「マイクロ・ナノスケールの熱物性とシステムデザイン」第３回研究会@Tokyo on Nov. 14, 2006

単層カーボンナノチューブの熱輸送に関する
単層カーボンナノチューブの熱輸送に関する
分子動力学
分子動力学

Single-Walled Carbon Nanotube, SWNT

Multi-Walled Carbon Nanotubes
MWNT

丸山 茂夫
http://www.photon.t.u-tokyo.ac.jp

Double-Walled Carbon Nanotubes
DWNT

Peapod東京大学 大学院工学系研究科
機械工学専攻Wrapping
Wrapping (10,10)(10,10) SWNTSWNT (armchair)(armchair)

CarbonCarbon Nanotubes
Nanotubes

Wrapping
Wrapping (10,5)(10,5) SWNTSWNT (chiral)(chiral)

(0,0)

(0,0)

Ch = (10,5)

a1a2

Ch = (10,10)yx

ChiralityChirality andand Radius
Radius ofof SWNT
SWNT

ya1a2xHexagonal
Hexagonal Lattice
Lattice (n,m)(n,m) nanotubes
nanotubes

a1a2(10,10)

(8,8)

(5,5)

(0,0) (1,0)(1,1)(10,0) Zigzag(2,0) (3,0)(4,0) (5,0)(2,1) (3,1) (4,1)(2,2)(3,2) (4,2)(3,3)(6,0) (7,0) (8,0)(5,1)(6,1) (7,1)(5,2) (6,2) (7,2)

(4,3)

(5,3)

(4,4)

a1

(6,3)

(5,4)

(9,1)

(8,2)

(7,3)

(6,4)

(5,5)

y(7,4)(6,5)(10,1)

(9,2)

(8,3)

(10,2)

(9,3)

(8,4)

(7,5)

(6,6)

(9,4)

(8,5)

(7,6)

(8,6)

(7,7)

a2

Zigzag(9,0) (10,0) (11,0)(8,1)

Armchairx(10,10)Armchair(10,5) Chiraln - m = 3q (q: integer): metallicn - m ≠ 3q (q: integer): semiconductor

1Electric
Electric DOS
DOS ofof Carbon
Carbon Nanotube
Nanotube

KΓ

M

kxb2

Γ

K

K’

b1

M

K’

π* (conduction)

4442222

0–2–2–4

wave vector

00–4010energy(eV)Γπ–5–10MΓ

s = 0 (symmetric)

04energy(eV)

E (eV)

5xa2K

π*

10a1M

s = 0.129

energy(eV)

15yK’energy(eV)

2-D2-D Energy
Energy dispersion
dispersion relation
relation forfor graphite
graphite–2–2–42wave vector

00–4012K

Γ

MKM

Γ

kyΜ

Μ

K’Κ

Brillouin Zone

ΚΚ

Μ

Μ

Κ

Κ

Κ

π (valence)

Reciprocal Lattice Vector

From: R. Saito et al., Physical Review B (2000).

Molecular Dynamics Simulation for Mechanism

Alcohol CCVD on Catalysts Supported with Zeolite
Alcohol
Ethanol
Methanolvacuum

Simple, High-Purity
Low-Temperature(550-900°C)
As-Grown

Shinohara& Nagy

2.5/2.5 : Fe/Co (wt%) on
USY Zeolite 30mg

7.4[Å]

500 Carbon & Ni108 : 2500K

1.2 nmElectric Furnace

G bandRBM

Most Used Carbon Sourcein NT06 Conference 2006
D bandBWF(b) 700°C(c) 800°C(d) 900°C(e) Laser–oven
500

1000

1500

S. Maruyama et al., Chem. Phys. Lett. 360 (2002) 229.

–1

Raman shift (cm )Photoluminescence
Photoluminescence Spectroscopy
Spectroscopy ofof SWNTs
SWNTsc1 c2

Y. Shibuta & S. Maruyama, Chem. Phys. Lett. 382 (2003) 381.

Comparison
Comparison ofof ACCVD
ACCVD andand HiPco
HiPcoe1

Energy (eV)Intensity (arb.units)(a) 600°C

E220(10,2)hν 11h–1(8,6)

E11hν 22(9,4)

(11,3)
(8,7)
(8,6)

(8,3)

(9,5)
(7,5)

v1

(10,5)(12,1)

(7,5)

(7,6)

(7,5)

(7,6)

(10,3)(11,1)v2(6,5)

hν 22 (E22)

D2O 1.1g /cm3SDS(8,4)

M. J. O’Connell et al., Science 297 (2002) 593
S. M. Bachilo et al., Science 298 (2002) 2361hν 11(E )111.0g / cm3Strong sonication
Super centrifuge 380,000g x 1h(a) ACCVD 850℃(b) HiPco

InGaAs Detector2

CVD
CVD Temperature
Temperature Dependence
Dependence

Room
Room Temperature
Temperature Blinking
Blinking inin PL
PL fromfrom Individual
Individual Nanotube
Nanotube
Laser off

Resasco’s
CoMoCAT7,6
8,7
6,5
S. M. Bachilo
7,5 et al.,8,620
9,5JACS,8,4
10,59,47,3125 (2003)
11186.
8,3
11,49,58,4
7,3

10,59,48,311,410,310

9,211,110,00.7

11,310,29,10

Chiral angle (deg.)

8,6

7,510,3

10

9,29,112,111,0

013,07,38,3

9,211,310,211,110,00.7

Most nanotubes: stable emission

11,4
10,3

10

010,59,49,113,00.8
1
Tube diameter (nm)

650℃

9,58,412,111,08,6

7,5

12,211,110,00.8
0.9
1
Tube diameter (nm)11,310,212,27,6

6,5

2.5μm

8,720

Single-molecule infrared photon
source12,212,113,011,00.8
0.9
1Tube diameter (nm)750℃

Some nanotubes: PL intensity fluctuation

850℃

PLPL peaks
peaks other
other thanthan Eii
Eii

12C

(6,5)

(6, 5)(8,4)

Ph(7, 5)

(7,5)

(7,6)

?

(8,3)

(9,5)

(8, 3)?
(9,4)

50
Time (sec)100

50Time (sec)

100

200

100

00Phononsideband

2.5

Ph(10,0)
(9,2)

(8,6)

0100

PL
PL peaks
peaks other
other thanthan Eii
Eii ~~ Phonon
Phonon Sideband
Sideband

(7,3)

(8,7)

0100

K. Matsuda, Y. Kanemitsu, K. Irie, T. Saiki, T. Someya,Y. Miyauchi and S. Maruyama, Appl. Phys. Lett., 86 (2005) 123116

Y. Miyauchi et al., Chem. Phys. Lett. 387 (2004) 198.

(11,1)(10,3)

100

0

Emission wavelength (nm)30

Chiral angle (deg.)

6,5

20

Intensity (counts/10msec)

(8,4)

30

8,7

7,6

(7,6)

(7,5)

Intensity (counts/10msec)(7,6)

Emission wavelength (nm)Emission wavelength (nm)30

Chiral angle (deg.)

(7,5)

(6,5)

(8,6)

(10,2)(10,2)(11,0)

III(10,5) (11,3)(12,1)

Excitation energy (eV)

(7,6)

(6,5)

Excitation wavelength (nm)(7,5)

Excitation wavelength (nm)

Excitation wavelength (nm)Carbon source : Ethanol, CVD time : 10minx5(7, 5) E 222?
(8, 3) E 22?
(10, 2) E 22

1.5

V. Perebeinos, J. Tersoff, P. Avouris,
PRL 94, 027402 (2005).

Phonon sideband
and Raman

Ph(14,0)(6, 5) E 22

Phononsideband

12C
13C

Emission intensity (arb. units)
ACCVD850 sample(NaDDBS-D2O dispersion)

Y. Miyauchi and S. Maruyama, Phys. Rev. B, 74 (2006) 35415

Vertically Aligned SWNTs on Quartz Substrate

Comparison
Comparison ofof pure
pure D//D// andand D⊥
D⊥ PL
PL Maps
Maps (1.15-1.3(1.15-1.3 eV)
eV)

IVV(b)(a)

Diameter (nm)
2I⊥

I //

(6,5)

(7,5)

(6,5)

Intensity (arb. units)Experiment100

0

1 0.9 0.8

200

500

300

0.7

400

1000
1500
–1
Raman Shift (cm )

(7,5)

IVH

(6,5)
(6,5)

(7,5)
(7,5)

(10,2)

G’G’

GG

10 μm
Y. Miyauchi, M. Oba, S. Maruyama, PRB in press (cond-mat/0608073) Sample: ACCVD SWNTs (TORAY)

Y. Murakami, S. Chiashi, Y. Miyauchi, M. Hu, M. Ogura, T. Okubo, S. Maruyama, Chem. Phys. Lett. 385 (2004) 298

3

TEM
TEM Image
Image fromfrom Top
Top ofof Carpet
CarpetDetachment of VA-SWNT films with hot water(a) As-grown on quartz(b) Re-attached on Sia

0.5 mmHidetsugu Shiozawa & Thomas Pichler (IFW-Dresden e.V.),
Erik Einarsson, Shigeo Maruyama

Y. Murakami and S. Maruyama, Chem. Phys. Lett. 422(2006) 575.

Experimental
Experimental challenges
challenges
Measurement
Measurement ofof isolatedisolated MWNTs
MWNTs

Berber,Berber,S.,
S.,Kwon,Kwon, Y.-K.,
Y.-K.,Tomanek,
Tomanek,D.,
D.,
Phys.
Phys. Rev.
Rev.Lett.,
Lett., 84-20
84-20(2000),(2000),4613-4616.
4613-4616.
400 Atoms per unit cell

Diameter dependence

6600 W/mK at 300K (10,10)

Temperature dependence

Fujii, etal, PRL 95 065502 (2005).

Kim, etal, PRL 87 215502(2001).

Experimental
Experimental measurements
measurements ofof SWNT
SWNT
thermalthermal conductivity
conductivity
Temperature dependence of suspended SWNTs

Total Energy Eb:

Tersoff-Brenner
Tersoff-BrennerPotential
Potential

Eb = ∑ ∑ {VR ( rij ) − B *ijVA (rij )}ij ( <i )VR ( r ) = f ( r )B *ij =Bij + B ji
2

From D. W. Brenner: Phys. Rev. B, 42, 9458(1990){}

De
exp − β 2 S ( r − Re )S −1

VA ( r ) = f (r )⎡⎤, Bij = ⎢1 + ∑ {Gc (θ ijk ) f (rik )}⎥⎣ k ( ≠i , j )⎦De =6.325 eV
0.80469&
1 .7 A

Potential Iδ =Potential II

De = 6.0 eV
δ = 0 .5&
R1 = 1.7 A

R1 =S=a0 =

1.290.011304&
R 2 = 2 .0 AS=
1.22
a 0 = 0.00020813&
R2 = 2 . 0 A

Single bonds

C. Yu et al, Nano Lett., 5 (2005) 18422⎞⎛ c2c0⎟
Gc (θ ) = a0 ⎜⎜1 + 0 2 − 2
d 0 + (1 + cosθ ) 2 ⎟⎠⎝ d0−1&&β = 1 .5 A
Re = 1.315 A
c0 = 19
d 0 = 2.5

β = 2.1A& −1
c0 = 330

Double bonds&
Re = 1.39 A
d 0 = 3 .5

Potential I260

1.56450

Triple bonds
Fc(N/m)
1.33610Potential II

500

1.55

870

1.381190

1.29

Experimental

450

1.54950

1.33

1600

1.20

Fc(N/m)E. Pop et al, Nano Lett., 6 (2006) 96⎧⎫
2De Sexp⎨− β( r − Re ) ⎬S −1S⎩⎭−δRe (Å) Fc (N/m)

Re (Å)

Re (Å)
1.20

4

Temperature
Temperature Distribution
Distribution along
along aa Nanotube
NanotubeMD
MD Simulation
Simulation Technique
Technique

310

Without the Periodic Boundary Condition!!

Temperature (K)290K

phantom (310K)

310K

y0.7nm6 ∼ 404 nm

Temperature Control by Phantom TechniquedTq = −λdzπσ = 2αk BTC / Δt S , α = m ω D , ωD = k B θ D / h6
θ D = 2500K (Diamond)

EffectEffect ofof Nanotube
Nanotube Length
LengthThermal conductivity λ (W/mK)

500

64.5 K/μm
phantom (290K)
290

–1000

0

Thermal Conductivity
Better than Diamond?
1000

Position (Å)

Diverging
Diverging Thermal
Thermal Conductivity
Conductivity ofof 1D
1D lattice
lattice

700
(5,5)
(8,8)
(10,10)2

q = 25.0 GW/mFourier’s Law

m &x& = f Pot + f Rand (σ) − α x&

600

300

FPU (Fermi-Pasta-Ulam) model0.27λ∝LS. Lepri, R. Livi and A. Politi,
On the anomalous thermal conductivity of one-dimensional lattices,
Europhys. Lett, 43 (3), 271-276 (1998)
2/5
Long wave length Fourier modes λ ∝ L

400

Diatomic Toda Lattice
300

0.11

λ∝L300K

Due to long time tail in Green-Kubo Integrands

0.15

λ∝L
200
10100
Length of nanotube L (nm)

Hatano T, Heat conduction in the diatomic Toda lattice revisited,
Phys. Rev. E, 59, R1-R4

Ring Area of 3.4 Åλ ∝ L0.35

Zhang, Fei, Isbister, Dennis J., Evans, Denis J.,
Nonequilibrium molecular dynamics simulations ofheat flow in one-dimensional lattices.
Evance NEMD makes solitons
Phys. Rev. E, (2000), 61(4-A), 3541-3546.
With high heat flux

Heat
Heat inin One
One Dimension
Dimension inin Nature
NatureLengthLengthdependence
dependenceof
offinite-length
finite-length SWNT
SWNT
thermalthermalconductivity
conductivity(2)(2)

Linearized Boltxmann-Peierls
phonon transport eqn.

“Divergence disappears if
considering second order (orhigher) 3-phonon scatteringprocesses”
Mingo and Broido, Nano Letters5(7) 1221 (2005)

5Dispersion
Dispersion relations
relations

Dispersion
DispersionRelations
Relations(low(lowfrequencyfrequencybands)
bands)(10,10)(10,10)SWNT
SWNT(5,5) SWNT

1800
600

5045

15100025800

cm-130ν (THz)1200
cm-1

ν[THz]

400

35

Tn :nth Transverse

10200

5

TW n :nth Twisting

400

L0(A)

L1

10200

5T3

20600

150400100

Ln :nth Longitudinal

TW0(A)

15

1400

40

5600

1600

0200

Mahan & Jeon,
PRB 70, (2004)
075405

1exp( in φ )

(c)
(b)
(a)
00 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
kkk
Current MD

0.5k

T20T1(Flexure)
001

0.5

exp(ikz )0Quantum numbers (k,n)
k: wave vector along the tube.

R. Saito etal,
PRB 57,(1998) 4145

n: angular wavenumber(Acoustic phonons have n=0)

Influence
Influenceof
ofBoundary
BoundaryCondition
Condition(Thermal(Thermalboundary
boundaryresistance,resistance,TBR)
TBR)

Data
Data convergence
convergence

Temperature control
4000
0-3ns
3-6ns
6-9ns
9-12ns12-15ns
15-18ns

1000

LLc

Lc

TBR310

Thermal boundary resistanceΔt=0.5fs

Reflection

305
T (K)

λ (W/mK)

2000

500

300

Transmission
Maruyama (2002)
200
10

Excited phonons

295

100
L (nm)

1000

Z=-1000Å

4000

290

-100

-50

050

100z[nm]TuningTuning thethe thermostat
thermostat length
length LLcc

Optimizing
Optimizingboundary
boundaryconditions
conditions

Temperature gradientNose-Hoover thermostat
with relaxation time τ290K• Lc :Length of thermostats

1 v& 2
η = 2 ( − 1)
τ T

t =4E5 (ps)t =4E3 (ps)t =40 (ps)t =0.4 (ps)310K

τ= 40psLc(variable)

Lc(variable)

L

120

310

• τ Thermostat relaxation time100
Lc = 0.01Lb
Lc = 0.05Lb
L = 0.25Lbc
Lc = Lb

T (K)

305

80
Q [nW]

Optimization parameters

Force = − mvη&Heat flux

60
40

300

20295

Lc

L

Lc

Increasing Lc

000.5

1

L c /L290-10-50z (nm)510Parameter with (sufficiently)
minimized TBR effect

Lc = 0.5L

6Effect
Effect ofof relaxation
relaxation time
time onon TBR
TBR (2)(2)

EffectEffect ofof relaxation
relaxation time
time onon TBR
TBR (1)(1)

Spectral mismatch

1x 10-10

10.8

0.8

0.8

0.6

0.6

0.6

x 10

0.4

0.4

0.20.20.2010010

τ (ps)

0510010τ (ps)52Transverse

31010010τ (ps)0-118
12(THz)3

SWNT
Thermostat

0

Lc
x 10

arb. unit

arb. unit

1

900.4

Longitudinal

Lc

0-10

TBR (mK/W)

TBR (mK/W)

1

Rh

Rc

-10

x 10

Temperature control

Longitudinalarb. unit

Rh
Rc
Rtot

310K

12Transverse

3

12(THz)3arb. unit

Temperature control

290K

07654540 ps

3010
τ (ps)

105Length
Length dependence
dependence ofof SWNTSWNT thermal
thermalconductance
conductanceTuningTuning thethe thermostat
thermostat relaxation
relaxation timetime ττ550
L=25 (nm)L=50 (nm)
L=101 (nm)

L=25 (nm)L=50 (nm)
L=101 (nm)

1.4

400
350
300

110

1.2
1.1

250

Red: SWNT+ TBR (thermal boundaryresistances)

Nose Hoover (TBR minimized)010
τ (ps)

1050.9010
τ (ps)

105-110
Parameter with (sufficiently)
minimized TBR effect10001000

Altering balance ofPhantom
Optimized Nose Hoover
231010L (nm)λ∝LCharility independent (L<100 nm)

Significant contribution from phonons
with finite circumferential wave
number. (2D phonons)500
λ (W/mK)λ (W/mK)

Deviation from
200

310

Chirality
Chirality (diameter)(diameter) dependence
dependence•Phonon mean free path longer
than 3.2 μm

500

210

τ= 40ps•λ~900 [W/mK] at L=3.2 μm

λ∝L110L (nm)

LengthLength effecteffect ofof SWNTSWNT Thermal
Thermal conductivity
conductivity

100 110Phantom technique

0101200150Blue: SWNT

1.32λ/ λ(τ=40ps)

λ(W/mK)450

K (GW/m K)

500

Chirality dependent (L>100 nm)

200(5,5)(10,10)Diffusive & Ballistic phonons
Contribution from a wide rangeof phonon branches
Shiomi and Maruyama (2006)

100 1
102

10L (nm)

310

Contribution from limited keyphonons, possibly the acousticphonons

Shiomi and Maruyama (2006)

7

Non-Fourier
Non-Fourier heat
heat conduction
conduction inin SWNT
SWNTSummaries
Summaries∂ 2T ∂T+= α∇ 2 T∂t∂t 2∂ 2 T ∂T∂
2. τ q 2 + = α ⎛⎜ ∇ 2T + τ θ ∇ 2T ⎞⎟∂t∂t∂t⎝⎠

1. τT/Tp• Thermal conductivity was measurement by minimizing the influence oftube-ends boundary resistances for comparisons with theories andexperiments.

10.5•Thermal conductivity was measured up to 3.2μm and the maximum
value was about 800 W/mK, about 1/4 of the experimentally observedvalue.

MDPeriodic
boundarycondition• Length dependence of SWNT was identified.

0L/2◊ Short nanotubes: diffusive-ballistic transport of a range of phonon branches

Eq. (1)0-1 -0.5 0 0.5 1 1.5 2t (ps)Heatedregion munitcells

Eq. (2)L/2z
t* = 0.48

Spatio-temporal temperature profilest (ps)

3LongitudinalTransverse
3

3222

11

1000

Circumferential

0.8

t* = 0.32

0.6θr◊ Long nanotubes: Ballistic transport of the key phonons• Diameter independence of the thermal conductivity in (L<200 nm)
suggests active contribution from 2D phonons in the regime

1

10.2

0105z (nm)0105z (nm)00510z (nm)1t* = 0.16

0.4

t* = 000.5z*

1

Shiomi and Maruyama, PRB, 73 (2006) 205420

Professor
ProfessorRichard
Richard E.
E.Smalley
Smalleyat
atRiceRiceUniversity
University(1943-2005)(1943-2005)

Heat
Heat conduction
conduction ofofSWNTSWNT isotopeisotopesuperlattices
superlattices

λ/λ00.65

Δz

0.6
12C

13C

L = 50 nm
L = 101 nm
L0.55

Increasing Δz

ν[THz]

Pure 12C SWNT
666655554

444

33

3

3

2222

111

1

00

0.2
0.4
ka/π

00.6 00.2

0.4ka/π00.6 00.2

0.4ka/π00.6 00.2

00.4ka/π51015Δz (nm)

0.6

20

Highlighting
molecular dynamics
by installingisotopes withdouble the mass1990
1943: Born in Akron, Ohio on June 6
1976: PhD from Princeton,Assistant Professor at Rice
Laser Vaporization Cluster Beam
1984: Discovery of C60
1996: Nobel Prize in Chemistry with Kroto and Curl
1996: Bulk Production of SWNTs
Then HiPco Process
2000: Testified for National Nanotechnology Initiative
2000: Found Carbon Nanotechnology Inc.
Application of Nanotechnology to Energy Problem
2005: Died at 62 on October 28, 2005
2001

2005

2001

2005

Shiomi and Maruyama, PRB 74 (2006) 155401.

8