11 Phonons and Thermal Prop erties
James Hone
DepartmentofPhysics
UniversityofPennsylvania
Philadelphia, PA 19104-6317, USA
Abstract. The thermal prop erties of carb on nanotub es displaya wide range of
b ehaviors which are related b oth to their graphitic nature and their unique struc-
ture and size. The sp eci c heat of individual nanotub es should b e similar to that of
two-dimensional graphene at high temp eratures, with the e ects of phonon quan-
tization b ecoming apparentatlower temp eratures. Inter-tub e coupling in SWNT
rop es, and interlayer coupling in MWNTs, should cause their low-temp erature sp e-
ci c heat to resemble that of three-dimensional graphite. Exp erimental data on
SWNTs show relatively weak inter-tub e coupling, and are in go o d agreementwith
theoretical mo dels. The sp eci c heat of MWNTs has not b een examined theoret-
ically in detail. Exp erimental results on MWNTs show a temp erature dep endent
sp eci c heat which is consistentwithweak inter-layer coupling, although di erent
measurements show slightly di erent temp erature dep endences. The thermal con-
ductivity of b oth SWNTs and MWNTs should re ect the on-tub e phonon structure,
regardless of tub e-tub e coupling. Measurements of the thermal conductivityofbulk
samples show graphite-like b ehavior for MWNTs but quite di erent b ehavior for
SWNTs, sp eci cally a linear temp erature dep endence at low T which is consis-
tent with one-dimensional phonons. The ro om-temp erature thermal conductivity
of highly-aligned SWNT samples is over 200 W/m-K, and the thermal conductivity
of individual nanotub es is likely to b e higher still.
202 Hone
1 Sp eci c Heat
Because nanotub es are derived from graphene sheets, we rst examine the
sp eci c heat C of a single such sheet, and how C changes when manysuch
sheets are combined to form solid graphite. We then in x1.2 consider the
sp eci c heat of an isolated nanotub e [1], and the e ects of bundling tub es
into crystalline rop es and multi-walled tub es (x1.3). Theoretical mo dels are
then compared to exp erimental results(x1.4).
1.1 Sp eci c Heat of 2-D Graphene and 3-D Graphite
In general, the sp eci c heat C consists of phonon C and electron C contri-
ph e
butions, but for 3-D graphite, graphene and carb on nanotub es, the dominant
contribution to the sp eci c heat comes from the phonons. The phonon con-
tribution is obtained byintegrating over the phonon densityof states with
aconvolution factor that re ects the energy and o ccupation of each phonon
state:
~!
Z
2
!
max
k T
B
(! ) d! ~! e
C = k ; (1)
ph B
2
~!
k T
B
0
k T
B
e 1
where (! ) is the phonon densityofstatesand! is the highest phonon
max
energy of the material. For nonzero temp eratures, the convolution factor is 1
at ! = 0, and decreases smo othly to a value of 0:1at~! = k T=6, so that
B
the sp eci c heat rises with T as more phonon states are o ccupied. Because
(! ) is in general a complicated function of ! , the sp eci c heat, at least at
mo derate temp eratures, cannot b e calculated analytically.
Atlow temp erature (T ), however, the temp erature dep endence of
D
the sp eci c heat is in general much simpler. In this regime, the upp er b ound
in Eq. (1) can b e taken as in nity,and(! ) is dominated by acoustic phonon
mo des, i.e., those with ! ! 0as k ! 0. If we consider a single acoustic mo de
in d dimensions that ob eys a disp ersion relation ! / k , then from Eq. (1) it
follows that:
(d= )
C / T (T ): (2)
ph D
Thus the low-temp erature sp eci c heat contains information ab out b oth the
dimensionality of the system and the phonon disp ersion.
A single graphene sheet is a 2-D system with three acoustic mo des,
twohaving a very high sound velo city and linear disp ersion [a longitudinal
(LA) mo de, with v =24 km/s, and an in-plane transverse (TA) mo de, with
v =18 km/s] and a third out-of-plane transverse (ZA) mo de that is describ ed
2 7 2
by a parab olic disp ersion relation, ! = k , with 6 10 m /s [2,3]. From
Eq. (2), we see that the sp eci c heat from the in-plane mo des should display
2
a T temp erature dep endence, while that of the out-of plane mo de should
Phonons and Thermal Prop erties 203
b e linear in T . Equation (1) can b e evaluated separately for each mo de; the
contribution from the ZA mo de dominates that of the in-plane mo des b elow
ro om temp erature (although the approximation is not valid ab ove 300 K).
The phonon contributions to the sp eci c heat can be compared to the
exp ected electronic sp eci c heat of a graphene layer. The unusual linear k
dep endence of the electronic structure E (k ) of a single graphene sheet at E
F
(see Fig. 2 of [4]) pro duces a low-temp erature electronic sp eci c heat that is
quadratic in temp erature, rather than the linear dep endence found for typical
metals [5]. Benedict et al. [1] show that, for the in-plane mo des in a graphene
sheet,
C v
ph F
2 4
) 10 : (3) (
C v
e
The sp eci c heat of the out-of plane mo de is even higher, and thus phonons
dominate the sp eci c heat, even at low T , and all the wayto T =0.
Combining weakly interacting graphene sheets in a correlated stacking
arrangement to form solid graphite intro duces disp ersion along the c-axis, as
the system b ecomes three-dimensional. Since the c-axis phonons have very
low frequencies, thermal energies of 50 K are sucient to o ccupy all ZA
phonon states, so that for T > 50 K the sp eci c heat of 3-D graphite is
essentially the same as that of 2-D graphene. The crossover b etween 2-D and
interplanar coupled b ehavior is identi ed as a maximum in a plot of C
ph
2
vs. T [6,7]. We will see belowthat this typ e of dimensional crossover also
exists in bundles of SWNTs. The electronic sp eci c heat is also signi cantly
changed in going from 2-D graphene to 3-D graphite: bulk graphite has a
small but nonzero density of states at the Fermi energy N (E ) due to c-axis
F
disp ersion of the electronic states. Therefore 3-D graphite displays a small
linear C (T ), while C has no such term. For 3-D graphite, the phonon
e ph
contribution remains dominantabove 1K [8,9].
1.2 Sp eci c Heat of Nanotub es
Figure 1 shows the low-energy phonon disp ersion relations for an isolated
(10,10) nanotub e. Rolling a graphene sheet into a nanotub e has twomajor
e ects on the phonon disp ersion. First, the two-dimensional band-structure
of the sheet is collapsed onto one dimension; b ecause of the p erio dic b oundary
conditions on the tub e, the circumferential wavevector is quantized and dis-
crete `subbands' develop. From a zone-folding picture, the splitting b etween
the subbands at the p oint is of order [1]
~v
E = k ; (4)
B subband
R
where R is the radius of the nanotub e and v is the band velo city of the rele-
vant graphene mo de. The second e ect of rolling the graphene sheet is to re-
arrange the low-energy acoustic mo des. For the nanotub e there are now four,
204 Hone
Fig. 1. Low-energy
phonon disp ersion re-
lations for a (10,10)
nanotub e. There are
four acoustic mo des: two
degenerate TA mo des
v = 9 km/s), a `twist'
20 (
mo de (v = 15 km/s),
and one TA mo de
(v = 24 km/s) [3]. The
15
inset shows the low-
energy phonon density
of states of the nan-
(solid line) and 10 otub e
E (meV)
that of graphite (dashed
line) and graphene
(dot-dashed line). The
5 Phonon DOS
nanotub e phonon DOS is
constant below 2.5 meV,
04812
increases stepwise
E (meV) then ter; 0 as higher subbands en
0 0.1 0.2 0.3 0.4 y
k(1/Å) there is a 1-D singularit
at each subband edge
rather than three, acoustic mo des: an LA mo de, corresp onding to motion
of the atoms along the tub e axis, two degenerate TA mo des, corresp onding
to atomic displacements p erp endicular to the nanotub e axis, and a `twist'
mo de, corresp onding to a torsion of the tub e around its axis. The LA mo de
is exactly analogous to the LA mo de in graphene. The TA mo des in a SWNT,
on the other hand, are a combination of the in-plane and out-of-plane TA
mo des in graphene, while the twist mo de is directly analogous to the in-
plane TA mo de. These mo des all show linear disp ersion (there is no nan-
otub e analogue to the ZA mo de) and high phonon velo cities: v = 24 km/s,
LA
v = 9 km/s, and v = 15 km/s for a (10,10) tub e [3]. Because all of the
TA twist
acoustic mo des have a high velo city, the splitting given in Eq. (4) corresp onds
to quite high temp eratures, on the order of 100 K for a 1.4 nm-diameter tub e.
In the calculated band structure for a (10,10) tub e, the lowest subband enters
at 2.5 meV (30 K), somewhat lower in energy than the estimate given by
Eq. (4).
The inset to Fig. 1 shows the low-energy phonon densityofstates(! )of
a (10,10) nanotub e (solid line), with (! ) of graphene (dot-dashed line) and
graphite (dashed line) shown for comparison. In contrast to 2-D graphene
and 3-D graphite, which show a smo othly-varying (! ), the 1-D nanotub e
has a step-like (! ), which has 1-D singularities at the subband edges. The
Phonons and Thermal Prop erties 205
markedly di erent phonon density of states in carb on nanotub es results in
measurably di erent thermal prop erties at low temp erature.
At mo derate temp eratures, many of the phonon subbands of the nanotub e
will b e o ccupied, and the sp eci c heat will b e similar to that of 2-D graphene.
Atlow temp eratures, however, b oth the quantized phonon structure and the
sti ening of the acoustic mo des will cause the sp eci c heat of a nanotub e to
di er from that of graphene. In the low T regime, only the acoustic bands will
b e p opulated, and thus the sp eci c heat will b e that of a 1-D system with a
linear ! (k ). In this limit, T ~v=k R , Eq. (1) can b e evaluated analytically,
B
yielding a linear T dep endence for the sp eci c heat [1]:
2 2
3k T
B
C = ; (5)
ph
~v 3
m
where is the mass per unit length, v is the acoustic phonon velo city,
m
and R is the nanotub e radius. Thus the circumferential quantization of the
nanotub e phonons should b e observable as a linear C (T ) dep endence at the
lowest temp eratures, with a transition to a steep er temp erature dep endence
ab ove the thermal energy for the rst quantized state.
Turning to the electron contribution, a metallic SWNT is a one-dimensional
metal with a non-zero density of states at the Fermi level. The electronic sp e-
ci c heat will b e linear in temp erature [1]:
2
4k T
B
C = ; (6)
e
3~v
F m
for T ~v =k R , where v is the Fermi velo city and is again the mass
F B F m
p er unit length. The ratio b etween the phonon and the electron contributions
to the sp eci c heat is [1]
C v
ph F
2
10 ; (7)
C v
e
so that even for a metallic SWNT, phonons should dominate the sp eci c heat
all the waydown to T = 0. The electronic sp eci c heat of a semiconducting
tub e should vanish roughly exp onentially as T ! 0 [10], and so C will b e
e
even smaller than that of a metallic tub e. However, if such a tub e were dop ed
so that the Fermi level lies near a band edge, its electronic sp eci c heat could
b e signi cantly enhanced.
1.3 Sp eci c Heat of SWNT rop es and MWNTs
As was mentioned ab ove, stacking graphene sheets into 3-D graphite causes
phonon disp ersion in the c direction, which signi cantly reduces the low-T
sp eci c heat. A similar e ect should o ccur in b oth SWNT rop es and MWNTs.
In a SWNT rop e, phonons will propagate b oth along individual tub es and
206 Hone
10 SWNT Rope
8
k||
Calculated dis-
6 k⊥ Fig. 2.
p ersion of the acous-
(meV)
tic phonon mo des in
E
in nite rop e of 1.4-
4 an
nm diameter SWNTs [6].
The phonon velo city is
in the longitudinal
2 high
direction and lower in
the transverse direction.
The rst two higher-order
(dashed lines)
0 subbands 0.2 0 0.2
k⊥(1/Å) k|| (1/Å)
are shown for comparison
between parallel tub es in the hexagonal lattice, leading to disp ersion in b oth
the longitudinal (on-tub e) and transverse (inter-tub e) directions. The solid
lines in Fig. 2 show the calculated disp ersion of the acoustic phonon mo des
in an in nite hexagonal lattice of carb on nanotub es with 1.4 nm diameter [6].
The phonon bands disp erse steeply along the tub e axis and more weakly in the
transverse direction. In addition, the `twist' mo de b ecomes an optical mo de
b ecause of the presence of a nonzero shear mo dulus between neighb oring
tub es. The net e ect of this disp ersion is a signi cant reduction in the sp eci c
heat at low temp eratures compared to an isolated tub e (Fig. 3). The dashed
lines show the disp ersion relations for the higher-order subbands of the tub e.
?
In this mo del, the characteristic energy k of the inter-tub e mo des is
B
D
( 5 meV), which is larger than the subband splitting energy, so that 3-D
disp ersion should obscure the e ects of phonon quantization. Wewilladdress
the exp erimentally-measured inter-tub e coupling b elow.
The phonon disp ersion of MWNTs has not yet b een addressed theoreti-
cally. Strong phonon coupling b etween the layers of a MWNT should cause
roughly graphite-like b ehavior. However, due to the lack of strict registry
between the layers in a MWNT, the interlayer coupling could conceivably
be much weaker than in graphite, esp ecially for the twist and LA mo des,
which do not involve radial motion. The larger size of MWNTs, compared to
SWNTs, implies a signi cantly smaller subband splitting energy [Eq. (4)], so
that the thermal e ects of phonon quantization should b e measurable only
well b elow1K.
Phonons and Thermal Prop erties 207
1000 Graphene Isolated (10,10) SWNT Graphite (phonon) 100 SWNT rope Measured SWNTs
10
600 SWNTs 1 C (mJ/gK)
400
Fig. 3. Measured sp e-
0.1 C (mJ/gK) 200
ci c heat of SWNTs,
to predictions 0 compared
0 100 200 300 for the sp eci c heat of
0.01 T (K)
isolated (10,10) tub es,
10T (K) 100 SWNT rop es, a graphene
layer and graphite [11]
1.4 Measured Sp eci c Heat of SWNTs and MWNTs
The various curves in Fig. 3 show the calculated phonon sp eci c heat for
isolated (10,10) SWNTs, a SWNT rop e crystal, graphene, and graphite. The
phonon contribution for a (10,10) SWNT was calculated by computing the
phonon density of states using the theoretically derived disp ersion curves
[3] and then numerically evaluating Eq. 1; C (T ) of graphene and graphite
was calculated using the mo del of Al-Jishi and Dresselhaus [12]. Because of
the high phonon density of states of a 2-D graphene layer at low energy
due to the quadratic ZA mo de, 2-D graphene has a high sp eci c heat at
low T . The low temp erature sp eci c heat for an isolated SWNT is however
signi cantly lower than that for a graphene sheet, re ecting the sti ening
of the acoustic mo des due to the cylindrical shap e of the SWNTs. Below
5K, the predicted C (T ) is due only to the linear acoustic mo des, and
C (T ) is linear in T , a b ehavior which is characteristic of a 1-D system. In
these curves, we can see clearly the e ects of the interlayer (in graphite) and
inter-tub e (in SWNT rop es) disp ersion on the sp eci c heat. Below 50 K,
the phonon sp eci c heat of graphite and SWNT rop es is signi cantly b elow
that of graphene or isolated (10,10) SWNTs. The measured sp eci c heat
of graphite [8,9] matches the phonon contribution ab ove 5K, below which
temp erature the electronic contribution is also imp ortant.
The lled p oints in Fig. 3 represent the measured sp eci c heat of SWNTs
[11]. The measured C (T ) agrees well with the predicted curve for individual
(10,10) nanotub es. C (T ) for the nanotub es is signi cantly smaller than that
of graphene b elow 50 K, con rming the relative sti ness of the nanotub es to b ending. On the other hand, the measured sp eci c heat is larger than that
208 Hone
exp ected for SWNT rop es. This suggests that the tub e-tub e coupling in a
rop e is signi cantly weaker than theoretical estimates [6,11].
5 E || 4 kBΘD kBΘsubband Θ⊥ 3 kB D
k⊥ k || 2 Measured C (mJ/gK) Acoustic Band 1st Subband 1 Total
0
04812T (K)
Fig. 4. Measured sp eci c heat of SWNTs compared to a two-band mo del with
k
?
=50 = 960 K; transverse disp ersion (inset). The tting parameters used are
D
D
K; and = 13 K [11]
subband
Figure 4 highlights the low-temp erature b ehavior of the sp eci c heat. The
exp erimental data, represented by the lled points, show a linear slop e b e-
low 8K, but the slop e do es not extrap olate to zero at T = 0, as would b e
exp ected for p erfectly isolated SWNTs. This departure from ideal b ehavior
is most likely due to a weak transverse coupling b etween neighb oring tub es.
The measured data can b e t using a two-band mo del, shown in the inset.
The dashed line in Fig. 4 represents the contribution from a single (four-fold
k
degenerate) acoustic mo de, which has a high on-tub e Debye temp erature
D
?
andamuch smaller inter-tub e Debye temp erature . The dot-dashed line
D
represents the contribution from the rst (doubly-degenerate) subband, with
?
minimum energy k . Because > , the subband is as-
B subband subband
D
sumed to be essentially one-dimensional. The solid line represents the sum
of the twocontributions, and ts the data quite well. The derived value for
k
the on-tub e Debye temp erature is = 960 K (80 meV), which is slightly
D
lower than the value of 1200 K (100 meV) which can be derived from the
?
calculated phonon band structure. The transverse Debye temp erature is
D
13 K (1.1 meV), considerably smaller than the exp ected value for crystalline
Phonons and Thermal Prop erties 209
MWNTs
100
10
C (mJ/gK) Yi et al.
Measured sp e-
Mizel et al. Fig. 5.
heat of MWNTs
Graphite ci c
compared to the
Graphene [6,13],
phonon sp e-
1 Isolated nanotube calculated
ci c heat of graphene,
graphite, and isolated 10(K) 100
T nanotub es
rop es (5 meV) or graphite (10 meV). Finally, the derived value of is
subband
50 K (4.3 meV), which is larger than the value of 30 K (2.5 meV) given bythe
calculated band structure [3].
Figure 5 shows the two rep orted measurements of the sp eci c heat of
MWNTs, along with the theoretical curves for graphene, isolated nanotub es,
and graphite. Yi et al. [13] used a self-heating technique to measure the sp e-
ci c heat of MWNTs of 20-30 nm diameter pro duced byaCVD technique,
and they nd a linear b ehavior from 10 K to 300 K. This linear behavior
agrees well with the calculated sp eci c heat of graphene b elow 100 K, but is
lower than all of the theoretical curves in the 200-300 K range. Agreement
with the graphene sp eci c heat, rather than that for graphite, indicates a rel-
atively weak inter-layer coupling in these tub es. Because the sp eci c heat of
graphene at low T is dominated by the quadratic ZA mo de, the authors p os-
tulate that this mo de must also b e present in their samples. As was discussed
ab ove(x1.1,1.2), such a band should should not exist in nanotub es. However,
the phonon structure of large-diameter nanotub es, whose prop erties should
approach that of graphene, has not b een carefully studied. Mizel et al. rep ort
a direct measurement of the sp eci c heat of arc-pro duced MWNTs [6]. The
sp eci c heat of their sample follows the theoretical curve for an isolated nan-
otub e, again indicating a weak inter-layer coupling, but shows no evidence
of a graphene-like quadratic phonon mo de. At present the origin of the dis-
crepancy between the two measurements is unknown, although the sample
morphologies may b e di erent.
210 Hone
2 Thermal conductivity
Carb on-based materials (diamond and in-plane graphite) display the highest
measured thermal conductivityofanyknown material at mo derate temp er-
atures [14]. In graphite, the thermal conductivity is generally dominated by
phonons, and limited by the small crystallite size within a sample. Thus the
apparent long-range crystallinity of nanotub es has led to sp eculation [15] that
the longitudinal thermal conductivity of nanotub es could p ossibly exceed the
in-plane thermal conductivity of graphite. Thermal conductivityalsoprovides
another to ol (b esides the sp eci c heat) for probing the interesting low-energy
phonon structure of nanotub es. Furthermore, nanotub es, as low-dimensional
materials, could have interesting high-temp erature prop erties as well [16].
In this section, we will rst discuss the phonon and electronic contributions
to the thermal conductivityin graphite. Then we will examine the thermal
conductivityofmulti-walled and single-walled nanotub es.
The diagonal term of the phonon thermal conductivity tensor can be
written as:
X
2
= Cv ; (8)
zz z
where C , v ,and are the sp eci c heat, group velo city, and relaxation time
of a given phonon state, and the sum is over all phonon states. While the
phonon thermal conductivity cannot b e measured directly, the electronic con-
tribution can generally b e determined from the electrical conductivityby
e
the Wiedemann{Franz law:
e
L ; (9)
0
T