Pressure-induced interlinking of carbon nanotubes
T. Yildirim,
1
O. Gu
¨
lseren,
1,2
C¸. Kılıc¸,
3
and S. Ciraci
3
1
NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
2
Department of Materials Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104
3
Physics Department, Bilkent University, Ankara 06533, Turkey
共Received 1 August 2000兲
We predict new forms of carbon consisting of one- and two-dimensional networks of interlinked single-wall
carbon nanotubes, some of which are energetically more stable than van der Waals packing of the nanotubes
on a hexagonal lattice. These interlinked nanotubes are further transformed with higher applied external
pressures to more dense and complicated stable structures, in which curvature-induced carbon sp
3
rehybrid-
izations are formed. We also discuss the energetics of the bond formation between nanotubes and the electronic
properties of these predicted novel structures.
Carbon nanotubes, originally discovered as by-products
of fullerene synthesis,
1,2
are now considered to be the build-
ing blocks of future nanoscale electronic and mechanical de-
vices. It is therefore desirable to have a good understanding
of their electronic and mechanical properties and the interre-
lations between them. In particular, single-wall carbon nano-
tubes 共SWNT’s兲 provide a system where the electronic prop-
erties can be controlled by the structure of the nanotubes and
by various deformations of their geometries.
3–5
The physical
properties can also be altered by intertube interactions be-
tween nanotubes packed in hexagonal lattices, as so called
‘‘nanoropes.’’
The intertube interactions in nanoropes can be probed by
applying external pressure to vary the intertube distance.
6–8
For fullerenes, such high-pressure studies have yielded many
interesting results including new compounds such as the
pressure-induced polymeric phases of C
60
.
9
It is, therefore,
of interest to inquire if similar covalent bonding can occur
between the nanotubes in a rope. This could have important
consequences for nanoscale device applications and compos-
ite materials that require strong mechanical properties since
nanoropes consisting of interlinked SWNT will be signifi-
cantly stronger than nanoropes composed of van der Waals
packed nanotubes.
10
A recent Raman study on SWNT ropes carried out up to
25.9 GPa 共Ref. 7兲 showed that the mode intensities and en-
ergies are not completely reversible upon pressure cycling,
suggesting irreversible pressure-induced changes in the
structure. In another high-pressure study Chesnokov et al.
8
observed a very large volume reduction and high compress-
ibility, signaling the presence of a microscopic volume-
reducing deformation other than van der Waals compression.
Some of these pressure-induced effects are tentatively attrib-
uted to possible crushing or flattening the nanotube cross
section from circular to elliptical or hexagonal.
8
Motivated
by these reports, we investigated possible new pressure-
induced ground-state structures for (n,0) nanotube ropes
11
from first-principles total energy calculations using the pseu-
dopotential method within the generalized gradient
approximation
12
共GGA兲. We observed an elliptical distortion
of the nanotubes under pressure and subsequent curvature-
induced carbon rehybridization, giving rise to one- or two-
dimensional interlinked networks of nanotubes. This is
somewhat reminiscent to the pressure-induced polymeriza-
tions of C
60
.
9
The first-principles total energy and electronic structure
calculations were carried out using the pseudopotential plane
wave code
CASTEP.
13
We used plane waves with an energy
cutoff of 500 eV. With this cutoff and using ultra soft pseu-
dopotentials for carbon,
14
the total energy converges within
0.5 meV/atom. For the Brillouin zone integration, we used
between 3⫻ 3⫻ 6to5⫻ 5⫻ 7 k points according to the
Monkhorst-Pack special k-point scheme.
15
This method has
already been applied to many carbon systems, including
fullerenes and cubane with remarkable success.
16
In this report, we present calculations on nanoropes con-
sisting of 共5,0兲, 共6,0兲, 共7,0兲, 共9,0兲, and 共6,6兲 nanotubes. For
simplicity, we model the nanoropes as a hexagonal lattice of
nanotubes. We further assume that we have only one nano-
tube per unit cell. 共6,0兲 nanotubes are perfectly compatible
with the hexagonal lattice 共i.e.,
␥
⫽ 120°). However in the
optimized structure, hexagonal carbon rings of one nanotube
face hexagonal rings from the neighboring tubes 关Fig. 1共a兲兴.
From studies of fullerenes, we know that the energy can be
reduced by rotating every other tube such that their C–C
bonds face the center of the hexagonal faces of the neighbor-
FIG. 1. Optimized structures
of the vdW 共a兲共6,0兲, 共b兲共7,0兲, and
one-dimensional interlinked 共c兲
共7,0兲 nanotube lattices. The inter-
linked structure shown in 共c兲 has
lower energy than vdW packed
共7,0兲 nanotubes shown in 共b兲.
PHYSICAL REVIEW B 15 NOVEMBER 2000-IVOLUME 62, NUMBER 19
PRB 62
0163-1829/2000/62共19兲/12648共4兲/$15.00 12 648 ©2000 The American Physical Society
ing nanotubes. This would double the unit cell for 共6,0兲
nanotubes. We will not pursue this here because we are
mainly interested in the potential for covalent bonding be-
tween nanotubes.
共9,0兲 nanotubes are also compatible with hexagonal sym-
metry. Furthermore the relative orientation of the nanotubes
are optimal energetically 共i.e., C–C bonds face the center of
the hexagonal faces of adjacent nanotubes兲. 共5,0兲 and 共7,0兲
nanotubes are not compatible with hexagonal symmetry and
therefore
␥
is expected to deviate from the ideal value of
120°. We find that when one bond is along the a axis, the
total energy is minimized when the b axis is aligned so that
it brings the C–C bonds of one nanotube to the center of
hexagonal faces of adjacent nanotubes. For 共5,0兲 and 共7,0兲
nanotubes this occurs when
␥
⫽ (360/5)⫻ 1⫽ 72.0° and
(360/7)⫻ 2⫽ 102.857°, respectively. These values are very
close to the values
␥
⫽ 72.46° and 102.35° obtained from the
first-principles structural optimization.
The pressure dependence of these lattices of nanotubes
was determined by calculating the total energy as a function
of nanotube separation 共i.e., a and b) while the other param-
eters, including atom positions, c, and
␥
are optimized. We
observe that 共7,0兲 nanotubes become elliptically distorted
with applied pressure 共i.e., decreasing nanotube-nanotube
distance兲. At a critical pressure, we observe a structural
phase transformation from the van der Waals nanotube lat-
tice 关as shown in Figs. 1共a兲 and 1共b兲兴 to a new lattice in
which the nanotubes are interlinked along the 关110兴 direc-
tion, where the strain of the nanotube is largest 关Fig. 1共c兲兴.
The covalent bonding between nanotubes is therefore the
result of curvature-induced rehybridization of the carbon or-
bitals. The same structural transformation was observed for
the other (n,0) nanoropes and the results are summarized in
Table I. The structure of each of these one-dimensionally
interlinked nanoropes is orthorhombic 共space group Cmcm),
with two nanotubes per unit cell. The relationship between
the conventional orthorhombic cell and the primitive one is
shown in Fig. 2共a兲.
Figure 2 also shows the local environment of the carbon
atoms involved in the interbonding of the nanotubes for 共6,0兲
and 共7,0兲. In the first case, the covalent bonding occurs be-
tween carbon atoms on two hexagonal rings 关Fig. 2共b兲兴.We
believe this process is slightly less favorable than covalent
bonding between carbons on a hexagonal ring and those in-
volved in an intratube C–C bond which occurs for the 共7,0兲
case 关Fig. 2共c兲兴. In both cases, the bond distances are com-
parable to those in diamond, indicating sp
3
hybridization.
The bond angles vary from about 100° to 120°, indicating
some strain. From Table I, we see that the energy of the
interlinked phase is actually lower than the van der Waals
共vdW兲 lattices for 共5,0兲 and 共7,0兲 nanotropes. For the 共9,0兲
nanorope, the energy difference is relatively small. Regard-
less of the energies relative to that of unlinked nanotubes, an
interlinked phase is stable once it is formed, because break-
ing the intertube bonds requires jumping over a significant
energy barrier.
To quantitatively study the bonding mechanism, we cal-
culated the total energies of the different phases as a function
of the lattice constant 共i.e., applied pressure兲. The result for
共7,0兲 nanotubes is summarized in Fig. 3. The energies of the
vdW and the one-dimensional interlinked 共1DI兲 phases cross
TABLE I. Various structural parameters and the total energies of the optimized structures of (n,0) vdW lattices and one-dimensional
interlinked 共1DI兲 nanoropes as shown in Fig. 1. The band gaps of all the structures listed here are found to be zero, indicating metallic
behavior.
Properties 共5,0兲共5,0兲共1DI兲共6,0兲共6,0兲共1DI兲共7,0兲共7,0兲共1DI兲共9,0兲共9,0兲共1DI兲
Formula C
20
C
20
C
24
C
24
C
28
C
28
C
36
C
36
Space group Cmcm Cmcm P6/mmm Cmcm Cmcm Cmcm P63/mcm Cmcm
a⫽ b 共Å兲 7.408 7.079 8.364 7.762 9.250 8.432 10.389 9.532
c 共Å兲 4.208 4.190 4.212 4.223 4.218 4.205 4.219 4.209
␥
72.46 125.39 120.00 121.96 102.35 119.15 120.00 111.97
Density (g/cm
3
) 1.8119 2.331 1.8713 2.2176 1.5849 2.1415 1.8213 2.0245
Energy/C 共eV兲 ⫺ 155.694 ⫺ 155.802 ⫺ 155.843 ⫺ 155.868 ⫺ 155.946 ⫺ 155.969 ⫺ 156.049 ⫺ 156.042
FIG. 2. 共a兲 A view along the c axis of the 1D interlinked (n,0)
nanotube lattice. The shaded rectangular region is the orthorhombic
unit cell. Local structure of carbon atoms involved in the intertube
bonding 共dotted lines兲 between two 共b兲共6,0兲 and 共c兲共7,0兲 nano-
tubes. The sp
3
hybridization occurs between two hexagonal faces
for 共6,0兲 nanotubes and between a hexagonal face and a C–C bond
for 共7,0兲 nanotubes.
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12 649BRIEF REPORTS
each other at about a⫽ 9.0 Å with an energy barrier of only
46 meV/unitcell 共552 K兲. The pressure required to attain this
lattice constant is only about 0.3 GPa for the vdW phase,
indicating that polymerization of vdW 共7,0兲 nanoropes could
occur at modest pressures and temperatures.
17
Once the in-
terlinked phase is reached, the energy barrier required to
break the bonds and obtain free nanotubes is about 0.7 eV
共25 meV/atom兲, which is comparable to that of 1D polymer-
ized C
60
molecules 共20 meV/atom兲.
18
Figure 3 also shows that another interlinked phase of 共7,0兲
nanotubes becomes the ground state for lattice parameter
smaller than 8.0 Å. In this new phase the nanotubes are in-
terlinked along both a and b axes 关see Fig. 4共a兲兴. This two-
dimensional interlinked 共2DI兲 structure is about four times
stiffer 共i.e., d
2
E/da
2
⫽ 13.7 eV/Å
2
) than the 1D interlinked
phase (d
2
E/da
2
⫽ 3.3 eV/Å
2
) and sixteen times stiffer than
the vdW nanoropes (d
2
E/da
2
⫽ 0.8 eV/Å
2
).
We observe that applying even higher pressures yields
more complicated and denser phases for many of the nano-
ropes studied here 共see Fig. 4兲. For 共9,0兲 nanoropes, we find
that the nanotubes are interlinked along three directions
forming a hexagonal network. The length of the intertube
bond, d
C–C
⫽ 1.644 Å, is significantly elongated for an sp
3
C–C bond. The two-dimensional interlinked phase of 共7,0兲
nanotubes is further transformed to a denser structure at 30
GPa 关Fig. 4共c兲兴. By comparison, 共6,6兲 nanotubes do not form
an interlinked structure up to a pressure of 60 GPa. Rather
the nanotubes are hexagonally distorted such that the local
structure of the nanotube faces is similar to that in graphite
sheets 关Fig. 4共c兲兴. Furthermore, releasing the pressure yields
the original structure, indicating that the distortion is purely
elastic. Similar calculations are currently underway for other
(n,m) nanotubes. However, we expect similar results for
other types of tubes.
A detailed discussion of the electronic band structure and
density of states of the predicted structures will be presented
elsewhere, however all the structures reported here 关except
one in Fig. 4共c兲兴 are found to be metal. The dispersion of the
bands near Fermi level in a direction perpendicular to the
tube axis is found to be around 0.5 eV, making the vdW
nanoropes metallic even though the individual tubes are in-
sulating 关such as 共7,0兲 nanotubes兴 or semiconducting with a
small band gap. The structural changes clearly have strong
effects on the electronic properties and therefore should be
detected in the pressure dependence of various transport
properties of nanoropes. From our calculations we expect
two effects to be observed. First, on the initial pressure cycle
one should observe an irreversible effect in the resistivity
due to the polymerization of the nanotubes. Second, the ap-
plied pressure will induce a reversible deformation of the
circular cross section, which will change the metallic behav-
ior of the nanotubes.
5
For example, the 共7,0兲 nanorope,
which is found to be metallic at ambient pressure, transforms
to a high-density phase as shown in Fig. 4共c兲 at 30 GPa,
FIG. 3. Planar lattice constant variation of the total energy of
共7,0兲 nanotube ropes in three different phases. Inset shows the view
of the structures along the c axis. The zero of energy was taken to
be the energy of vdW packing of the nanotubes.
FIG. 4. Various high density phases of carbon nanotubes. 共a兲
Two-dimensional interlinked 共2DI兲 structure of 共5,0兲 nanotubes,
consisting of rectangularly distorted nanotubes interlinked on a 2D
network. 共b兲 A hexagonal network of 共9,0兲 nanotubes, in which
共9,0兲 nanotubes are interlinked along a, b, and 关110兴 directions. 共c兲
A very dense structure of 共7,0兲 nanotubes obtained under 30 GPa
pressure. 共d兲 The optimized structure of 共6,6兲 nanotubes under P
⫽ 53 GPa. Nanotubes are distorted in such a way that the local
nearest-neighbor structure is somewhat similar to graphite sheets.
d
C–C
indicates the smallest distance between two carbon atoms of
nearest neighbor nanotubes.
12 650 PRB 62
BRIEF REPORTS
which is found to be a band insulator with a band gap of 2
eV.
The new pressure-induced, high-density phases reported
here can also be important host lattices for intercalation and
sorption. For instance, the high-density phase of 共7,0兲 nano-
ropes 关Fig. 4共c兲兴 has an interesting structure in which some
parts of the lattice are very strained containing sp
3
bonded
square carbon rings, while other parts consist of graphitelike
sp
2
bonded carbon atoms 共28% of all atoms兲. In this struc-
ture, there are still interstitial sites that would accommodate
other species such as alkali metals.
In summary, we have presented first-principles calcula-
tions of the structures and electronic properties of various
nanoropes. We find that small nanotubes in a rope are dis-
torted elliptically with applied pressure and then are co-
valently bonded to each other at the positions of highest
curvature point of nanotubes by sp
3
hybridization of the car-
bon orbitals. For small nanotubes, the resulting one-
dimensional chains of elliptic carbon nanotube structure are
found to be energetically more stable than the circular van
der Waals nanoropes. Higher applied pressures resulted in
more dense and complicated structures. Thus pressure in-
duced polymerization of the nanotubes may provide a way of
synthesizing novel carbon base materials with interesting
physical properties. For example interlinking of the nano-
tubes may improve the mechanical performance of compos-
ites based on these materials. It will be an experimental chal-
lenge to confirm the structures predicted here. A difference
NMR spectrum of two identical samples; one treated with
pressure and the other not, may give some evidence for the
new phases. Similar difference measurements by ESR and
Raman could be equally valuable.
We thank D. A. Neumann, R. L. Cappelletti, and J. E.
Fischer for many fruitful discussions and critical reading of
the manuscript. This work is partially supported by the Na-
tional Science Foundation under Grant No. INT97-31014
and TU
¨
BI
˙
TAK under Grant No. TBAG-1668共197 T 116兲.
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