Keywords: doped clusters of silicon, cage and basket structures, total energy calculations. 1. ... clusters have been done since the fullerene like structure was.
Materials Transactions, Vol. 45, No. 5 (2004) pp. 1429 to 1432 Special Issue on Advances in Computational Materials Science and Engineering III #2004 The Japan Institute of Metals
Growth and Magic Behavior of Metal Encapsulated Silicon Clusters Hiroaki Kawamura1 , Vijay Kumar1;2 and Yoshiyuki Kawazoe1 1 2
Institute for Materials Research (IMR), Tohoku University, Sendai 980-8577, Japan Dr. Vijay Kumar Foundation, 45 Bazaar Street, Chennai 600 078, India
Metal encapsulated silicon clusters M@Sin (M = Ti and Cr and n = 8–16) have been studied using ab-initio ultrasoft pseudopotential method. Several structures for each cluster have been optimized to obtain the lowest energy isomers. Our results show that cage structures begin to form at the size of n ¼ 12 for Cr@Sin and 13 for Ti@Sin . For Ti@Sin our results are in excellent agreement with the available experimental results. In smaller size, metal doped silicon clusters have basket structures to be of the lowest energy. The bonding nature in these clusters is discussed from the electronic charge distribution. (Received December 11, 2003; Accepted February 16, 2004) Keywords: doped clusters of silicon, cage and basket structures, total energy calculations
1.
Introduction
Clusters of silicon are interesting for the design of nanodevices. Their properties change with size and there are exciting possibilities of luminescence in the visible range. Self-assembly of clusters is a key way to build nano-devices and therefore searching for stable clusters that could serve as building blocks, is an important task. Silicon is the most important material for semiconductor devices. Therefore its clusters have attracted much attention. However, these often have a size distribution. The recent finding1) of metal doped silicon clusters has opened up a new direction. These clusters have higher stability, high symmetry (in some cases) and size selectivity which can lead to their mass production. Fullerene structures for metal encapsulated silicon clusters have been predicted by Kumar and Kawazoe prior to experiments. In that report various metal atoms were doped in Sin (n ¼ 14–16) clusters and exceptionally large highest occupied-lowest unoccupied molecular orbital (HOMOLUMO) gaps were obtained for Ti@Si16 and Hf@Si16 with Frank-Kapper (FK) structure. After that report, further possibilities of doping of other transition metal atoms, such as Cr, Mo and W2,3) and using Ge or Sn instead of Si clusters4,5) as well as adsorption of hydrogen,6) have been studied. Recently experiments by Ohara et al.7) have confirmed high stability and cage structures of metal doped silicon clusters. Their experimental results obtained from the timeof-flight mass spectra show peaks at n = 15 and 16 for M@Sin (M ¼ Ti, Mo, Hf and W) as predicted by theoretical calculations. Further in order to determine the size at which cage structures would be formed, experiments have also been performed on the reactivity of H2 O molecules with Ti@Sin clusters. In experiments it is assumed that the reactivity of H2 O on titanium doped clusters is small if the metal atom is fully covered by silicon. It is reported that H2 O molecule could adsorb on Ti@Sin clusters up to the size of n = 12 so that Ti@Sin clusters with (n > 12) are thought to form cage structures. Although not so much experimental work has been reported yet, many theoretical studies on metal doped silicon clusters have been done since the fullerene like structure was
suggested. Lu and Nagase,8) Hagelberg et al.9) and Sen and Mitas10) have studied various kinds of metal atoms for doping Si clusters. Fe doped clusters have been investigated by Mpourmpakis et al.11) and Khanna et al.12) Khanna et al. also reported studies on chromium doped silicon clusters.13) However, all these reports treat only a few specific cluster sizes for M@Sin and it is difficult to investigate general trends of the properties of these clusters. Therefore in order to understand the effects of changing the doping metal atom and increasing the cluster size, the study of a wide range of cluster size for each kind of doped metal is indispensable. Here we present results of our studies of Ti and Cr doped silicon clusters in the range of n ¼ 8 to 16. Treating such a wide range of sizes enables us to discuss the general trends of structures, properties and growth behavior of metal doped silicon clusters. In Sec. 2 we explain briefly the calculation method. The results are presented in Sec. 3. Section 4 contains conclusions of the paper. 2.
Computational Method
The calculations on neutral clusters have been performed using the ab initio ultrasoft pseudopotential14,15) plane wave method, within the spin-polarized generalized gradient approximation for the exchange-correlation energy.16) A simple cubic supercell with size 1.5 nm for n ¼ 8–13 and 1.8 nm for larger clusters is used with periodic boundary conditions and the point for the Brillouin zone integrations. For Ti we treat 3p atomic core states also as valence states. Selected atomic structures of clusters are optimized using the conjugate gradient method to find the lowest energy structures. 3.
Results and Discussion
3.1 Titanium doped silicon clusters Titanium doped silicon clusters are interesting system because titanium is the only metal for which experiments on adsorption of H2 O molecules on M@Sin have been reported. Figure 1 shows the most stable and some interesting structures for Ti@Sin (n ¼ 8–16). Binding energies per atom, the second order difference of total energies and
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H. Kawamura, V. Kumar and Y. Kawazoe
n=8
n=9
n=10
(i)
n=10
(ii)
( +0.037 ) n=11
(i)
n=12
(ii)
n=12
(iii)
( +0.273 ) n=13
n=13
(i)
n=14
n=12
( +0.680) n=15
n=16
(ii)
( +0.878 ) Fig. 1 Optimized structures of Ti@Sin (n ¼ 8–16). The ball located inside each cluster shows Ti atom while all other balls show Si atoms. The difference of the total binding energy of an isomer with respect to the most stable one is given in eV.
Table 1 Binding energies (BE) per atom, the second order difference of the total energies (E) and HOMO-LUMO gaps for Ti@Sin (n = 8–16). E is defined as E(n 1) + E(n þ 1) 2E(n), where E(n) is the total energy of the most stable isomer of the Ti@Sin cluster. Positive values of E mean that Ti@Sin clusters are stable against fragmentation to Ti@Sinþ1 and Ti@Sin1 . Cluster size 8 9
Structure
BE/atom (eV)
E (eV)
gap (eV)
basket basket
3.798 3.822
0:073
1.233 0.986
10(i)
basket
3.848
0:617
1.187
10(ii)
basket
3.845
0:692
0.732
11
basket
3.921
0.386
1.299
12(i)
basket
3.953
0:282
1.257
12(ii)
cage
3.932
0:829
0.763
12(iii)
cage
3.901
1:642
0.160
13(i) 13(ii)
cage basket
4.001 3.938
0.322 1:160
1.569 0.718
14
cage
4.021
0.252
1.451
15
cage
4.077
0:136
16
cage (FK)
4.135
1.578 2.358
HOMO-LUMO gaps are given in Table 1. For Ti@Sin up to n ¼ 12 basket like structures are most favorable. These are shown in two different views in Fig. 1 to point out that Ti atom is not fully covered with Si atoms and has a bare part. At n ¼ 10, two stable isomers are found and the difference in the total binding energy is only 0.037 eV so that these two isomers are nearly degenerate and are likely to be present simultaneously. Among the basket isomers, only Ti@Si11 has positive E. In this size range the binding energy increases
nonlinearly and thus E has negative values. However n ¼ 12 is not suitable in both the basket and the cage structures, because in the basket structure one Si atom is located at the outside of the basket so that this Si atom does not make strong bond with Ti atom and in the cage structure one Si-Si bond is too elongated to be bonded (Fig. 1). Therefore n ¼ 12 is relatively unstable and as a result of it n ¼ 11 has positive E. In other words n ¼ 11 is the most favorable size for basket structure when Ti is doped. Though hexagonal prism structure seems to be stable for M@Si12 , Si atoms could not form a symmetric prism structure because of the large ionic radius of Ti. The elongated (not bonded in Fig. 1) Si-Si bond length is 0.300 nm so that the interaction between these two Si atoms is quite weak. The other Si-Si bonds in this cluster are about 0.237 nm. This value is close to the covalent bond length in bulk silicon (0.234 nm). Considering Si-Si bond lengths in other smaller Ti@Sin (0.247 nm for n ¼ 11 and 0.245 for n ¼ 12 basket isomers, for example), the length of 0.237 nm is quite short and this is the reason why one bond is elongated in the prism isomer. Cutting one bond enables to make such short Si-Si bonds and the tight bonds stabilize the cluster as compared to the case in which all Si-Si bonds are kept close to the values as in other clusters. We have also calculated anti-hexagonal prism isomer. However, it lies 0.680 eV higher than the basket isomer. n ¼ 13 is the first size in the growth process which has a cage structure as the most stable isomer. Ti@Si13 with a capped hexagonal prism structure has significantly high HOMO-LUMO gap (1.569 eV) and E (0.322 eV) is positive. Therefore, this cluster should show magic behavior. But in the reported mass spectrum n ¼ 13 does not show a high
Growth and Magic Behavior of Metal Encapsulated Silicon Clusters
n=8
n=9
n=13
Fig. 2
n=10
n=14
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n=11
n=15
n=12
n=16
Optimized structures of Cr@Sin (n ¼ 8–16). The ball located within each cluster shows Cr while others show Si atoms.
Table 2 Binding energies (BE) per atom, second order difference of the total energies (E) and the HOMO-LUMO gaps for Cr@Sin (n ¼ 8–16). The definition of E is the same as in Table 1. Cluster size
Structure
BE/atom (eV)
E (eV)
gap (eV)
8
basket
3.528
9
basket
3.615
0:911
0.997
10
basket
3.770
1.649
1.328
11 12
basket cage
3.761 3.835
1:061 0.858
1.382 0.847
0.863
13
cage
3.837
1:175
0.711
14
cage
3.918
0.600
1.536
15
cage
3.950
0.768
1.538
16
cage
3.934
1.244
peak. There could be some difference in the calculated behavior due to the different charge state of the clusters. In the experiment clusters are ionized while in the calculation clusters are neutral. However our result that Ti@Sin could form a cage structure from n ¼ 13 is in excellent agreement with the experiments on H2 O adsorption which show significant decrease of reaction ratio from n ¼ 13 onwards. 3.2 Chromium doped silicon clusters Although chromium doped silicon clusters have already been reported for specific sizes,6,13) we have calculated many isomers for different sizes. Figure 2 shows the lowest energy structures for Cr@Sin (n ¼ 8–16) among all the calculated isomers. Their BE/atom, E and HOMO-LUMO gaps are given in Table 2. Because Cr has smaller atomic radius as compared to Ti, Cr atom could be surrounded by fewer Si atoms. This tendency is clearly seen in Cr@Si9 in which 9 Si atoms cover half of the Cr atom, while only a part of Ti is covered in Ti@Si9 (see Fig. 1). Cr@Si10 has exceptionally large positive E. This high stability is explained by tight SiCr bonds. The average distance between Cr and Si is 0.244 nm for n = 10 and this is the shortest average Cr-Si bond length among Cr@Sin (n ¼ 8–16) clusters. On the other hand Cr@Si10 has the longest average Si-Si bonds (0.251 nm)
among them. This indicates that the stability of basket structures depends on M-Si bonds, otherwise Si would try to take more high coordination. This idea gives reasonable understanding of Ti@Si11 . The lowest energy structure of Cr@Si11 is obtained by adding a Si atom on the Cr@Si10 cluster and this extra Si atom has only 2 coordination so that this Si atom should be stabilized by the Cr-Si bonding. Cr@Si12 could form hexagonal prism structure due to smaller ionic radius of Cr in contrast to Ti. This prism isomer has the shortest average Si-Si bond length (0.235 nm) and beacuse of the tight bonding it has a large positive E. This high stability of Cr@Si12 makes Cr@Si13 less stable (E ¼ 1:175 eV). As it can be seen in Fig. 2, the capping Si atom on the hexagonal prism isomer causes distortions. The high coordination of the capping Si atom does not favor covalent bonding with the other Si atoms. A comparison of our results with the reported ones13) shows that the reported structures of Cr@Si11 and Cr@Si12 are similar to ours, but Cr@Si13 and Cr@Si14 are quit different. About Cr@Si13 we tried the reported structure and obtained the similar isomer. However it lies 0.235 eV higher in energy than the capping hexagonal prism isomer obtained by us. In the case of Cr@Si14 we couldn’t obtain their structure which is quite distorted and has one capping Si atom. 3.3 Bonding nature To understand the bonding nature in these clusters, constant electronic charge density surfaces have been shown for Ti@Sin in Fig. 3. For Ti@Si11 basket isomer two different views (Figs. 3(a-i) and (a-ii)) are shown, because the bonding character of this cluster consists of two parts. In Figs. 3(a-i) except for the bottom 4 atoms, the charge is attracted from the bonds between the Si atoms to Ti. Figures 3(a-ii) shows the bottom side of the cluster and the charge is localized on the Si-Si bonds consisting of 4 bottom Si atoms. Therefore, at the bottom side covalent character is strong, while in the upper side both metallic and covalent characters are present. The contrast between metallic and covalent characters is clearly shown in Figs. 3(b) and (c). In the hexagonal prism
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H. Kawamura, V. Kumar and Y. Kawazoe
(a-i)
(a-ii)
atom. Ti doped Si clusters could form a cage structure at n = 13 and this size is in excellent agreement with experiments. Cr has smaller atomic radius than Ti so that it could form cage at n ¼ 12. Bonding nature depends on structures of clusters. Hexagonal and anti-hexagonal prism structures have covalent and metallic character, respectively. Basket isomers have both characters and M-Si bondings have large contribution to their stability. Acknowledgments
(b)
(c)
VK thankfully acknowledges the kind hospitality at the Institute for Materials Research and the support from Japan Society for Promotion of Science (JSPS). We are grateful to the staff of the Center for Computational Materials Science at IMR for making Hitachi SR8000/64 supercomputer available. REFERENCES
Fig. 3 Constant electronic charge density surfaces at 410 eV/nm3 for (a)Ti@Si11 basket isomer from two different views, (b)Ti@Si12 hexagonal antiprism isomer and (c) Ti@Si13 capped hexagonal prism isomer.
isomer (Fig. 3(b)), strong covalent Si-Si bonds could be found except for the capping Si atom. Anti-hexagonal prism isomer (Fig. 3(c)) has completely different charge distribution from others. Because of the high coordination of Si, Si atoms could not form covalent bonding so that the charge of Si atoms is distributed around Si more uniformly. Thus antihexagonal prism isomer has strong metallic bonding character and this is the reason why hexagonal prism isomer is energetically more favorable than anti-hexagonal one. 4.
Summary
In summary, we have presented results of our studies on growth behavior of titanium or chromium doped silicon clusters. We find that the size where cage structure could be formed depends on the atomic radius of the doped metal
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