Calculation of optical matrix elements in carbon nanotubes
S. V. Goupalov,
1,2
A. Zarifi,
3
and T. G. Pedersen
4
1
Department of Physics, Jackson State University, Jackson, Mississippi 39217, USA
2
A.F. Ioffe Physico-Technical Institute, 26 Polytechnicheskaya, 194021 St. Petersburg, Russia
3
Department of Physics, Yasouj University, Yasouj, Iran
4
Department of Physics and Nanotechnology, Aalborg University, Aalborg, Denmark
Received 23 December 2009; revised manuscript received 5 February 2010; published 2 April 2010
Analytical expressions for dipole matrix elements describing interband optical transitions in carbon nano-
tubes are obtained for arbitrary light polarization and nanotube chiralities. The effect of the symmetry with
respect to the time reversal on the dependences of the optical matrix elements on the quantum numbers of
electronic states in carbon nanotubes is studied.
DOI: 10.1103/PhysRevB.81.153402 PACS numbers: 78.67.Ch, 78.30.Na, 78.40.Ri
The dependences of matrix elements for optical transi-
tions in carbon nanotubes on quantum numbers of the elec-
tronic states and on the nanotube chiral indices are essential
for both the study of the fundamental optical phenomena in
carbon nanotubes and the application of optical methods for
their characterization. Two different approaches have been
used to calculate the dipole optical matrix elements within
the tight-binding method. The first approach proposed by
Jiang et al.
1
is based on the introduction of the so-called
atomic dipole vectors. Jiang et al. derived an analytical ex-
pression for the optical matrix element for the case of the
parallel polarization valid for all allowed optical transitions
between various subbands of the valence and conduction
bands of carbon nanotubes. They also tried to extend their
treatment to transitions excited by light polarized perpen-
dicular to the nanotube cylindrical axis in armchair nano-
tubes. However, as it was recently pointed out
2
and is dis-
cussed below, this attempt was not quite correct. The method
of Jiang et al. was further developed in Ref. 2 resulting in the
calculation of the optical matrix elements for perpendicular
polarization. However, the derivation of Ref. 2 was quite
laborious while the final result was obtained in a rather com-
plicated form.
An alternative method was proposed in Ref. 3. There it
was applied to calculate matrix elements of optical transi-
tions within the effective-mass scheme accounting for a few
lowest uppermost subbands of the conduction valence
band of a carbon nanotube. In the present paper we show that
extension of this method to account for all possible interband
transitions yields an elegant and straightforward derivation
of the optical matrix elements for arbitrary polarizations and
nanotube chiralities. We obtain an original analytical expres-
sion for the case of the perpendicular polarization. We then
compare two different schemes accounting for the electron
states in carbon nanotubes and get further insight into their
optical properties.
The findings of Ref. 3 can be summarized as follows.
Calculation of optical matrix elements in carbon nanotubes
within the nearest-neighbor tight-binding method can be di-
vided into two tasks: 1 calculation of the k dependences of
the column coefficients C
ˆ
s , k of the tight-binding method,
and 2 expression of the optical matrix elements in terms of
these coefficients. While for the first task the graphene band
structure and zone folding are essential, it is the nanotube’s
cylindrical geometry which is relevant for an accomplish-
ment of the second task.
In Ref. 3 the first task was performed using the effective-
mass scheme and the results were formulated in terms of k
counted out from the K or K' points of the graphene’s Bril-
louin zone. However, the derivation used to accomplish the
second task is insensitive to the choice of the origin in the k
space and remains valid when a more traditional zone-
folding scheme
4
is applied to find the coefficients C
ˆ
s , k.
For example, the interband coordinate matrix element for
parallel polarization was found in Ref. 3 in the form we use
the notations of Refs. 1, 2, and 4
v, k'|z|c, k = i
,
'
K
2
,K
2
'
b=A,B
C
b
v, k'
C
b
c, k
K
2
. 1
Here k refers to the electron two-dimensional wave vector in
graphene and has the components K
1
along the nanotube’s
circumference and K
2
along the nanotube’s cylindrical axis.
In a nanotube the wave vector K
1
becomes quantized: K
1
= / R, where R is the nanotube’s radius. The key idea in
deriving Eq. 1and its counterpart for the perpendicular
polarization, Eq. 7 was to use the fact that the periodic in
z Bloch functions belonging to the same K
2
but different
band indices s s = c , v and
s
form a complete set of func-
tions.
Within the zone-folding scheme the coefficients C
ˆ
c , k
and C
ˆ
v , k are found to be
1,2,4
C
ˆ
c, k
C
A
c, k
C
B
c, k
=
1
2
e
i
c
1
, C
ˆ
v, k
C
A
v, k
C
B
v, k
=
1
2
- e
i
v
1
, 2
where
c,v
=
arctan
B
A
A 0
arctan
B
A
+ A 0,
PHYSICAL REVIEW B 81, 153402 2010
1098-0121/2010/8115/1534024 ©2010 The American Physical Society 153402-1