PHYSICAL REVIEW B 77, 245441 共2008兲

Curvature-induced optical phonon frequency shift in metallic carbon nanotubes
Ken-ichi Sasaki,1 Riichiro Saito,1 Gene Dresselhaus,2 Mildred S. Dresselhaus,3,4 Hootan Farhat,5 and Jing Kong4
1Department of Physics, Tohoku University and CREST-JST, Sendai 980-8578, Japan
2Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
4Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,

Cambridge, Massachusetts 02139-4307, USA
5Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA

共Received 21 March 2008; revised manuscript received 25 April 2008; published 27 June 2008兲
The quantum corrections to the frequencies of the ⌫ point longitudinal-optical 共LO兲- and transverse-optical
共TO兲-phonon modes in carbon nanotubes are investigated theoretically. The frequency shift and broadening of
the TO-phonon mode strongly depend on the curvature effect due to a special electron-phonon coupling in
carbon nanotubes, which is shown by the Fermi energy dependence of the frequency shift for different
nanotube chiralities. It is also shown that the TO mode near the ⌫ point decouples from electrons due to local
gauge symmetry and that a phonon mixing between LO and TO modes is absent due to time-reversal symmetry. Some comparisons between theory and experiment are presented.
DOI: 10.1103/PhysRevB.77.245441

PACS number共s兲: 61.46.⫺w, 63.22.⫺m

I. INTRODUCTION

of density-functional theory. Recently Nguyen et al.17 andFarhat et al.18 observed the phonon-softening effect of
SWNTs as a function of EF by field effect doping and electrochemical doping, respectively, and their results clearlyshow that the LO-phonon modes become soft as a functionof EF. Ando19 discussed the phonon softening of metallic
SWNTs as a function of the EF position, in which the phonon
softening occurs for the LO-phonon mode and for a special
range of EF, that is, for 兩EF兩 ⬍ ប␻LO / 2.
In this paper, we show that the ⌫ point TO-phonon mode
becomes hard when 兩EF兩 ⱗ ប␻TO / 2 and has a considerable
broadening for metallic zigzag nanotubes. The occurrence ofthe phonon hardening for the TO mode is due to the curvature effect, a special character of the el-ph coupling, and abasic consequence of second-order perturbation theory. We
show using a gauge symmetry argument that the electrons
completely decouple from the TO mode near the ⌫ point.
Besides, we show that for a chiral nanotube, both the LO andTO modes are softened due to the fact that the direction of
the TO-phonon vibration is not parallel to the nanotube circumferential direction.20 Another interest of ours is the mixing of LO and TO phonons to form degenerate phonon frequencies. When the LO-phonon mode becomes soft, acrossing of the LO mode with the TO mode occurs at a
certain value of EF. For such a mode crossing, we should
generally consider the el-ph coupling for degenerate phononmodes to sense the crossing or anticrossing of the two phonon frequencies as a function of EF. We show by an analytical calculation that there is no mixing between the LO- andTO-phonon modes for any case due to time-reversal symmetry.
The organization of the paper is as follows. In Sec. II weshow our method of calculation and present the results forarmchair and metallic zigzag SWNTs. In Sec. III using
effective-mass theory, we show how the el-ph interaction depends on the chiral angles of SWNTs, and in Sec. IV a discussion based on gauge symmetry and time-reversal symmetry for the el-ph coupling is given. In Sec. V, a comparison
with the experiments and a summary are given.

In the Raman spectra of a single-wall carbon nanotube
共SWNT兲, the two in-plane optical-phonon modes, that is, thelongitudinal-optical 共LO兲- and transverse-optical 共TO兲phonon modes at the ⌫ point in the two-dimensional Brillouin zone 共2D BZ兲, which are degenerate in graphite andgraphene, split into two peaks, G+ and G− peaks,
respectively.1–3 The splitting of the two peaks for SWNTs isinversely proportional to the square of the diameter dt of
SWNTs due to the curvature effect, in which G+ does notchange with changing dt, but the G− frequency decreases
with decreasing dt.4,5 In particular, for metallic SWNTs, theG− peaks appear at a lower frequency than the G− peaks forsemiconducting SWNTs with a similar diameter.6 The spectra of G− for metallic SWNTs show a much larger spectral
width than that of semiconducting SWNTs. Further, the spectral G− feature shows an asymmetric line shape as a functionof frequency, which is known as the Breit-Wigner-Fano
共BWF兲 line shape.7 The origin of the BWF line shape isconsidered to be due to the interaction of discrete phononstates with continuous free-electron states.
It has been widely accepted that the frequency shift of the
G band is produced by the electron-phonon 共el-ph兲interaction.8–13 An optical phonon changes into an electronhole pair as an intermediate state by the el-ph interaction.
This process gives the phonon a self-energy. The phonon
self-energy is sensitive to the Fermi energy, EF. In the case ofgraphite intercalation compounds in which the charge transfer of an electron from a dopant to the graphite layer can becontrolled by the doping atom and its concentration, Eklund
et al.14 observed a shift of the G-band frequency with anincrease in the spectral width. The frequency shifted spectra
show that not only the LO mode but also the TO mode areshifted in the same fashion by a dopant. For a graphenemonolayer, Lazzeri et al.9 calculated the EF dependence ofthe shift of the G-band frequency. The LO mode softening in
metallic SWNTs was shown by Dubay et al.15,16 on the basis
1098-0121/2008/77共24兲/245441共8兲

245441-1©2008 The American Physical Society

PHYSICAL REVIEW B 77, 245441 共2008兲

SASAKI et al.

E(k)
EF

h̄ω

(0)

(a)

(b)
excludeelectron╳ E

0h̄ω (0)holekh(E) =1h̄ω (0) − E + iΓ

Γ = 5 (meV)0

2|EF |

0.2

0.4

E (eV)

0.6

TO

1600
1560

LO

1520

(10,10)-0.3 -0.2 -0.1

00.1

EF (eV)

FIG. 1. 共Color online兲 共a兲 An intermediate electron-hole pairstate that contributes to the energy shift of the optical phononmodes is depicted. A phonon mode is denoted by a zigzag line andan electron-hole pair is represented by a loop. The low-energy
electron-hole pair satisfying 0 ⱕ E ⱕ 2兩EF兩 is forbidden at zero temperature by the Pauli principle. 共b兲 The energy correction to thephonon energy by an intermediate electron-hole pair state, especially the sign of the correction, depends on the energy of the intermediate state as h共E兲. 共See text.兲(b)h̄ω(cm−1 )

1600

TO

1560

LO

1520

The frequency shift of the ⌫-point LO- and TO-phononmodes for metallic SWNTs is calculated by second-order
perturbation theory. The phonon energy, including the el-phinteraction, becomes ប␻␭ = ប␻␭共0兲 + ប␻␭共2兲 共␭ = LO, TO兲, where␻␭共0兲 is the original phonon frequency without the el-ph interaction and the correction term ប␻␭共2兲 is given by兩具eh共k兲兩Hint兩␻␭典兩2
共0兲
k ប␻␭ − 关Ee共k兲 − Eh共k兲兴 + i⌫␭ប␻␭共2兲 = 2 兺

共1兲

The factor 2 in Eq. 共1兲 comes from spin degeneracy. ប␻␭共2兲 isthe quantum correction to the phonon energy due to theelectron-hole pair creation, as shown in Fig. 1共a兲. In Eq. 共1兲,具eh共k兲兩Hint兩␻␭典 is the matrix element for creating anelectron-hole pair at momentum k by the el-ph interaction,
Hint, Ee共k兲 关Eh共k兲兴 is the electron 共hole兲 energy, and ⌫␭ is the
decay width. In Fig. 1共a兲, an intermediate electron-hole pairstate that has the energy of E = Ee共k兲 − Eh共k兲 is shown. We
need to sum 共兺k 兲 over all possible intermediate electronhole pair states in Eq. 共1兲, which can have a much larger
energy than the phonon 共E Ⰷ ប␻␭共0兲兲.
Since 具eh共k兲兩Hint兩␻␭典 is a smooth function of E = Ee共k兲
− Eh共k兲 in the denominator of Eq. 共1兲, the contribution toប␻␭共2兲 from an electron-hole pair depends on its energy. In
Fig. 1共b兲, we plot the real part and imaginary part of thedenominator of Eq. 共1兲, h共E兲 = 1 / 共ប␻共0兲 − E + i⌫兲 as a functionof E in the case of ប␻共0兲 = 0.2 eV and ⌫ = 5 meV. HereRe关h共E兲兴 has a positive 共negative兲 value when E ⬍ ប␻共0兲 共E⬎ ប␻共0兲兲 and the lower 共higher兲 energy electron-hole pairmakes a positive 共negative兲 contribution to ប␻␭共2兲. Moreover,an electron-hole pair satisfying E ⬍ 2兩EF兩 cannot contributeto the energy shift 关shaded region in Figs. 1共a兲 and 1共b兲兴
because of the Fermi distribution function f in Eq. 共1兲. Thus,
the quantum correction to the phonon energy by an intermediate electron-hole pair can be controlled by changing the

0.2

0.3

T = 10 K

1640

II. PHONON-FREQUENCY SHIFT⫻兵f关Eh共k兲 − EF兴 − f关Ee共k兲 − EF兴其.

T = 300 K

1640

Re(h(E))−Im(h(E))

h̄ω(cm−1 )(a)(10,10)-0.3 -0.2 -0.1

00.1

EF (eV)

0.2

0.3

FIG. 2. 共Color online兲 The EF dependence of the LO 共red curve兲
and TO 共black curve兲 phonon energy in the case of the 共10,10兲
armchair nanotube. 共a兲 is taken at room temperature and 共b兲 is at 10
K. Only the energy of the LO mode is shifted, with the TO mode
frequency being independent of EF. The decay width 共⌫␭兲 is plotted
as an error bar.

Fermi energy, EF. For example, when 兩EF兩 = ប␻共0兲 / 2, thenប␻␭共2兲 takes a minimum value at zero temperature since all
positive contributions to ប␻␭共2兲 are suppressed in Eq. 共1兲.
Since Re关h共E兲兴 ⬇ −1 / E for E Ⰷ ប␻共0兲, all high-energy intermediate states contribute to phonon softening if we include
all the electronic states in the system.10 Here we introduce acutoff energy at Ec = 0.5 eV as 兺kEe共k兲⬍Ec in order to avoid
such a large energy shift in Eq. 共1兲. The energy shift due tothe high-energy intermediate states 兺kEe共k兲⬎Ec can be neglected by renormalizing ប␻共0兲 so as to reproduce the experimental results of Raman spectra18 since the contribution
from Ee共k兲 ⬎ Ec just gives a constant energy shift to ប␻共2兲.
We have checked that the present results do not depend on
the selection of the cutoff energy since Ec is much larger thanប␻共0兲.
Im关h共E兲兴 is nonzero only very close to E = ប␻共0兲, whichshows that the phonon can resonantly decay into an electronhole pair with the same energy. It is noted that when 兩EF兩⬎ ប␻共0兲 / 2, ⌫␭ ⬇ 0 at zero temperature while ⌫␭ may take afinite value at a finite temperature. In this paper, we calculate⌫␭ self-consistently by calculating ⌫␭ = −Im共ប␻␭共2兲兲 in Eq. 共1兲.
In Fig. 2, we show calculated results for ប␻␭ as a functionof EF for a 共10,10兲 armchair nanotube. Here we take 1595
and 1610 cm−1 for ប␻␭共0兲 for the TO and LO modes, respectively. The energy bars denote ⌫␭ values. We have used theextended tight-binding scheme to calculate Ee共k兲 and Eh共k兲
and the electron wave function for 具eh共k兲兩Hint兩␻␭典.21 As for245441-2

PHYSICAL REVIEW B 77, 245441 共2008兲

CURVATURE-INDUCED OPTICAL PHONON FREQUENCY…

(b)

K

k2
k1

(a)
k2

T = 300 K

1640h̄ω(cm−1 )

(a)

θ(k)

K

k1

TO

1560

LO

1520

Cutting line

具eh共k兲兩Hint兩␻LO典 = − igu sin ␪共k兲,
共2兲where u is the phonon amplitude and g is the el-ph coupling
constant. Here ␪共k兲 is the angle for the polar coordinate
around the K 共or K⬘兲 point in the 2D BZ, in which the k= 共k1 , k2兲 point is taken on a cutting line for a metallic energy
subband. The k1 共k2兲 axis is taken in the direction of thenanotube circumferential 共axis兲 direction 共see Fig. 3兲. Equation 共2兲 shows that 具eh共k兲兩Hint兩␻␭典 depends only on ␪共k兲 but
not on 兩k兩, which means that the dependence of this matrix
element on E is negligible. Since the armchair nanotube is
free from the curvature effect,24 the cutting line for its metallic energy band lies on the k2 axis. Thus, we have ␪共k兲= ␲ / 2 共−␲ / 2兲 for the metallic energy subband, which has
k1 = 0 and k2 ⬎ 0 共k2 ⬍ 0兲. Then, Eq. 共2兲 tells us that only the

-0.2

-0.1

0

0.1

(18,0) (21,0) (24,0)h̄ω(cm−1 )

0.2

0.3

EF (eV)(b) 1640

zigzag

1600

1560

LO
TO

T = 300 K

1520

1

Egap (meV)

the el-ph matrix element,22 we adopted the deformation potential derived on the basis of density-functional theory by
Porezag et al.23 To obtain the phonon eigenvector, we used
the force-constant parameters calculated by Dubay andKresse16 for the dynamical matrix. We show the resulting
ប␻␭ as a function of EF at the room temperature 共T= 300 K兲 and T = 10 K in Figs. 2共a兲 and 2共b兲, respectively. It
is shown that the TO mode does not exhibit any energychange while the LO mode shows an energy shift and broadening. As we have mentioned above, the minimum energy isrealized at 兩EF兩 = ប␻共0兲 / 2 共⬇0.1 eV兲. There is a local maximum for the spectral peak at 兩EF兩 = 0. The broadening for the
LO mode occurs within 兩EF兩 ⱗ ប␻共0兲 / 2 for the lower temperature, while the broadening has a tail at room temperaturefor 兩EF兩 ⲏ ប␻共0兲 / 2.
An effective-mass theory for electrons in a carbon nanotube is adopted in this paper to explain the lack of an energy
shift of the TO modes for armchair nanotubes. As we willshow in Sec. III, the el-ph matrix element for the electronhole pair creation by the 共A1g兲 LO and TO modes is given by(15,0)

Egap = 0.045 eV-0.3

FIG. 3. 共Color online兲 共a兲 Cutting line near the K point. The k1
共k2兲 axis is selected as the nanotube circumferential 共axis兲 direction.
The amplitude for an electron-hole pair creation depends strongly
on the relative position of the cutting line from the K point. 共b兲 If
the cutting line crosses the K point, then the angle ␪共k兲
共⬅arctan共k2 / k1兲兲 takes ␲ / 2 共−␲ / 2兲 values for k2 ⬎ 0 共k2 ⬍ 0兲. In
this case, the LO mode strongly couples to an electron-hole pair,while the TO mode is decoupled from the electron-hole pair according to Eq. 共2兲.

具eh共k兲兩Hint兩␻TO典 = − igu cos ␪共k兲,

1600

1.4

50

0

1

1.4

1.8

2.2

1.8

2.2

dt (nm)FIG. 4. 共Color online兲 共a兲 The EF dependence of the LO 共red
curve兲 and TO 共black curve兲 phonon frequency for a 共15,0兲 zigzag
nanotube. Not only the frequency of the LO mode but also that of
the TO mode is shifted due to the curvature effect. 共b兲 The diameterdt dependence of the phonon frequency for zigzag nanotubes, including zigzag semiconducting tubes. Egap denotes the curvatureinduced minienergy gap.

LO mode couples to an electron-hole pair and the TO mode
is not coupled to an electron-hole pair for armchair SWNTs.
In Fig. 4共a兲, we show calculated results for ប␻␭ as a function of EF for a 共15,0兲 metallic zigzag nanotube. In the caseof zigzag nanotubes, not only the LO mode but also the TO
mode couples with electron-hole pairs. The spectrum peak
position for the TO mode becomes harder for EF = 0, sinceRe关h共E兲兴 for E ⬍ ប␻TO contributes to a positive frequency
shift. It has been shown theoretically24 and experimentally25
that even for “metallic” zigzag nanotubes a finite curvature
opens a small energy gap. When the curvature effect is taken
into account, the cutting line does not lie on the K point but
is shifted from the k2 axis. In this case, cos ␪共k兲 = k1 / 共k21
+ k22兲1/2 is nonzero for the lower energy intermediate electronhole pair states due to k1 ⫽ 0. Thus, the TO mode can couple
to the low energy electron-hole pair which makes a positiveenergy contribution to the phonon energy shift. The highenergy electron-hole pair is still decoupled from the TO
mode since cos ␪共k兲 → 0 for 兩k2兩 Ⰷ 兩k1兩. Therefore, when兩EF兩 ⱗ ប␻TO / 2, then ប␻TO increases by a larger amount thanប␻LO. The TO mode for small diameter zigzag nanotubes

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PHYSICAL REVIEW B 77, 245441 共2008兲

SASAKI et al.

couples strongly with an electron-hole pair because of thestronger curvature effect. In Fig. 4共b兲, we show the diameter
共dt兲 dependence of the ប␻␭ of zigzag nanotubes for EF = 0
not only for metallic SWNTs, but also for semiconducting
SWNTs. In the case of the semiconducting nanotubes, theLO 共TO兲 mode appears at around 1600 共1560兲 cm−1 without
any broadening. Only the metallic zigzag nanotubes show anenergy shift, and the energy of the LO 共TO兲 mode decreases
共increases兲 as compared to the semiconducting tubes. In thelower part of Fig. 4共b兲, we show the curvature-induced energy gap Egap as a function of dt. The results show that higher
共lower兲 energy electron-hole pairs contribute effectively to
the LO 共TO兲 mode softening 共hardening兲 in metallic nanotubes. In the case of semiconducting nanotubes, we may expect that there is a softening for the LO and TO modes according to Eq. 共2兲. However, the softening is small ascompared with that of the metallic nanotubes because theenergy of intermediate electron-hole pair states is muchlarger than ប␻␭共0兲 in this case.
III. CHIRALITY DEPENDENCE OF THE ELECTRONPHONON INTERACTION

Here, we derive Eq. 共2兲 on the basis of an effective-mass
theory for ␲ electrons near the K point in graphene. In acontinuous model, the local modulation of the hopping integral due to lattice vibrations appears as a deformationinduced gauge field, A共r兲 = 关Ax共r兲 , Ay共r兲兴, in the Weyl
equation.26 The Weyl equation for ␲-electrons with energy
eigenvalue E is written byvF␴ · 关p̂ + A共r兲兴⌿共r兲 = E⌿共r兲,uB (r)

R1



冑3
2

关␦␥20共r兲 − ␦␥30共r兲兴.

共4兲

共5兲

where ui共r兲 ⬅ u共r兲 · ei, 共i = x , y兲, and R1 − 共R2 + R3兲 / 2 = ᐉeyand 冑3 / 2共R2 − R3兲 = ᐉex have been used 共see Fig. 5兲. Then,the el-ph interaction for an in-plane lattice distortion u共r兲 canbe rewritten as the vector product of ␴ and u共r兲,10Hint = vF␴ · A共r兲 = g关␴ ⫻ u共r兲兴 · ez .

共6兲We consider the LO and TO phonon modes with q = 0
共i.e., ⌫ point兲. Then, an electron-hole pair is excited by a

δγ2

δγ3

R2 R3

x

constant u = 共ux , uy兲, which corresponds to the A1g phononmodes. The el-ph matrix element for the electron-hole pairgeneration is given by
具eh共k兲兩Hint兩␻典 =

冕

ⴱ
⌿c,k
共r兲Hint⌿v,k共r兲d2r

冉0
uy + iuxg= 共e+i⌰共k兲/2 e−i⌰共k兲/2 兲uy − iux
0
2
⫻

冉

冊

e−i⌰共k兲/2,− e+i⌰共k兲/2

冊
共7兲

where ⌿c,k共r兲 关⌿v,k共r兲兴 denotes an energy eigenstate ofvF␴ · p in the conduction 共valence兲 energy band with energy
eigenvalue E = vF兩p兩 共E = −vF兩p兩兲,冉冑 冉⌿c,k共r兲 =⌿v,k共r兲 =Here ␦␥a0共r兲 for the LO and TO modes is given by ␦␥a0共r兲
= 共g / ᐉ兲u共r兲 · Ra, where Ra denotes the nearest-neighbor vectors 共Fig. 5兲 and u共r兲 is the relative displacement vector of aB site from an A site 关u共r兲 = uB共r兲 − uA共r兲兴 and g is the el-phcoupling constant. We rewrite Eq. 共4兲 asvF关Ax共r兲,Ay共r兲兴 = g关uy共r兲,− ux共r兲兴,uA (r)

FIG. 5. A hexagonal unit cell of graphene consists of A 共closed
circle兲 and B 共open circle兲 sublattices. Ra 共a = 1 , 2 , 3兲 are vectors
pointing to the nearest-neighbor B sites from an A site 关R1 = accey,
R2 = −共冑3 / 2兲accex − 共1 / 2兲accey, and R3 = 共冑3 / 2兲accex − 共1 / 2兲accey兴.
Local modulations of the hopping integral are defined by ␦␥a0共r兲
共a = 1 , 2 , 3兲. The modulation is given by optical phonon modes as␦␥a0 = 共g / ᐉ兲u共r兲 · Ra, where u共r兲 关=uA共r兲 − uB共r兲兴 is a relative displacement vector of a B site relative to the nearest A site.

1
vFAx共r兲 = ␦␥10共r兲 − 关␦␥20共r兲 + ␦␥30共r兲兴,2
vFAy共r兲 =

δγ1

A

z

共3兲

where vF 共⬅␥0ᐉ / ប兲 is the Fermi velocity and ␥0
共⬇2.7 eV兲 is the nearest-neighbor hopping integral, ᐉ⬅ 共3 / 2兲acc, p̂ = −iបⵜ is the momentum operator, and ␴= 共␴x , ␴y兲 is the Pauli matrix. A共r兲 is given in Eq. 共3兲 of Ref.
27 by a small change ␦␥a0共r兲 共a = 1 , 2 , 3兲 in the hopping integral from −␥0 共see Fig. 5兲 as

B

y

冊
冊

eik·r e−i⌰共k兲/2
冑2S e+i⌰共k兲/2 ,e−i⌰共k兲/2+i⌰共k兲/2 ,2S − eeik·r

共8兲

where S denotes the total area of the system, and kx − iky⬅ 兩k兩e−i⌰共k兲.
We first consider the case of a zigzag nanotube in Fig. 5.
Then, we denote x 共y兲 as a coordinate around 共along兲 the axis
关so that ⌰共k兲 = ␪共k兲兴, and ux共r兲 关uy共r兲兴 are assigned to the
TO 共LO兲 phonon mode. The corresponding A共r兲 for ux共r兲
and uy共r兲 are Ay共r兲 and Ax共r兲, respectively. By calculating
Eq. 共7兲 for the TO mode with 共ux , uy兲 = 共u , 0兲 and for the LOmode with 共ux , uy兲 = 共0 , u兲, we get Eq. 共2兲. Next, we consider
the case of an armchair nanotube. Then, x 共y兲 is the coordinate along 共around兲 the axis 关so that ⌰共k兲 = ␪共k兲 + ␲ / 2兴, andux共r兲 关uy共r兲兴 is assigned to the LO 共TO兲 phonon mode. The
direction of the gauge field A共r兲 is perpendicular to the phonon eigenvector u共r兲 and the LO mode shifts the wave vector around the tube axis, which explains how the LO mode
may induce a dynamical energy-band gap in metallic
nanotubes.15,28 By calculating Eq. 共7兲 for the TO mode with
共ux , uy兲 = 共0 , u兲 and for the LO mode with 共ux , uy兲 = 共u , 0兲, we

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CURVATURE-INDUCED OPTICAL PHONON FREQUENCY…
Egap = 0.045 eV

IV. DISCUSSION

1600

LO

1560

TO

1520

(10,4)The phonon frequencies with some broadening as shownin Fig. 2 are not directly related to the Raman spectra of the
G band but are related to the phonon density of states of the
LO and TO phonons at q = 0. In previous papers,29 we haveshown that the G-band intensity depends on chiral angle forwhich for zigzag nanotubes the LO 共TO兲 phonon mode intensity is strong 共suppressed兲, while for armchair nanotubes
the TO 共LO兲 phonon mode intensity is strong 共comparable兲.
This chirality dependence of the G-band intensity is also
observed in single nanotube Raman spectroscopy.30,31 Thischirality dependence of the G-band mode intensity is exactly
opposite to the chirality dependence of the phonon-softening
for which the TO phonon mode is suppressed in armchair
nanotubes. These observations clearly show that the corresponding el-ph interaction for phonon-softening and Ramanprocesses are independent of each other. In the following, we
will show that not only the LO mode but also the TO mode
becomes Raman-active for armchair nanotubes due to thetrigonal warping effect.
In the Raman process, the phonon is emitted by scattering
an electron 共or an exciton兲 from one conduction-band state toanother conduction-band state, while in the phonon softening, the electron is scattered 共or excited兲 from a valence-bandstate to a conduction-band state. Thus the corresponding matrix element for the Raman process is expressed by substituting ⌿c,k共r兲 in Eq. 共11兲 for ⌿v,k共r兲 in Eq. 共7兲 as follows:h̄ω(cm−1 )T = 300 K-0.3 -0.2 -0.1

0

0.1 0.2 0.3

EF (eV)

FIG. 6. 共Color online兲 The EF dependence of the LO 共red curve兲
and TO 共black curve兲 phonon energy for the 共10,4兲 metallic chiralnanotube. Egap denotes the curvature-induced minienergy gap which
has the same value as that for the 共15,0兲 tube in Fig. 4共a兲. The
difference between the behavior of the 共10,4兲 and 共15,0兲 nanotubes
comes from the fact that the phonon eigenvector of the LO 共TO兲mode is not pointing along the tube axis in the case of the 共10,4兲
tube.

get Eq. 共2兲, too. Equation 共2兲 is valid regardless of the tube
chirality if the phonon eigenvector of the LO 共TO兲 phononmode is in the direction along 共around兲 the tube axis. This isbecause p̂ and u共r兲 are transformed in the same way as wechange the chiral angle. As a result, there would be no chiralangle dependence for the el-ph coupling on Eq. 共2兲.
However, the phonon eigenvector depends on the chiralangle. Reich et al.20 reported that, for a chiral nanotube, atoms vibrate along the direction of the carbon-carbon bonds
and not along the axis or the circumference. In the case of achiral nanotube, the phonon eigenvector may be written as冉 冊冉

cos ␾ sin ␾
uTO
=
uLO
− sin ␾ cos ␾

冊冉 冊

u1
,
u2

具e共k兲, ␻兩Hint兩e共k兲典 =where u1 共u2兲 is in the direction around 共along兲 a chiral tube
axis, and ␾ is the angle difference between the nanotube axis
and the vibration. This modifies Eq. 共2兲 as

具eh共k兲兩Hint兩␻TO典 = − igu cos 共␪共k兲 + ␾兲.

共10兲

In Fig. 6, we show numerical results for ប␻␭ as a function of
EF for a 共10,4兲 metallic chiral nanotube for T = 300 K. Theenergy difference between the minimum point at EF= ប␻共0兲 / 2 and at EF = 0 shows that the LO mode couples
more strongly to the low energy electron-hole pair than the
TO mode. Since the curvature-induced minienergy gap for
共10,4兲 and 共15,0兲 tubes are almost the same 共Egap= 0.045 eV兲, we may expect a similar energy shift of the LO
and TO modes. However, the results of the 共15,0兲 zigzag
tube show that the TO mode couples more strongly for the
low-energy electron-hole pair than the LO mode. This is explained not by Eq. 共2兲 but by Eq. 共10兲 with an appropriate ␾
共␾ ⬇ 23.4°兲, which is given by the phonon eigenvector calculation. Since the chiral angle for 共10,4兲 is 16.1°, ␾ is notdirectly related to the chiral angle. The identification of ␾ in
Eq. 共10兲 as a function of nanotube chirality would be useful
to compare theoretical results and experiments, which will be
explored in the future.

ⴱ⌿c,k共r兲Hint⌿c,k共r兲d2rg= 共e+i⌰共k兲/2 e−i⌰共k兲/2 兲
2共9兲

具eh共k兲兩Hint兩␻LO典 = − igu sin 共␪共k兲 + ␾兲,

冕

⫻

冉

0

uy + iuxuy − iux

0冊冉

冊

e−i⌰共k兲/2.
e+i⌰共k兲/2
共11兲Then the electron-phonon matrix element for the Ramanscattering process becomes
具e共k兲, ␻LO兩Hint兩e共k兲典 = gu cos ␪共k兲,
具e共k兲, ␻TO兩Hint兩e共k兲典 = − gu sin ␪共k兲.

共12兲

It is stressed again that the el-ph interactions of Eqs. 共2兲 and共12兲 are for phonon-softening and Raman intensity, respectively, and that they are independent of each other. When we
consider the Raman-scattering processes for carbon nanotubes, the k vector which is relevant to the resonance Ramanprocess is the k vector at the van Hove singular point, kii,
which corresponds to the touching points of the equienergy
surface to the cutting line 共one-dimensional Brillouin zone
plotted in the 2D BZ兲,1,32 which are shown in Figs. 7共a兲 and7共b兲 for zigzag and armchair nanotubes, respectively.
In the case of the zigzag nanotubes, the kii point corresponds to ␪ = 0 which is not the chiral angle but is defined inFig. 3共b兲. Thus from Eq. 共12兲, we see that the TO mode is not
excited. On the other hand, in the case of the armchair nanotube, the value of ␪ is not zero because of the trigonally245441-5

PHYSICAL REVIEW B 77, 245441 共2008兲

SASAKI et al.

k2

(a)

⫻ A共r兲兴 but the TO mode does yield a divergence of A共r兲instead because

(b)
k1
θ

k ii
k ii
Armchair (θ = 0 )

k ii

k ii

Bz共r兲 = −k2

k1

ⵜ · A共r兲 =Zigzag (θ = 0 )FIG. 7. Trigonal warping effect of an equienergy surface in the
2D Brillouin zone and cutting lines for 共a兲 armchair and 共b兲 zigzag
nanotubes. In the case of 共a兲 armchair nanotubes, the ␪ value defined from the k1 axis in Fig. 5 is not zero while in the case of 共b兲
zigzag nanotubes, ␪ is always zero, which explains the chirality
dependence of the relative Raman intensity of LO and TO phononmodes.

warped equienergy surface, as shown in Fig. 7共b兲.1,32 Thus
not only the LO but also the TO phonon modes can be observed, which is consistent with the previous experimental
results30,31 and the theories.1,32 Thus when we see the G-band
Raman spectra for metallic nanotubes, we can expect thefollowing features: for armchair nanotubes, we can see boththe LO and TO modes in which the LO mode appears at alower frequency region with some spectral width, while the
TO mode appears without spectral broadening because of theabsence of phonon softening. For zigzag nanotubes, we cansee only a broadened LO phonon mode but we cannot see a
TO phonon mode. For a general chiral nanotube, we generally observe both the LO and TO phonon modes with somebroadening as a function of ␪. Further, we expect some phonon hardening effect for the TO phonon mode. The relative
intensity between the LO and TO phonon modes is determined by the chirality dependent Raman intensity and spectral width. The detailed Raman spectral features for all metallic 共n , m兲 nanotubes will be presented elsewhere.
The gauge-field descriptions for the lattice deformation
关Eq. 共4兲兴 and for the el-ph interaction of the LO and TO
modes 关Eq. 共5兲兴 are useful to show the appearance of thecurvature-induced minienergy gap in metallic zigzag carbon
nanotubes and the decoupling between the TO mode with afinite wave vector and the electrons, as shown in the following. For a zigzag nanotube, we have ␦␥10 = 0 and ␦␥20 = ␦␥30
from the rotational symmetry around the tube axis 共see Fig.
5兲. Then, Eq. 共4兲 shows that for Ax ⫽ 0 and Ay = 0, the cutting
line of kx = 0 for the metallic zigzag nanotube is shifted by afinite constant value of Ax because of the Aharanov-Bohm
effect for the lattice distortion-induced gauge field A. This
explains the appearance of the curvature-induced minienergy
gap in metallic zigzag carbon nanotubes33 and of the phonon
broadening for the TO mode as a function of EF. The TO
phonon mode with q ⫽ 0 does not change the area of thehexagonal lattice but instead gives rise to a shear deformation. Thus, the TO mode 关uTO共r兲兴 satisfies
ⵜ · uTO共r兲 = 0,ⵜ ⫻ uTO共r兲 ⫽ 0.

ⵜ · uTO共r兲 = 0,

共ⵜ ⫻ uTO共r兲兲 · ez ⫽ 0.

共14兲

Thus, we can set A共r兲 = ⵜ␸共r兲 where ␸共r兲 is a scalar function. Since we can set A共r兲 = 0 in Eq. 共3兲 by a redefinition of
the phase of the wave function 共a local gauge symmetry兲 as⌿共r兲 → exp关−i␸共r兲 / ប兴⌿共r兲,26 and thus A共r兲 in Eq. 共3兲 disappears for the TO mode with q ⫽ 0. This explains why the
TO mode with q ⫽ 0 is completely decoupled from the electrons and that only the TO mode with q = 0 couples withelectrons. This conclusion is valid even when the graphenesheet has a static surface deformation. In this sense, the TO
phonon mode at the ⌫ point is anomalous since the el-phinteraction for the TO mode cannot be eliminated by thephase of the wave function. It may be difficult to include thelocal gauge symmetry in a numerical analysis. On the other
hand, the LO phonon mode with q ⫽ 0 changes the area ofthe hexagonal lattice, while it does not give rise to a shear
deformation. Thus, the LO mode 共uLO共r兲兲 satisfies
ⵜ · uLO共r兲 ⫽ 0,

ⵜ ⫻ uLO共r兲 = 0.

共15兲Using Eqs. 共5兲 and 共15兲, we see that the LO mode gives rise
to a deformation-induced magnetic field since
Bz共r兲 ⫽ 0,

ⵜ · A共r兲 = 0.

共16兲

Since a magnetic field changes the energy-band structure ofelectrons, the LO mode 共with q ⫽ 0兲 couples strongly to theelectrons.
In the case of 2D graphene, Eq. 共2兲 tells us that the ⌫point TO and LO modes give the same energy shift because
the integral over ␪共k兲 gives the same ប␻␭共2兲 in Eq. 共1兲 forboth TO and LO modes. This explains why no G-band splitting is observed in a single layer of graphene.34 Even when
we consider the TO and LO modes near the ⌫ point, we do
not expect any splitting between the LO and TO phonon
energies because the TO mode with q ⫽ 0 is completely decoupled from the electrons. Thus, for q ⫽ 0, only the LOmode contributes to the G band.
In our numerical results for ប␻␭, we see a crossing between the LO and TO modes at some values of EF. If there isa finite transition amplitude between the LO and TO phononmodes, we need to diagonalize the following matrix according to second-order perturbation theory for a degenerate
case,

冉

共13兲

Using Eqs. 共5兲 and 共13兲, we see that the TO mode does notyield the deformation-induced magnetic field 关B共r兲 = ⵜ

g
vF

g
vF

具␻LO兩H共2兲兩␻LO典 具␻LO兩H共2兲兩␻TO典
具␻TO兩H共2兲兩␻LO典 具␻TO兩H共2兲兩␻TO典

冊

,

共17兲

where H共2兲 is the effective Hamiltonian in second-order perturbation theory given by

245441-6

PHYSICAL REVIEW B 77, 245441 共2008兲

CURVATURE-INDUCED OPTICAL PHONON FREQUENCY…

(b)

(a)

1650

1650

Raman Shift [cm−1]−1

Raman Shift [cm ]

1600
1550
1500
1450
1400
−2

1600
1550
1500
1450

0
2
Gate Voltage [V]

−1

−0.5
0
0.5
Gate Voltage [V]FIG. 8. 共Color online兲 Experimental results for the Raman
G-band intensity as a function of applied gate voltage for two metallic SWNTs 共a兲 and 共b兲. A strong 共weak兲 intensity peak is denotedby the yellow 共black兲 color.

H共2兲 = 兺kHint兩eh共k兲典具eh共k兲兩Hint
ប␻␭共0兲 − 共Ee共k兲 − Eh共k兲兲 + i⌫

.

共18兲

Using Eq. 共2兲, we see that cos ␪共k兲 sin ␪共k兲 which is an oddfunction of k2 appears in the numerator for the off-diagonalterms in Eq. 共17兲. Thus, the mode coupling through theelectron-hole pair at k2 is canceled by that at −k2, i.e., due tothe time-reversal symmetry of the system. Thus, even though
the LO and TO modes cross each other, there is no mixing
between the LO and TO phonon modes for any chirality. If
there is a breaking of time-reversal symmetry for k2, we
expect some mixing between the LO and TO phonon modes.
V. COMPARISON WITH EXPERIMENT AND
SUMMARY

The first comparison we make is to experimental resultsin the literature. Here we try to assign the chirality of thenanotube that is given in Fig. 2 of Ref. 18. This figure shows
two strong intensity peaks with different phonon frequencies.
The higher-frequency peak does not depend on the gate voltage and the lower-frequency peak shows a frequency shift
with a strong broadening near the Dirac point. The existenceof a flat intensity peak as a function of Fermi energy for a
metallic SWNT indicates that the electron-phonon couplingis very weak for the phonon mode. Thus, the nanotube canbe thought of as an armchair SWNT 共or close to an armchair
SWNT with a chiral angle near 30°兲 since the el-ph coupling
between the TO mode and electron-hole pairs is negligible inthis case. Further investigation comparing theory with experiment is strongly necessary for tubes of different chiralities.
In Figs. 8共a兲 and 8共b兲, we show experimental results of
the G-band intensity as a function of applied gate voltage fortwo different isolated SWNTs. For a given excitation energyof 1.91 eV and the observed radial breathing mode 共RBM兲
frequencies of 196 and 193 cm−1 for the sample of Figs. 8共a兲and 8共b兲, respectively, we can assign 共n , m兲 by using theconventional assignment technique that we used for single
nanotube Raman spectroscopy.35 As a result, we assign the
共n , m兲 values as 共12,6兲 for Fig. 8共a兲 and 共15,3兲, 共16,1兲, or
共11,8兲 for Fig. 8共b兲.
Although 共12,6兲 is not so close to the chiral angle for anarmchair nanotube, we expect that Fig. 8共a兲 exhibits a similar
behavior to that of Fig. 2 of Ref. 18. Thus the assignment of
共12,6兲 from the RBM spectra is consistent with the present
phonon spectra. In Fig. 8共b兲, the intensity near the Diracpoint of the TO 共LO兲 mode is weaker 共stronger兲 than shownin Fig. 8共a兲. This indicates that the tube is not an armchair
SWNT but close to a metallic zigzag SWNT 共or a metallic
chiral SWNT兲. Thus we can assign this SWNT either to
共15,3兲 or to 共16,1兲 for the SWNT for Fig. 8共b兲 and we can
exclude 共11,8兲. A further comparison with more experimental
data will be required, which will be carried out in the future.
A chirality-dependent Raman intensity was observed formetallic SWNTs as shown in Fig. 1 of Ref. 36. Although the
EF positions for the observed metallic SWNTs are unclear,
the results are consistent with our calculations in the following way. For example, the TO mode in a 共15,15兲 SWNT
gives a sharp line shape and the LO mode shows a broadfeature for a 共24,0兲 SWNT. It is noted that the Raman intensity is proportional to the el-ph coupling for an optically
excited electron via Eq. 共12兲. Since this el-ph coupling depends on the chiral angle due to the trigonal warping effect,
one of the two optical modes may be invisible in the Raman
intensity.36,37
In summary, we have calculated the EF dependence ofប␻␭ 共␭ = LO and TO兲 on metallic carbon nanotubes. The results show a strong dependence of the phonon frequency
shift on the chirality of single-walled carbon nanotubes because of the curvature-induced shift of the wave vector
around the tube axis. This is explained by the general property of second-order perturbation theory and the characteristic electron-phonon coupling for the ⌫ point LO and TO
phonon modes 关Eqs. 共2兲 and 共6兲兴. Since the frequency shiftof the LO/TO phonon modes depends on the value of EF, we
think that the assignment of the LO/TO phonon modes to theG− / G+ bands should be defined for EF = 0 without any ambiguity. For the LO and TO phonon modes near the ⌫ point,
we showed the absence of an electron-phonon coupling for
the TO mode for q ⫽ 0 due to a local gauge symmetry andthat the LO mode works as a deformation-induced magnetic
field. The phonon mixing between LO and TO phononmodes is absent in second-order perturbation theory due totime-reversal symmetry.
ACKNOWLEDGMENTS

R.S. acknowledges a Grant-in-Aid 共Nos. 16076201 and
20241023兲 from MEXT. MIT authors acknowledge support
from NSF under Grant No. DMR 07-04197.

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