Download Optical Phonon Interacting with Electrons in Carbon Nanotubes

Optical Phonon Interacting with Electrons
in Carbon Nanotubes
Kohta ISHIKAWA and Tsuneya ANDO
Department of Physics, Tokyo Institute of Technology
2–12–1 Ookayama, Meguro-ku, Tokyo 152-8551
Effects of interactions with electrons on optical phonons are studied in an effective-mass
approximation. The longitudinal mode with displacement in the axis direction is lowered in its
frequency, while the transverse mode with displacement in the circumference direction is raised, in
metallic nanotubes. The shifts are opposite but their absolute values are smaller in semiconducting
nanotubes. Only the longitudinal mode has a considerable broadening in metallic nanotubes. In
the presence of an Aharonov-Bohm magnetic flux, the broadening appears for the transverse mode
and diverges when the induced gap becomes the same as the frequency of the optical phonon.
Keywords: carbon nanotube, graphite, optical phonon, softening, hardening, effective-mass theory,
Aharonov-Bohm effect
§1. Introduction
Carbon nanotubes are quasi-one-dimensional materials made of sp2 -hybridized carbon networks.1) The
electronic states change from metallic to semiconducting depending on the tubular circumferential vector
characterizing a nanotube. The characteristic properties were predicted by calculations in tight-binding
models2−11) and also in a k·p scheme or an effective-mass
approximation.12−14) Electron-phonon interactions can
play an important role in the behavior of optical phonons
in carbon nanotubes. In fact, recent first-principles calculations reported lowering of phonon energy compared
to that in the two-dimensional graphite.15−17) The purpose of this paper is to study effects of electron-phonon
interactions on long-wavelength optical phonons based
on the k·p scheme.
Although electronic properties have been understood by those of the graphite plane using a periodic
boundary condition, the phonon modes of nanotubes are
not simply given by the zone-folded modes of planes
because they fail to give breathing modes.7) In a previous work, a continuum model suitable for a correct
description of long-wavelength acoustic phonons was
constructed.18) In this paper we shall introduce a similar
continuum model of optical phonons and derive the
Hamiltonian for electron-phonon interactions. Then, we
calculate the self-energy of phonon Green’s function in
the lowest order approximation. The real part of the
self-energy gives an energy shift and the imaginary part
provides a lifetime.
As is expected, the most important feature comes
from the bands in the vicinity of the Fermi level. In
metallic nanotubes a narrow gap can open due to effects of a finite curvature2,14,19,20) and possible lattice
strain.14,18,19,21−24) This gap can be controlled by a magnetic flux passing through the cross section due to the
Aharonov-Bohm effect.14,25,26) Recently, splitting of optical absorption and emission peaks due to the AharonovBohm effect was observed.27,28) It will be shown that this
small gap can manifest itself as considerable broadening
of optical phonon.
This paper is organized as follows: In §2, the k·p
Submitted to Journal of Physical Society of Japan
scheme for the description of energy bands is reviewed
very briefly and a continuum model of optical phonons is
introduced. The phonon Green’s function is calculated
and shifts and broadening of phonon modes are discussed
in §3. The results are discussed in §4 and a short
summary is given in §5.
§2. Formulation
2.1 Effective-mass description
Figure 1 shows the lattice structure of the twodimensional graphite, the first Brillouin zone, and the
coordinate system in nanotubes. In a graphite sheet
the conduction and valence bands consisting of π orbitals cross at K and K’ points of the Brillouin zone,
where the Fermi level is located.29,30) Electronic states
of the π-bands near a K point are described by the k·p
equation:12,14,31,32)
H0 F (r) = εF (r),
with
H0 = γ
0
k̂x +ik̂y
k̂x −iky
0
(2.1)
= γ(σx k̂x +σy k̂y ), (2.2)
where γ is a band parameter, σx and σy are the Pauli
spin matrices, and k̂ = (k̂x , k̂y ) = −i∇ is a wave-vector
operator.
The structure of a nanotube is specified by a chiral
vector L corresponding to the circumference as shown in
Fig. 1(a). It is written as
L = na a+nbb,
(2.3)
in terms of two integers na and nb , where a and b are
the primitive translation vectors of a graphite sheet with
|a| = |b| = a (= 2.46 Å). In the following we shall choose
the x axis in the circumference direction and the y axis
in the axis direction, i.e., L = (L, 0), where L is the
circumference. The angle η between L and the horizontal
axis is called the chiral angle.
Electronic states of a nanotube with a sufficiently
large diameter are obtained by imposing the boundary
Page 2
K. Ishikawa and T. Ando
conditions around the circumference direction:12,14)
ν ,
(2.4)
F (r+L) = F (r) exp 2πi ϕ−
3
where ϕ = φ/φ0 with φ being a magnetic flux passing
through the cross section and φ0 being the flux quantum
given by φ0 = ch/e, and ν is an integer (ν = 0 or ±1)
determined by
na +nb = 3M +ν,
K2 are the force constants for the bond stretching and
bond-angle change, respectively. The resulting phonons
are isotropic within the plane of the two-dimensional
graphite.
The longitudinal phonon has the frequency ωl (|q|)
and the eigen vector el (q) given by
ωl (|q|)2 = ω02 (1−η1 a2 |q|2 ),
i
qx
el (q) =
,
|q| qy
(2.5)
with integer M . Metallic and semiconducting nanotubes
correspond to ν = 0 and ±1, respectively.
The energy bands are specified by s = ±1 (s = −1
and +1 for the valence and conduction band, respectively), integer n corresponding to the discrete wave
vector along the circumference direction, and the wave
vector k in the axis direction. The wave function for a
band associated with the K point is written as
1
bνϕ (n, k)
Fs,n,k (x, y) = √
exp[iκνϕ (n)x+iky]
,
s
2AL
(2.6)
where A is the length of the nanotube,
ν
2π κνϕ (n) =
n+ϕ− ,
(2.7)
L
3
(2.13)
with
ω02 =
3(K1 +6K2 )
,
M
(2.14)
η1 =
1 K1 −12K2
.
8 M ω02
(2.15)
and
The transverse phonon has the frequency ωt (|q|) and the
eigen vector et (q) given by
ωt (|q|)2 = ω02 [1−(η1 +η2 )a2 |q|2 ],
i
qy
et (q) =
,
|q| −qx
(2.16)
with
and
κνϕ (n)−ik
bνϕ (n, k) = .
κνϕ (n)2 +k 2
The corresponding energy is given by
εsνϕ (n, k) = sγ κνϕ (n)2 +k 2 .
(2.8)
(2.9)
For the K’ point, the k·p Hamiltonian is obtained by
replacing k̂y with −k̂y in eq. (2.2) and the boundary
conditions by replacing ν with −ν. Therefore, the energy
band is given by eq. (2.9) in which κνϕ (n, k) is replaced
with κ−νϕ (n, k), and the wave function is given by eq.
(2.6) in which bνϕ (n, k) is replaced with b−νϕ (n, k)∗ .12,14)
2.2 Long Wavelength Optical Phonon
An equation of motion for optical phonons of the
two-dimensional graphite in the long wavelength limit
has been derived previously based on a valence-force-field
model as 18)
M ω(q)2 u(q) = Hop (q)u(q),
(2.10)
where q is the wave vector and u = (ux , uy ) is the relative
displacement of two sub-lattice atoms A and B, defined
by
1
u(q) = √ [uA (q)−uB (q)].
(2.11)
2
The effective Hamiltonian is given by
1 0
Hop (q) = 3(K1 +6K2)
0 1
2
K
3K2 2
qx
qx qy
1
+ −
+
a
(2.12)
qy qx
qy2
8
2
2 2
3 K1 +3K2
qx +qy
0
−
K 2 a2
,
0
qx2 +qy2
2 K1 +6K2
where M is the mass of a carbon atom and K1 and
η2 =
3 3K1 +3K2 K2
.
2 2K1 +6K2 M ω02
(2.17)
The dispersion of the optical phonons is small for |q|a 1 and therefore will be neglected completely in the
following.
In the representation of the second quantization, the
phonon displacement u(r) is given by
h̄
u(r) =
(ag,q + a†g,−q )eg (q)eiq·r , (2.18)
2N
M
ω
0
g,q
where N is the number of unit cells, g denotes the
modes (t for transverse and l for longitudinal), and
a†g,q and ag,q are the creation and destruction operators,
respectively. In carbon nanotubes, the wave vector in
the circumference direction becomes discrete, i.e., qx =
(2π/L)j with integer j, and that in the axis direction
remains continuous, i.e., qy = q.
In the following, we shall confine ourselves to the
long wavelength limit, j = 0 and |q|a 1, i.e., q = (0, q).
In this case we have
0
i
,
et (q) =
,
(2.19)
el (q) =
i
0
which means that the longitudinal mode has lattice displacement along the axis y direction and the transverse
mode along the circumference x direction.
2.3 Electron-Phonon Interaction
The optical phonon distorts the distance between
neighboring carbon atoms and therefore modifies the
band structure through the change in the resonance integral between carbon atoms. The corresponding effective
Hamiltonian can be obtained easily in a manner similar
to the case of acoustic phonons. Details are discussed
Optical Phonon in Carbon Nanotubes
in Appendix A. When we use a simple nearest-neighbor
tight-binding model, the interaction Hamiltonian for the
K point is given by
√ βγ
0
uy (r)+iux(r)
K
Hint = − 2 2
, (2.20)
uy (r)−iux(r)
0
b
and for the K’ point
√ βγ
0
uy (r)−iux(r)
K
=− 2 2
, (2.21)
Hint
uy (r)+iux(r)
0
b
√
where b = a/ 3 is the equilibrium bond length, γ0 is
the resonance integral between nearest neighbor carbon
atoms appearing
√ in a tight-binding model related to γ
through γ = ( 3a/2)γ0 , and
β=−
d ln γ0
.
d ln b
(2.22)
This means that the lattice distortion gives rise to a shift
in the origin of the wave vector, i.e., ux in the y direction
and uy in the x direction.
We have derived the optical phonons starting with
the simplest valence-force-field model and the effective
Hamiltonian describing their interaction with electrons
in the two-dimensional graphite. It should be noticed,
however, that the phonon modes are exact in the longwavelength limit if we use an appropriate value of the
frequency ω0 and also that the explicit form of the
obtained Hamiltonian is much more general and is valid
although the coupling parameter β can be different from
that estimated above.
2.4 Phonon Green’s Function
The phonon Green’s function D(q, iωl ) is given by
D(q, iωl ) =
2ω0
,
2
2
(iωl ) −ω0 −2ω0 Π(q, iωl )
(2.23)
where Π(q, iωl ) is the self energy and ωl = (2πkB T /h̄)l
is the Matsubara frequency with integer l. We shall
consider the lowest order self-energy given in Fig. 2. By
making an analytic continuation iωl → ω+i0, we have the
retarded Green’s function
Dret (q, ω) =
2ω0
.
(ω+i0)2 −ω02 −2ω0 Πret (q, ω)
(2.24)
Page 3
from states in the vicinity of the K point,
dk βγ 2 h̄
(q,
ω)
=
−2
ΠgK
ret
2π b2 N M ω0
n
s,s
×
ss [κνϕ (n)2 −k(k−q)]
1± 2
κνϕ (n)2 +k 2 κνϕ (n)2 +(k−q)2
×
f [εsνϕ (n, k)]−f [εsνϕ(n, k−q)]
,
h̄ω −εsνϕ(n, k)+εsνϕ
(n, k−q)+i0
1
(2.28)
where f (ε) is the Fermi distribution function and the
upper and lower sign correspond to g = l and t, respectively. The factor two comes from the electron spin. The
contribution from states in the vicinity of the K’ point
ΠgK
ret (q, ω) is obtained by appropriate replacement of the
wave vectors and the energies and the total self-energy
is given by the sum
gK
Πgret (q, ω) = ΠgK
ret (q, ω) + Πret (q, ω).
(2.29)
In this paper, we shall calculate the self-energy of
optical phonons starting with the known phonon modes
in the two-dimensional graphite. Therefore, the direct
evaluation of the above self-energy causes a problem
of double counting.33) In fact, if we apply the above
formula to the case of infinitely large circumference,
we get the frequency shift due to virtual excitations of
electrons in the π bands in the two-dimensional graphite.
However, this contribution is already included in the
definition of ω0 . In order to avoid such a problem, we
have to subtract the contribution in the two-dimensional
graphite, which is obtained by the sum over discrete wave
vector in the circumference direction κνϕ (n) replaced
with a continuous integration. Because the frequencydependence of the phonon self-energy is negligible, the ω
dependence can be ignored completely for this term.
Therefore, the final expression of the self-energy is
given by
ΠgK
ret (q, ω) = −2
s,s
n
1/2
dt
−1/2
dk βγ 2 h̄
2π b2 N M ω0
1
ss [κνϕ (n)2 −k(k−q)]
1± ×
2
κνϕ (n)2 +k 2 κνϕ (n)2 +(k−q)2
The phonon frequency is determined by the pole of
Dret (q, ω) as
ω 2
2
−1 =
ReΠret (q, ω)
(2.25)
ω0
ω0
As will become clear in the following, the phonon selfenergy is small. In this case, the shift of the phonon
frequency is given by
Δω = Re Πret (q, ω0 ),
(2.26)
and the broadening is given by
Γ = Im Πret (q, ω0 ).
(2.27)
A straightforward calculation gives the contribution
f [εsνϕ (n, k)]−f [εsνϕ(n, k−q)]
h̄ω −εsνϕ(n, k)+εsνϕ
(n, k−q)+i0
ss [κνϕ (n+t)2 −k(k−q)]
1
1± −
2
κνϕ (n+t)2 +k 2 κνϕ (n+t)2 +(k−q)2
f [εsνϕ (n+t, k)]−f [εsνϕ(n+t, k−q)]
×
.
(2.30)
−εsνϕ (n+t, k)+εsνϕ
(n+t, k−q)+i0
×
This self-energy shows that the renormalization of phonon
frequencies in nanotubes from the two-dimension graphite
is determined mainly by electronic states in the vicinity
of the Fermi level where the discreteness of the bands
plays important roles. The contributions from states
away from the Fermi level are the same between the
nanotube and the two-dimensional graphite and therfore
Page 4
K. Ishikawa and T. Ando
cancel each other in the self-energy.
§3. Frequency Shift and Lifetime
3.1 Long Wavelength Limit
In the following, we shall consider phonons with
q = 0 at zero temperature. Then, the self-energy is
simplified considerably
1/2 1
t
dt dk̃
Πret (ω) = − α(L)ω0
2
−1/2
n
(n+ϕ−ν/3)2
4
×
[(n+ϕ−ν/3)2 + k̃ 2 ]1/2 4[(n+ϕ−ν/3)2 + k̃ 2 ]− ω̃ 2
(n+t)2
−
+ (ν → −ν) ,
(3.1)
[(n+t)2 + k̃ 2 ]3/2
1/2 1
Πlret (ω) = − α(L)ω0
dt dk̃
2
−1/2
n
4
k̃ 2
×
2
2
1/2
[(n+ϕ−ν/3) + k̃ ]
4[(n+ϕ−ν/3)2 + k̃ 2 ]− ω̃ 2
k̃ 2
−
+ (ν → −ν) ,
(3.2)
[(n+t)2 + k̃ 2 ]3/2
where (ν → −ν) represents terms in which ν is replaced
with −ν, we have introduced the dimensionless quantities,
2πγ −1
2π −1
ω̃ = h̄ω ×
, k̃ = k ×
,
(3.3)
L
L
and α(L) is the coupling parameter defined by
a
α(L) = λ ,
L
(3.4)
with the dimensionless parameter in the two-dimensional
graphite, given by
λ=
h̄2 1 2
27 2
β γ0
.
π
2M a2 h̄ω0
(3.5)
For the parameter h̄ω0 = 0.196 eV corresponding to 1583
cm−1 , γ0 = 2.63 eV, and β = 2, we have λ = 0.08.
The above shows that the self-energy depends on the
circumference L or the diameter d = L/π only through
the universal form given by a/L and is independent of the
chirality or the individual values of na and nb . Further,
except in extremely thick nanotubes, the phonon energy
h̄ω0 is much smaller than the typical electronic energy
2πγ/L of the order of band gaps, i.e., ω̃ 1. In the
following, we shall obtain analytic expressions of the
phonon self-energy using this fact explicity.
3.2 Metallic Nanotubes
First, we consider a metallic nanotube in the absence of an Aharonov-Bohm magnetic flux, i.e, ν = 0 and
ϕ = 0. Expanding the terms with n = 0 in terms of ω̃ in
eqs. (3.1) and (3.2), we immediately obtain
π
Πt (ω) = α(L)ω0 2− ω̃ 2 ,
9
2π ω̃ π 2 2
Πl (ω) = α(L)ω0 2 ln
− ω̃ +iπ .
e
18
2
(3.6)
(3.7)
Because ω̃ 1, the frequency is shifted to the higher
energy side and no imaginary part appears in the case
of the transverse mode. In the case of the longitudinal
mode, on the other hand, the frequency is shifted to the
lower energy side and exhibits a logarithmic divergence
for small ω̃. Further, the self-energy has an imaginary
part comparable or larger than the real part, meaning
that the phonon has a finite life time due to spontaneous
emission of an electron-hole pair. This width is qualitatively in agreement with that obtained previously.17)
The effective Hamiltonian describing the electronphonon interaction shows that the lattice displacement
ux for the transverse mode causes a shift in the wave
vector along the axis direction. This shift does not give
rise to any change in the energy of the metallic linear
bands n = 0. The frequency lowering due to interactions
with electrons with wave vector close to kx = 0 (but not
exactly kx = 0) present in the two-dimensional graphite
disappears in nanotubes because of quantization of kx
into multiples of 2π/L. This quantization is the origin of
the frequency increase obtained for the transverse mode.
In the case of the longitudinal mode, the displacement uy causes a shift in the wave vector along the
circumference direction, leading to opening of a band gap
reducing the energy of electrons in the metallic linear
bands. As a result, the phonon frequency is lowered
and at the same time the imaginary part appears due
to excitations of electrons in the linear bands. The logarithmic divergence for ω̃ = 0 corresponds to an instability
toward a spontaneous lattice deformation accompanying opening of a small gap obtained in the adiabatic
approximation.3,33−35) This instability is known to be
unimportant except in extremely thin nanotubes.33)
Because of the logarithmic divergence for the longitudinal mode, the phonon Green’s function gives a pole
in the vicinity of zero frequency. In fact (2.25) has a pole
at
1 eγ
exp −
,
(3.8)
ω∗ =
L
4α(L)
which is extremely small for conventional single-wall
nanotubes L/a >
∼ 10. Its spectral weight, i.e., the integral
of the imaginary part of the Green’s function around
ω ∗ , is calculated as πω ∗ /α(L)ω0 . This turns out to be
negligibly small.
Figure 3 shows the frequency shift of the transverse
and longitudinal modes as a function of the circumference. The dependence of these shifts on the circumference is determined essentially by the effective coupling
parameter α(L) = (a/L)λ. For the longitudinal mode,
the broadening is comparable or even larger than the
shift.
3.3 Semiconducting Nanotubes
In semiconducting nanotubes, to the leading order
in ω̃, we have
4π 2 ω̃ 2 ,
Πtret (ω) = α(L)ω0 −
9
1
2π 2 2 Πlret (ω) = α(L)ω0
ln 3 −
ω̃ .
2
9
(3.9)
(3.10)
This shows that in contrast to the case of metallic
nanotubes, the transverse mode is shifted to the lower
Optical Phonon in Carbon Nanotubes
energy side and the longitudinal mode is shifted to
the higher energy side. However, the amount of the
shifts is very small. Figure 3 contains also the shifts
in semiconducting nanotubes.
3.4 Aharonov-Bohm Effect
In metallic nanotubes with an Aharonov-Bohm
magnetic flux ϕ, we have
8ϕ2
ω̃
+2
Πtret (ω) = α(L)ω0 − sin−1
2
2
2|ϕ|
ω̃ 4ϕ − ω̃
1 π2
1 2
−
ω̃ ,
(3.11)
−
3 sin2 πϕ ϕ2
for ω̃ < 2|ϕ|, and
Πtret (ω)
ω̃+ ω̃ 2 −4ϕ2 = α(L)ω0 ln +2
2
2
2
2
ω̃ ω̃ −4ϕ
ω̃− ω̃ −4ϕ
1 π2
4πϕ2
1 −
, (3.12)
− 2 ω̃ 2 + i 2
3 sin πϕ ϕ
ω̃ ω̃ 2 −4ϕ2
4ϕ2
for ω̃ > 2|ϕ|. The real part diverges when ω̃ approaches
the gap 2|ϕ| due to the Aharonov-Bohm effect from the
high frequency side, while it converges a finite value from
the low frequency side. Therefore, the phonon Green’s
function has in principle an extra pole below 4πγ|ϕ|/L,
but its intensity is extremely small and can be neglected
practically.
The most notable feature is the appearance of the
imaginary part for 2|ϕ| < ω̃ giving rise to the broadening
of the phonon spectrum. With the increase of the gap
4πγ|ϕ|/L due to the flux, the imaginary part increases
in proportion to ϕ2 , diverges when the gap reaches
the optical-phonon energy h̄ω0 , and vanishes when the
gap exceeds the phonon energy. Figure 4 shows this
broadening as a function of the Aharonov-Bohm gap.
For the longitudinal mode, we have
2
2 sin πϕ 2 4ϕ − ω̃ 2
ω̃
sin−1
+ 2 ln Πlret (ω) = α(L)ω0
ω̃
2|ϕ|
e
1 π2
1 (3.13)
−
− 2 ω̃ 2 ,
2
6 sin πϕ ϕ
for ω̃ < 2|ϕ| and
Πlret (ω)
2
ω̃ −4ϕ2 ω̃ + ω̃ 2 −4ϕ2 ln = α(L)ω0
ω̃
ω̃ − ω̃ 2 −4ϕ2
2 sin πϕ 1 π 2
1 + 2 ln − 2 ω̃ 2
−
2
e
6 sin πϕ ϕ
2ϕ 2
+ iπ 1−
,
(3.14)
ω̃
for ω̃ > 2|ϕ|. The real part remains continuous and does
not exhibit divergence at ω̃ = 2|ϕ|. The most notable
feature lies in the broadening of the phonon spectrum
again. As shown in Fig. 4 the imaginary part gradually
decreases with the magnetic flux and vanishes when the
gap reaches the phonon energy h̄ω0 .
§4. Discussion
In the Raman spectra of single-wall nanotubes the
Page 5
so-called G band is usually fit with two components G+
and G− arising from transverse and longitudinal optical
phonons at the Brillouin-zone center. Semiconducting
tubes have sharp G+ and G− , while metallic tubes have a
broad down-shifted G− and a sharp G+ ,36−44) although
some experiments seem to show that the G− peak appears only in nanotube bundles.45) The G− band shows
a strong diameter dependence, being lower in frequency
for smaller diameters. This suggested its attribution
to a circumferential mode, whose atomic displacements
would be most affected by a variation in diameter. Thus,
the G+ and G− peaks were often assigned to longitudinal (axial) and transverse (circumferential) modes,
respectively.36,37,39−41) It was suggested also that the
G band can be understood in terms of double-resonant
scattering involving phonon wave vectors throughout the
whole Brillouin zone.46)
Optical phonon modes in nanotubes were studied
theoretically in various methods such as the zone-folding
scheme,47) force-constant models,48,49) first-principles
methods,15,16,50−55) and some combinations.17) In refs.
15 and 16, for example, the obtained longitudinal frequency is lowered considerably in metallic tubes, while
it is not shifted in semiconducting tubes. Further, the
transverse frequency is lowered but its amount is less
than the longitudinal mode in metallic nanotubes and
is almost independent of whether the tube is metallic
or semiconducting. In ref. 17, on the other hand, the
calculated longitudinal modes are lowered considerably
and the transverse modes remain near the frequency of
the zone-center phonon of the two-dimensional graphite
independent of whether the nanotube is metallic or semiconducting.
The considerable lowering of the longitudinal mode
in metallic nanotubes and the small lowering of the transverse mode in semiconducting nanotubes obtained in this
paper are qualitatively in agreement with the results
of refs. 15 and 16 although being different quantitatively. The present calculations show that the transverse
modes also become different between metallic and semiconducting nanotubes and the difference between the
longitudinal and transverse frequencies is much smaller
in semiconducting cases than in metallic cases. These
features are quite different from those of refs. 15 and 16
and of ref. 17.
Figure 5 shows the comparison of the present results
with the G+ and G− peaks experimentally obtained in
ref. 40. We have assumed ω0 = 1570 cm−1 and λ = 0.17,
corresponding to β slightly larger than 2. The amount of
the splitting of the G+ and G− peaks seems to be in good
agreement with the experiments for metallic nanotubes,
but is smaller than the experiments for semiconducting
nanotubes. The bond strength of neighboring carbon
atoms due to bonding through σ orbitals is likely to
be lowered in nanotubes due to nonzero curvature and
as a result the frequency of the optical mode itself can
be lowered from that of the two-dimensional graphite
depending on the diameter. The actual estimation of
such effect is out of the scope of this paper but the
renormalization of ω0 depending on the curvature has
Page 6
K. Ishikawa and T. Ando
a tendency to reduce the disagreement in particular for
nanotubes with a small diameter.
Figure 6 compares the width of the phonon spectrum with some of existing experiments for the G− peak.
In this comparison the parameters same as in Fig. 5 are
used for ω0 and λ. The broadening predicted theoretically has a right order of magnitude as that observed
experimentally.
§5. Summary and Conclusion
An effective Hamiltonian has been derived for describing the interaction between long-wavelength optical
phonons and electrons in carbon nanotubes within the
continuum approximation for phonons and the effectivemass approximation for electrons. It has been used for
the evaluation of the frequency shift and broadening of
phonons due to interactions with electrons. The results
are summarized as follows:
The longitudinal mode with displacement in the axis
direction is lowered in its frequency, while the transverse
mode with displacement in the circumference direction
is raised, in metallic nanotubes. The shifts are opposite
but their absolute values are smaller in semiconducting
nanotubes. These shifts increase almost in proportion
to the inverse of the diameter with the decrease of the
diameter.
The longitudinal mode has a considerable broadening or a spectral width due to interactions with electronic
excitations. For the transverse mode, the broadening
appears in the presence of an Aharonov-Bohm magnetic
flux and diverges when the induced gap becomes the
same as the frequency of the optical phonon for the
transverse mode. For the longitudinal mode, on the other
hand, the broadening decreases gradually and disappears
as a function of the flux. Therefore, the Aharonov-Bohm
effect manifests itself also in optical phonons in metallic
carbon nanotubes
that of the carbon pz orbital. Then, we have
εψA (RA ) = −γ0
The authors thank Dr. Mikito Koshino for valuable
discussion. This work was supported in part by a 21st
Century COE Program at Tokyo Tech “Nanometer-Scale
Quantum Physics” and by Grant-in-Aid for Scientific
Research from Ministry of Education, Culture, Sports,
Science and Technology Japan.
Appendix A: Electron-Phonon Interaction
The method to obtain the effective Hamiltonian
for electron-phonon interaction is essentially the same
as that for acoustic phonons discussed in refs. 14 and
18. We shall start with a nearest-neighbor tight-binding
model and then obtain the effective-mass equation. The
resulting Hamiltonian itself is much more general, however.
Let −γ0 be the resonance integral between nearestneighbor carbon atoms and choose the energy origin at
ψB (RA −τl ),
l=1
εψB (RB ) = −γ0
3
(A1)
ψA (RB +τl ),
l=1
where RA and RB denote the positions of the A and
B site, respectively, and ψA (R) and ψB (R) are the
amplitude of the pz orbital, and τl (l = 1, 2, 3) is the
vector connecting neighboring carbon atoms as shown in
Fig. 1.
In the following we shall consider the coordinates
rotated by η. In the vicinity of the Fermi level ε = 0, we
can write
ψA (RA ) = eiK·RA FAK (RA ) + eiη eiK
·RB
ψB (RB ) = −ωeiη eiK·RB FBK (RB ) + eiK
FAK (RA ),
·RB
FAK (RB ),
(A2)
in terms of the slowly-varying envelope functions FAK ,
FBK , FAK , and FBK . This can be written as
ψA (RA ) = a(RA )† FA (RA ),
(A3)
ψB (RB ) = b(RB )† FB (RB ),
with
a(RA )† = ( eiK·RA
†
eiη eiK
iη iK·RB
b(RB ) = ( −ωe e
and
FA =
FAK
FAK
e
,
·RA
),
iK ·RB
),
FB =
FBK
FBK
.
(A4)
(A5)
Introduce a smoothing function g(r) varying smoothly
in the range |r| <
∼ a and decaying rapidly for |r| a. It
should satisfy the condition
g(r−RA ) =
g(r−RB ) = 1,
(A6)
RA
and
Acknowledgments
3
RB
g(r)dr = Ω0 ,
(A7)
where
√ 2 Ω0 is the area of a unit cell given by Ω0 =
3a /2. This function g(r) can be replaced by a delta
function when it is multiplied by a smooth function such
as envelopes, i.e., g(r) ≈ Ω0 δ(r). In order to obtain
equations for F , we first substitute eq. (A3) into eq. (A1).
Multiply the first equation by g(r−RA )a(RA ) and then
sum it over RA . With the full use of the slowly-varying
nature of the envelope function, the equation is obtained
as14)
k̂x −ik̂y
0
εFA (r) = γ
(A8)
FB (r).
0
k̂x +ik̂y
The lattice displacement corresponding to optical
phonons causes a change in the distance between nearest
neighbor carbon atoms and in the resonance integral.
Let uA (RA ) and uB (RB ) be a lattice displacement at
A and B site, respectively. Then the resonance integral
between an atom at RA and RA −τl changes from −γ0
Optical Phonon in Carbon Nanotubes
by the amount
∂γ0 (b) τl +uA (RA )−uB (RA −τl ) − b
∂b
∂γ0 (b) 1 =−
τl · uA (RA )−uB (RA −τl ) ,
(A9)
∂b b
√
where b = |τl | = a/ 3. Therefore, the extra term
appearing in the right hand side of eq. (A8) is calculated
as
∂γ (b) 0
g(r−RA )a(RA )b(RA −τl )† −
∂b
l RA
1 × τl · uA (RA )−uB (RA −τl ) FB (r)
b
−ωeiη e−iK·τl
∂γ0 (b) 0
−
=
0
e−iη e−iK ·τl
∂b
l
1
× τl · uA (r)−uB (r−τl ) FB (r).
(A10)
b
−
For optical phonons, we have
uA (r)−uB (r−τl ) ≈
√
2u(r),
(A11)
and therefore the extra term becomes
√ βγ uy (r)+iux(r)
0
− 2 2
FB (r).
0
−uy (r)+iux (r)
b
(A12)
A similar term can be derived from the second equation
of (A1), giving the effective Hamiltonian (2.20) and
(2.21).
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Figure Captions
Fig. 1 (a) A schematic illustration of the lattice structure of the two-dimensional graphite and the lattice
displacement for transverse and longitudinal optical
phonons. (b) The coordinate system in the nan-
otube with an Aharonov-Bohm magnetic flux. (c)
The first Brillouin zone and K and K’ points.
Fig. 2 A Feynman diagram for the self-energy for
optical phonons with a wave vector q along the axis
(y) direction.
Fig. 3 Calculated frequency shifts of the optical phonons.
The transverse and longitudinal modes are denoted
by TO and LO, respectively. The shadowed region
shows the broadening Γ. The longitudinal mode is
lowered in the frequency and has large broadening
due to interactions with electrons in metallic nanotubes.
Fig. 4 The broadening of optical phonons in metallic
nanotubes as a function of the band gap induced
by an Aharonov-Bohm flux. When the gap reaches
the optical-phonon energy h̄ω0 , the broadening disappears for the longitudinal mode and becomes
infinitely large for the transverse mode.
Fig. 5 Comparison of the calculated frequency shifts
and experimental results of the Raman G+ and G−
peaks (ref. 40).
Fig. 6 Comparison of the calculated broadening of the
longitudinal mode and the half-width-half-maximum
for experimentally observed G− peaks in metallic
nanotubes. The experiments are those of Brown et
al.,39) Jorio et al.,40) Maultzsch et al.,42) Oron-Carl
et al.,43) and Doorn et al.44)
Optical Phonon in Carbon Nanotubes
Axis
y
(a)
q
Page 9
y
x
φ
LO (Longitudinal)
B
τ2
b
(b)
K
A
τ1
TO (Transverse)
τ3
s ,n,k−q,εm−ωl
L
K
Γ
x
η Circumference
a
(c)
ωl, q
ωl, q
s,n,k,εm
Fig. 1
Fig. 2
10
Transverse
Longitudinal
LO
Broadening (units of (a/L)λω0)
Frequency Shift (units of λω0)
TO
0.0
TO
LO
-0.5
5
Semiconducting
Metallic ( Width)
-1.0
10
15
20
25
Circumference (units of a)
Fig. 3
30
0
0.0
0.5
1.0
Energy Gap (units of hω0)
Fig. 4
Page 10
K. Ishikawa and T. Ando
Diameter (nm)
1.0
0.5
1.5
2.0
TO
1550
LO
1500
Metallic
Semiconducting
Experiments (Jorio et al)
ω0 = 1570 cm-1 λ = 0.17
-0.5
Broadening (units of λω0)
LO
Frequency (cm-1)
Frequency Shift (units of λω0)
TO
0.0
1.5
2.0
Experiments
Brown et al
Jorio et al
Maultzsch et al
Oron-Carl et al
Doorn et al
ω0 = 1570 cm-1
λ = 0.17
0.6
1650
1600
1.0
0.7
1700
0.5
0.4
150
100
0.3
0.2
50
1450
0.1
1400
0.0
10
15
20
25
Circumference (units of a)
Fig. 5
30
0
10
15
20
25
Circumference (units of a)
Fig. 6
30
Broadening (cm-1)
Diameter (nm)