Hone Thermal Ency Nano

822019 Hone Thermal Ency Nano

httpslidepdfcomreaderfullhone-thermal-ency-nano 19

Carbon Nanotubes Thermal Properties

J HoneColumbia University New York New York USA

INTRODUCTION

As nanoscale graphitic structures carbon nanotubes are

of great interest not only for their electronic and

mechanical properties but also for their thermal proper-

ties Because of their small size quantum effects are

important and the low-temperature specific heat and

thermal conductivity show direct evidence of 1-D

quantization of the phonon bandstructure Modeling of

the low-temperature specific heat allows for determina-

tion of the on-tube phonon velocity the splitting of

phonon subbands on a single tube and the interaction

between neighboring tubes in a bundle The thermal

conductivity of nanotubes has been examined both

theoretically and experimentally Theoretical work pre-

dicts a room-temperature thermal conductivity that is

larger than graphite or diamond Measurements show a

room-temperature thermal conductivity over 200 Wm K

for bulk samples of single-walled nanotubes (SWNTs)and over 3000 Wm K for individual multiwalled

nanotubes (MWNTs) Addition of nanotubes to epoxy

resin can double the thermal conductivity for a loading of

only 1 showing that nanotube composite materials may

be useful for thermal management applications

The first part of this manuscript discusses theoretical

and experimental work on the specific heat of nanotubes

The section lsquolsquoSpecific Heatrsquorsquo provides an introduction to

specific heat In the section lsquolsquoPhonon Density of Statesrsquorsquo

the theoretically derived phonon density of states of na-

notubes and nanotube bundles is compared to that of 2-D

graphene and 3-D graphite In lsquolsquoTheoretically DerivedSpecific Heatrsquorsquo the measured specific heat of nanotubes is

compared to theoretical models

The second part of this manuscript reviews the thermal

conductivity of nanotubes The first section provides

an introduction to thermal conductivity The section

lsquolsquoThermal Conductivity Theoryrsquorsquo discusses theoretical

treatments of the thermal conductivity lsquolsquoMeasured K (T )

of SWNTsrsquorsquo reviews measurements of the thermal con-

ductivity of single-walled nanotubes and lsquolsquoMeasured

K (T ) of MWNTsrsquorsquo reviews measurements of the ther-

mal conductivity of multiwalled nanotubes Finally lsquolsquoAp-

plicationsrsquorsquo describes thermal conductivity measurements

of nanotube-based composites

Specific Heat

The specific heat C (T ) of a material is a sensitive probe of

the low-energy excitations In 3-D graphite 2-D graphene

and nanotubes phonons are the dominant excitations and

the phonon specific heat C ph

dominates C (T ) at most

temperatures C ph depends on the phonon density of states

r(o) and can be obtained by integrating r(o) together

with a temperature-dependent convolution factor account-

ing for the temperature-dependent occupation of each

phonon state[1]

C ph frac14

Z k B

ho

k BT

2e

hok BT eth THORN

rethoTHORNdo

ehok BT Agrave1

2eth1THORN

At a given temperature T the convolution factor decreases

from a value of 1 at o=0 to a value of $ 01 at ho=k BT 6

so that phonons above this energy do not appreciablycontribute to the specific heat In general Eq 1 must be

evaluated numerically

At low temperatures C ph probes only the lowest energy

phonons These are the acoustic modes whose dispersion

can often be expressed as a power law r(o)k a For a

single such mode Eq 1 simplifies to

C ph T ethd =aTHORNeth2THORN

where d is the dimensionality of the system For a linearly

dispersing mode (a=1) the specific heat is linear in T for

a 1-D system and shows the familiar Debye T 3 behavior

for a 3-D system

Phonon Density of States

The phonon bandstructure of isolated nanotubes has been

calculated by Saito et al[23] and by Sanchez-Portal et al[4]

From the bandstructure it is straightforward to calculate

the phonon density of states Fig 1 shows the phonon

density of states of a (1010) nanotube compared to the

density of states of a single 2-D sheet of graphene When a

graphene sheet is lsquolsquorolledrsquorsquo into a nanotube the 2-D

bandstructure folds into a large number of 1-D subbands

For a (1010) tube for instance the six phonon bands

(three acoustic and three optical) of the graphene sheet



ORDER REPRINTS

become 66 separate 1-D subbands A direct result of this

folding is that the nanotube density of states has a number

of sharp peaks as a result of 1-D van Hove singularities

which are absent in graphene and graphite In spite of the

presence of these singularities the overall density of

states is similar at high energies so that the high-

temperature specific heat should be roughly equal as well

This is to be expected the high-energy phonons are more

reflective of carbonndashcarbon bonding than the geometry of

the graphene sheet

At low energies the geometry of the nanotube causes

the phonon structure to substantially differ from that of the

parent graphene sheet Fig 2 shows the theoreticallyderived low-energy phonon bandstructure of an isolated

(1010) nanotube There are four acoustic modes (those

with o0 as k 0) All four have a linear dispersion

o=nk near the zone center The longitudinal (LA) mode

has n=24 kmsec the (doubly degenerate) transverse (TA)

mode has n=9 kmsec and the lsquolsquotwistrsquorsquo mode has n=15

kmsec The first (doubly degenerate) optical mode enters

at 27 meV

The inset to Fig 2 shows the nanotube phonon density

of states (solid line) derived from the bandstructure

shown Only the four acoustic modes are present below

27 meV producing a constant density of states At the

band edge of each optical mode the density of statesdisplays a van Hove singularity characteristic of 1-D

dispersion and increases stepwise By comparison 2-D

graphene and 3-D graphite have very different low-energy

phonon structure The dotndashdashed line shows the phonon

density of states for an isolated graphene sheet[3] r(o)

is large and roughly constant at low energy and does not

extrapolate to zero at zero energy This is because a

graphene sheet has an out-of-plane acoustic mode

(corresponding to a sheet-rolling motion) that is quadratic

in energy For a 2-D system this corresponds to a constant

r(o) The phonon density of states for 3-D graphite

shown as the dashed line in the inset to Fig 2 is sig-

nificantly smaller than that of 2-D graphene and ap-

proaches zero at k =0 This is because interlayer coupling

introduces dispersion in the z direction and moves low-

energy states upward in energy The characteristic energy

scale for this process is the Debye energy E D

of the

interlayer modes which is roughly 10 meV

Fig 1 Phonon density of states of an isolated SWNT (solid

line) compared to a 2-D graphene sheet (From Ref [2])

Fig 2 Low-energy phonon bandstructure of an isolated (1010)

SWNT The inset shows the low-energy phonon density of

states compared to that of 2-D graphene and 3-D graphite

Fig 3 Calculated low-energy phonon bandstructure of a

SWNT bundle for the case of graphite-like (lsquolsquostrongrsquorsquo)

coupling (From Ref [5])

604 Carbon Nanotubes Thermal Properties



ORDER REPRINTS

By analogy to the case of graphite intertube interaction

in bundles should depress the low-energy density of states

in SWNT bundles The characteristic energy for this

process is the Debye energy of intertube modes Because

of the heterogeneous nature of nanotube bundles it is

difficult to calculate these modes theoretically Mizel

et al[5] considered the case in which the coupling is

graphite-like By adjusting standard coupling constants

for graphite for the case of tubular structures they derived

a bandstructure that has E D$5 meV However because

of issues of incommensurability between neighboring

nanotubes the true intertube coupling is likely to be

weaker than this Fig 3 shows the 3-D dispersion of the

nanotube acoustic modes calculated by Mizel et al

Theoretically Derived Specific Heat

For a given r(o) it is straightforward to calculate C (T )

using Eq 1 The lines in Fig 4 show the specific heatcalculated from the known phonon bandstructure of

graphene and graphite from the predicted bandstructure

of an isolated SWNT and from the predicted bandstruc-

ture of a SWNT bundle in the case of graphite-like

(lsquolsquostrongrsquorsquo) coupling At temperatures above $100 K the

specific heat of all four materials is quite similar

However at lower temperatures the specific heats diverge

substantially The isolated graphene sheet displays the

largest specific heat which is roughly linear in T because

of the 2-D quadratic sheet-rolling mode The isolated

nanotube specific heat is smaller as a result of the absence

of the rolling mode At the lowest temperatures the

isolated nanotube C (T ) is linear in T The contribution of

each acoustic mode to C (T ) is can be analytically

expressed[6]

C ph frac14pk 2BT

hnrm

eth3THORN

where rm is the linear mass density Above $8 K the

slope of C (T ) increases as the optical subbands begin tocontribute The linear behavior at low T is a direct

signature of the 1-D quantized nature of the nanotube

phonon bandstructure Interlayer coupling (in graphite)

and intertube coupling (in strongly coupled bundles)

depresses the C (T ) at low T In real samples the tem-

perature at which the measured C (T ) diverges from

the single-tube curve provides measure of the actual in-

tertube coupling

Electronic Specific Heat

A metallic SWNT is a one-dimensional metal with a

constant density of states near the Fermi level At lowtemperatures it will have an electronic heat capacity that

is linear in temperature with a magnitude given by[6]

C el frac144pk 2BT

3hnFrm

eth4THORN

where nF is the Fermi velocity Because there are four

acoustic phonon modes the ratio between the phonon and

electron specific heat is

C ph

C el

nF

nph

100 eth5THORN

so that as expected the phonon contribution will domi-nate the electron contribution

Measured Specific Heat of SWNTs

The solid circles in Fig 4 represent the measured specific

heat of a bulk sample of highly purified single-walled

nanotubes over the range 2ndash300 K[7] The measured C (T )

data agree with all of the theoretical curves at high T as is

to be expected this is a good confirmation that experi-

mental errors such as sample contamination were not

significant At low T the measured C (T ) follows the

predicted curve for isolated SWNTs over almost the entire

temperature range diverging below the single-tube curveonly below $5 K This surprising result indicates that the

tubendashtube coupling in real samples is substantially weaker

than would be expected from a straightforward analogy

to graphite

Fig 5 highlights the specific heat data at low tem-

perature The measured C (T ) shows a linear temperature

dependence from 2 to 8 K with an upturn above 8 K This

behavior is a direct signature of the 1-D quantized nature

Fig 4 Calculated specific heat (lines) of graphene isolated

SWNTs graphite and strongly coupled SWNT bundles The

solid points represent the measured specific heat of a bulk

sample of SWNTs (From Ref [7])

Carbon Nanotubes Thermal Properties 605

C



ORDER REPRINTS

of the SWNT phonon bandstructure as discussed above

The lines in Fig 5 represent the results of fitting the

measured data to a simplified theory that takes into

account intertube coupling The low-energy phonon

structure is simplified to include two phonon modes a

fourfold-degenerate acoustic mode with (high) on-tube

Debye energy E D and (low) transverse Debye energy E D

and a doubly degenerate optical subband with a band

minimum at E sub The lines shown are obtained after

varying all three parameters to obtain the best fit to the

data The dotndashdashed line represents the contribution fromthe acoustic modes with E D=92 meV and E D

=12 meV

Its contribution is roughly cubic at low temperatures then

roughly linear above $ 2 K corresponding to the sat-

uration of the intertube modes The dashed line represents

the contribution from the optical mode with E sub=41

meV This mode begins to contribute at $ 7 K and the

sum of the two contributions (solid line) fits the data well

over the entire range from 2ndash12 K above which it can be

expected that other optical subbands will begin

to contribute

The values for the fitting parameters in this model are

directly related to the mechanical properties of nanotube

bundles The measured value of E D (92 meV) agrees wellwith the value of 103 meV that can be derived from theory

confirming the high phonon velocity in the tubes which is

a direct result of their high Youngrsquos modulus and low

density On the other hand the value of E D

is much

weaker than would be expected from a simple analogy to

graphite This indicates that coupling between the tubes in

bundles is very weak an issue that will need to be care-

fully considered for applications such as high-strength

composites The measured value of E sub (41 meV) is

slightly larger than the theoretically derived value of 27

meV which may be a result of mode stiffening due to

radial tubendashtube interaction In fact the measured value

is in good agreement with theoretical calculations for

nanotube bundles[8] An unresolved issue is the seeming

contradiction between the weak tubendashtube coupling im-

plied by the low transverse Debye energy and the stif-

fening of the first optical mode It may be that bundles

are relatively well coupled radially but are weak in

shear Detailed theoretical investigation of this matter is

still needed

THERMAL CONDUCTIVITY

Because of the high thermal conductivity of diamond

and graphite it is interesting to examine whether nano-

tubes exhibit high thermal conductivity a property that

might complement their extraordinary electrical and me-

chanical properties This property has been addressed

theoretically for single tubes experimentally for bulk

samples of SWNTs and experimentally for individual

multiwalled nanotubes

In general the thermal conductivity K is a tensor

quality but for this discussion it is only important to

consider the diagonal elements

K zz frac14X

C n2 zt eth6THORN

where C is the specific heat and n z and t are the group

velocity (n=do dk ) and relaxation time of a given phonon

state At low temperatures (T (YD) the relaxation time isdetermined by scattering off of fixed impurities defects

sample boundaries etc and is roughly constant Therefore

in ordinary materials the low-temperature thermal con-

ductivity has the same temperature dependence as the

specific heat However in anisotropic materials this

relationship does not strictly hold Because the contribu-

tion of each state is weighted by the scattering time and

the square of the velocity the thermal conductivity

preferentially samples states with high v and t For

instance in graphite the thermal conductivity parallel to

the basal planes is only weakly dependent on the

interlayer phonons In SWNT bundles it is likely that

K (T ) depends only on the on-tube phonons rather than the

intertube modes

Thermal conductivity is of particular interest in low-

dimensional systems For a 1-D ballistic electronic

channel the electronic conductance is quantized with a

universal value of

G0 frac142e2

heth7THORN

Fig 5 Measured specific heat of SWNTs at low temperature

fit with a simplified model The fitting parameters are given in

the text (From Ref [7])




ORDER REPRINTS

Similarly for a single ballistic 1-D channel the thermal

conductance is independent of materials parameters and

there exists a quantum of thermal conductance which is

linear in temperature

Gth frac14p

2k 2BT

3heth8THORN

Conditions for observation of this quantum were first

examined in detail by Rego and Kirczenow[9] Using

lithographically defined nanostructures Schwab et al[10]

confirmed this value experimentally

At high temperatures three-phonon Umklapp scatter-

ing begins to limit the phonon relaxation time Therefore

the phonon thermal conductivity displays a peak and

decreases with increasing temperature Umklapp scatter-

ing requires production of a phonon beyond the Brillouin

zone boundary because of the high Debye temperature of

diamond and graphite the peak in the thermal conductiv-

ity of these materials is near 100 K significantly higher

than for most other materials In less crystalline forms of graphite such as carbon fibers the peak in K (T ) occurs at

higher temperatures because defect scattering remains

dominant over Umklapp scattering to higher tempera-

ture[11] In low-dimensional systems it is difficult to

conserve both energy and momentum for Umklapp

processes[12] and so it may be possible that Umklapp

scattering is suppressed in nanotubes relative to 2-D or 3-

D forms of carbon

A measurement of K (T ) yields the combined contribu-

tion of the electrons and phonons However a simulta-

neous measurement of the electrical conductivity s

provides a measure of the electron thermal conductivity

K e from the WiedemannndashFranz law[1]

K e

sT L 0 frac14 245Acirc 10Agrave8 ethV=K THORN2

eth9THORN

In this way the phonon contribution can be deduced by

subtracting the electronic contribution from the total

measured thermal conductivity

Thermal Conductivity Theory

Berber et al[13]

have calculated the phonon thermalconductivity of isolated nanotubes Fig 6 shows the

results of theoretical calculations of the phonon thermal

conductivity of an isolated SWNT K (T ) peaks near 100 K

and then decreases with increasing temperature The value

of K at the peak (37000 Wm K) is comparable to the

highest thermal conductivity ever measured (41000 Wm

K for an isotopically pure diamond sample at 104 K)

Even at room temperature the thermal conductivity is

quite high (6600 Wm K) exceeding the reported room-

temperature thermal conductivity of isotopically pure

diamond by almost a factor of 2

Fig 7 shows the calculated nanotube thermal conduc-

tivity compared to the calculated thermal conductivity of asingle plane of graphene and that of 3-D graphite In

graphite the interlayer interactions quench the thermal

conductivity by nearly 1 order of magnitude It is likely

that the same process occurs in nanotube bundles Thus it

is significant that the coupling between tubes in bundles is

weaker than expected It may be that this weak coupling

which is problematic for mechanical applications of

nanotubes is an advantage for thermal applications

Measured K (T ) of SWNTs

Fig 8 shows the measured K (T ) of a bulk sample of SWNTs[14ndash16] K (T ) increases with increasing temperature

to 300 K the decreasing slope at high temperature may

indicate the onset of Umklapp scattering It is difficult to

ascertain the intrinsic thermal conductivity of an individ-

ual tube from these measurements although they point

strongly to a very high value In disordered lsquolsquomatrsquorsquo

samples the room-temperature thermal conductivity is

$35 Wm K[15] However the nanotubes in such a sampleFig 6 Calculated thermal conductivity of an isolated SWNT

as a function of temperature (From Ref [13])

Fig 7 Calculated nanotube thermal conductivity (solid line)

compared to the thermal conductivity of a 2-D graphene sheet

(dotndashdashed line) and 3-D graphite (dotted line) (From

Ref [13])


C



ORDER REPRINTS

are highly tangled and the thermal path is considerably

longer than the direct distance between points This effect

can be reduced by aligning the nanotubes in samples

where the nanotubes have been aligned by filtration in a

magnetic field the thermal conductivity is significantly

higher above 200 Wm K[14] which is comparable to that

of a good metal Even in these samples the thermal

conductivity is likely to be limited by tubendashtube junctions

so that the intrinsic single-tube thermal conductivity is

certainly higher Significantly the temperature depen-

dence of the thermal conductivity is roughly the same for

both types of samples suggesting that the measured K (T )reflects the intrinsic temperature dependence of the single-

tube K (T )

Using Eq 7 it is possible to calculate the electronic

contribution to the thermal conductivity In all samples

simultaneous measurement of the electrical and thermal

conductivity shows that the electronic contribution to the

thermal conductivity is only $1 of the total so that

phonons dominate K (T ) at all temperatures

At low temperature SWNT samples exhibit a linear

K (T ) strongly suggesting quantum effects Because of the

large number of nanotubes in a bulk sample it is not

possible to directly observe the thermal conductivity

quantum measured by Schwab et al[10] However it ispossible to measure the K (T ) of SWNT samples with

varying diameters the phonon subband splitting is higher

in smaller-diameter tubes so that the linear K (T ) behavior

should extend to higher temperature Fig 9 shows the

thermal conductivity divided by temperature K T of two

nanotube samples one with average diameter 12 nm and

the other with average diameter 14 nm[16] In both

samples K T approaches a constant value at low T just as

is expected for 1-D channels At higher temperatures K T

increases as more phonon modes contribute In the 12-nm

diameter sample the upturn in K T occurs $5 K higher

than in the 14-nm diameter sample This shift provides

additional evidence that the low-T linear behavior is true

1-D thermal conductivity However one unresolved issue

is the different temperature ranges of the 1-D regime in

heat capacity vs thermal conductivity For constant

scattering time the temperature ranges should be approx-

imately identical One possible explanation is that the

phonons in the optical bands are much more strongly

scattered and so do not begin to contribute to the thermal

conductivity until higher temperatures

Measured K (T ) of MWNTs

Because of the large diameter of MWNTs the temperature

scale for quantum effects should be quite small and their

thermal conductivity should be that of a 2-D system withlinear acoustic phonons The K (T ) of such a 2-D sheet

should follow a T 2 temperature dependence Graphite

shows a temperature dependence closer to T 23 because of

the effect of the quadratically dispersing out-of-plane

mode[17] As was discussed above interlayer effects can be

ignored when considering the thermal conductivity

Yi et al[18] have measured K (T ) for bulk samples of

MWNTs They found a roughly T 2 temperature depen-

dence up to 100 K as expected The room-temperature

thermal conductivity of these samples is only $25 Wm

K possibly as a result of the effects of tubendashtube contacts

or also of the incomplete graphitization in their samples

Fig 8 Temperature-dependent thermal conductivity of a bulk

sample of SWNTs which have been aligned by filtration in a

high magnetic field (From Ref [14])

Fig 9 Thermal conductivity divided by temperature for

SWNT samples with different average diameters The smaller-

diameter tubes exhibit linear K (T ) up to higher temperature

consistent with quantization effects (From Ref [16])




ORDER REPRINTS

Recently Kim et al[19] have used a microfabricated

structure (inset to Fig 10) to directly measure the thermal

conductivity of individual MWNTs The data in Fig 10

show the measured thermal conductivity of one MWNTK (T ) increases as T

2 up to $100 K peaks near 300 K and

decreases above this temperature Again the quadratic

temperature dependence is exactly what would be

expected for large-diameter nanotubes that essentially

act as 2-D sheets The room-temperature value of K (T ) is

over 3000 Wm K

Applications

The high thermal conductivity of nanotubes may be useful

for a number of thermal management applications such as

heat sinking of silicon processors or to increase the

thermal conductivity of plastics in such areas as housingfor electric motors Although many groups have studied

nanotubendashepoxy composite materials for their mechanical

properties their possible thermal properties have only

recently attracted attention

Biercuk et al[20] have measured the thermal conduc-

tivity of epoxy resin loaded with SWNTs Fig 11 shows

the enhancement in the thermal conductivity for loadings

up to 1 SWNTs and the enhancement for identical

loadings of graphitic carbon fibers Addition of 1

SWNTs doubles the thermal conductivity of the epoxy

while the same loading of carbon fibers provides only a

$40 increase This initial result is quite promising forthe development of composites for thermal management

Fig 10 Measured thermal conductivity of a single MWNT (From Ref [19])

Fig 11 Enhancement of the thermal conductivity of epoxy

resin as a function of loading by SWNTs and by carbon fibers

(From Ref [20])


C



ORDER REPRINTS

CONCLUSION

The thermal properties of carbon nanotubes are dominated

by phonons The measured specific heat of SWNTs

closely matches calculations based on the phonon band-

structure of isolated nanotubes and shows direct evidence

of 1-D quantization of the phonon bandstructure This

shows that coupling between nanotubes in a bundle is

relatively weak detailed modeling permits direct mea-

surement of the tubendashtube coupling strength as well as the

low-energy phonon structure Theoretical work predicts a

room-temperature thermal conductivity of 6600 Wm K

for individual nanotubes Measurements show a room-

temperature thermal conductivity over 200 Wm K for

bulk samples of single-walled nanotubes and over 3000

Wm K for individual multiwalled nanotubes Addition of

nanotubes to epoxy resin can double the thermal conduc-

tivity for a loading of only 1 showing that nanotube

composite materials may be useful for thermal manage-

ment applications

REFERENCES

1 Ashcroft NW Mermin ND Solid State Physics

Harcourt Brace New York 1976

2 Saito R Takeya T Kimura T Dresselhaus G

Dresselhaus MS Raman intensity of single-wall

carbon nanotubes Phys Rev B 1998 57 4145ndash

4153

3 Saito R Dresselhaus G Dresselhaus MS

Physical Properties of Carbon Nanotubes ImperialCollege Press London 1998

4 Sanchez-Portal D Artacho E Solar JM Rubio

A Ordejon P Ab initio structural elastic and

vibrational properties of carbon nanotubes Phys

Rev B 1999 59 12678ndash12688

5 Mizel A Benedict LX Cohen ML Louie SG

Zettl A Budraa NK Beyermann WP Analysis

of the low-temperature specific heat of multiwalled

carbon nanotubes and carbon nanotube ropes Phys

Rev B 1999 60 3264ndash3270

6 Benedict LX Louie SG Cohen ML Heat

capacity of carbon nanotubes Solid State Commun

1996 100 177ndash1807 Hone J Batlogg B Benes Z Johnson AT

Fischer JE Quantized phonon spectrum of single-

wall carbon nanotubes Science 2000 289 1730ndash

1733

8 Kahn D Lu JP Vibrational modes of carbon

nanotubes and nanoropes Phys Rev B 1999 60

6535ndash6540

9 Rego LGC Kirczenow G Quantized thermal

conductance of dielectric quantum wires Phys Rev

Lett 1998 81 232ndash235

10 Schwab K Henriksen EA Worlock JM

Roukes ML Measurement of the quantum of

thermal conductance Nature 2000 404 974ndash977

11 Heremans J Beetz CP Thermal-conductivity and

thermopower of vapor-grown graphite fibers Phys

Rev B 1985 32 1981ndash1986

12 Peierls RE Quantum Theory of Solids Oxford

University Press London 1955

13 Berber S Kwon YK Tomanek D Unusually

high thermal conductivity of carbon nanotubes

Phys Rev Lett 2000 84 4613ndash461614 Hone J Llaguno MC Nemes NM Johnson

AT Fischer JE Walters DA Casavant MJ

Schmidt J Smalley RE Electrical and thermal

transport properties of magnetically aligned single

wall carbon nanotube films Appl Phys Lett 2000

77 666ndash668

15 Hone J Whitney M Piskoti C Zettl A Thermal

conductivity of single-walled carbon nanotubes

Phys Rev B 1999 59 R2514ndashR2516

16 Hone J Llaguno MC Biercuk MJ Johnson

AT Batlogg B Benes Z Fischer JE Thermal

properties of carbon nanotubes and nanotube-based

materials Appl Phys A Mater 2002 74 339ndash343

17 Kelly BT Physics of Graphite Applied Science

London 1981

18 Yi W Lu L Zhang DL Pan ZW Xie SS

Linear specific heat of carbon nanotubes Phys

Rev B 1999 59 R9015ndashR9018

19 Kim P Shi L Majumdar A McEuen PL

Thermal transport measurements of individual

multiwalled nanotubes Phys Rev Lett 2001

8721 art no 215502

20 Biercuk MJ Llaguno MC Radosavljevic M

Hyun JK Johnson AT Fischer JE Carbon

nanotube composites for thermal managementAppl Phys Lett 2002 80 2767ndash2769




Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081EENN120009128

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright

Law requires that you get permission from the articlersquos rightsholder before

using copyrighted content

All information and materials found in this article including but not limited

to text trademarks patents logos graphics and images (the Materials) are

the copyrighted works and other forms of intellectual property of Marcel

Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order

reprints quickly and painlessly Simply click on the Request Permission

Order Reprints link below and follow the instructions Visit the

US Copyright Office for information on Fair Use limitations of US

copyright law Please refer to The Association of American Publishersrsquo

(AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted

reposted resold or distributed by electronic means or otherwise without

permission from Marcel Dekker Inc Marcel Dekker Inc grants you the

limited right to display the Materials only on your personal computer orpersonal wireless device and to copy and download single copies of such

Materials provided that any copyright trademark or other notice appearing

on such Materials is also retained by displayed copied or downloaded as

part of the Materials and is not removed or obscured and provided you do

not edit modify alter or enhance the Materials Please refer to our Website

User Agreement for more details



ORDER REPRINTS

become 66 separate 1-D subbands A direct result of this

folding is that the nanotube density of states has a number

of sharp peaks as a result of 1-D van Hove singularities

which are absent in graphene and graphite In spite of the

presence of these singularities the overall density of

states is similar at high energies so that the high-

temperature specific heat should be roughly equal as well

This is to be expected the high-energy phonons are more

reflective of carbonndashcarbon bonding than the geometry of

the graphene sheet

At low energies the geometry of the nanotube causes

the phonon structure to substantially differ from that of the

parent graphene sheet Fig 2 shows the theoreticallyderived low-energy phonon bandstructure of an isolated

(1010) nanotube There are four acoustic modes (those

with o0 as k 0) All four have a linear dispersion

o=nk near the zone center The longitudinal (LA) mode

has n=24 kmsec the (doubly degenerate) transverse (TA)

mode has n=9 kmsec and the lsquolsquotwistrsquorsquo mode has n=15

kmsec The first (doubly degenerate) optical mode enters

at 27 meV

The inset to Fig 2 shows the nanotube phonon density

of states (solid line) derived from the bandstructure

shown Only the four acoustic modes are present below

27 meV producing a constant density of states At the

band edge of each optical mode the density of statesdisplays a van Hove singularity characteristic of 1-D

dispersion and increases stepwise By comparison 2-D

graphene and 3-D graphite have very different low-energy

phonon structure The dotndashdashed line shows the phonon

density of states for an isolated graphene sheet[3] r(o)

is large and roughly constant at low energy and does not

extrapolate to zero at zero energy This is because a

graphene sheet has an out-of-plane acoustic mode

(corresponding to a sheet-rolling motion) that is quadratic

in energy For a 2-D system this corresponds to a constant

r(o) The phonon density of states for 3-D graphite

shown as the dashed line in the inset to Fig 2 is sig-

nificantly smaller than that of 2-D graphene and ap-

proaches zero at k =0 This is because interlayer coupling

introduces dispersion in the z direction and moves low-

energy states upward in energy The characteristic energy

scale for this process is the Debye energy E D

of the

interlayer modes which is roughly 10 meV

Fig 1 Phonon density of states of an isolated SWNT (solid

line) compared to a 2-D graphene sheet (From Ref [2])

Fig 2 Low-energy phonon bandstructure of an isolated (1010)

SWNT The inset shows the low-energy phonon density of

states compared to that of 2-D graphene and 3-D graphite

Fig 3 Calculated low-energy phonon bandstructure of a

SWNT bundle for the case of graphite-like (lsquolsquostrongrsquorsquo)

coupling (From Ref [5])




ORDER REPRINTS































expressed[6]

C ph frac14pk 2BT

hnrm

eth3THORN









tertube coupling





C el frac144pk 2BT

3hnFrm

eth4THORN




C ph

C el

nF

nph

100 eth5THORN














to graphite










C



ORDER REPRINTS












=12 meV









to contribute







is much
















still needed













K zz frac14X

C n2 zt eth6THORN
















intertube modes




universal value of

G0 frac142e2

heth7THORN







ORDER REPRINTS





Gth frac14p

2k 2BT

3heth8THORN




















D forms of carbon






K e


eth9THORN





Berber et al[13]


































Ref [13])


C



ORDER REPRINTS













tube K (T )

























































ORDER REPRINTS










over 3000 Wm K

Applications



















(From Ref [20])


C



ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502



































ORDER REPRINTS































expressed[6]

C ph frac14pk 2BT

hnrm

eth3THORN









tertube coupling





C el frac144pk 2BT

3hnFrm

eth4THORN




C ph

C el

nF

nph

100 eth5THORN














to graphite










C



ORDER REPRINTS












=12 meV









to contribute







is much
















still needed













K zz frac14X

C n2 zt eth6THORN
















intertube modes




universal value of

G0 frac142e2

heth7THORN







ORDER REPRINTS





Gth frac14p

2k 2BT

3heth8THORN




















D forms of carbon






K e


eth9THORN





Berber et al[13]


































Ref [13])


C



ORDER REPRINTS













tube K (T )

























































ORDER REPRINTS










over 3000 Wm K

Applications



















(From Ref [20])


C



ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502



































ORDER REPRINTS












=12 meV









to contribute







is much
















still needed













K zz frac14X

C n2 zt eth6THORN
















intertube modes




universal value of

G0 frac142e2

heth7THORN







ORDER REPRINTS





Gth frac14p

2k 2BT

3heth8THORN




















D forms of carbon






K e


eth9THORN





Berber et al[13]


































Ref [13])


C



ORDER REPRINTS













tube K (T )

























































ORDER REPRINTS










over 3000 Wm K

Applications



















(From Ref [20])


C



ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502



































ORDER REPRINTS





Gth frac14p

2k 2BT

3heth8THORN




















D forms of carbon






K e


eth9THORN





Berber et al[13]


































Ref [13])


C



ORDER REPRINTS













tube K (T )

























































ORDER REPRINTS










over 3000 Wm K

Applications



















(From Ref [20])


C



ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502



































ORDER REPRINTS













tube K (T )

























































ORDER REPRINTS










over 3000 Wm K

Applications



















(From Ref [20])


C



ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502



































ORDER REPRINTS










over 3000 Wm K

Applications



















(From Ref [20])


C



ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502



































ORDER REPRINTS

CONCLUSION


















ment applications

REFERENCES






4153






Rev B 1999 59 12678ndash12688





Rev B 1999 60 3264ndash3270






1733



6535ndash6540



Lett 1998 81 232ndash235






Rev B 1985 32 1981ndash1986










77 666ndash668









London 1981



Rev B 1999 59 R9015ndashR9018




8721 art no 215502





























































Date post:	06-Apr-2018
Category:	Documents
Upload:	ravi-kr-yadav
View:	231 times
Download:	0 times

Hone Thermal Ency Nano

Documents