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Effective permittivity of single-walled carbon nanotube composites: Two-fluid model
Afshin Moradi, Hamid Reza Zangeneh, and Firoozeh Karimi Moghadam
Citation: Physics of Plasmas 22, 122104 (2015);
View online: https://doi.org/10.1063/1.4936872
View Table of Contents: http://aip.scitation.org/toc/php/22/12
Published by the American Institute of Physics
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PHYSICS OF PLASMAS 22, 122104 (2015)
Effective permittivity of single-walled carbon nanotube composites:
Two-fluid model
Afshin Moradi,1,2,a) Hamid Reza Zangeneh,3 and Firoozeh Karimi Moghadam3
1
Department of Engineering Physics, Kermanshah University of Technology, Kermanshah, Iran
Department of Nano Sciences, Institute for Studies in Theoretical Physics and Mathematics (IPM),
Tehran, Iran
3
Department of Photonics, Faculty of Physics, University of Kashan, Kashan, Iran
2
(Received 10 September 2015; accepted 18 November 2015; published online 7 December 2015)
We develop an effective medium theory to obtain effective permittivity of a composite of
two-dimensional (2D) aligned single-walled carbon nanotubes. Electronic excitations on each
nanotube surface are modeled by an infinitesimally thin layer of a 2D electron gas represented by
two interacting fluids, which takes into account different nature of the r and p electrons. Calculations
of both real and imaginary parts of the effective dielectric function of the system are presented, for
C 2015 AIP Publishing LLC.
different values of the filling factor and radius of carbon nanotubes. V
[http://dx.doi.org/10.1063/1.4936872]
I. INTRODUCTION
II. BASIC EQUATIONS
Since the discovery of carbon nanotube (CNT) by Iijima
in 1991,1 this material has been the subject of intensive
research efforts throughout the world due to its outstanding
mechanical, thermal, electrical, magnetic, and optical properties. Also, advanced composites, combined with aligned
CNTs are now fabricated by many research groups using
different techniques.2–5 For example, aligned CNTs can be
synthesized easily right now, using plasma enhanced hot filament chemical vapor deposition (PE-HF-CVD) method, and
arrays of aligned CNTs can be synthesized with controlled
diameter, length, and inter-tube distance.4
Recently, Kumar and Tripathi,6 studied the linear and
nonlinear interaction of electromagnetic waves with an array
of CNTs, using an effective medium theory based on the
fluid theory. On the other hand, we investigated the optical
properties of an isolated single-walled CNT,7 by means of
the vector wave function method and the two-fluid hydrodynamic model.8,9 We found that strong interaction between
the r and p fluids gives rise to the splitting of the extinction
spectra into two peaks in quantitative agreement with the p
and r þ p plasmon energies.10
In present work, inspired by the idea given in Ref. 6, we
extend our previous results7 to study the optical characteristics of a composite of 2D aligned single-walled CNTs. For
this purpose, we develop the standard effective medium
theory proposed by Wu et al.11 Huang and Gao.12,13 Then,
using the results obtained in Ref. 7, we derive the effective
permittivity of the system.
The paper is organized as follows: In Section II, we set
the basic equations concerning the problem. Numerical
results are discussed in Section III, and finally, Section IV
contains our conclusions.
Let us consider a composite consisting of infinitely long
single-walled CNTs (shown by 1) of radius a and volume
fraction f parallel to each other and randomly embedded in a
background medium (shown by 2) with dielectric permittivity 0 (i.e., in vacuum) as sketched in Fig. 1(a).
To study the effective permittivity eff of the whole composite, we invoke the standard effective medium theory,11–13
which deals with the scattering problem of a single-walled
CNT with radius a that is enclosed by a dielectric cylinder of
radius b within an effective medium with permittivity eff, as
seen in Fig. 1(b). The volume between a and b represents the
average envelope per CNT and f ¼ a3/b3. The effective
dielectric permittivity of the system eff may be determined
by requiring that this single-walled CNT causes no perturbation of the electromagnetic waves in the surrounding
medium.11–15
We take an electromagnetic wave normally incident on
the structure. For this propagation direction there are two different values of eff corresponding to different polarizations.
In the case of transverse magnetic (TM) waves, i.e., when
the electric field is parallel to the cylinders axis, no plasmon
resonance peak is found in the far-field extinction spectra of
an isolated single-walled CNT7 because the polarization
direction along the axis of CNTs cannot induce collective
motions of the conduction electrons. Therefore, we consider
only the case of transverse electric (TE) waves.
For TE waves, the magnetic field is parallel to the nanotubes axis and the solutions in the effective medium can be
written as
Hz ¼
þ1
X
½am ðeff ÞJm ðke rÞ þ bm ðeff ÞHm ðke rÞeim/ ;
(1)
m¼1
a)
Electronic mail: a.moradi@kut.ac.ir
1070-664X/2015/22(12)/122104/4/$30.00
pffiffiffiffiffiffiffipffiffiffiffiffi
where ke ¼ x ef f l0 and x is the frequency of the electromagnetic wave. Here Jm(x) and Hm(x) are the Bessel
22, 122104-1
C 2015 AIP Publishing LLC
V
122104-2
Moradi, Zangeneh, and Moghadam
Phys. Plasmas 22, 122104 (2015)
Hz ¼
þ1
X
½am ð0ÞJm ðk0 rÞ þ bm ð0ÞHm ðk0 rÞeim/ ;
(2)
m¼1
where k0 ¼ x/c and c is the light speed in vacuum.
The usual two boundary conditions at the dielectricdielectric interface require the continuity of the tangential
components of the electric and magnetic fields across the
interface. By matching the aforementioned boundary conditions on the interface of the background and effective
medium, we obtain
am ð0Þ
A11 A12
am ðeff Þ
¼F
;
(3)
bm ðeff Þ
bm ð0Þ
A21 A22
where
F ¼ ð0 ke Þ1 ½Jm ðke bÞHm0 ðke bÞ Jm0 ðke bÞHm ðke bÞ1 ;
A11 ¼ 0 ke Jm ðk0 bÞH 0m ðke bÞ ef f k0 J 0m ðk0 bÞHm ðke bÞ;
A12 ¼ 0 ke Hm ðk0 bÞH 0m ðke bÞ ef f k0 H 0m ðk0 bÞHm ðke bÞ;
A21 ¼ ef f k0 Jm ðke bÞJ 0m ðk0 bÞ 0 ke Jm ðk0 bÞJ 0m ðke bÞ;
A22 ¼ ef f k0 Jm ðke bÞH 0m ðk0 bÞ 0 ke Hm ðk0 bÞJ 0m ðke bÞ:
By using the condition b1(eff) ¼ 011–13 for the present system,
from Eq. (3) we obtain
A21
b1 ð0Þ
¼
¼ D1 ðxÞ;
A22
a1 ð0Þ
(4)
where D1(x) represent the Mie scattering coefficients of the
single-walled CNT and have the form of7
2
rr// þ rp// J10 ðk0 aÞ
D 1 ðx Þ ¼
: (5)
2=ðpxal0 Þ þ rr// þ rp// J 01 ðk0 aÞH 01 ðk0 aÞ
FIG. 1. (a) The single-walled CNTs in a dielectric host. (b) A single-walled
CNT plus surrounding dielectric shell embedded in the effective medium.
The single-walled CNTs are infinitely long in the z-direction.
function and Hankel function of the first kind, respectively.
Furthermore, we assume a harmonic dependence for all
quantities as a function of time [ expðixtÞ, where x is the
frequency of electromagnetic wave]. Also, we take cylindrical coordinate ðr; /; zÞ for an arbitrary point in space and use
the two-fluid theory. In this way, we consider each CNT to
be composed of p and r fluids superimposed at r ¼ a with
equilibrium densities (per unit area) n0p ¼ 38 nm2 and
n0r ¼ 3n0p ¼ 3 38 nm2 , respectively.3 We should note that
according to the Longe and Bose results,16 such classical
models give results in agreement with random phase approxpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
imation models when 1 kFa, where kF ¼ 2pðn0p þ n0r Þ is
the Fermi momentum. For the single-walled CNT composites that will be considered, we find 1 kFa for each CNT,
which justifies our use of the hydrodynamic model.
Similarly, the magnetic field in the background medium
of the coated single-walled CNT takes the form
We note that rk// is the conductivity of the electron fluid k
(the index k takes values r and p) along the / direction and
may be written as7
rk// ¼ i0 ax2pk
x
;
xðx þ ick Þ ak =a2
(6)
where xpk ¼ ðn0k e2 =0 mk aÞ1=2 . Also, mk and ck are the effective electron mass and the effective collision frequency of
the electron fluid k, respectively.
By considering the limit of keb 1, the Bessel and
Hankel functions of the first kind in A21 and A22 can be
approximated as J1 ðxÞ ffi x=2; J 01 ðxÞ ffi 1=2; H1 ðxÞ ffi ðx=2Þ
ið2=pxÞ, and H 01 ðxÞ ffi ð1=2Þ þ ið2px2 Þ with x ¼ keb.
Therefore, using these approximations, Eq. (4) reduces to
J1 ðk0 bÞ
D 1 ðx Þ
k0 bJ 01 ðk0 bÞ Y 01 ðk0 bÞ
;
¼ 0
Y1 ðk0 bÞ
iJ 1 ðk0 bÞ 1 þ D1 ðxÞ
ef f ðxÞ 0
k0 bY 01 ðk0 bÞ
ef f ðxÞ 0
(7)
where Ym(x) is the Neumamn function and Hm(x) ¼ Jm(x)
þ iYm(x). The above equation is the main result of the
122104-3
Moradi, Zangeneh, and Moghadam
Phys. Plasmas 22, 122104 (2015)
FIG. 2. Variation of Im[eff] and
Re[eff] vs. the frequency x [in the frequency range 1 eV < x < 10eV]. In
panels (a1) and (a2) f ¼ 0.11, and
a ¼ 4aB, 6aB, and 8aB, while in panels
(b1) and (b2) a ¼ 4aB, and f ¼ 0.081,
0.11, and 0.16.
present work, namely, the (complex) effective dielectric
function of a composite of aligned single-walled CNTs.
III. NUMERICAL RESULT AND DISCUSSION
Equation (7) can be used to obtain both real and imaginary parts of the effective dielectric function of the system, as
a function of the frequency x, volume fraction f, and CNTs
radius a. In calculations, we assume that r and p electrons
have equal effective masses, mr ¼ mp ¼ me , where me is the
electron mass. Also, we choose the friction coefficients to be
small but finite parameters cr ¼ 0.01xpr and cp ¼ 0.01xpp.
We divide the frequency regime into two regions: (1)
1 eV < x < 10 eV and (2) 10 eV < x < 25 eV. In Fig. 2, panels (a1) and (a2), we show both imaginary and real parts of
the effective dielectric function of the system as a function of
the frequency x, choosing volume fraction f ¼ 0.11 and different values of a ¼ 4aB, 6aB, and 8aB [aB is the Bohr radius],
respectively. Here, the sharp resonant peak in panel (a1) is
due to p plasmon excitations.7 One can see that by increasing
a, the resonant peak position of p plasmons exhibits a redshift, as can be observed from results of panels (b1) and (b2)
of Fig. 2 in Ref. 7. The similar results of resonant peak of
r þ p plasmons, in the frequency range 10 eV < x < 25 eV,
are presented in Fig. 3, panels (a1) and (a2).
To illustrate volume fraction effects, we plot in panels
(b1) and (b2) of Fig. 2 and 3, both the imaginary and real
parts of the effective permittivity of the system, respectively,
versus the frequency x, for the CNTs characterized by
a ¼ 4aB and different values of f ¼ 0.081, 0.11, and 0.16. We
FIG. 3. The corresponding results of
Fig. 2, but in the frequency range of
10 eV < x < 25 eV.
122104-4
Moradi, Zangeneh, and Moghadam
see as parameter f increases, the interaction among the CNTs
leads to a resonance at lower frequencies for resonant peak
of r þ p plasmons [this result corresponds to the result given
in Ref. 17 for the optical response of composite materials
with metallic nanoparticles], while the resonant peak of p
plasmons remains stable.
IV. CONCLUSION
To summarize, we have developed a standard effective
medium theory to obtain the effective permittivity of a composite of 2D aligned single-walled CNTs. We have presented
the calculations of both real and imaginary parts of the effective dielectric function of the system, for various values of
the filling factor and radius of CNTs.
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4
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