Theoretical study of quadratic electro-optic effect in semiconducting zigzag carbon nanotubes
Abbas Zarifi,* Christian Fisker, and Thomas Garm Pedersen
Department of Physics and Nanotechnology, Aalborg University, Skjernvej 4A, DK-9220 Aalborg East, Denmark
Received 14 March 2007; published 6 July 2007
Using the perturbation treatment developed by Aspnes and Rowe Phys. Rev. B 5, 4022 1972, an analytic
expression for the third-order nonlinear optical susceptibility
3
;0,0, is computed and analyzed for
single walled zigzag carbon nanotubes. By improving their method, our calculations based on a tight-binding
model take into account the transitions between all pairs of valence and conduction bands and thereby the
contributions to the third-order susceptibility associated with different energy bands are investigated. With
increasing radius of the nanotube, a nonmonotonous increase of the quadratic electro-optic effect has been
demonstrated except for the fundamental peak. The nonuniformity is a result of the overlap between two
energy bands as well as the reduced effective masses associated with each pair of conduction and valence
bands. A nonperturbative numerical calculation is applied to obtain the high-field response as well as to assess
the applicability of the low-field perturbation expression.
DOI: 10.1103/PhysRevB.76.045403 PACS numbers: 78.67.Ch, 78.20.-e
I. INTRODUCTION
The physical properties of carbon nanotubes CNs, as
quasi-one-dimensional systems, have been intensively stud-
ied theoretically and experimentally.
1–8
Of considerable in-
terest are the nonlinear optical NLO properties of semicon-
ductor CNs not only because the nonlinear spectrum gives
information on their electronic structure but also in view of
the possible device applications. Along with third-harmonic
generation, electro-optic EO or dc Kerr effect studies have
been used to investigate the origins of the optical nonlineari-
ties in CNs as well as CN-based composite materials. Several
experimental results on the NLO properties of CNs have
been reported so far
9–14
and some papers have studied theo-
retically the third-order nonlinear optical susceptibility and
EO effect in semiconducting CNs.
15–19
All previous theoret-
ical approaches to NLO properties in CNs are based on nu-
merical calculations and rely on the two-band approxima-
tion. In this paper, we derive an analytic expression for the
third-order nonlinear optical susceptibility of semiconducting
zigzag CNs in the presence of a uniform electric field di-
rected along the nanotube axis. Our calculations are based on
a tight-binding model and include all energy bands of the
semiconducting zigzag CNs. Therefore, it is possible to study
closely the contribution to the quadratic electro-optic QEO
effect from different energy bands. The obtained results gen-
erally agree with those previously reported for the funda-
mental resonance peak. However, for higher resonance
peaks, a more complicated behavior follows from our cal-
culations. We do not see a monotonous increase of
3
;0,0, for higher resonances.
The outline of the paper is as follows. In Sec. II, we
derive an analytic expression for the QEO function of semi-
conducting zigzag CNs. In Sec. III, the physical reason for a
nonuniform behavior of
3
;0,0, for higher resonances
is discussed. Furthermore, the apparent displacement of fun-
damental resonance peak between two groups of semicon-
ducting CNs reported in some papers is analyzed and shown
to be simply a question of taking an approximate rather than
exact value of the band gap for semiconducting zigzag CNs.
In order to obtain the high-field response and to assess the
applicability of the perturbation approach, we study numeri-
cally the response of a long but finite length CN placed in a
uniform electric field in Sec. IV before summarizing our con-
clusions in Sec. V.
II. THEORY AND ANALYTICAL DERIVATIONS
Single walled CNs, constructed by rolling up a graphite
sheet into a cylinder, are characterized by two integers
n , m. For a more detailed classification of zigzag CNs m
=0 as a subclass of CNs, we introduce two integer param-
eters p and q which are connected with n by means of the
relation n =3p + q. The zigzag tubes with q =0 are known as
narrow gap semiconductors metallic whereas the tubes with
q =1,2 and arbitrary p are moderate-gap semiconductors
MSs. Among these, the MS with index q =1 are defined as
MS1 and those with index q =2 are defined as MS2. In our
previous work
20
hereafter referred to as I using an orthogo-
nal -orbital tight-binding model, we obtained the electronic
structure, electric dipole matrix elements, and subsequently
linear susceptibility of zigzag CNs. In the present work, we
obtain an analytic expression for the QEO effect in zigzag
CNs. We utilize the perturbation expression obtained by As-
pnes and Rowe
21
in their derivation of the third-order non-
linear optical susceptibility caused by a uniform electric field
using time-dependent perturbation theory. The same method
has been applied in Ref. 22 to find the QEO effect in the
conjugated polymer polypara-phenylene. The zz compo-
nent of the EO function in the vicinity of each band gap is
given by
21
zz
3
;0,0, =
1
3
2
2
e
2
2
F
2
8m
*
3
2
2
zz
3
, 1
where e 0 is the elementary charge, m
*
the reduced effec-
tive mass, F the dc field directed along the nanotube axis, z ˆ ,
and where = + i includes the photon energy and the
phenomenological broadening parameter . As in I, we have
introduced the dimensionless “susceptibility”
zz
ob-
PHYSICAL REVIEW B 76, 045403 2007
1098-0121/2007/764/0454035 ©2007 The American Physical Society 045403-1