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Electron-phonon coupling mechanism in two-dimensional graphite
and single-wall carbon nanotubes
Ge. G. Samsonidze,1E. B. Barros,1,2 R. Saito,3J. Jiang,3G. Dresselhaus,4and M. S. Dresselhaus1,5
1Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139-4307, USA
2Departamento de Física, Universidade Federal do Ceará, Fortaleza 60455-760, Ceará, Brazil
3Department of Physics, Tohoku University and CREST JST, Aoba, Sendai 980-8578, Japan
4Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
5Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
共Received 19 January 2007; revised manuscript received 6 February 2007; published 19 April 2007兲
The electron-phonon coupling in two-dimensional graphite and metallic single-wall carbon nanotubes is
analyzed. The highest-frequency phonon mode at the Kpoint in two-dimensional graphite opens a dynamical
band gap that induces a Kohn anomaly. Similar effects take place in metallic single-wall carbon nanotubes that
undergo Peierls transitions driven by the highest-frequency phonon modes at the ⌫and Kpoints. The dynami-
cal band gap induces a nonlinear dependence of the phonon frequencies on the doping level and gives rise to
strong anharmonic effects in two-dimensional graphite and metallic single-wall carbon nanotubes.
DOI: 10.1103/PhysRevB.75.155420 PACS number共s兲: 63.20.Kr, 71.18.⫹y, 71.70.⫺d, 73.22.⫺f
I. INTRODUCTION
Phonon modes of certain symmetries in graphitic materi-
als exhibit a frequency softening, as observed by resonance
Raman scattering from metallic 共metallic armchair and mini-
band-gap semiconducting chiral and zigzag兲single-wall car-
bon nanotubes1共SWNTs兲and by inelastic x-ray scattering
from a graphite flake.2The frequency softening is attributed
to Peierls instabilities in metallic SWNTs 共Ref. 3兲and to
Kohn anomalies in two-dimensional 共2D兲graphite 共a single
graphene sheet兲.4The Peierls instability, analogous to the
Jahn-Teller effect in molecular systems, occurs when a pho-
non mode opens a dynamical 共oscillating with the phonon
frequency兲band gap at the Fermi level EFin a graphene
sheet5and in metallic SWNTs.3The Kohn anomaly occurs
when electrons at the Fermi surface screen the phonon mode
in a graphene sheet4and in metallic SWNTs.6–9The two
aforementioned phenomena are manifestations of the same
underlying electron-phonon coupling mechanism. When a
phonon mode opens a dynamical band gap, all the valence
electrons lie in states whose energy is lowered, thus reducing
the total energy and softening the phonon frequency. On the
other hand, the soft phonon mode induces electron scattering
at the Fermi surface, which in turn generates charge-density
waves, opening a dynamical band gap.
In Sec. II, we derive an analytic expression for the elec-
tronic response to the phonon perturbation. In Sec. III, we
study the effect of electronic distortion on the phonon fre-
quency. In both sections, we start our consideration with a
graphene sheet, and then we extend it to metallic SWNTs.
Our approach is based solely on the symmetry of the phonon
modes obtained from group theory 共GT兲, and it does not
involve the explicit phonon-dispersion relations. The present
analysis reveals the mechanism of the electron-phonon cou-
pling that is behind Kohn anomalies in a graphene sheet and
metallic SWNTs, which is not examined in the previous pa-
pers devoted to this subject.
II. PEIERLS INSTABILITY
A graphene sheet is defined by the translation vectors a1
and a2in the two-atom unit cell, as shown in Fig. 1共a兲in
light gray.10 The reciprocal-lattice vectors b1and b2are ob-
tained from a1and a2following the standard definition
ai·bj=2
␦
ij, where
␦
ij is the Kronecker delta.10 The first
Brillouin zone 共BZ兲is spanned by b1and b2, as shown in
Fig. 1共b兲in light gray, where its center and the two inequiva-
lent corners are labeled by the ⌫,K, and K⬘points,
respectively.10 The graphene sheet is a zero-gap semiconduc-
tor with the Fermi surface reduced to two points, kFand kF
⬘,
which appear, respectively, at the Kand K⬘points.10 The
electrons at the Fermi surface are thus scattered either within
the same Kor K⬘point by the phonon modes around the ⌫
point 共intravalley scattering兲, or between different Kand K⬘
points by the phonon modes near the Kor K⬘point 共inter-
valley scattering兲. Below, we consider the ⌫point phonon
modes first, and then we turn to the K共K⬘兲point phonon
modes.
The group of the wave vector at the ⌫point 共G⌫兲is iso-
morphic to the point group D6h. The longitudinal and in-
FIG. 1. 共a兲The two-atom unit cell of the graphene sheet 共in light
gray兲and the six-atom supercell at the Kpoint 共in dark gray兲.共b兲
The first Brillouin zone 共BZ兲of the graphene sheet 共in light gray兲
and the triple-folded BZ of the Kpoint supercell 共in dark gray兲.
PHYSICAL REVIEW B 75, 155420 共2007兲
1098-0121/2007/75共15兲/155420共8兲©2007 The American Physical Society155420-1
plane transverse optical phonon modes 共LO and iTO兲belong
to the irreducible representation 共IR兲with E2gsymmetry.11,12
The directions of the atomic displacements specified by IR
E2gare shown in Figs. 2共a兲and 共b兲, respectively.12 The
Hamiltonian of the graphene sheet distorted by the E2g⌫
point phonon mode takes the form
H=
冉
HAA HAB
HBA HBB
冊
,共1兲
where matrix elements HAA,HAB,HBA, and HBB are evalu-
ated within the framework of the nearest-neighbor
-band
orthogonal tight-binding model10 in the linear in u/aap-
proximation, thereafter referred to as a simple tight-binding
共STB兲model:
HAA =E0+兺
j
3
共uBj −uA0兲·共rBj −rA0兲/aCC,
HAB =兺
j
3
关t+
␣
共uBj −uA0兲·共rBj −rA0兲/aCC兴
⫻exp关ik·共rBj −rA0+uBj −uA0兲兴,共2兲
HBA =HAB
*, and HBB =HAA. Here, E0is the atomic-orbital en-
ergy set to zero for our energy scale, t=−2.56 eV is the
transfer or hopping integral,13 =39.9 eV is the on-site
electron-phonon coupling 共EPC兲coefficient,13
␣
=58.2 eV/nm is the off-site EPC coefficient,13 rAj and rBj
are the equilibrium atomic positions shown by the open and
solid dots in Fig. 1共a兲, respectively, uAj and uBj are the
atomic displacements associated with the E2g⌫point phonon
mode represented by arrows in Figs. 2共a兲and 共b兲, subscript
j=0 , ... ,3 labels the central atom and its three nearest neigh-
bors as illustrated in Fig. 1共a兲,aCC =0.142 nm is the inter-
atomic distance, and kis the electron wave vector.
Upon substituting uAj and uBj from Figs. 2共a兲and 共b兲into
Eq. 共2兲and setting the determinant of Eq. 共1兲to zero, we find
that kF共kF
⬘兲oscillates at the phonon frequency with displace-
ment amplitude ⌬kF共⌬kF
⬘兲given by
⌬kF=−⌬kF
⬘=−2冑3
␣
u
ta y
ˆfor ⌫LO,
⌬kF=−⌬kF
⬘=+2冑3
␣
u
ta x
ˆfor ⌫iTO, 共3兲
around the K共K⬘兲point.3Here, uis the amplitude of phonon
displacements, a=冑3aCC =0.246 nm is the lattice constant,
and 共x
ˆ,y
ˆ兲are the unit vectors shown in the inset of Fig. 1共b兲.
Note that ⌬kFand ⌬kF
⬘are determined by the off-site EPC
coefficient
␣
, since the terms in Eq. 共2兲that are linear in
u/acancel out for the uAj and uBj vectors shown in Figs. 2共a兲
and 共b兲.14
The group of the wave vector at the Kpoint 共GK兲is iso-
morphic to the point group D3h. Among the longitudinal and
in-plane transverse optical and acoustic phonon modes 共LO,
iTO, LA, and iTA兲,15 iTO belongs to IR A1
⬘of group D3h,LO
and LA to IR E⬘, and iTA to A2
⬘.11,12 The directions of the
atomic displacements specified by IRs A1
⬘,E⬘, and A2
⬘are
shown in Figs. 2共c兲,共d兲,共e兲, and 共f兲, respectively,12 as are the
C2,C3, and C6rotation axes. Note that the complex traveling
phonon modes at the K共K⬘兲point only have the C3rotation
axes, since the group GKis isomorphic to group D3h.12 Time-
reversal symmetry mixes the complex traveling phonon
modes at the Kand K⬘points into the real stationary phonon
modes that obey D6hsymmetry.12
Since the lattice distortions shown in Figs. 2共c兲,共d兲,共e兲,
and 共f兲are incommensurate with the two-atom unit cell, the
six-atom supercell must be introduced.5The supercell
spanned by the translation vectors c1and c2for which cj·x
ˆ
=aj·x
ˆand cj·y
ˆ=3aj·y
ˆ共j=1,2兲is shown in Fig. 1共a兲in dark
gray. The first BZ for the supercell generated by the recipro-
cal lattice vectors d1and d2for which dj·x
ˆ=bj·x
ˆand dj·y
ˆ
=bj·y
ˆ/3 共j=1,2兲is shown in Fig. 1共b兲in dark gray. One can
see from Fig. 1共b兲that the dark gray hexagon is obtained by
cutting the light gray hexagon along six M-Llines and fold-
ing it along six L-Llines into one-third of its actual size. The
first BZ of the supercell is therefore triple folded, with both
the Kand K⬘points 共kFand kF
⬘兲mapped to the ⌫point. The
electronic states at the ⌫point are therefore fourfold degen-
erate, but this degeneracy, however, is lifted by the lattice
distortions caused by the Kpoint phonon modes. To study
the degeneracy-lifting mechanism, we employ GT.
The group of the wave vector Gk共G⌫or GK兲is isomor-
phic to the group D2hwhen the graphene sheet is distorted by
the E2g⌫or E⬘Kpoint phonon modes shown in Figs. 2共c兲,
共d兲,共e兲, and 共f兲. The fourfold degenerate electronic state at
the ⌫point thus consists of the four one-dimensional 共1D兲
IRs of group D2h: two B1u共valence bands兲and two B2g共con-
duction bands兲. This state therefore splits into two twofold
degenerate states B1u+B2gbelow and above EF. Such a split-
ting shifts the band-crossing points kFand kF
⬘away from the
⌫point to states kand −k, respectively, maintaining the
SAMSONIDZE et al. PHYSICAL REVIEW B 75, 155420 共2007兲
155420-2
time-reversal symmetry requirement kF
⬘=−kF. This shift is
allowed by GT because the star of a general wave vector k
⫽0共the set of wave vectors generated from kby point-group
operations兲consists of two states, kand −k. For the E2g⌫
point phonon modes, the magnitude of this shift is deter-
mined by the off-site EPC coefficient
␣
, according to Eq. 共3兲.
In contrast, the magnitude of this shift is governed by the
on-site EPC coefficient for the E⬘Kpoint phonon modes,
for which ⌬kFand ⌬kF
⬘are given by Eq. 共3兲with /2 sub-
stituted for
␣
.14
The group of the wave vector GKis isomorphic to the
group D6h共C6h兲when the graphene sheet is distorted by the
A1
⬘共A2
⬘兲Kpoint phonon mode shown in Fig. 2共c兲关Fig. 2共f兲兴.
The fourfold degenerate electronic state at the ⌫point con-
sists of the two 2D IRs of group D6h共C6h兲:E2u共valence
bands兲and E1g共conduction bands兲. This state is therefore not
required to split by GT. If it splits, however, a band gap will
be opened at the ⌫point. Indeed, there are only two in-
equivalent Fermi points, kFand kF
⬘, while the star of a gen-
eral wave vector k⫽0 consists of six states. Thus, kFand kF
⬘
cannot move away from the ⌫point.
To check whether the A1
⬘共A2
⬘兲Kpoint phonon mode opens
a dynamical band gap at the ⌫point, we construct the 6
⫻6 STB Hamiltonian at k= 0 for the six-atom supercell.
Labeling atoms in the supercell as shown by numbers 1–6 in
Fig. 2关atoms 1 to 3 共4to6兲belong to the A共B兲sublattice兴,
the Hamiltonian takes the form of Eq. 共1兲, where HAA,HAB,
HBA, and HBB are 3⫻3 matrices. For an ideal graphene
sheet, we have
HAA =HBB =
冢
E000
0E00
00
E0
冣
,
HAB =HBA =
冢
ttt
ttt
ttt
冣
.共4兲
Substituting Eq. 共4兲into Eq. 共1兲and setting its determinant to
zero yields the following electronic states:
E=共E0+3t,E0,E0,E0,E0,E0−3t兲.共5兲
The four states Ej=E0with band index j=2,3,4,5 are de-
generate, in agreement with the previous discussion.
For the graphene sheet distorted by the A1
⬘symmetry K
point phonon mode, we construct the STB Hamiltonian con-
sidering the atomic displacements shown in Fig. 2共c兲. Keep-
ing only terms linear in u/a, the terms in HAA and HBB
cancel out, so that HAA and HBB are the same as in Eq. 共4兲,
while HAB and HBA become
HAB =HBA =
冢
t+2
␣
ut−
␣
ut−
␣
u
t−
␣
ut+2
␣
ut−
␣
u
t−
␣
ut−
␣
ut+2
␣
u
冣
.共6兲
Substituting Eqs. 共4兲and 共6兲into Eq. 共1兲and setting its de-
terminant to zero yields the following electronic states:
E=共E0+3t,E0−3
␣
u,E0−3
␣
u,E0+3
␣
u,E0+3
␣
u,E0−3t兲.
共7兲
The A1
⬘Kpoint phonon mode thus splits the fourfold degen-
erate state of Eq. 共5兲,Ej=E0共j=2,3,4,5兲, into the two two-
fold degenerate states of Eq. 共7兲,Ej=E0−3
␣
u共j=2,3兲and
Ej=E0+3
␣
u共j=4,5兲, opening a dynamical band gap of the
following amplitude:
Eg=E4−E3=6
␣
ufor K共K⬘兲iTO, 共8兲
which is determined by the off-site EPC coefficient
␣
.5
The interatomic distances in the graphene sheet are not
affected by the A2
⬘symmetry Kpoint phonon mode within
the linear in u/aapproximation 关see in Fig. 2共f兲兴. Thus, nei-
ther nor
␣
terms enter the STB Hamiltonian in Eq. 共4兲, and
we obtain the fourfold degenerate electronic state at the ⌫
point described by Eq. 共5兲. However, one of the three inter-
atomic distances in Fig. 2共f兲is slightly changed in the
second-order series expansion with respect to u/a. Such a
deformation opens a dynamical band gap of amplitude Eg
=4
␣
u2/a, which is negligible compared to Eq. 共8兲. Thus, the
only phonon mode associated with the dynamical band gap
in the graphene sheet is the A1
⬘Kpoint phonon mode.5
For a general phonon wave vector qaway from the ⌫and
K共K⬘兲points, the size of the supercell increases signifi-
cantly, thereby making the supercell method impractical. We
thus implement the linear-response method originally devel-
oped within the framework of density-functional perturba-
tion theory16 and further modified for the extended tight-
binding 共ETB兲model,9which operates within the original
two-atom unit cell of the graphene sheet. As qvaries from ⌫
to K共K⬘兲, the directions of the atomic displacements uAj and
uBj gradually change from those shown in Fig. 2共a兲to the
ones in Fig. 2共c兲. Substituting uAj and uBj into Eqs. 共2兲and
共6兲yields the q-dependent ⌬kF共⌬kF
⬘兲and Eginstead of Eqs.
共3兲and 共8兲. In the vicinity of the ⌫point, we have qaⰆ1.
Keeping only terms linear in qa yields
⌬kF=−⌬kF
⬘=−2冑3
␣
u
ta
冉
1−3qa
2
冊
y
ˆfor ⌫LO,
⌬kF=−⌬kF
⬘=+2冑3
␣
u
ta
冉
1−3qa
2
冊
x
ˆfor ⌫iTO. 共9兲
In the vicinity of the K共K⬘兲point, we have qKaⰆ1共qK⬘a
Ⰶ1兲. Keeping only terms linear in qKa共qK⬘a兲yields
Eg=6
␣
u
冉
1−3qKa
2
冊
for KiTO,
Eg=6
␣
u
冉
1−3qK⬘a
2
冊
for K⬘iTO, 共10兲
where qK共qK⬘兲is measured from the K共K⬘兲point. Thus, the
amplitudes ⌬kF共⌬kF
⬘兲and Egreach their maximum values at
the ⌫and K共K⬘兲points, vanishing halfway between the ⌫
and K共K⬘兲points, according to Eqs. 共9兲and 共10兲. The de-
tailed derivation of Eqs. 共9兲and 共10兲is given in the Appen-
dix.
ELECTRON-PHONON COUPLING MECHANISM IN TWO-…PHYSICAL REVIEW B 75, 155420 共2007兲
155420-3
The same approach can be applied to metallic SWNTs,
whose band structure consists of pairs of mirror valence and
conduction subbands along the 1D momentum quantization
lines in the 2D BZ of the graphene sheet.10 The A1
⬘Kpoint
phonon mode in metallic SWNTs opens a dynamical band
gap or induces oscillations of the mini-band-gap with ampli-
tude given by Eq. 共10兲. The E2g⌫point phonon mode in
metallic SWNTs splits into the LO and iTO components in-
volving atomic vibrations in the axial and circumferential
directions, respectively. The LO component shifts kFand kF
⬘
perpendicular to the momentum quantization lines, which in
turn opens a dynamical band gap or causes oscillations of the
mini-band-gap with amplitude given by Eq. 共10兲.3The iTO
component induces oscillations of kFand kF
⬘or the band
edges along the momentum quantization lines with ampli-
tudes given by Eq. 共9兲.3
Let us estimate the numerical values of ⌬kFand Eg.
Within the second quantization formalism, u=
冑
and
2
=冑3a2ប/共4M
兲, where uis the amplitude of phonon dis-
placements,
is the density of phonon states, Mis the mass
of a carbon atom, and
is the phonon frequency. The latter
is
共E2g兲=1582 cm−1 and
共A1
⬘兲⬇1300 cm−1 for the phonon
modes of interest.9,17–19 Integrating
over the first BZ gives
=1/Aper phonon mode, where A=冑3a2/2 = 0.052 nm2is
an area of the unit cell. On averaging the scaling factor 关1
−3qa /共2
兲兴 in Eqs. 共9兲and 共10兲over the first BZ, the effec-
tive density of phonon states contributing to ⌬kFand Egis
reduced by a factor of
/共18冑3兲=0.1 for each of the LO E2g
⌫,iTOE2g⌫,A1
⬘K, and A1
⬘K⬘point phonon modes. The
Bose-Einstein distribution at room temperature T=300 K
yields f共E2g兲=5⫻10−4 and f共A1
⬘兲=2⫻10−3. Putting all the
factors together gives
共E2g兲=10−4 nm−2 and
共A1
⬘兲=8
⫻10−4 nm−2. Using
共E2g兲=6.8⫻10−4 nm2and
共A1
⬘兲=7.5
⫻10−4 nm2,wegetu共E2g兲= 0.7 ⫻10−5 nm and u共A1
⬘兲=2.1
⫻10−5 nm. Substituting these values into Eqs. 共3兲and 共8兲
yields 兩⌬kF兩=1.3⫻10−4共⌫K兲along the y
ˆand x
ˆdirections for
the LO and iTO components of the E2g⌫point phonon
mode, and Eg=10 meV for the A1
⬘symmetry Kand K⬘point
phonon modes in the graphene sheet. Similarly, 兩⌬kF兩=1.3
⫻10−4共⌫K兲for the iTO E2g⌫phonon mode, and Eg
=10 meV for the LO E2g⌫and A1
⬘Kand K⬘phonon modes
in metallic SWNTs.
III. KOHN ANOMALY
The electronic dispersion relations of an ideal graphene
sheet and the graphene sheet distorted by the A1
⬘K共K⬘兲point
phonon mode at T=300 K are shown in Fig. 3共a兲by dashed
and solid curves, respectively. The dispersion relations are
calculated within the framework of the long-range
-band
nonorthogonal tight-binding model13 without making the ex-
pansion in a power series in u/a, and thereafter referred to as
an ETB model. Considering that the amplitude of the dy-
namical band gap Egis less than the thermal energy T
=26 meV, the former does not affect the transport properties
of the graphene sheet at T=300 K, though it softens the fre-
quency of the A1
⬘K共K⬘兲point phonon mode. The latter is
derived from the equation of motion M
2u=dE/du, where E
is the total energy of the graphene sheet per carbon atom. In
the harmonic approximation, E=
u2/2, where
=1.02
⫻104eV/nm2is the effective force constant for the A1
⬘K
共K⬘兲point phonon mode. The electronic contribution to Eat
T=0 K is given by the integral of the band energy of the
valence electrons over the 2D BZ of the graphene sheet.
Formation of the dynamical band gap of width Eglowers the
band energy of the valence
electrons and reduces E.By
approximating the valence
-band dispersion around the K
共K⬘兲point with a cone and integrating it over the 2D BZ of
the graphene sheet, we express the change in Eat T=0 K in
the following form:
⌬E=2冑3a2
16
2
冕
0
2
d
冕
0
kBZ
kdk关Ev共Eg兲−Ev共0兲兴,共11兲
where a factor of 2 stands for the Kand K⬘points, a circle of
radius kBZ =2
1/23−1/4a−1 bounds a half of the 2D BZ,
Ev共Eg兲=−冑3t2k2a2
4+Eg
2共12兲
is the valence
-band energy when there is a band gap of
magnitude Eggiven by Eq. 共10兲, while Ev共0兲is the case with
no phonon perturbation. Upon performing the integration in
Eq. 共11兲and keeping only the leading term in Eg/t, we obtain
⌬E=− Eg
2
2
1/231/4t.共13兲
The total energy is then given by
E=
冋
−
冉
1−3q
˜
a
2
冊
2
册
u2
2,共14兲
where
=−36
␣
2
−1/23−1/4t−1 =2.04⫻104eV/ nm2and q
˜
=qK
共q
˜
=qK⬘兲for the iTO A1
⬘K共K⬘兲point phonon mode. The
phonon frequency is expressed accordingly:
FIG. 3. 共a兲The electronic dispersion relations of an ideal
graphene sheet 共dashed curves兲and the graphene sheet distorted by
the A1
⬘Kpoint phonon mode at T=300 K 共solid curves兲calculated
within the STB 共black curves兲and ETB 共gray curves兲models. 共b兲
The phonon-dispersion relations of the graphene sheet calculated
within the ETB model 共Refs. 9and 20兲at T=0 K 共gray curves兲and
from Eq. 共15兲共solid black curves兲. The dashed black line shows the
leading term in Eq. 共15兲共with
set to zero兲.
SAMSONIDZE et al. PHYSICAL REVIEW B 75, 155420 共2007兲
155420-4
=冑1
M
冋
−
冉
1−3q
˜
a
2
冊
2
册
.共15兲
In the vicinity of the Kpoint, q
˜
aⰆ1, the leading term of Eq.
共15兲takes the form
=冑
−
M+冑
M
3q
˜
a
2
,共16兲
taking into account that
Ⰶ
. The Kohn anomaly thus ex-
hibits a linear dispersion around the K共K⬘兲point.4
The coefficient
in Eq. 共14兲is calculated analytically by
approximating the valence
band with Eq. 共12兲. However,
the valence
band starts to deviate from Eq. 共12兲away from
the K共K⬘兲point. The valence
bands also give a nonvan-
ishing contribution to
. By performing the numerical inte-
gration of the ETB valence
-band dispersion distorted by
the A1
⬘Kpoint phonon mode over the 2D BZ of the graphene
sheet, we find
=0.23⫻104eV/ nm2. The phonon-dispersion
relations of the graphene sheet calculated within the ETB
model9,20 and those given by Eq. 共15兲are shown in Fig. 3共b兲
by gray and black curves, respectively. The leading term of
Eq. 共15兲共with
set to zero兲is shown in Fig. 3共b兲by a dashed
line.
In a similar fashion, the E2g⌫point phonon mode in the
graphene sheet exhibits a Kohn anomaly4driven by the os-
cillations of kF共kF
⬘兲as described by Eq. 共9兲. There is no
simple analytical expression for the dispersion of the dis-
torted valence
band around the K共K⬘兲point, analogous to
Eq. 共12兲involving the dynamical band gap. We thus perform
the numerical integration of the ETB valence
-band dis-
persion distorted by the E2g⌫point phonon mode over the
2D BZ of the graphene sheet. This yields Eand
in the
form of Eqs. 共14兲–共16兲with
=1.31⫻104eV/ nm2and
=0.07⫻104eV/ nm2. The Kohn anomaly around the ⌫
point is indeed seen in the phonon-dispersion relations of the
graphene sheet calculated elsewhere.4,9,18 Note that the oscil-
lations of kF共kF
⬘兲only lower Edue to the two dimensionality
of reciprocal space. As follows from the ETB numerical cal-
culations, the softening of the E2g⌫point phonon mode is
dominated by the valence
-band states away from the K
共K⬘兲point in the 2D BZ of the graphene sheet.
The Kohn anomalies at the ⌫and K共K⬘兲points in the 2D
BZ of the graphene sheet are governed by the electronic
contribution to the total energy E, which in turn depends on
the doping level. As the Fermi level EFis moved into the
valence or conduction band, the dynamical band gap Egin-
duced by the A1
⬘K共K⬘兲point phonon mode has less contri-
bution to E, or in other words,
in Eqs. 共14兲–共16兲decreases,
so that the Kohn anomaly at the K共K⬘兲point is smeared out.
On the other hand, the oscillations of kF共kF
⬘兲induced by the
E2g⌫point phonon mode contribute to Eregardless of EF,so
that the Kohn anomaly at the ⌫point is not affected by EF.
Surely, the Kohn anomalies at the ⌫and K共K⬘兲points are
formed by the valence
-band states away from and close to
the K共K⬘兲point in the 2D BZ of the graphene sheet, respec-
tively. This is illustrated in Fig. 4共a兲, where we plot the fre-
quencies of the A1
⬘Kand E2g⌫point phonon modes as a
function of the doping level calculated within the ETB
framework. While the former frequency increases with
changing the doping level, the latter stays constant. However,
recent experiments on a graphene sheet show that the fre-
quency of the E2g⌫point phonon mode also increases by
changing the doping level.21 This behavior is attributed to
breaking the Born-Oppenheimer approximation.21–24 The lat-
ter is implicit in our ETB calculations, and so the frequency
of the E2g⌫point phonon mode in Fig. 4共a兲is independent
of the doping level. Once the Born-Oppenheimer approxima-
tion is broken, the electronic contribution to Eand, conse-
quently the frequency of the E2g⌫point phonon mode in-
crease by changing the doping level, as shown elsewhere.21
Note that a similar increase in the frequency of the A1
⬘K
point phonon mode shown in Fig. 4共a兲is induced by the
dynamical band gap opening and is not affected by breaking
the Born-Oppenheimer approximation.
The dynamical band gap Eginduced by the A1
⬘K共K⬘兲
point phonon mode in the graphene sheet gives rise to large
anharmonic terms proportional to u3and u4in the total en-
ergy Eof Eq. 共14兲. As shown in Fig. 4共b兲, the frequency of
the A1
⬘K共K⬘兲point phonon mode calculated within the ETB
framework has a strong dependence on the amplitude of the
phonon displacements u. In contrast, the frequency of the E2g
⌫point phonon mode is independent of u, according to Fig.
4共b兲, even though the E2g⌫point phonon mode undergoes a
Kohn anomaly. The anharmonicity suggests the importance
of the A1
⬘Kpoint phonon mode for thermal expansion and
thermal conductivity in the graphene sheet. A more formal
treatment of vibrational anharmonicity in the graphene sheet
requires calculation of the phonon-phonon scattering matrix
elements, which is beyond the scope of this paper.
In the case of metallic SWNTs, the LO E2g⌫and iTO A1
⬘
K共K⬘兲point phonon modes open a dynamical band gap or
induce a mini-band-gap oscillation at the K共K⬘兲point, ac-
cording to Sec. II, resulting in Kohn anomalies in the
phonon-dispersion relations at the ⌫and K共K⬘兲points. By
analogy with Eq. 共11兲for the graphene sheet, the variation of
the total energy Eat T=0 Kis obtained by integrating the
valence metallic
subbands:
FIG. 4. The frequencies of the E2g⌫and A1
⬘Kpoint phonon
modes in the graphene sheet calculated within the ETB framework
as functions of 共a兲doping level and 共b兲atomic displacement. The
frequency dependence on 共a兲doping and 共b兲displacement arises
from 共a兲the dynamical band gap Egand 共b兲anharmonicity in the
total energy E, which is in turn attributed to Eg.
ELECTRON-PHONON COUPLING MECHANISM IN TWO-…PHYSICAL REVIEW B 75, 155420 共2007兲
155420-5
⌬E=T
4
N2
冕
−
/T
/T
dk关Ev共Em+Eg兲−Ev共Em兲兴,共17兲
where Tis the length of the translational unit cell, Nis the
number of hexagons in the translational unit cell, Emis the
mini-band-gap, Egis given by Eq. 共10兲, and Evis the same as
Eq. 共12兲. Integration of Eq. 共17兲yields
⌬E=tT
冑3
Na
冋
−F
冉
Em+Eg
t,冑3
a
2T
冊
+F
冉
Em
t,冑3
a
2T
冊册
,
共18兲
where we define the following function:
F共⌬,K兲=
冕
−K
K冑x2+⌬2dx
=K冑K2+⌬2+⌬2
2关ln共冑K2+⌬2+K兲
−ln共冑K2+⌬2−K兲兴.共19兲
The mini-band-gap Emin Eq. 共18兲is zero for metallic
armchair SWNTs and is on the order of room temperature
T=300 K for mini-band-gap semiconducting chiral and zig-
zag SWNTs.25 Upon expanding Eqs. 共18兲and 共19兲in a
power series in Eg/tup to the second order for mini-band-
gap semiconducting chiral and zigzag SWNTs, we find that
the total energy is expressed by Eq. 共14兲with different coef-
ficients
and
for each 共n,m兲SWNT. For metallic armchair
SWNTs, however, the expansion of Eqs. 共18兲and 共19兲con-
tains a logarithmic term:
⌬E=tT
冑3
Na
冉
−Eg
2
2t2+Eg
2
t2ln Eg
t
冊
.共20兲
The total energy is then given by
E=
冠
−6冑3T
␣
2
Nat
冉
1−3q
˜
a
2
冊
2
⫻
再
1−2ln
冋
6
␣
u
t
冉
1−3q
˜
a
2
冊
册
冎
冡
u2
2,共21兲
where q
˜
=qand q
˜
=qK共q
˜
=qK⬘兲for the LO E2g⌫and iTO A1
⬘
K共K⬘兲point phonon modes. Once again, the nonlinearity of
the electronic dispersion away from the K共K⬘兲point and the
contribution of the valence nonmetallic
and
subbands to
the total energy influence the numerical coefficients in Eq.
共21兲. The numerical integration of the ETB valence
-band
dispersion over the 1D BZ of SWNTs yields
E=
再
+
冉
1−3q
˜
a
2
冊
2
ln
冋
6
␣
u
t
冉
1−3q
˜
a
2
冊
册
冎
u2
2,共22兲
where coefficients
and
are different for each 共n,m兲
SWNT. The phonon frequency
is not simply expressed by
the second derivative of Eq. 共22兲because of its nonanalytic
dependence on u. A detailed consideration of the lattice dy-
namics yields
=冑1
M
冉
+
ln 3q
˜
a
2
冊
.共23兲
Taking into account the inequality
Ⰶ
, the leading term of
Eq. 共23兲takes the following form:
=冑
M+1
2冑
Mln 3q
˜
a
2
.共24兲
The LO E2g⌫and iTO A1
⬘K共K⬘兲point phonon modes thus
exhibit a logarithmic divergence6,8,9,26 for metallic armchair
SWNTs, which in turn gives rise to the static Peierls distor-
tions at low T.7On the other hand, the iTO E2g⌫point
phonon mode that causes oscillations of kFand kF
⬘or the
band edges along the momentum quantization lines does not
induce Kohn anomalies in metallic armchair SWNTs. We
omit the analytical integration because of the complexity of
the expression for the distorted band structure. However, the
numerical integration of the distorted band structure with the
displaced kFand kF
⬘shows that the total energy of the 1D
system is independent of the distortion, while the total en-
ergy of the 2D system shows a quadratic dependence with
the distortion amplitude. The iTO E2g⌫point phonon mode
thus exhibits a Kohn anomaly in the graphene sheet but not
in metallic armchair SWNTs.
The numerical integration of the ETB valence
-band
dispersion over the 1D BZ of the 共7,7兲SWNT yields
=0.98⫻104eV/ nm2and
=0.27⫻104eV/ nm2for the LO
E2g⌫point phonon mode, while
=0.76⫻104eV/ nm2and
=0.32⫻104eV/ nm2for the iTO A1
⬘K共K⬘兲point phonon
mode. The phonon-dispersion relations of the 共7,7兲SWNT
calculated within the ETB model and those given by Eq. 共23兲
with the aforementioned coefficients
and
are shown in
Fig. 5.
IV. SUMMARY
In summary, we analyze the electron-phonon coupling in
a graphene sheet and in metallic SWNTs by combining GT
with a tight-binding approach. While most of the phonon
FIG. 5. The phonon-dispersion relations of the 共7, 7兲SWNT
calculated within the ETB model 共Refs. 9and 20兲at T=0 K 共gray
curves兲and from Eq. 共24兲共solid black curves兲. The dashed black
line shows the leading term in Eq. 共24兲共with
set to zero兲.
SAMSONIDZE et al. PHYSICAL REVIEW B 75, 155420 共2007兲
155420-6
modes in the graphene sheet induce oscillations of the Fermi
points in the first BZ, the highest-frequency phonon mode at
the Kpoint opens a dynamical band gap at EF. Both the
Fermi point oscillation and the dynamical band gap opening
give rise to Kohn anomalies in the phonon spectrum of the
graphene sheet, while the dynamical band gap opening also
yields strong anharmonic effects. Similar phenomena take
place in metallic SWNTs, except that both Kohn anomalies
are induced by the dynamical band gaps and not by the
Fermi point oscillations. In metallic armchair SWNTs, the
dynamical band gap results in a logarithmic divergence of
the phonon frequencies and in static Peierls deformations at
low T. The dynamical band gap opening discussed in this
paper is equivalent to the electron-phonon scattering at the
Fermi surface reported in the literature.4,9
ACKNOWLEDGMENTS
G.G.S. and E.B.B. thank A. Jorio and L.G. Cançado for
helpful discussions about the GT of the Kpoint. The MIT
authors acknowledge financial support under NSF Grant No.
DMR 04-05538. E.B.B. acknowledges support from CAPES,
Brazil. R.S. acknowledges a Grant-in-Aid 共No. 16076201兲
from the Ministry of Education, Culture, Sports, Science and
Technology, Japan.
APPENDIX: THE CASE OF A GENERAL
PHONON WAVE VECTOR
As the phonon wave vector qvaries from the ⌫point to
the K共K⬘兲point, the directions of the atomic displacements
uAjand uBjgradually change from those shown in Fig. 2共a兲
to the ones in Fig. 2共c兲This gradual change is illustrated in
Fig. 6. While Figs. 6共a兲and 6共d兲are, respectively, identical
to Figs. 2共a兲and 2共c兲, Figs. 6共b兲and 6共c兲correspond to some
intermediate wave vectors along the ⌫Kdirection. The direc-
tions of uAjand uBjin Figs. 6共b兲and 6共c兲are defined by
angles
=qa/2 and
K=qKa/2, given the rotation of uAjand
uBjfrom Fig. 6共a兲to Fig. 6共d兲by angle 2
/3 and the dis-
tance of 4
/共3a兲between the ⌫and Kpoints.
The Hamiltonian of the graphene sheet distorted by the
phonon mode in the vicinity of the ⌫point is obtained upon
substituting the atomic displacements uAjand uBjshown in
Fig. 6共b兲into Eq. 共2兲:
HAA =E0+
再
2u−2
冋
1
2+ cos
冉
3−
冊
册
u
冎
,
HAB =关t+2
␣
u兴exp关ikx共aCC +2u兲兴
+2
冋
t−
␣
冉
1
2+ cos
冉
3−
冊冊
u
册
exp
冋
ikx
冉
−aCC
2+2u
冊
册
⫻cos
冋
ky
冉
−冑3aCC
2−
u
冊册
.共A1兲
In the vicinity of the ⌫point, qⰆ4
/共3a兲and thus
Ⰶ
.
Also taking into account the inequality uⰆaCC, Eq. 共A1兲can
be linearized:
HAA =E0−冑3
u,
HAB =关t共1+2ikxu兲+2
␣
u兴exp关ikxaCC兴
+2关t共1+2ikxu兲−
␣
u兴exp
冋
−ikxaCC
2
册
cos
冋
冑3kyaCC
2
册
+2t
kyuexp
冋
−ikxaCC
2
册
sin
冋
冑3kyaCC
2
册
.共A2兲
Upon substituting Eq. 共A2兲into Eq. 共1兲and setting its deter-
minant to zero, we find the Fermi point near the Kpoint in
the form kFx =⌬kFx and kFy =−4
/共3a兲+⌬kFy, where ⌬kFx
and ⌬kFy are given by Eq. 共9兲.
In a similar fashion, the 6⫻6 Hamiltonian of the
graphene sheet distorted by the phonon mode in the vicinity
of the K共K⬘兲point is constructed using the atomic displace-
ments uAjand uBjshown in Fig. 6共c兲. To derive the magni-
tude of the dynamical band gap, it is essential to consider the
6⫻6 Hamiltonian at k= 0, by analogy with Eqs. 共4兲and 共6兲.
The 6⫻6 Hamiltonian at k= 0 can be linearized with respect
to
KⰆ
and uⰆaCC in the same way as Eq. 共A2兲. Finally,
we obtain
HAA =HBB =
冢
E0+冑3
2
Ku00
0E0−冑3
Ku0
00
E0+冑3
2
Ku
冣
,
FIG. 6. The arrows show directions of the atomic displacements
for the highest-frequency optical phonon mode of the graphene
sheet 共a兲at the ⌫point, 共b兲along the ⌫Kdirection near the ⌫point,
共c兲along the ⌫Kdirection near the Kpoint, and 共d兲at the Kpoint.
Here, 共a兲and 共d兲are equivalent to Figs. 2共a兲and 共c兲, respectively.
The angles indicated in 共b兲and 共c兲are given by
=qa/2 and
K
=qKa/2.
ELECTRON-PHONON COUPLING MECHANISM IN TWO-…PHYSICAL REVIEW B 75, 155420 共2007兲
155420-7
HAB =HBA =
冢
t+2
␣
ut−
␣
冉
1+冑3
2
K
冊
ut−
␣
共1−冑3
K兲u
t−
␣
冉
1+冑3
2
K
冊
ut+2
␣
ut−
␣
冉
1+冑3
2
K
冊
u
t−
␣
共1−冑3
K兲ut−
␣
冉
1+冑3
2
K
冊
ut+2
␣
u
冣
.共A3兲
Upon setting the determinant of the Hamiltonian given by
Eq. 共A3兲to zero, we find the magnitude of the dynamical
band gap in the form of Eq. 共10兲.
It should be pointed out that the directions of the atomic
displacements in Figs. 6共b兲and 6共c兲are rotated by an integer
number of angles
and 2
/3−
K, respectively, when mov-
ing to different unit cells in the graphene sheet. For these unit
cells, the Hamiltonians can be constructed by analogy with
Eqs. 共A1兲–共A3兲. Upon diagonalizing these Hamiltonians, one
obtains ⌬kF共⌬kF
⬘兲and Egthat only differ from Eqs. 共9兲and
共10兲in the second order with respect to u/aCC,
/
, and
K/
, in accordance with the linear-response method.9,16
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SAMSONIDZE et al. PHYSICAL REVIEW B 75, 155420 共2007兲
155420-8
