carbon nanotube antennas- modeling and simulation 2013
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V
List of Contents
List of Contents .......................................................................................................... III
List of Abbreviations ................................................................................................... VI
List of Symbols ........................................................................................................ VIII
Chapter One: Introduction ............................................................................................1
1.1 General Introduction ......................................................................................1
1.2 Geometry of a CNT ........................................................................................2
1.3 Literature Survey ............................................................................................5
1.4 Aim of the Work .......................................................................................... 10
1.5 Thesis Outline .............................................................................................. 10
Chapter Two: Theory of Carbon Nanotube Antenna .................................................12
2.1 Introduction ..................................................................................................12
2.2 Structures of Carbon Nanotubes .................................................................... 12
2.2.1 Structure of SWCNT ....................................................................... 12
2.2.2 Structure of MWCNT....................................................................... 15
2.2.3 Structure of Bundle CNT ................................................................. 17
2.3 Conductivity Model of a SWCNT ................................................................. 18
2.4 Two-CNT Transmission Line Properties ....................................................... 23
2.5 CNT Antenna ............................................................................................... 25
2.5.1 Resonance Frequency of a SWCNT Antenna ..................................25
2.5.2 CNT Antenna Analysis Based on TL Method ..................................27
2.5.3 CNT Antenna Analysis Based on HIE Method................................. 28
Chapter Three: Modeling of Electromagnetic Properties of a CNT Antenna 43
3.1 Introduction ..................................................................................................43
3.2 Effective Conductivity of CNT .................................................................... 43
3.2.1 SWCNT .......................................................................................... 44
VI
3.2.2 MWCNT .......................................................................................... 46
3.3 Effective Hollow CNT Conductivity ............................................................. 48
3.3.1 SWCNT ........................................................................................... 49
3.3.2 MWCNT ......................................................................................... 50
3.4 Analysis Approaches for CNT....................................................................... 51
3.4.1 CNT Complex Permittivity .............................................................. 51
3.4.2 Surface Impedance of a CNT ........................................................... 54
3.5 Wave Propagation in CNT ............................................................................ 56
3.5.1 Complex Propagation Constants in CNT .......................................... 57
3.5.2 Effective Skin Depth in CNT ........................................................... 61
3.6 CNT Synthesis .............................................................................................. 63
Chapter Four: Investigation of CNT Dipole Antenna Using Complex Permittivity
Approach .....................................................................................................................66
4.1 Introduction ..................................................................................................66
4.2 SWCNT Antenna Parameters ........................................................................ 66
4.3 Modeling of SWCNT Antenna Using Complex Permittivity Approach
68
4.3.1 Resonance Frequencies ...................................................................69
4.3.2 S11 Parameter.................................................................................... 73
4.3.3 Input Impedance............................................................................... 75
4.3.4 Radiation Pattern, Gain, and Efficiency............................................ 81
4.4 Simulation of MWCNT Antennas ................................................................. 86
4.5 Simulation of Bundle CNT Antennas ........................................................... 95
4.6 CNT Contact ............................................................................................... 101
4.7 Feeding CNT Antenna with CNT Transmission Line .................................. 105
Chapter Five: Simulation of Advanced CNT Antenna Configurations ..................... 108
5.1 Introduction ................................................................................................ 108
5.2 Loop CNT Antennas ................................................................................... 108
VII
5.2.1 Square Loop CNT Antenna ............................................................ 108
5.2.2 Circular Loop CNT Antenna .......................................................... 115
5.3 Helical SWCNT Antenna ............................................................................ 117
5.3.1 Geometry Generation ..................................................................... 118
5.3.2 S11 Characteristics and Resonance Frequencies ............................. 119
5.3.3 Directivity and Gain ....................................................................... 120
Chapter Six: Conclusions and Suggestions for Future Work.................................... 123
6.1 Conclusions ................................................................................................ 123
6.2 Suggestions for Future Work ...................................................................... 124
References .................................................................................................................. 125
VIII
List of Abbreviations
BCNT Bundle Carbon Nanotube
CLCNT Circular Loop Carbon Nanotube
CNT Carbon Nanotube
CST Computer Simulation Technology
DWCNT Double-Wall Carbon Nanotube
EM Electromagnetic
FIT Finite Integration Technique
HIE Hallén's Integral Equation
HRTEM High Resolution Transmission Electron Microscopy
HSWCNT Helical Single-Wall Carbon Nanotube
IE Integral Equation
IR Infrared Radiation
MFWNT Metal Filled Multi-Wall Nanotubes
MoM Method of Moments
MWCNT Multi-Wall Carbon Nanotube
MWs Microwave Studio
PEC Perfect Electric Conductor
RBCNT Rectangular Bundle Carbon Nanotube
RF Radio Frequency
SLCNT Square Loop Carbon Nanotube
SWCNT Single-Wall Carbon Nanotube
THz Tera Hertz
TL Transmission Line
IX
List of Symbols
Symbol Definition
, Primitive lattice vectors
Underlying Bravais lattice constant
Magnetic vector potential
Carbon carbon bond length
Magnetic flux density vector
Speed of light in free space
Chiral vector
Electrostatic capacitance
Quantum capacitance
Delta gap of the CNT antenna
Diameter of a single-wall carbon nanotube
D Directivity
D0 Diameter of SLCNT or CLCNT antenna
Electron charge
Electric field intensity vector
Incident field on the surface of CNT antenna
Scattering field from the surface of CNT antenna
Frequency
Maximum gain frequency
Plasma frequency
Resonance frequency
Equilibrium Fermi distribution function in the conduction band
X
Hollow factor
Equilibrium Fermi distribution function in the valence band
Conductivity of the CNT
G Gain of the CNT antenna
Magnetic field intensity vector
Line current density vector
Volume current density vector
Boltzmann's constant
Plasmon wave number
Free space wave number
Surface current density vector
Length of the CNT antenna
Contact length
CNT transmission line length
Total length of the HSWCNT antenna wire
Kinetic inductance
Magneto-static inductance
First radius index of SWCNT
Electron mass
Second radius index of SWCNT
Equivalent electron density
Quasi-momentum in the axial direction
Kernel function
Radius of SWCNT
XI
Interlayer spacing between two shells of MWCNT
Radius of MWCNT with N layers
Distance between the source and the observation point.
Matrix element for the CNT in the conduction band
Loss resistance
Radiation resistance
Surface resistance of the CNT
Matrix element for the CNT in the valence band
Quantum resistance
S11 Reflection coefficient
Inner radius of the hollow cylinder
Translation vector
Temperature
Effective temperature of the CNT
Incident voltage
Surface reactance of the CNT
Input impedance of CNT
Input impedance of the discrete port
Input impedance of the CNT antenna
Characteristic impedance
Surface impedance of the CNT
Effective attenuation constant in CNT
Effective phase constant in CNT
Effective propagation constant in CNT
XII
Effective skin depth in SWCNT
Permittivity
Relative complex permittivity
Real part of the relative complex permittivity
Imaginary part of the relative complex permittivity
Relative permittivity of the CNT material
Permittivity of free space
Electron dispersion relation for armchair CNT in the conduction band
Electron dispersion relation for armchair CNT in the valence band
Intrinsic impedance in free space
Total efficiency
Radiation efficiency
Reflection efficiency
R =
Plasma wavelength
Plasmon wavelength
Free space wavelength
Permeability
Chemical potential in graphite
Relaxation frequency
Collision frequency
Angular frequency
Angular plasma frequency
(i) Volume charge density
(ii) Cylindrical radial distance coordinate
XIII
Conductivity of metals
Conductivity of the hollow cylinder
` Imaginary part of the conductivity
Real part of the conductivity
Relaxation time
Fermi velocity for the CNT
Phase velocity
Electric scalar potential
Chiral angle
1
Chapter One
Introduction
1.1 General Introduction
Carbon nanotubes (CNTs) were discovered in 1991 and have since led to an
enormous amount of research into their fundamental properties [1]. They have been
widely used as the building blocks for biosensors, gas sensors, pressure sensors,
transistors, light emitters, and photodetectors due to their excellent physical, electrical,
and mechanical properties [2]. In particular, CNTs have been proposed to fabricate
several different integrated-circuit elements and electromagnetic devices, such as
transmission lines, interconnects, and nanoantennas [3]. Nowadays, there is a great
interest in the development of nanoantennas for terahertz (THz), infrared (IR) and
optical ranges. Such antennas would allow a very desirable modality of communications
between nanoelectronic devices and the macroscopic world as well as wireless
interconnection layout in nanocircuits [4]. Also, bundles of CNTs are applicable as
interconnections and antennas in radio communication [5]. Furthermore, CNTs come in
two major flavors, the single-wall (SWCNT) and multi-wall (MWCNT). The newer
forms of carbon have significantly contrasting properties compared with the older forms
of carbon, which are graphite and diamond [6]. Many researchers have been intrigued by
mechanical, chemical, optical, and electrical properties and
their fabrication advantages (low cost and defect-free crystalline structure). Indeed,
CNTs have already been influential in widespread areas as sensitive chemical sensors,
high-resolution imaging probes, field emission displays, supercapacitors, conductive
flexible electrodes, and more [7].
Research on carbon nanotubes has been primarily focused on MWCNTs in 1991
and SWCNTs in 1993 [7]. Since the discovery of the unique antenna from CNTs in
2004, there has been more strong interest in nanoantenna, and much concerning their
2
lengths from nanometer scale to centimeters, naturally. This leads to a topic considering
CNTs for sub-millimeter, millimeter and centimeter wave antenna applications [8].
Therefore, CNTs have been investigated nowadays as antennas in various areas-
communication between nanodevices and macroscopic world, nanointerconnect, fiber
communication, aircraft and space communication systems. Their advantages are the
small sizes, light weight, and remarkable electric properties [9].
This work investigates theoretically and by simulation the main features behind
different types and configurations of CNT antenna.
1.2 Geometry of a CNT
Figure 1.1 shows the geometry representation of a SWCNT which results after
rolling up a graphene sheet according to the synthesis environment that gives a SWCNT
with radius as [1]
where nm is the interatomic distance in graphene and , is called the dual
index. Thus SWCNTs can be characterized by the dual index , where for
zigzag, for armchair, and , for chiral. Electrically, SWCNTs
have good properties. For example, they can be either metallic or semiconducting,
depending on their geometry (i.e., on and ). Armchair SWCNTs are always metallic,
as are zigzag SWCNTs with , where is an integer.
4
(a)
(b)
Fig. 1.3 HRTEM image of (a) SWCNT [11] (b) MWCNT consisting of 20 walls [12].
Three experimental techniques have so far become available for the growth of
carbon nanotubes. These are the arc-discharge technique, the laser ablation technique
and, recently, the chemical vapor deposition technique [13].
5
1.3 Literature Survey
In 2003, Burke [14] developed an RF circuit model for SWCNTs for both dc and
capacitively contacted geometries. The nanotube was modeled as a nanotransmission
line with distributed kinetic and magnetic inductance as well as distributed quantum and
electrostatic capacitance.
In 2004, Dresselhaus et al. [15] presented the unusual structure and properties of
CNTs, with particular reference to SWNTs and nanotube properties that differ from
those of their bulk counterparts. The atomic structure, electronic structure, and
vibrational, optical, mechanical, and thermal properties were discussed, with reference
made to nanotube junctions, nanotube filling, and double-wall nanotubes (DWNTs).
Special attention was given to resonance Raman spectroscopy at the single nanotube
level.
In 2005, Hanson [1] investigated fundamental properties of dipole transmitting
antennas formed by CNTs. Dipole CNT antennas were investigated via a classical
Hallén's integral equation (HIE), based on a quantum mechanical conductivity.
In 2006, Lan et al. [16] suggested a novel THz antenna structure, made of CNT
arrays. The capabilities of CNT THz antenna arrays were simulated. The dependence of
gain upon geometrical factors, e.g., nanotube diameter, nanotube length and the inter-
tubes distance, were investigated. The directivity patterns of antenna arrays and the
surface current distribution of an antenna were simulated.
In 2006, Hao and Hanson [17] presented a model for electromagnetic scattering
from infinite planar arrays of finite-length metallic CNTs, and isolated nanotubes, in the
lower infrared radiation (IR) bands. The scattered field was predicted using a
quantum conductance function for the CNTs. The finite length of the tubes was
accounted for electromagnetically by imposing a boundary condition on the tube ends.
Scattering characteristics were investigated for various armchair CNT array
6
configurations, as well as for isolated nanotubes.
In 2006, Hanson [18] studied the current on an infinitely-long CNT antenna fed
by a delta-gap source using Fourier transform technique. The CNT was modeled as an
infinitely-thin tube characterized by a semi-classical conductance. The CNTs current
was compared with the current on solid and tubular copper antennas having similar or
somewhat larger radius values.
In 2006, Hao and Hanson [19] investigated the characteristics of armchair CNT
dipole antennas in the infrared and optical regime. The analysis is based on a classical
electromagnetic HIE, and an axial quantum mechanical conductance function for the
tube.
In 2006, Burke et al. [20] presented quantitative predictions of the performance
of nanotubes and nanowires as antennas, including the radiation resistance, the input
reactance and resistance, and antenna efficiency, as a function of frequency and
nanotube length. They developed models for both far-field antenna patterns as well as
near-field antenna-to-antenna coupling. Also, a circuit model was developed for a
transmission line made of two parallel nanotubes. Finally, an analog of HIE appropriate
for SWCNT antennas was derived.
In 2007, Hao and Hanson [21] developed a model for optical scattering from
planar arrays of finite-length single-wall metallic CNTs. The scattered field was
electromagnetic interactions, and a quantum conductance function for the CNTs.
In 2007, Wang et al. [8] investigated by simulation the geometric structure and
the THz/IR characteristics of CNTs dipole antenna arrays using finite integral methods.
In terahertz and infrared frequency span, the antenna properties such as electrical field
distributions, scattering parameters, standing wave ratio, gain, and two dimensional
directivity patterns were discussed. The results show that N×N antenna arrays have
higher radiation efficiency than a single CNT dipole antenna.
7
In 2007, Maksimenko et al. [4] presented a theory of the metallic chiral CNT as a
vibrator antenna. The Leontovich Levin IE method was extended to the case of CNTs.
Integral equations for the finite-length CNT and CNT bundles were solved numerically
in the integral operator quadrature approximation with the subsequent transition to the
finite-order matrix equation.
In 2008, Ying and Baoqing [22] studied the properties of CNT as an optical
antenna. The equation of current distribution on a single antenna was obtained by using
conventional transmission line theories. The re-radiation lobe pattern of a single antenna
and an antenna arrays were gained by simulation.
In 2008, Fichtner et al. [23] investigated the characteristics of small CNT dipole
antennas on the basis of the thin wire HIE. A surface impedance model for the CNT was
adopted to account for the specific material properties resulting in a modified kernel
function for the IE. A numerical solution for the IE gave the current distribution along
the CNT. From the current distribution the antenna, driving point impedance and the
antenna efficiency were computed.
In 2008, Huang et al. [24] carried out a theoretical investigation on predicting
radiation characteristics of SWCNT bundle dipole antennas based on the distributed
circuit parameters and the model of a SWCNT. The current distributions in such novel
antennas were predicted to investigate the effects of bundle cross-sectional size, tube
diameter, tube length, and operating frequency.
In 2009, Attiya [25] used the slow wave property to introduce resonant
dipole antennas with dimensions much smaller than traditional half-wavelength
dipole in THz band. The physical interpretation of this property was introduced
based on the relation between the resonance frequency and the surface wave
propagation constant on a carbon nanotube.
In 2009, Yue et al. [26] investigated the electromagnetic radiation characteristics
of a metallic, large aspect ratio SWCNT antenna in the THz frequency region below
12.5 THz. The key features of THz pulse were revealed on the CNT antenna in
8
comparison with conventional photoconductive switching. The THz waveforms,
radiation power and their field distributions were evaluated and are analyzed. The
Fourier transformed spectra over the whole frequency range demonstrate that the CNT
antenna can be used as radiation source for broadband THz applications.
In 2009, Nasis et al. [27] investigated the behavior of an array of parallel CNTs
when they are illuminated by a plane wave. Transmission and scattering properties were
quantified. This analysis may be used in the future as a reference for shielding
applications based on CNT arrays.
In 2010, Nemilentsau et al. [28] examined lateral resolution of CNT tip with
respect to an ideal electric dipole representing an elementary detected object. A
Fredholm IE of the first kind was formulated for the surface electric current density
induced on SWCNT by the electromagnetic field due to an arbitrarily oriented electric
dipole located outside the tube.
In 2010, Mehdipour et al. [29] explored SWCNT composite materials for the
design of multiband antennas. An accurate electromagnetic (EM) model of the modified
Sierpinski fractal composite antenna was developed using simulation.
In 2010, Pantoja et al. [30] introduced a novel procedure for the simulation of
CNT antennas directly in the time domain. This formulation is based on the time-domain
electric-field IE for thin-wires, including external loads. Appropriate loads were derived
to match the physical response of single-walled CNT media to external electromagnetic
fields.
In 2011, Berres and Hanson [31] analyzed isolated, infinitely long MWCNTs
interacting with electromagnetic waves in the microwave and far-infrared frequency
regime using a semi-classical approach. An expression for the bulk effective
conductivity of MWCNTs was obtained, valid up to THz frequencies. Comparisons
between metallic MWCNTs, metallic SWCNTs, and metal nanowires were made.
In 2011, Choi and Sarabandi [32] evaluated the performance of bundled carbon
nanotubes (BCNTs) as a conducting material for the fabrication of antennas in the THz
9
frequency range and above. The performance was compared with that of gold film. The
macroscopic behavior of BCNTs was modeled by an anisotropic resistive sheet model
which is extracted from the discrete circuit model of a SWCNT. Numerical simulations
using the method of moments (MoM) and the mixed potential IE were performed to
quantify radiation efficiencies of resonant strip antennas composed of BCNTs and thin
gold films.
In 2011, Miano et al. [3] proposed a model for the signal propagation along
SWCNTs of arbitrary chirality. They first studied an SWCNT, disregarding the wall
curvature, in the frame of a semiclassical treatment based on the Boltzmann equation in
the momentum independent relaxation time approximation. This allows expressing the
longitudinal dynamic conductivity in terms of the number of effective conducting
channels. Next, the dependence of this number with nanotube radius and its relation with
the kinetic inductance and quantum capacitance were investigated.
In the previous survey, the CNT antennas were studied in different ways. All
these ways don't give an adequate solution for the problems associated with this type of
antennas because these papers depend on inefficient numerical electromagnetic methods
such as MoM (when it is compared with another method such as a finite integration
technique (FIT) which is considered as a developed generation with respect to others). In
addition, some papers assume the conductivity is measured in Siemens (S) instead of
(S/m) and this will lead to limit the area of applying more efficient numerical methods
and at the same time this means that there is no conjunction with available three-
dimensional electromagnetic software packages. In this work, all these problems will be
manipulated when a CNT material obeys in such a way that it becomes more applicable
in CST which is based on FIT. This opens the door to deal with all configurations of
CNTs antennas not just the dipole one. Further, to the author's knowledge, only CNT
dipole antennas are investigated in the literature. Therefore, it is important to investigate
other CNT antenna structures such as loop and helical ones. This issue is also addressed
in this work.
10
1.4 Aim of the Work
The aim of this work is to develop an accurate modeling for the carbon nanotube
material when it is used as an electromagnetic (EM) radiating element. The model
should take into account the frequency dependent complex permittivity and modify the
problem of the conductivity of the CNT material in addition to various geometric
parameters and adapted easily to antenna theory and EM software package solver. All
physical properties must be manipulated such that the model stays near to the reality.
The model is to be used to predict the performance characteristics of various CNT
antenna configurations.
1.5 Work Outline
Chapter one presents an introduction to the concepts of a CNT material,
properties, and applications. A literature survey of the references which refer to a CNT
as antenna or related to this issue has been given. Finally, the aim of the work is stated.
Chapter two talks about the theory of a CNT material from the point of view
of construction, electrical properties, and functionalization of this material as a
nanoantenna. This chapter also is concerned with the new concepts such as quantum
resistance, quantum capacitance, and kinetic inductance. These parameters are usually
related to the transmission line which is necessary for understanding the behavior or
modeling of CNT antennas. This is not enough for representing a CNT
electromagnetically and therefore, the conductivity of a CNT needs to be clarified for
antenna applications as discussed here.
Chapter three discusses the modeling of electromagnetic properties of a CNT
material. A model of the effective conductivity for SWCNT and MWCNT is introduced.
Then, the concepts of complex permittivity, plasma, and surface impedance are
generalized to the CNT such that this material can be simulated directly using CST. This
11
chapter ends with a review of a CNT material synthesis so that one can become familiar
with this nano material from all aspects.
Chapter four contains simulation results for CNT antennas. SWCNT,
MWCNT, and BCNT antennas which are simulated using CST MWs.
Chapter five deals with new configurations of CNT antennas, namely loop
and helical CNT antennas. These antennas are simulated using CST MWs and the results
are compared with these of the SWCNT dipole antenna.
Chapter six summarizes the main results drawn from this study and gives
suggestions for future work.
12
Chapter Two
Theory of Carbon Nanotube Antenna
2.1 Introduction
Carbon nanotubes (CNTs) are tubular structures typically of nanometer diameter
and many microns in length. They are unusual because of their very small diameters,
which can be as small as 0.4 nm and contain only 10 atoms around the circumference,
and because the tubes can be only one atom in thickness. The aspect ratio
(length/diameter) can be very large (greater than 104), thus leading to a prototype one-
dimensional system [15]. This aspect ratio can be accepted when one talks about the
CNT application in antenna field. Therefore it is necessary to study the electromagnetic
properties of this material. This chapter introduces the main concepts about CNT
materials and discusses the electrical properties as related to antenna applications.
2.2 Structures of Carbon Nanotubes
2.2.1 Structure of SWCNT
First, it is necessary to know the origin of CNT material which is started from the
graphene. Graphene sheet is constructed from graphite after removing one of the two-
dimensional planes as shown in Fig. 2.1. A carbon nanotube can be constructed from the
graphene sheet after rolling-up to form a closed cylinder. This rolling-up will change the
properties of carbon and create a new material having different electrical and mechanical
properties. The result is a CNT having its own electromagnetic properties according to
the approach of synthesis.
13
Fig. 2.1 Illustration of the graphite structure, showing the parallel stacking of two-dimensional planes, called graphene sheet, the colored atoms are under observation [33].
Figure 2.2a shows the honeycomb lattice of graphene and the primitive lattice
vectors and , defined on a plane with unit vectors and [6]
where is the underlying Bravais lattice constant, , and is the
carbon carbon bond length . With reference to Fig. 2.2a, a SWCNT can be
conceptually conceived by considering folding the dashed line containing primitive
lattice points A and C with the dashed line containing primitive lattice points B and D
such that point A coincides with B, and C with D to form the nanotube shown in Fig.
2.2b. The CNT is characterized by three geometrical parameters, the chiral vector ,
the translation vector , and the chiral angle , as shown in Fig. 2.2a. The chiral vector
is the geometrical parameter that uniquely defines a CNT, and | | = is the CNT
circumference. is defined as the vector connecting any two primitive lattice points of
graphene such that when folded into a nanotube these two points are coincidental or
indistinguishable. For the particular example of Fig. 2.2, the chiral vector is the vector
14
from point A to B, . In general:
and the resulting
CNT is described as an CNT.
(a) (b)
Fig. 2.2 The conceptual construction of a CNT from graphene. (a) Wrapping or folding the dashed line
containing points A and C to the dashed line containing points B and D resulting in the (3, 3) armchair
carbon nanotube in (b) with = 30 . The CNT primitive unit cell is the cylinder formed by wrapping
line AC onto BD and is also highlighted in (b) [6].
Expressions for the main geometric parameters of a CNT are given in Ref. [6] and
the results are summarized in the following paragraphs.
The diameter of a carbon nanotube is derived from its circumference | |
The other two geometrical parameters ( and ) can be derived from the chiral
vector. For instance, the chiral angle is the angle between the chiral vector and the
15
primitive lattice vector
In order to determine the primitive unit cell of the CNT, it is necessary to consider
the translation vector which defines the periodicity of the lattice along the tubular axis.
Geometrically, is the smallest graphene lattice vector perpendicular to and is
given by
where is the greatest common divisor of and . The length of the
translation vector is
The number of hexagons per unit cell is the surface area divided by the area of one
hexagon
2.2.2 Structure of MWCNT
Multi-wall carbon nanotube (MWCNT) is composed of multiple concentric shells
of SWCNTs with an outer diameter that is anywhere from several nanometers to about a
16
100 nm. The inner diameter is the diameter of the smallest shell din, the outer diameter is
the diameter of the largest shell dout, and the shell-to-shell spacing is approximately
equal to the interlayer spacing in graphite ( ) [6]. Other values of this
interlayer spacing, in the range 0.342 nm to 0.375 nm, have also been reported, with the
spacing increasing with a decrease in the nanotube diameter [13]. Figure 2.3 shows
MWCNT consisting of three shells of a SWCNT.
In the simplest model, the walls or shells of an MWCNT are considered to be
non-interacting, which implies that each shell can be treated as an independent SWCNT.
Therefore, the electrical conductance of an MWCNT is a linear sum of the conductances
of each shell. In general, multi-wall nanotubes consisting of more than three shells
exhibit metallic properties. One can arrive at this conclusion from a variety of arguments
of varying sophistication. For example, based on the standard statistical distribution of
SWCNTs, one-third of all the shells is metallic and, hence, MWCNTs with greater than
three shells have an overall metallic character [6].
Fig. 2.3 Representation of a three-shell MWCNT.
17
The dual wall carbon nanotube (DWCNT) is the simplest form of the MWCNT
which consists of two SWCNTs. To construct a DWCNT for example, the chiral vector
of the inner SWCNT should be related, as characterized by the indices , to the
chiral vector of the outer nanotube, as characterized by the indices . These
indices are related to each other, by noticing that the radius of the outer nanotube is
related to the radius of the inner nanotube via [13]
where which is the interlayer spacing between the two shells. The radius
of the inner nanotube is given by eqn. (2.2), therefore one can find after choosing
a value for and then solving for to the nearest whole number. Bigger MWCNTs,
composed of a larger number of shells, can be constructed by following a procedure
similar to that for the DWCNT [13].
2.2.3 Structure of Bundle CNT
SWNTs are typically found as aggregated bundles, as illustrated in Fig. 2.4 [7].
These bundle CNTs (BCNTs) can take circular or rectangular geometry as proposed in
[24]. The individual tubes in the bundle are attracted to their nearest neighbors via Van
der Waals interactions, with typical distances between nanotubes being comparable to
inter planar distance of graphite which is 0.34 nm. The cross section of an individual
nanotube in a bundle is circular if the diameter is smaller than 1.5 nm and deforms to a
hexagon as the diameter of the individual tubes increases [34].
18
Fig. 2.4 Model of (6, 6) SWNTs assembled into a closest packed bundle. The nanotubes on the
right are shown with their Van der Waals contact surfaces [7]
2.3 Conductivity Model of a SWCNT
The electrical conductivity is considered as an important parameter in the
electromagnetic problems because it controls the electric field or current density such as
in the electric circuit when the resistance controls the voltage or current. Therefore it is
necessary to seek this quantity since it is directly related to Maxwell's equations which
are considered the basis of the antenna problems.
There are several methods for the derivation of the conductivity of materials. The
familiar model is the Drude method which benefits from the uniform distribution of
carriers (constant carrier density). Then from Ohm's law , Drude was capable of
finding an expression for the conductivity after calculating the current density and
electric field in certain formulas. In CNT, the problem is different somehow because
of the nature of this material which seems to be one dimensional. The CNT has carriers
on the surface but its inside is hollow. In addition, the time and frequency parameters are
different with respect to the classical materials such as copper. Literature on CNT has
tried to find a formula for the CNT own conductivity.
19
equation with the axial current density (in two dimensions) and some modification result
in the conductivity of the SWCNT [1], [19].
For an armchair or zigzag carbon nanotube the quantum conductance is given by
[17], [19]
where is the electron charge, is the
, is the phenomenological relaxation frequency ( ) , is
temperature in Kelvin [31] and is the relaxation time. Furthermore,
are the equilibrium Fermi distribution function in the conduction and valence band,
respectively, is the chemical potential in graphite and is Boltzmann's constant.
The electron dispersion relation for armchair CNT in the conduction and valence bands
is given, respectively, by [19] [35]
20
where is the approximate range of the overlap integral, , is the quasi-
momentum in the axial direction (z direction). Finally, the matrix element for the tube in
the conduction and valence bands is given, respectively, by
where is the dual index in SWCNT. The integral in eqn. (2.8) is performed over the
first Brillouin zone (BZ), e.g., from to [19].
In the low-frequency regime, below optical interband transitions ,
where is the Fermi velocity for the CNT (at low and middle IR
frequencies ), the conductivity of armchair or zigzag SWCNT is given by
[17]
The conductivity for armchair SWCNT tubes with various values is shown
in Fig. 2.5. Equation (2.12) indicates that the conductivity is a complex quantity and
therefore can be expressed as
22
Fig. 2.5 Conductivity of the armchair SWCNT for various values (i.e., various radius
values) at T=300oK.
One can note that the maximum values of and occur at and
respectively. At zero frequency, the conductivity is real and is given by
For an armchair SWCNT with , and assuming ,
the quantum resistance is at . The distributed impedance
of the SWCNT structure is given by [36]
0 200 400 600 800 10000
0.01
0.02
0.03
Re(
g)(s
)
0 200 400 600 800 1000-0.015
-0.01
-0.005
0
Im(g
)(s
)
m=n=25m=n=50m=n=75m=n=100m=n=125m=n=150
23
where
and Here are the per unit length (p.u.l) quantum resistance and
kinetic inductance, respectively. Note that is inversely proportional to CNT radius and
proportional to temperature (when the effect of temperature on Fermi velocity is
neglected). The value of for is which is large
value with respect to magnetic inductance in transmission line which has values near
to . This emphasizes the domination of the inductance on the
parameters of transmission line such as phase velocity, propagation constant, phase
constant, and so on. This fact makes the CNT more attractive than the other materials
since it has properties which are not present in the other materials.
2.4 Two-CNT Transmission Line Properties
Most antennas need to be connected to the source through the transmission line
and this type of feeders must match properly with the involving antenna. The radio
frequency (RF) circuit model for differential mode of two-nanotube transmission line
(TL) is shown in Fig. 2.6, where are the p.u.l quantum capacitance and
electro-static capacitance, respectively. In CNT, the value of is much greater
24
than . Also, the value of is much greater than the magneto-static inductance .
The characteristic impedance and wave velocity are given by [20]
with
and
Fig. 2.6 RF circuit model for differential mode of two-nanotube transmission line [20].
25
In the case of non- dispersive CNT, the quantum capacitance vanishes and only the
electric capacitance affects the calculation of eqns. (2.19) and (2.20).
The physical origin of the quantum capacitance comes from the finite density of
states at the Fermi energy. In a quantum particle in a box, the spacing between allowed
energy levels is finite. Because of this, to add an extra electron to the system takes a
finite amount of energy above the Fermi energy. The kinetic inductance has a simple
physical origin as well. It is due to the charge-carrier inertia, that the electrons do not
instantaneously respond to an applied electric field; there is some delay. For periodic
electric fields, the electron velocity lags the electric field in phase, i.e., the current lags
the voltage in phase [20].
2.5 CNT Antenna
One interesting area of potential applications is in the use of nanotubes as
antennas. So far in the RF and microwave frequencies, no experiments have been
reported on this topic. However, there have been some theoretical developments [24].
The geometry which was firstly considered is that of a thin-wire center-fed antenna
where the wire is made of a single-walled metallic carbon nanotube [20] as shown in
Fig. 2.7. Here the antenna is placed along the z-axis, in which is the radius, is the
total length, and is the delta gap generator. The symbol of A.C generator is denoted
for the feeding of SWCNT antenna which consists of two rods of SWCNT wire (upper
and lower).
2.5.1 Resonance Frequency of a SWCNT Antenna
It was predicted that SWCNTs will exhibit longitudinal resonances when
, where is a plasmon wavelength ( is the free space wavelength), with
[17]. Let a SWCNT antenna have , then , and the
26
exact value of resonance frequency must be found from the numerical result and
subsequently it helps to find the phase velocity according to [1]. Also, the
resonance frequency can be associated with plasmon using the transmission line model
developed in Ref. [20], where the propagation velocity on the antenna was found to be
( is taken here to be m/s for SWCNT [1]). Thus, the transmission-
line model predicts that , where is the speed of light in free space.
Therefore, the wavelength on the antenna should be approximately ,
( , and the expected value of . For the same length of a SWCNT
antenna but using copper material, the resonance frequency is calculated to be 75 THz.
Fig. 2.7 Representation of nanotube antenna.
27
2.5.2 CNT Antenna Analysis Based on TL Method
The knowledge of current distribution along CNT antenna means that a part of the
problem of electromagnetic is solved. Until now, the problem has two approaches of
solution. The first approach was used by Ref. [20], where the TL which consists of two
wires of SWCNT is flared to become as a SWCNT antenna. The current distribution
along the TL is considered as the same current distribution of the CNT antenna placed
along z-axis with length and is given by
where is the plasmon wave number defined by , the phase velocity is defined
by eqn. (2.20), and can be computed from where is the characteristic
impedance and is defined by eqn. (2.19). The incident voltage can be computed after
knowing the voltage of the terminals on the CNT antenna as
Now, all properties of the CNT antenna are computed from eqn. (2.23), where the
scalar quantity of the current can be transformed into the vector quantity in z
direction as . As seen from eqn. (2.23), a new wave number is used instead
of , where is the wave number in free space. The parameter is not finish and still
has an effect on the properties of CNT antenna such as the electric field. The new
expression of the electric field in the far field region and in the direction is given by
[20]
28
where is the distance between the source and the observation points and is the
intrinsic impedance. If then eqn. (2.25) reduces to
This is the same equation of the finite length dipole reported in Ref. [37]. For a SWCNT
antenna, and this ratio will make the radiation resistance very small,
therefore, bundle CNT antenna has been chosen to overcome this problem [24].
2.5.3 CNT Antenna Analysis Based on HIE Method
The previous analysis depends on the estimation of the current distribution on the
surface of the CNT antenna which is assumed to be sinusoidal. However, the fact says
the current distribution is not exactly sinusoidal and it must be calculated from certain
methods. This leads to a second approach for calculating the current distribution which
was followed by several researchers when dealing with conventional antennas. Hanson
[1] used the HIE, without derivation, to investigate the performance of a simple CNT
dipole. In the rest of this subsection, a comprehensive mathematical derivation of the
HIE equation for CNT dipole is represented for reference purposes.
A- Framework
Maxwell's equations can be applied to the CNT antenna and they are written as
[37]
29
where , , , , and are the electric flux intensity, magnetic field intensity, volume
current density, magnetic field density, and volume charge density, respectively, all
related to the CNT antenna. The constitutive parameters and are the permittivity and
permeability of the medium, respectively. Define the magnetic vector potential , such
that
or
Substituting eqn. (2.28b) into eqn. (2.27a) yields
From the vector identity
30
where is a mathematical electric scalar potential. By equating eqns. (2.29) and 2.30,
the result is
or
Taking the curl for eqn. (2.28b) gives
Now, substitute eqn. (2.27b) into eqn. (2.32) and make use of the vector identity to get
where the used vector identity is
Substitute eqn. (2.31b) into eqn. (2.33) to yield
31
or
where is the free space wave number. Equation (2.35b) can be rewritten as
Since and are mathematical quantities, then one can assume [37]
Therefore, eqn. (2.36) becomes
B-
Equation (2.38) can be solved using the Green's function [38]
where the Green's function is given by
32
with
where are the source and observation position vectors, respectively,
are the source and observation points, respectively, and is the cylindrical radial
distance coordinate as shown in Fig. 2.8. The expression of can be obtained from eqn.
(2.37) as
Substituting eqn. (2.42) into eqn. (2.31b) yields
where the longitudinal component in z direction is assumed such that .
33
Fig. 2.8 Thin wire model of CNT antenna.
C- Current Density
The volume current density for the CNT antenna can be written in term of its x, y,
and z components as . Since the CNT antenna is a one
dimensional system, then and therefore
The delta function is used here because the current density only exists at the surface of
CNT antenna ( ). The current along the CNT can be found using the surface
integral of
35
where
D- Field Distribution
On the surface of CNT antenna, Ohm's law says [1]
where is the conductivity of CNT and is the total electric field on the surface of
CNT antenna which is a combination of the incident and scattering electric
field and given by
From eqns. (2.45b), (2.48), and (2.49), the scattering field on the surface of CNT
antenna is computed as
Substituting eqn. (2.50) into eqn. (2.46) yields
37
E- Scaled Version of the Vector Field
Instead of working with the vector potential , it is convenient to work with a
scaled version of it that has units of volts and is defined as [39]
where is a complex number and c is the speed of light in free space.
When eqn. (2.42) is multiplied by and is noted,
Also when eqns. (2.53b) and (2.55) are multiplied by , then
38
where is the intrinsic impedance. From eqns. (58a) and (58b)
Equation (2.58b) can be solved using Green's function , where the particular
solution is obtained from [38]
where is the delta function. The general solution of eqn. (2.58b) is obtained by
adding the most general solution of the homogeneous equation ,
to the Green's function solution and the result will be
where , and are arbitrary constants. There are several types of the Green's
function, for example [40] Substituting this expression and
eqn. (2.54) into eqn. (2.61b) yields
39
F- Source Excitation
Assume the source of the CNT antenna is a delta gap generator which is modeled
as
Substituting eqn. (2.63) into eqn. (2.62) yields
Substituting (from eqn. (2. 59)) into eqn. (2.64b) and multiplying the result by
41
Equation (2.66) is called Hallén's integral equation (HIE) which can be solved for the
current by several methods such as the method of moments (MoM) [40].
In the case of , eqn. (2.66) reduces to the HIE of a thin wire made from a
perfect electric conductor (PEC) [39]
G- Kinetic Inductance and Quantum Resistance
Using the definition of from eqn. (2.17a), then eqn. (2.67) can be written in
terms of kinetic inductance and quantum resistance as [20]
From eqn. (2.69), the main parameters of the CNT seem to govern the properties of CNT
antenna over the conventional parameters such as free space wave number.
In the high frequency range, eqns. (2.8) and (2.17a) predict the following relation
42
where , , and are named as the quantum resistance, kinetic inductance, and
quantum capacitance, respectively, which can be obtained from eqn. (2.8).
The previous TL and HIE methods may be considered the main methods of
prediction of the characteristics of the CNT antenna. It is noted that the first method
takes the effect of the plasmon wavelength on all properties but this quantity is omitted
in the second method. The kernel function seems to be the oscillation function with
the plasmon wavelength which is controlled by the kernel function with free space
wavelength.
43
Chapter Three
Modeling of Electromagnetic Properties of a CNT Antenna
3.1 Introduction
The purpose of this chapter is to construct the electromagnetic properties of CNT
material in more functionality as related to the direction of the antenna problems. In
conventional antenna investigation, the material is assumed to be perfect electric
conductor (PEC) where the conductivity approaches infinity. The CNT cannot be
assumed to be a PEC since its conductivity is complex and frequency dependent. The
CNT conductivity formula should be modified in order to be applied in more powerful
way to the antenna problems. The modified expression of the conductivity of CNT
material is denoted as the effective conductivity. This issue is addressed in this chapter
and yields new analytical expressions describing the electromagnetic properties of CNT
material.
3.2 Effective Conductivity of CNT
The unit of CNT conductivity (as a hollow tube) is measured in (S), but the
conductivity in Maxwell's equations is measured in (S/m). The reason for this
difference is the geometry of the CNT which leads to considering it as a longitudinal
geometry and the derivation of the CNT conductivity is based on this fact. In contrast,
the metal wires are usually assumed to be a bulk and the physical geometry is assumed
to be a transverse then the unit of the conductivity appears to be (S/m). To solve this
problem, the CNT is considered here as having an equivalent cylinder with the effective
parameters in order to manipulate it simply with Maxwell's equations. This modification
will encompass SWCNT and MWCNT antennas and can also be applied to the BCNT
antenna.
44
3.2.1 SWCNT
As mentioned before, the conductivity of a SWCNT model was derived as one
dimensional system, where the current is passing on the surface only. Now, assuming a
uniform surface current density for the SWCNT with radius and line current density
passing through it, then is defined as
One can model the CNT as a solid cylinder with radius and line current density I to
give the volume current density
From eqns. (3.1) and (3.2) . to the effective
solid cylinder and to the SWCNT leads to
where eqn. (2.12) has been used. Here, the electric field is assumed to be the same on
the surfaces of hollow and solid cylinders and this assumption will give more confidence
to this modeling since the electric field and the line current density are assumed
constant. Thus, the factor may be considered as a transformation factor from the
old to the new version.
The Drude model of the metal conductivity may be written as [1]
45
where is the number of electrons per and is the mass of electron. Hence, the
equivalent expresion for in a SWCNT may be found as
Equation (3.5) shows that the electron density in the solid cylinder is inversely
proportional to . This inverse proportion may make a constraint on the
electromagnatic properties of a SWCNT when the radius becomes large enough and the
SWCNT seems to be empty of carriers which are mainly responsible for the electric
conduction. To explain this, the values of are calculated for a SWCNT with several
radii as shown in Table 3.1. The results are to be compared with electron density of
copper which equals to electron/m3. It is seen from the table that all values
of are very far from and this may explain the high resistance of a SWCNT when
it is compared with the resistance of metals. Thus the enough carriers for the conduction
case do not exist in CNT.
According to this modeling, the input impedance of a SWCNT is equal to the
input impedance of the effective cylinder [36]
where is the length of the CNT. This is not the input impedance of the SWCNT
antenna as seen later.
46
Table 3.1 Dependence of effective carrier density of SWCNT on radii parameters.
m=n r (nm) (electron/m3) m=n r (nm) (electron/m3)
10 0.6780 1.9489×109 60 4.0680 5.9023×108
20 1.3560 1.2277×109 70 4.7460 5.3258×108
30 2.0340 9.3693×108 80 5.4240 4.8722×108
40 2.7120 7.7342×108 90 6.1020 4.5043×108
50 3.3900 6.6651×108 100 6.7800 4.1988×108
3.2.2 MWCNT
The MWCNT consists of multiple co-centric SWCNTs, where the distance
between each tube wall is approximately 0.34 nm, which is the distance between
interatomic layers of graphite (i.e., graphene sheets). The number of tube walls for an
MWCNT can vary anywhere from 2 to several hundred. From the emerging literature it
becomes clear that for far-infrared applications, individual SWCNTs have losses that are
too large (associated with their extremely small radius) to serve as antennas or
interconnects [31].
As in a SWCNT, the MWCNT is suffering from the approach of measuring the
conductivity which is considered as the main problem when these materials are applied
for the antenna applications.
If only the longitudinal current is assumed, then the current in each CNT shell of
the MWCNT (Fig. 3.1) is approximated as
where , , and are the surface longitudinal electric field, conductivity, and radius,
respectively, for the nth wall and is the number of tubes. Since the distance between
47
neighboring shells is very small and assuming no current passing between the
neighboring shells, then for all values of and the total current is
For the effective solid cylinder of radius , the longitudinal current is given by
which is assumed to be an equivalent to the current in eqn. (3.8). Therefore,
Generally, and therefore . If the relaxation frequency is
assumed for simplicity to be the same for all tubes, then the conductivity of MWCNT
can be simplified to
where is the effective conductivity of the Nth tube. In the view of electrical circuit,
48
the effective solid cylinder of a MWCNT behaves as an equivalent resistance for
parallel equal resistances.
Fig. 3.1 Schematic representation of the MWCNT cross section.
3.3 Effective Hollow CNT Conductivity
It is worth mentioning here that CNT has a hollow geometry but the concept of
effective conductivity has been adopted in the previous section using equivalent solid
cylindrical geometry in order to connect with Maxwell's equations. The solid effective
cylinder can be made equivalent to effective semi-hollow cylinder. The reason behind
this idea is trying to converge to the real world representation because the solid cylinder
is not the same as the tubed cylinder during the antenna simulation. Here this concept
will be applied to the SWCNT and MWCNT.
2
(N)
N
49
3.3.1 SWCNT
In the same manner one can consider the effective solid cylinder as a hollow and
rewrite eqn. (3.3) as
where is called hollow factor and is given by
Here and are the inner and outer radii of the hollow cylinder, respectively. The
hollow factor can make the real CNT nearer from the effective CNT on the side of
geometry. The best values of must be chosen such that the cylinder becomes more
hollow and on the other hand does not affect the simulation results. Whenever the
cylinder is hollow, then the current is guaranteed to pass on the surface of the CNT as
shown in Fig.3.2. The main purpose for this modification is the effect of electromagnetic
field being not the same when the hollow part exists or not. This is because of the larger
conductivity of a hollow cylinder with respect to a solid by a factor of which is
greater than one since in eqn. (3.13).
50
Fig. 3.2 Modeling of CNT as a (a) solid and (b) hollow cylinder.
3.3.2 MWCNT
Equation (3.11) can be modified for the hollow MWCNT model as follows
where the hollow factor of a MWCNT can be given as
51
This factor will modify the electromagnetic result as in a SWCNT with the assumption
that the distance between the adjacent walls stays small in order to avoid the high
difference of the electric field between them which may not give accurate results during
the simulation.
3.4 Analysis Approaches for CNT
This section presents two useful approaches for analyzing and investigating the
properties of various CNT-based antennas using EM software package.
3.4.1 CNT Complex Permittivity
The concept of the effective conductivity can be generalized to other quantities
such as permittivity and permeability which are considered as main parameters in
Maxwell's equations. If the relative permeability is assumed to be unity in all types of
CNT then the concern is around the permittivity. The effective conductivity of the CNT
is generally a complex quantity and can be expressed for SWCNT or MWCNT as
where and are, respectively, the real and imaginary parts of the effective
conductivity measured in (S/m). Therefore, from Maxwell's equations one can show that
the real and imaginary parts of the relative complex permittivity
can be expressed as
52
Here is the relative permittivity of the CNT material and is the permittivity of the
free space. If the imaginary part of the effective conductivity is assumed to be zero then
the real part of the complex permittivity is equal to the relative permittivity as in
metals such as copper and gold (low frequency regime).
The Drude model in terms of the relative permittivity of the metals can be
expressed as [41]
where is the plasma frequency which in the CNT is found to be
and is the collision frequency in metals and assumed to be the same as the
relaxation frequency in CNTs. For the CNT material with r=2.71 nm (m=n=40), the
values of and are found to be 1212 and 0.77 THz, respectively. This
means that the CNT will receive the signal of this frequency and beyond as known from
the plasma theory. If one assumes a large SWCNT radius with r= 67 nm (m=n=1000),
then plasma frequency of a SWCNT material is 48.481 THz which is equivalent to a
53
receiving antenna of length of .
In conjunction of complex permittivity with plasma frequency, eqns. (3.18a) and
(b) can be rewritten in terms of as
As a result, the CNT material can be modeled with either eqn. (3.18) or (3.20)
where both equations give the same results. Figure 3.3 shows the values of and for
a SWCNT having with .
Fig. 3.3 Real and imaginary parts of the complex permittivity of SWCNT with (m=n=40).
0 1000 2000 3000 4000 5000-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1x 10
6
Re(c)
Im(c)
54
3.4.2 Surface Impedance of a CNT
Maxwell's equations can be controlled in indirect method by the important
parameters which play the main role in indication of the surface impedance of the CNT
materials.
The effective propagation constant in materials is given by [41]
where is the permeability, is the permittivity, and and are the effective
attenuation and phase constant, respectively. From eqns. (3.16) and (3.22), and
are obtained as
The depth of penetration (skin depth) of the effective CNT is defined by
The values of for several values of are calculated using eqns. (3.19a) and (20)
and the results are shown in Fig. 3.4. It is shown from the mentioned figure that the
value of is an increasing function of . Note that is much greater than the
radius of the SWCNT. Since then the definition of surface impedance
55
[42] cannot be applied here. Therefore another expression for the surface
impedance is necessary.
The input impedance of SWCNT in term of can be expressed as [36]
From the antenna laws, the input impedance can be written in term of surface impedance
as [37]
Fig. 3.4 Skin depth of SWCNT according to eqns. (3.19a) and (3.20).
10 20 30 40 50 60 70 80 90 1000
200
400
600
800
1000
1200
m=n=10m=m=20m=n=30m=n=40m=n=50m=n=60m=n=70
56
Thus, the surface impedance of the SWCNT can be expressed as
where the unit denotes Ohms per square.
The surface impedance is the reciprocal of which gives the surface resistance
and reactance as follows
where
and . One can note that the surface resistance in CNT is frequency independent,
but it is frequency dependent in metals such as copper.
3.5 Wave Propagation in CNT
In this section, all quantities that are related to the wave propagation in CNT are
57
investigated. Initially, the concern is on the effective propagation constant and what is
generated from it. After that, the skin depth of the CNT will be considered.
3.5.1 Complex Propagation Constants in CNT
The concept of the wave equation in mediums can be generalized to the CNT
material, where the wave equation in terms of the electric field is now became to be
expressed as
The CNT is placed along the z-axis which is considered as the propagation variable. The
time factor is suppressed from eqn. (3.30)
The effective propagation constant of the CNT can be written in the following
form
where
The ratio of the imaginary part to the real one of the conductivity can be determined
58
from eqns. (3.32a) and (3.32b) as
This ratio is responsible for the difference in characteristic between the propagation
constant in the CNT and conventional materials, where it vanishes in the latter.
At the same time, can be written in terms of and as
also, the expression of can be written in terms of and as
The ratio of the plasma to relaxation frequency in SWCNT can be expressed as
The frequency ranges are chosen such that the low, middle, and high ranges are
covered for the purpose of simplicity and showing the behavior of this material in these
ranges as presented in the following subsections. Also, all parameters related to the
wave propagation in the CNT medium such as attenuation and propagation constant, and
phase velocity will be discussed.
59
A- Low Frequency Range ( )
In the case of ( ), eqn. (3.34a) reduces to
and
The wavelength in CNT can be found according to and eqn. (3.37)
where denotes the plasma wavelength which is a very small value since is very
large. Here, is dependent on , , and , this is different from the propagation in
metals where it is only dependent on . In order to examine eqn. (3.38), let
, the wavelength in CNT is computed to be .
B- Middle Frequency Range ( )
In the middle frequency range ( ), eqn. (3.34a) becomes
60
and
The phase velocity of the propagated wave in CNT can be expressed (according
to the relation and with aid eqn. (3.35)) as
In order to express the phase velocity of the SWCNT in terms of Fermi velocity,
substitute eqn. (3.35) into eqn. (3.41)
where is the effective temperature which can be defined by
61
As seen from eqn. (3.42), the phase velocity of the propagated wave in SWCNT
approaches the speed of light when .
C- High Frequency Range ( )
In the high frequency range ( ), eqn. (3.34a) can be expressed as
and
In this case, the phase constant does not exist and only the attenuation constant
can affect magnitude of the propagated wave through the CNT material. The value of
is large when it is compared with the first and second cases ( ).
Therefore, the hope of the propagation through the CNT ( ) is weak.
3.5.2 Effective Skin Depth in CNT
The skin depth of the CNT can be expressed as . In the frequency
ranges mentioned in Sec. 3.5.1, the value of is not equal to zero, therefore, it is
possible to find an expression for the attenuation constant at all the frequency ranges
62
except the zero frequency (DC) where the electric field is equally distributed in all
surface regions.
In the low frequency range, the effective skin depth is a function of frequency
and is given by
Equation (3.46) seems to be similar to the skin effect in metals at least from the point of
view of frequency dependence.
In the middle and high frequency ranges, the expressions of the effective skin
depth can be written, respectively, as
Here, the matter of the frequency dependent vanishes and only the plasma frequency
governs the effective skin depth.
From the previous equations, the effective skin depth depends on the radius of
the CNT material. The effective skin depth for a SWCNT (at a typical values of
, , where
, , and are low, medium, and high frequencies, respectively) are
63
for the low, medium, and high frequency ranges, respectively.
These values are still greater than the radius of the SWCNT that
have . Therefore, the effective skin depth must be
calculated at a frequency and radius such that no one falls in the problem of .
3.6 CNT Synthesis [43]
Various techniques have been developed for the synthesis of CNTs, including
arc discharge, laser vaporization, and chemical vapor deposition. This section will
concentrate on the arc discharge technique since it is widely reported in the literature as
a simple synthesis technique.
The basic concept of the arc discharge method is based on applying DC or AC
voltage to the two electrodes of graphite at a special environment as shown in Fig. 3.5.
The arc discharge technique generally involves the use of two high-purity graphite
electrodes. The anode is either pure graphite or contains metals. In the latter case, the
metals are mixed with the graphite powder and introduced in a hole made in the anode
center. The electrodes are momentarily brought into contact and an arc is struck. The
synthesis is carried out at low pressure (30-130 torr or 500 torr) in controlled atmosphere
composed of inert and/or reactant gas. The distance between the electrodes is reduced
until the flowing of a current (50 150 A). The temperature in the inter-electrode zone is
so high that carbon sublimes from the positive electrode (anode) that is consumed. A
constant gap between the anode and cathode is maintained by adjusting the position of
the anode. A plasma is formed between the electrodes. The plasma can be stabilized for
a long reaction time by controlling the distance between the electrodes by means of the
voltage (25 40 V) control. The reaction time varies from 30 60 seconds to 2 10
minutes.
64
Fig. 3.5 Arc discharge scheme.
Various kinds of products are formed in different parts of the reactor: (1) large
quantities of rubbery soot on the reactor walls; (2) web-like structures between the
cathode and the chamber walls; (3) grey hard deposit at the end of cathode; and (4)
spongy collaret around the cathodic deposit. The metals usually utilized are Fe, Ni, Co,
Mo, Y either alone or in mixture. Better results are obtained using bimetallic catalysts.
Amorphous carbon, encapsulated metal nanoparticles, polyhedral carbon are also
present in the product. When no catalyst is used, only the soot and the deposit are
formed. The soot contains fullerenes while MWNTs together with graphite carbon
nanoparticles are found in the carbon deposit. The inner diameter of the MWNTs varies
from 1 to 3 nm, the outer diameter varies in the range of 2 25 nm, the tube length does
not exceed 1 , and the tubes have closed tips. When metal catalysts are co-evaporated
with carbon in the DC arc discharge, the core of the deposit contains MWNTs, metal
filled MWNTs (MFWNTs), graphitic carbon nanoparticles, metal filled graphite carbon
nanoparticles and metal nanoparticles, while the powder-like or spongy soot contains
MWNTs, MFWNTs and SWNTs. The SWNTs have closed tips, are free of catalyst and
are either isolated or in bundles. Most of the SWNTs have diameters of 1.1 1.4 nm and
are several microns long. The collarette is mainly constituted of SWNTs (80%), isolated
or in bundles, but it is only formed in the presence of certain catalysts.
65
The physical and chemical factors influencing the arc discharge process are the
carbon vapour concentration, the carbon vapour dispersion in inert gas, the temperature
in the reactor, the composition of catalyst, the addition of promoters and the presence of
hydrogen. These factors affect the nucleation and the growth of the nanotubes, their
inner and outer diameters and the type of nanotubes (SWNTs, MWNTs). The amount of
carbon nanoparticles was found to diminish when pure hydrogen was used during the
reaction.
Table 3.2 summarizes the methods of production of a CNT by arc discharge
technique under different synthesis conditions.
Table 3.2 CNTs synthesized by arc discharge method applying different synthesis conditions.
Product Comments Conditions CNTs Deionized water MWNTsCNTs Metal filledMWNTs, SWNTs
NaCl solution
SWNTs Liquid N2CNTs Continuous productionSWNTs Diameter= 0.9 1.4 nm Ni and CaC2/Ni catalyst,
HeSWNTs High yield Ni catalyst
MWNTsThe product type depends on thecatalyst composition
Fe catalysts, various Fesources
DWNTs
Large quantity, high quality, diameter= 2 6 nm
KCl/FeS catalyst, H2
Mixture of Ni/Co/Fe smallamount of SFeS, CoS, NiS catalysts, H2
Bundles of high qualityMWNTs Optimization process Graphite electrodes, H2
66
Chapter Four
Investigation of CNT Dipole Antenna Using Complex
Permittivity Approach
4.1 Introduction
In this chapter, the complex permittivity approach mentioned in chapter three is
applied to CNT dipole antennas by means of a CST software package. The SWCNT and
MWCNT antennas are modeled to assess antenna properties such as radiation pattern,
efficiency, and input impedance.
4.2 SWCNT Antenna Parameters
There are several methods for the investigation of the properties of the CNT
antenna. The transmission line method [20] assumes the SWCNT antenna as a flared
transmission line. The parameters of a transmission line are generalized to the SWCNT
antenna. This method is capable of simple understanding of the effect of plasmon
wavelength in CNT properties which is attributed to the existing kinetic inductance and
quantum capacitance. Another method considers the SWCNT as a tubed cylinder with a
longitudinal current only and the method of moments is applied as a solver for the
Maxwell's equations [1]. The mentioned references agree with the fact that SWCNT
antenna does not resonate at and its multiples where is the free space
resonance wavelength. The SWCNT will resonate at and its multiples, where
is the plasmon wavelength. Then, the resonance frequency is computed from
instead of where , c, and are the phase velocity, speed of light
in free space, and resonance frequency, respectively. Usually, the phase velocity has
values close to the Fermi velocity but not equal to it and it is estimated here because
67
there is no exact formula in SWCNT. The previous explanation will support the results
when the complex permittivity is used for modeling the CNT antenna shown in Fig. 4.1.
Fig. 4.1 Geometry of the effective SWCNT or MWCNT antenna.
68
4.3 Modeling of SWCNT Antenna Using Complex Permittivity
Approach
In this approach the numerical analysis depends on knowing the constitutive
parameters of the antenna material. These parameters have been modeled in chapter
three for CNT material and accordingly the CNT material becomes ready for the
simulation. Because the CNT is non-magnetic material [31], the relative permeability is
assumed to be unity such that where is the permeability of CNT and is the
free space permeability. In this work, the CNT material is set up as a new material in
CST environment with normal type. Discrete values of and are tabulated in
dielectric dispersion fit window and computed using MATLAB. The frequency range is
set up from to GHz with GHz frequency separation between two successive
samples.
In fact, there is another way for dealing with the complex permittivity approach
using CST software package, where the CNT material is set as a new material with
normal type and Drude dispersion model. The value of epsilon infinity is one for all
types of CNT material according to eqn. (3.19), where the term after the minus on the
RHS will be finished when . The value of plasma frequency is computed from
eqn. (3.20), and the collision frequency is considered as the same value of the relaxation
frequency.
Equations (3.21a) and (b) state the relations between the real and imaginary parts
of the relative complex permittivity, respectively, and both plasma and relaxation
frequencies. Therefore, no difference between the two ways is expected, but the input
data of the relative complex permittivity which is required by the CST needs high
accuracy and this may not be adjusted quietly. Therefore, feeding the input data of the
plasma and relaxation frequencies when asked by the CST seems to be healthier.
69
By using of the concept of the solidity in chapter three, one can convert the
electric and magnetic fields on the surface of original CNT (hollow cylinder) to the
electric field and magnetic field in all regions of the solid cylinder model. The
direction of and depends on the location of the CNT antenna in simulation
environment. Using Maxwell's equations (eqns. (2.27a) and (b)) after replacing eqn.
(2.27b) by
will solve the CNT antenna problem, where is the relative complex permittivity
defined by eqn. (3.17). Maxwell's equations may be solved by several methods.
Choosing CST program as a numerical solver for these equations (after adjusting all
parameters properly) will give more efficient results. The flowchart in Fig. 4.2 shows the
process applied to CNT antenna using CST package.
4.3.1 Resonance Frequencies
The SWCNT antenna is modeled with ,
and as shown in Fig. 4.3. The plasma and relaxation frequencies
are computed to be and , respectively. The delta gap width is
chosen to be 2r. The CNT is meshed tetrahedrally with mesh cells such that the
convergence of S-parameters is satisfied with avoiding the long computational time.
71
Fig. 4.3 Modeled SWCNT antenna with and .
The simulation results reveal that resonance frequencies occur at ,
and GHz. According to the relation and GHz, the
phase velocity is calculated to be and this value is near to
the expected phase velocity which is calculated according to the transmission line model
as , where c is the speed of light in free space. The phase velocities at the
second, third, and fourth resonance frequencies are equal to , , and
, respectively. The four phase velocities (at resonance frequencies) seem to
be close. Therefore, these values predict the proper working of the simulated SWCNT
antenna.
The first resonance frequency of this antenna type is close to the collision
frequency which may be the natural frequency that leads to the resonance case. In
conventional antenna, the relaxation frequency is very high and this leads to the
conclusion that the oscillation is independent of the collision frequency.
Figure 4.4 shows the variation of the relaxation and plasma frequencies with
72
radius index. Both frequencies decrease sharply with for . When increases
beyond , the variation of these two frequencies with decreases and both frequencies
approach asymptotic values when tends to . At and oK,
THz and THz. These values are to be compared with and
THz, respectively, when .
Fig. 4.4 (a) Plasma frequency of a SWCNT versus m (b) relaxation frequency of a SWCNT
versus m at difference temperature values.
The first resonance frequency is found to be a decreasing function of m as shown
in Fig. 4.5. The resonance frequency decreases from to GHz as increases
from to . One of the main parameters that affects the resonance frequency is the
50 100 150 2000
100
200
300
400
500
600
700
800
m
f p(T
Hz)
50 100 150 2000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
m
(TH
z)
T=250oK
T=300oK
(b)(a)
73
relaxation frequency and this will be supported by where stays constant but
increases according to (see Fig. 4.4b for variation of with ).
Fig. 4.5 First resonance frequency versus m index for a -armchair SWCNT antenna.
4.3.2 S11 Parameter
The S11 parameter or reflection coefficient of SWCNT antenna is shown in Fig.
4.6. The very small values of the reflection coefficient at the first and second resonance
frequencies (< dB) mean the SWCNT antenna has low reflection losses.
In this work, the adjustment of the input impedance of the discrete port
depends on getting a minimum value of the reflection coefficient or S11 parameter. In the
20 40 60 80 100 120 140 160 180 200800
820
840
860
880
900
920
940
960
980
1000
m
f r1(G
Hz)
74
first time, the initial value of has been estimated from and in each simulation run
the seeking is toward the minimum value of S11. The S11 parameter values of the
simulated CNT antenna at are
dB, respectively. These small values are encouraging because more incident power will
be transmitted and a little of it will be reflected. The computation time of the CST
program depends on the optimal value of S11 parameter. This property will accelerate the
final result. In testing the complex permittivity approach, it is noted that any wrong
change in the input data will make it impossible to get S11 parameter in this good range
and this will give a good hoping for this approach.
From Fig. 4.6, the minimum values of the S11 parameter of the SWCNT antenna
which occur at its resonance frequencies increase at high resonance frequencies. This
does not exist in the conventional dipole antennas. This problem is due to the nature of
the CNT conductivity, where the decrease in the CNT conductivity at high frequencies
makes this material deviate from the state of good conductivity. This problem is
naturally solved when the point of operation is far toward more high frequencies. The
general expression of the conductivity (eqn. (2.8)) declares this fact, where the
conductivity is capacitively and inductively oscillated for all frequency ranges.
75
Fig. 4.6 Magnitude of S11 parameter of a SWCNT designed with m=n=40 and
Figure 4.7 shows the S11 parameter of a SWCNT antenna for several values of m.
These results are estimated at the first resonance frequency and show that dB S11
dB for the range of considered here. The estimated value may be
considered acceptable when dealing with this type of antennas where the source power
must be small enough since it has a very low efficiency.
4.3.3 Input Impedance
As it is well known, the input impedance is considered as the main parameter in
antenna design. A SWCNT antenna has input impedance which is quite different
from the classical antenna. The standard value of this impedance is , but the
0 1000 2000 3000 4000 5000-35
-30
-25
-20
-15
-10
-5
0
Frequency (GHz)
S11
par
amet
er(d
B)
76
classical antenna has a standard value of . Therefore, the task of adjustment of this
quantity is very important since any error will lead to destroy the transmitting or
receiving signal. The matter of the power is sensitive here due to the very small values
of the source power and high value of the input impedance of the CNT antenna with
respect to the classical antenna.
Fig. 4.7 S11 parameters of a SWCNT antenna estimated at first resonance frequency for
several values of m when .
The simulated real and imaginary parts of the input impedance are shown in Fig.
4.8. The first value of the antiresonance occurs at GHz. The resonance
frequencies occur at the transition points from the capacitive to the inductive properties,
20 40 60 80 100 120 140 160 180 200-40
-35
-30
-25
-20
-15
m
S11
(dB
)
77
but antiresonance frequencies occur at the transitions from inductive to capacitive
region. Therefore, in order to design a SWCNT antenna the resonance frequencies
should be chosen over antiresonance frequencies where the latter have large and rapid
changes in values of the impedance. The values of the input resistance of the discrete
port seem to be in the range of the quantum resistance and this value is compatible with
the quantum resistance in the nanoelectronics [1], [19] where the CNT antenna is to be
matched with it. Therefore, in order to make the SWCNT antenna work properly the
impedance matching issue must be taken carefully to avoid the high input impedance
which may not be satisfied in nanoelectronics, and at the same time, the large values
may make deviation in the optimal initial values which speed up the running of CST. It
is noted during the simulation of the CNT antenna by CST that one of the main
parameters that accelerates the solution is the initial value of the input impedance. For
example, if the input impedance of the discrete port is chosen to be greater than
then no good results are obtained. The reason is that the CST is a numerical solver
whose convergence speed depends on the good estimation of the initial values.
Therefore, after choosing the initial value of the input impedance of the discrete port,
depending on the calculated input impedance of the SWCNT transmission line in
chapter three, the input impedance of the discrete is found to be . One of the
main advantages of this approach over referring references is its capability of computing
the input impedance of the discrete port that is connected to the CNT antenna because
this feature is very important during the designing of any type of antennas.
78
Fig. 4.8 Input impedance of a SWCNT antenna designed with and
The input impedance of the discrete port versus several values of the
armchair SWCNT diameters is shown in Fig. 4.9. Here, a SWCNT of length is
simulated using the complex permittivity approach. Figure 4.9 shows decrease in the
values of as the value of m increases. Therefore, as in the classical antennas,
SWCNT must operate under matched condition which can be achieved by varying of
as well as the CNT length.
0 1000 2000 3000 4000 5000
-100
-50
0
50
100
150
Frequency (GHz)
Rea
lan
dim
agin
ary
par
tsof
Z in(k
) Re(Zin
)
Im(Zin
)
79
Fig. 4.9 Input impedance of the discrete port versus index for a -armchair
SWCNT antenna.
Figure 4.10 shows the input resistance of the SWCNT antenna for
, and simulated using the complex permittivity approach where the peaks
change increasingly with different values of m. The input reactance is shown in Fig. 4.11
and it states the increasing in the reactance peaks as in the input resistance. The increase
in the peaks of input resistance and reactance with is due to the wide bandwidth
which is set by CST, where the frequency domain solver needs in narrow bandwidth
setting in order to give more accurate results. The wide bandwidth has been chosen for
seeing at least four resonance frequency regions.
20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
12
m
Zin
p(k
)
80
Fig. 4.10 Input resistance of a SWCNT antenna for several values and
Fig. 4.11 Input reactance of a SWCNT antenna for several values and
0 1000 2000 3000 4000 5000-50
0
50
100
150
200
250
300
Frequency (GHz)
Inp
ut
resi
stan
ce(k
)
m=40m=50m=60
0 1000 2000 3000 4000 5000-150
-100
-50
0
50
100
150
Frequency (GHz)
Inp
ut
reac
tan
ce(k
)
m=40m=50m=60
81
4.3.4 Radiation Pattern, Gain, and Efficiency
The radiation pattern (directivity) of the SWCNT antenna at the resonance
frequencies is depicted in Fig. 4.12. On the other hand, the maximum values of the
directivity as a function of frequency are shown in Fig. 4.13. The low values of the
directivity of Fig. 4.12 are interpreted by Fig. 4.13 where the maximum value of the
directivity at frequency GHz is . In moment, the directivity is enhanced in the
other resonance frequencies as shown in Fig. 4.12b, c, and d. The best value of the
directivity is found at which is equal to . This value can be
compared with the directivity of the conventional dipole antenna which is .
Therefore in order to obtain a CNT working with a maximum directivity, the first
resonance frequency does not satisfy this requirement, i.e. another frequency must be
sought such that the directivity and matching should be satisfied.
Figure 4.13 explains that the maximum directivity is not obtained at one of the
resonance frequencies. The reason for this phenomenon is the conductivity which makes
the SWCNT a propagation filter with high pass filter mode, where there is a relation
between the directivity and electric field which is related to the conductivity by Ohm's
law. At the same time, the high frequency values of the directivity are about which is
considered as the maximum value of the directivity of a classical dipole antenna.
The realized gain of the simulated antenna is shown in Fig. 4.14. Four peak gain
values are shown whose frequency location coincides to some extent, with the four
resonance frequencies shown in Fig. 4.6. The gain of a SWCNT is very low as expected
in Ref. [20] since the antenna is designed with a very small radius with respect to the
length. In other words, this antenna is electrically very thin according to
condition. Therefore, an agreement about this problem must be obeyed.
In order not to waste more time, the efficiency is considered as a copy of gain
because of unity value of the directivity. The efficiency parameter is considered as the
master feature in the antenna properties if the matter of power is considered. Since the
82
total efficiency is approximately equal to the radiation efficiency (see Table 4.1),
then the SWCNT antenna does not suffer from the reflection efficiency where
and . Here and are dielectric and conduction efficiencies,
respectively However, the antenna suffers from the low radiation efficiency as
mentioned before. Also, the radiation efficiency can be written in the form
, where are the radiation and loss resistances, respectively.
(a)
Fig. 4.12 Radiation pattern for a SWCNT designed with and at (a)
(b) (c) (d) .
84
(d)
Fig. 4.12 (Continued).
Fig. 4.13 Directivity versus frequency for a SWCNT antenna designed with m=n=40 and
L= .
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Frequency (GHz)
Dir
ectiv
ity
85
Fig. 4.14 Realized gain versus frequency for a SWCNT antenna having and
L= .
Table 4.1 Gain, radiation efficiency, and total efficiency at the resonance frequencies for a SWCNT
antenna designed with and
Resonance frequency
(GHz)
Gain
G
Radiation efficiency Total efficiency
891
2400
3680
4761
0 1000 2000 3000 4000 50000
1
2
3
4
5
6x 10
-6
Frequency (GHz)
Rea
lized
gain
86
4.4 Simulation of MWCNT Antennas
As in SWCNT antenna, the complex permittivity approach is used to simulate a
MWCNT antenna using CST software package. This type of antennas can be
constructed according to the number of shells in each MWCNT material. In real world, a
MWCNT antenna consists of many parts and each part is called a SWCNT. In the
environment of simulation, the MWCNT antenna is simulated by considering one
equivalent part which is called an effective MWCNT antenna. In other words, the
conductivity of the overall MWCNT is modified to be an effective conductivity (eqn.
(3.10)). Therefore, the final geometry is a MWCNT dipole antenna that is similar to the
SWCNT dipole antenna but with mostly different parameters. Figure 4.15 shows the
geometrical representation of a MWCNT dipole antenna positioned along z-axis and
built by CST package.
Fig. 4.15 Geometrical representation of a MWCNT dipole antenna built by CST package.
87
A number of MWCNT dipole antennas are simulated here, each with a different
number of shells (N). The shell number means that the MWCNT dipole antenna
consists of SWCNTs each inside the other. The inner SWCNT starts with dual
index , the dual index of the second SWCNT is 10, and so on. Finally, the
outer SWCNT will have . The same thing is valid
for . This scenario of increasing the tube shells has been
made by [31]. The results of simulated MWCNT dipole antennas of length with
several shell numbers are shown in Table 4.2. It is clear from this table that as the
number of shells increases then the first resonance frequency shifts forward since the
conductivity is increasing with N. On other hand, the plasmon wave number
decreases. Therefore, the free space wave number becomes as the dominant factor
in term. Whenever , then the unusual low resonance frequency will
be terminated and a MWCNT antenna seems to be a classical antenna. Therefore, a
compromised solution must be made through designing a MWCNT antenna. The sharp
decrease in the input impedance of the discrete port with is deduced from eqn.
(3.10) or (3.11). The summation in eqn. (3.10) will increase the conductivity as
increases. The low effect of radiation resistance in CNT is due to the small radius which
makes the input impedance of CNT line nearly alike to the input impedance of the CNT
antenna.
Figure 4.16a and b shows the input resistance and input reactance, respectively,
versus frequency for a MWCNT antenna simulated with and . At
the peak values, the resonance state is achieved. As in SWCNT antenna, the values of
the input impedance of the MWCNT antenna are still very large. This makes the
thinking with the antiresonance regions impossible. The justification which has been
mentioned about the antiresonance in SWCNT antenna can be also applied to the
MWCNT antennas.
88
Table 4.2 Results of simulated MWCNT dipole antenna of length and several shell numbers.
Number of
shells
N
Resonance
frequency
(GHz)
Discrete
port impedance
Directivity
D
Total
efficiency
The S11 parameter is considered as one of the main parameters used to
characterize antenna performance. This performance is shown in Fig. 4.17. In this
figure, S11 spectra are plotted for . When the number of shells increases,
the first resonance moves gradually to upper frequencies. When , the lowest first
resonance frequency is attained and within the frequency range there are three others
resonance frequencies. In the case of N=10, 15, and 20, the number of resonance
frequencies is two. Finally, there is one resonance frequency when N=25 and 30. Thus,
the preferred region of operation depends on the demand and this must be confirmed by
the number of shells.
89
(a)
(b)
Fig. 4.16 Input resistance (a) and input reactance (b) of a MWCNT dipole antenna length
with various values of number of shells (N).
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-50
0
50
100
150
200
250
300
Frequency (GHz)
Inp
ut
resi
stan
ce(k
)
N=5N=10N=15N=20N=25N=30
0 1000 2000 3000 4000 5000-150
-100
-50
0
50
100
150
Frequency (GHz)
Inp
ut
reac
tan
ce(k
)
N=5N=10N=15N=20N=25N=30
90
Fig. 4.17 S11 parameter of a MWCNT dipole antenna length with various values of number of
shells (N).
The directivity D and efficiency of MWCNT antenna appear to be adjusted to the
ideal case as N is enlarged. This will keep away from the problem of low directivity at
the first resonance frequency. More details on the directivity are depicted in Fig. 4.18
which shows the directivity does not efficiently change with . As long as the directivity
stays constant and at the same time the efficiency increases then the gain will be
increased as given by .
The radiation patterns for the MWCNT antenna simulated at the first resonance
frequency for different values of and the results are shown in Fig. 4.19. In the case
of , the maximum value of the directivity is . As it is expected, the
maximum value of the directivity is improved when N increase to become at
as shown in Table 4.2.
0 1000 2000 3000 4000 5000-60
-50
-40
-30
-20
-10
0
Frequency (GHz)
S11
Par
amet
er(d
B)
N=5N=10N=15N=20N=25N=30
91
Fig. 4.18 Directivity versus frequency of a MWCNT dipole antenna length with various
values of number of shells (N).
(a)
Fig. 4.19 Radiation pattern for a MWCNT designed with and (a) (b)
(c) (d) (e) (f) .
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Frequency (GHz)
Dir
ectiv
ity
N=5N=10N=15N=20N=25N=30
94
(f)
Fig. 4.19 (Continued).
For the purpose of comparison between the SWCNT and MWCNT antennas,
consider a SWCNT antenna having and MWCNT antenna having
where the outer SWCNT has . Assume that both antennas have the same
length which is equal to . Table 4.3 summarizes the main performance results
deduced at the first resonance frequency for each antenna.
As a simple comparison with a SWCNT antenna, the MWCNT offers best
performance if the issue of the low resonance frequency with respect to the electrical
length is not taken into consideration. The prominent feature over a SWCNT antenna is
the modification in efficiency and subsequently the gain is improved.
95
Table 4.3 Results of comparison between SWCNT and MWCNT antennas having the same outer
radius.
Parameters SWCNT MWCNT
(GHz) 831 2041
Zinp (k ) 0.992 0.409
S11 (dB) -20 -49
D 0.2375
G
4.5 Simulation of BUNDLE CNT Antennas
The Bundle CNT antenna that is modeled in CST can be described as
aggregation of SWCNT lines positioned along z-axis and arranged in a rectangular form
to become a rectangular bundle CNT (RBCNT) antenna as illustrated in Fig. 4.20. The
distance between adjacent SWCNTs is chosen to be 1nm and the length of each SWCNT
is set to and dual index of . Two square contacts are placed in the
feeding region each with thickness equal to and the length of the discrete port is .
The contact is placed in this shape to make a discrete port touch all tubes. The
simulation frequency range is set to in order to cover at least the first
resonance frequency band. Several values of the number of bundles are chosen in the
simulation in order to inspect carefully the properties of a RBCNT antenna.
97
Table 4.4 Performance parameters of RBCNT antenna designed with different values of N.
Numberof SWCNTs
N
Resonancefrequency
(GHz)
Maximum gainfrequency
(GHz)
Impedance ofthe discrete port
RealizedGain
Fig. 4.21 S11 parameter of the RBCNT antenna for several numbers of SWCNTs, N.
98
The radiation pattern of a RBCNT antenna having is shown in Fig. 4.22.
This figure shows that the directivity stays in the range of 1.64 (directivity of the
conventional antenna). Therefore, no problem about the directivity is of concern.
The maximum value of realized gain at is calculated by taking N as
independent parameter and the results are shown in Fig. 4.23. Note that the gain is
modified as long as N is large. The frequencies at maximum gain and are listed
in Table 4.4.
Fig. 4.22 Radiation pattern of the RBCNT antenna for and GHz.
The simulation results related to the case of varying the dual index of each
SWCNT in the RBCNT antenna of , while keeping the same separation distance
between two SWCNTs as used in the previous RBCNT antenna, is illustrated in Fig.
4.24. This figure shows the dependence of S11 parameter on the frequency. The value of
first resonance frequency decreases when m increases. This makes the entire SWCNT
99
governs the properties of a RBCNT as well as N. Also, the input impedance of the
discrete port decreases with increasing the peak of the realized gain as shown in
Table 4.5. From this table, seems to approach the value of the input impedance of
the classical antenna and makes of the RBCNT unaffected by the quantum
resistance. At the same time, the first resonance frequency is still in the range of
relaxation frequency. Hence, the RBCNT antenna may be considered as a compromise
solution between the CNT and the classical antennas. Note that the peak of maximum
gain at increases without increasing N, therefore, in order to modify the
efficiency of a RBCNT antenna, a little of bundle element is enough with increasing the
radius of each SWCNT element for the purpose of enhancement of gain and efficiency.
Table 4.5 First resonance frequency, input impedance of the discrete port, and peak of maximum gain
of a RBCNT antenna with various m=n values and N=16.
Dual
index
m=n
Resonance
frequency
(GHz)
Input impedance of
the discrete port
Gain
Peak
100
Fig. 4.23 Maximum value of realized gain at =0, for various N.
Fig. 4.24 S11 parameter of a RBCNT antenna with various m.
0 1000 2000 3000 4000 50000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5x 10
-3
Frequency (GHz)
Rea
lized
Gai
n
N=4
N=9
N=16
N=25
101
4.6 CNT Contact
One of the main problems of CNT antenna is the contact of this type of antenna
with the external world [44], [2], [45]. The small hollow geometry of CNT makes the
connection more difficult. Therefore another solid material is necessary as an interface
and should be chosen carefully such that it does not affect the main properties of the
SWCNT antenna. First, gold and copper are chosen in this work as metal contacts
between the CNT rod and the feeding point. The length of the contact must be as
short as possible in order to make the current on the CNT having enough time to
oscillate sinusoidally with respect to the contact material. To address this issue, the
SWCNT is modeled in CST environment using the complex permittivity approach with
the following parameters: The
reason for changing the length of the CNT antenna from to is to make the
ratio of the contact length to antenna length as small as possible. Several values of
contact length are taken and the results are compared with a CNT antenna designed
without contact (the discrete port is connected directly to the CNT rods). Figure 4.25
shows the effective SWCNT antenna with a metal contact which is modeled in CST
environment.
Fig. 4.25 Geometical representation of the simulated SWCNT antenna with a metal contact.
102
When the length of SWCNT antenna increases from to , the first
resonance frequency decreases. Therefore, the frequency band is set from
instead of . Within this frequency range, the input
impedance of the discrete port is computed for several values of the contact
length . It is expected that when increases then the first resonance frequency also
increases due to the effect of copper (or gold) which has relaxation frequency much
greater than that of the SWCNT. Several values of are chosen and the related input
impedance of the discrete port and the first resonance frequencies for each case
are computed as shown in Figs. 4.26 and 4.27, respectively. The results indicate
that decreases with the contact length. This is because of the effect of quantum
resistance which has large value with respect to the reciprocal of the conductivity for
copper or gold contacts. The cause of increase in the values of the first resonance
frequency is the effect of relaxation frequency in SWCNT where it is much smaller than
the relaxation frequency in nanowires (copper or gold). As a result, CNT antenna can
change its properties according to the design environment, for example by adding an
external material to the contact point with different lengths.
It is worth noting here that there is a small effect variation between using copper
or gold contact on the values of but there is no change is noticed in the resonance
frequency. This makes a free choose of the material type. Because of the effect of
copper or gold contact, the two materials are used to design an antenna having a length
of and radius nm. Figure 4.28 shows the S11 parameters for the copper and
gold antenna which seem to be near to each other with respect to CNT antenna. The
copper- and gold-contact antennas resonate at and THz, respectively. The
SWCNT antenna with the same geometry parameters but without contact resonates at
GHz. The input impedances of the discrete port of copper-and gold-contact
antennas are found to be and , respectively, and for a contactless SWCNT
103
antenna is . Therefore, in each case the contacted CNT tries to change its
properties toward that of an antenna designed from the contact material.
Fig. 4.26 Input impedance of the discrete port versus contact length of a SWCNT antenna with
different material contacts.
104
Fig. 4.27 First resonance frequency of a SWCNT antenna versus contact length.
Fig. 4.28 S11 parameter of copper and gold antennas.
0 200 400 600 800 1000 1200 1400400
500
600
700
800
900
1000
1100
1200
1300
Contact length (nm)
Fir
stre
son
ance
freq
uen
cy(G
Hz)
Copper contactGold contact
50 51 52 53 54 55 56 57 58 59 60-80
-70
-60
-50
-40
-30
-20
-10
Frequency (THz)
S11
Par
amet
er(d
B)
CopperGold
105
4.7 Feeding CNT Antenna with CNT Transmission Line
One of the main design specifications is connecting the CNT antenna with the
external world. In this work, the CNT antenna is proposed to be connected with a two-
wire transmission line (TL) and this transmission line is assumed to have the same
properties of the antenna (i.e., CNT TL) as shown in Fig. 4.29. In this figure, the length
of the TL ( ) is equal to L/32 for the purpose of viewing. Several values of have
been chosen as: L, L/2, L/4, L/8, L/16, and L/32. The length and radius of a SWCNT
antenna are set to L= and r=2.71 nm, respectively.
Fig. 4.29 SWCNT antenna connected with SWCNT transmission line.
Figure 4.30 shows the S11 characteristics of the simulated SWCNT antenna with
different TL lengths. For TL length = L, the lowest first resonance frequency is
obtained and the following is at = L/2, and so on. Therefore, the resonance
frequency of the SWCNT antenna depends on the length of the feeding TL, where
GHz for a SWCNT antenna simulated without TL and GHz at =
106
L. This high difference makes the issue of matching the CNT antenna with the TL more
essential. Also, can be changed but not rapidly (decreasing or increasing) as
increases where its values alternate at nearest values as shown in Table 4.6. Here, let the
input impedance of the SWCNT antenna be termed and the characteristic
impedance of the SWCNT TL as , where . Thus, the
characteristic impedance of the SWCNT TL can be calculated as for
= L, where for all TL lengths.
Fig. 4.30 S11 parameter of a SWCNT antenna connected with a SWCNT transmission line having various lengths.
107
Table 4.6 First resonance frequency and input impedance of the discrete port of a SWCNT
antenna connected with a SWCNT transmission line of various lengths.
Transmission
line length
( )
First resonance
frequency
Input impedance
of the discrete port
2.0000 291 4.512
1.0000 516 3.215
0.5000 686 3.523
0.2500 770 4.250
0.1250 791 4.225
0.0625 810 4.579
Without 815 4.706
108
Chapter Five
Simulation of Advanced CNT Antenna Configurations
5.1 Introduction
This chapter introduces new configurations for the CNT antenna which are
denoted as a loop and helical CNT antennas. The design of these antennas depends on
the geometry which consists of one or more CNT elements. The pending of CNTs exists
and it is assumed it does not affect the electromagnetic properties because of the high
ratio between the radius of the loop and the radius of CNT element. CST MWs is used
as a simulator for these antennas in conjunction with complex permittivity approach
which has been mentioned in chapter four. All properties of these antennas are inspected
such as S11 parameter, input impedance, and resonance frequency.
5.2 Loop CNT Antennas
Two loop antennas are introduced here, square loop CNT (SLCNT) and circular
loop CNT (CLCNT) antennas as shown in Fig. 5.1. Each type of these two antennas is
assumed to have a dual index of and =2500K. Also, all
results are compared with those of a SWCNT dipole antenna having the same tube
radius and a length equal to the length of the diameter D0 of SLCNT or CLCNT antenna.
The frequency range is set from GHz such that the first resonance frequencies
occur obviously.
5.2.1 Square Loop CNT Antenna
The diameter D0 of the SLCNT antenna is related to the length of each SWCNT
element as D0 . The center of the SLCNT antenna is located at the origin
109
and the discrete port is positioned at ( ) with delta gap length
( ).
(a)
(b)
Fig. 5.1 Geometry representation of (a) SLCNT and (b) CLCNT antennas.
A- S11 Parameter
Figure 5.2 shows the S11 parameter of SLCNT and SWCNT dipole antennas.
Note that the first resonance frequency of the SLCNT antenna (= 586 GHz) is lower
than the first resonance frequency of the SWCNT dipole antenna (= 850 GHz). At the
110
same time, SLCNT antenna has eight resonance frequencies within the frequency range
from 0-3000 GHz while the SWCNT dipole antenna has two resonance frequencies in
this range. This makes the SLCNT antenna useful in the field of the multi-band
antennas. Also, there is a main feature where the first resonance frequency occurs near
to the zero frequency. This makes this antenna be used in the baseband application and
this can be satisfied when of the discrete port is set at the counterpart of this
frequency.
Fig. 5.2 S11 parameter of a SLCNT and SWCNT dipole antennas.
B- Input Impedance
Figure 5.3a and b shows the input resistance and input reactance, respectively,
with frequency for the SLCNT and SWCNT dipole antennas. As in the S11 parameter,
there are eight peaks for the input resistance and reactance compared with two peaks for
0 1000 2000 3000 4000 5000-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Frequency (GHz)
S11
par
amet
er(d
B)
SLCNT
Dipole
111
the SWCNT dipole antennas within the frequency range (0-3000 GHz). Note that the
peaks of input resistance and reactance decrease in height with increasing the resonance
frequency for both antennas.
C- Radiation Pattern, Gain, and Efficiency
It is expected that one can modify the directivity in some regions when a SLCNT
antenna is used and this does occur as shown in Fig. 5.4. Here a maximum directivity of
is achieved at GHz which has a counterpart value equal to for the
SWCNT dipole antenna. In certain frequency range, the curve of the directivity of the
SLCNT antenna seems to have an oscillating behavior around the curve of the SWCNT
dipole antenna. This may be considered as a good advantage for investigating various
approaches of simulation of CNT antenna since the SLCNT antenna is as a result of a
SWCNT dipole antenna, i.e., no complication in geometry is acquired as in the more
complex antennas such as a helix.
Figures 5.5 and 5.6 show the values of radiation pattern of the SWCNT dipole
and SLCNT antennas, respectively. The simulation is carried out at the first resonance
frequency for each antenna. The maximum directivity of the SLCNT antenna is 0.419 at
compared with 0.248 at for the SWCNT antenna.
In addition to the increase in the directivity, SLCNT antenna has an increasing
gain as shown in Fig. 5.7. The total efficiency is modified from for the
SWCNT dipole antenna to for the SLCNT antenna. This change in the total
efficiency is due to the small increase in the electrical radius of the SLCNT antenna.
112
(a)
(b)Fig. 5.3 Input resistance (a) and input reactance (b) of SLCNT and SWCNT dipole antennas.
0 1000 2000 3000 4000 50000
50
100
150
200
250
300
350
400
450
Frequency (GHz)
Inp
ut
resi
stan
ce(k
)SLCNTDipole
0 1000 2000 3000 4000 5000-250
-200
-150
-100
-50
0
50
100
150
200
250
Frequency (GHz)
Inp
ut
reac
tan
ce(k
)
SLCNTDipole
113
Fig. 5.4 Directivity of SLCNT and SWCNT dipole antennas.
Fig. 5.5 Radiation pattern for SWCNT dipole antenna at GHz.
114
Fig. 5.6 Radiation pattern for SLCNT antenna at GHz.
Fig. 5.7 Realized gain of SLCNT and SWCNT dipole antennas.
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
1.2
1.4x 10
-5
SLCNTDipole
115
5.2.2 Circular Loop CNT Antenna
The circular loop CLCNT antenna seems to have properties more nearly similar
to those of the SLCNT antenna as shown in Figs. 5.8 - 5.12. Therefore the general
comment will be similar to those reported in subsection 5.2.1.
Fig. 5.8 S11 parameter of CLCNT and SWCNT dipole antennas.
Fig. 5.9 Input resistance of SLCNT and SWCNT dipole antennas.
0 1000 2000 3000 4000 5000-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
CLCNT
Dipole
0 1000 2000 3000 4000 50000
50
100
150
200
250
CLCNTDipole
116
Fig. 5.10 Directivity of CLCNT and SWCNT dipole antennas.
Fig. 5.11 Realized gain of CLCNT and SWCNT dipole antennas.
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8x 10
-5
CLCNT
Dipole
117
Fig. 5.12 Radiation pattern for CLCNT antenna at GHz.
5.3 Helical SWCNT Antenna
In addition to the loop CNT antennas, the helical single-wall carbon nanotube
(HSWCNT) antenna is added to the simulated CNT antenna group as shown in Fig.
5.13. The purpose of this section is to study the properties of this type of antennas and
show the possibility of realization it depending on the constraint that does not affect the
physical properties. One of these constraints is the curvature of the CNT which may
affect the electrical properties of the helical antenna. In fact, the presence of bending
may change the conducting SWCNT into semiconducting SWCNT [13]. Therefore, the
ratio of radius of the HSWCNT antenna to the radius of SWCNT element is
chosen to be large enough.
118
Fig. 5.13 Geometry representation of the HSWCNT antenna.
5.3.1 Geometry Generation
Initially, the simplest form of a HSWCNT antenna is manipulated. The
geometrical configuration of the HSWCNT antenna consists of turns,
diameter D0 = 2 and spacing between each turn. The total
length of the antenna is while the total length of the SWCNT
wire is where
is the length of the SWCNT wire between each turn and
is the circumference of the HSWCNT antenna. The discrete port is located in
the middle of the antenna with delta gap distance and (
119
). CST MWs with the aid of the complex permittivity approach is used here to
simulate this type of SWCNT antennas. The used constraints are directly related to the
CNT materials such as radius and length without going to geometry complexity or
discrete port location. The results are compared with a SWCNT antenna with varying
length and radius.
5.3.2 S11 Characteristics and Resonance Frequencies
Figure 5.14 shows S11 characteristics of the HSWCNT and SWCNT antennas
where the length of the SWCNT antenna is chosen to be . On the
other hand, another SWCNT antenna is simulated but with for the
purpose of seeing which one is close to the HSWCNT antenna. From the values of S11
parameter, the input impedance of the discrete port of each antenna is set by going
to the minimum value of S11 parameter at first resonance frequency until the
convergence is satisfied. The calculated input impedances of the discrete port are
for HSWCNT and SWCNT with
antenna, respectively. The counterparts of the first resonance frequency
for the three antennas are computed and found to be .
The first resonance frequency of the HSWCNT antenna is very close to the
longer SWCNT antenna than to the short SWCNT antenna. This makes the effect of the
height H dominant over the length of the SWCNT wire. In addition, the HSWCNT
antenna offers a multiple resonance frequencies in the range of 0 -2000 GHz where this
antenna can be classified as a multi-band antenna.
120
Fig. 5.14 S11 parameter of HSWCNT antenna and SWCNT antenna with L=12.57 and 1 m.
5.3.3 Directivity and Gain
No more change in the directivity of the HSWCNT over the directivity of the
SWCNT antenna is noticed as shown in Fig. 5.15. The peak of maximum directivity is
about 1.67 for all types.
The key property in the HSWCNT antenna is the modification in the gain as
shown in Fig. 5.16, where for the same wire radius, HSWCNT offers a maximum gain
of which is considered as a good result with respect to the SWCNT antenna
having the same radius. Therefore, the problem of the effective small radius on the
antenna gain or efficiency will be solved by changing antenna type to the HSWCNT.
0 500 1000 1500 2000-40
-35
-30
-25
-20
-15
-10
-5
0
121
Fig. 5.15 Directivity versus frequency of HSWCNT antenna and SWCNT antenna with
.
Fig. 5.16 Realized gain versus frequency of HSWCNT antenna and SWCNT antenna with.
0 1000 2000 3000 4000 50000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
HSWCNTSWCNT with L=12.75 mSWCNT with L=1 m
0 1000 2000 3000 4000 50000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1x 10
-4
HSWCNTSWCNT with L=12.75 mSWCNT with L=1 m
122
In summary, the HSWCNT antenna is proposed and simulated in order to solve
some problems that are related to the SWCNT antenna. At the same time, the issue of
dimensions in nanoarea is very important. This means that, a HSWCNT antenna with
length will functionalize the role of length with some modification in
some properties.
123
Chapter Six
Conclusions and Suggestions for Future Work
6.1 Conclusions
Theoretical investigation has been carried out for predicting the radiation
characteristics of various configurations of carbon nanotube antenna. Simulation results
related to single - wall, multi - wall, and bundle CNT dipole antennas have been
presented and the results have been compared with loop and helical counterparts. The
investigation reveals the following main findings
(i) The concept of the effective conductivity of CNT material, which takes into
account the frequency dependent complex permittivity, is an efficient tool to
model various CNT antenna configurations using commercial software package.
(ii) CNT antennas exhibit multiband operation and this property is more emphasized
when loop or helical configurations are used.
(iii) The gain and efficiency of the SWCNT dipole antenna are very small which can
be enhanced when the dipole is redesigned using MWCNT or bundle CNT
structures. The improvement increases with number of shells and number of
SWCNT elements incorporated in the design.
(iv) Metal contact affects the performance of the CNT antennas and may lead to
performance degradation. This degradation can be overcome by feeding the
antenna with CNT transmission line.
(v) The helical SWCNT offers one order of magnitude gain enhancement compared
with the corresponding SWCNT dipole.
(vi) The radius parameters m=n play a major role in determining the radiation
characteristics of the CNT antennas such as input impedance, reflection parameter
S11, and radiation patterns.
124
6.2 Suggestions for Future Work
The work presented in this work can be extended in the future to cover the
following issues
(i) To develop a comprehensive finite - difference time - domain (FDTD) modeling for
CNT antennas and compare the results with those obtained using reported numerical
models and commercial softwares. The nonlinear and polarization properties of the
CNT material should be taken into account to get accurate results.
(ii) To address the problem of feeding the CNT antennas carefully by taking into
account various materials and geometry parameters. The presence of feeding circuit
may affect the radiation properties of the antenna due to its (antenna) short length.
(iii) Recently, there is increasing interest in composite nanostructured materials
incorporating CNTs where significant progress has been achieved in the
characterization of their mechanical, thermal, and electrical properties. A good
research point is to investigate the potential RF applications of these materials to
design new patch antennas or broadband shields.
(iv) To investigate the performance of other CNT antenna configurations based on
fractal or phased array geometry.
(v) To study the matter of contacting of the metallic and semiconductor CNTs in order
to assure the optimal interconnecting between the nanoelectronics and
nanoantennas.
125
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