Analysis of the low-temperature specific heat of multiwalled carbon nanotubes and carbon nanotube ropes

Article abstract of DOI:10.1103/PhysRevB.60.3264

We analyze specific-heat measurements in the temperature range 1a specific-heat not unlike that of graphite, which we maintain is reasonable given the structural similarities between the two materials. In contrast, the low-temperature specific-heat data for ropes are surprisingly large in magnitude and have surprisingly strong temperature dependence. We present model calculations that high-light the puzzling nature of these results and then suggest explanations. 1999 The American Physical Society.

Full text of DOI:10.1103/PhysRevB.60.3264

PHYSICAL REVIEW B
VOLUME 60, NUMBER 5
1 AUGUST 1999-I
Analysis of the low-temperature specific heat of multiwalled carbon nanotubes and carbon
nanotube ropes
Ari Mizel
Department of Physics, University of California at Berkeley, and Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720
Lorin X. Benedict
Department of Physics, University of California at Berkeley, and Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720
and Optical Technology Division, Physics Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899
Marvin L. Cohen, Steven G. Louie, and A. Zettl
Department of Physics, University of California at Berkeley, and Materials Sciences Division, Lawrence Berkeley National Laboratory,
Berkeley, California 94720
Nasser K. Budraa and W. P. Beyermann
Department of Physics, University of California at Riverside, Riverside, California 92521
͑
Received 14 December 1998͒
We analyze specific-heat measurements in the temperature range 1ϽTϽ200 K for two types of carbon
nanotube samples: multiwalled tubes and ropes of single-walled tubes. The multiwalled tube sample has a
specific-heat not unlike that of graphite, which we maintain is reasonable given the structural similarities
between the two materials. In contrast, the low-temperature specific-heat data for ropes are surprisingly large
in magnitude and have surprisingly strong temperature dependence. We present model calculations that high-
light the puzzling nature of these results and then suggest explanations. ͓S0163-1829͑99͒00129-0͔
I. INTRODUCTION
little changed when flat graphene sheets are rolled into cyl-
inders with diameters in the 40–300 Å range. In contrast, the
measured C(T) for the rope sample is larger than that of
graphite and of multiwalled tubes for TϽ50 K and has
stronger temperature dependence. Above 50 K, the two
samples are virtually indistinguishable. To analyze the be-
havior of the ropes, we develop a model calculation with no
fitting parameters. The calculation shows that the rope C(T)
data are substantially larger than expected, and we indicate
possible explanations.
There is presently a great deal of interest in the physical
properties of carbon nanotubes. Much of this interest derives
from the development of a means of growing ropes of single-
1
walled tubes with nearly uniform diameters. These ropes are
two-dimensional 2D hexagonal lattices of parallel carbon
nanotubes with diameters Ϸ13.8 Å . They are grown with a
laser vaporization technique. In contrast, the earliest produc-
2
tion of carbon nanotubes with an arc-discharge method
yielded concentric multiwalled tubes with wide variations in
size ͑diameters ranging from 40–300 Å͒. In researching car-
bon nanotubes, it is important to explore and to understand
differences in the physical properties of the two sample
types. For instance, the small diameters of tubes in the ropes
may lead to electronic and vibrational densities of states
markedly different from those of the larger multiwalled
tubes. Also, the coupling between adjacent tubes in ropes
should be quite different from the coupling between concen-
tric tubes in a multiwalled sample.
II. EXPERIMENT
In this section, we present the measurements of C(T) for
multi-walled tube and rope samples. Sample preparation and
characterization is outlined, followed by a discussion of the
experimental technique. Results are then presented.
A. Sample preparation and characterization
1
. Multiwalled carbon nanotubes
In this work, we investigate these issues by studying the
low-temperature specific heat of both sample types. The T
dependence of C(T) provides us with information about the
low-energy excited states. There are two major kinds of ex-
citations in these systems: electronic excitations and phonon
excitations. Measurements of C(T) yield insight into the
densities of states associated with both kinds of excitations.
Experimentally, we find that C(T) of multiwalled tubes is
similar to that of graphite. We argue that indicates the low-
energy electronic and phonon densities of states are probably
Multiwalled carbon nanotubes were synthesized in an arc-
discharge chamber under 600 Torr helium pressure. A
1.25-cm diameter graphite electrode was arced against a
water-cooled copper cathode, using 100-A dc current. With a
gap between electrodes of about 1–2 mm, the electrode dif-
ferential voltage was 18 V. Nanotube-rich material from the
resulting boule was heated in air at 600 °C for 30 min to burn
off undesirable nontube components. The nanotube samples
were characterized by high-resolution transmission electron
0
163-1829/99/60͑5͒/3264͑7͒/$15.00
PRB 60
3264
©1999 The American Physical Society

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ANALYSIS OF THE LOW-TEMPERATURE SPECIFIC . . .
3265
microscopy; tubes were typically multiwalled with outer di-
ameters of order 10–20 nm and lengths exceeding 10 ␮m.
2. Carbon nanotube ropes
Single-walled carbon nanotubes were synthesized using
two different methods. In the arc-synthesis method, the pro-
cedure was similar to that used in multiwall carbon nanotube
production; the most important difference was that the
graphite anode was first drilled out and loaded with catalyst
͑either Co/Ni or Y/Ni͒. A sample of single-walled nanotubes,
obtained from Rice University, was produced using a dual
1
Nd/YAG laser vaporization technique. The tube-rich mate-
rial was purified by heating in vacuum for 30 min at
1
000 °C. High-resolution transmission electron microscopy
performed on the tubes revealed the well-known bundles or
‘ropes’’ of closely packed nanotubes with each rope con-
‘
taining on the order of 100 parallel tubes, each tube approxi-
mately 1.3 nm in diameter.
B. Experimental technique
A thermal relaxation technique was used to measure the
3
specific heat from 0.6 to 210 K. For each sample, ϳ1 kbar
of pressure was used to press 10–20 mg of powder into a
small cylindrical pellet with a diameter of ϳ3.2 mm. The
pellet was attached to the sapphire platform of the calorim-
eter with a small amount of Apiezon grease. In order to
cover the whole temperature range, two calorimeters with
FIG. 1. Measured C(T) versus T for ropes of single-walled
nanotubes ͑SWNT͒ and for multiwalled nanotubes ͑MWNT͒. Lines
connecting data points are a guide to the eye. Calculated graphite
C(T) from Nicklow, Wakabayashi, and Smith is included for com-
parison.
4
different Cernox thermometers were used. The final data in-
3
clude corrections for ␶ effects, which arise because internal
A. Specific heat and density of states
2
thermal impedances in the sample/platform system influence
the approach to thermal equilibrium. Such effects are exac-
erbated by the porosity of the sample.
For a metallic system, the density of electronic states is
nonzero at the Fermi energy, and this gives an electronic
contribution to the specific heat that is linear in temperature.
C. Results
In Fig. 1, C(T) is plotted as a function of T for the dif-
ferent nanotube samples. On the same graph, a calculated
5
specific heat for graphite is included for comparison pur-
poses. The three curves are similar, although the rope sample
shows a larger C(T) at small temperatures. When the data
are plotted as C(T)/T versus T in Fig. 2, it is evident that the
specific heat at low temperatures for the rope sample is larger
than that for the multiwalled nanotubes and graphite. In ad-
dition, the rope sample shows an interesting shoulder near 5
2
K. In Fig. 3, where C(T)/T is plotted against T, the rope
sample exhibits much stronger temperature dependence than
the other two materials. The rope data show a pronounced
peak around 2 K. Above approximately 50 K, in all three
figures, the three samples appear essentially the same.
III. MODEL
In this section, we analyze the essential features of the
results presented above. We first review some basic facts
about the relationship between the specific heat and the den-
sity of states. Then, we comment on the similarity between
graphite and the multiwalled tube sample. A model of the
low-temperature phonon contribution to C(T) for ropes is
developed.
FIG. 2. Experimental C(T)/T versus T for ropes of SWNT and
for MWNT. Graphite calculation included.

3266
ARI MIZEL et al.
PRB 60
preciable (Ͼ0.1) for ប␻Ͻ6kT. For ប␻Ͼ6kT, the factor
dampens out the contribution of D(␻) to the specific-heat
integral. In this paper, we will be mainly concerned with
features in C(T) below 20 K, so we will consider the PDOS
below 6ϫ20 K ϳ10 meV. This is the energy range of
long-wavelength acoustic phonons. Features in the excitation
spectrum above this energy, like those arising from high-
energy optical modes, will have little or no impact on the
temperature region of interest. We will neglect them.
From the form of Eq. ͑1͒, it follows that, at low tempera-
pϩ1
tures, C(T) is proportional to T
when D(␻) is propor-
p
tional to ␻ . Thus, by plotting C(T) divided by various pow-
ers of T, we gain insight into the frequency dependence of
the PDOS.
B. Graphite and multiwalled tubes
The specific heat of graphite has been well characterized.
One of the major contributions to this is the work of Nick-
5
low, Wakabayashi, and Smith. These authors use inelastic
neutron-scattering data to fit force constants in a Born-von
Karman force-constant model. From this, they calculate the
PDOS and the resulting phonon contribution to the specific
ph
heat, C (T), where the ph superscript refers to phonons.
2
FIG. 3. Experimental C(T)/T versus T for ropes of SWNT and
We have used the force-constant model of Ref. 5 to calculate
for MWNT. Graphite calculation included.
ph
C (T) for graphite in the temperature range 1ϽT
Ͻ200 K. The results are plotted in Figs. 1 –3 along with our
Even for good metals, however, this behavior is only observ-
able below a few Kelvin because phonon excitations domi-
nate at higher temperatures. In this paper, we are concerned
with graphitelike samples. Graphite is a semimetal with a
small density of electronic states at the Fermi energy and has
ph
2
nanotube data. The main feature in C (T)/T is the maxi-
mum at ϳ40 K. This results from a transition in the PDOS
2
0
from ␻ ˜␻ behavior at ប␻ϭ16.3 meV.
nanotubes exhibit a broad peak in C(T)/T below 50 K.
Figure 3 shows that both graphite and multiwalled carbon
2
6
C(T)ϳT below Tϳ1.5 K. Thus the phonon contribution to
Although the peak associated with the multiwalled nanotube
sample is somewhat sharper and at slightly lower tempera-
ture, the two curves are generally quite similar. The fact that
the specific heat dominates even at relatively low tempera-
tures. For this reason, we will focus on the vibrational spe-
cific heat in attempting to understand our data.
2
C(T)/T for multiwalled tubes can be compared favorably to
In the case of phonons, C(T) typically follows T raised to
some power higher than unity. The precise power depends
on the detailed phonon dispersion relations and the dimen-
sionality of the system. As is evident from our plots of the
experimentally measured C(T)/T and C(T)/T2 versus T,
complex behavior is seen in both multiwalled tube and rope
samples for TϽ20 K. To explain this behavior, it is neces-
sary to study the low-energy vibrational modes of these
samples; once the phonon density of states ͑PDOS͒ associ-
ated with these modes is determined, the vibrational specific
heat is evaluated as follows.
ph
2
C (T)/T for graphite implies two things. First, the elec-
tronic contribution to the multiwalled tube specific heat,
e
C (T), is small at these temperatures. This is to be expected,
as remarked in the beginning of Sec. III A. It is supported
experimentally by the nearly zero intercept of C(T)/T versus
T in the multiwalled nanotube curve of Fig. 2. Second, the
2
similarity between C(T)/T for multiwalled tubes and
ph
2
C (T)/T for graphite indicates that the PDOS of the two
materials are similar. This is also not surprising, since the
average radius of a tube in the sample is so large
(
ϳ100 Å ). Deviations from the flat graphene sheet PDOS
The specific heat is defined as the temperature derivative
of the energy density u:
have been predicted for isolated small radius single-walled
7,8
carbon nanotubes, but such sharp radius-dependent fea-
tures should be averaged out in a sample with wide varia-
ץu
ץ
ប␻
tions in tube radii. In short, C(T) for the multiwalled tube
C͑T͒ϭ
ϭ
͵
d␻D͑␻͒
ph
ץT ץT
exp͑ប␻/kT͒Ϫ1
sample looks much like C (T) for graphite, and this is to be
ប␻ 2
expected.
ប␻
ͩ
ͩ
ͪ
expͩ ͪ
kT
kT
ϭk
͵
d␻D͑␻͒
,
͑1͒
2
C. Carbon nanotube ropes
ͫ
ប␻
ͬ
expͩ ͪϪ1
ͪ
In this section, we model the PDOS of ropes of single-
walled nanotubes. Since the tubes in ropes have roughly the
kT
1
where D(␻) is the phonon density of states. The factor in
brackets is convolved with the density of states to obtain the
specific heat. This ‘‘heat-capacity convolution factor’’ is ap-
same diameter, 13.8 Å , we first consider a single isolated
tube of this size. Then, intertube forces are taken into ac-
count.

PRB 60
ANALYSIS OF THE LOW-TEMPERATURE SPECIFIC . . .
3267
FIG. 5. Cross-section of 2D hexagonal array of carbon nano-
tubes. Seven tubes are pictured to illustrate lattice structure. An
actual rope includes on the order of 100 tubes.
tube; by neglecting the internal structure of the rings, we
avoid the distraction of the tube’s high-energy optical modes.
The ring parameters are chosen to take the values R
Ϫ22
ϭ7 Å , Mϭ40m ϭ8ϫ10
g, and cϭ2.5 Å , based
c
FIG. 4. Coarse-grained model of a carbon nanotube.
. Isolated carbon nanotube
upon the properties of a (10,10) nanotube. Each ring has four
degrees of freedom, expressed as coordinates (x,y,z,␪). The
first three are the Cartesian positions of the center of mass of
the ring, and ␪ is the angle of rotation about the tube axis
1
An ideal, infinitely long nanotube has the following four
phonon modes with ␻(q)˜0 as q˜0: A longitudinal
acoustic ͑LA͒ mode, in which the tube is stretched and com-
pressed along its axis; two transverse acoustic ͑TA͒ modes,
in which the tube vibrates like a plucked string ͑the tube can
be ‘‘plucked’’ in any direction perpendicular to its axis, so
this mode is doubly degenerate͒; and a twist mode, in which
portions of the tube are rotated by varying amounts about the
tube axis. At exactly qϭ0, the LA mode is a rigid translation
along the tube axis, the TA modes are translations perpen-
dicular to the axis, and the twist mode is a rigid rotation
about the axis. In order to understand the low-temperature
specific heat of a rope sample, we do not need to consider the
numerous optical modes that do not have ␻(q)˜0 as q
ˆ
(
z). We define these coordinates to be displacements away
from equilibrium. The Hamiltonian takes the form
2
1
1
1
2
v
TA
2
z 2
lc
͑x(lϩ1)cϪx ͒2
lc
Hϭ
p ϩ
L
ϩ
M
͚
lc
2MR2
c2
l
2
M
2
2
1
v
1
v
LA
TA
2
͑z(lϩ1)cϪzlc͒2
ϩ M
͑y(lϩ1)cϪylc͒ ϩ M
c2
c2
2
2
2
1
v
twist
ϩ
MR2
͑␪(lϩ1)cϪ␪ ͒ .
2
͑2͒
lc
c2
2
˜
0 ͑see Sec. III A͒. Indeed, the calculations of Ref. 8 show
that the lowest energy for optical modes of tubes of this size
is greater than 10 meV, which must be inactive for low tem-
peratures.
ˆ
Here, l is an integer, and cϵcz is a vector pointing from one
ring to the next, so that lc indexes a particular ring in the
tube. L refers to the angular momentum of a ring about the
z axis. This Hamiltonian yields four acoustic phonon
branches, with sound speeds vLA , vTA , vTA , and vtwist
z
Because the LA and twist modes involve atoms moving
along the cylindrical surface of the undistorted tube, these
modes can be approximately derived by folding the graphite
.
9
in-plane LA and TA modes, respectively. The neutron-
2. Hexagonal array of carbon nanotubes
scattering measurements of Nicklow, Wakabayashi, and
5
A carbon nanotube rope is a collection of roughly 100 to
500 parallel tubes arranged in a 2D hexagonal array and
separated by a center-to-center distance of aϭ17 Å . Here,
we will neglect the rope surface and approximate the rope by
an infinite lattice ͑see Fig. 5͒. The neglect of the surface
introduces little error into the PDOS.10 Each tube in the rope
is modeled by the linear chain of rings described above. The
equilibrium location of each ring in a rope is specified by a
Smith provide an estimate of the sound velocities of these
modes, v ͑graphite͒ Ϸ2.5ϫ106 cm/s, and vTA ͑graphite͒
LA
1
6
Ϸ1.5ϫ10 cm/s. We would expect these velocities to be
similar to vLA ͑tube͒ and vtwist ͑tube͒. The tube LA mode
sound velocity has been calculated for tubes in the size range
of interest with a reliable tight-binding molecular-dynamics
7
scheme. The result for 13–14 Å diameter tubes is v ͑tube͒
LA
6
Ϸ1.7ϫ10 cm/s, which is smaller than, but comparable to,
ˆ
ˆ
vector rϭja ϩka ϩlc, where a ϵax, a ϵa͓(1/2)x
the neutron-scattering result for vLA ͑graphite͒. In addition,
1
2
1
2
ˆ
Ref. 7 finds the tube TA mode sound velocity to be v
ϩ( 3/2)y͔, and j, k, and l are integers. The integers j and k
identify a tube in the rope, and the integer l identifies a ring
within a given tube.
ͱ
TA
5
͑
tube͒ Ϸ7.5ϫ10 cm/s. The sound velocities computed by
8
Saito et al. are in reasonable agreement with these values.
For our simple force-constant model, we shall take the three
There are two additional kinds of potential energy terms
that must be added to the Hamiltonian ͑2͒ to account for
intertube coupling. The first is a compression term, which
enforces the tendency for tubes to be a distance a apart. The
second is a shearing term, which imposes an energetic cost
when tube surfaces are sheared past each other. The Hamil-
tonian becomes
tube sound velocities to be v ͑tube͒ ϭ 2ϫ106 cm/s, vtwist
LA
6
͑
tube͒ ϭ 1ϫ10 cm/s, and vTA ͑tube͒ ϭ 8ϫ105 cm/s.
To introduce the simplest picture that will include four
acoustic branches, we model a tube as a 1D array of rings of
radius R, and mass M, spaced a length c apart ͑see Fig. 4͒.
Each ring is intended to represent a full unit cell of the nano-

3268
ARI MIZEL et al.
PRB 60
2
2
2
1
1
1
2
v
1
v
1
v
TA
TA
LA
2
z2
2
2
͑zrϩcϪz ͒2
Hϭ
p ϩ
L ϩ
M
͑xrϩcϪx ͒ ϩ M
͑yrϩcϪy ͒ ϩ M
r
͚
r
r
r
r
2MR2
c2
c2
c2
rϭja ϩka ϩlc 2M
2
2
1
2
2
1
2
v
1
1
1
twist
ϩ
MR2
͑␪rϩcϪ␪ ͒ ϩ ␬comp͑xrϩa1Ϫx ͒ ϩ ␬comp͑xrϩa2Ϫx ͒ ϩ ␬comp͑x
2
2
2
Ϫx ͒2
rϩa Ϫa
1
r
r
r
r
c2
2
2
2
2
1
2
1
1
1
1
2
2
2
͒2ϩ ␬shear͑u
͒2
ϩ
ϩ
␬comp͑yrϩa1Ϫy ͒ ϩ ␬comp͑yrϩa2Ϫy ͒ ϩ ␬comp͑y
Ϫy ͒ ϩ ␬shear͑u
rϩa Ϫa r rϩa ,r
1
r
r
rϩa ,r
2
1
2
2
2
2
2
1
2
2
␬shear͑u
͒ .
͑3͒
rϩa Ϫa ,r
2
1
In the last few terms, we introduce the vector u
, the
2
The calculated low-frequency phonon band structure is
plotted in Fig. 6. Note that the bands disperse much more for
q along the tube axis than for q perpendicular to the axis.
This is because intratube forces are much stronger than in-
tertube forces. Note also that there are three modes with
r ,r
1
shear displacement between neighboring points on rings r1
and r when the rings shift and rotate. Shear can result from
2
a combination of translations and rotations, so u
involves
r ,r
1
2
(
x,y,z) coordinated as well as ␪ coordinates:
␻
(q˜0)ϭ0, rather than the four modes of an isolated tube.
The twist mode is no longer acoustic, because of the pres-
ence of a nonzero ␬shear . The three lowest modes for q
perpendicular to the tube axis are all modes in which the
tubes are rigidly displaced and are ‘‘bumping into’’ adjacent
tubes. These low-frequency, nondispersive modes give rise
to prominent structure in the low-energy PDOS.
2
R
2r
1
2,y
u
ϭ
r12,y͑␪ ϩ␪ ͒ϩ
͑x Ϫx ͒
r r
1 2
r ,r
ͩ
r
r
1
2
1
2
2
͉
r12
͉
͉
r12͉
2
r
2,xr
12,y
1
ˆ
Ϫ
ϩ
Ϫ
͑y Ϫy ͒ x
r
r
2
ͪ
2
1
͉
r12͉
3
. Cph(T) for hexagonal array of carbon nanotubes
2
R
2r1
2,x
Ϫr12,x͑␪ ϩ␪ ͒ϩ
͑y Ϫy ͒
The calculated PDOS for the hexagonal array can be sub-
ͩ
r
r
r
r
2
1
2
2
1
͉
r12
͉
ph
͉
r12
͉
stituted into Eq. ͑1͒ to obtain C (T), the temperature-
dependent phonon contribution to the specific heat. We plot
2
r1 r12,y
2,x
ˆ
ˆ
r
2
͑x Ϫx ͒ yϩ͑z Ϫz ͒z,
͑4͒
r
r
ͪ
r
2
1
2
1
͉
r12͉
where we have abbreviated r ϭr Ϫr . Equation ͑3͒ pro-
1
2
1
2
vides a reasonable phenomenological representation of the
low-energy phonons in a carbon nanotube rope. By including
the important binding forces in a standard fashion as two-
body energy terms and choosing parameters appropriately, a
quantitative modeling of the specific heat should be possible
as in Ref. 5. Two new force constants enter into Eq. ͑3͒,
␬comp and ␬shear . They are each chosen by fitting to analo-
gous graphite modes measured by Nicklow, Wakabayashi,
5
and Smith. The compression mode is analogous to the c axis
LA mode in graphite. For the c axis LA mode in graphite,
the spring connecting a unit cell in one sheet to an adjacent
sheet has a spring constant of 5.8ϫ103 dyn/cm. Assuming
that the same type of spring connects rings in two adjacent
tubes, we approximate ␬compϭ6ϫ103 dyn/cm. This choice
is physically reasonable, and the resulting value of ␬comp
agrees well with the calculations of Charlier, Gonze, and
Michenaud for the energetics of a hexagonal lattice of 4.1 Å
1
1
radius tubes. The shearing mode force constant ␬shear is
similarly chosen. This mode is analogous to the c axis TA
mode in graphite. Referring to Nicklow, Wakabayashi, and
Smith, we take ␬shearϭ8ϫ102 dyn/cm. These choices for
␬comp and ␬shear are physically reasonable, and our theoret-
ical heat capacity does not depend sensitively upon them ͑see
Fig. 7͒.
FIG. 6. Calculated low-frequency phonon band structure for in-
finite hexagonal lattice of carbon nanotubes with radii ϭ 7 Å .
Wave vectors along ⌫A are parallel to the tube axis. All other wave
vectors are perpendicular to the tube axis.

PRB 60
ANALYSIS OF THE LOW-TEMPERATURE SPECIFIC . . .
3269
correlation in nanotubes,12 one might entertain the possibility
that electronic effects are responsible for the disagreements
between model and experiment. As remarked in the begin-
ning of Sec. III A, one expects the electronic contribution to
the specific heat of a graphitelike sample to be minor. This
expectation appears to be borne out in the case of multi-
walled tubes. It therefore seems unlikely that electronic ef-
fects are making appreciable contributions to our rope
sample measurements, especially at relatively high tempera-
tures like 20 K. However, the magnitude of such effects
could be checked in the future by extending our specific-heat
measurements to lower temperatures. Finally, we note that
some of the sample may include material other than carbon
nanotubes. The admixture of nontubelike material would
change the density of excited states, and complicate the
analysis. For example, molecules like methane possess spe-
cific heats which exceed that of graphite by more than a
factor of 4 even at room temperature. If a large fraction of
the rope sample were include such molecules, the sample
would exhibit a markedly enhanced low-temperature heat ca-
pacity.
IV. CONCLUSION
FIG. 7. Calculated C(T) versus T for model of infinite hexago-
We present measurements of the specific heat of two
kinds of carbon nanotube samples. The data for multiwalled
nanotubes are found to agree generally with the behavior of
graphite. For ropes of single walled tubes, the specific heat is
found to exceed that of graphite at low temperatures, and
nal lattice of carbon nanotubes with radii ϭ 7 Å compared to ex-
perimental data. Kϭ0 curve gives results of model calculation with
␬comp and ␬shear reduced to nearly zero.
ph
C (T) in Fig. 7, along with the experimental result. The
2
C(T)/T is found to peak at a lower temperature than for
low-temperature rope specific heat is predicted to be substan-
tially lower than that of graphite, because tubes should lack
the low-energy ‘‘carpet-ripple’’ modes that exist in a graph-
ite sheet. This prediction does not agree with experiment; the
experimental heat capacity for ropes substantially exceeds
that for graphite.
graphite. A model calculation of the long-wavelength vibra-
tions of a rope of carbon nanotubes highlights the surprising
nature of the experimental results.
ACKNOWLEDGMENT
To test the robustness of the model, we compute the spe-
cific heat with ␬comp and ␬shear both set infinitesimally
small. It is important to assess the sensitivity of our results to
these parameters in particular because they are difficult to
estimate accurately. The increase in the specific heat that
results from softening the intertube springs is shown in Fig.
This work was supported by National Science Foundation
Grants Nos. DMR-9520554 and DMR-9624778, and the Of-
fice of Energy Research, Office of Basic Energy Services,
Materials Sciences Division of the U.S. Department of En-
ergy under Contract No. DE-AC03-76SF00098. We thank
the group of R. E. Smalley for providing us with a sample of
carbon nanotube ropes. P. Vashishta graciously supplied us
with details about Ref. 7. We are grateful to V. H. Crespi and
M. Cote for many helpful discussions. A. M. acknowledges
the support of the National Science Foundation.
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