowest energy isomer for CuSi 15. Next, we consider AuSi 16, the biggest cluster included in the present work. Its lowest energy isomer, displayed in Fig. 8(a), consists of the lowest energy Si 13 isomer adjoining a smaller cluster consisting of an Au and three Si atoms. The next lowest energy isomer shown in Fig. 8(b), on the other hand, is formed by capping the lowest energy AuSi 15 isomer (Fig. 7(a)) with an Si atom. Alternatively, one may regard it as an Si (pointed to by the red arrow) and an Au atom embedded on Si 15 in which these two atoms are slightly distorted. Moreover, this isomer shown in Fig. 8(b) is the lowest energy isomer for AgSi 16. The next isomer shown in Fig. 8(c) is similar to that in Fig. 8(b), except that the Si atom (pointed to by the red arrow) is found at the bottom of the structure. It also can be

16 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 97 generated from the lowest energy isomer for AuSi 15 in Fig. 7(a) by placing the Si atom at the bottom. Additionally, Fig. 8(d) is essentially a TTP unit bonded to an Si 7 -like cluster wherein a corner Si atom in the latter has been shifted to the surface. A metal atom at the tip completes the structure. We have also calculated the zero point vibrational energies, E vib, of the MSi n clusters (M= Au and Cu; n 10) using the VASP package. We did not observe any significant change in the lowest energy isomers when the stability of isomers are evaluated according to E tot = E 0 + E vib. In general, the zero point vibrational energies of the clusters increase with increasing cluster size, n. Furthermore, we report that the uncertainty due to the zero point vibrational energies for the different motifs considered here can be up to 13 mev per atom. However, for isomers of the same size and sharing a similar motif, this uncertainty is only less than 4 mev. The only change resulting from this re-recalculation is that the isomer for CuSi 13 shown in Fig. 5(k) now has a lower energy than that in Fig. 5(l). Here we should mention that these values compare well with the corresponding uncertainties due to the zero point vibrational energies obtained for CuSi n (n 10) using the Gaussian 03 package, which are all less than 5.2 mev per atom. Our recalculations reveal that the energy orderings for CuSi 10, CuSi 12, and CuSi 13 are affected, specifically those which are degenerate in energy. The new ordering in ascending total energy are reflected in the isomers shown in Fig. 3(l), Fig. 3(i), and Fig. 3(j). For CuSi 12, the isomer in Fig. 5(e) is lower than that in Fig. 5(g) by 0.1 mev per atom, whereas for CuSi 13 the isomer in Fig. 5(k) is lower than that in Fig. 5(i) by 0.4 mev per atom. Overall, we found that for sizes n = 1 16, the MSi n clusters, in which M is a Group IB metal, tends to be exohedral, except for CuSi n (n 12) in which both the lowest energy exohedral and endohedral isomers are energetically degenerate. Our studies further suggest that for exohedral isomers the Group IB metal generally favors capping a pure silicon cluster rather than substituting a silicon atom in the cluster. This latter tendency is in sharp contrast with Group VI-B transition metals (W and Cr) behavior in which endohedral atoms is the preferred structure. Furthermore, for n = 6, 7, 10, 11, 12, 14, and 15, doping does not alter the inner core structure in any significant way. However, for the relatively unstable Si clusters (n = 5, 8, 9, and 13) substitution may compete with capping as the dominant mode in forming the lowest energy structure in certain cases, and which unlike the latter, may cause significant structural changes in the clusters. Moreover, our studies suggest that the TTP unit is the dominant motif for the exohedral MSi n (n > 9) clusters. For the largest clusters considered here (n = 14 16), a pattern in which the Au atom prefers to cap the clusters emerges. Also, we found evidence that MSi n clusters behavior (as regards the lowest energy structures preferred for a particular size) shows some uniformity for certain sizes (2, 3, 5, 7, 10, 14, and 15). However, for n = 4, 6, 8, 9, 11, 12, 13, and 16, we found that three Group IB metals have different lowest energy structures. This might be attributed to the difference in mass spectral data in the range n > 10. We note that in some cases, the lowest energy isomers of CuSi n found using the VASP package are not identical to those obtained from the Gaussian 03 package. This might be attributed to two possible reasons. One is that the use of the PAW pseudopotential is not fully capable of reproducing the results of an all-electron calculation. The other is that the

17 98 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 specific exchange-correlation functionals employed in the two packages are not identical. III-2. Exohedral versus endohedral structures To illustrate the structural preferences of MSi n (n = 10 16) clusters, we identified both the lowest energy endohedral and exohedral isomers for each cluster size and summarized the results in Table I. The relative energy per atom of the lowest endohedral isomer (in mev/atom) with respect to the lowest exohedral isomer as a function of the number of Si atoms in the clusters is plotted in Fig. 9. For AgSi n and AuSi n, the endohedral isomers have much higher relative energies than their exohedral counterparts. Here we notice a pattern in which the relative energy gets smaller as the cluster size increases, signifying that the critical size beyond which endohedral structures are favored over exohedral ones may not be smaller than 17. In contrast, our studies indicate that for CuSi n there is a critical size n = 12. In fact, for cluster size larger than 12, the energy difference between the endohedral and exohedral isomers for each size is small and less than 16 mev per atom, as shown in Fig. 9. Notice that the lowest energy endohedral and exohedral isomers are energetically degenerate. As these are general trends, certain nuances are revealed by our VASP calculations. Specifically, we report that for CuSi n, where n = 12 and 14, exohedral isomers are slightly lower in energy than endohedral ones by 1.84, and 2.55 mev per atom, respectively. For n = 16, the caged isomer is preferred over exohedral structures by 7.60 mev per atom. As mentioned previously, we also recalculated the CuSi n (n = 10 16) with the Gaussian 03 package. We found that for n = 12, 13, 14, 15, and 16, the endohedral isomers have lower energies than their exohedral counterparts by 26.44, 7.07, 56.83, 14.96, and mev per atom, respectively. This is reminiscent of a recent experimental study by Janssens et al. [16] which demonstrated that there is a critical size beyond which the argon-complex formation is unlikely, suggesting that Cu metals are caged inside the Si clusters. III-3. Relative Stability of AuSi n A commonly employed method to evaluate the relative stability of clusters is to focus on the binding energy per atom, defined here as and E b (MSi n ) = [E(MSi n ) E(M) n E(Si)]/(n + 1) (1) E b (Si n ) = [E(Si n ) n E(Si)]/n, (2) where E(M) and E(Si) are the single atom energies and E(MSi n ) and E(Si n ) are the total energies of clusters. The binding energies (per atom) for metal-doped Si clusters and pure Si clusters as a function of cluster size are plotted in Fig. 10(a). This shows that for pure silicon clusters, those with size n = 7, 10, 12, and 15 are stable, whereas for Group IB metal doped Si clusters only n = 10 appears to be stable. Our current analysis points toward slightly higher binding energies for pure Si n clusters compared to their doped (MSi n ) counterparts and that doping does not necessarily lead to more stable clusters.

18 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 99 Another indicator of the relative stability of clusters is the second difference in binding energy per atom defined by and 2 E(MSi n ) = 2 E b (MSi n ) + E b (MSi n+1 ) + E b (MSi n 1 ) (3) 2 E(Si n ) = 2 E b (Si n ) + E b (Si n+1 ) + E b (Si n 1 ). (4) In Fig. 10(b), we plotted 2 E(n) as a function of cluster size. Stability analysis based on this plot suggests that among the various sizes for pure Si n, those with n = 7, 10, 12, and 15 are relatively stable. For AuSi n, the same analysis points toward stability of size n = 5, 10, and 13, noting that we already know n = 10 to be stable based on an earlier binding energy curve analysis. We remark that the spike in the curve observed in the case of AgSi n, a feature indicative of the stability of the cluster with n = 7, is missing in both the AuSi 7 and CuSi 7 plots. Additionally, the plot reveals that metal clusters with n = 5, 10, and n = 13 are stable for both AuSi n, whereas for CuSi n only n = 5 and 10 appear to be stable. For comparison, in the case of AgSi n the stable clusters are those with n = 5, 7, 10, and 13. Lastly, we point out that the conclusions regarding the stability of AuSi n (n = 5, 10, and 13) and CuSi n (n = 5 and 10) but excluding n = 7 partly agree with the mass spectral data obtained by Jaeger et al. [14], whose group studied Group IB metal-silicon clusters. In order to determine specific fragmentation pathways of clusters, as well as evaluate the stabilities of the resulting products, it is often useful to turn to the dissociation energy, or the energy needed to dissociate or break up a neutral cluster. The dissociation energy of an MSi n cluster needed to break it up into MSi m and Si n m clusters is provided below, E n,m = E(MSi m ) + E(Si n m ) E(MSi n ), (5) where E(MSi m ) is the total energy of the cluster with m silicon atoms (0 m n 1). We have evaluated the fragmentation energies of the neutral clusters for all possible pathways and provided plots for key pathways in Fig. 11. Analysis of fragmentation energies reveals AuSi n s propensity toward dissociation into an Au atom and Si n for n < 11 and n = 14, excepting n = 8, as AuSi 8 prefers to split into Si 7 and an AuSi dimer instead. For n = 11, 15, and 16, AuSi n clusters tend to dissociate into an Si 10 and either a AuSi dimer, AuSi 5, or AuSi 6. For n = 12 and 13, the clusters both break up into the stable Si 7 cluster and either AuSi 5 or AuSi 6. In contrast, for AgSi n clusters the preferred fragmentation channel is via evaporation of an Ag atom up to n= 15. For n = 16, clusters tend to break up into Si 10 and AgSi 6. Next, we have also calculated all the possible fragmentation pathways for CuSi n. We found that the primary pathway for the smaller CuSi n (n 11) clusters is via evaporation of one Cu atom, which is similar to AgSi n. For the next largest clusters examined, n 12, CuSi n fragmentation behavior is likewise similar to that in AuSi n, as discussed in the preceding paragraph.

19 100 ATOMIC AND ELECTRONIC STRUCTURES... VOL. 48 III-4. Embedding energy We also point out an alternative way of measuring the endohedral cluster s stability via the so-called embedding energy [36]. This quantity can be obtained from the fragmentation energy introduced in Eq. 5 above by setting m = 0 and letting E(MSi n ) = E(M@Si n ), defined as the total energy of the encapsulated isomers. The equation is now rewritten as follows: EBE 1 = E(M) + E(Si n ) lowest E(M@Si n ), (6) where E(Si n ) lowest is the lowest energy of the Si n cluster. The same definition was adopted in Ref. [10]. The embedding energy of the aforementioned endohedral isomers in Table I for MSi n (M= Cu, Ag, and Au; n = 10 16) are plotted as a function of the number of Si atoms in the clusters in Fig. 12. As we can see, embedding energies of CuSi n are higher than that of the other two metals, signifying that the MSi n (n = 10 16) clusters are most stable when the particular metal M is Cu. III-5. Electronic properties of MSi n (M = Au, Ag, and Cu) Certain cluster stability analyses invariably rely on the HOMO-LUMO gap (the difference between the highest occupied molecular orbital and the lowest unoccupied molecular orbital) being the prototypical electronic property and thus a staple in first-principles calculations. In this work, we set out to evaluate the HOMO-LUMO gaps of the lowest energy isomers for each cluster size using spin-polarized DFT calculations. Analysis of the data in our previous study indicates that the AgSi n clusters with n = 1, 5, and 12 have relatively wider HOMO-LUMO gaps, contrasting with our current results which indicate that the same is true only for AuSi n and CuSi n with n = 1 and 5. The new result for n = 12 could simply be attributed to the fact that all three metals have different stable configurations for this size, as discussed in Section III. Fig. 10(c) also shows that doping a Group IB atom (Au and Cu) significantly decreases the HOMO-LUMO gaps for n > 6. Furthermore, for pure Si n clusters, the results strongly indicate a correlation between the HOMO-LUMO gaps and the energetic stability, whereas for the MSi clusters (M= Cu, Ag, and Au) our data is inconclusive. IV. CONCLUSION In this paper, we have calculated the energetics of neutral AuSi n clusters (n = 1 16) corresponding to their equilibrium geometries, and whenever possible, compared and contrasted the results with other Group IB metals such as Ag and Cu. Firstly, we observed that for AuSi n and AgSi n, the exohedral isomers are dominant for the sizes that we studied. We also found that for CuSi n exohedral isomers are slightly lower in energy than endohedral ones for sizes n = 12 15, whereas the caged isomer is preferred over exohedral structures for n = 16. Furthermore, optimizations using the Gaussian 03 package (as opposed to the VASP package primarily used in this study) confirm that the critical size for CuSi n is 12.

20 VOL. 48 FENG-CHUAN CHUANG, CHIH-CHIANG HSU, et al. 101 This result is consistent with the recent experimental study by Janssens et al. [16] which demonstrated that there is a critical size beyond which the argon-complex formation is unlikely, suggesting that Cu metals are caged in the Si clusters. Our studies point toward an Au atom tendency to cap the Si clusters rather than be encapsulated inside Si clusters, a behavior reminiscent of Ag-doped clusters. We also found that among different cluster sizes, AuSi 5, AuSi 10, AuSi 13, and AuSi 15 are relatively stable. We note that the spike which we saw earlier in our previous work (see Fig. 9(a) in Ref. [15]) on AgSi n, indicative of n = 7 being stable, is missing in both the AuSi 7 and CuSi 7 plots shown in Fig. 10. We also found that for these clusters doping appears not to have much impact on the inner core structure, leaving it mostly unchanged. However, the same cannot be said of substitution, as it causes structural changes in the inner structure especially for the relatively unstable clusters. It also competes with capping as the primary mechanism for achieving the most stable configuration. Moreover, analysis of fragmentation energies reveals AuSi n s propensity toward dissociation into an Au atom and Si n for n < 11 (except n = 8) and n = 14. We also observed AuSi 8 s tendency to split into Si 7 and an AuSi dimer. We also observed that clusters with size n = 11, 15, and 16 tend to dissociate into an Si 10 and either an AuSi dimer, AuSi 5, or AuSi 6. On the other hand, those with n = 12 and 13 favor splitting into a stable Si 7 and either AuSi 5 or AuSi 6. Here we remark that CuSi n has fragmentation pathways that are very similar to AgSi n for n 11. Likewise, CuSi n fragmentation behavior is similar to that in AuSi n for n 12. Given the similarity, we are able to conclude that this fragmentation behavior is characteristic of Group IB metals in general. Acknowledgment We acknowledge support from the National Science Council of Taiwan under Grant No. NSC M MY3 and the Taiwan National Center for Theoretical Sciences (NCTS). We are also grateful to the Taiwan National Center for High-performance Computing (NCHC) for computer time and facilities. We thank Dr. K.-M. Ho and C.-Z. Wang at Iowa State University for fruitful discussions. References [1] S. M. Beck, J. Chem. Phys. 87, 4233 (1987); 90, 6306 (1989). [2] H. Hiura, T. Miyazaki, and T. Kanayama, Phys. Rev. Lett. 86, 1733 (2001). [3] V. Kumar and Y. Kawazoe, Phys. Rev. Lett. 87, (2001). [4] A. K. Singh, T. M. Briere, V. Kumar, and Y. Kawazoe, Phys. Rev. Lett. 91, (2003); H. Kawamura, V. Kumar, and Y. Kawazoe, Phys. Rev. B 70, (2004); H. Kawamura, V. Kumar, and Y. Kawazoe, Phys. Rev. B 71, (2005). [5] V. Kumar, A. K. Singh, and Y. Kawazoe, Phys. Rev. B 74, (2006). [6] J. J. Scherer, J. B. Paul, C. P. Collier, and R. J. Saykally, J. Chem. Phys. 102, 5190 (1995); 103, 113 (1995); J. J. Scherer, J. B. Paul, C. P. Collier, A. O Keefe, and R. J. Saykally, J.

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