単層カーボンナノチューブSingle-Walled Carbon Nanotubes

１．幾何学と電子構造Geometry and Electronic Structure

単層カーボンナノチューブSingle-Walled Carbon Nanotubes

１．幾何学と電子構造Geometry and Electronic Structure

丸山 茂夫Shigeo Maruyama東京大学大学院工学系研究科

機械工学専攻

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

分子熱流体工学(Molecular Thermo-Fluid Engineering) 2013

Contents

１．幾何学と電子構造Geometry & Electronic Structure２．電子顕微鏡観察と分光Electron Microscopy and Spectroscopy３．合成と応用Growth and Applications４．ナノチューブの伝熱Heat Transfer５．生成メカニズムとカイラリティ制御Growth Mechanism and Chirality Control

from NNI Home Page: http://www.nano.gov

Nanometer Scale

1-D: Carbon Nanotube

Allotropes of Carbon

3-D: Diamond2-D (3-D) Graphite

0-D: Fullerene

2-D: Graphene

Graphite Diamond (from CHAUMET Paris HP)

(a) C60 (c) La@C82(b) C70

(e) C240

PVWin

(d) Sc2@C84

Fullerene Structures

C60のアイデア: 大澤(1975)

フラーレンの発見: Smalley, Kroto & Curl (1985)

フラーレンの量的生成: Krätschmer & Huffman(1990)

フラーレンの超伝導の発見: Hebard(1991)

ナノチューブの生成: 飯島(1991)

金属内包フラーレンの量的生成: Smalley (1991)

単層ナノチューブの量的生成: Smalley (1996)

電子ドープ超伝導: Batlog (2000)

ノーベル化学賞（１９９６）

フラーレンの発見

？？

BuckminsterFullerene

BuckminsterFullerene

Euler’s Theorem: 2 evff: faces, v: vertices, e: edges

Usual Explanation of Even Numbered Positive Spectra

65

65

65

653652

ffvffe

fff

6

5

22012

fvf

5f6f

Euler’s Theorem

C10

C8

25001000 1500 2000500Time (ps)

C60C49

C28C26

C12 C15

C8

C330

20

40

60

C70C53

C8 Clu

ster

Siz

e

500 carbon atoms 342 Å cubic boxTc = 3000 K

PVWin

GrowthProcess

of Fullerene

Y.Yamaguchi & S.Maruyama, Chem. Phys. Lett., 286, 336 (1998).

74

7

77

4 8

77

7

77

77 7

initial

215.81 ns 215.82 ns 216.35 ns 216.40 ns

216.45 ns 217.18 ns 218.39 ns 220.56 ns 221.70 ns (Ih C60)

A

215 ns

B

–6.72

–6.68

–6.64

–6.6

0

4

# of

dam

glin

g bo

nds

ND

B

Ep

NDB

pote

ntia

l ene

rgy

Ep

(eV)

time (ns)190 200 210 220 230

Ih C60

PVWin

Annealing Process to perfect C60S.Maruyama & Y.Yamaguchi, Chem. Phys. Lett., 286, 343 (1998).

Chain

Ring FlatC10 C20 C30

Tangledpoly-cyclic

Closed cage Stone-Walestransformations

C50

C70C60

Fullerene(stable)

Opencage

Graphitic sheetToo low temperature

Chaotic 3-dimensionalstructure

Randomcage

Higherfullerene

Fullerene Formation Model

S. Maruyama & Y. Yamaguchi, Chem. Phys. Lett. 286 (1998) 343.

Single-Walled Carbon Nanotube, SWNT

Multi-Walled Carbon NanotubesMWNT

Carbon NanotubesPeapod

Double-Walled Carbon NanotubesDWNT

Individual tube diameter: 1.3 nmSpacing: 0.34 nmMisalignments and Terminations

TEM from Smalley et al. at Rice University

About 100 SWNTs

TEM Pictures of SWNT Ropes

5 nm

By ACCVD

Peapods

Suenaga et al., PRL 2003

Peapod with Sc2@C84

http://vortex.tn.tudelft.nl/~dekker/nanotubes.html

STM Image of Individual Atoms

(0,0)Ch = (10,0)

Wrapping (10,0) SWNT (zigzag)

a1a2 x

y

(0,0)Ch = (10,0)

Wrapping (10,0) SWNT (zigzag)

a1a2 x

y

(0,0)

Ch = (10,10)

Wrapping (10,10) SWNT (armchair)

a1a2 x

y

(0,0)

Ch = (10,10)

Wrapping (10,10) SWNT (armchair)

a1a2 x

y

(0,0)

Ch = (10,5)

Wrapping (10,5) SWNT (chiral)

a1a2 x

y

(0,0)

Ch = (10,5)

Wrapping (10,5) SWNT (chiral)

a1a2 x

y

Chirality and Radius of SWNT

(10,10)Armchair

(10,0) Zigzag

(10,5) Chiral

a1

a2

(10,10) (8,8)(5,5)

Hexagonal Lattice (Definition of Vectors)

Chiral vector

21 aaC mnh a1a2

O

(4,-5)

Ch

T

x

y

(6,3)

)23,

23(

)23,

23(

2

1

cccc

cccc

aa

aa

a

a

aacc 321 aa

a

a

)21,

23(

)21,

23(

2

1

a

a

Hexagonal Lattice (n,m) nanotubes

a1a2

x

y

(0,0) (1,0) (2,0) (3,0)

(1,1) (2,1)

Zigzag

Armchair

(2,2)

(4,0) (5,0) (6,0)

(3,1) (4,1) (5,1)

(3,2) (4,2) (5,2)

(7,0) (8,0) (9,0)

(6,1) (7,1) (8,1)

(6,2) (7,2) (8,2)

(10,0) (11,0)

(9,1) (10,1)

(9,2) (10,2)

(3,3) (4,3) (5,3) (6,3) (7,3) (8,3) (9,3)

(4,4) (5,4) (6,4) (7,4) (8,4) (9,4)

(5,5) (6,5) (7,5) (8,5)

(6,6) (7,6) (8,6)

(7,7)

n - m = 3q (q: integer): metallicn - m 3q (q: integer): semiconductor

(n,m) Symmetry

Diameter of Tube 223 mnmnaCd ccht

Chiral vector 21 aaC mnh

Chiral angle )2/(3tan 1 nmm Lattice Vector Rdmnnm /)2()2( 21 aaT

Rh dCT /3

dofmultipleaismnifd

dofmultipleanotismnifddR 33

3

d: highest common divisor of (n,m)

RdnmnmN )(2

22 Number of hexagons per unit cell:

cct and

3Armchair

Electric DOS of Graphite

幾何学構造と同様に，SWNTの電子構造はグラフェン（グラファイト１層）の電子構造を基礎として理解できる．そこで，最初にグラフェンの電子構造について復習する．

炭素のπ電子の挙動が問題となる．電子の波動関数を波数(kx, ky)の平面波で展開し，６角形のブリリアンゾーンにおける分散関係を求める．グラフェンは，ゼロバンドギャップ半導体であり，K点とM点でのみ，π電子とπ＊電子の分散関係が接する．

ReferenceP. R. Wallace, Phys. Rev, 71 622 (1947).

Reciprocal Lattice Vector

逆格子ベクトル

2,2/2,/2

2211

2211

baba

abab

aa

aa

34)

23,

21(2)1,

31(

34)

23,

21(2)1,

31(

2

1

b

b

aaPer

aaPer

ccy

ccx

3

33

a

a

)21,

23(

)21,

23(

2

1

a

a aacc 321 aa

Brillouin Zone

aaa 32)0,1(

32

32

2

2

1

1 kbbk

bbk

y

a2

a1

x

kx

ky

M

K

b2

b1

475.1322

31

ccaa

703.133

4342

32

ccaaa

554.22 a

2121 bb

Reciprocal Lattice Vector

Brillouin Zone

逆格子ベクトル

Brillouin Zone

y

a2

a1

x

kx

ky

M

K

b2

b1

475.1322

31

ccaa

703.133

4342

32

ccaaa

554.22 a

2121 bb

Reciprocal Lattice Vector

波長kx, kyで表現した位相空間を逆格子空間という．電子の平面波の高波数の上限は（π／格子定数）で表せる．このような上限波数範囲を逆格子空間で表したものをブリリアンゾーンとよぶ．６角格子の場合には，ブリリアンゾーンも６角形となる．方向が９０度ずれていることに注意！

Plane Wave Representation and Tight-Binding Wave Function

EHSchrödinger EquationkriePlane Wave

rkk r

)()( GiG

GeC

G: reciprocal vector

Plane Wave Representation

Fourier Transform of wave function

),()( rkrk ii

iC Tight-binding wave function

R

kR Rrrk )(1),( iu

i eN Bloch orbital

Tight-Binding Method

EHInstead of Solving Schrödinger Equation

Find best which minimize

HE

With Tight-binding wave function

Functional Method

jiijji

jiijji

SCC

HCCHE

,

*,

*

jiij HH jiijS Hamiltonian Matrix Overlap Integral

Here,

Tight-Binding Method 2

j

jj

ijjij CSECH )(k

0)(*

iCE k

0

,

*

,

*

ij

jj

jiijji

ijji

ji

ijj

j SCSCC

HCCHC

0)( 2

,

*

,

*

,

**

jiijji

ijj

jijji

ji

jiijji

ijj

j

iSCC

SCHCC

SCC

HC

CE k

2-D Electronic Energy Dispersions of Graphite

)(1)(

)( 022 kk

ksw

wE pDg

2cos4

2cos

23cos41)()( 22

akakakfw yyx kk

1*)()(1

*)()(

20

02

ksfksf

S

kfkf

Hp

p

H: (2x2) Hamiltonian

S: (2x2) Overlap integral matrix2p: Site Energy of 2p atomic orbital

2cos2)( 32/3/

akeekf yakak xx

0)det( ESHSecular equation (永年方程式)

where CCaa 3

where

2-D Energy dispersion relation for graphite

From: R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Trigonal warping effect of carbon nanotubes, Physical Review B, vol. 61, no. 4, 2981 (2000).[Color picture was from Professor R. Saito]

)(1)(

)( 022 kk

ksw

wE pDg

2cos4

2cos

23

cos41)( 2akakak

w yyx k

Overlap integral: s=0.129C-C interaction energy: 0=2.9eV

2p = 0

Energy dispersion relation for and * bands

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

kx

ky

0.000

15.000

M

K

K’

M

MK

MK’

M

M

K’

K

)(1)(

)( 022 kk

ksw

wE pDg

2cos4

2cos

23cos41)( 2

akakakw yyx ks=0.129Gamma=2.9eV

CCaa 3

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

kx

ky

-10.000

0.000

Electric DOS of Nanotube

グラフェンを巻いたSWNTの場合には，円周方向に周期境界条件を満たす電子の波動関数しか許されなくなる．このため，グラフェンの場合の６角形のブリリアンゾーン（平面）は，有限数の線となってしまう．この線が，K点かM点を通過すると金属，そうでないと半導体となる．

Reference最初の理論予測：R. Saito et al., Phys. Rev. B46, 1804 (1992).

詳細かつわかりやすい論文：R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Trigonal warping effect of carbon nanotubes, Physical Review B, vol. 61, no. 4, 2981 (2000).

M

K

K’

MK’

M

K

K’

MK’

Electric DOS of Carbon Nanotube

M

K

K’

MK’

0–4

–2

0

2

4

wave vector

ener

gy(e

V)

0 1 2–4

–2

0

2

4

ener

gy(e

V)

0–4

–2

0

2

4

wave vector

ener

gy(e

V)

0 1 2–4

–2

0

2

4

ener

gy(e

V)

1D Dispersion

Lattice Vector Rdmnnm /)2()2( 21 aaT

RNdmnnm /)2()2( 211 bbK

Nnm /)( 212 bbK

Discrete unit vector along the circumferential direction

Reciprocal lattice vector along the nanotube axis

1

2

22)( KK

K kEkE Dg

Tk

T

N

,...,2,1

Rh dCT /322 mnmnaCh

h

R

C

mmnna

mmnnmmnna

Ndmmnna

2

12

)(2/22

/22

22

2222

221

K

RdnmnmN )(2

22

Td

C

mmnnd

a

mmnnmmnnda

Nmmnna

Rh

R

R

23

123

12

)(2/3

22

/3

22

22

2222

222

K

Summary

12

2 KKK

k

Tk

T

N

,...,2,1

1

2

22)( KK

K kEkE Dg

where

Slice

-2 -1 0 1 2-2

-1

0

1

2

kx

ky

0.000

3.000

-2 -1 0 1 2-2

-1

0

1

2

kx

ky

0.000

3.000

(10,0)K1=(0.221239,0.127732)K2=(-0.737463,1.277323)

-2 -1 0 1 2-2

-1

0

1

2

kx

ky

0.000

3.000

(10,10)K1=(0.147493,0.000000)K2=(0.000000,2.554647)

-2 -1 0 1 2-2

-1

0

1

2

kx

ky

0.000

3.000

(10,5)K1=(0.189633,0.036495)K2=(-0.105352,0.547424)

van Hove Singularity

ブリリアントゾーンを積分するといわゆる状態密度（Density of States, DOS)が求まることになる．金属か半導体かという点以外にも，周期境界条件によって，ブリリアンゾーンが線となるために，一次元物質に特有のvan Hove特異点と呼ばれる発散するDOSとなる．

ReferenceDresselhaus, M. S. & Dresselhaus, G., Science of Fullerenes and Carbon Nanotubes, Academic Press (1996).Saito, R., ほか2名, Physical Properties of Carbon Nanotubes, Imperial College Press (1998).

点線はグラフェンのDOS

Comparison of DOS for Armchairs

–2 0 20

2

4

(5,5)

(10,10)

(15,15)

(20,20)

Energy (eV)

Den

sity

of S

tate

s (s

tate

s/1C

–ato

m/e

V)

Comparison of DOS for Zig-zag

–2 0 20

2

4(10,0)

(20,0)

(30,0)

(40,0)

Energy (eV)

Den

sity

of S

tate

s (s

tate

s/1C

–ato

m/e

V)

2-D Energy dispersion relation for graphite

y

a2

a1

x

kx

ky

M

K

b2

b1

Reciprocal Lattice VectorFrom: R. Saito et al., Physical Review B (2000).

MK

K’

MK’

Brillouin Zone

* (conduction)

(valence)

–10

–5

0

5

10

15

E (e

V)

K M K

*

s = 0.129

s = 0 (symmetric)