JOURNAL OF CHEMICAL PHYSICS
VOLUME 119, NUMBER 7
15 AUGUST 2003
ARTICLES
The structure of silicon doped intermediate size carbon clusters
Eva González Noyaa)
Departamento de Fı́sica de la Materia Condensada, Facultad de Fı́sica, Universidad de Santiago
de Compostela, E-15782 Santiago de Compostela, Spain
Madhu Menonb)
Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055
and Center for Computational Sciences, University of Kentucky, Lexington, Kentucky 40506-0045
共Received 20 February 2003; accepted 23 May 2003兲
The lowest energy configurations of silicon doped carbon clusters of intermediate size (Cn Sim , n
⫹m⫽11,12, m⫽1,2,3) are investigated using generalized tight binding molecular dynamics
scheme and ab initio calculations. Our results favor low dimensional structures over
three-dimensional arrangements for these clusters. This trend is in agreement with photolysis
experiments that suggest linear chains to be more stable isomers. © 2003 American Institute of
Physics. 关DOI: 10.1063/1.1592152兴
that the spectra of the Sin C⫺
1 (3⭐n⭐7) are similar to those
⫺
clusters, and attributed it to being of similar
of pure Sin⫹1
geometry. However, this similarity is not present between
⫺
⫺
and Si1 Cm
, and this they attribute to structural change
Cm⫹1
from chains to rings. More recently, another study by
Pellarin et al.22,23 covered different composition regimes of
binary Sin Cm clusters by means of photolysis experiments.
For clusters containing n⭐20 carbon atoms, they reported an
enhanced stability for the most likely Cn Si⫹
2 clusters and
evidences for carbon chains terminated by one Si atom at
each end rather than more compact structures.
There are several theoretical studies dealing with the
topic of small Sim Cn clusters.24 –32 Presilla-Márquez
et al.26 –29 have combined measurements of frequencies, relative intensities, and isotopic shifts of vibrational fundamentals with ab initio calculations to identify new species or new
modes of known species including SiC2 , Si2 C, Si2 C2 , Si3 C,
SiC4 , Si2 C3 , Si3 C2 , and Si2 C4 . Froudakis et al.31 have performed an ab initio study of Si2 C4 , Si3 C3 , and Si4 C2 clusters and point out that, in the building-up mechanism of
silicon–carbon clusters, strong C–C bonds are favored over
Si–C bonds, while Si–Si bonds are much less important and,
in addition, multicenter bonding plays an important role.
More recently, Bertolus et al.32 made a systematic study of
Sim Cn with m⫹n苸 具 3,6典 using the density functional theory.
Their results showed that gradient corrections are ineffective
when applied to this system. Furthermore, they predicted a
segregation between silicon and carbon atoms, an increase in
dimension as the silicon content increases, and the existence
of a large number of isomers and transition states.
Despite these studies, to the best of our knowledge, no
theoretical work has been reported for clusters of intermediate sizes in the range n⫽10 to n⫽20. This is a challenging
issue since the lack of symmetry, that we might find for the
SiC hetero-clusters, would make the ab initio calculations
computationally prohibitive. Also, the high number of sta-
I. INTRODUCTION
The structures and properties of mixed silicon–carbon
hetero-clusters have attracted much attention over the past
several years. One of the main reasons for this being that
SiC, SiC2 , SiC3 , and SiC4 have been detected in the interstellar space and in the atmosphere of carbon stars.1–5 It is
quite probable that bigger clusters are also present in these
stars. Their detection, therefore, is important for an understanding of chemical processes in interstellar environments,
and their discovery has inspired several works which reported the production of such clusters in the laboratory.6 –9
Moreover, silicon carbide is an attractive material from a
technological point10 due to its high potential in electrical11
and high temperature mechanical12 devices.
Even though carbon and silicon are contiguous in the
same column 共Group IV兲 of the periodic table, their chemical
and bonding properties are rather different. While carbon exhibits huge flexibility by forming single and multiple bonds,
this characteristic is not shared by silicon which prefers multidirectional single bonds. The different behavior in the
bonding is reflected in the structure of small pure clusters.
Small carbon clusters (n⭐10) form planar structures, linear
for odd n and cyclic for even n. 13–15 By contrast, silicon
clusters are known to prefer three-dimensional structures
from clusters as small as n⫽5.16 –19 These striking differences are also evident in the bulk phase, since no graphitic
structure exists for Si. In the intermediate size silicon–
carbon hetero-clusters it is reasonable to expect to find a
transition from carbonlike to siliconlike behavior as we go
from carbon-rich to silicon-rich clusters. This trend has been
experimentally confirmed in previous works.20–23 For example, Nakajima et al.21 measured the photoelectron spectra
⫺
anion clusters (1⭐n⭐7 and 1⭐m⭐5), and found
of Sin Cm
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
b兲
0021-9606/2003/119(7)/3594/5/$20.00
3594
© 2003 American Institute of Physics
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J. Chem. Phys., Vol. 119, No. 7, 15 August 2003
tionary points in the potential energy surface 共PES兲 and their
proximity to each other would make the exploration of the
surface formidable. Consequently, the use of a computationally efficient method that allows us to sweep the PES as
exhaustively as possible, is crucial. In this respect, the generalized tight-binding molecular dynamics scheme
共GTBMD兲33 provides us with a powerful tool to study the
SiC hetero-clusters. This method has been proven to yield
geometries for silicon and carbon clusters34 –36 in agreement
with ab initio results. Recently it has also been successfully
applied to the study of small size Sim Cn clusters and SiC
hetero-fullerenes.33
II. THEORETICAL PROCEDURE
In order to effectively explore the PES of the SiC heteroclusters we use a combination of methods that includes
GTBMD scheme in conjunction with ab initio methods. The
first step involves scanning the surface using the GTBMD
scheme,33 thereby allowing us to go through a large number
of configurations in a reasonable time and make a first guess.
The structures found with this technique are then further relaxed using Hartree–Fock 共HF兲 method using the 6-31G*
basis set. The stability of the clusters was examined by
evaluating the harmonic frequencies with the same method
and basis set. Even though correlation effects are very significant in this system and are not taken into account with
HF, it can help us approach the minima and eliminate some
of the geometries considered. However, a careful look at the
vibrational analysis is necessary. For low frequencies HF is
often unable to distinguish real frequencies from imaginary
ones, confusing minima with saddle points and vice versa.
In the final step, the structures were optimized using the
density functional theory 共DFT兲 within the local spin density
approximation 共LSDA兲. The functional used was a combination of Slater exchange functional37 and Vosko–Wilk–Nusair
correlation functional 共SVWN兲38 using the 6-31G * basis set.
This method has been found to yield the best results for
small SiC clusters in accordance with experiment and accurate ab initio calculations among several functionals, including gradient-corrected functionals.32 All the HF and DFT calculations were performed using the GAUSSIAN 98 program.39
When the final geometry corresponds to a saddle point, a
small displacement is applied along the eigenvectors of the
normal mode associated with the imaginary frequency. Further relaxing of the system would then yield a true minima
connected to this stationary point.
III. GEOMETRIES OF THE LOW-ENERGY
CONFIGURATIONS
Following the techniques discussed in Sec. II, we have
examined the structures and energies of carbon clusters substitutionally doped with up to three silicon atoms (Cn Sim ,
n⫹m⫽11,12, m⫽1,2,3). The initial geometries were based
on the lowest-lying structures of C11 共Ref. 40兲 and C12 共Ref.
41兲, including in both cases linear, planar, and threedimensional configurations. In the search of the PES, many
minima and saddle points were found very close in energy
among themselves, supporting the trend previously reported
Silicon doped carbon clusters
3595
TABLE I. Energy of the most stable silicon doped C11 clusters, where the
energy is given in eV/atom. ␦ E denotes energy relative to the lowest energy
isomer found and N Imag is the number of imaginary frequencies.
SVWN
Cluster
C10Si
C9 Si2
C8 Si3
Isomer
C10Si
C10Si
C10Si
C10Si
C9 Si2
C9 Si2
C9 Si2
C9 Si2
C8 Si3
C8 Si3
C8 Si3
C8 Si3
共a兲
共b兲
共c兲
共d兲
共a兲
共b兲
共c兲
共d兲
共a兲
共b兲
共c兲
共d兲
HF
EB
␦E
N Imag
E B 共eV兲
␦E
N Imag
⫺9.27
⫺8.94
⫺8.89
⫺8.83
⫺8.64
⫺8.55
⫺8.38
⫺8.35
⫺8.05
⫺7.99
⫺7.93
⫺7.92
0
0.15
0.44
0.57
0
0.09
0.26
0.29
0
0.06
0.12
0.13
0
0
0
0
0
1
0
2
0
0
0
0
⫺6.22
⫺6.11
⫺5.88
⫺5.90
⫺5.94
⫺5.99
⫺5.37
⫺5.71
⫺5.36
⫺5.54
⫺5.08
⫺5.35
0
0.11
0.67
0.69
0.05
0
0.62
0.28
0.18
0
0.46
0.19
0
0
1
1
0
0
2
0
2
0
1
2
in the calculations of SiC clusters of smaller sizes.32 The
three-dimensional structures, however, were found to be sufficiently higher in energy when compared to linear and planar structures and can be ruled out as candidates for ground
state. We, therefore, limit our discussion mainly to the linear
and planar structures and a few cage structures. Also, only
the final structures resulting from the relaxation with the sequence of methods described are shown. All the molecular
states corresponding to the structures discussed are singlets,
except for the chains with 12 atoms, where the triplets were
found to be more stable.
It is worth noting that qualitative differences exist in the
frequencies and order of energies predicted by HF and
SVWN. As mentioned earlier, HF is only used as a tool to
approach the minima in the relaxation process. The omission
of correlation effects, extremely important in this system,
makes any predictions using this method inaccurate and unreliable, and this task is performed using the more accurate
SVWN LDSA method. Nevertheless, the use of the HF
method is justified here. Mühlhäuser et al.42 have studied the
effects of correlation in the Si3 C3 cluster by relaxing it using
the HF and calculations based on the Møller–Plesset 共MP兲
theory, which does include correlation. Change in the resulting geometry was found to be minimal 共around 1% in distances and 3–5 degrees in angles兲 and correlation was found
to affect only the relative energies of the different isomers
studied. This provides proof that HF, even with its deficiencies, can still be a useful tool to perform relaxation for obtaining different starting configurations and to locate stationary points in the PES.
A. Substitutional doping of the C11 cluster with
silicon: C10Si, C9 Si2 , and C8 Si3 clusters
The search results for the minima of the C11 cluster substitutionally doped with up to three silicon atoms is summarized in Table I and Fig. 1. In the three situations many
minima with energies very close among each other were
found. Consequently, the accuracy of the method used does
not allow us to determine exactly the true minima of the
PES. However, it appears that both linear and planar struc-
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3596
J. Chem. Phys., Vol. 119, No. 7, 15 August 2003
E. González Noya and M. Menon
TABLE II. Energy of the most stable silicon doped C12 clusters, where the
energy is given in eV/atom and ␦ E denotes energy relative to the lowest
energy isomer found. The linear isomer C9 Si3 was not found to be a stationary point using HF.
SVWN
Cluster
C11Si
C10Si2
C9 Si3
FIG. 1. Structures of the lowest energy isomers of the C11 cluster doped
with one, two, and three silicon atoms.
tures are competitive as most stable isomers, while threedimensional arrangements lie higher in energy, even when up
to three atoms are substituted 共0.51 eV/atom for C10Si, 0.54
eV/atom for C9 Si2 , and 0.12 eV/atom for C8 Si3 ).
Another characteristic worth noting is that the presence
of silicon atoms in a carbon cluster introduces some degree
of asymmetry. The original geometries are deformed to accommodate the silicon atom, which has a longer bond length
and prefers the directional single bonds. As a result, many of
the relaxed structures show very low degree of symmetry.
These binding preferences of silicon are clear in the linear
chain C8 Si3 共Fig. 1兲 where a bending at the edge containing
two silicons atoms is seen. The angle between the Si–C and
Si–Si bonds is ⬇150°.
B. Substitutional doping of the C12 cluster with
silicon: C11Si, C10Si2 , and C9 Si3
In Fig. 2 we show low energy geometries of Cn Sim clusters with n⫹m⫽12. Table II contains a summary of results.
As seen in the table, the various isomers lie very close in
energies to each other, making it virtually impossible to list
one single structure as the most stable.
FIG. 2. Structures of the lowest energy isomers of the C12 cluster doped
with one, two, and three silicon atoms.
HF
Isomer
EB
␦E
N Imag
E B 共eV兲
␦E
N Imag
C11Si 共b兲
C11Si 共a兲
C11Si 共c兲
C11Si 共d兲
C10Si2 共a兲
C10Si2 共b兲
C10Si2 共c兲
C10Si2 共d兲
C9 Si3 共a兲
C9 Si3 共b兲
C9 Si3 共c兲
C9 Si3 共d兲
⫺9.01
⫺9.00
⫺8.98
⫺8.86
⫺8.77
⫺8.70
⫺8.56
⫺8.56
⫺8.35
⫺8.24
⫺8.22
⫺8.19
0
0.01
0.03
0.15
0
0.07
0.21
0.21
0
0.06
0.15
0.16
0
0
0
1
0
0
0
1
0
1
0
0
⫺6.23
⫺6.17
⫺6.23
⫺5.86
⫺6.08
⫺6.06
⫺5.94
⫺5.77
⫺5.87
⫺5.62
⫺5.68
0
0.06
0
0.37
0
0.02
0.18
0.01
0
0.25
0.19
1
1
4
1
0
2
0
3
0
0
0
In this case, planar or nearly planar single and double
rings and linear chains appear to be likely candidates for the
ground state. As in Sec. III A, three-dimensional structures
can be ruled out as possible candidates in this range of composition, except perhaps for the C9 Si3 cluster, where a bowl
structure 共Fig. 2兲 is among the lowest in energy.
These results lead us to conclude that the ground state of
clusters with these compositions are geometries with low
dimensions, consisting of either chains or single and double
rings. This may seem counter intuitive because silicon atoms
prefer single bonds (sp 3 hybridization兲 which is not compatible with the sp 2 hybridization found in linear and planar
structures. However, we note that this behavior was also
found in experiments. Pellarin et al.22,23 proposed linear
chains in this regimen of composition and found an enhanced
stability for the Cn Si⫹
2 clusters.
IV. BONDS AND COORDINATION
It is well known that silicon and carbon behave quite
differently when forming bonds. With this in mind we analyze the different types of bonds formed by the two elements
and present a histogram with the interatomic distances of 37
clusters studied 共see Fig. 3兲. This group of clusters includes
the more stable isomers of all the stochiometries considered
in this work.
For the C–C bonds, there are four distributions centered
at 1.28 Å, 1.35 Å, 1.41 Å, and 1.54 Å. As expected, the
geometry of the clusters determines, for the most part, the
range of the bonds. The smallest set of bonds arises mainly
from one-dimensional chains, and it is coincident with the
double bond in allenes. Two-dimensional structures contribute mainly to the maximum at 1.35 Å. The distribution at
1.41 Å 共distance between carbons in a graphene layer兲 derives contribution from both; two- and three-dimensional
structures, the latter showing another peak at 1.54 Å 共the
single bond length for carbon兲.
For the Si–C interdistances, the histogram shows two
distributions: one at 1.70 Å and a fairly broad distribution
centered at 1.84 Å, which is the bond length of the single
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J. Chem. Phys., Vol. 119, No. 7, 15 August 2003
Silicon doped carbon clusters
3597
to the complexity of the problem, it is not possible to propose one single structure as the global minimum in the PES.
Additionally, we have also presented a study of bond
length distribution, analyzing both neighbor distances and
the coordination. The trend found is similar to the one proposed by Bertolus et al.32 in a similar study of smaller Si–C
clusters.
We believe that the method would be equally valid for
the Si clusters doped with carbon. The candidates for ground
state geometries, however, will be dominated by threedimensional structures. This is because orbitals in Si atoms
prefer to form multidirectional single bonds. The ability of C
atom to form various bonding configurations makes substitutional doping relatively easy. The mass spectroscopy experiments of Pellarin et al.22,23 suggests that in clusters containing up to 11 Si atoms a simple substitution of carbon
atoms seems to be highly plausible.
FIG. 3. Bond length distribution for all the SiC clusters studied in this work
in the form of a histogram.
Si–C bond. As before, it can be seen that the dimensionality
of the structures is closely related to the distances between
neighbors in the molecule. In fact, only linear structures contribute to bond lengths of 1.70 Å. For the Si–Si bond the
lack of statistics does not allow us to make a reliable analysis.
It is worth noting that hypervalence 共more than four
bonds per atom兲 was not found for any clusters and that most
of the atoms contained a coordination of two. This finding is
consistent with the results reported by Bertolus et al.32 in a
similar study of small Si–C clusters, where up to six atoms
were considered. The results for both the coordination and
the bond lengths are fairly close to the values obtained in our
analysis.
V. SUMMARY AND CONCLUSIONS
In this work we have used the GTBMD scheme to search
for possible ground state candidates for silicon doped carbon
clusters, Cn Sim , n⫹m⫽11,12, m⫽1,2,3. The efficiency of
this technique allowed us to analyze up to hundred of configurations and make a first guess. This was followed by
more accurate calculations, based on the DFT and the LSDA
approximation, to further relax the geometries and get the
binding energies. As a result, it was found that Si–C heteroclusters present a fairly complex PES with numerous minima
and saddle points that lie very close in energies, making the
task difficult in the search for the minima and defying the
resolution of the theoretical methods used. Moreover, a
strong competition between linear chains and planar or
nearly planar structures occurs, while three-dimensional
structures are not seen as possible candidates for the ground
state, even when three silicon atoms are inserted in the cluster. This is quite surprising as silicon is known to prefer
single bonds which would form three-dimensional geometries. Nevertheless, our finding is consistent with photolysis
experiments performed by Pellarin et al.22,23 However, due
ACKNOWLEDGMENTS
This work was supported by NSF 共NER-0165121, ITR0266277兲, DOE Grant 共00-63857兲, NASA Grant 共00463937兲, and the University of Kentucky Center for
Computational Sciences. E.G.N. wishes to thank L. J. Gallego and C. Rey for useful discussions and acknowledges
support from the CICYT, Spain 共Projects Pb98-0368-C02
and MAT 2002-03142兲 and the Xunta de Galicia 共Projects
PGIDT01PXI20605PR and PGIDT00PXI20611PN兲.
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