PHYSICAL REVIEW B
VOLUME 62, NUMBER 23
15 DECEMBER 2000-I
Solid C36 as hexaclathrate form of carbon
Madhu Menon*
Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055
and Center for Computational Sciences, University of Kentucky, Lexington, Kentucky 40506-0045
Ernst Richter†
DaimlerChrysler AG FT3/SA, Wilhelm-Runge-Strasse 11, 89081 Ulm, Germany
Leonid Chernozatonskii‡
Institute of Biochemical Physics, Russian Academy of Sciences, Moscow 117977, Russian Federation
共Received 10 August 2000兲
The structure and stability of the newly synthesized C36 solid is investigated using a generalized tightbinding molecular-dynamics method. Symmetry and electronic structure analysis point to a new form of carbon
clathrate containing C36 fullerenes as building blocks. All of its atoms have sp 3 bonding and the structure has
exact D 6h symmetry. This is consistent with the electron-diffraction pattern of the new solid.
Cluster-based solids illustrate the importance of local order in determining global properties in solids. These solids
add a new dimension to material science. The stronger intracluster bonding in these materials allows them to keep their
individual identity while forming part of the bulk material.
The most striking example of this is the C60 molecule.1–3
Bulk carbon crystals made of C60 are highly stable and form
a metastable phase of carbon in addition to the energetically
most stable bulk graphite. C60 molecules form a facecentered-cubic 共fcc兲 crystal lattice at room temperature with
weak van der Waals type bonding between the molecules.4
Furthermore, Raman-scattering investigations of visible or
ultraviolet irradiated solid C60 film has revealed yet another
solid form in which photopolymerized fullerene molecules
were observed with covalent intermolecular bonding in contrast to the weak van der Waals type bonding found in the
crystal.5
The solid forms of other fullerenes can also be expected
to be stable when synthesized under optimal conditions. Indeed very recently, successful synthesis of a solid form composed of C36 fullerene molecules has been reported using the
arc-discharge method.6 The C36 solid films were found to be
insoluble in toluene and benzene indicating that the bonding
is covalent and not van der Waals type. Furthermore,
electron-diffraction patterns suggested very high symmetry
(D 6h ) and an intermolecular separation of 6.68 Å. Note that
an earlier work on the C36 ‘‘barrel’’ using the Hückel approach had indicated the formation of the C36 dimer to be an
exothermic reaction.7
Soon after the synthesis of the C36 solid was reported, a
number of theoretical calculations on various plausible structures were carried out by some groups.8–11 In particular, Cote
et al.8 using the density-functional theory calculations within
the local-density approximation 共LDA兲 proposed a rhombohedral structure for this solid. Optimizations were performed
by keeping the symmetry fixed. They found this solid to be
metallic with a half-filled conduction band. However, at
room temperature, the pure C36 samples were found to be
insulating with a resistivity in excess of 107 ⍀ cm, 6 and the
0163-1829/2000/62共23兲/15420共4兲/$15.00
PRB 62
scanning tunneling spectroscopy measurements showed the
C36 solid to be semiconducting with an energy gap of 0.8
eV.9 This was based on the observed highest occupied molecular orbital 共HOMO兲 and lowest unoccupied molecular
orbital 共LUMO兲 features in the density of states for C36 islands. The authors of Ref. 9 also observed similarities between the experimental electronic spectrum and those calculated for an isolated C36 molecule as well as for one of the
C36 dimer configurations 共dimer C in Ref. 9兲. This led them
to conclude that this dimer may be considered the basic
structural unit for the solid, although they did not provide the
exact structure.
Fowler et al.,10 employing density-functional tightbinding optimization, have considered various solid forms,
including a ‘‘super-benzene’’ oligomer, a ‘‘super-graphite’’
layer, and a hexagonal closed-packed 共hcp兲 solid and suggested the hcp solid with a lattice spacing of 6.82 Å to be
the candidate compatible with experiment. The hcp structure
was found to be more stable than either rhombohedral or
supergraphite. The authors of Ref. 10 also did not perform
symmetry unrestricted molecular-dynamics optimizations.
Menon and Richter,11 on the other hand, using a generalized tight-binding molecular-dynamics 共GTBMD兲 scheme
proposed two classes of solids, including polymeric and
clathrate forms of C36 . Although the starting configurations
for both classes had the perfect D 6h symmetry, only the
clathrate forms retained this symmetry on full relaxation.
Although using different methods, all these schemes predict the C36 solid to be covalent. Furthermore, they also predict relatively small energy differences between competing
structures. In order to gain a further improvement in the understanding a more detailed study of all the proposed geometries using symmetry unconstrained optimization based on
molecular-dynamics methods is needed. Furthermore, the
molecular-dynamics method must also allow for full symmetry unconstrained relaxation of the lattice degrees of freedom. It is the purpose of this paper to provide these.
In this work we report results of theoretical investigations
of the structure of C36 solid by investigating the following
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©2000 The American Physical Society
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BRIEF REPORTS
structures; a hexagonal closed-packed solid, a rhombohedral
solid, two pseudo ‘‘super-graphite’’ solids and four
hexaclathrates. In our search we also use the available experimental results as a guide.
The theoretical method used in the present work is the
generalized tight-binding molecular dynamics 共GTBMD兲
scheme of Menon and Subbaswamy12 that allows for full
relaxation of covalent systems with no symmetry constraints.
The GTBMD has been found reliable in obtaining good
agreement with experimental and local-density approximation 共LDA兲 results for the structural and vibrational properties of fullerenes and nanotubes.12 The GTBMD gives a
graphite-C60 energy difference of ⫺0.41 eV/atom, in very
good agreement with the experimental estimate of ⫺0.44
eV/atom.13 The efficacy of the scheme has been further enhanced by the incorporation of a constant pressure ensemble
method into the GTBMD scheme to allow for simultaneous
relaxation of lattice and basis degrees of freedom when applied to bulk solids. The constant pressure MD method was
introduced by Andersen14 and subsequently extended by Parrinello and Rahman.15 Its usefulness in applications to structural changes in the solid-state phases has been amply demonstrated in recent works.15–17 In all our molecular-dynamics
simulations of the C36 solids using supercells, a large sampling of k points 共a uniform grid consisting of 343 points in
the full zone兲 was considered in the calculation of the forces
and further checked for convergence. As a further check for
stability, vibrational frequencies were computed for each of
these relaxed structures within the GTBMD scheme.12 None
of the structures proposed here had any imaginary frequencies, indicating them to be true local minima of the total
energy. The electronic band structures of all the relaxed geometries were obtained using a s p 3 s * tight-binding model18
that correctly reproduces the band gap for bulk carbon in the
diamond structure.
Euler’s formula dictates that all fullerene molecules contain 12 pentagonal arrangements of atoms. Furthermore, stability criteria favors minimal contact between these pentagons. Only two of the isomers of C36 have minimal
pentagonal contacts; the D 6h and D 2d isomers and, therefore,
most stable.10,19 The D 6h isomer was found to be unstable on
GTBMD relaxation, transforming into a D 2d isomer. The
elongation along the symmetry axis gives both isomers ‘‘barrel’’ like shapes and interesting dimer geometries. One of the
most stable dimer configurations obtained on MD relaxation
共Ref. 11兲 consisted of hexagonal faces of the dimers facing
each other in an ‘‘eclipsed’’ geometry, when the symmetry
axis is common. The other consisted of bonds common to
two adjacent hexagons facing a similar bond on the neighboring fullerene, when the ‘‘barrel’’ surfaces face each other.
The two dimer geometries were obtained regardless of which
isomers were used in the starting configuration. The HOMOLUMO gaps of these two structures are 0.76 and 0.77 eV,
respectively. This range of values for the dimers are consistent with the values obtained in Ref. 9 using LDA methods if
one makes allowance for the underestimation of energy gaps
in LDA.
We next study the solid forms of the C36 fullerene. In
view of the D 6h symmetry observed in the experimental
electron-diffraction pattern of the C36 solid, we restrict our
initial starting configurations in the GTBMD relaxation to
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FIG. 1. Supergraphite C36 solid isomers 共a兲 and 共b兲 obtained
using the GTBMD relaxation.
have perfect hexagonal structures. The relaxation is then performed with no symmetry constraints. Interestingly, both
hexagonal closed-packed solid as well as the rhombohedral
solid did not retain their original D 6h symmetry on relaxation
and distorted to lower symmetry structures.
We also considered two ‘‘super-graphite’’ solid structures
constructed from the C36 fullerene units. In the first structure
the intermolecular connectivity is via the hexagonal faces of
the C36 cages facing each other in ‘‘eclipsed’’ geometry. The
GTBMD relaxed structure is shown in Fig. 1共a兲. The interfullerene bridging bond length is 1.61 Å and the intermolecular separation is 5.59 Å. The second super-graphite
structure was obtained by using the super-graphite structure
in Ref. 10 as the starting configuration in GTBMD simulations. The relaxed geometry is shown in Fig. 1共b兲. The intermolecular connectivity in the plane involved four-membered
carbon rings between adjacent C36 units. The relaxation,
however, resulted in ‘‘flattening’’ of the faces with additional intercage bond formations 关Fig. 1共b兲兴. The interfullerene bridging bond lengths range from 1.52 to 1.57 Å,
while the intermolecular separation is 5.86 Å. This structure
was found to be more stable than the former by 0.05 eV/
atom. The band gap for this structure is obtained to be 3.0
eV. Both the super-graphite structures showed considerable
cage distortions on relaxation for the individual fullerene
units and neither of the relaxed super-graphite solids had the
D 6h symmetry.
All the solid structures considered so far contain carbon
atoms with mixed sp 2 and sp 3 bondings. By adding carbon
atoms in the interstitial regions between C36 fullerenes in
strategic locations, another class of solids can be obtained in
which all atoms have sp 3 bonding. This is called the clathrate form20 in which all the atoms have fourfold coordination. Clathrates differ from diamond in that their bond angles
deviate from the ideal tetrahedral value found in the diamond
structure. Several clathrate forms can be obtained by varying
the locations of the interstitial carbon atoms. We obtained
three distinct clathrates, each with 40 atoms in the unit cell.
They may readily form during arc-discharge production of
C36 material,6 where many C2 molecules and ions are always
present. Two of these clathrate forms have been reported in
Ref. 11. Figure 2 shows the most stable clathrate form obtained by us. The other two are less stable than this one by
0.24 and 0.39 eV/atom, respectively. Clathrates with larger
number of atoms in the unit cell can be obtained by introducing more C atoms in the interstitial regions between C36
fullerenes. One such clathrate with 54 atoms in the unit cell
is shown in Fig. 3. This structure is less stable than the struc-
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BRIEF REPORTS
FIG. 2. Relaxed C36 hexaclathrate structure with sp 3 bonding
for all atoms obtained using the GTBMD scheme. The structure,
containing a pair of C atoms in each ‘‘hole’’ between C36 cages,
has perfect D 6h symmetry and is the most stable of all clathrates
considered in this work. For clarity of illustration only one layer is
shown.
ture shown in Fig. 2 by 1.54 eV/atom. Clathrate forms of Si
have been investigated in detail by several groups.21–24 Both
C and Si belong to group IV in the periodic table and their
clathrate forms, much like their diamond structures, should
be similar.
A remarkable feature of these clathrate structures is that
they all retain their perfect D 6h symmetry on GTBMD relaxation and the theoretical electron-diffraction patterns for
them are in excellent agreement with those reported in Ref.
6. Moreover, the individual C36 fullerene units also retained
their perfect D 6h symmetry. The clathrate structures tend to
have wide band gaps in general. The structure shown in Fig.
2 has an indirect band gap of 7.0 eV, comparable to bulk
diamond. The cohesive energy of this structure is 6.60 eV/
atom. The intermolecular separation in the symmetry plane is
6.91 Å, while intermolecular separation along the symmetry
FIG. 3. Another GTBMD relaxed hexaclathrate structure
formed with C36 cages containing 54 atoms in the unit cell. This
structure also has perfect D 6h symmetry. Only one layer is shown
for clarity.
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FIG. 4. Local densities of states 共LDOS兲 for the surface atoms
of the clathrate structure shown in Fig. 2.
axis is 6.23 Å. We note that the intermolecular separation
for the C36 solid deduced from the electron-diffraction pattern, 6.68 Å in Ref. 6, lies between these two values. This
structure is more stable than the most stable super-graphite
structure shown in Fig. 1共b兲 by 0.2 eV/atom.
The lower symmetry of mixed sp 2 and sp 3 structures
must be understood on general grounds by a consideration of
the special nature of the chemical bonding in carbon materials. Carbon shows a large variation in bond lengths when
going from sp 2 type solid to sp 3 type. For example, the C-C
bond length is 1.42 Å in graphite 共all sp 2 ), while the same
bond has a length of 1.54 Å in the diamond structure 共all
sp 3 ). 2 Consequently, a mixed sp 2 and sp 3 system suffers
considerable strain in trying to accommodate the two types
of bonds that differ significantly in length. The system lowers its energy by strain release through a decrease in symmetry on relaxation. The lower symmetries obtained for hexagonal closed-packed solid, rhombohedral solid as well as
the super-graphite structures containing both s p 2 and sp 3
types of bonding should, perhaps, be not surprising. Likewise, the high symmetry of the clathrate forms are to be
expected.
Thus, although several solid forms are possible with C36
fullerene units as building blocks, symmetry consideration
and the experimental electron-diffraction pattern favor the
clathrate forms. Furthermore, the theoretical electrondiffraction pattern for these structures are in very good
agreement with the experimental pattern reported in Ref. 6.
The nonmetallic nature of these structures is also consistent
with the experimental findings.
Possibilities still exist for other clathrate structures
formed with the C36 fullerene units, some even more stable
than that shown in Fig. 2. Because of the smaller radius of
curvature, the C36 molecule is known to be very reactive;
more so than either the C60 or C70 . As a result, when optimum conditions exist for their production, they can be expected to readily coalesce to form solids with strong covalent
intermolecular bonding. In the mixed sp 2 and sp 3 phase, the
sp 2 sites will be reactive, while the sp 3 sites will remain
inert. The clathrate phase with sp 3 bonding for all atoms,
therefore, contains no reactive sites. This may explain the
greater stability of the clathrate form. It should be noted that
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while all fullerenes are capable of polymerizations, clathrate
formation requires relatively high-symmetry capability for
individual units on account of the s p 3 bonding for all atoms
in the solid. The C36 fullerene may thus be one of a very few
candidates meeting this criteria.
Quantitative discrepancy remains, however, with respect
to the values of electronic band gap. We note that even
though, regardless of the preparation methods, the C36 molecules tend to aggregate into islands, the islands formed are
neither well ordered nor smooth.9 The authors of Ref. 9 also
suggest that crystalline ordering of the islands may be frustrated both by random orientation of the molecules on the
substrate and by the low temperature of the substrate during
sample preparation. This may explain the lower value of the
gap obtained in experiment. In order to explore the surface
effects on electronic structure we calculate local density of
states 共LDOS兲 on the surface of a C36 clathrate structure.25
Figure 4 shows the LDOS on the surface of the structure
*Email address: [email protected]
†
Email address: [email protected]
‡
Email address: [email protected]
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The present work was supported through grants by the
NSF 共No. OSR 99-07463, MRSEC Program under Grant No.
DMR-9809686兲, DEPSCoR 共No. OSR 99-63231 and No.
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The local electron DOS is obtained by projecting the Green’s
function G(E)⫽ 关 E⫺H⫹i ␦ 兴 ⫺1 onto the set of the basis functions of the given atomic site; ␦ ⫽0.05 eV was used.
11