Graphene antennas for THz radiation - DTU Fotonik

Graphene antennas for THz radiation

(a) Antenna layout (b) 3D �gure of directivity

Figure 1: (a) Half wave dipole in graphene. (b) Directivity of half wave dipoleantenna

MA thesis

Author: Bjørn Langbjerg Larsen s960728

Supervisors:

supervisor Professor Andrei Lavrinenkoassistant supervisor Postdoc Andrei Andryieuski

© June 4, 2013, 0:27 by Bjørn Larsen, Kgs. Lyngby

Technical University of Denmark

Abstract

The communication data rates are constantly increasing. They will, in the near future,continue to increase to the Tb/s rates. The operating frequency of these communicationssystems should correspond and with it move to THz. Optical �bers allow people to trans-mit huge amount of information with light. For wireless free space communication lowerfrequencies are more desirable because the optical free space devices only work for theline-in-sight. Metallic antennas for THz frequencies can be used but they do not allowtunability for di�erent resonance frequencies.On this background I decided to investigategraphene antennas for THz communication. In this paper I will present an overview ofTHz antennas and graphene antennas. A de�nition of graphene with its surface conduc-tivity and the surface impedance. I have set up the simulations in CST MW Studio. I havedesigned and analysed the properties of half wave dipole, patch and Yagi-Uda antennas forshort range communication. I have shown that the design of the half wave dipole antennawas the best of the three antennas since it had the highest e�ciency. From the resultsI have shown that it is possible to tune the graphene antenna to a di�erent frequency.Besides this, that it is possible to reach a high gain by making arrays of these designs.Some key words are graphene antennas, THZ communication, THz antennas, antennaarrays, CST MW Studio.Keywords are:- Graphene antennas- Terahertz communication- Terahertz antennas- Antennas array- CST-Microwave Studio 2012,

© Bjørn L. Larsen

DTU Photonics

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Resumé

Mængden af datainformation i kommunikations systemer stiger konstant. Disse mængderaf data vil blive ved med at stige, voksende til terabit/sek. Frekvensen der bruges til disseskal følge med op til teraHerz. Optiske �bre forbindelser gør det muligt at overføre storemængder data ved hjælp af lys. For trådløs kommunikation er lavere frekvenser ønskeligtda optiske trådløse forbindelser kun virker i ?line-in-sight? systemer. Metalliske antennertil THz frekvenser kan bruges, men kan ikke indstilles til �ere resonans frekvenser. På denbaggrund har jeg valgt at undersøge grafen antenner til THz kommunikation. I denne rap-port vil jeg præsentere et overblik over THz antenner og graphen antenner. En de�nitionaf grafen ud fra dets over�ade ledningsevne og over�ade impedansen. Jeg har brugt CSTMW Studio til mine simuleringer. Jeg har designet og analyseret egenskaberne af en ?halfwave? dipol, en mikro strip og en Yagi-Uda antenne til brug over korte afstande. Jeg harvist at dipol antennen var den bedste af de tre simulerede antenner, da den havde denhøjeste e�ektivitet. Fra resultaterne har jeg vist at det er muligt at tilpasse antennernetil andre frekvenser. Desuden, at det er muligt ved hjælp af en række af et antenne designkan øge antennens ?gain?. Vigtige ord- Grafen antenner- THz kommunikations systemer- THz antenner- Antenne array- CST MW studio

Contents

Abstract ii

Resumé ii

Contents iii

List of Figures v

Preface 1

1 Introduction 1

1.1 THz communication systems . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Metallic antennas for THz communication . . . . . . . . . . . . . . . . . . 21.3 Graphene antennas for THz communication . . . . . . . . . . . . . . . . . 31.4 Goal of the project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Antenna theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Methodology 6

2.1 The unique properties of graphene . . . . . . . . . . . . . . . . . . . . . . 62.2 Graphene in CST Microwave studio . . . . . . . . . . . . . . . . . . . . . 72.3 Modelling of antennas in CST Microwave studio . . . . . . . . . . . . . . 82.4 Veri�cation of the graphene model with Llatser, Kremers et al. . . . . . . 9

3 The antenna designs and results 12

3.1 Introduction to the antenna designs . . . . . . . . . . . . . . . . . . . . . 123.2 Dipole antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Discussion of dipole antenna . . . . . . . . . . . . . . . . . . . . . 15

3.3 Patch antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.1 Patch antenna theory . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3.3 Discussion of patch antenna . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Yagi-Uda antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4.2 Discussion of Yagi-Uda antenna . . . . . . . . . . . . . . . . . . . . 21

4 Conclusion and perspective 22

Appendices 25

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DTU Photonics

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Contents

A Drude-model 25

B GSmacro, used in CST 27

Bibliography 29

June 4, 2013, 0:27

List of Figures

1 (a) Half wave dipole in graphene. (b) Directivity of half wave dipole antenna 1(a) Antenna layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1(b) 3D �gure of directivity . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2.1 (a) Semiconductor bandgap (b) Zero bandgap for graphene, Fermi-level = 0 7(a) Energy level for semi conductor with respect to momentum P . . . . 7(b) Energy level for Graphene . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Electromagnetic wave crossing a graphene patch, simulation setup in CST 9(a) Waveguide model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 (a) The graph is showing the CST values and the Matlab calculated val-ues.(b) Is the di�erence between the two graphs/methods . . . . . . . . . 10(a) CST-simulation and Matlab calculation . . . . . . . . . . . . . . . . 10(b) The di�erence when the two graphs are subtracted . . . . . . . . . . 10

2.4 (a) This my result of the Conductivity/area cross section in CST.(b) Thisis Llatser and Kremers result of the conductivity for di�erent thickness ofgraphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10(a) CST-Area cross section . . . . . . . . . . . . . . . . . . . . . . . . . 10(b) Extinction cross section of unit width. Dashed line surface impedance 10

2.5 My result and Kremers result of the radar cross section(RCS) . . . . . . . 11(a) My Result of the RCS . . . . . . . . . . . . . . . . . . . . . . . . . . 11(b) Kremers result of the RCS . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Balanis's eksample 14.1, a metallic antenna with a return loss of -10 dB.(b) e�ciency about 80% . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11(a) Metal antenna simulated in CST . . . . . . . . . . . . . . . . . . . . 11(b) E�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Return loss for dipole antenna without substrate and groundplane . . . . 13(a) Return Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 (a) The dipole. (b) The dipole's rad. e�ciency) . . . . . . . . . . . . . . . 14(a) Set up of the dipole antenna on a substrate . . . . . . . . . . . . . . 14(b) The radiation e�ciency of the dipole with a substrate . . . . . . . . 14

3.3 (a) Directivity (b) Radiation e�ciency . . . . . . . . . . . . . . . . . . . . 14(a) The directivity of the dipole . . . . . . . . . . . . . . . . . . . . . . 14(b) The Radiation e�ciency of the dipole . . . . . . . . . . . . . . . . . 14

3.4 ((a) The dipole's rad. e�ciency). b) The dipole's directivity . . . . . . . 14(a) New feed line there is a change in the radiation e�ciency . . . . . . 14(b) Directivity is more or less constant . . . . . . . . . . . . . . . . . . . 14

3.5 (a) The dipole. (b) The dipole's rad. e�ciency) . . . . . . . . . . . . . . . 15(a) Set up of the dipole antenna on a substrate . . . . . . . . . . . . . . 15

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Contents

(b) The radiation e�ciency of the dipole with a substrate . . . . . . . . 153.6 (a) The return loss for 3 × 3 array of dipole antennas. (b) The Array's

radiation e�ciency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15(a) Return loss, antenna array with a fermi level= 0.1 . . . . . . . . . . 15(b) The radiation e�ciency array fermi level = 0.1 . . . . . . . . . . . . 15

3.7 (a) The return loss for the square micro strip antenna. (b) Radiatione�ciency, here impedance is 50 Ohms) . . . . . . . . . . . . . . . . . . . . 17(a) Return loss for square microstrip antenna . . . . . . . . . . . . . . . 17(b) The radiation e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.8 (a) The return loss for square patch antenna, impedance 93 Ohms (b)Radiation e�ciency at impedance 93 Ohms and center fed . . . . . . . . . 17(a) Return loss, 176× 176 micro strip antenna . . . . . . . . . . . . . . 17(b) The radiation e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.9 (a) The return loss for square patch antenna, impedance 100 Ohm. (b)Radiation e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18(a) Return loss, 165.47× 136.76 microstrip antenna . . . . . . . . . . . 18(b) The radiation e�ciency rect. patch . . . . . . . . . . . . . . . . . . . 18

3.10 (a) The return loss for reduced size rectangular patch antenna, impedance100 Ohm. (b) Radiation e�ciency for reduced size antenna . . . . . . . . 18(a) Return loss for the reduced size, 126.61× 115.61 rect. patch antenna 18(b) The radiation e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.11 (a) The return loss rect. patch antenna, impedance 100 Ohm. (b) Shift infrequency at di�erent Fermi level . . . . . . . . . . . . . . . . . . . . . . . 18(a) Return loss at Fermi level 0.3 electron volt rect. patch antenna . . . 18(b) The return loss is shifted in frequency . . . . . . . . . . . . . . . . . 18

3.12 (a) The directivity patch antenna, Fermi level 0.4 eV. (b) Radiation e�-ciency improved about 4 times to 2.2% . . . . . . . . . . . . . . . . . . . . 19(a) Directivity rect. patch antenna Fermi Level 0.4 eV . . . . . . . . . . 19(b) The radiation e�ciency at 0.4 electron volt . . . . . . . . . . . . . . 19

3.13 (a) The return loss for Yagi-Uda antenna. (b) radiation e�ciency) . . . . 20(a) Return loss, Yagi-Uda antenna . . . . . . . . . . . . . . . . . . . . . 20(b) The radiation e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.14 (a) The return loss. (b) Radiation e�ciency at impedance 2500 Ohm) . . 20(a) Return loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20(b) The radiation e�ciency . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.15 Radiation e�ciency at impedance 2500 Ohm and a substrate thickness of20 micrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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Chapter 1

Introduction

Communication in the world today takes place at many various levels. People use longrange communication between satellites or across continents, or short range communica-tion between cell phones, computers or other electronic devices in an o�ce or at home.One thing is certain the demand for transferring large amount of data from one side ofthe globe to the other and back is increased everyday. Most of the solution today isdone by cables, wires and optical �bres and this is also very convenient when transferringinformation under the Atlantic ocean from Europe to America. There is though also anincreasing demand for transferring high number of bits wireless. This is meant to be usedin short range communication in WLAN and WPAN's in the THz range. At the momentthere is Wi-Fi working up to 5 GHz. In order to have data rates above 10 Gbit/s thefrequency has to be increased to 100 to 500 GHz or even higher. For this to succeed thereis a need for antennas working in the THz range. This project was selected to explore anddesign these small antennas to transfer data between electronic devices on a short rangeterm with in a few meters, or even use them as inter connection between chips. The inves-tigation of wireless data transfer in the THz spectre could have been done with metallicantennas but graphene was selected above metal because of graphene's tunability.

1.1 THz communication systems

"Moores law" state that transistors in integrated circuits double every 18 month. Ina similar trend the requirement for bandwidth in wireless communication systems hasdoubled every 18 month from 1984 to 2009. Going from less than 1 kb/s to more than100 Mb/s. If this continues it will be within the next 10 years that there is a need for5-10 Gb/s. There has been suggestions that THz communication systems will replaceor supplement WLANs from 2017-2023. Right now wireless communication is operatingbelow 5 GHz which therefore will be limited in bandwidth. At the moment only the lowerpart of the THz band from 0.1 to 1 THz is used for research. This is mainly because in thisregion there exist so called atmospheric transparency windows(300 GHz, 350 GHz, 410GHz, 670 GHz and 850 GHz), where the free space attenuation is small. There are manyadvantages of THz communications compared to infra red transmission which is alreadyat use today. Because of the smaller wavelength in infra red transmission this is vulnerableto even small objects blocking the way. THz communication due to higher wavelength canprovide the abilities of a higher bandwidth. It is more directional due to less di�raction

© Bjørn L. Larsen

DTU Photonics

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Chapter 1 Introduction

of the Waves. It can provide more secure transmission and less attenuations under specialcircumstances. There is of cause also challenges with these systems. The more directionalalso means it has to have a higher gain to reduce the attenuation, for this is needed aline in sight detection. This can be analysed by one of the most important formula forwireless communication systems, the Friis Formula:

PR =PTGTGRλ

2

(4πR)2(1.1.1)

Up to this point not so much research has been done in THz communication and mainlybelow 77 GHz. This is mostly related to the fact that there is a need for compactequipment such as ampli�ers and antenna arrays and THz generators which at the momentonly exist up to 125 GHz. Alternative equipment is used complicating the test andmeasurements in the THz spectrum. This has to change before easy and faster data bittransfer can become a reality for ordinary people, otherwise this will only take place inthe laboratory..

Right now the infra red free space communication is used to transfer data over large dis-tances(up to 10 km), but the data rate is small about 150 Mbit/s and there is problemswith transceiver alignment because of atmospheric losses and interferences such as hu-midity in the air. These disturbances is minor problems for THz communication. This iswhy most people predict that THz transmission will be most use full in short range indoorcommunication systems. There are lots of proposals of how to make the THz communi-cation system work and how to get around some of the problems an considerations whichhas to be taken into account. This is done both for in- and outdoor communication. Thefocus here will be on short range indoor systems. The estimate for a 350 GHz link with a10% bandwidth and a 5 meter range would have a gain of about 30 dB. The value dB(deciBell)is a logarithmic scale referring to how much power [watt] is transferred, where 1 wattequal to 0 dB this means 10 · log(1) = 0. Here 30 dB is 10 · log(1000) = 30 [5] [11]

1.2 Metallic antennas for THz communication

Horn antennas and parabolic antennas can been fabricated with very high directivitybetween 15 dB and 60 dB but for the horn antennas it will require large amount of ad-justment, they have to be aligned more or less perfect for the receiver to detect the signal.Planar devices could be a good solution due to cheap and easy fabrication, �exibility andgood integration. Here unfortunately the natural drawbacks are high loss, low gain andlow power. If a thin �lm of conducting material is located on top of a dielectric substrate,then the radiation will rather take place in the the dielectric than in the free space makingthe antenna ine�cient and with a poor directivity. Small metallic structures has the dis-advantage of having low electron mobility, because of this metallic antennas has a lowerlimit. In the end they will stop resonating and will not transmit any power. Anotherdrawback are that metallic antennas are not tunable. If you change the frequency youwould also have to change the size. Some antennas could though work at more than onefrequency. There are articles which has done research and development in this area. Thisarticle claims to have made 2 antennas which has achieved a 6 dBi gain at 122 GHz andanother antenna at 140 GHz with a gain of 11 dBi. [8] Other designs have been madeto meet the demands for high gain or directivity. The people behind this article wereexperimenting with a promising Yagi-Uda antennas. They manage to simulate a small 4by 4 mm Yagi-Uda antenna with a directivity of 12.5 dB. [4]

June 4, 2013, 0:27

Graphene antennas for THz communication Section 1.5

Wireless communication in micrometer systems can not be achieved by just reducingmetallic antennas in size.

1.3 Graphene antennas for THz communication

What do we know about graphene antennas and what research has been done in this �eld.Graphene is a fairly new material discovered in 2004 and with some highly interestingelectrical/electromagnetic properties. This was announced to be a material that couldrevolutionize several areas of the electronic industry and therefore caught attention forscienti�c research. Because of graphene's conductive properties, it could be assumed thatmore people would have tried to create antenna designs. This is not the case. Mostscienti�c papers and articles only simulate graphene sheets, very few people really havesuggestions for making and simulating antennas. I have managed to �nd a few articlesabout antennas which seems to be working.[15][6] The article [15], have done a lot ofwork on de�ning the conductivity of graphene and trying to explain what happens on thesurface of graphene when applying an external electric �eld.

1.4 Goal of the project

With graphenes ability to support surface plasmon polaritons(SPP) with high e�ectiveindex, there is an opportunity to create micrometer size patch antennas which resonatesin the THz band. These antennas will also have the possibility to be tuned to di�erentfrequencies. This can be done by doping the graphene sheet or applying an externalelectromagnetic �eld in order to change the Fermi level. This was the the major reasonfor choosing graphene as the material over metallic antennas. The goal was to explorethe possibility of using graphene instead of metal as the material for antennas. To beable to model thin sheets of graphene in CST Microwave studio. Then develop a designof graphene antennas for THz communication and simulate it in CST Microwave studio.Next see that the performance were as good as metallic antennas and at the same timelive up to the requirements for short range wireless communications within 5 meters. Therequirements for short range data transmission at THz is about a gain of 20 dB to supportenough bandwidth to transfer more than 10 Gbit/s data. [4]

1.5 Antenna theory

When talking about antennas in general it is di�cult to get around one of the mostfundamental equation, The Friis Transmission equation or formula. This is used to cal-culate the power received at antenna 1, when transmitted from antenna 2, separated bya distance and operating at a given frequency. This formula can look di�erently takingdi�erent things into account. The Frequency dependent Friis Formula is:

PR =PTGTGRc

2

(4πRf)2(1.5.1)

From this it is observed that more power is lost at higher frequencies. This is a funda-mental result. The di�erence between power transmitted and power received is known as

© Bjørn L. Larsen

DTU Photonics

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Chapter 1 Introduction

the path loss and this loss is higher at higher frequencies. This is an important result.There may be more frequency spectrum available at higher frequencies but as the pathloss also will bigger the quality of reception will be poor. If we take 60 GHz antennashere the path loss will normally be too high for long range communication. Here onlypoint to point communication is possible. This means that the receiver and transmitterhas to be in the same room facing each other. There could also be some problems aboutthe polarisation of the signals but I will not get into this .

Some important de�nitions for antenna design.

Most things is given in dB when it comes to power, because this is easier to operate with.The value dB is a logarithmic scale and power is given by

PdB = 10 · log10(PW ) (1.5.2)

Another variation on dB in antenna theory is dBi. This means "decibels relative to anisotropic antenna". This just speci�es the gain of an antenna relative to the isotropicgain, an this is 1 watt. So really nothing changes.

The return loss, m,thematically given by:

RL(dB) = 10log(Pi

Pr) (1.5.3)

Where Pi is the incident power over Pr re�ected power from the feed line. The easiestway to determine an antennas best performance is by looking at the its return loss orS11. This value is normal easiest to look at in dB. This tells the viewer how much ofthe energy is re�ected and this value should be as low as possible. This is why -30 dB isbetter than -10 dB. -30 dB says that only 0.001 watt is re�ected and 99 % transmitted.If the value was -10dB then only 90% was transmitted. The shape of the graph in dBmakes it easy to see both quality of the performance and the bandwidth. Normally agood metallic antenna would have a return loss around -10 dB or less.

Directivity can be described as peak value of power over average power. This means theantenna can have a peak value of power in one direction and less in another. The typicaldirectivity for a half wave dipole antenna is about 2.15 dB and for a micro strip the typicalvalue is between 5 and 8 dB.

The Antenna e�ciency or radiation e�ciency is:

εR =Pradiated

Pinput(1.5.4)

Antenna e�ciency is a very important antenna factor and can be up to a 100% for horn anddish antennas, even for half wavelength dipoles if there is no lossy material around them.For mobile or Wi-Fi antennas the e�ciency is typically 20% to 70%. The surroundingelectronics and material has a tendency to absorb some of the radiated power and turn itin to heat, this results in loss that e�ects the e�ciency. Total e�ciency is the loss due toimpedance mismatch multiplied with the radiation e�ciency, therefore the total e�ciencyis always smaller than the radiation e�ciency.

Antenna gain is:Gain = εR ·Directivity (1.5.5)

June 4, 2013, 0:27

Antenna theory Section 1.5

This is how much power is received from far than that transmitted in the direction ofpeak radiation. This is the radiation e�ciency multiply by the directivity at the resonancefrequency, for optimal performance.

The voltage standing wave ratio is:

V SWR =1 + Γ

1− Γ(1.5.6)

is related to the matching of the antenna to the transmission line. Here Γ is the re�ectioncoe�cient and indicates how much power is re�ected. The minimum value is 1 this meansthat no power is re�ected from the antenna. This is related to how good the antenna ismatched. Perfect match is 1. [3]

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Chapter 2

Methodology

2.1 The unique properties of graphene

Graphene is a one atom thick layer of carbon atoms covalent bonded with 3 othercarbon atoms, arranged in a honeycomb(hexagonal)lattice structure. Graphene has a2D-structure, since it is only 1 atom thick and it's the ultimate 2-dimensional "carbonmolecule" because of the carbon-atoms binding together in one large molecule. When itwas discovered in 2004 scientist also showed it had some extraordinary and remarkableproperties. Graphene has a isotropic thermal conductivity at room temperature which ismuch higher than other carbon allotrope's like fullerenes and diamonds. It was measuredto be > 5000 W/m/K at room temperature. This is about 5 times larger than graphitewhich is the 3D version of graphene since this is build of multiple layers of graphene.This knowledge of heat conductivity can be important when building graphene basedelectronics. Mechanical properties of graphene is also very interesting. It is very light,strong and �exible. Actually it was found to be stronger than a diamond and about 300times stronger than steel. At the same time it only weighs 0.77 milligram/m2. This couldbe illustrated as a 4 kg cat carried by a 1 m2 hammock of graphene weighing 0.77 gramapproximately the same as one of the cats whiskers. The illustration with the hammockI think is good because graphene can be stretch 20%, before it breaks. Even thoughgraphene sheets only consist of 1 atom and is about 0.3 nanometer thick it is visible tothe human eye. The electronic properties of graphene makes it absorb 2,3% of the lightthat passes through it. This can be enhanced by deposit it on a silicon wafer. Thisleads to the light been partially transmitted and partially re�ected. This is very complexsince it also depends on the thickness of the material but it is the same physics thatcauses the "rainbow e�ect" if you have some oil on top of water. This is the refraction ofwhite light splitting it into colours caused by the di�erent wavelength. Chemical prop-erties for graphene is also very unique. Weakly attached atoms can act as acceptors forother atoms keeping a high conductivity, which can be used as bio-sensors In the nextparagraph I will explain more about the electrical properties of graphene and how thisis di�erent from other materials.[2][14] In 2004 Professor Andrew Geim and Ph.d. stu-dent Konstantin Novoselov from University of Manchester discovered and demonstratedthe unique electronic properties of graphene. In 2005 the same group showed that quasiparticles in graphene behaves as massless Dirac fermions. One of the most importantelectronic properties of graphene is that electrons �ow more easily through this material,making it a great conductor, better than copper. Graphene has a high electron mobility

June 4, 2013, 0:27

Graphene in CST Microwave studio Section 2.2

µ , weakly depended on temperature, between 15.000 and 40.000 cm2 · V −1 · s−1, even ata n(carrier density) up to 1013cm−2 at room temperature and can be tuned up to 100.000cm2 ·V −1 ·s−1. At the same time it has a low resistivity of 1.0×10−6ohm ·cm which is lessthan silvers resistivity. Graphene is a zero-gap semiconductor. In graphene it is possibleto control the chemical potential µ or Fermi level by doping with holes or by applying anexternal electric �eld. The electrons in graphene behave as massless Dirac fermions witha remarkably speed of approximately νF = 1.0 × 106m/s, known as the Fermi velocity.Apparently in condensed matter physics the Schrôdinger equation is dominant but ingraphene it is easier to use Dirac equations to de�ne. The spin of electrons around thecarbon atoms in graphene give rise to new quasi particles that can be described by the(2+1)Dirac equation. These quasi particles depend on there momentum.[12][7] I don'twant to get too deep into this part of physics, I am not quali�ed for that. I just want toshow Figure 2.1, which illustrate the Fermi level of graphene and this also indicates thezero band gap which can be change with a electric �eld.[9]

(a) Energy level for semi conductor withrespect to momentum P

(b) Energy level for Graphene

Figure 2.1: (a) Semiconductor bandgap (b) Zero bandgap for graphene, Fermi-level= 0

The surface conductivity of graphene is linked to the Fermi level or chemical potential.The total surface conductivity has a inter band and intra band contribution and is con-nected to the frequency dependent complex permittivity and can be described throughthe Drude formula model. Taking the article by Llatser, Kremers et al. In order to bet-ter understand this I derived the equation for the plasma frequency, by combining theconductivity with the Drude-formula. See Appendix A

2.2 Graphene in CST Microwave studio

In the scienti�c article by Llatser, Kremers et al. they state that you can do two thingsto describe the surface conductivity of graphene. The numerically methods. One isdescribing the surface conductivity with plasma frequency ωp and the collision time τ−1 =γ I derived an equation for the plasma frequency with the Drude-formula. When youcreate your own material in CST the material can be de�ne out of one of the propertiesthat the material has. Here I choose to use the the Drude-formula and write a value for

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Chapter 2 Methodology

gamma and then de�ning the plasma frequency from the derived formula.

ωp =

√2e2kBT

ε0∆π~2ln[exp(

µce

2kBT) + exp(

−µce

2kBT)] (2.2.1)

Here all the parameters in the formula are given values. The thickness ∆ of the graphenesheet in the designed �gure, should be connected to ∆ in the formula. It is also possibleto change the Fermi level in the simulation you just change the value in the parameter listof CST. Of course, this is normally zero but if it is change it is like tuning the grapheneto another frequency. As they say in the article the challenge is to de�ne a in�nitely thinlayer of graphene using a �nite-size. This requires a very dense mesh and takes extracomputational power.

The other way is to model the graphene sheet with its surface impedance Zs connect-ing the tangential component of electric �eld on the surface with the electrical surface

current. Then we have that Zs =1

σsIn CST their is an option to use a macro to set

the environment for your designs. See this in Appendix B. Now the macro has alreadyassigned graphene to a speci�ed Fermi Level. If you change the Fermi-Level you have torecalculate the Impedance of graphene in the macro. You can say the macro is presettingthe environment for the antenna design. This also meant that the drawing of in�nitelythin layer was done by "assigning" the face of a "brick" to a new part with a materialconnected.

2.3 Modelling of antennas in CST Microwave studio

CST is a numerical tool to do the simulation of antennas performance in free space.When modelling antennas for simulation in CST, there are some things that has to beset up before you start. The main thing is that CST gives you the environment to do thesimulations in. Before the simulation you set the SI units for the design. In this case it'sin Frequency, time, distance and temperature. It is possible to change the surroundingspace but is pre-set as normal. After this you draw the design you want to simulate, usingthe shapes given in the menu. It is possible to choose randomly where the �gure shouldstart but for practical reasons every new shape should be around the origin. Every shapeshould be given a size in x,y,and z direction including material value.The di�erent valueswill be saved in a list so the parameter can be changed and not the whole �gure. It ispossible to create your own material or choose from a library of materials. Simulationof the electromagnetic waves can be done in a di�erent ways and here there is a greathelp function with tutorials and examples which can be use full, for which ports to usein the simulations of di�erent antennas. In CST it is possible to set up extra monitors. Iuse this to get and simulation of how the far �eld of the antenna would look like. Thiscould also be other monitors for example the the radar cross sections which I also use.Some things can be calculated after the simulations is done. Before the simulation youcan choose either time or frequency domain. Sometimes the frequency domain can be usefull but i mainly used the time domain solver. It is also important to be aware of theboundary conditions which is set separately before you start. As in all compilers thereis help lines to give you information about things happening during the simulation. Lastthing I will mention here is the mesh. CST is a numerical tool and divides everythinginto polygons which the program then adds up and get a value for the calculation of the

June 4, 2013, 0:27

Veri�cation of the graphene model with Llatser, Kremers et al. Section 2.4

�eld in all regions. If there is a boundary the it sometimes can be necessary to make adenser mesh. CST have a tutorial with examples. Go to CST.com for more information.You probably have to have the programme to see the tutorial but this can be helpful forexplanations.

2.4 Veri�cation of the graphene model with Llatser, Kremers etal.

The scienti�c work with graphene has taken its origin mainly from one article by Llatser,Kremerset al.[12] Now that I have used the Drude-Formula to model graphene in CST. I will verifythe graphene transmission and re�ection at normal incidence. Then, before I start makingmy own designs, I will check graphene simulation and graphene antenna simulations . The�rst veri�cations is of the transmission and re�ection at normal incidence of a graphenepatch. See Figure 2.2

(a) Waveguide model

Figure 2.2: Electromagnetic wave crossing a graphene patch, simulation setup inCST

We have the formulas for transmitted and re�ected energy. Transmission: t =2n

2n+ σsurfZ0

Re�ection: r =−σsurfZ0

2n+ σsurfZ0These formulas derived in an aticle by Andryieuski et

al.[?]

To verify my results I took the data from CST and transferred them into Matlab andfrom these data I drew a �gure. Then I made the calculations with Matlab and thetransmission formula. I made an interpolation to get a more accurate curve I couldcompare with the curve made from CST data. They two curves were very much alike.In the end I subtracted one from the other and looked at the di�erence between the two.The fault was insigni�cant in the area of 10−4 for the transmitted part. See Figure 2.3

The same thing was done for the re�ection, here the di�erence was bigger but you haveto take into account that only 2% of the overall energy is re�ected and about 98% istransmitted.

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Chapter 2 Methodology

(a) CST-simulation and Matlab calculation (b) The di�erence when the two graphs are sub-tracted

Figure 2.3: (a) The graph is showing the CST values and the Matlab calculatedvalues.(b) Is the di�erence between the two graphs/methods

To verify that I can model/draw graphene sheets correctly in CST I tried to recreate theresults that Llatser and Kremers got from their example 1. This set up is done withouta "wave guide" to simulate electromagnetic waves in free space. They of course have tomake a �nite model assuming that the only contribution to the conductivity is intra band.Llatser and Kremers example 1 is a graphene sheet with a width and length of a 5 by 10micrometers. They present a �gure, showing the conductivity per area with regards to thefrequency. I redrew the experiment in CST and came up with a similar graph. see picture.I am displaying the radar cross section of the graphene patch which is an expression forthe conductivity per area. See Figure2.4 for comparing the two graphs.

(a) CST-Area cross section (b) Extinction cross section of unit width.Dashed line surface impedance

Figure 2.4: (a) This my result of the Conductivity/area cross section in CST.(b)This is Llatser and Kremers result of the conductivity for di�erent thickness ofgraphene

June 4, 2013, 0:27

Veri�cation of the graphene model with Llatser, Kremers et al. Section 2.4

As with the transmission and re�ection experiment the two graph are almost identical.When trying to use a thin sheet of graphene as an antenna it is also interesting to knowabout the scattering and absorption cross section. Llatser and Kremers et al's. example 2look into these observations. This involves a silicone substrate with a dielectric constantε = 11.9 and the dimension of the graphene slap is changed to a width of 100 micrometerand a length of 1 micrometer. See Figure2.5

(a) My Result of the RCS (b) Kremers result of the RCS

Figure 2.5: My result and Kremers result of the radar cross section(RCS)

This results I did not manage to reproduce. I tried to change the dielectric of the substrate.That did not help. I tried di�erent things, but it was not indicated in the article how thichthe substrate should be. There was a link between the scattering and the the absorption.I couldn't though get them both to be right at the same time.

I decided to redo an example from Balanis [?] book ,of a normal metallic patch antennaat 10 GHz and this was same results as in the book. See Figure 2.6

(a) Metal antenna simulated in CST (b) E�ciency

Figure 2.6: Balanis's eksample 14.1, a metallic antenna with a return loss of -10dB. (b) e�ciency about 80%

From these results I concluded that nothing was wrong with the set up and I was capableof modelling graphene patches in CST from the conductivity model and the resistivitymodel using the impedance of in�nitely thin sheets of Graphene. Besides this I couldmodel antennas and Graphene antennas in CST

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Chapter 3

The antenna designs and results

3.1 Introduction to the antenna designs

Looking at the requirements for transmitting a high amount of data the antennas hasto have a gain of about 20 dB then of course a bandwidth but this should be severalGHz and should not be a problem in the THz frequency band. I ended up working onthree antenna designs which are quite di�erent in both performance an how the designis executed. Because graphene is an in�nitely thin material the antennas are all madeas micro strip antennas even though the antennas is normally in a di�erent categoryof antennas. I know that it is possible to make nano carbon tubes, which are smallhollow cylinders made from graphene which can be conducting wires but it was never theintention to shape graphene before use.

3.2 Dipole antenna

The �rst design is a a dipole antenna. A This also means that the antenna will need asubstrate to be mounted on, which will have some in�uence on the antennas performance.More on that later. Antenna theory says that a normal half-wave dipole has an impedanceof 73 + i42.5 ohm if you make the antenna little less than λ/2, about 0.48λ̇ then theimaginary part will become zero. This is very useful when I make the design in CST.It should then be possible to match the antenna to the right impedance. In CST this isdone by changing the impedance to the value it has where the imaginary part is zero. Tomake the antenna perform at the right frequency i changed the size.

3.2.1 Simulation results

I have designed a half wave dipole antenna to 0.850 THz. The dipole antenna is normallymade from conducting wires, but since I am using graphene sheets as the building blocs,the antenna is designed like a micro strip antenna where the graphene sheets will be theconducting "wires". Between the two graphene sheets there is a gap. In this gap I feed theantenna. This creates the radiating �eld. The ideal free space resonant dipole antennahas a theoretical impedance Zin = 73 + i42.5 Ohms. According to theory it is bene�cialto make the dipole a little less than half a wavelength, about 0.48λ̇, then the imaginary

June 4, 2013, 0:27

Dipole antenna Section 3.2

part disappears and you have a Zin = 70 ohms which turned out to very practical in CSTdesign. Matching the impedance of the feed and the antenna is of major importance inorder to reduce losses. The gap between the 2 antenna parts is recommended to be assmall as possible. If this distance is change, the impedance match will change as well.I decided the distance to be 5 µm, taking future fabrication issues into account. Stepone, I drew a model of the antenna in CST with no substrate but with a �ctive block ofvacuum around it, to simulate free space. The dipole antenna is 185 micrometer long and15 micrometer wide. It took several time-domain simulation, changing the size and theimpedance several times before getting a decent result which was close to the 850 GHz.The directivity is 2.34 dBi and a e�ciency close to 67% below is a �gure of the returnloss. See Figure 3.1

(a) Return Loss

Figure 3.1: Return loss for dipole antenna without substrate and groundplane

Next I added a substrate since I can not have a antenna �oating in free space. I againoptimized the antenna only by changing the size and the impedance at the port. Thesubstrate used was a material with dielectric constant of ε = 2.33. The size of the antennawas reduced a little to a total of 177 micrometer and the impedance was cut from 780.97to 702.18 Ohm at the port.The substrate size a little larger 181 micrometer and abouthalf in width. The antenna still has a directivity of about 2.34 dBi and radiates in alldirections. See Figure 3.2 for set up and e�ciency.

I want the antenna to be directive in one direction only so I add a ground plane in thenext step. This turn out to radically change the antenna design. I ended up doing it overtwo steps where I �rst change the substrate to vacuum and enhanced it to a 180 × 180micrometer patch. Before it had been a rectangle under the antenna been little longerthan the antenna and only about half the size in width. This was done just to avoid anyproblems with the boundaries. The antenna is now reduced in length to 104 micrometers.Idid not change the thickness of the substrate keeping it at 58 micrometer. This had theright e�ect on the directivity, been in the desired direction and an increased value of 6.981dBi. The e�ciency fell from 66% to just under 50%. I also had to reduce the impedanceto 653.397 Ohm to keep the good performance. See Figure 3.3 for the new set up, thegood directivity and e�ciency.

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Chapter 3 The antenna designs and results

(a) Set up of the dipole antennaon a substrate

(b) The radiation e�ciency of the dipole with asubstrate

Figure 3.2: (a) The dipole. (b) The dipole's rad. e�ciency)

(a) The directivity of the dipole (b) The Radiation e�ciency of the dipole

Figure 3.3: (a) Directivity (b) Radiation e�ciency

So far I have used a discrete port to simulate the feed of my antenna mainly to adjust theimpedance of the feed. I don't say that this wrong but I tried a di�erent approach to thefeeding of the antenna. So in this step I redrew the model with some copper lines to actas transmission lines to feed my antenna design. I chose a wave guide port as the currentsource. A wave guide port simulate a electric �eld propagating along the transmissionlines. This change the size of the antenna to a length of 74.7 micrometer. This turnedout to work as well but the design lost a little in e�ciency to around 30% and gained alittle in directivity to 7.5 dBi. See Figure 3.4

(a) New feed line there is a change in the radiatione�ciency

(b) Directivity is more or less constant

Figure 3.4: ((a) The dipole's rad. e�ciency). b) The dipole's directivity

June 4, 2013, 0:27

Dipole antenna Section 3.2

To use the antenna to transfer data up to a 5 meter distance I earlier stated that theantenna needs a gain of 30dB. This can be solved by making an array of the same antenna,that will increase the directivity.See Figure 3.5. Two other things of importance is thatthe e�ciency stays at 30% and the return loss stays at -30 dB this is a re�ection of just1 milliwatt out of 1 watt.

(a) Set up of the dipole antenna on a substrate (b) The radiation e�ciency of the dipole with asubstrate

Figure 3.5: (a) The dipole. (b) The dipole's rad. e�ciency)

The last thing I simulated was the tunability with the Fermi level. In the previoussimulations the Fermi level was set to 0.3 electron volt. In the last part here I did asimulation with the Fermi level at 0.1 electron volt. From See Figure 3.6 it is clearlyseen that the return loss is shifted to 0.57 THz and the e�ciency is also change. Thedirectivity is more or less the same but a little higher at 0.57 THz.

(a) Return loss, antenna array with a fermilevel= 0.1

(b) The radiation e�ciency array fermi level =0.1

Figure 3.6: (a) The return loss for 3× 3 array of dipole antennas. (b) The Array'sradiation e�ciency)

3.2.2 Discussion of dipole antenna

This antenna have almost met the desired goal at a gain of 20 dB. It could maybe beoptimized more with the change of Fermi level but is use full. The focus was to get thematching correct and that result was not so bad. This antenna can be change in manyways with the gap and the width of the sheet, you could say it is �exible and it works.With the opportunity to make arrays of these antennas the directivity comes second in

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focus. It is just a matter of making enough small antennas a match them correctly andthe focus on optimizing the e�ciency. Next is the production. The last design where Imade the transmission lines through the substrate is not so easy to fabricate. I came upwith a new model where i made 3 identical antennas, having the transmission lines locatedon the substrate crossing the graphene sheets, simulating an array of antennas . It worksbut there is of course some problems with the impedance match to all the antennas. Thiscould be solved by a common point were the distance to all the antennas are the same,matching it to this point and then feed the point with a matched transmission line.

3.3 Patch antenna

The patch or micro strip antenna would be the ideal design from a production point ofview. Since you would just have to create the right size of the sheet. the next step wouldbe to �nd the optimal way to feed the antenna. Even though the micro strip antennais widely use in electronics, it has poor radiation e�ciency and directivity but it is usedbecause it is cheap to fabricate.When in present of a dielectric substrate the radiationwill rather take place in the substrate than in the material it self and this will lead tolosses. The size of a patch antenna should be as with the dipole little less than half awave length.

3.3.1 Patch antenna theory

The patch work as a transmission line, which has to be slightly shorter than half a wavelength to resonate. I worked with two approaches. One was a calculated one where Iused formulas and examples from Constantin Balanis book "Antenna theory - Analysisand design", chapter 14.[1] I here used the formulas for a rectangular patch antenna. Theother was where I used a square patch. Trying to optimize it focus on the right size, feedand radiation e�ciency. For this optimization I used the summary "Microstrip antennasoverview" [16] I used this in the process of optimizing both antennas. Two things areimportant, the dependence of the substrate and the matching of the feed line. I knewbeforehand that the antenna was not so e�cient and only used in electronics for the sizeand the cheap fabrication. So I started out with the basic square or rectangular patch witha size little under half a wave length. Then the antenna was suppose to resonate. Thiswas case but at one particular frequency and not good compare to the dipole. I decidedto try a sub feed. I thought it would be easier to �nd the right impedance match and sizeof the antenna. There was not big di�erence in the performance, while I moved the feedaround changing the impedance. The best performance even though so extraordinary wasat the center.


The square patch is designed to half a wave length which is 176 micrometers in bothx-direction and z-direction. I use y-direction for thickness, which for substrate is 58micrometers. I tried to �nd the right impedance, size and the right way to feed it. The�rst part here is at 50 Ohms. Figure 3.7 Shows a e�ciency of 11% but a poor return lossat 0.85 THz. change of substrate thickness.

June 4, 2013, 0:27

Patch antenna Section 3.3

(a) Return loss for square microstrip antenna (b) The radiation e�ciency

Figure 3.7: (a) The return loss for the square micro strip antenna. (b) Radiatione�ciency, here impedance is 50 Ohms)

I did try to look at the impedance as I have done with the dipole antenna to �nd thegood match. The only problem here was that the imaginary part of the impedance wasvery stable above or close zero without really been zero. I was using the discrete port asdid in the dipole so I easily could change the impedance. I tried di�erent impedances butwith very little improvement. See Figure 3.8 The �gure shows the e�ciency and returnloss at 93 Ohms which was the best. Directivity was 7.45, but too small to reach a gainof 30 dB together with the e�ciency. I also change the substrate thickness.

(a) Return loss, 176× 176 micro strip antenna (b) The radiation e�ciency

Figure 3.8: (a) The return loss for square patch antenna, impedance 93 Ohms (b)Radiation e�ciency at impedance 93 Ohms and center fed

If I increase the substrate thickness then e�ciency can be increased,but both the returnloss and the directivity becomes really poor. Then I tried to move the feed position, thishad no or little e�ect on the e�ciency of the antenna. It did though change the directionof radiation/directivity so I decided to keep the feed at the center.

The other part was simulation of the calculated rectangular patch where I used the for-mulas from Balanis's book Here I did the same thing including the feed move and theresult were almost the same. The feed move for the most part only resulted in a di�erentdirection, not a higher directivity. I used here a impedance of 100 Ohms, close to 93Ohms which I found was okay to use. See Figure 3.9

Here the antenna is calculated from a constant value for thickness of the substrate. Ifyou want to change this thickness, then the whole antenna changes. At the end I tried to

reduce the length L with the formula: L =

λ02√εSee Figure 3.10

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(a) Return loss, 165.47 × 136.76 microstrip an-tenna

(b) The radiation e�ciency rect. patch

Figure 3.9: (a) The return loss for square patch antenna, impedance 100 Ohm. (b)Radiation e�ciency

(a) Return loss for the reduced size, 126.61×115.61rect. patch antenna

(b) The radiation e�ciency

Figure 3.10: (a) The return loss for reduced size rectangular patch antenna,impedance 100 Ohm. (b) Radiation e�ciency for reduced size antenna

I will here present the tunability with the Fermi energy. I have done with the original cal-culated rectangular antenna. I have used the return loss, to see the shift in frequency.SeeFigure 3.11

(a) Return loss at Fermi level 0.3 electron voltrect. patch antenna

(b) The return loss is shifted in frequency

Figure 3.11: (a) The return loss rect. patch antenna, impedance 100 Ohm. (b)Shift in frequency at di�erent Fermi level

This had a positive e�ect on both the directivity and the e�ciency. So the antenna wasindeed tuned.See Figure 3.12

June 4, 2013, 0:27

Yagi-Uda antenna Section 3.4

(a) Directivity rect. patch antenna Fermi Level 0.4eV

(b) The radiation e�ciency at 0.4 electron volt

Figure 3.12: (a) The directivity patch antenna, Fermi level 0.4 eV. (b) Radiatione�ciency improved about 4 times to 2.2%

3.3.3 Discussion of patch antenna

You could say that the return loss was as good as the one for the metallic antenna but itwas di�cult to maintain it close to our resonance frequency. The radiating e�ciency wasworse than expected up to 3% but was very di�erent in the whole frequency spectre. Thisdepended a lot on the substrate thickness. This means that at our resonance frequencyit was under 0.5% I would say this is partly due to substrate dielectric and of course thethickness, but other things could be the reason. But before I started the simulation. I setthe resonance frequency and the substrate material and these parameters was to this way.This is also not a dipole so there is no formula saying that the imaginary part will be zeroif i change the size Maybe the sub feed is not a good solution but it gives me a chance totest di�erent impedances. There are other ways to feed the antenna and this is probablynot the best one production. The Fermi energy is a tool to optimize your antenna to therequired resonance frequency, this could not be done with a metallic antenna.

3.4 Yagi-Uda antenna

They layout for a Yagi-Uda antenna is set up in a table. Every length between the partsare depended on the wavelength or frequency that you want to make the antenna for.This you can look up in a table for di�erent length of antennas. They should be verydirective and the design have thought of making should have a gain of about 11 dB. Thiscan be increased by adding more directors, making the antenna longer.


Before this I haven't worked so much with yagi-uda antennas. I knew how they lookedand what they were used fore. So I am starting at gorund zero. I just wanted to makesimple antenna with 3 directors a re�ector and of course the driven element. The drivenelement is designed as a half wave dipole antenna. I haven't had this much time workingwith the antenna so I have only been working on changing the thickness of the substrateand the impedance match. The �rst part I made with out a substrate just �oating in airand it seems to be working. Then I added a substrate with a dielectric at 2.33. This isthe same substrate material as the one I used for the half wave dipole antenna. I had

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no expectation to the impedance match so i started at 50 Ohm.Figure 3.13 shows the�rst results. Since we are dealing with a dipole antenna I looked at the impedance at

(a) Return loss, Yagi-Uda antenna (b) The radiation e�ciency

Figure 3.13: (a) The return loss for Yagi-Uda antenna. (b) radiation e�ciency)

the resonance frequency and found out It needed to be higher it indicated 2500 Ohms,so I tried that and this improved the e�ciency for the better. See Figure 3.14 There is

(a) Return loss (b) The radiation e�ciency

Figure 3.14: (a) The return loss. (b) Radiation e�ciency at impedance 2500 Ohm)

not so much you can do to improve the antenna since all size on the antenna is �xed. SoI experimented with the thickness of the substrate the �rst results from Figure 3.13 arewith a substrate of 100 micrometers. The results from Figure 3.14 I did that simulationboth with 100 and 58 micrometers the best e�ciency was at the �gures I presented here.These �gures at Figure 3.15 show the results for a substrate of 20 micrometer.

Figure 3.15: Radiation e�ciency at impedance 2500 Ohm and a substrate thicknessof 20 micrometers

June 4, 2013, 0:27

Yagi-Uda antenna Section 3.4

3.4.2 Discussion of Yagi-Uda antenna

The directivity of the antenna is not very good. It should be about 11 dBi but is onlyabout half and not in the right direction. It is more towards the substrate and not awayor in the x-direction as expected. I was thinking "What am I doing Wrong?" All thespacing is �xed, what else can I change? At this point I was out of time I could notdo more simulations, I kinda spend most of the time trying to improve the micro stripantenna. Then I realised that I designed the antenna is too short. The inspiration for theYagi-Uda antenna I found in this article[13]. I just wish I had more time to work on thisantennas since it showed a lot of potential. I can not imagine that the tunability with theFermi level should be much di�erent with this antenna than for the other two. Even if Imade the antenna too short it did showed that it had a high directivity and a e�ciencythat could match the half wave dipole antenna.

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Chapter 4

Conclusion and perspective

The main focus of this thesis was on the development of graphene antennas for THz com-munication. The aim of the project was to design a graphene-based antenna competitivein main characteristic to a metal antenna. One of the di�erences between metal andgraphene is that graphene is tunable, metal is not. Besides this, also some knowledge andresearch behind the formulas used to characterise graphene as a conductor. It also coversan introduction to CST a widely used simulation program for scienti�c research. CSTis the tool for the development and design of these graphene antennas. They are meantfor short range communication in the THz frequency domain. The simulation program isused to specify and optimise the antenna designs.

In CST, graphene as a material is not speci�ed, but it is possible to de�ne the materialfrom di�erent properties which is relevant for my research. In classical electromagnetictheory it is possible to de�ne the surface properties of a material by the Drude-Formula.The Drude-formula can also be used in CST because of the possibility to apply your ownformulas an parameters. The surface impedance of graphene is another way to characterisegraphene. The surface impedance is inversely proportional to the surface conductivity aswe saw in section 2.2.

In the beginning of my investigations, the requirements were set for these antennas. Theantennas should be able to transfer data rates over a range of about 5 meter. For thisto work, the antenna would need a a gain of about 20 dBi. The bandwidth for datatransfer up to Tbit/s should be available in the THz spectrum from 0.1 to 10 THz. Imade designs of three di�erent antennas, a half wave dipole, a micro strip antenna anda Yagi-Uda antenna. Each antenna work di�erently and have a di�erent outline. Theyall have di�erent strong sides and di�erent weak sides. One of the challenges is that allthree antennas should be outlined as a micro strip antenna.

The substrate and the impedance match of an antenna, has in�uence on the performance.In order to deal with some of this in�uence, the design of the dipole antenna was done insteps. The antenna design proved to be quite e�cient. I was able to match it by changingthe impedance. I also manage to make the antenna perform good at the resonancefrequency, by changing the length of the antenna. The radiation e�ciency started closeto 70%. At the beginning I found out that you can't make the width of the graphenesheets of the antenna too small. If you do, there is too little material to radiate theelectromagnetic waves. Adding the substrate and the ground plane was very crucial.Here it was important to take notice of the thickness of the substrate. If it is too thin you

June 4, 2013, 0:27

Section 4.0

will get a kind of a mirror e�ect on both sides of the ground plane, that will cancel outthe electric �eld in graphene and the antenna will not radiate. The radiation e�ciencyended at 35% and a directivity D of about 7.98 dBi. As we recall the gain = εr ·D andthe directivity for a dipole antenna is between 5-8 dBi. So the directivity could not havebeen much higher. Even with a metallic antenna at 70% e�ciency in the GHz spectre, itwould not have been possible to get a gain of more than about 5 dBi.

By making an 3 × 3 array of this antenna I improved the directivity by a factor 3.5 toabout 27 dBi and by that the gain increased to above 10 dBi. The array is a powerfultool to improve your antennas performance. There is still some production challenges thatneeds further consideration. The transmission lines need to be match to the antenna. Ifyou make an array of the antenna the transmission lines to all the antennas in the arrayhas to be matched. This could be a di�cult task, but not impossible. Since the antennaarray only has a gain of 10 dBi, it may not work over a range of 5 meters, but maybe 1or 2 meters. To improve the gain, you could make the array larger or better yet optimizethe single antenna. This is still a good result. It indicates that there is a possibility tomake a good short range graphene antenna, which can work in the THz spectrum. Thereis another way to optimize or tune the antenna. This can be done by changing the Fermilevel of graphene. This can be used to change the resonance frequency to another value.I suppose this can both be used to optimize the antenna at given frequency or the sameantenna can be used, without change, to work at di�erent frequency. This is all doneby changing the Fermi level. The possibility to change the Fermi level is a useful andpowerful tool.

The micro strip antenna is facing a little more challenge. The �rst thing is that changingthe size did not seem to have any e�ect on where the return loss was located. If was quitedi�cult to get return loss located over the resonance frequency. The next thing was thefeed from the bottom through the substrate. The position of the feed line, at the centerseem to be the best. This position of the feed did not show any real advantaged over theother positions. It only changed the direction of the radiation. The radiation e�ciencywas still very poor under 0.04% at 0.85 THz. The radiation e�ciency �gure was notvery uniform. It had a better e�ciency in other areas than at the resonance frequency.The e�ciency were up to 1.3% in some areas. The rectangular antenna, has a calculatedratio between the width and the length based on the substrates thickness and resonancefrequency. This micro strip antenna had a better e�ciency at about 0.2%. At otherfrequencies the e�ciency were up to 3%. The high re�ection at the resonance frequencyis probably the main reason for the low e�ciency of the antenna. Changing the thicknessof the substrate for the square patch had an e�ect on the e�ciency. If I made the substrate10 times thicker to 100 micrometers I had an e�ciency around 20%. The negative e�ectwas that the return loss and the directivity did not improve. To improve the micro stripantennas it is necessary to �nd the right balance between the substrate thickness andthe antenna performance. For the rectangular patch with a constant substrate thickness,this can only be done by impedance matching. For the square micro strip antenna allthree parameters can be changed, impedance matching, substrate thickness and size ofthe antenna. The design do needs optimization to reach the required gain. If not thearrays will increase beyond realistic size.

The Yagi-Uda antenna was made as supplement to the dipole and micro strip antenna.This type of antenna normally has high directivity. Before I started designing the Yagi-Uda antenna, I didn't know how e�ective it could be. There are also some challengeswith this design. How do you control the directivity, it normally has no ground plane?

© Bjørn L. Larsen

DTU Photonics

June 4, 2013, 0:27

Chapter 4 Conclusion and perspective

How do you mount, it needs a substrate? Unfortunately there was not so much time, inthis project, to work with this antenna. The results I produced did show an radiatione�ciency of about 35%. This is, as good as the dipole antenna design. It believe theantenna has potential for a high gain since the antenna normally has a high directivity.There is a problem, since I made a mistake and made the design too short. I don't knowif the data I required, are valid. So there is work to continued here.

I believe that the perspective for THz transmission is there. All the devices and antennasfor THz communication are still tested in laboratories. So the perspective maybe longerthan expected. Before wireless transmission in the THz frequency becomes some whatcommercialised, there is a need for more compact electrical devices. I think that thisproject shows that it is indeed possible to make antennas of graphene. That they cancompete with metallic antennas. The di�erent antennas may not be fully developed, butthey all have potential. The project showed some new ideas about di�erent materialfor antennas. I demonstrated three possible designs for graphene antennas. I foreseethat in the near future there is a marked for further development and production forthese antennas. This is one of many steps towards realising graphene antennas for THzcommunication. Lately I discovered that Ian F. Akyildiz from a group also investigatinggraphene antennas, said that they would have an antenna ready to show data transfer in1 meter, at THz frequency later this year.[10] I will await this release with excitement tosee what they have come up with.

June 4, 2013, 0:27

Appendix A

Drude-model

The research originates from the Article by Llatzer,Kremers et. al

The Drude Formula: In order to explain the connection between the permittivity and theconductivity in a simple source free medium where the medium is not conducting we haveρ = 0, ~J = 0 and σ = 0. Then the time harmonic Maxwells equations becomes:

∇× ~E = −iωµ ~H

∇× ~H = iωε ~E

∇ · ~E = 0

∇ · ~H = 0

If the medium is a conducting media. Meaning that σ 6= 0 and a current ~J = ~E will�ow, then the 2nd Maxwell equation will change to

∇× ~H = (σ + iωε) ~E = iω(ε+σ

iε) ~E = iωεc ~E

With complex permittivity as εc = ε− i σω

The other three Maxwell equations will stay unchanged. Equations for non conductingmedia will apply for conducting media if ε is replaced with εc. When an external time-varying electric �eld is applied to material bodies, then small displacements of boundcharges results in a volume density of polarization. This polarization-vector vary withthe same frequency as the applied �eld. When the frequency increases the inertia of thecharged particles makes the particle displacements go out of phase. This results in a

frictional damping mechanism which causes a power loss. The out of phase polarizationcan be described with a complex susceptibility and hence a complex permittivity. If thematerial has a appreciable amount of charge carries as the electrons in a conductor,

there will also be ohmic losses. It is customary when dealing with this type of media toinclude the e�ects of both damping and the ohmic losses in the imaginary part of a

complex permittivity equal to εc.

Complex permittivity εc = ε′ − iε′′(F/m) both ε′ and ε′′ may be functions of frequency.We could also de�ne a new equivalent conductivity representing all losses σ = ωε′′

(S/m).

© Bjørn L. Larsen

DTU Photonics

June 4, 2013, 0:27

Chapter A Drude-model

The 2nd Maxwell equation can be written in the time domain and then change it to thefrequency dependency as follows:

∇× ~E =∂ε ~E

∂t+ ~J = −iωεε0 ~E + σ ~E

Since we know that ~J = σ ~E

The equation then becomes:−iωε0(ε+ iσ

ε0ω) ~E

where σ =σsurf

∆⇒ σ(ω) =

σsurf (ω)

∆

Then we have a model for the frequency dependent complex permittivity, known as theDrude-formula:

−ε(ω) = 1−ω2p

ω(ω + iγ)= 1 + i

σ(ω)

ε0ω= 1 + i

σsurf (ω)

ε0ω∆(A.0.1)

From this we see that

−ω2p

ω(ω + iγ)= i

σsurf (ω)

ε0ω∆(A.0.2)

This can be reduced to:

−ωp2

(ω + iγ)= i

σsurf (ω)

ε0∆(A.0.3)

Looking into the scienti�c article of Kremers et al. They state that you can calculatethe surface conductivity for an in�nite graphene �lm using the Kubo formalism. Sincethis is �nite and not in�nite structure they do a random phase approximation and

within this they represent the conductivity in a local form with a Drude-like intrabandcontribution. The representation of the conductivity:

σ(ω) =2e2

π~kBT

~ln[2cosh[

µc

2kBT]]

i

ω + iτ−1(A.0.4)

This surface conductivity can be put in the Drude-formula above:

−ω2p

(ω + iγ)=

i

ε0∆

2e2kBT

π~2i

ω + iτ−1ln[2cosh

EF

2kBT], γ = τ−1 = 2 · 1013s−1 (A.0.5)

We can rewrite this a little:

ωp =

√[2e2kBT

ε0∆π~2ln[2cosh

EF

2kBT]] =

√[2e2kBT

ε0∆π~2ln[exp(

EF

2kBT) + exp(

−EF

2kBT)]] (A.0.6)

ωp =

√2e2kBT

ε0∆π~2ln[exp(

µce

2kBT) + exp(

−µce

2kBT)] (A.0.7)

The variable e is the charge of an electron = 1.602× 10−19 The variable EF is the Fermienergy. The variable µc is the Chemical potential.

June 4, 2013, 0:27

Appendix B

GSmacro, used in CST

De�nes graphene's surface impedance.

' GSmacro

Sub Main ()Dim Factor,f,fx,Zre,Zimf As Double

Factor=164.65/(19.324*Ef+Log(1+Exp(-38.647*Ef)))Zre=gammaMacro*Factor

Zimf=Factor*2*PiWith Material

.Reset.Name "GSmacro"

.Folder "".FrqType "all"

.Type "Lossy metal".SetMaterialUnit "THz", "um"

.SetTabulatedSurfaceImpedanceModel "Transparent"For fx=fmin To fmax STEP 0.1

.AddTabulatedSurfaceImpedanceFittingValue fx, Zre, Zimf*fx, "1.0"Next

- .ReferenceCoordSystem "Global"- .CoordSystemType "Cartesian"

- .DispersiveFittingSchemeTabSI "Nth Order"- .MaximalOrderNthModelFitTabSI "10"- .ErrorLimitNthModelFitTabSI "0.1"

- .UseOnlyDataInSimFreqRangeNthModelTabSI "True"- .NLAnisotropy "False"- .NLAStackingFactor "1"- .NLADirectionX "1"- .NLADirectionY "0"- .NLADirectionZ "0"

- .Rho "0"- .ThermalType "Normal"- .ThermalConductivity "0"

- .HeatCapacity "0"

© Bjørn L. Larsen

DTU Photonics

June 4, 2013, 0:27

Chapter B GSmacro, used in CST

- .MetabolicRate "0"- .BloodFlow "0"

- .VoxelConvection "0"- .MechanicsType "Unused"

- .Colour "1", "0", "0"- .Wireframe "False"- .Re�ection "False"- .Allowoutline "True"

- .Transparentoutline "False"- .Transparency "0"

- .Create- End With

-- End Sub

June 4, 2013, 0:27

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[1] Antenna Theory Analysis and Design - Constantine A_ Balanis - Google Bøger.

[2] K.Novoselov A.Geim, http://www.graphene.manchester.ac.uk/story/properties/, â¢.

[3] Joseph J. Carr, http://www.antenna-theory.com/basics/main.html, â¢.

[4] Motorola Labs Esr, Elliot Rd, and Kinnear Rd, Antenna Requirements for Short, 2(2006), no. 1, 6�9.

[5] John Federici and Lothar Moeller, Review of terahertz and subterahertz wireless com-munications, Journal of Applied Physics 107 (2010), no. 11, 111101.

[6] R Filter, M Farhat, M Steglich, R Alaee, C Rockstuhl, and F Lederer, TunableGraphene Antennas for Selective Enhancement of THz-Emission, 1�6.

[7] A K Geim and K S Novoselov, The rise of graphene., Nature Materials 6 (2007),no. 3, 183�191.

[8] Pablo Herrero, Martin Jacob, and Joerg Schoebel, Planar antennas and interconnec-tion components for 122 GHz and 140 GHz future communication systems, Frequenz62 (2008), no. 5-6, 5�6.

[9] http://www.physics.upenn.edu/ kane/pedagogical/295lec3.pdf, Quantum theory ofgraphene.

[10] Josep Miquel Jornet and Ian F Akyildiz, http://www.technologyreview.com/news/511726/graphene-antennas-would-enable-terabit-wireless-downloads/, 2013.

[11] Thomas Kleine-Ostmann and Tadao Nagatsuma, A Review on Terahertz Communi-cations Research, Journal of Infrared, Millimeter, and Terahertz Waves 32 (2011),no. 2, 143�171.

[12] Ignacio Llatser, Christian Kremers, Albert Cabellos-Aparicio, Josep Miquel Jornet,Eduard Alarcón, and Dmitry N. Chigrin, Graphene-based nano-patch antenna forterahertz radiation, Photonics and Nanostructures - Fundamentals and Applications10 (2012), no. 4, 353�358.

[13] Lukas Novotny and Niek van Hulst, Antennas for light, Nature Photonics 5 (2011),no. 2, 83�90.

[14] T H E Royal, Swedish Academy, and O F Sciences, compiled by the Class for Physicsof the Royal Swedish Academy of Sciences Graphene, 50005 (2010), no. October.

© Bjørn L. Larsen

DTU Photonics

June 4, 2013, 0:27

Bibliography

[15] M. Tamagnone, J. S. GoÌmez-DiÌaz, J. R. Mosig, and J. Perruisseau-Carrier, Anal-ysis and design of terahertz antennas based on plasmonic resonant graphene sheets,Journal of Applied Physics 112 (2012), no. 11, 114915.

[16] title = Microstrip Antennas Overview �le = :C\:/Documents and Set-tings/s960728/Desktop/Bjoern Masterproject/antenna design/microstrip anten-nas overview.pdf:pdf www0.egr.uh.edu/courses/ece/ECE5318/microstrip= Jackson,David R, url =http://www0.egr.uh.edu/courses/ece/ECE5318/microstrip

June 4, 2013, 0:27