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
8 2
where the Lorenz number L =2:45 10 (V/K) .Thus it is in principle
0
straightforward to separate the electronic and lattice contributions to (T ).
In graphite, phonons dominate the sp eci c heat ab ove 20 K [17], while in
MWNTs and SWNTs, the phonon contribution dominates at all temp era-
tures.
In highly crystalline materials and at high temp eratures (T > =10), the
D
dominantcontribution to the inelastic phonon relaxation time is phonon-
phonon Umklapp scattering. Atlow temp eratures, however, Umklapp scat-
tering disapp ears and inelastic phonon scattering is generally due to xed
sample b oundaries or defects, yielding a constant .Thus at low temp erature
(T < =10), the temp erature dep endence of the phonon thermal conductiv-
D
ity is similar to that of the sp eci c heat. However, in an anisotropic material,
2
the weighting of each state by the factor v b ecomes imp ortant. The thermal
conductivityismostsensitive to the states with the highest band velo cityand
scattering time. In graphite, for instance, the ab-plane thermal conductivity
can b e closely approximated by ignoring the inter-planar coupling [17]. From
this argument, wewould exp ect that the temp erature-dep endence of the ther-
mal conductivity of SWNT rop es and MWNTs should b e close to that of their
Phonons and Thermal Prop erties 211
constituent tub es. However, bundling individual tub es into rop es or MWNTs
mayintro duce inter-tub e scattering, which could p erturb somewhat b oth the
magnitude and the temp erature dep endence of the thermal conductivity.
2.1 Thermal Conductivity of MWNTs
25 κ∝T2 )
-1 20 K -1 15
(W m 10 MWNTs κ 5
0 0 100 200 300
T (K)
Fig. 6. Measured thermal conductivity of MWNTs [13] from 4 K to 300 K
2:3
In highly graphitic b ers, (T ) follows a T temp erature dep endence
until 100 K , then b egins to decrease with increasing T ab ove 150 K [18].
This decrease in (T ) ab ove 100 K is due to the onset of phonon-phonon
Umklapp scattering, which grows more e ective with increasing temp erature
as higher-energy phonons are p opulated. In less graphitic b ers, the magni-
tude of is signi cantly lower, and the Umklapp peak in (T )is not seen,
b ecause grain-b oundary scattering dominates (T ) to higher temp eratures.
Figure 6 shows the thermal conductivity of CVD-grown MWNTs, on a
linear scale, from 4 K to 300 K [13]. Because of the large diameter of these
tub es, we exp ect them to act essentially as 2-D phonon materials. Indeed, at
2 2:3
low temp erature (T < 100 K), (T ) increases as T , similar to the T
b ehavior in graphite. The ro om-temp erature thermal conductivityissmall,
comparable to the less-graphitic carb on b ers, and the MWNTs do not show
amaximum in (T ) due to Umklapp scattering; b oth prop erties are consistent with a small crystallite size.
212 Hone
1 SWNTs
0.8
0.6 (arb. units) (arb. 0.4 κ
0.2
Thermal conduc-
0 Fig. 7. y of a bulk sample of 0 100 200 300 tivit
T(K)
SWNTs [19]
2.2 Thermal Conductivity of SWNTs
Figure 7 represents the measured (T ) of a bulk sample of laser-vap orization
pro duced SWNTs, with 1.4-nm diameter [20]. The di erent temp erature-
dep endence of (T ) re ects the much smaller size of SWNTs compared to
MWNTs. (T ) increases with increasing T from 8K to 300 K, although a
gradual decrease in the slop e ab ove 250 K may indicate the onset of Umk-
lapp scattering. Most striking is a change in slop e near 35 K: below this
temp erature, (T ) is linear in T and extrap olates to zero at T =0. We will
discuss the low-temp erature b ehavior in detail b elow.
Although the temp erature dep endence of the thermal conductivityisthe
same for all 1.4-nm diameter SWNT samples, the magnitude of (T ) is
sensitive to sample geometry. In disordered `mat' samples, the the ro om-
temp erature thermal conductivityis35 W/m-K. However, in samples con-
sisting of aligned SWNTs, the ro om-temp erature thermal conductivity is
ab ove 200 W/m-K [21], within an order of magnitude of the ro om-temp erature
thermal conductivity of highly crystalline graphite. Because even such an
aligned sample contains many rop e-rop e junctions, it is likely that a single
tub e, or a rop e of continuous tub es, will have signi cantly higher thermal
conductivity than the bulk samples.
Simultaneous measurement of the electrical and thermal conductance of
bulk SWNT samples yields a Lorenz ratio = T which is more than two
orders of magnitude greater than the value for electrons at all temp eratures.
Thus the thermal conductivity is dominated by phonons, as exp ected.
Figure 8 highlights the low-T behavior of the thermal conductivity of
SWNTs [19]. As discussed ab ove, the linear T dep endence of (T )likely re-
ects the one-dimensional band-structure of individual SWNTs, with linear
acoustic bands contributing to thermal transp ort at the lowest temp eratures
and optical subbands entering at higher temp eratures. (T ) can b e mo deled
using a simpli ed two-band mo del (shown in the inset to Fig. 8), considering
Phonons and Thermal Prop erties 213
0.4 SWNTs Acoustic band v=20 km/s 1st subband Total 0.3 Measured K(T)
0.2 ω ω
(arb. units) (arb. 0 κ
0.1 k
Fig. 8. Measured Low-
temp erature Thermal
yof SWNTs, 0 Conductivit
0 20 40 60 80 100
compared to a two-band
T(K)
mo del [19]
a single acoustic band and one subband. In a simple zone-folding picture, the
acoustic band has a disp ersion ! = vk and the rst subband has disp ersion
2 2 2 2
! = v k + ! ,where! = v=R. The thermal conductivityfromeachband
0
0
can then b e estimated using Eq. (8) and assuming a constant scattering time
.Thus provides an overall scaling factor, and v sets the energy scale ~!
0
of the splitting b etween the two bands.
Figure 8 shows the measured (T ) of SWNTs, compared to the results
of the two-band mo del discussed ab ove, with v chosen to b e 20 km/s, which
is b etween that of the `twist' (v = 15 km/s) and LA (v = 24 km/s) mo des.
The top dashed line represents (T ) of the acoustic band: it is linear in T ,
as exp ected for a 1-D phonon band with linear disp ersion and constant .
The lower dashed line represents the contribution from the optical subband;
it is frozen out at low temp eratures, and b egins to contribute near 35 K. The
solid line is the sum of the twocontributions, and is quite successful in tting
the exp erimental data b elow 100 K. The phonon energies, and temp erature
scale of the observed linear b ehavior, are higher in the thermal conductivity
measurements than in the heat capacity measurements (x1.4). This maybe
due to the preferential weighting of higher-velo city mo des in thermal conduc-
tivity, although more detailed mo deling is needed to resolve this issue. The
measured linear slop e can b e used to calculate the scattering time, or, equiv-
alently, a scattering length. A ro om-temp erature thermal conductivityof200
W/m-K (as is seen in the bulk aligned samples) implies a phonon scattering
length of 30 nm, although this value is likely to b e higher for single tub es.
Wehaveseenabove that the small size of nanotub es causes phonon quan-
tization which can be observed b oth in the heat capacity and in the ther-
mal conductivityatlow temp eratures. The restricted geometry of the tub es
may also a ect the thermal conductivity at high temp erature since Umklapp
214 Hone
scattering should be suppressed in one dimensional system b ecause of the
unavailability of states into which to scatter [16,22]. Extension of these mea-
surements to higher temp eratures, as well as additional theoretical mo deling,
should proveinteresting.
References
1. L. X. Benedict, S. G. Louie, and M. L. Cohen, Solid State Commun. 100,
177{180 (1996).
2. D. Sanchez-Portal, E. Artacho, J. M. Soler, A. Rubio, and P. Ordejon, Phys.
Rev. B 59, 12678{12688 (1999).
3. R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon
Nanotubes (Imp erial College Press, London, 1998).
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5. C. Kittel, in Introduction to Solid State Physics, 6th ed., (John Wiley and Sons,
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D. G. Onn, Synth. Met. 2, 203 (1980).
10. Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt Brace
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Physics.
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Comm. 59, R9015 (1999).
14. G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants,
16th ed. (Longman, London, 1995).
15. R. S. Ruo and D. C. Lorents, Carb on 33, 925 (1995).
16. D. T. Morelli, J. Heremans, M. Sakamoto, and C. Uher, Phys. Rev. Lett. 57,
869 (1986).
17. B. T. Kelly,inPhysics of Graphite, (Applied Science (London), 1981).
18. J. Heremans and C. P. Beetz, Jr., Phys. Rev. B 32, 1981 (1985).
19. J. Hone, M. Whitney, C. Piskoti, and A. Zettl, Phys. Rev. B Rapid Comm. 59,
R2514 (1999).
20. A. Thess, R. Lee, P. Nikolaev, H. Dai, P.Petit, J. Rob ert, C. Xu, Y. H. Lee,
S. G. Kim, A. G. Rinzler, D. T. Colb ert, G. E. Scuseria, D. Tomanek, J. E.
Fischer, and R. E. Smalley, Science 273, 483{487 (1996).
21. J. Hone et al., (unpublished).
22. R. E. Peierls, in Quantum Theory of Solids, (Oxford University Press, London, 1955).
Index
{ electronic, 8, 317 abstract, 31, 133, 135, 215
{ electronic device, 288, 315, 316 acceptor doping
{ electronic devices, 96, 290 { b oron, 114
{ EMI shielding, 308 acoustic phonon mo des, 150, 208, 213,
258
{ energy storage, 301
adsorption
{ eld emission, 218, 263, 291
{ graphite, 19 { gas
{{ hydrogen, 301
{ hydrogen storage, 290, 302
Aharonov{Bohm
{ inorganic fullerenes, 89
{ intercalated nanotub es, 28 { e ect, 218, 226, 235, 253
{ light emission, 217, 263 { ux, 235
{ p erio d, 247
{ lithographic techniques, 8
{ phase, 246
{ lubrication, 90
{ materials science, 291 Al O ,73
2 3
{ mechanical, 269, 306 Al O
2 3
{ whiskers, 24
{ mechanical reinforcement, 276
{ medical, 310 aligned single wall carb on nanotub es, 3
{ medical catheter, 312 aligned tub es
{ micro electronics, 89 { bundles, 156
{ MWNT lms, 292
{ microprob e tips, 96
{ microwave ampli ers, 294 { MWNTs, 292
{ nano-lithography, 310 { SWNT bundles, 307
{ nano-technology, 312 angle resolved photo emission sp ec-
{ nanolithography,90 troscopy (ARPES)
{ ARPES, 176
{ nanoprob es, 304
anti-Stokes Raman , 160
{ nanotub e , 316
{ ,160
{ nanotub es, 3, 9, 286, 287
{ optical, 308 { sp ectra, 160{162, 168
applications
{ opto-electronic, 86
{ conductors (1D), 259
{ PAN b er, 24
{ photo-voltaic, 308 { actuator
{ photo catalysis, 91 {{ electromechanical, 312
{ actuators, 312
{ reinforcement, 64, 316
{ battery, 28, 301, 315
{ scanning prob e tip, 218
{ scanning prob e tips, 64, 71 { biological, 311
{ sensors, 312 { carb on b er, 24, 318
{ catalysis, 73, 315
{ structural, 303
{ surface prop erties, 288 { chemistry,311
{ thin-screen technology,220 { commercial, 317
{ trib ological, 91 { comp osites, 290, 291, 303, 306
{ CVD b er, 24
{ vacuum electronics, 291
{ vap or phase growth, 220 { display,294
arc discharge, 19, 20 { electro chemical, 297
arc metho d, 152 { electron emission, 263
Index 329
Breit{Wigner{Fano plot, 159 arc-discharge
Brillouin zone (BZ) { growth, 223, 225, 227, 230, 256, 272,
{ K points, 142 273
{ metho d, 263, 270
buckling, 76, 304
buckling ARIPES, 176
{ MWNT, 305 armchair
{ nanotub e, 311 { (n; n) tub es, 98
{ SWNT, 304 armchair nanotub es, 4{6, 139, 192
bucky pap er, 220 asymmetric-digital-signal-line (ADSL)
bulk mo dulus, 21 { telecom network, 297
atomic force microscop e (AFM), 122
bundle, 8
bundle e ect, 161 atomic force microscop e (AFM)
bundles, 272 { exp eriments, 275
BZ { image, 120, 271, 274, 275, 277
{ K -p oint, 18, 141, 146, 155 { instruments, 311
{ measurements, 271{273, 275
C , 179, 180, 184, 186, 196
{ metho d, 272, 275
60
carb on
{ nano-lithography,311
2
{ sp b onds, 275
{ technique, 271
{ liquid, 15, 26
{ tip, 290, 310, 311, 317, 318
{ phase diagram, 15, 26
{ tips, 275
carb on b onding , 14
Auger p eak
2
{ sp b ond, 16
{ OKLL, 184
{ -electrons, 175
{ b onds, 175 B C N , 75, 91
x y z
2
{ sp b ond, 14, 27 ball-milling, 299
3
{ sp ,14 bamb o o structure, 68, 222, 223, 225
3
{ sp b ond, 16 band
{ b ond length, 6 { gap, 19, 23
{ chemical, 176 { overlap, 18, 23
{ covalent, 14 band-folding, 98, 100, 106
{ hybridization, 14, 15, 175 BC N, 66, 75, 77, 80
2
{ hybridization BC , 66, 77
3
n
Bernal stacking, 17, 18
{{ sp ,14
carb on b ers, 13, 14, 20, 21, 35 BET characterization, 299
carb on b ers blackbody, 154
{ bundles, 21 BN, 62, 64, 66, 74, 75
{ comp osites, 24 BN
{ diameter, 3 { nanotub e, 66, 74, 75
{ history,23 { nanotub es, 75
{ mechanical prop erties, 22 { nanotub es
{ mechanical strength, 20 {{ rop es, 76
{ melting, 27 { p olyhedra, 80
{ PAN, 20, 21 BN nanotub e, 105
{ p erformance, 20 BN particles
{ pitch, 20, 21 { nano-co co ons, 76
{ p olymer-based, 24 b ond p olarization theory, 151
{ thermal conductivity,25 b oron, 113
{ vap or grown, 2, 14, 20{26 breaking strength, 22
330 index
current-voltage, 109 carb on materials
{ liquid, 14
dangling b ond, 62, 65
{ whiskers, 13, 14, 19, 20, 24
dangling b onds, 16
{ whiskers
DC breakdown voltage, 297
{{ mechanical, 24
Debye temp erature, 208
{{ synthesis, 20
defect
carb on nano b ers, 299
{ pentagon-heptagon pair, 95, 105{110
carb on nanotub e
defect pair, 106
{ parameters, 7
density functional
{ pro duction, 74
{ lo cal, 99
catalyst, 152
{ metho d, 112, 120
catalyst sp ecies, 8
density of rop e states
catalytic chemical vap or dep osition, 24
{ electronic, 118, 119
catho de ray lighting element, 294
density of states
catho de raytube(CRT)
{ electronic, 109, 117, 127
{ thermionic, 295
{ joint, 119
catho de strip es, 296
{ lo cal, 113, 115, 116
CdCl structure, 73
2
{ phonon, 202, 204, 207
charge transfer, 27, 230
density of states (DOS)
charge-discharge, 301
{ electronic, 144
chemical sensor, 312
Department of Energy
chemical vap or dep osition (CVD)
{ benchmark, 302
{ dep osition, 24
dep olarization e ect, 145
chiral angle, 3{7, 139
diameter
chiral nanotub e, 139
{ d ,139
t
chiral nanotub es, 6
diameter distribution, 8, 152
chiral vector, 4{7
diameter-selective formation, 151
chiral vector
diamond
{ C , 139
3
h
{ sp b ond, 15
chirality, 80, 81, 138
dielectric function, 155, 177, 181, 182,
collapsed tub es, 100
191, 194, 196
concentric structure, 167
dielectric tensor, 177, 181
conductance
Dirac
{ intratub e, 120
{ hamiltonian, 124
{ junction, 120
{ points, 117
{ quantum, 95, 115, 121
disorder p otential, 125
{ tub e, 96, 101, 116, 121, 122
disp ersion relations, 25
conductivity, 197
dop e, 156
con guration
dop e
2
{ sp ,15
{ Br, 156
conjugated p olymers, 308
{ Br , 161
2
core electrons, 14, 176
{ Cs, 156
Coulomb blo ckade, 122, 242, 243
doping
covalent
{ alkali metals, 315
2
{ sp b ond, 64
{ BC N tub es, 81
2
cross-links, 274
{ BN tub es, 81
crossed-tub e junctions, 96
{ carb on materials, 301
{ external, 18, 230 current density, 294
Index 331
{ rop e, 96 { functionalization of tub es, 308
{ SWNTs, 97 { halogens, 315
{ theoretical, 97 { inorganic tub es, 91
electronic sp eci c heat, 205 { intercalation tub es, 315
electronic structure, 16, 64, 178, 186, { internal, 18, 224
189 { interstitial, 18
electronic structure { K, 197, 198
{ turb ostratic graphite, 18 { metallic tub es, 122
{ E , 146 { morphology,28
pp
{ carb on b ers, 23 { nanotub es, 174
{ graphene, 18
{ p olymers with nanotub es, 309
{ symmetry gap, 98, 109, 119 { rop es, 118
electrons { semiconducting tub es, 96, 122, 127
{ -electrons, 184 { substitutional, 18
Elliott relation, 258 Drude resistance, 229, 236, 247
emission site density,294 Drude{Lorentz mo del, 197
emission tips, 291 Drude-like tail, 181
energy capacity, 299
Edison, 23
energy contour plot, 141
EELS, 176
energy disp ersion
elastic prop erties, 20
{ graphite, 139
electric p olarizability, 101
energy disp ersion relations
electro chemical intercalation, 299
{ phonon, 148
electromagnetic induction (EMI)
energy loss sp ectrum, 154
{ shielding, 308
energy storage, 297
electron
facets, 21, 23, 88, 89 { -electrons, 174, 191
Fermi level, 27 electron density of states, 178, 180, 184,
Fermi liquid, 227, 250, 254, 256
186, 188, 189
eld emission, 218, 263{267, 291 electron energy loss sp ectroscopy
eld emission (EELS), 74, 75, 77, 177, 180, 189,
{ current, 266 192, 195, 196, 198, 299
{ guns, 263 electron energy loss sp ectroscopy
{ individual nanotub es, 264
(EELS)
eld enhancement factor, 291, 292 { graphite, 180, 181
lled nanotub es, 314 { nanotub es, 189
lling nanotub es, 314 { theoretical, 189
lling nanotub es electron eld emitters, 290
{ liquid metals, 225 electron sp ectroscopies, 174
at panel display,295 electron spin resonance, 257
at-panel screen, 174
electron spin resonance (ESR)
force constants, 16 { g -factor, 257
Fourier transform infrared (FTIR), 302 { g -shift, 257
Fowler{Nordheim { signal, 259
{ behavior, 293 { technique, 218
{ equation, 263, 291 electronic density of states, 288
{ mo del, 265, 267 electronic prop erties, 2, 3, 95, 96, 101,
{ plot, 292 105
fracture, 304 electronic prop erties
332 index
highly oriented pyrolytic graphite fuel cells, 297
(HOPG), 19
fullerene-like caps, 3, 5, 7
history
fullerenes
{ carb on b ers, 3
{ discovery,3,16
{ carb on nanotub es, 2{4
{ isolated p entagon rule, 3, 7
{ fullerenes, 8
functionalization, 315
{ nanotub es, 25
furnace temp erature, 152
hollow core, 21, 22, 28, 224
honeycomb lattice, 4{6
gas discharge tub e (GDT)
honeycomb lattice
{ nanotub e-based, 297
{ unit vectors, 4
{ nonlinear shunt, 297
{ unit vectors, 5
{ protectors, 297
HOPG, 17, 19
GaSe nanotub es, 82
hybridization e ect, 167
gate voltage, 101
hydrogen storage, 290, 301, 302
Gaussian p otential, 123
graphene, 4, 6, 14, 202
Iijima, 26
graphite, 63, 66, 74, 77, 79, 81, 177{181,
impurities, 112
186, 189, 191, 192, 198, 203, 210
impurities
graphite
{ b oron, 113, 114, 116
2
{ sp b ond, 19, 79
{ substitutional, 112
{ arcing, 74
indium-tin-oxide (ITO)
{ crystalline 3D, 14
{ strip es, 295
{ honeycomb lattice, 14
infrared sp ectra, 23
{ kish, 19
innermost nanotub e, 167
{ resistivity,231
inorganic
{ single-crystal, 231
{ fullerene, 62, 63
graphite intercalation comp ound
{ nanotub e, 62, 63
(GIC), 196, 197
inorganic fullerene, 87, 91
greatest common divisor
inorganic nanotub e, 63, 86, 88, 90
{ d ,6,142
R
inter-layer interaction, 165
growth, 33
intercalated nanotub es, 196
growth
intercalation, 91, 299
{ catalytic growth, 46
intercalation
{ close-ended mechanism, 36
{ Li, 71
{ lip-lip interaction mo del, 42
{ Na, 78
{ op en-ended mechanism, 37
interface b onding, 306
{ ro ot growth, 53
interlayer
{ rop es growth, 52
{ force, 27
{ separation, 17, 18
helicity, 98, 105, 113, 192
{ spacing in MWNTs, 222
hetero junctions
interstitial channels, 315
{ semiconductor, 105, 107
IPES, 176
hexagons/unit cell
{ N , 141
junction
HfB ,74
{ metal-metal, 96, 105, 108{110, 121,
2
highly ordered pyrolytic graphite
126
(HOPG), 24
{ metal-semiconductor, 96, 105{108,
110, 126 highly oriented pyrolytic graphite, 24
Index 333
MoS , 62, 84 { semiconductor-semiconductor, 121
2
MoS inorganic nanotub e, 66 junctions, 95, 96, 98, 105{107, 126
2
MoS nanotub e, 83 junctions
2
multi-walled carb on nanotub es, 163 { conductance, 121
multiwall carb on nanotub es, 3, 8
{ crossed, 121, 122
{ crossed-tub e, 119
nano comp osite, 315
{ metal-metal, 105, 127
nano b ers, 22
{ metal-semiconductor, 127
nano b ers
{ semiconductor-metal, 106
{ vap or grown, 25
nanoswitch, 109
kinked nanotub e, 110, 112, 116
nanotub e
kinked nanotub e
{ chiral angle, 6
{ I-V, 110
{ chirality,6
{ junctions, 126
{ curvature, 14, 16
Kramers-Kronig analysis, 177, 181
{ diameter, 6
Landauer formalism, 108
{ zigzag, 82, 86
laser vap orization metho d, 151
nanotub e hollows, 314
lattice constant, 17
nanotub e indices
light emission, 263
{ (n; m), 139
Lindhardt formula, 191
nanotub e tip, 51
linear disp ersion, 141
nanotub es
lip-lip interaction, 76
{ excited states, 189
lithium battery,299
{ intercalated SWNTs, 196
loss function, 177, 181, 182, 189, 195,
{ metallic, 16
196
{ semiconducting, 16
Luttinger liquid, 184, 226, 250, 253,
{ uno ccupied states, 189
256, 259
Nernstian b ehavior, 298
nitrogen, 114
magnetoresistance, 235, 236, 246, 247,
nuclear magnetic resonance (NMR)
249, 250
{ proton, 303
materials prop erties, 270
nucleation, 33
mean free path, 112, 122, 126, 127
Nyquist noise, 249
mechanical deformation, 109
mechanical prop erties, 105, 269, 271
one-dimensionality, 2{4
mechanical prop erties
onion morphology, 20, 61, 73, 75, 91,
{ disorder e ect, 272
219, 221, 228
optical, 83 mechanical stress, 109
mechanical stress
optical
{ prop erties, 89 { axial, 275
{ compressive, 22
{ device, 86
{ measurements, 63 { MWNTs, 270
metal hydrides, 301
{ prop erties, 83
{ transitions, 86 metallic window, 157
{ tuning, 82, 86 metallic wires, 224
micro electro des, 298
optical absorption, 153
optical absorption sp ectroscopy, 191, mo dulus
192 { carb on b ers, 21
optical density,157 molecular prob es, 311
334 index
{ graphite, 178 organic light emitting dio des (OLED),
{ nanotub es, 184 309
plasma breakdown, 297 over-voltage protection, 297
plasmon, 154, 177, 181, 189, 191, 192,
pentagon-heptagon
195, 196
{7 pair, 57
plasmon
pentagons and heptagons, 74
{ + plasmon, 182
p erturbations, 123
{ -plasmon, 182, 184, 189, 191, 192,
p erturbations
194{198
{ asymmetric, 119
{ + ,184
{ external, 95, 122
{ + -plasmon, 189, 192, 195, 198
{ long range, 117, 123, 127
{ energy,183
PES, 179, 180, 184, 186, 188, 189
{ free electron, 183
phase diagram, 15, 16, 26, 62, 65, 66,
plasmon excitation sp ectrum, 190
78
p olarizability
phase shifts, 113{116
{ static, 101, 104
phase velo city,150
{ tensor, 102
phonon
{ unscreened, 102
{ K point, 151
polyacrylonitrile, 24
{ disp ersion, 87
polyacrylonitrile (PAN)
{ twist mo de, 213
{ b er, 24
{ zone-b oundary,88
p olyhedra, 62
phonon disp ersion
p olymer comp osite, 306
{ SWNTs, 206
p otassium, 196, 198
phonon disp ersion of MWNTs, 206
PPV, 308
phonon disp ersion relations, 148
pseudo-spin, 124
phonon mo des
pseudop otential, 99, 112
{ acoustic, 202, 204
puri cation, 153
{ twist mo de, 204
phonon scattering length, 213
quantized conductance, 234
phonon states, 87
quantum size e ect, 83
phonon thermal conductivity, 210
quantum conductance, 108
phonon velo city, 204
quantum con nement, 16
phonons
quantum con nement
{ heat capacity, 103
{ Coulomb blo ckade, 101
{ twistons, 103
quantum dots, 122, 125
phonons
quantum size e ect, 64, 83, 85
{ subbands, 203
quantum wire
{ zone-folding, 203
{ metallic tub es, 101
photo thermal de ection sp ectrum,
radial breathing mo de, 23
153, 154
radial breathing mo de (RBM), 138,
photo-thermal de ection sp ectrum
151{153, 155, 158{161, 163{168 (PDS), 154
radial breathing mo de (RBM)
photo electron sp ectroscopy (PES), 175
{ frequency, 152, 166 photo emission sp ectroscopy (PES)
{ Raman intensity, 157
{ angle resolved (ARPES), 178
{ Raman sp ectra, 164 { angle resolved inverse (ARIPES), 178
{ sp ectra, 157 { data, 184
radiation dose, 273 { fullerenes, 188
Index 335
solid lubricant, 89, 90 Raman sp ectra, 87
Raman intensity
solution sample, 152
S
~
{ I (d ), 160
space group, 18
0
1540
Raman p eak
sp eci c heat, 202, 205
{ D -band, 151
sp eci c heat
Raman sp ectra, 23, 28
{ electronic, 203
Raman sp ectra
{ graphite, 203, 207
{ G-band, 157
{ SWNTs, 207
rehybridization
sp eci c heat of MWNTs, 209
{ - , 100
spin
Resonance Raman, 88
{ di usion length, 259
resonance Raman e ect, 157
{ relaxation time
resonant inelastic x-ray scattering
{{ Elliott theory, 257
(RIXS)
{ susceptibility, 261
{ SWNTs, 186, 187
{ susceptibility
rhomb ohedral phase, 17
{{ Pauli, 261
RIXS, 176
spin susceptibility, 263
rop e, 17, 302, 307
splitting of the DOS p eaks, 146
rop e
spray metho d, 152
{ (10,10) nanotub es, 117
STM, 191
{ diameter, 273
STM
{ MWNT, 227, 238
{ image, 101, 104
{ SWNT, 238, 271, 272, 276, 290, 303,
STM tip, 267
306
Stokes, 160
rop es, 7, 8
Stone{Wales defect, 112, 115, 116, 289,
304
sample preparation, 152
Stone{Wales defects, 96
scanning prob e, 69, 79
strain energy,100
scanning prob e
strength
{ microscopy,90
{ carb on b ers, 21, 24
{ tips, 64
strong lo calization, 243, 244
scanning prob e microscop e tip, 311
STS, 101, 147, 188, 191
scanning tunneling microscop e, 71
subband, 208
scanning tunneling microscopy (STM),
substrate, 120
4
substrate
scanning tunneling sp ectroscopy, 147
{ attraction, 120
scanning tunneling sp ectroscopy (STS),
{ contact force, 127
234
surface sensitive, 174
Schottky
SWNT
{ barrier, 126
{ diameter, 139
{ barriers, 105, 122
SWNT lms, 294
{ behavior, 127
sword-in-sheath failure, 21, 22
{ dio de, 121
symmetry, 5, 102, 109, 126
scroll, 19, 20, 222, 223
symmetry
selection rule, 144
{ axis, 105
Si
{ breaking, 109
{ whiskers, 24
{ broken, 117{119
single-electron tunneling, 242
{ group, 6 SiO ,73 2
336 index
tree ring morphology, 20, 23, 25 { mirror, 118
triangular lattice, 17 { rotation, 6
{ rotational, 109
trigonal warping e ect, 146
trio de-typ e, 295 { Stone{Wales, 115
{ symmetry vector, 6
turb ostratic, 26
{ vector, 7
turb ostratic
synthesis
{ nanotub es, 18
{ chemical vap or phase dep osition, 8
turb ostratic graphite, 18
{ arc metho d, 8
turb ostratic graphite, 18, 222
{ laser vap orization metho d, 7, 8
twist mo de, 206
twisting acoustic mo de (TW), 150
tangential mo de, 158
telecom, 297
Umklapp p eak, 211
temp erature-programmed desorption
Umklapp scattering, 210{212, 214
(TPD)
unit vectors, 139
{ SWNTs, 302
universal conductance, 236
tensile strength, 21, 303, 304
thermal conductivity, 213
V O ,62
2 5
thermal conductivity
vacuum micro electronic devices, 290
{ MWNTs, 211
valence orbitals, 14
{ SWNTs, 212
van der Waals force, 18, 273
{ carb on-based materials, 210
van der Waals interaction, 179
{ graphite, 26
van der Waals spaces, 299
{ pitch b ers, 21
van Hove singularities, 144, 234, 235,
thermal gravimetric analysis, 314
251, 252
three p eaks, 154
van Hove singularity, 114, 179, 186,
threshold elds, 292
188, 189, 191, 192, 195, 196, 198
tight binding molecular dynamics, 148
weak lo calization, 235, 236, 247, 250, tight-binding mo del, 109, 178, 191, 194
256 time reversal symmetry,146
TiO , 62, 73
weak lo calization
2
{ logarithmic dep endence, 237 tip radius, 265
top ological defects, 96
{ low-T conductance, 238
{ magnetoresistance, 247 transfer energy
{ ,140
{ WL theory (2D), 236
0
transistor, 174
whiskers
translation vector, 4, 7
{ carb on, 24
translational vector
{ graphite, 14, 19, 20, 230
{ T,141
{ iron, 24
transmission electron microscop e
{ nonmetals, 24
(TEM), 2, 4, 7, 25, 64, 74{76, 84, 85,
WKB approximation, 263
89, 219, 222{226, 254, 256, 265, 272,
work function, 292
275, 276, 278
WS ,82
2
transp ort
WS
2
{ nonlinear, 109
{ nanotub es, 64
{ prop erties, 115
WS nanotub e, 83
2
transp ort exp eriments, 101, 197
x-ray absorption sp ectroscopy (XAS) transverse acoustic (TA) mo des, 150
{ SWNTs, 189 trap states, 124
Index 337
x-ray electron sp ectroscopy (XES), 186 Young's mo dulus
x-ray emission sp ectroscopy (XES) { MWNTs, 271
{ graphite, 180
zero gap semiconductor, 18
XAS, 177
zigzag nanotub es, 4{6, 139
XES, 176
zone folding, 25
Young's mo dulus, 22, 64, 89, 105, 271, zone-folding, 213
272, 275, 303 ZrO ,73 2