fdtd modelingof graphene-basedrfdevices: · pdf fileacknowledgements i would like to express...

FDTD Modeling of Graphene-Based RF Devices: Fundamental Aspectsand Applications

by

Xue Yu

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Edward S. Rogers Sr. Department of Electrical and Computer

Engineering

University of Toronto

Copyright c© 2013 by Xue Yu

Abstract

FDTD Modeling of Graphene-Based RF Devices: Fundamental Aspects and Applications

Xue Yu

Master of Applied Science

Graduate Department of Edward S. Rogers Sr. Department of Electrical and Computer

Engineering

University of Toronto

2013

Graphene is a single atomic layer of graphite and has many extraordinary properties. Many

graphene based applications have been proposed in recent years and the need of a time domain

simulation tool for studying graphene based devices emerges. This thesis focuses on developing

a simulation framework for graphene based devices using finite-difference time-domain (FDTD)

method. Formulation for a perfectly matched layer (PML) for the sub-cell FDTD method for

thin dispersive layers has been derived and implemented. Such a PML is useful when thin layers

extend to the boundaries of the computational domain. Using the sub-cell PML formulation to

model the graphene thin layers significantly reduces the computational cost compared to using

the conventional FDTD. The proposed formulation is accompanied by detailed validation and

error analysis studies. Several graphene applications are simulated using the new framework

and the results show good agreement with the respective analytical models.

ii

Dedication

To my parents

iii

Acknowledgements

I would like to express my deepest gratitude to my supervisor, Prof. Costas D. Sarris, for his

guidance and patience throughout the two years of my master’s study. He has been generous

with his time and has provided many invaluable advices that helped to shape up this work. His

commitment to excellence has been motivational to me, now and in my future career.

I want to thank my friends and colleagues of EM group for all the stimulating discussions

we had about research and all the fond memories we shared over the past two years. Thanks

for making my graduate school experience memorable.

I would also like to thank the professors and classmates I had for my undergraduate study.

Thanks for teaching me to strive for excellence, you are the reason that I’ve made it this far.

Lastly, I want to thank my parents for their love and support. They are always there to

encourage me when I go through tough times. Thank you for everything!

iv

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Graphene Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Graphene as a Novel Material . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Graphene Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Current State of the Simulation Tools for Graphene Devices . . . . . . . . 10

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 FDTD Modeling of Graphene 12

2.1 Graphene Conductivity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 The Finite-difference Time-domain Method . . . . . . . . . . . . . . . . . . . . . 16

2.3 FDTD Update Equations for Graphene at Microwave Frequencies . . . . . . . . . 19

2.4 Dispersion and Stability Study: Intra-band Conductivity . . . . . . . . . . . . . . 21

2.5 Study of the Numerical Wave Number in 1-D . . . . . . . . . . . . . . . . . . . . 23

2.6 Modeling of Inter-band Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Theory of Sub-cell Dispersive Perfectly Matched Layer (PML) 27

3.1 Introduction to the Sub-cell Technique . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 FDTD Update Equations for 2-D Sub-cell Scheme . . . . . . . . . . . . . . . . . 28

3.3 Review of Dispersive PML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Sub-cell Dispersive PML in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

v

3.5 Sub-cell Dispersive PML in 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Numerical Results for Sub-cell PML 39

4.1 Study of PML Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Sub-cell PML Error Test with Dielectric Slab Structure . . . . . . . . . . . . . . 42

4.3 Dielectric Slab Transmission Coefficient Test . . . . . . . . . . . . . . . . . . . . 44

4.3.1 Theoretical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Handling of Material Interface in Graphene Simulations . . . . . . . . . . . . . . 48

4.5 Graphene Slab Test in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52



4.6 Graphene Slab Test in 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.7 Graphene Parallel Plate Waveguide Test . . . . . . . . . . . . . . . . . . . . . . . 56



4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Study of Graphene Antennas in 2-D 63

5.1 Analytical Model of A Graphene Patch Antenna . . . . . . . . . . . . . . . . . . 63

5.2 Simulations of Graphene Patch Antennas in 2-D . . . . . . . . . . . . . . . . . . 66

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Conclusion 70

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

A Derivation of the Dispersion Relation and Stability Analysis 73

A.0.1 Derivation of the Dispersion Relation . . . . . . . . . . . . . . . . . . . . 73

vi

A.0.2 Detailed Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Bibliography 79

vii

List of Tables

1.1 Graphene’s main properties [1], [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Routh table for equation (2.40). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.1 Normalized wave impedance for the graphene parallel plate waveguide. . . . . . . 59

A.1 Routh table for equation (A.16). . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

viii

List of Figures

1.1 Graphitic materials in various dimensionalities. . . . . . . . . . . . . . . . . . . . 3

1.2 A graphene sample at the millimeter scale. . . . . . . . . . . . . . . . . . . . . . 5

1.3 A graphene field-effect transistor [3]. . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Schematics and optical images of the graphene mixer circuit. . . . . . . . . . . . 7

1.5 A graphene CPW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 (a) CNT nano-patch antenna (b) Graphene nano-patch antenna. . . . . . . . . . 9

1.7 Controlling of graphene conductivity. . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 Graphene surface conductivity for µc = 0.5eV, γ = 1012 and T = 0. . . . . . . . . 13

2.2 Graphene surface conductivity for µc = 0− 0.5eV, γ = 1012 and T = 0. . . . . . . 14

2.3 Graphene relative permittivity for µc = 0.5eV , γ = 1012 and T = 0. . . . . . . . . 16

2.4 Yee’s cell with E and H fields positions in the unit cell. . . . . . . . . . . . . . . 18

2.5 Comparison of theoretical and simulated β and α vs. frequency. . . . . . . . . . . 24

3.1 (a) A 3-D sub-cell model (b) A 2-D sub-cell model in the xz plane. . . . . . . . . 28

3.2 A sub-cell FDTD domain in 2-D. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Integral contour of 3-D Maxwell’s equations. . . . . . . . . . . . . . . . . . . . . . 30

3.4 Simulation domain with PML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Update flow of dispersive PML equations. . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Simulation domain with sub-cell dispersive PML. . . . . . . . . . . . . . . . . . . 34

4.1 Simulation domain for PML error contour plot. . . . . . . . . . . . . . . . . . . . 40

4.2 Contour plot for graphene PML error in dB (a) at point A (b) at point B. . . . 41

ix

4.3 Geometry of the three dielectric layer test. . . . . . . . . . . . . . . . . . . . . . . 42

4.4 Error of sub-cell PML in time domain for (a) α = 0.5 (b) α = 0.1. . . . . . . . . 43

4.5 Computational domain for plane wave incidence on a dielectric slab. . . . . . . . 45

4.6 Geometry of a 3-slab structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.7 Transmission coefficient for a dielectric slab terminated with sub-cell PML. . . . 46

4.8 Transmission coefficient for a dielectric slab terminated with air PML. . . . . . . 47

4.9 Geometry of the one plate test structure. . . . . . . . . . . . . . . . . . . . . . . 48

4.10 Default node assignment at the material interface. . . . . . . . . . . . . . . . . . 49

4.11 Node assignment using averaged values at the material interface. . . . . . . . . . 49

4.12 A conflicting node assignment for the averaging method. . . . . . . . . . . . . . . 50

4.13 A new node assignment at the material interface. . . . . . . . . . . . . . . . . . . 51

4.14 The actual location of the material interface using the new node assignment. . . 51

4.15 Comparison of late time stability conditions using different node assignments. . . 52

4.16 Computational domain for plane wave incidence on a graphene slab. . . . . . . . 53

4.17 Transmission coefficient for a graphene slab in 2-D. . . . . . . . . . . . . . . . . . 54

4.18 A 3-D computational domain for plane wave incidence on a graphene slab. . . . . 55

4.19 Transmission coefficient for a graphene slab in 3-D. . . . . . . . . . . . . . . . . . 57

4.20 Geometry of the parallel plate waveguide. . . . . . . . . . . . . . . . . . . . . . . 58

4.21 Computational domain for the graphene PPWG simulation. . . . . . . . . . . . . 59

4.22 Normalized wave impedance at a cross section of the PPWG. . . . . . . . . . . . 60

4.23 Normalized phase constant (β/k0) for graphene PPWG. . . . . . . . . . . . . . . 60

4.24 Attenuation constant (α) for the graphene PPWG. . . . . . . . . . . . . . . . . . 61

5.1 Schematic of a graphene based patch antenna. . . . . . . . . . . . . . . . . . . . . 64

5.2 Graphene conductivity at T=300 K, µc = 0 and γ = 5× 1012. . . . . . . . . . . . 64

5.3 Graphene conductivity at T=300K, µc = 0 and γ = 5× 1012. . . . . . . . . . . . 65

5.4 Simulation domain for graphene patch antenna in 2-D. . . . . . . . . . . . . . . . 66

5.5 Theoretical current distribution along the length of a linear wire antenna. . . . . 67

5.6 Normalized ∆H value vs. frequency for different antenna lengths L. . . . . . . . 68

x

5.7 First resonance of a 2-D graphene patch antenna vs. antenna length. . . . . . . . 69

xi

Chapter 1

Introduction

1.1 Motivation

Graphene is a novel material made of a single atomic plane of graphite, which is a form of

carbon. Graphene has many extraordinary properties, mechanically and electrically. It is the

thinnest and strongest material sheet known, and its conductivity can be tuned to be metal-like

or semiconductor-like, which makes it an excellent candidate for high frequency electronics.

The field of graphene study has been growing quickly since the discovery of the material in

2004. Many electronic and radio frequency (RF) applications, such as the graphene transistor [3]

and the graphene integrated circuit (IC) [4], have been proposed and studied. As researchers

investigate the possibilities of graphene based electromagnetics (EM) related applications, such

as waveguides or nano-antennas, the need for an EM simulation tool for graphene based devices

emerges.

The complicated graphene conductivity model (macroscopic) and the electronic/quantum

transport occurring in graphene (microscopic) make modeling of graphene challenging. This

is especially true for time domain simulations because the graphene conductivity is frequency

dependent. As such, the development of an accurate and efficient time domain simulation tool

for graphene is important to further study graphene based devices.

1

Chapter 1. Introduction 2

1.2 Graphene Overview

Graphene is a new material with a history of less than 10 years and yet it has already attracted

appreciable attention in fields ranging from material science, nanotechnology to physics and

electrical engineering. Researchers across disciplines have made many discoveries on graphene’s

material properties and potential applications. In the following section, an overview of graphene

and its modeling is provided.

1.2.1 Graphene as a Novel Material

Graphene is a one-atom-thick layer of graphite, a form of carbon, and is considered as a 2-

D material. It was discovered by K. S. Novoselov and A. K. Geim from the University of

Manchester in 2004. The two scientists were awarded the 2010 Nobel Prize in Physics “for

groundbreaking experiments regarding the two-dimensional material graphene”.

Graphitic materials come in a variety of forms. The 3-D graphite is the most commonly seen

form, and the 1-D carbon nanotube (CNT) has been another recent research interest, especially

in composite materials. The newly discovered graphene is the 2-D graphitic material, and is

essentially the building block of the other forms as illustrated in Fig. 1.1.

The discovery of the 2-D graphene sheet has been a surprise to the scientific community

because a 2-D crystal was predicted to be thermodynamically unstable. A 2-D crystal is in

general hard to grow because as the lateral size of the crystal grows, the thermal vibration also

rapidly grows and diverges on a macroscopic scale, which forces the 2-D crystallites to morph

into a stable 3-D structure [6].

As briefly mentioned in section 1.1, graphene has many outstanding properties. Its electrical

properties include its high carrier mobility, which is measured in various devices as 8000-10000

cm2V−1s−1 and could reach 200000 cm2V−1s−1 in suspended graphene [1]. Graphene’s non-

electronic properties bring a new dimension to graphene research. It was found to have a

breaking strength of 40 N/m, reaching the theoretical limit, as well as a Young’s modulus

of 1.0 TPa, which is a record value [7]. The one-atom-thick graphene is also found to be

impermeable to gases, which could be of interest in bio-molecular and ion transport research.


Figure 1.1: Graphitic materials in various dimensionalities: 0-D buckyball, 1-D carbon nan-otube, 2-D graphene and 3-D graphite [5].


Table 1.1 summarizes the main properties of graphene.

Parameter Value and Units

Thermal Conductivity 5000W/mK

Young’s Modulus 1.0TPa

Mobility (maximum) 200000cm2V−1s−1

Saturation Velocity 4− 5× 107cm/Sec

Table 1.1: Graphene’s main properties [1], [2].

The technology of manufacturing graphene has advanced in the past few years and both

the size and quality of the graphene made have been significantly improved. Knowledge of

the state-of-the-art manufacturing capability allows researchers to design new applications that

were unimaginable before, therefore a brief review on graphene manufacturing is presented in

the following paragraphs.

The experimental discovery of graphene shows that while 2-D crystals do not grow naturally,

they can be made artificially. There are two principal ways of manufacturing graphene: one is

through mechanically splitting the 3-D material into layers and the other is through growing

graphene epitaxially on top of other crystals. One of the mechanical methods, called the scotch-

tape technique was used when graphene was first isolated. Basically it is repeatedly peeling

graphite with adhesive tape [6]. The graphene sheet is hidden in thick graphite flakes, and can

only be found under an optical microscope while being placed on top of SiO2. This method is

time-consuming but the resultant graphene crystal reaches millimeter size in lateral directions

and has high structure and electronic quality. Figure 1.2 shows a millimeter-sized graphene

sample. Other mechanical methods such as ultrasonic cleavage [8], and sonification [9] can

be used to produce smaller graphene crystallites (at a submicrometer scale) on an industrial

scale [6].

The second family of methods is to grow graphitic layers epitaxially on top of other crystals,

and after the entire structure is cooled down, the substrate is removed by chemical etching [10]

[11] [12]. This category of methods includes the epitaxial growth of graphene on a silicon carbide

(SiC) substrate [13], the graphene oxide based method [14] and chemical vapour deposition

(CVD) method [11]. The CVD technique is the most promising method so far, from a yield


and quality point of view. A 76.2cm (30 in) graphene film has been fabricated on a copper

substrate using the CVD technique and has been further processed to make a graphene based

touch screen [15]. The aforementioned graphene properties and current level of manufacturing

capability make graphene a good candidate for many novel devices as will be introduced in the

next section.

Figure 1.2: A graphene sample at the millimeter scale.

1.2.2 Graphene Applications

As silicon-based technology approaches its fundamental limits, the electronics industry is look-

ing for an alternative material to replace silicon in order to further shrink the device size.

Graphene’s ultra-thin nature and its electrical properties make it a good candidate and this

brings more attention and input to graphene research. Graphene field-effect transistors (FET)

and graphene wafer scale integrated circuits have been fabricated and versatile graphene appli-

cations in EM and transformation optics has been developed in recent years. Several designs

emerged from graphene’s unique properties will be presented in the following sections.

A Graphene Field-effect Transistor (FET)

Graphene’s high carrier mobility and saturation velocity make it a promising candidate for

high-speed electronics and radio-frequency applications. Since the transistor is the most basic

element of electronic circuits, building graphene-based transistors opens up the possibilities for

a wider range of graphene based applications.


Several publications about the graphene FET have appeared in the past few years [3], [2],

[16]. The performance of the graphene FET has been raised from a few gigahertz to a cutoff

frequency (fT ) of 100 GHz, and fMAX of 14 GHz [3]. Both fT and fMAX are important figures

of merit for a transistor: fT is the frequency at which the current gain is 1, and it is the highest

theoretical operation frequency of the transistor; fMAX also depends on factors such as device

layout.

Figure 1.3: A graphene field-effect transistor [3].

Figure 1.3 shows the graphene FET fabricated on a wafer. The cross-sectional view of the

graphene FET shows the top-gated configuration where the graphene sheet is located on top

of the SiO2 substrate and is the channel of the transistor. The graphene transistor can be

integrated with other components to build more complex circuits based on graphene.

A Graphene Based Wafer-scale Integrated Circuit

A wafer-scale graphene integrated circuit (IC) has been developed by the same research group

from IBM [4]. The integrated circuit consists of a graphene FET and two inductors and the


integrated circuit operates as a broadband radio-frequency mixer at frequencies up to 10 GHz.

The gate length for the graphene transistor is 550nm; the inductors are made of 1µm thick

aluminium. The schematic and optical images of the circuit are shown in Fig. 1.4.

(a) (b)

Figure 1.4: (a) Schematics of the graphene mixer circuit (b) Optical images of graphene ICs. [4]

The challenges in making a graphene based IC come from the distinct material properties

of graphene comparing to the conventional semiconductors, for example, its poor adhesion with

metals and oxides, and its vulnerability to damage in plasma processing. The final circuit is

fabricated on a SiC substrate, and is less than 1mm2 including the contact pads. The graphene

mixer has a 27 dB conversion loss at 4 GHz, compared to a 7 dB conversion loss at 1.95 GHz

for GaAs-based mixer. The authors suggest that by using high-k dielectrics or highly scaled

gate lengths, for example a 40nm gate, it is possible to improve the conversion gain by more

than 20 dB.

Graphene Coplanar Waveguide (CPW)

Unlike in carbon nanotubes, a 50Ω characteristic impedance can be obtained in graphene, and

this fact opens up the perspective to the graphene based RF applications. As an example, a

graphene based coplanar waveguide (CPW) has been developed in [17]. The CPW operates up

to 65 GHz. One of the challenges encountered in making of the CPW comes from the limited


size of the graphene flake for this experiment. The CPW has a dimension of 25µm × 80µm

and the graphene flake is large enough to pattern the CPW but not the probe tip which has a

size of 150µm. Therefore the electrodes are extended and enlarged on the SiO2 substrate to fit

the probe tips.

(a) (b) (c)

Figure 1.5: (a) The graphene sample used to fabricate the CPW [18] (b) The position of theCPW electrodes over graphene [18] (c) Optical images of graphene CPW [17].

Fig. 1.5(a) shows the graphene sample used to fabricate the graphene CPW, Fig. 1.5(b)

shows the positioning of the CPW electrodes over the graphene flake and Fig. 1.5(c) presents

the optical image of the graphene CPW, illustrating the extended electrodes; the graphene layer

is not visible on this photo.

Graphene Nano-antenna

Earlier research on nano-antenna was focusing on the carbon nanotube (CNT) based antennas.

CNTs support slow-wave propagation, whereby the phase velocity of electromagnetic waves

propagating in CNT is on the order of c0/100 to c0/50 (c0 being the speed of light in a vaccum)

[19], therefore CNT nano-antennas are usually much shorter than the free-space wavelength and

are electrically small. This feature enables the miniaturization of the structures. One drawback

of the CNT antenna is its high resistance. Due to its extremely high aspect ratio (length/cross

sectional area) of the CNT, the resistance reaches several kilo-ohm per micrometer. As a result

of the high resistance, the CNT nano-antenna often has a low efficiency. Fig. 1.6 shows a CNT

antenna and a graphene nanoribbon (GNR) antenna.


Figure 1.6: (a) CNT nano-patch antenna (b) Graphene nano-patch antenna.

Similar wave propagation properties exist in graphene. The slow-wave effect in the plasma

mode is on the order of c0/100. A few graphene-based nano-antennas have been investigated.

In [20] and [21], a graphene patch antenna of 5µm× 1µm on top of silicon substrate is studied

and the resonant frequency of the antenna is found to be 1.3 THz. This study also confirms

that the radiation pattern of a graphene antenna is similar to that of an equivalent metallic

antenna. Graphene can also be used as the substrate for metallic antennas as shown in [22],

and the radiation pattern of the dipole array can be controlled by switching the high- and low-

resistivity state of graphene via an external bias voltage.

Graphene Transformation Optics

(a) (b)

Figure 1.7: (a) Graphene conductivity controlled by bias voltages [23]. (b) Graphene conduc-tivity controlled by uneven ground plane [23].


Graphene has been investigated for its potential in metamaterial and transformation optics

recently [23]. Graphene’s conductivity is a function of the chemical potential, which depends

on gate voltage, and/or chemical doping. Its tunable conductivity allows it to tailor electro-

magnetic fields into desired spatial patterns which is an advantage over the noble metals. Fig.

1.7(a) shows it is possible to have part of the graphene sheet supporting a surface plasmon

polariton (SPP) wave while not on the other part, and this is done by varying the sign of the

imaginary part of the conductivity via applying different gate voltages. The tuning can be

done in real time by varying the gate voltage, and it can be done inhomogeneously to form a

conductivity pattern on a single graphene sheet. Fig. 1.7(b) shows alternatively, an uneven

ground plane underneath the graphene can be implemented to design the conductivity profile

on the graphene. With the above mentioned techniques, transformation optical devices such as

graphene-based Luneburg lens can be designed.

1.2.3 Current State of the Simulation Tools for Graphene Devices

Some of the studies mentioned in the previous section involve simulation work. The nano-

patch antenna in [21] has been simulated using FEKO, and the graphene transformation optics

structures in [23] have been simulated using CST Microwave Studio. In both studies, the

simulations are in the frequency domain, where the graphene conductivity is a complex number

at a particular frequency. One can enter the conductivity value into the commercial software

and run the simulation without making further modification to the software.

Simulating graphene-based devices in time domain has several advantages: the transient

behaviour of the device can be observed and studied and the result for a wide frequency band

can be obtained from a single simulation in time domain. However, modeling graphene in

time domain requires more work, as the graphene’s electrical property is frequency dependent.

There are two major approaches to model graphene: the microscopic model and the macroscopic

model. The transmission line matrix (TLM) method developed in [24] and [25] used the micro-

scopic model, where the Schrodinger’s equation was the governing equation for the electronic

transport in graphene and the combined Maxwell’s equation and Schrodinger’s equation were

solved by the simulator. Such simulator is designed for modeling graphene transistors and is not


applicable to study macroscopic devices such as graphene-based waveguides at current stage.

The macroscopic model uses a conductivity model to represent graphene property. It has been

used in some of the analytical studies on graphene [26], [27], as well as in graphene simulation

based on FDTD method [28], because it is easier to incorporate the macroscopic model into the

general FDTD scheme. In [28], a sub-cell model was used to model the graphene sheet, and a

periodic boundary condition (PBC) was applied to terminate the simulation domain. With the

PBC in place, only simulations with normal incidence waves can be supported and the use of

point sources or oblique incidence waves was not possible with the existing scheme. The limi-

tation of the simulation scheme resulted from the use of PBC can be removed by introducing

a perfectly matched layer (PML) to terminate the simulation domain.

In this thesis, a novel sub-cell PML technique is developed to simulate graphene-based

devices using the FDTD method. The use of sub-cell model reduces computational cost of

the FDTD simulations compared to using regular FDTD mesh, and using PML for boundary

condition allows a wider selection of applications to be studied compared to the FDTD scheme

based on PBC [28]. In addition, it is a general technique and can be applied to study a variety

of dispersive or dielectric thin layer structures.

1.3 Outline

This thesis starts with the study of the FDTD modeling of graphene. In Chapter 2, the

graphene conductivity model is introduced, and the FDTD update equations for graphene are

developed. The dispersion and stability study for intra-band conductivity is presented, and

such analysis serves as a basis for the simulation work presented in later chapters. Chapters

3 and 4 present the novel sub-cell PML framework developed for simulating graphene-based

devices: the background of the techniques involved is included, the detailed implementation

of the method is presented and the test results of the framework with various 2-D and 3-D

dielectric and graphene structures are shown and discussed. In Chapter 5, a graphene-based

patch antenna is studied in 2-D. Chapter 6 concludes this thesis, with a summary of work done

and contributions along with possible future work related to this thesis.

Chapter 2

FDTD Modeling of Graphene

As mentioned in section 1.2.3, the macroscopic graphene conductivity model is used when sim-

ulating graphene based devices using the FDTD method. In this chapter, the macroscopic

graphene conductivity model is introduced, the FDTD update equations for graphene are de-

rived and the dispersion and stability analysis for the update scheme are provided.

2.1 Graphene Conductivity Model

The graphene surface conductivity (in unit of [S]) is given by the Kubo formula [29]:

σ(ω, µc, γ, T ) =je2(ω − j2γ)

πh2

[

1

(ω − j2γ)2

∫

∞

0ǫ

(

∂fd(ǫ)

∂ǫ− ∂fd(−ǫ)

∂ǫ

)

dǫ

−∫

∞

0

fd(−ǫ)− fd(ǫ)

(ω − j2γ)2 − 4(ǫ/h)2dǫ

]

. (2.1)

In equation (2.1), fd(ǫ) = (e(ǫ−µc)/kBT + 1)−1 is the Fermi-Dirac distribution, ω is the angular

frequency in radians and γ is the scattering rate in s−1. Also, µc is the chemical potential

in eV, which can be controlled by chemical doping or by applying a bias voltage, T is the

temperature in Kelvin, −e is the electron charge, h is the reduced Planck’s constant, and kB is

the Boltzmann constant. The first term in equation (2.1) is due to the intra-band contribution,

and the second term is from the inter-band contribution.

12

Chapter 2. FDTD Modeling of Graphene 13

The Kubo formula is in an integral form which makes it hard to evaluate as well as to

integrate with the FDTD scheme. The Kubo formula can be simplified [26], and the intra-band

conductivity is:

σintra(ω, µc, γ, T ) =je2kBT

πh2

(

µc

kBT+ 2 ln(e(−µc/kBT ) + 1)

)

1

ω − j2γ, (2.2)

and the inter-band conductivity is as follows:

σinter(ω, µc, γ, 0) =−je2

4πhln

(

2|µc|−(ω − j2γ)h

2|µc|+(ω − j2γ)h

)

. (2.3)

The intra-band conductivity mainly accounts for the low frequency electrical transport and

the inter-band conductivity is for the optical excitations. The work presented in this thesis is

mainly in the microwave frequency range. Taking the parameters used in the graphene parallel

plate waveguide study [27] as an example, µc = 0.5eV, γ = 1012 and T = 0K, the corresponding

conductivity values are plotted in Fig. 2.1. Fig. 2.2 shows the conductivity value for a range

of chemical potential.

0 200 400 600 800 1000 1200−0.02

0

0.02

0.04

σ intr

a (S

)

Re(σintra

) Im (σintra

)

0 200 400 600 800 1000 12000

0.5

1

1.5

2x 10

−7

frequency (GHz)

σ inte

r (S

)

Re(σinter

) Im(σinter

)

Figure 2.1: Graphene surface conductivity for µc = 0.5eV, γ = 1012 and T = 0.

From Fig. 2.1 and 2.2, we can see that in the frequency range up to 800 GHz, the value of


0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

σintra

real part (S)

frequency (GHz)

chem

ical

pot

entia

l µc (

eV)

−0.01

0

0.01

0.02

0.03

0.04

0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

σintra

imaginary part (S)

frequency (GHz)

chem

ical

pot

entia

l µc (

eV)

−0.01

0

0.01

0.02

0.03

0.04

0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

σinter

real part (S)

frequency (GHz)

chem

ical

pot

entia

l µc (

eV)

0.5

1

1.5

2

2.5x 10

−6

0 200 400 600 8000

0.1

0.2

0.3

0.4

0.5

σinter

imaginary part (S)

frequency (GHz)

chem

ical

pot

entia

l µc (

eV)

1

2

3

4

5

6

x 10−6

Figure 2.2: Graphene surface conductivity for µc = 0− 0.5eV, γ = 1012 and T = 0.


σintra dominates over that of σinter. Therefore, σinter can be ignored for most simulations below

optical frequencies.

If we take a closer look at the formula for σintra, the expression is actually a Drude model:

σintra,s =Q′

jω + 2γ, (2.4)

where

Q′ =e2kBT

πh2

(

µc

kBT+ 2 ln(e(−µc/kBT ) + 1)

)

. (2.5)

The surface conductivity (in unit of [S]) can be converted to the volumetric conductivity (in

unit of [S/m]) by dividing equation (2.4) by the thickness of graphene, which is assumed to

be 10−9m or 1 nm here. Using a different thickness value around 1 nm would not result in

a significant change in simulation result as stated in [23]. If the thickness value used in a

simulation is deviated from 1 nm by a large margin, the accuracy of the simulation result could

be affected because the geometry of the structure is not modeled properly. The stability of the

updating scheme is not affected by the thickness value used. The volumetric conductivity is

then given by:

σintra,v =σintra,s10−9

, (2.6)

σintra,v =Q

jω + 2γ, (2.7)

where

Q(µc, T ) =Q′

10−9=

e2kBT

πh2

(

µc

kBT+ 2 ln(e(−µc)/(kBT ) + 1)

)

/10−9. (2.8)

To find the relative permittivity (ǫr) from the conductivity, the following equations are used:

ǫr = ǫ′ − jǫ′′, (2.9)

ǫ′′ =σ

ωǫ0, (2.10)

ǫr = 1− jσ

ωǫ0, (2.11)

ǫr = 1 +Q

jωǫ0(jω + 2γ)= 1 +

Q/ǫ0−ω2 + 2γjω

. (2.12)


The ǫr value is plotted in Fig. 2.3.

0 100 200 300 400 500 600 700 800−2

−1.5

−1

−0.5

0x 10

6

Re

ε r (u

nitle

ss)

0 100 200 300 400 500 600 700 800−3

−2

−1

0x 10

7

frequency (GHz)

Imε

r (u

nitle

ss)

Figure 2.3: Graphene relative permittivity for µc = 0.5eV , γ = 1012 and T = 0.

In summary, equations (2.4), (2.7) and (2.12) are different ways of presenting graphene’s

material property. In the following work, equation (2.7) and (2.12) are used as the graphene

model in the regular FDTD scheme and the sub-cell PML FDTD scheme respectively.

2.2 The Finite-difference Time-domain Method

The finite-difference time-domain (FDTD) method is a numerical method to solve Maxwell’s

equations in time domain based on Yee’s work in 1966 [30]. Faraday’s Law (equation (2.13))

and Ampere’s Law (equation (2.14)) are solved to update the electric fields E and magnetic

fields H in space and time.

∇× E = −∂B

∂t−M (2.13)

∇×H =∂D

∂t+ J (2.14)


The constitutive relations for the electric flux density D and the magnetic flux density B are

as follows:

D = ǫE = ǫ0ǫrE, (2.15)

B = µH = µ0µrH. (2.16)

Assuming J = 0 and M = 0, the two Maxwell’s curl equations (2.13), (2.14) in Cartesian

coordinates can be written as:

∂Ex

∂t=

1

ǫ

[

∂Hz

∂y− ∂Hy

∂z

]

(2.17a)

∂Ey

∂t=

1

ǫ

[

∂Hx

∂z− ∂Hz

∂x

]

(2.17b)

∂Ez

∂t=

1

ǫ

[

∂Hy

∂x− ∂Hx

∂y

]

(2.17c)

∂Hx

∂t=

1

µ

[

∂Ey

∂z− ∂Ez

∂y

]

(2.18a)

∂Hy

∂t=

1

µ

[

∂Ez

∂x− ∂Ex

∂z

]

(2.18b)

∂Hz

∂t=

1

µ

[

∂Ex

∂y− ∂Ey

∂x

]

. (2.18c)

In [30], the Yee’s cell of dimension ∆x×∆y×∆z is introduced as a unit cell of the discretized

spatial domain as shown in Fig. 2.4. The E fields and H fields are defined in an interlinked

way such that each E node is surrounded by 4 H nodes and vice versa, and the E nodes and

H nodes are dislocated by half a cell spatially.

All fields are initially set to zero in the simulation domain. With a source excitation, all E

fields are updated at the integer time step n, and H fields are updated at half time steps n+ 12

using the E values at time step n. Repeating this process allows the simulation to march in

time.

A second order accurate, centered finite-difference approximation is applied to discretize the


x

y

z

Δx

Δy

Δz

Hx

Hy

Hx

Hx

Hy

Hy

Hz Hz

Hz

Ez

Ey

Ex

Figure 2.4: Yee’s cell with E and H fields positions in the unit cell.

partial derivatives in equations (2.17) and (2.18) as follows:

∂Fn(i, j, k)

∂x=

Fn(i+ 0.5, j, k) − Fn(i− 0.5, j, k)

∆x+O(∆x2), (2.19a)

∂Fn(i, j, k)

∂t=

Fn+0.5(i, j, k) − Fn−0.5(i, j, k)

∆t+O(∆t2), (2.19b)

where i, j, k are space indices in x−, y−, z− directions respectively and n is the temporal index;

Fn(i, j, k) = F (i∆x, j∆y, k∆z, n∆t). Following the discretization method in equation (2.19),

the FDTD update equations for (2.17a) and (2.18a) can be obtained as:

En+1x,(i,j,k) =En

x,(i,j,k) +∆t

ǫ

(

1

∆y

(

Hn+0.5z,(i+0.5,j+0.5,k) −Hn+0.5

z,(i+0.5,j−0.5,k)

)

− 1

∆z

(

Hn+0.5y,(i+0.5,j,k+0.5) −Hn+0.5

y,(i+0.5,j,k−0.5)

)

)

,

(2.20)

Hn+0.5x,(i,j+0.5,k+0.5)

=Hn−0.5x,(i,j+0.5,k+0.5)

− ∆t

µ

(

1

∆y(En

z,(i,j+1,k+0.5) − Enz,(i,j,k+0.5))

− 1

∆x(En

y,(i,j+0.5,k+1) − Eny,(i,j+0.5,k))

)

.

(2.21)

The update equations for other E and H components can be derived in a similar way.


2.3 FDTD Update Equations for Graphene at Microwave Fre-

quencies

The FDTD update equations for graphene at microwave frequencies involve σintra only. The

conduction current (J) can be updated as:

J = σintra,vE =Q

jω + 2γE, (2.22a)

∂

∂tJ + 2γJ = QE, (2.22b)

Jn+1 − J

n

∆t+ 2γ

Jn+1

+ Jn

2= Q

En+1

+ En

2, (2.22c)

Jn+1

=1− γ∆T

1 + γ∆TJn+

Q∆t

2(1 + γ∆t)(E

n+1+ E

n). (2.22d)

The Jn+0.5

term will be needed when updating En:

Jn+0.5

=1

2(J

n+1+ J

n) =

1

1 + γ∆tJn+

Q∆t

4(1 + γ∆t)(E

n+1+ E

n). (2.23)

The E fields can be updated as:

En+1

= En − ∆t

ǫ∇×H

n+0.5 − ∆t

2ǫ(J

n+1+ J

n), (2.24a)

En+1

=

ǫ∆t −

Q∆t4(1+γ∆t)

ǫ∆t +

Q∆t4(1+γ∆t)

En − 1

(1 + γ∆t)(

ǫ∆t +

Q∆t4(1+γ∆t)

)Jn+∇×H. (2.24b)

The update equation for H fields follow as:

Hn+0.5

= Hn−0.5 − ∆t

µ∇× E

n+0.5. (2.25)

The discretized 3-D update equations for graphene are listed below:

Jn+1x,(i+0.5,j,k) =

1− γ∆T

1 + γ∆TJnx,(i+0.5,j,k) +

Q∆t

2(1 + γ∆t)(En+1

x,(i+0.5,j,k) + Enx,(i+0.5,j,k)) (2.26)

Jn+1y,(i,j+0.5,k) =

1− γ∆T

1 + γ∆TJny,(i,j+0.5,k) +

Q∆t

2(1 + γ∆t)(En+1

y,(i,j+0.5,k) + Eny,(i,j+0.5,k)) (2.27)


Jn+1z,(i,j,k+0.5) =

1− γ∆T

1 + γ∆TJnz,(i,j,k+0.5) +

Q∆t

2(1 + γ∆t)(En+1

z,(i,j,k+0.5) + Enz,(i,j,k+0.5)) (2.28)

En+1x,(i+0.5,j,k) =

(

ǫ∆t −

Q∆t4(1+γ∆t)

)

(

ǫ∆t +

Q∆t4(1+γ∆t)

)Enx,(i+0.5,j,k) −

1

(1 + γ∆t)(

ǫ∆t +

Q∆t4(1+γ∆t)

)Jnx,(i+0.5,j,k)

+1

(

ǫ∆t +

Q∆t4(1+γ∆t)

)

(

1

∆y

(

Hn+0.5z,(i+0.5,j+0.5,k) −Hn+0.5

z,(i+0.5,j−0.5,k)

)

− 1

∆z

(

Hn+0.5y,(i+0.5,j,k+0.5) −Hn+0.5

y,(i+0.5,j,k−0.5)

)

)

(2.29)

En+1y,(i,j+0.5,k) =

(

ǫ∆t −

Q∆t4(1+γ∆t)

)

(

ǫ∆t +

Q∆t4(1+γ∆t)

)Eny,(i,j+0.5,k) −

1

(1 + γ∆t)(

ǫ∆t +

Q∆t4(1+γ∆t)

)Jny,(i,j+0.5,k)

+1

(

ǫ∆t +

Q∆t4(1+γ∆t)

)

(

1

∆z

(

(Hn+0.5x,(i,j+0.5,k+0.5) −Hn+0.5

x,(i,j+0.5,k−0.5)))

− 1

∆x

(

Hn+0.5z,(i+0.5,j+0.5,k) −Hn+0.5

z,(i−0.5,j+0.5,k)

)

)

(2.30)

En+1z,(i,j,k+0.5) =

(

ǫ∆t −

Q∆t4(1+γ∆t)

)

(

ǫ∆t +

Q∆t4(1+γ∆t)

)Enz,(i,j,k+0.5) −

1

(1 + γ∆t)(

ǫ∆t +

Q∆t4(1+γ∆t)

)Jnz,(i,j,k+0.5)

+1

(

ǫ∆t +

Q∆t4(1+γ∆t)

)

(

1

∆x

(

Hn+0.5y,(i+0.5,j,k+0.5) −Hn+0.5

y,(i−0.5,j,k+0.5)

)

− 1

∆y

(

Hn+0.5x,(i,j+0.5,k+0.5) −Hn+0.5

x,(i,j−0.5,k+0.5)

)

)

(2.31)

Hn+0.5x,(i,j+0.5,k+0.5) =Hn−0.5

x,(i,j+0.5,k+0.5) −∆t

µ

(

1

∆y(En

z,(i,j+1,k+0.5) − Enz,(i,j,k+0.5))

− 1

∆x(En

y,(i,j+0.5,k+1) − Eny,(i,j+0.5,k))

) (2.32)

Hn+0.5y,(i+0.5,j,k+0.5) =Hn−0.5

y,(i+0.5,j,k+0.5) −∆t

µ

(

1

∆z(En

x,(i+0.5,j,k+1) − Enx,(i+0.5,j,k))

− 1

∆x(En

z,(i+1,j,k+0.5) − Enz,(i,j,k+0.5))

) (2.33)


Hn+0.5z,(i+0.5,j+0.5,k) =Hn−0.5

z,(i+0.5,j+0.5,k) −∆t

µ

(

1

∆x(En

y,(i+1,j+0.5,k) − Eny,(i,j+0.5,k))

− 1

∆y(En

x,(i+0.5,j+1,k) − Enx,(i+0.5,j,k))

) (2.34)

2.4 Dispersion and Stability Study: Intra-band Conductivity

To derive the dispersion relation, the von Neumann method is used here. Substituting a discrete

travelling wave solution u = ej(ω∆t−kd∆d) (d is direction: x, y or z) into equations (2.26)-(2.34),

the dispersion relation is:

(

λe−j0.5ω∆t − ej0.5ω∆t)

(

4 sin2(

ω∆t

2

)

−4∆t2

ǫµ∆x2sin2

(

kx∆x

2

)

− 4∆t2

ǫµ∆y2sin2

(

ky∆y

2

)

− 4∆t2

ǫµ∆z2sin2

(

kz∆z

2

))

+ (4j)P cos2(

ω∆t

2

)

sin

(

ω∆t

2

)

= 0,

(2.35)

where λ = 1−γ∆T1+γ∆T and P = P∆t

ǫ = Q∆t2

2ǫ(1+γ∆t) . The dispersion relation can be further simplified

by substituting g = ejω∆t into (2.35). g is the growth factor and needs to be less than or equal

to 1 for the updating scheme to be stable [31]. After doing some algebra, the dispersion relation

is as follows:

Wg3 +Xg2 + Y g + Z = 0, (2.36)

where

W =P

2+ 1, (2.37a)

X =P

2− λ+ (D − 2), (2.37b)

Y = 1− (D − 2)λ− P

2, (2.37c)

Z = −λ− P

2, (2.37d)

and

D = 4

(

∆t2c2

∆x2sin2

(

kx∆x

2

)

+∆t2c2

∆y2sin2

(

ky∆y

2

)

+∆t2c2

∆z2sin2

(

kz∆z

2

))

. (2.38)


To derive a closed form stability criterion, the Routh-Hurwitz criterion is evaluated [31]. Apply

the following transformation mapping to the dispersion equation:

g =r + 1

r − 1. (2.39)

This transforms the exterior of the unit circle for g in the Z−plane to the right half of the

r−plane, and the stability condition is met if there is no roots in the right half plane of the

r−plane. Applying the transformation in (2.39) to (2.36):

a3r3 + a2r

2 + a1r + a0 = 0, (2.40)

where

a3 = W +X + Y + Z = D(1− λ), (2.41a)

a2 = 3W +X − Y − 3Z = 4P +D(λ+ 1), (2.41b)

a1 = 3W −X − Y + 3Z = (D − 4)(λ − 1), (2.41c)

a0 = (4−D)(λ+ 1). (2.41d)

A Routh table [31] for (2.40) is constructed:

a3 a1a2 a0

a1 − a3a0a2

0

a0 0

Table 2.1: Routh table for equation (2.40).

To achieve stability, the first column of the Routh table needs to be non-negative. Given

γ > 0 and Q > 0, this translates to:

(

∆t2c2

∆x2sin2

(

kx∆x

2

)

+∆t2c2

∆y2sin2

(

ky∆y

2

)

+∆t2c2

∆z2sin2

(

kz∆z

2

))

< 1. (2.42)

To simplify the expression, assume the worst case where sin2(

kx∆x2

)

= sin2(

ky∆y2

)

= sin2(

kz∆z2

)

=


1. Then, equation (2.42) becomes

S =∆t2c2

∆x2+

∆t2c2

∆y2+

∆t2c2

∆z2< 1, (2.43)

which is the CFL limit of the regular 3-D FDTD update equations.

The above analysis shows that the stability condition for the graphene update equations is

the same as the CFL limit for the conventional FDTD method. The stability condition for the

1-D and 2-D case can be derived following the same method.

A more detailed derivation of the dispersion relation and stability condition is provided in

Appendix A.

2.5 Study of the Numerical Wave Number in 1-D

The analysis of the accuracy of the numerical scheme is carried out by comparing the numerical

wavenumber k with the analytical k in 1-D case. The graphene parameters used are µc = 0.5eV ,

γ = 1012 and T = 0K. The analytical wave number for wave propagating in graphene is

calculated as

kz = ω√ǫ0µ0ǫr, (2.44)

where ǫr = 1− j σωǫ0

. The numerical wave number kz can be found from the dispersion relation

(2.35):

kz =1

∆zcos−1

(

1− 2

S2

(

jP cos2(πS/Nλ) sin(πS/Nλ)

Ce−jπS/Nλ − e−jπS/Nλ+ sin2

(

πS

Nλ

)))

. (2.45)

The real part of kz is the propagation constant βz, and the imaginary part is the attenuation

constant αz. Fig. 2.5 shows the comparison of the wave number extracted from a 1-D graphene

simulation with the analytical wave number as a function of frequency using ∆z = 5nm. The

result shows good agreement between the simulation and the analytical solution.


100 200 300 400 500 600 700 8001

2

3

4

β (r

ad/µ

m)

β (theory)β (FDTD)

100 200 300 400 500 600 700 8000

5

10

frequency (GHz)

α (1

/µ m

)

α (theory)α (FDTD)

Figure 2.5: Comparison of theoretical and simulated β and α vs. frequency.

2.6 Modeling of Inter-band Conductivity

The inter-band conductivity for graphene at µc = 0.5eV , γ = 1012 and T = 0K has been plotted

in Fig. 2.1. The conductivity can be fit with a linear model as follows:

σinter(ω, µc,Γ, 0) =−je2

4πhln

2|µc|−(ω − j2γ)h

2|µc|+(ω − j2γ)h, (2.46)

σinter(ω) =−e2a

4πh(jω) +

−e2b

4πh, (2.47)

where

a = −1.31666 × 10−15, (2.48)

b = −4.4× 10−4 + j9.6116 × 10−8. (2.49)


Equation (2.47) can be written as:

σinter(ω) = C(jω) +D (2.50)

C =−e2a

4πh, D =

−e2b

4πh. (2.51)

This model works for the frequency range where the conductivity can be fit as a straight

line. For a wider frequency range, one way to model it is to use a piecewise linear model, which

models a curve with several linear segments. One can find the different coefficients for each of

the segment and therefore form a model for a wide frequency range.

Aside from the piecewise linear model, a Pade approximation can also be applied to fit

the inter-band conductivity for graphene as introduced in [32]. A Pade approximation fits a

rational polynomial of the form:

a0 + a1ω + ...+ aMωM

1 + b1ω + ...+ bNωN= σinter(ω) (2.52)

to a set of data points. Since polynomials of order M = N = 2 are easy to fit in to the FDTD

scheme, it is chosen to model the graphene inter-band conductivity:

a0 + a1jω + a2(jω)2

1 + b1jω + b2(jω)2= σinter(ω). (2.53)

The optimal coefficients are found to be:

a0 = −9.114 × 10−28 (2.54a)

a1 = 1.674 × 10−20 (2.54b)

a2 = 1.343 × 10−36 (2.54c)

b1 = 8.082 × 10−17 (2.54d)

b2 = 2.148 × 10−31 (2.54e)

for a frequency range up to 2.2 × 1014 Hz [32].


Either the linear model or the Pade approximation can be applied to model the graphene

inter-band conductivity, depending on the frequency range of interest and the complexity level

of a particular problem.

2.7 Summary

This chapter started with presenting the graphene conductivity model in various forms, with

a focus on the intra-band conductivity. A brief introduction to FDTD has been given and the

FDTD update equations for graphene in the microwave frequency range were derived. The von

Neumann method was used to derive the dispersion relation in the FDTD update equations for

graphene, and the Routh-Hurwitz criterion was evaluated to find the stability criterion. It has

been found that the stability condition for the graphene update equations was the CFL limit,

regardless of the value of chemical potential (µc) or temperature (T ), which affect the graphene

conductivity value. The numerical wave number extracted from a 1-D FDTD simulation was

compared with the analytical wave number for waves traveling in graphene. The matching

result confirmed that the FDTD model was accurate. The modeling of inter-band conductivity

has also been briefly discussed in this chapter. There are two approaches to model the inter-

band conductivity: the linear model and the Pade approximation. The linear model is easier

to implement and the Pade approximation covers a wider range of frequencies. The decision

on which model to use would depend on the nature of individual problems.

Chapter 3

Theory of Sub-cell Dispersive

Perfectly Matched Layer (PML)

The modeling of graphene using the regular FDTD mesh is computationally intensive, in terms

of both the memory requirement and the simulation time. Because of the ultra-thin nature

of graphene, the geometry of the test structure instead of the maximum frequency of interest

becomes the limiting factor of the spatial grid. An ultra-fine mesh in space and time would

result in a large simulation domain and long simulation time.

The sub-cell method [33] can be applied to reduce the computational cost by introducing a

sub-layer inside a Yee’s cell, and the spatial grid is not limited by the smallest physical feature

any more. This chapter presents a framework to simulate graphene based structures using the

sub-cell method and the dispersive PML.

3.1 Introduction to the Sub-cell Technique

The sub-cell model was first introduced in [33] to model thin material sheets that were perpen-

dicular to one of the major axes of the FDTD lattice. Fig. 3.1(a) shows a 3-D Yee’s cell with

a sub-layer in the yz plane. A sub-layer could be a material that has a different permittivity

and/or a different permeability from the rest of the cell. The sub-layer is inserted in x− direc-

tion with a thickness of α ×∆x, where ∆x is the x− dimension of the Yee’s cell and α is the

27

Chapter 3. Theory of Sub-cell Dispersive Perfectly Matched Layer (PML) 28

fraction of the cell occupied by the sub-layer. The sub-cell α needs to be less than 0.5. A 2-D

sub-cell model is shown in Fig. 3.1(b). The inserted layer becomes a slot in the 2-D model.

x

z

-y

Yee's cellSub-cell

layer

α·dxdx

(a)

x

z

Yee's cell

Sub-cell

layer

α·dx

dx

(b)

Figure 3.1: (a) A 3-D sub-cell model (b) A 2-D sub-cell model in the xz plane.

To find the update equations for the sub-cell scheme, new field nodes are introduced to

model the sub-layer. For the 2-D model in Fig. 3.1(b) with a sub-layer of different permittivity,

the electric field normal to the sheet, Ex, is split into two parts: Ex,in is the Ex field inside

the sub-layer and Ex,out is the Ex field for the rest of the cell. This allows the modeling of the

discontinuous permittivity on the material interface. The tangential electric field Ez and the

tangential magnetic field Hy are continuous across the material interface and therefore do not

need to be split. The field nodes to be updated are shown in Fig. 3.2.

Insertion of the sub-layer introduces uncertainty in the geometry of the structure because

one can not choose the exact location of the sub-layer within a Yee’s cell. The inserted layer

usually has an α value less than 0.5, and the lower limit of α depends on the geometry of the

structure and the accuracy level required for the simulation.

3.2 FDTD Update Equations for 2-D Sub-cell Scheme

The update equations for the 2-D sub-cell scheme involving a dispersive sub-layer can be derived

from the integral form of Maxwell’s equations as follows:

∮

E · dl = − ∂

∂t

∫∫

B · dA−∫∫

M · dA, (3.1)


Figure 3.2: A sub-cell FDTD domain in 2-D.

∮

H · dl = ∂

∂t

∫∫

D · dA+

∫∫

J · dA. (3.2)

To update the Hy field, consider the Hyi+0.5,j+0.5 node, which is updated with the neigh-

bouring nodes at Ex,ini+0.5,j, E

x,outi+0.5,j, E

x,ini+0.5,j+1, E

x,outi+0.5,j+1, E

zi,j+0.5 and Ez

i+1,j+0.5, which in turn

form a contour enclosing Hyi+0.5,j+0.5, as shown in Fig. 3.3(d). Applying equation (3.1) to this

path/area:

(

Ez,n(i,j+0.5) − Ez,n

(i+1,j+0.5)

)

∆z +(

Ex,in,n(i+0.5,j+1) −Ex,in,n

(i+0.5,j)

)

α∆x

+(

Ex,out,n(i+0.5,j+1) − Ex,out,n

(i+0.5,j)

)

(1− α)∆x = − µ0

∆t

(

Hy,n+0.5(i+0.5,j+0.5) −Hy,n−0.5

(i+0.5,j+0.5)

)

.

(3.3)

Converting equation (3.3) to differential form gives:

∂Ez

∂x− α

∂Ex,in

∂x− (1− α)

∂Ex,out

∂x= −jωµ0Hy. (3.4)

The update equations for E fields can be worked from the 3-D Maxwell’s equations and

then be simplified to 2-D.

To update Ex,out:


x

y

z

Hy Hy

Hz

Hz

Ex,out

x

y

z

Hy Hy

Hz

Hz

Ex,in

x

y

z

Ez Ez

Ex,out

Ex,out

Hy

Ex,in

Ex,in

z

x

y

Hx Hx

Hy

Hy

Ez

a) Updating Ex,out b) Updating Ex,in

c) Updating Ez d) Updating Hy

Figure 3.3: Integral contour of 3-D Maxwell’s equations.


(

Hy,n+0.5(i+0.5,j,k−0.5) −Hy,n+0.5

(i+0.5,j,k+0.5)

)

∆y +(

Hz,n+0.5(i+0.5,j+0.5,k) −Hz,n+0.5

(i−0.5,j−0.5,k)

)

∆z

=ǫ0∆t

Ex,out,n+0.5(i+0.5,j,k) ∆y∆z

(3.5)

To convert equation (3.5) into 2-D differential form, divide ∆y∆z on both sides of the

equation, and eliminate terms containing ∆y and Hz, which do not exist in the 2-D TM

domain.

1

∆z

(

Hy,n+0.5(i+0.5,j,k−0.5) −Hy,n+0.5

(i+0.5,j,k+0.5)

)

=ǫ0∆t

Ex,out,n+0.5(i+0.5,j,k) (3.6)

∂

∂zHy = jωǫ0Ex,out (3.7)

To update Ex,in:

(

Hy,n+0.5(i+0.5,j,k−0.5) −Hy,n+0.5

(i+0.5,j,k+0.5)

)

∆y +(

Hz,n+0.5(i+0.5,j+0.5,k) −Hz,n+0.5

(i−0.5,j−0.5,k)

)

∆z

=ǫ0ǫr∆t

Ex,in,n+0.5i+0.5,j,k ∆y∆z

(3.8)

1

∆z(Hy,n+0.5

(i+0.5,j,k−0.5) −Hy,n+0.5(i+0.5,j,k+0.5)) =

ǫ0ǫr∆t

Ex,in,n+0.5(i+0.5,j,k) (3.9)

∂

∂zHy = jωǫ0ǫrEx,in (3.10)

To update Ez:

(

Hx,n+0.5(i,j−0.5,k+0.5) −Hx,n+0.5

(i,j+0.5,k+0.5)

)

∆x+(

Hy,n+0.5(i,j+0.5,k+0.5) −Hy,n+0.5

(i,j+0.5,k−0.5)

)

∆y

=ǫ0ǫr∆t

Ez,n+0.5(i+0.5,j,k)α∆y∆z +

ǫ0∆t

Ez,n+0.5(i+0.5,j,k)(1− α)∆y∆z

(3.11)

1

∆x

(

Hn+0.5y,(i+0.5,j,k−0.5) −Hn+0.5

y,(i−0.5,j,k+0.5)

)

=ǫ0(αǫr + 1− α)

∆tEz,n+0.5

(i,j,k+0.5) (3.12)

∂

∂xHy = jωǫ0(αǫr + 1− α)Ez (3.13)

The equations to update E fields, namely equation (3.7), (3.10) and (3.13), are presented in a


compact form below:

−∂Hy

∂z

−∂Hy

∂z∂Hy

∂x

=jωǫ0

1 0 0

0 ǫr 0

0 0 αǫr + 1− α

Ex,out

Ex,in

Ez

(3.14)

with ǫr = ǫr(ω). Equations (3.4) and (3.14) form a complete set of update equations for the

2-D sub-cell scheme.

3.3 Review of Dispersive PML

In order to simulate infinite structures, it is necessary to extend the existing scheme to include

a PML. The dispersive PML is needed for graphene simulations because graphene is modeled

as an equivalent dispersive medium with an effective permittivity ǫr(ω). The dispersive PML

provides a broadband absorption for highly dispersive materials. Fig. 3.4(a) shows the layout

of a simulation domain with a regular air PML as the absorbing boundary condition and Fig.

3.4(b) shows the different regions of a simulation domain with a dispersive layer terminated by

a dispersive PML.

(a) (b)

Figure 3.4: (a) Simulation domain with air PML (b) Simulation domain with dispersive PML.


The dispersive PML introduced in [34] is summarized as follows for the 2-D TM domain:

−∂Hy

∂z∂Hy

∂x

= jωǫ0ǫr

Sz

Sx0

0Sx

Sz

Ex

Ez

, (3.15)

(

∂Ex

∂z− ∂Ez

∂x

)

= −jωµ0SxSzHy, (3.16)

where Sx = 1 +σxjωǫ0

and Sz = 1 +σzjωǫ0

. σx and σz represent the PML conductivities in the

x− and z− direction respectively.

Auxiliary variables for the dispersive PML scheme are introduced as follows: Px =1

Sxǫ0ǫrEx,

and Dx =1

ǫrPx, and the same for z− component of the fields; By = µSzHy. The update flow

involving the auxiliary variables is shown in Fig. 3.5.

H E

P D

B

Figure 3.5: Update flow of dispersive PML equations.

With the auxiliary variables, the update equations are given in equations (3.17) - (3.21).

Px

Pz

=

1

jω

1

Sz0

01

Sx

−∂Hy

∂z∂Hy

∂x

(3.17)

Dx

Dz

=

1

ǫr(ω)0

01

ǫr(ω)

Px

Pz

(3.18)

Ex

Ez

=

1

ǫ0Sx 0

01

ǫ0Sz

Dx

Dz

(3.19)


By = − 1

jω

(

∂Ez

∂x− ∂Ex,in

∂x

)

(3.20)

Hy =1

µSzBy. (3.21)

3.4 Sub-cell Dispersive PML in 2-D

The sub-cell dispersive PML is designed based on the dispersive PML in section 3.3 to match

the dispersive sub-layer. Fig. 3.6 shows the layout of the simulation domain with the sub-cell

dispersive PML.

Figure 3.6: Simulation domain with sub-cell dispersive PML.

To apply the sub-cell method to the dispersive PML, we follow the field splitting method

presented in section 3.1. The changes in the formulation are as follows: 1) both Dx and Ex need

to be split into Dx,in, Dx,out, Ex,in and Ex,out, respectively; 2) the updates of the D-components

from the P -components have to be modified in accordance to equations (3.4) and (3.14) and 3)

the update equation for B involves Ex,in and Ex,out now. The modified update equations are

as follows:

Dx,in

Dx,out

Dz

=

1

ǫr(ω)0

1 0

01

αǫr(ω)) + 1− α

Px

Pz

, (3.22)


Ex,in

Ex,out

Ez

=

1

ǫ0Sx 0 0

01

ǫ0Sx 0

0 01

ǫ0Sz

Dx,in

Dx,out

Dz

, (3.23)

By = − 1

jω

(

∂Ez

∂x− α

∂Ex,in

∂x− (1− α)

∂Ex,out

∂x

)

. (3.24)

To model graphene, replace ǫr(ω) in equation (3.22) with the relative permittivity of graphene

in equation (3.25).

ǫr(ω) = 1 +Q

jωǫ0(jω + 2γ)= 1 +

Q/ǫ0−ω2 + 2γjω

(3.25)

The discretization of Dx,in is trivial since ǫr = 1 in this case:

Dn+1x,out = Pn+1

x . (3.26)

The update equation for Dx,in is as follows:

ǫr(ω)Dx,in = Px, (3.27)

(

1 +Q/ǫ0

(jω)2 + jω2γ

)

Dx,in = Px, (3.28)

jω(jω + 2γ)Px = jω(jω + 2γ)Dx,in +Q

ǫ0Dx,in. (3.29)

Converting the expression from frequency domain to time domain via jω → ∂∂t :

∂2

∂t2Px +

∂

∂t(2γ)Px =

∂2

∂t2Dx,in +

∂

∂t(2γ)Dx,in +

Q

ǫ0Dx,in. (3.30)

The discretization in time domain is as follows:

Pn+1x − 2Pn

x + Pn−1x

∆t2+ 2γ

Pn+1x − Pn−1

x

2∆t=

Dn+1x,in − 2Dn

x,in +Dn−1x,in

∆t2

+ 2γDn+1

x,in −Dn−1x,in

2∆t+

Q

ǫ0

Dn+1x,in + 2Dn

x,in +Dn−1x,in

4.

(3.31)

Center averaging is used for discretizing each term. Notice that Dn has been discretized as


Dn+1

x,in+2Dnx,in+Dn−1

x,in

4 for improved stability as mentioned in [35]. The final update equation for

Dx,in is:

Dn+1x,in =

(

Q∆t2

2ǫ0− 2

)

Dnx,in −

(

1− γ∆t+Q∆t2

4ǫ0

)

Dn−1x,in

+(1 + γ∆t)Pn+1x − 2Pn

x + (1− γ∆t)Pn−1x

/

1 + γ∆t+Q∆t2

4ǫ0

.

(3.32)

Similar approach can be applied to find the update equation for Dz:

(αǫr(ω) + 1− α)Dz = Pz, (3.33)

and

αǫr(ω) + 1− α = α+αQ/ǫ0

(jω)2 + jω2γ+ 1− α = 1 +

αQ/ǫ0(jω)2 + jω2γ

. (3.34)

Comparing equation (3.34) with equation (3.25), the only difference is the extra α factor ap-

pearing in front of Q in equation (3.34) that does not present in equation (3.25). Following the

same discretization, the update equation for Dz is as follows:

Dn+1z =

(

αQ∆t2

2ǫ0− 2

)

Dnz −

(

1− γ∆t+αQ∆t2

4ǫ0

)

Dn−1z

+(1 + γ∆t)Pn+1z − 2Pn

z + (1− γ∆t)Pn−1z

/

1 + γ∆t+αQ∆t2

4ǫ0

.

(3.35)

To complete the set, the other update equations are shown below:

Pn+1x =

1− ∆tσz,i+0.5,j

2ǫ0

1 +∆tσz,i+0.5,j

2ǫ0

Pnx − ∆t∆x

∆z(1 +∆tσz,i+0.5,j

2ǫ0)(Hn+0.5

y,i+0.5,j+0.5 −Hn+0.5y,i+0.5,j−0.5) (3.36)

Pn+1z =

1− ∆tσx,i+0.5,j

2ǫ0

1 +∆tσx,i+0.5,j

2ǫ0

Pnz +

∆t∆z

∆x(1 +∆tσx,i+0.5,j

2ǫ0)(Hn+0.5

y,i+0.5,j+0.5 −Hn+0.5y,i−0.5,j+0.5) (3.37)

En+1x,in = En

x,in +1

ǫ0((1 +

∆tσx,i+0.5,j

2ǫ0)Dn+1

x,in − (1− ∆tσx,i+0.5,j

2ǫ0)Dn

x,in) (3.38)


En+1x,out = En

x,out +1

ǫ0((1 +

∆tσx,i+0.5,j

2ǫ0)Dn+1

x,out − (1− ∆tσx,i+0.5,j

2ǫ0)Dn

x,out) (3.39)

En+1z = En

z +1

ǫ0((1 +

∆tσz,i+0.5,j

2ǫ0)Dn+1

z − (1− ∆tσz,i+0.5,j

2ǫ0)Dn

z ) (3.40)

Bn+0.5y,i+0.5,j+0.5 =

1− ∆tσx,i+0.5,j

2ǫ0

1 +∆tσx,i+0.5,j

2ǫ0

Bn−0.5y,i+0.5,j+0.5 +

∆t

∆z∆x(1 +∆tσx,i+0.5,j

2ǫ0)(En

z,i+1,j − Enz,i,j

− Enx,i,j+1 + En

x,i,j)

(3.41)

Hn+0.5y,i+0.5,j+0.5 =

1− ∆tσz,i+0.5,j

2ǫ0

1 +∆tσz,i+0.5,j

2ǫ0

Hn−0.5y,i+0.5,j+0.5 +

1

µ0(1 +∆tσx,i+0.5,j

2ǫ0)(Bn+0.5

y,i+0.5,j+0.5 −Bn−0.5y,i+0.5,j+0.5).

(3.42)

Equations (3.26),(3.32),(3.35) and equations (3.36) - (3.42) form a complete set of update

equations for the 2-D sub-cell dispersive PML.

3.5 Sub-cell Dispersive PML in 3-D

The 3-D sub-cell dispersive PML can be derived in the same way as the 2-D scheme developed

in section 3.4. The derivation process is skipped and the update equations in differential form

are presented below:

Px

Py

Pz

=1

jω

1

Sy0 0

01

Sz0

0 01

Sx

∂Hz

∂y− ∂Hy

∂z∂Hx

∂z− ∂Hz

∂x∂Hy

∂x− ∂Hx

∂y

(3.43)

Dx,in

Dx,out

Dy

Dz

=

1

ǫr(ω)0 0

1 0 0

01

αǫr(ω) + 1− α0

0 01

αǫr(ω) + 1− α

Px

Py

Pz

(3.44)


Ex,in

Ex,out

Ey

Ez

=

1

ǫ0

Sx

Sz0 0 0

01

ǫ0

Sx

Sz0 0

0 01

ǫ0

Sy

Sx0

0 0 01

ǫ0

Sz

Sy

Dx,in

Dx,out

Dy

Dz

(3.45)

Bx

By

Bz

= − 1

jω

1

Sy0 0

01

Sz0

0 01

Sx

0 0 − ∂

∂z

∂

∂y

α∂

∂z(1− α)

∂

∂z0 − ∂

∂x

−α∂

∂y−(1− α)

∂

∂y

∂

∂x0

Ex,in

Ex,out

Ey

Ez

(3.46)

Hx

Hy

Hz

=1

µ0

Sx

Sz0 0

0Sy

Sx0

0 0Sz

Sy

Bx

By

Bz

. (3.47)

Equations (3.43) - (3.47) form a complete set of update equations for the 3-D sub-cell

dispersive PML.

3.6 Summary

In this chapter, the sub-cell technique has been introduced and applied to model graphene.

Using the sub-cell technique to model graphene allows the spatial cell size to be larger than

the smallest physical feature (thickness) of graphene, thereby reducing both the memory re-

quirement and the simulation time. The 2-D sub-cell FDTD update equations for graphene

have been derived first, and the dispersive PML has been integrated into the sub-cell FDTD

scheme and the resulting update equations have been presented. The sub-cell dispersive PML

can be used to terminate infinite structures such as waveguides that have graphene sub-layers

extended into the PML. The framework has been extended to 3-D to allow applications with

more complicated geometries to be studied.

Chapter 4

Numerical Results for Sub-cell PML

The sub-cell dispersive PML framework developed in Chapter 3 is validated in this chapter.

The PML error studies characterize the performance of the sub-cell PML. Several test cases of

dielectrics and graphene structures further demonstrate the functionality and accuracy of the

sub-cell PML framework.

4.1 Study of PML Parameters

To study the performance of the PML, a test designed to optimize the PML parameters [36]

is performed on the graphene PML. Two parameters are studied: the PML polynomial grade

number (m), as defined in σx(x) = (x/d)mσx,max, is varied within the range of 1 ≤ m ≤ 6 with

a 0.1 increment and the σmax value is varied between 0.1 × σopt to 4 × σopt, with a 0.1 × σopt

increment, where

σopt = −(m+ 1)(−16)

2ηd(4.1)

as defined in [34] for general UPML, with η = 120π and d = 50 ×∆z being the width of the

PML region.

The simulation domain is 160 × 160 cells, with a 50 cell PML on each side of the 2-D TMz

domain as shown in Fig. 4.1. The grid is 7.5µm and dt = 1.6794 × 10−14, which corresponds

to SlL = 0.95× CFL limit. The test is run for 20000 steps, because it takes a long time for

the entire waveform to reach probe B, due to the highly dispersive nature of graphene. The

39

Chapter 4. Numerical Results for Sub-cell PML 40

Figure 4.1: Simulation domain for PML error contour plot.

simulation domain is divided into two half spaces, the top half is free space and the bottom

half is graphene. Although graphene is in a thin sheet form in nature, using a half space here is

to probe the field inside graphene for testing purpose. The termination PML on all four sides

are set to match the corresponding material at the boundaries. The source is a point source

carrying a modulated Gaussian waveform, for frequency range between 50 to 800 GHz, and is

located at two cells above the air graphene interface. The probe A is in the air region and is

located two cells away from the PML and two cells away from the material interface. Probe B

is totally in the graphene region and is located 2 cells away from PMLs from both directions.

The test results are plotted as contour plots in Fig. 4.2(a) and Fig. 4.2(b). The contour

lines are the maximum relative error in time domain in dB at probe A and B. The relative

error in the time domain:

E(t) = |y(t)− yreference(t)|max|yreference(t)|

(4.2)

is computed following the PML benchmarking studies in [36], where y(t) is the FDTD-simulated

fields with dispersive PML termination, and yreference(t) is extracted via a simulation of a large

domain, where the probe point is far away from the boundary.

Fig. 4.2 shows that the optimal PML parameters are slightly different when probing at point

A and point B. However, the default values for the regular PML, m = 4 and σmax = 1 × σopt,

are very close to the optimal values in both plots. Therefore, these two values are used for all

the simulations involving sub-cell PML for graphene in the following sections.


σmax

/ σopt

m

0.5 1 1.5 2 2.5 3 3.5 41

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

−130

−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

−20

(a)

σmax

/ σopt

m

0.5 1 1.5 2 2.5 3 3.5 41

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

−90

−80

−70

−60

−50

−40

−30

−20

(b)

Figure 4.2: Contour plot for graphene PML error in dB (a) at point A (b) at point B.


4.2 Sub-cell PML Error Test with Dielectric Slab Structure

As an initial test for the time domain reflection error in sub-cell PML, a stack of three sub-cell

layers terminated into a PML is studied. The test is on a dielectric structure because it is

easier to quantify the performance of the sub-cell PML and compare to the air PML on such

structures. The parallel dielectric layers of ǫr = 12 are modeled with sub-cell PML framework

as shown in Fig. 4.3. The layers are separated by 75 µm in the x-direction and terminated by

sub-cell PML in the z-direction.

Figure 4.3: Geometry of the three dielectric layer test structure terminated into a sub-cell PMLindicating the position of the field probe.

The PML region has 50− cells in the x−direction and 16 or 25 cells in the z-direction. The

main simulation domain is 100× 800 cells. The cell size is 7.5µm in both the x− and the z−

direction, which is λmin/50 and ∆t = 1.6794 × 10−14 which corresponds to SCFL = 0.95 times

the FDTD stability limit and 8192 steps are run. Moreover, a second simulation is performed

using a 16 or 25 cells air PML, ignoring the inserted sub-cell dielectric layer, to measure the

impact of the sub-cell layer on the overall performance of the absorber. The source for all tests

is a point source carrying a modulated Gaussian waveform, for frequency range between 50 to

800 GHz. It is located in the air region above all three dielectric plates, as shown in Fig. 4.3.

Field data are recorded at point A, which is located two cells away from the PML and two cells

away from the third layer, in the air region. The relative error in the time domain is calculated

with equation (4.2).


200 400 600 800 1000 1200 1400 1600−150

−100

−50

0E

rror

(dB

)

16−cell sub−cell PML 16−cell regular PML

200 400 600 800 1000 1200 1400 1600−150

−100

−50

0

Time steps

Err

or (

dB)


(a)

200 400 600 800 1000 1200 1400 1600−150

−100

−50

0

Err

or (

dB)


200 400 600 800 1000 1200 1400 1600−150

−100

−50

0

Time steps

Err

or (

dB)


(b)

Figure 4.4: Error of sub-cell PML in time domain for (a) α = 0.5 (b) α = 0.1.


The relative error of the sub-cell PML is shown in Fig. 4.4. The maximum relative error

occurs within the first 1600 time steps. For the α = 0.5 case, the maximum error for the air

PML case goes up to -18 dB, whereas with the sub-cell PML in place, the maximum error

reduces to -51 dB for the 16-cell sub-cell PML. Similarly, for the α = 0.1 case, the maximum

error for the sub-cell PML is -57 dB for the 16-cell sub-cell PML, compared to -46 dB of the

air PML. Error is further reduced when 25-cell PML is in place. A few points to be made with

the results are as follows. First, the errors for all cases with α = 0.1 are smaller than those for

α = 0.5 because when α = 0.1, the inserted sub-layer is thinner, hence a smaller area is covered

by the dielectric-PML interface. Second, the error for the regular PML increases significantly

with the dielectric contrast between the thin layer and the surrounding medium. While the air

PML still provides a poor matching condition for the dielectric layers, no meaningful result can

be obtained with the air PML for a graphene sub-layer like the example in section 4.5.

For the above structure with the dielectric layer modeled by an α = 0.1 sub-cell, the run

time of the simulation for 8192 steps is 391s in comparison to a run time of 5383s using a regular

FDTD mesh; while ∆z = 7.5µm for both tests. The sub-cell method reduces the execution

time by more than 10 times, because for the same size of simulation domain, the number of

cells is reduced by a factor of 10 when using a sub-cell method at α = 0.1. Also, substantially

less memory is required when the sub-cell method is applied.

4.3 Dielectric Slab Transmission Coefficient Test

A dielectric slab structure as shown in Fig. 4.5 is simulated using the same parameters as the

test case in section 4.2, to show the accuracy of the sub-cell framework on dielectric structures.

The sub-cell PML has 10 cells in the x direction and 25 cells in the z direction. A sub-cell

α = 0.5 is used in this simulation.

4.3.1 Theoretical Solution

The analytical transmission coefficient for a dielectric slab in the air is found as follows:

First, find the impedance at the interface of first 2 slabs, at z = 0, as in Fig. 4.6:

Z2(0) = η2η1 cos(β2d) + jη2 sin(β2d)

η2 cos(β2d) + jη1 sin(β2d), (4.3)


Figure 4.5: Computational domain for plane wave incidence on a dielectric slab.

z=0 z=d

ε1 ε1

z

ε2

Figure 4.6: Geometry of a 3-slab structure.

where material 1 is the air and material 2 is the dielectric with ǫr = 12. In equation (4.3)

β2 = ω√ǫr√

(µ0ǫ0) and d is the thickness of the dielectric layer.

Then, the reflection coefficient and transmission coefficient can be found as:

Γ =Z2(0) − η0Z2(0) + η0

, (4.4)

T = 1 + Γ. (4.5)

4.3.2 Simulation Results

The transmission coefficient for the dielectric slab from the simulation is shown in Fig. 4.7.

It is clear that the simulation result matches very well with the analytical solution, for both

the magnitude and the phase. A reference case where the dielectric structure terminated with

a regular air PML is run, and the result is shown in Fig. 4.8. The discrepancy between the

simulation result and the analytical solution shows that even when the value of ǫr is small, the

sub-cell PML is needed to get an accurate simulation result.


1 2 3 4 5 6 7 8

x 1011

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

frequency (Hz)

mag

nitu

de o

f tra

nsm

issi

on c

oeffi

cien

t

simulated transmission coeff.analytical transmission coeff.

(a)

1 2 3 4 5 6 7 8

x 1011

−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

frequency (Hz)

phas

e of

tran

smis

sion

coe

ffici

ent


(b)

Figure 4.7: Dielectric slab terminated with sub-cell PML (a)transmission coefficient magnitudeand (b) transmission coefficient phase.


1 2 3 4 5 6 7 8

x 1011

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

frequency (Hz)

mag

nitu

de o

f tra

nsm

issi

on c

oeffi

cien

t


(a)

1 2 3 4 5 6 7 8

x 1011

−20

−15

−10

−5

0

5

frequency (Hz)

phas

e of

tran

smis

sion

coe

ffici

ent


(b)

Figure 4.8: Dielectric slab terminated with regular air PML (a)transmission coefficient magni-tude and (b) transmission coefficient phase.


4.4 Handling of Material Interface in Graphene Simulations

The first graphene-based structure under test is shown in Fig. 4.9, where a graphene slab is

terminated by the dispersive PML. The initial simulation goes unstable at a late time and the

fields at the intersection of graphene and PML grow exponentially. This problem is not related

to the sub-cell as it also exists in a regular FDTD simulation with dispersive PML. A similar

problem has been reported in [37]. In this section, the stability issue is studied and a solution

is provided and discussed.

Figure 4.9: Geometry of the one plate test structure.

Zooming in at the interface of the air and graphene, the updating nodes are shown in Fig.

4.10, where the darker shade represents graphene and the lighter shade represents air. For a

graphene layer at x = i, the default node assignment would be updating Ex,i and Ez,i with

ǫr(ω) 6= 1 and all other nodes with ǫr = 1 as in the air region. The superscript (2) in E(2)x is

referring to the graphene region, and (1) would be the air region which is the default case and

not labeled in the figure.

Taking a closer look at the geometry, we can see that Ez node of the cells at x = i and

x = i+1 are both tangential to the interface of the two materials. If we apply the integral form

of the Maxwell’s equations for this region, it can be found that the permittivity for both Ez

nodes should be the same. However, this is not the case in Fig. 4.10, therefore, modifications

need to be done to the default node assignment.

A conventional way to handle the nodes on the interfaces is to take the average of the

two material properties [35]. For example, as in Fig. 4.11, Ex,(i+ 1

2,j) is updated with ǫ2 and


x

z

i

i+1

i+2

i-1

j j+1j-1

Ex(2) Ex

(2)

Ez(2) Ez

(2)

Figure 4.10: Default node assignment at the material interface.

x

z

i

i+1

i+2

i-1

j j+1j-1

Ex(2) Ex

(2)

Ez(avg) Ez

(avg)

Ez(avg) Ez

(avg)

Figure 4.11: Node assignment using averaged values at the material interface.


Ex,(i− 1

2,j) and Ex,(i+ 3

2,j) is updated with ǫ1, where ǫ1 is for air and ǫ2 is for graphene. Ez,(i,j+ 1

2)

and Ez,(i+1,j+ 1

2) are both updated with ǫavg = (ǫ1+ǫ2)

2 . However, when the averaging technique

is applied, the simulation still suffers from the late time instability. (Refer to Fig 4.15 (b).)

Looking into the node assignment for the averaging method, we find that there is a conflict

in updating the Ex node and the nearby Ez nodes, as illustrated in Fig. 4.12. To update the

Ex,(i+ 1

2,j) node by applying the Maxwell’s equations with ǫ = ǫ2, it is implying the graphene

layer is located between x = i and x = i+1 and has a relative permittivity of ǫ = ǫ2. However,

to update Ez,(i,j+ 1

2) and Ez,(i+1,j+ 1

2) with ǫ = ǫavg , it is implying the material between x = i− 1

2

and x = i+ 32 has a relative permittivity of ǫ = ǫavg. Assigning different ǫr values to the same

spatial region when updating Ex and Ez could potentially be the reason of having a late time

instability. This discrepancy does not raise a stability issue for the dielectric simulations because

the ǫr value is very small in dielectrics (order of 10) comparing to that of graphene or silver (in

the order of 107).

x

z

i

i+1

i+2

i-1

j j+1j-1

Ex(2) Ex

(2)

(a)

x

z

i

i+1

i+2

i-1

j j+1j-1

Ez(avg) Ez

(avg)

Ez(avg) Ez

(avg)

(b)

Figure 4.12: A conflicting node assignment for the averaging method (a) for updating Ex (b)for updating Ez.

In order to improve the stability condition and to maintain the accuracy level of the simu-

lation, a new node assignment scheme is proposed. As shown in Fig. 4.13, for a graphene layer

at x = i, Ex,(i+ 1

2,j) as well as both of Ez,(i,j+ 1

2) and Ez,(i+1,j+ 1

2) are updated using ǫ2. For the

cells at x = i + 1, the Ex,(i+ 3

2,j) node is assumed to be just above x = i + 3

2 and is entirely in

air. Hy,(i+ 3

2,j+ 1

2) is updated using Ez,(i+1,j+ 1

2) which is in the graphene and Ez,(i+2,j+ 1

2), which


is in the air. Notice that in this configuration, the change of permittivity has been naturally

taken care of when updating Hy and an explicit averaging is not needed. Similarly, for cells at

x = i− 1, Ex,(i− 1

2,j) is assumed to be just below the interface and in the air, and Hy,(i− 1

2,j+ 1

2)

is updated in a similar way.

x

z

i

i+1

i+2

i-1

j j+1j-1

Ex(2) Ex

(2)

Ez(2) Ez

(2)

Ez(2) Ez

(2)

Figure 4.13: A new node assignment in which both of the Ez nodes on the boundaries are setto graphene nodes.

x

z

i

i+1

i+2

i-1

j j+1j-1

Ex(2) Ex

(2)

Ez(2) Ez

(2)

Ez(2) Ez

(2)

Figure 4.14: The actual location of the material interface using the new node assignment.

If we analyze the permittivity value at each node, we can see from Fig. 4.14 that because


both Ez,(i,j+ 1

2) and Ez,(i+1,j+ 1

2) are updated using ǫ2, essentially the region between x = i− 1

2

and x = i+ 32 is the graphene region, and the boundaries between air and graphene have been

pushed to x = i − 12 and x = i + 3

2 . With the shifting of boundaries in place, the effective

thickness of the graphene layer in the simulation is 2×∆x. It is important to know the exact

effective graphene layer thickness because it is involved in finding the volumetric graphene

conductivity.

0 1 2 3 4 5 6 7

x 104

−1

0

1

Regular node assignment

0 1 2 3 4 5 6 7

x 104

−1

0

1

Averaging permittivity at material boundary

0 1 2 3 4 5 6 7

x 104

−1

0

1

time steps

Ez

field

shift material interface by half grid at boundary

Figure 4.15: Comparison of late time stability conditions using different node assignments insimulations.

A comparison of late time stability performance is presented in Fig. 4.15. All three tests

are run with same parameters except the interface node handling. It is shown that with both

the regular node assignment and the averaging method, the simulations go unstable around 60k

time steps, and only the boundary shifting method remains stable in late time.

4.5 Graphene Slab Test in 2-D

After resolving the late time instability issue in section 4.4, the sub-cell dispersive PML frame-

work can be applied to simulate the graphene slab structure.



The analytical transmission coefficient for a lossy slab between air can be found in a similar

way as presented in section 4.3.1. Referring to Fig. 4.6, the impedance at the interface of first

2 slabs, at z = 0, is:

Z2(0) = η2η1 cosh(γ2d) + η2 sinh(γ2d)

η1 sinh(γ2d) + η2 × cosh(γ2d). (4.6)

Here, material 1 is air and material 2 is graphene. In equation (4.6) γ = jω√

(µǫ) = j ωc

√

(ǫr),

d is the thickness of graphene layer.

Then, the reflection coefficient and transmission coefficient can be found as:

Γ =Z2(0) − η0Z2(0) + η0

, (4.7)

T = 1 + Γ. (4.8)


Figure 4.16: Computational domain for plane wave incidence on a graphene slab.

The geometry of the test structure is shown in Fig. 4.16. The graphene sheet is modeled as a

sub-cell layer in the x-direction. A line source excites the Ez field with a modulated Gaussian

pulse of 2 THz bandwidth, producing a plane wave incident normally onto the graphene sheet.

The spatial mesh is ∆x = ∆z = λmin/1000. The thickness of the graphene plate is d =

2× α×∆x = 3 nm, with α = 0.01. The parameters for graphene in this test are µc = 0.2 eV,

γ = 0.065 meV and T = 300 K as in [38].

To record the data, a probe is set at point A, which is one cell above the graphene sheet and


500 1000 1500 20000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency (GHz)

|S21

|

Simulated with Sub−cell PMLAnalytical Solution

500 1000 1500 2000

−4

−2

0

Simulated with regular PMLAnalytical Solution

(a)

400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

Frequency (GHz)

∠ S

21 (

degr

ee)


500 1000 1500 2000−400

−200

0

200

Simulated with regular PMLAnalytical Solution

(b)

Figure 4.17: (a)Transmission coefficient magnitude, inset: simulation with air PML and (b)transmission coefficient phase, inset: simulation with air PML.


in the middle of the domain in the z− direction. The incident field is obtained from running a

simulation in the same domain without the graphene sub-layer and the reflected field is found

by subtraction of the incident field from the total field. Also, the transmission coefficient of

the slab is found analytically. The simulation results are compared with the analytical solution

and shown in Fig. 4.17.

From Fig. 4.17, it is clear that with the sub-cell scheme in place, both the magnitude

and phase of the transmission coefficient obtained from the simulation match well with the

analytical solution. This shows that the sub-cell PML scheme can provide accurate simulations

for graphene-based devices while reducing the simulation time and memory requirements. In

fact, the same simulation with a regular FDTD mesh is not even feasible due to an extremely

high computational cost. The inset of Fig. 4.17 shows the graphene transmission coefficient

simulated with the regular air PML in comparison to the analytical solution. The significant

discrepancies between the two solutions attest to the fact that the sub-cell PML is necessary

to properly terminate the graphene sub-layer.

4.6 Graphene Slab Test in 3-D

Figure 4.18: A 3-D computational domain for plane wave incidence on a graphene slab.

A similar graphene slab structure as in section 4.5 has been used to test the 3-D framework.

The structure is shown in Fig. 4.18. The graphene sheet is modeled as a sub-cell layer in


the x-direction. Instead of using a line source, a plane of source excitation is located above

the graphene layer and produces a plane wave incident normally onto the graphene sheet. The

spatial mesh is ∆x = ∆y = ∆z = λmin/20 and α = 0.01. Although the thickness of the graphene

plate is not exact with this mesh, it serves the purpose of numerical demonstration, as using a

fine mesh in 3-D test requires a large memory and long simulation time. The parameters for

graphene in this test is µc = 0.2 eV, γ = 0.065 meV and T = 300 K as in [38] which is the same

as that of the 2-D simulation.

The simulation result is shown in Fig. 4.19. The analytical solution has been solved in

section 4.5 and not repeated here. The simulation result matches with the analytical solution

as in the case of 2-D simulation scheme, confirming the functionality and accuracy of the 3-D

scheme.

4.7 Graphene Parallel Plate Waveguide Test

Many graphene applications can be investigated using the sub-cell PML framework. One of the

applications of our interest is a 2-D parallel plate waveguide (PPWG) using graphene sheets

instead of metallic ones as the plates [27]. In this study, the dominant quasi-transverse electro-

magnetic (TEM) mode of the PPWG is excited and parameters such as the wave impedance,

propagation constant and attenuation constant of the waveguide are studied and compared to

the analytical solutions.


Figure 4.20 shows the geometry of the graphene PPWG. It consists of two laterally infinite

graphene plates in the yz plane separated by a distance of 1000 nm in the x− direction. When

modeled in a 2-D domain, a cross-section of the waveguide in the xz plane is taken. This way it

is implicitly defined that the plates are infinite in the y− direction. The plates extend infinitely

in the z− direction, and they are terminated by the sub-cell PML in simulations.

In [27], the modal fields guided by the graphene PPWG in Fig. 4.20 have been studied


500 1000 1500 20000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency (GHz)

|S21

| (dB

)


(a)

500 1000 1500 200015

20

25

30

35

40

45

50

55

Frequency (GHz)

∠ S

21 (

degr

ee)


(b)

Figure 4.19: (a)Transmission coefficient magnitude, and (b) transmission coefficient phase.


x

z

y

x

z

d

ε1

ε1

ε1

Figure 4.20: Geometry of the parallel plate waveguide.

analytically, and the wave constant for the quasi- TEM mode has been found as follows:

γ/k0 ≃√

1 +1

η20σ2

[

1− j2ση0k0d

]

. (4.9)

Following equation (4.9), the wave impedance can be found as:

Z = | Ex

η0Hy|= | γ

k0|. (4.10)

The propagation constant can be found as:

β = Re(γ), (4.11)

and the attenuation constant is:

α = 8.686Im(γ). (4.12)


The simulation domain for the graphene PPWG study is shown in Fig. 4.21. The two graphene

plates are modeled with sub-layers with sub-cell αsub−cell = 0.1, and they are separated by 1000

nm. The spatical mesh is ∆ = 50 nm which is mainly limited by the geometry of the problem,

and ∆z = 500 nm. The source is a line source carrying a modulated Gaussian waveform, for

frequency range between 50 to 800 GHz, with Hy polarization. The fields are probed at two

spatial locations between the plates, with 1 cell separation in the z− direction between the two


Figure 4.21: Computational domain for the graphene PPWG simulation.

probes.

The normalized wave impedance varies along the x direction, as shown in Fig. 4.22, in-

dicating the quasi-TEM wave being confined between the two plates. The normalized wave

impedance at f = 100 GHz and f = 800 GHz can be read from Fig. 4.22 and shows good

agreement with the analytical solution as presented in Table 4.1.

f=100 GHz f=800 GHz

Simulation (αsub−cell = 0.1) 9.45 5.434

Analytical solution 9.55 5.6

Table 4.1: Normalized wave impedance for the graphene parallel plate waveguide.

The normalized phase constant (β/k0) and attenuation constant (α) are plotted as a function

of frequency in Fig. 4.23 and Fig. 4.24. In both plots, the simulation results match with the

analytical solution very well.

In the PPWG test, a few sources of error are introduced when implementing the sub-cell

method to model the graphene thin layers: 1) The sub-cell αsub−cell introduces uncertainty

in the exact location of each plate in the x− direction, therefore the plate separation is not

exactly 1000 nm in the simulation. Comparing the simulation result of such structures with

the analytical solution that assumed 1000 nm plate separation, small error would show up;

2) Even with the sub-cell formulation, the modeled graphene thickness is not exactly 1 nm

in this case. Although this difference has been taken care of when calculating the volumetric


−1 −0.5 0 0.5 10

5

10

15

x/d (the graphene planes lie at x=± d/2)

Ex/

η 0/Hy

f=100GHzf=800GHz

Figure 4.22: Normalized wave impedance at a cross section of the PPWG.

0 200 400 600 8004

6

8

10

12

14

16

Frequency (GHz)

beta

nor

mal

ized

(be

ta /

k0)

simulation resultanalytical solution

Figure 4.23: Normalized phase constant (β/k0) for graphene PPWG.


0 200 400 600 8000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Frequency (GHz)

alph

a (d

B/m

icro

n)

simulation resultanalytical solution

Figure 4.24: Attenuation constant (α) for the graphene PPWG.

conductivity model for each case, and should not make a significant impact in the simulation

result as discussed in [23], it remains a source of error in this simulation.

In general the choice of the sub-cell αsub−cell is a trade off between the computational

cost and simulation accuracy. While it is desirable to use a smaller αsub−cell to reduce the

computational cost to a greater extent, the uncertainly in the geometry of the structure can

affect the accuracy of the results significantly. Therefore, in most studies, αsub−cell values

between 0.01 and 0.5 are chosen.

4.8 Summary

In this chapter, the sub-cell dispersive PML framework has been validated through a series of

tests. The PML parameters were optimize through constructing a PML contour plot and the

PML reflection error in time domain were found consistently below -50 dB, for various PML

widths and sub-cell α values. A stability issue was encountered when developing the dispersive

PML for graphene. It was found that the problem was caused by the E field nodes on the

interface of the graphene and the air. The issue was resolved by modifying the node assignment


at the material interface. The simulation framework was then validated through a dielectric slab

test in 2-D and graphene slab tests in both 2-D and 3-D. In all cases, the transmission coefficients

found from the simulations matched well with the analytical solutions. Furthermore, a parallel

plate waveguide was simulated and both the wave impedance and wave number extracted from

the simulation matched with the analytical solutions. These tests validated the functionality

and accuracy of the newly developed framework.

Chapter 5

Study of Graphene Antennas in 2-D

Graphene antennas support Surface Plasmon Polaritons (SPP) modes, and this desirable feature

allows miniaturization of antenna to micrometer scale without operating the antenna in optical

frequency range. In this chapter, the graphene patch antenna introduced in [21] and [39] is

studied using the sub-cell scheme in a 2-D domain.

5.1 Analytical Model of A Graphene Patch Antenna

Graphene antennas attract much attention from the graphene research community in recent

years because it offers the possibility of miniaturizing the antenna size without going into

very high resonant frequencies. Downsizing classical metallic antennas to the micrometer scale

would result in resonant frequencies in optical frequency range. Transceiver systems at such

frequency spectrum are infeasible to implement due to high channel attenuation. Graphene,

on the other hand, supports transverse-magnetic (TM) SPP waves, which travel at speeds

around c/50. This feature allows graphene antennas of micrometer size to resonate in the

terahertz frequency range, where practical transceiver systems can be implemented, therefore,

gives graphene antennas advantage over their metallic counterparts.

A graphene patch antenna is shown in Fig. 5.1. The antenna has a length L, and width

W . It is illuminated by a plane wave normal to the patch length. The antenna is assumed

to operate at room temperature (T=300 K), with µc = 0 and γ = 5 × 1012. The graphene

63

Chapter 5. Study of Graphene Antennas in 2-D 64

conductivity is plotted in Fig. 5.2 based on equations (2.2), (2.3), for a frequency range below

3 THz. Fig. 5.2 shows that in the frequency range of interest, the intra-band conductivity

dominates over the inter-band conductivity. Therefore, in the following work, only intra-band

conductivity is used to model graphene.

Figure 5.1: Schematic of a graphene based patch antenna.

0 0.5 1 1.5 2 2.5−4

−2

0

2

4

6x 10

−4

σ intr

a (S

)

Re(σ

intra) Im (σ

intra)

0 0.5 1 1.5 2 2.50

2

4

6

8x 10

−5

frequency (THz)

σ inte

r (S

)

Re(σinter

) Im(σinter

)

Figure 5.2: Graphene conductivity at T=300 K, µc = 0 and γ = 5× 1012.

A free-standing graphene layer supports TM SPP waves, and the dispersion relation is given


by [23]:

β2 = k20

[

1−(

2

η0σg

)2]

, (5.1)

where β is the wave number of the guided modes, k0 is the free space wavenumber, η0 is the

intrinsic impedance of free space, and σg is the graphene surface conductivity. The effective

mode index can be derived from equation (5.1) as:

neff (ω) =

√

1− 4ǫ0µ0

1

(σg(ω))2. (5.2)

The plasmon wavelength (λspp) is found as:

λspp =λ

neff. (5.3)

0 0.5 1 1.5 2 2.5 3 3.5 410

−6

10−5

10−4

10−3

10−2

Frequency (THz)

λ spp[m

]

Figure 5.3: Graphene conductivity at T=300K, µc = 0 and γ = 5× 1012.

In Fig. 5.3, the plasmon wavelength as a function of frequency is plotted on a semi-

logarithmic scale. Since the resonant patch antenna is expected to have a length L around

half of the plasmon wavelength, according to Fig. 5.3, the antenna length is expected to be


on the order of a few micrometers in terahertz frequency range. The resonance condition for a

plasmonic patch antenna is [21]:

m1

2

λ

neff= L+ 2δL, (5.4)

where m is the order of the resonance, λ is the wavelength of incident wave, L is the antenna

length and δL is the field penetration length of the patch antenna. Equation (5.4) serves as a

guideline in designing graphene patch antennas.

5.2 Simulations of Graphene Patch Antennas in 2-D

Figure 5.4: Simulation domain for graphene patch antenna in 2-D.

Graphene patch antennas of various lengths L are simulated in a 2-D domain and their respective

resonant frequencies are found and compared to the analytical model. The 3-D graphene

antenna shown in Fig. 5.1 is in the yz plane with the thickness (t) of the graphene sheet in

the x− direction. When modeling the antenna in a 2-D domain, a cross section of the antenna

is taken in the xz plane, as shown in Fig. 5.4, and the width (W ) of the antenna is assumed

to be infinite in the y− direction. The length L is in the z− direction and the thickness t is

in the x− direction, which is modeled as a sub-cell layer. The spatial mesh is ∆x = 0.15µm


and ∆z = 0.25µm. With α = 0.01, the thickness of the graphene plate is d = 2× α×∆x = 3

nm. Using this mesh, a L = 5µm patch is modeled by 20 cells along the z− direction and 1

cell with sub-layer in the x− direction. The source excitation is a modulated Gaussian pulse of

2 THz width, exciting a plane wave with Ez polarization, incident normally onto the graphene

antenna. A receiving antenna is modeled here, and by reciprocity, a transmitting antenna would

have the same resonant condition.

L/2

L/2

Js

Js,center

L=λ/4

L=λ/2

L=λ

Figure 5.5: Theoretical current distribution along the length of a linear wire antenna.

In order to find the resonant frequency of the antenna, the surface current density Js in the

time domain have been recorded at each cell along the length of the antenna. The Js value can

be found by subtracting the Hy fields around the antenna, in a similar way as how the current

I is found in 3-D. The time domain field values at each spatial point are then transformed to

frequency domain through Fast Fourier Transform (FFT). At each frequency, the Js distribution

along the antenna can be plotted, and the Js value at the center of the antenna (Js,center), as

labeled in Fig. 5.5 can be found. Theoretically, Js,center peaks at the fundamental resonant

frequency. In this simulation setting, since the plane wave excitation is a modulated Gaussian


in the time domain, it is a shifted Gaussian in the frequency domain. Therefore the power

spectral density of the Js in the incident waveform is not uniform across frequency, and this

needs to be taken into consideration when searching for the peak value of Js over frequency. A

separate simulation is run in the same simulation domain but without the antenna, so that the

incident Js,inc field is recorded. The normalized Js is found as:

Js,norm =Js,centerJs,inc

(5.5)

The simulations are run for 6 different antenna lengths ranging from 3µm to 18µm. In Fig.

5.6, the normalized Js,center values obtained from simulations are plotted over frequency for

various antenna lengths. The fundamental resonant frequencies of the graphene patch antennas

can be found at the peak of each curve from the plot.

0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency (THz)

Nor

mal

ized

Js,

cent

er

L=3 µ m

L=5 µ m

L=8 µ m

L=12 µ m

L=16 µ m

L=18 µ m

Figure 5.6: Normalized ∆H value vs. frequency for different antenna lengths L.

In Fig. 5.7, the simulated resonant frequencies of graphene antennas are compared to those

from the analytical model. The first resonance of an infinitely wide graphene patch antenna

as a function of antenna length is plotted based on equation (5.4) using δL = 15% × L as

mentioned in [40]. One can see from Fig. 5.7 that the simulation results are matching closely to

the analytical solutions. The good agreement shows that the graphene antennas at micrometer


0 5 10 15 200.5

1

1.5

2

2.5

3

Resonator Length (µ m)

Fre

quen

cy (

TH

z)

Analytical modelSimulation results

Figure 5.7: First resonance of an infinitely wide graphene patch antenna as a function of antennalength.

scale resonate in terahertz range as predicted by the analytical model.

5.3 Summary

In this chapter, a graphene patch antenna has been studied. The theory of the SPP mode

supported on the graphene sheet and the resonant condition for the graphene antenna have

been presented. Simulations of graphene patch antennas of various lengths in the micrometer

scale have been run in a 2-D domain. The first resonant frequencies found from the simulations

are in the terahertz range, consistent with the prediction from the analytical model. The free-

standing graphene patch antenna in 2-D is a simplified model of a 3-D patch antenna with a

finite width. The results in 2-D simulations show that graphene antennas have the potential to

be used as terahertz antennas. Further studies of the antenna parameters such as the antenna

width and substrate material or size can be carried out to fine tune the antenna resonant

frequency.

Chapter 6

Conclusion

6.1 Summary

With growing interest in the newly discovered 2-D material graphene, many applications have

been developed and EM simulation tools for graphene based devices would be necessary to

study the complicated devices where analytical solutions do not exist. This thesis focuses on

developing a simulation framework for modeling graphene based devices in the time domain

using the FDTD method.

A systematic approach has been applied to develop the simulation framework. Firstly, a

study of the various forms of the graphene conductivity model has been done. The graphene

conductivity can be represented as a sum of two parts: the intra-band conductivity and the

inter-band conductivity, and it is found that in the microwave frequency range, only the intra-

band conductivity is needed in the model. The FDTD update equations for modeling graphene

have been derived using the auxiliary differential equation (ADE) method. The dispersion and

stability study for the intra-band model have been performed and the stability condition for

the graphene update equations is found to be the CFL limit. The modeling of the inter-band

conductivity has been briefly discussed, a linear model is sufficient for the microwave frequency

range, and a piecewise linear model or a Pade approximation can be used for a wider frequency

range.

To develop a framework for simulating graphene based structures, a 2-D sub-cell scheme has

70

Chapter 6. Conclusion 71

been studied. The need for having the PML as the terminating boundary condition is rather

natural, since many waveguiding applications are infinite structures that need to be terminated

by PML in simulations. The dispersive PML for graphene has been developed and integrated

with the sub-cell scheme to form a sub-cell dispersive PML framework in 2-D and 3-D.

The newly developed sub-cell dispersive PML framework has been characterized and val-

idated on several cases. First of all, the time domain reflection error of the sub-cell PML is

found significantly lower than that of the air PML. Secondly, the transmission coefficient tests

for a dielectric slab in 2-D and graphene slabs in 2-D and 3-D are carried out and the simu-

lation results and analytical solutions show good agreement. In addition, a 2-D parallel plate

waveguide is characterized with the simulation framework, and both the wave impedance and

wave number are found to match with the analytical solution.

Following the validation of the framework, a graphene patch antenna is studied in a 2-

D domain. A free-standing graphene sheet supports TM SPP wave, which allows graphene

antennas of micrometer size to resonant at terahertz frequency. Graphene patch antennas of

lengths ranging from 3µm to 18µm have been simulated and their resonant frequencies are

found to be in the terahertz frequencies and match well with the analytical model based on the

SPP wavelength. Such results show that the graphene antenna is a good candidate for terahertz

applications.

6.2 Contributions

There are two major contributions resulted from this thesis:

Journal Publication:

X. Yu, C. D. Sarris, “A Perfectly Matched Layer for Sub-cell FDTD and Applications to the

Modeling of Graphene Structures”, IEEE Antennas and Wireless Propagation Letters, Vol. 11,

p1080-p1083, 2012

Conference Proceeding:

X. Yu, C. D. Sarris, “FDTD Modeling of Graphene-Based RF Devices: Fundamental Aspects

and Applications” Antennas and Propagation (APS/URSI), 2012 IEEE International Sympo-

Chapter 6. Conclusion 72

sium on, 8-14 July 2012

6.3 Future Work

The sub-cell dispersive PML framework has been proven working in both 2-D and 3-D, and it

forms a foundation for future studies on more graphene based applications. Here are some of

the possible extensions of this thesis:

Based on the existing framework, graphene patch antennas with finite width can be sim-

ulated and the resonant frequencies for different antenna widths can be compared. Other

applications such as the graphene coplanar waveguide in [17] and graphene microstrips can also

be characterized with the 3-D code.

Beside using the sub-cell scheme to model graphene, the surface impedance boundary condi-

tion (SIBC) [41], [42] can also be applied to model graphene. The SIBC method further relaxes

the constraint on the unit cell size comparing to the sub-cell method, because with SIBC, the

unit cell size is not limited by the smallest physical feature of the structure. However, as a

trade off, SIBC is more complicated to implement. Also, modification to the original SIBC

formulation is needed, if the graphene sheet is not located at the boundary but in the middle

of the simulation domain, as in the case of the parallel plate waveguide studied in section 4.7.

In order to extend the existing framework to model devices up to optical frequency, the inter-

band conductivity need to be taken into consideration. This can be done by applying the Pade

approximation for graphene inter-band conductivity from [32], as mentioned in Chapter 2. For

devices operating at the high terahertz frequency range, where the intra-band conductivity and

the inter-band conductivity have comparable magnitudes, both models are needed for accurate

simulation results.

Appendix A

Derivation of the Dispersion

Relation and Stability Analysis

A.0.1 Derivation of the Dispersion Relation

To derive the dispersion relation, substitute a discrete travelling wave solution u = ej(ω∆t−kd∆d)

(d is direction: x, y or z) into equations (2.26)-(2.34) from section 2.3, the 9 equations for

dispersion relation calculation are:

Let λ = 1−γ∆T1+γ∆T and P = Q∆t

2(1+γ∆t)

Jx0ej(0.5ω∆t) = λJx0e

j(−0.5ω∆t) + P (Ex0ej(0.5ω∆t) + Ex0e

j(−0.5ω∆t)) (A.1)

Jy0ej(0.5ω∆t) = λJy0e

j(−0.5ω∆t) + P (Ey0ej(0.5ω∆t) + Ey0e

j(−0.5ω∆t)) (A.2)

Jz0ej(0.5ω∆t) = λJz0e

j(−0.5ω∆t) + P (Ez0ej(0.5ω∆t) + Ez0e

j(−0.5ω∆t)) (A.3)

( ǫ

∆t

)

Ex0ej(0.5ω∆t) =

( ǫ

∆t

)

Ex0ej(−0.5ω∆t) − 1

2(Jn

x0ej(−0.5ω∆t) + Jn

x0ej(−0.5ω∆t))

+

(

1

∆y(Hz0e

j(−0.5ky∆y) −Hz0ej(0.5ky∆y))− 1

∆z(Hy0e

j(−0.5kz∆z) −Hy0ej(0.5kz∆z))

) (A.4)

73

Appendix A. Derivation of the Dispersion Relation and Stability Analysis 74

( ǫ

∆t

)

Ey0ej(0.5ω∆t) =

( ǫ

∆t

)

Ey0ej(−0.5ω∆t) − 1

2(Jn

y0ej(−0.5ω∆t) + Jn

y0ej(−0.5ω∆t))

+

(

1

∆z(Hx0e

j(−0.5kz∆z) −Hx0ej(0.5kz∆z))− 1

∆x(Hz0e

j(−0.5kx∆x) −Hz0ej(0.5kx∆x))

) (A.5)

( ǫ

∆t

)

Ez0ej(0.5ω∆t) =

( ǫ

∆t

)

Ez0ej(−0.5ω∆t) − 1

2(Jn

z0ej(−0.5ω∆t) + Jn

z0ej(−0.5ω∆t))

+

(

1

∆x(Hy0e

j(−0.5kx∆x) −Hy0ej(0.5kx∆x))− 1

∆y(Hx0e

j(−0.5ky∆y) −Hx0ej(0.5ky∆y))

) (A.6)

Hx0ej(0.5ω∆t) =Hx0e

j(−0.5ω∆t) − ∆t

µ∆y(Ez0e

j(−0.5ky∆y) − Ez0ej(0.5ky∆y))

+∆t

µ∆z(Ey0e

j(−0.5kz∆z) − Ey0ej(0.5kz∆z))

(A.7)

Hy0ej(0.5ω∆t) =Hy0e

j(−0.5ω∆t) − ∆t

µ∆z(Ez0e

j(−0.5kz∆z) − Ez0ej(0.5kz∆z))

+∆t

µ∆x(Ez0e

j(−0.5kx∆x) − Ez0ej(0.5kx∆x))

(A.8)

Hz0ej(0.5ω∆t) =Hz0e

j(−0.5ω∆t) − ∆t

µ∆x(Ey0e

j(−0.5kx∆x) − Ey0ej(0.5kx∆x))

+∆t

µ∆y(Ex0e

j(−0.5ky∆y) − Ex0ej(0.5ky∆y))

(A.9)

Re-organize the 9 equations in the Ax = 0 format:


a 0 0 b 0 0 0 0 0

0 a 0 0 b 0 0 0 0

0 0 a 0 0 b 0 0 0

c 0 0 d 0 0 0 f −g

0 c 0 0 d 0 −f 0 h

0 0 c 0 0 d g −h 0

0 0 0 0 −l m e 0 0

0 0 0 l 0 −n 0 e 0

0 0 0 −m n 0 0 0 e

Jx0

Jy0

Jz0

Ex0

Ey0

Ez0

Hx0

Hy0

Hz0

= 0 (A.10)

The elements of matrix A are as follows:

a = λe−j0.5ω∆t − ej0.5ω∆t (A.11a)

b = P(

e−j0.5ω∆t − ej0.5ω∆t)

= 2P cos

(

ω∆t

2

)

(A.11b)

c =1

2

(

e−j0.5ω∆t − ej0.5ω∆t)

= cos

(

ω∆t

2

)

(A.11c)

d = (2j)( ǫ

∆t

)

sin

(

ω∆t

2

)

(A.11d)

e = e−j0.5ω∆t + ej0.5ω∆t = (−2j) sin

(

ω∆t

2

)

(A.11e)

f =1

∆z

(

e−j0.5kz∆z − ej0.5kz∆z)

= − 2j

∆zsin

kz∆z

2(A.11f)

g =1

∆y

(

e−j0.5ky∆y − ej0.5ky∆y)

= − 2j

∆ysin

ky∆y

2(A.11g)

h =1

∆x

(

e−j0.5kx∆x − ej0.5kx∆x)

= − 2j

∆xsin

kx∆x

2(A.11h)

l = − ∆t

µ∆z

(

e−j0.5kz∆z − ej0.5kz∆z)

=2j∆t

µ∆zsin

kz∆z

2(A.11i)

m = − ∆t

µ∆y

(

e−j0.5ky∆y − ej0.5ky∆y)

=2j∆t

µ∆ysin

ky∆y

2(A.11j)

n = − ∆t

µ∆x

(

e−j0.5kx∆x − ej0.5kx∆x)

=2j∆t

µ∆xsin

kx∆x

2(A.11k)

The dispersion relation can be found by setting det(A) = 0

Det(A) = e(ad − bc)(bce − ade+ afl + agm+ ahn)2 = 0 (A.12)


The simplified dispersion relation is:

bce = a(de − (fl+ gm+ hn)) (A.13)

equation (A.13) can be re-written as

a(de − (fl + gm+ hn))− bce = 0 (A.14)

Substitute in each of the variables, the dispersion relationship is:

(

λe−j0.5ω∆t − ej0.5ω∆t)

(

4 sin2(

ω∆t

2

)

−4∆t2

ǫµ∆x2sin2

(

kx∆x

2

)

− 4∆t2

ǫµ∆y2sin2

(

ky∆y

2

)

− 4∆t2

ǫµ∆z2sin2

(

kz∆z

2

))

+ (4j)P∆t

ǫcos2

(

ω∆t

2

)

sin

(

ω∆t

2

)

= 0

(A.15)

A.0.2 Detailed Stability Analysis

As mentioned in section 2.3, the dispersion relation in the r−plane is:

a3r3 + a2r

2 + a1r + a0 = 0 (A.16)

where

a3 = W +X + Y + Z = D(1− λ) (A.17a)

a2 = 3W +X − Y − 3Z = 4P +D(λ+ 1) (A.17b)

a1 = 3W −X − Y + 3Z = (D − 4)(λ− 1) (A.17c)

a0 = (4−D)(λ+ 1) (A.17d)

The stability condition is achieved when each element in the first column of the Routh table is

non-negative. A Routh table for equation (A.16) is constructed:


a3 a1a2 a0

a1 − a3a0a2

0

a0 0

Table A.1: Routh table for equation (A.16).

Now we evaluate each of the four conditions:

a3 = D(1− λ) (A.18)

D = 4

(

∆t2c2

∆x2sin2

(

kx∆x

2

)

+∆t2c2

∆y2sin2

(

ky∆y

2

)

+∆t2c2

∆z2sin2

(

kz∆z

2

))

(A.19)

(1− λ) = 1− 1− γ∆t

1 + γ∆t=

2γ∆t

1 + γ∆t(A.20)

since γ is always greater than zero (as it has a unit of 1/second), both D and (1−λ) are always

greater than zero, therefore a3 > 0 always.

a2 = 4P +D(λ+ 1) = 4Q∆t2

2ǫ(1 + γ∆t)+D(λ+ 1) (A.21)

4Q∆t2

2ǫ(1 + γ∆t)> 0 (A.22)

D > 0 (A.23)

λ+ 1 =2

1 + γ∆t> 0 (A.24)

since Q is always a positive number regardless the µc and T chosen, a2 > 0 always.

a0 = (4−D)(λ+ 1) (A.25)

λ+ 1 =2

1 + γ∆t> 0 (A.26)

we need to have (4−D) > 0 in order to achieve stability, divide both sides by 4, and substitute

in the expression for D from equation (A.19), the resulted expression is:

(

∆t2c2

∆x2sin2

(

kx∆x

2

)

+∆t2c2

∆y2sin2

(

ky∆y

2

)

+∆t2c2

∆z2sin2

(

kz∆z

2

))

< 1 (A.27)


To simplify the expression, assume the worst case where sin2(

kx∆x2

)

= sin2(

ky∆y2

)

= sin2(

kz∆z2

)

=

1, equation (A.27) becomes

S =∆t2c2

∆x2+

∆t2c2

∆y2+

∆t2c2

∆z2< 1 (A.28)

which is the CFL limit of the regular 3-D FDTD update equations. Therefore a0 > 0 is

conditionally true, subject to the CFL limit.

The last criteria is

a1 −a3a0a2

> 0 (A.29)

since we already showed that a2 > 0, we can convert equation (A.29) to a2a1 − a3a0 > 0

a2a1 − a3a0 = 4P (4−D)(1− λ) (A.30)

P > 0 (A.31)

1− λ > 0 (A.32)

4−D > 0 corresponding to the CFL limit as in the case for a0, and a2a1 − a3a0 > 0 always.

In conclusion, the stability condition for the graphene update equations is the same as the

CFL limit for the regular 3-D FDTD update equations.

References

[1] M. Dragoman, D. Neculoiu, D. Dragoman, G. Deligeorgis, G. Konstantinidis, A. Cis-

maru, F. Coccetti, and R. Plana, “Graphene for microwaves,” Microwave Magazine, IEEE,

vol. 11, no. 7, pp. 81 –86, Dec. 2010.

[2] J.-S. Moon and D. Gaskill, “Graphene: Its fundamentals to future applications,” Mi-

crowave Theory and Techniques, IEEE Transactions on, vol. 59, no. 10, pp. 2702 –2708,

Oct. 2011.

[3] Y.-M. Lin, C. Dimitrakopoulos, K. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill, and

P. Avouris, “100-GHz transistors from wafer-scale epitaxial graphene,” Science, vol. 327,

no. 5966, p. 662, 2010.

[4] Y.-M. Lin and et al, “Wafer-scale graphene integrated circuit,” Science, vol. 332, no. 6035,

pp. 1294–1297, June 2011.

[5] A. Geim and K. Novoselov, “The rise of graphene,” Nature Materials, vol. 6, pp. 183–191,

Mar. 2007.

[6] A. Geim, “Graphene: Status and prospects,” Science, vol. 324, no. 5934, pp. 1530–1534,

Jun 2009.

[7] C. Lee, X. Wei, J. Kysar, and J. Hone, “Measurement of the elastic properties and intrinsic

strength of monolayer graphene.” Science, vol. 321, p. 385388, 2008.

[8] Y. Hernandez and et al, “High-yield production of graphene by liquid-phase exfoliation of

graphite,” Nature Nanotechnology, vol. 3, pp. 563 – 568, 2008.

79

REFERENCES 80

[9] D. Dikin and et al, “Preparation and characterization of graphene oxide paper,” Nature,

vol. 448, pp. 457–460, 2007.

[10] J. Meyer, A. Geim, M. Katsnelson, K. Novoselov, T. Booth, and S. Roth, “The structure

of suspended graphene sheets,” Nature, vol. 446, pp. 60–63, Mar. 2007.

[11] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong,

“Few-layer graphene films on arbitrary substrates by chemical vapor deposition,” Nano

Letters, vol. 9, no. 1, pp. 30–35, Jan. 2009.

[12] K. Kim and et al, “Large-scale pattern growth of graphene films for stretchable transparent

electrodes,” Nature, vol. 457, pp. 706–710, Feb. 2009.

[13] J. Haas, W. de Heer, and E. Conard, “The growth and the morphology of epitaxial mul-

tilayer graphene,” J. Phys. Matter, vol. 20, no. 32, p. 323202/127, Aug. 2008.

[14] V. C. Tung, M. Allen, Y. Yang, and R. Kaner, “High-throughput solution processing of

large-scale graphene,” Nat. Nanotechnol., vol. 4, no. 1, p. 2528, Jan. 2009.

[15] S. Bae and et al, “Roll-to-roll production of 30 inch graphene films for transparent elec-

trodes,” Nat. Natechnol., vol. 5, p. 574578, Jun. 2010.

[16] G. Deligeorgis, M. Dragoman, D. Neculoiu, D. Dragoman, G. Konstantinidis, A. Cismaru,

and R. Plana, “Microwave switching of graphene field effect transistor at and far from the

dirac point,” Applied Physics Letters, vol. 96, no. 10, pp. 103 105–3, Mar 2010.

[17] D. Neculoiu, G. Deligeorgis, M. Dragoman, D. Dragoman, G. Konstantinidis, A. Cisman,

and R. Plana, “Electromagnetic propagation in graphene in the mm-wave frequency range,”

in Microwave Integrated Circuits Conference (EuMIC), 2010 European, Sept. 2010, pp. 377

–380.

[18] G. Deligeorgis and et al, “Microwave propagation in graphene,” Appl. Phys. Lett., vol. 95,

no. 073107, 2009.

REFERENCES 81

[19] P. Russer, N. Fichtner, P. Lugli, W. Porod, J. Russer, and H. Yordanov, “Nanoelectronics-

based integrate antennas,” Microwave Magazine, IEEE, vol. 11, no. 7, pp. 58 –71, Dec.

2010.

[20] I. Llatser, C. Kremers, A. Cabellos-Aparicio, J. Jornet, E. Alarcon, and D. Chigrin, “Scat-

tering of terahertz radiation on a graphene-based nano-antenna,” in AIP Conference Pro-

ceedings, no. 1398, 2011, pp. 144 – 147.

[21] I. Llatser, C. Kremers, D. Chigrin, J. Jornet, M. Lemme, A. Cabellos-Aparicio, and E. Alar-

con, “Characterization of graphene-based nano-antennas in the terahertz band,” in Anten-

nas and Propagation (EUCAP), 2012 6th European Conference on, March 2012, pp. 194

–198.

[22] M. Dragoman, A. Muller, D. Dragoman, F. Coccetti, and R. Plana, “Terahertz antenna

based on graphene,” Journal of Applied Physics, vol. 107, no. 10, p. 104313, May 2010.

[23] A. Vakil and N. Engheta, “Transformation optics using graphene,” Science, vol. 332, pp.

1291–1294, Jun. 2011.

[24] L. Pierantoni, D. Mencarelli, and T. Rozzi, “Modeling of the electromagnetic/coherent

transport problem in nano-structured materials, devices and systems using combined TLM-

FDTD techniques,” in IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2011.

[25] D. Mencarelli, L. Pierantoni, and T. Rozzi, “Graphene modeling by TLM approach,” in

Microwave Symposium Digest (MTT), 2012 IEEE MTT-S International, June 2012, pp. 1

–3.

[26] G. W. Hanson, “Dyadic greens functions and guided surface waves for a surface conduc-

tivity model of graphene,” Journal of Applied Physics, vol. 103, no. 064302, Mar. 2008.

[27] ——, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate

waveguide,” Journal of Applied Physics, vol. 104, no. 084314, 2008.

[28] G. Bouzianas, N. Kantartzis, C. Antonopoulos, and T. Tsiboukis, “Optimal modeling

REFERENCES 82

of infinite graphene sheets via a class of generalized FDTD schemes,” Magnetics, IEEE

Transactions on, vol. 48, no. 2, pp. 379 –382, Feb. 2012.

[29] V. Gusynin, S. Sharapov, and J. Carbotte, “Magneto-optical conductivity in graphene,”

Journal of Physics: Condensed Matter, vol. 19, no. 026222, 2007.

[30] K. Yee, “Numerical solution of initial boundary value problems involving maxwells equa-

tions in isotropic media,” IEEE Transactions on Antennas and Propagation, vol. 14, no. 3,

pp. 802–807, 1966.

[31] A. Pereda, L. Vielva, A. Vegas, and A. Prieto, “Analyzing the stability of the FDTD

technique by combining the von neumann method with the routh-hurwitz criterion,” Mi-

crowave Theory and Techniques, IEEE Transactions on, vol. 49, no. 2, pp. 377 –381, Feb

2001.

[32] A. Mock, “Pade approximant spectral fit for FDTD simulation of graphene in the near

infrared,” Optical Materials Express, vol. 2, no. 6, pp. 771 –781, 2012.

[33] J. G. Maloney and G. S. Smith, “The efficient modeling of thin material sheets in the finite-

difference time-domain (FDTD) method,” Antennas and Propagation, IEEE Transactions

on, vol. 40, no. 3, pp. 323–330, Mar. 1992.

[34] S. D. Gedney, “An anisotropic PML absorbing media for the FDTD simulatiohn of fields

in lossy and dispersive media,” Electromagnetics, vol. 16, no. 4, 1996.

[35] Y. Zhao, P. Belov, and Y. Hao, “Accurate modelling of left-handed metamaterials using

finite-difference time-domain method with spatial averaging at the boundaries,” J. Opt.

A: Pure Appl. Opt., vol. 9, pp. S468–S475, 2007.

[36] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-

Domain Method. Artech House, June 2000.

[37] C. C. Chao, S. H. Tu, H. I. Huang, C. C. Chen, and J. Y. . Chang, “Impedance-matching

surface plasmon absorber for FDTD simulations,” Plasmonics, vol. 5, pp. 51–55, 2010.

REFERENCES 83

[38] G. D. Bouzianas, N. V. Kantartzis, and T. D. Tsiboukis, “Consistent analysis and rigor-

ous characterization of infinite graphene layers via a subcell frequency-dependent FDTD

technique,” in General Assembly and Scientific Symposium, 2011 XXXth URSI, Aug. 2011.

[39] I. Llatser, C. Kremers, A. Cabellos-Aparicio, J. Jornet, E. Alarcon, and D. Chigrin,

“Graphene-based nano-patch antenna for terahertz radiation,” Photon Nanostruct: Fun-

damentals and Applications, vol. 10, no. 4, pp. 353–358, 2012.

[40] I. Llatser, C. Kremers, A. Cabellos-Aparicio, E. Alarcon, and D. Chigrin, “Comparison of

the resonant frequency in graphene and metallic nano-antennas,” in AIP Conf. Proc., no.

1475, Oct. 2012, pp. 143 –145.

[41] W. Thiel, “A surface impedance approach for modeling transmission line losses in FDTD,”

Microwave and Guided Wave Letters, IEEE, vol. 10, no. 3, pp. 89 –91, Mar 2000.

[42] J. Beggs, R. Luebbers, K. Yee, and K. Kunz, “Finite-difference time-domain implementa-

tion of surface impedance boundary conditions,” Antennas and Propagation, IEEE Trans-

actions on, vol. 40, no. 1, pp. 49 –56, Jan 1992.

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