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

21

where

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

34

or

Substituting eqn. (2.45b) into eqn. (2.39) and using the result into eqn. (2.43b),

one gets

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

36

where

Equation (2.51) can be rewritten as

or

where

Equation (2.53b) can be re-expressed as

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

40

gives [1]

Equation (2.65b) can be written in suitable form as

where is the kernel function

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) .

83

(b)

(c)

Fig. 4.12 (Continued).

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

92

(b)

(c)

Fig. 4.19 (Continued).

93

(d)

(e)

Fig. 4.19 (Continued).

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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