propagation of electromagnetic waves in graphene waveguides · propagation of electromagnetic waves...

Propagation of ElectromagneticWaves in Graphene Waveguides

Bachelorarbeitvon

Christoph Helbig

vorgelegtam 20. Oktober 2011

Lehrstuhl fur Theoretische Physik IIUniversitat Augsburg

Propagation of ElectromagneticWaves in Graphene Waveguides

Bachelorarbeitvon

Christoph Helbig

vorgelegtam 20. Oktober 2011

Name: Christoph HelbigMatrikelnummer: 1071660Studiengang: Bachelor Physik

Erstprufer: Prof. Dr. Ulrich EckernZweitprufer: Prof. Dr. Arno Kampf

5

Contents

1 Motivation and Introduction 7

2 Electromagnetic Waves and Graphene Properties 112.1 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Graphene Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Electronic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Charge Carrier Density . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Gate Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.6 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 General Parallel Plate Waveguides (PPWG) 253.1 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Single Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Solution of Differential Equation . . . . . . . . . . . . . . . . . . . . . 273.4 Equal Conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 Optical Plasmon . . . . . . . . . . . . . . . . . . . . . . . . . 303.4.2 Acoustical Plasmon . . . . . . . . . . . . . . . . . . . . . . . . 32

3.5 Arbitrary Conductivities . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Graphene Waveguides 374.1 Dimensionless Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Single Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Equal Conductivities . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.1 Analytical Solution for Small Frequencies with Scattering . . . 464.3.2 Numerical Solution for All Frequencies Without Scattering . . 49

4.4 Arbitrary Conductivities . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Conclusion 63

List of figures 65

Bibliography 67

Lists of used symbols and constants 69

Acknowledgements 71

7

1 Motivation and Introduction

The element carbon is an unusual element. Though quasi-2-dimensional structures

have occupied physicists around the globe, real 2-dimensional states of matter have

been assumed not to exist. Indeed carbon is the first element found which truly

forms 2-dimensional states. With its 3-dimensional states diamond and graphite well

studied, fullerenes have caught the attention in the late 20th century, culminating in

a Nobel Prize in chemistry for Curl, Kroto and Smalley in 1996 [1]. These fullerenes

are quasi-0-dimensional globes of carbon, for example formed by 60 carbon atoms. In

the meantime also quasi-1-dimensional carbon nanotubes have found its way into the

world of recent physics. Their single walled form which “can be formed from graphene

sheets which are rolled up to form tubes has been known since 2003.”[2] Finally, with

the work of K. S. Novoselov and A. K. Geim published in 2004, the 2-dimensional state

of carbon, so called graphene has been compounded and experimentally analysed for

the first time [3]. Novoselov and Geim were awarded with a Nobel Prize in physics in

2010 [1], which made graphene popular even in public.

Nevertheless, graphene’s electronic properties have already been studied theoretically

since the middle of the 20th century, considered as a monolayer of graphite [2, 4]. At

that time though, it was assumed that the isolated 2-dimensional state of carbon, a

monolayer of carbon atoms, is unstable. Novoselov and Geim have proved it’s not [5].

With their work being published, experimental groups all over the world have started

searching for faster or cheaper ways of creating larger and better graphene flakes.

Theoretical physicists in the meantime have searched for interesting electromagnetic

or mechanic characteristics and possible appliances for graphene. A major summary

of the theoretical electronic properties of graphene has been given by Neto et al.

in 2009 [6]. According to Geim, this summary “is unlikely to require any revision

soon.”[7] Nevertheless with the electronic properties of graphene monolayers studied,

we can still search for possible appliances of graphene in electronic devices. Carbon

is available in huge quantities, so if we find appliances it could be possible to replace

8 Chapter 1 Motivation and Introduction

rare elements, for example in semiconductor devices. Perhaps even completely new

possibilities may emerge from the unique properties of graphene.

Graphene is considered special because of both its mechanic and electronic properties.

Graphene has relatively high conductivity although conducting band and valence

band only touch each other at six points [6]. Due to that electronic configuration, the

charge carrier density is much lower than in metals, but can be tuned easily. It was

shown that concerning quasi-TEM modes, graphene waveguides can be tuned and

the “energy loss is similar to many thicker-walled metal structures.”[10] Recently, the

properties of graphene in waveguides when used as a frequency multiplier have also

been studied [8, 9].

Intention of this work

This work is about one of the possible applications of graphene. Graphene shall make

it possible to change the velocity of waveguide modes during operation. In this work

I will assume two infinitely large monolayers of graphene with a fixed separation. I

will consider an electromagnetic wave propagating within the space in between both

monolayers, thus setting up a 2-dimensional waveguide. I will then derive dispersion

equations for different cases using always p-polarized waves. It will be shown that the

dispersion equation depends on the charge carrier density in the graphene layers.

This means the propagation speed of the electromagnetic wave is determined not

solely, but decisively by the charge carrier density which can be controlled by a gate

voltage. This voltage can be manipulated easily during the operation of the waveguide.

Of course, the difficult part of our waveguide would be to actually assemble the

two monolayers with fixed separation in an extent large enough as necessary for the

eventual use they would be designed for. Nevertheless, the problem of feasibility of

graphene waveguides will not be discussed in this work. I want to set up the dispersion

equation and for that I will just assume the required waveguide exists.

Such a graphene waveguide could possibly be used as a delay line for waves. The

velocity of the wave inside the retarder could be changed during operation just by

manipulating the gate voltage. Thus the amount of delay can be adjusted for every

application of the device.

9

Outline of this work

In chapter 2 I will discuss some general physical basics needed for the interpretation

of the results discussed in this work as well as basic properties of graphene for which

there is a lot of literature.

Chapter 3 is devoted to the basics of general parallel plate waveguides (PPWG).

That part consists mostly of analytical work and is a continuation of the studies

by Dr. S. Mikhailov from the Institute of Theoretical Physics II at the University of

Augsburg who set up the topic for this Bachelor’s thesis. It will introduce the different

kinds of waveguide modes which are investigated in this thesis. For all analytical

calculations, the Gaussian unit system will be used.

After that I will discuss the specialities of parallel plate waveguides made out of

graphene in chapter 4 by using a suitable conductivity. Apart from further deriving

analytical solutions which will be only possible for low frequencies, I will derive nu-

merical solutions of the dispersion equations. For that I will ignore scattering effects

though. For the numerical work I used Wolfram Mathematica for Students, version

number 8.0.1.0. Chapter 5 will give a short summary and interpretation of the results.

11

2 Electromagnetic Waves and

Graphene Properties

In this chapter a small introduction about electromagnetic radiation will be given.

Questions answered will be: Which kind of electromagnetic waves are we using?

What about polarization, what about frequency? After that we will discuss the basic

properties of graphene: The graphene lattice and the electronic configuration resulting

from that lattice. We will discuss charge carrier density in graphene and how gate

voltage can change that density. We will finish with the conductivity in the Drude

model and especially in graphene.

2.1 Electromagnetic Waves

We consider a structure consisting of two parallel conducting layers occupying the

planes z = −d/2 and z = d/2 and infinite in x and y directions. The layers are

assumed to be infinitely thin with the surface conductivity σ(ω) and can be made out

of metal or graphene. We assume that the electromagnetic wave propagates in the

x-direction. So the electric and magnetic field of the propagating waveguide mode

will be:~E(~r, t) = ~E(z)ei(qx−ωt) , (2.1)

~B(~r, t) = ~B(z)ei(qx−ωt) . (2.2)

In this geometry, two types of waveguide modes can propagate: The modes with the

B-vector in y-direction and the E-vector in the x-y-plane, so called TM- or p-polarized

modes, and the modes with the E-vector in the y-direction and the B-vector in the

x-y-plane, so called TE- or s-polarized modes. In this work we will only discuss p-

12 Chapter 2 Electromagnetic Waves and Graphene Properties

Figure 2.1: Geometry of the graphene waveguide layers and the p-polarized waveguide mode.

polarized waves propagating within the waveguide. Figure 2.1 shows the geometry of

the waveguide and his p-polarized mode as used in this work.

~E(z) =

Ex(z)

0

Ez(z)

, (2.3)

~B(z) =

0

By(z)

0

. (2.4)

So our electromagnetic waves will be defined by the wave number q and the frequency

ω. The link between these two values q and ω is called the dispersion relation q(ω)

or its inverse function ω(q). This dispersion equation is determined by the geometry

of the problem and the properties of matter the wave is propagating in. Normally,

the waveguide is entered by a wave with fixed frequency ω which is completely real,

which means the wave doesn’t decay in time. The wave number q however can be

complex: q = q′ + iq′′. Then if q′′ > 0, it expresses the decay in space concerning the

propagation direction x. Also the dispersion relation q(ω) doesn’t need to have one

unique solution for q to every frequency: Often different waveguide modes exist with

2.2 Graphene Lattice 13

different wave length and different decay parameter. The propagation velocity of the

wave can be obtained from the dispersion relation:

v =ω

q′(ω). (2.5)

In most cases, different waveguide modes will have different propagation velocities.

Now we need to know what kind of radiation we are talking about. Electromagnetic

waves, even linear polarized, can be of totally different kind, from radio waves to

gamma rays. And the different kinds will of course interfere with matter completely

differently. For this work we will stick with approximately microwaves.

2.2 Graphene Lattice

Graphene is considered as an infinitely thin and infinitely large 2-dimensional layer

of carbon. In this configuration all carbon atoms exist in the so called sp2 hybrid

electronic configuration. As those sp2 hybrid orbitals form 120 angles, all carbon

atoms have three closest neighbours forming a symmetrical lattice. In short we can

say: carbon atoms are bound to each other in a honeycomb lattice. That honeycomb

lattice though is not a 2-dimensional Bravais lattice, but we can find a 2-dimensional

hexagonal lattice with a 2-atom-basis to describe the 2-dimensional crystal as a Brav-

ais lattice.

Then the two lattice vectors of the hexagonal Bravais lattice are for example

~a1 = a(1/2,√

3/2) and ~a2 = a(−1/2,√

3/2). With the lattice vectors defined as ~a1 and

~a2 as well as a being the lattice constant, the vectors to the three closest neighbours

can be written as ~b1 = a(0, 1/√

3), ~b2 = a(1/2,−1/2√

3) and ~b3 = a(−1/2,−1/2√

3)

[4, 12]. Figure 2.2 displays the honeycomb lattice structure of graphene with the two

hexagonal sublattices.

The Brillouin zone of graphene then is a hexagon just like the honeycombs. The

basis vectors of the reciprocal lattice then can be written as ~G1 = 2πa−1(1, 1/√

3)

and ~G2 = 2πa−1(1,−1/√

3). This means that every second corner of the hexagon

is equivalent to the others, since they differ only by the addition or subtraction of

complete basis vectors of the reciprocal lattice [13]. We will from now on call these

kinds of corner points K and K ′. K and K ′ are equivalent concerning charge carrier

density and density of states. Figure 2.3 shows the first Brillouin zone of graphene.


Figure 2.2: The honeycomb lattice of graphene. Two Bravais sublattices can be identified. All pointsof the sublattice A (black circles) are given by n1~a1 + n2~a2 with n being integers and~a being the lattice vectors. All points of the sublattice B (open circles) are given by

n1~a1 + n2~a2 + ~b with ~b being the one vector to a closest neighbour atom of which everyatom in a honeycomb lattice has three in total. Dashed lines show the boundaries of theelementary cell. a is the lattice constant. Picture taken from Tudorovskiy [11].

2.2 Graphene Lattice 15

Figure 2.3: The Brillouin zone of graphene. The basis vectors of the reciprocal lattice are ~G1 and~G2. The vectors Kj , j = 1, . . . , 6, correspond to the corners of the Brillouin zone.These corners are called the Dirac points. Every second corner is equivalent, sincethey only differ by addition or subtraction of a basis vector. Here K1 = −K4 =2πa−1(1/3, 1/

√3),K2 = −K5 = 2πa−1(2/3, 0),K3 = −K6 = 2πa−1(1/3,−1/

√3). We

will call odd numbered corners K while even numbered corners K ′. Picture taken fromTudorovskiy [11].


2.3 Electronic Configuration

Now with a tight-binding model we can derive the electronic band structure of graph-

ene. The result will be the relation for the energy eigenvalues in dependence of the

wave vector [4, 12].

E(~k) = ±2~Va√

3|ei~k·~b1 + ei

~k·~b2 + ei~k·~b3| . (2.6)

Here ~b1, ~b2 and ~b3 are the vectors to the closest neighbours of one carbon atom as

defined in chapter 2.2. We can see that the two energy bands are symmetrical around

energy 0 and they form cones in the six corners of the first Brillouin zone [12]. Figure

2.4 displays the valence band and the conduction band for the whole first Brillouin

zone. Since the cones touch each other, the total band gap is zero.

Figure 2.4: Energy band structure of graphene for whole first Brillouin zone. k is given in units of1/a. Valence band and conduction band are symmetrical around zero-energy and theband gap is zero. Touching points are the six corners K and K ′ of the first Brillouinzone. For µ = 0 and T = 0 all negative energy states are filled, all positive energy statesare empty.

At this point it is important to show that the Dirac cones really are cones in first

approximation for small values of q. We show this just as an example for wave vectors

2.3 Electronic Configuration 17

close to ~K ′ = ~K2. If we define a relative wave vector ~q by ~k = ~K + ~k for corners K

and respectively ~k = ~K ′ + ~k for corners K ′ both with k K respectively k K ′

then we get in the calculation of the band energies:

ei((~K′+~k)·~b1) + ei((

~K′+~k)·~b2) + ei((~K′+~k)·~b3) ≈

= (1 + i~k · ~b1)ei~K′·~b1 + (1 + i~k · ~b2)ei

~K′·~b2 + (1 + i~k · ~b3)ei~K′·~b3 =

= a

√3

2(kx − iky) .

(2.7)

As for complex values counts |z| =√=(z)2 + <(z)2 we get the linear relation between

energy eigenvalue and relative wave vector. Similar calculation works with wave

vectors close to ~K = ~K5 and we get [14]:

ei((~K+

~k)·~b1) + ei((

~K+~k)·~b2) + ei((

~K+~k)·~b3) ≈ a

√3

2(−kx − iky) . (2.8)

The total energy relation near Brillouin corner points gets:

E±(~k) = ±~V k −O((k/K)2) . (2.9)

This means that around Fermi energy level, which is 0, both energy bands are linear,

not parabolic, and they touch exactly at Fermi level at just 6 discrete points: the

corners K and K ′ [6]. This structure of energy somehow looks similar to the band

structure of relativistic particles with vanishing mass. This is the reason why we can

also call electrons in graphene quasi-relativistic electrons. This also results in the

corners of the Brillouin zone being called Dirac points [6]. The Fermi velocity V in

graphene is 108 cm/s or 1/300 of the speed of light [3]. The cone structure of energy

bands near the Dirac point is displayed in figure 2.5.


Figure 2.5: Energy band structure of graphene around Dirac point K ′ = K2. k is given in units of1/a. Cone structure for low energies is clearly visible.

2.4 Charge Carrier Density 19

2.4 Charge Carrier Density

For proper calculation of the conductivity of graphene we need to know the depend-

encies of charge carrier density.

ns =1

S

∑~k,σ,v

f0(~k) =gsgv(2π)2

∫dkx

∫dky f0(~k) . (2.10)

Here we introduce gs as spin degeneracy and gv as degeneration for the Dirac cones of

which exist six in total, but each only contributes with one third to the first Brillouin

zone [15, 6]. So both gs and gv have the value of 2. With f0(~k) = Θ(µ−E) at T = 0

and E = ~V k as well as cylindrical coordinates used this transforms into [6]:

ns =4

(2π)22π

µ∫0

dE1

~VE

~V, (2.11)

⇔ ns =µ2

π~2V 2. (2.12)

2.5 Gate Voltage

For further analysis we need to get to know the link between gate voltage applied

and the change of the chemical potential. Gate voltage, that means the voltage

between both conducting layers or, if suitable for application, voltage between a third

external layer and one or both graphene layers. As already known, the chemical po-

tential defines the density of charge carriers and thus the conductivity of the graphene

monolayer. We can look on the graphene layers as part of a capacitor. The density

of electrons on the two surfaces is defined by the amount of voltage applied. For

plane-parallel capacitors following basic rule applies:

Q = C · U . (2.13)

Here C is the capacity of the constructed capacitor. Given a surface area A we get

with C = S/4πd:

ensS =SU

4πd. (2.14)


We get the following dependency of charge carrier density from gate voltage:

ns =U

4πed. (2.15)

Using equations (2.12) and (2.15) we get:

U

4πed= ns =

µ2

π~2V 2. (2.16)

This gives us the relation between chemical potential and the voltage applied.

µ = ~V√

U

4ed. (2.17)

2.6 Conductivity

For calculating the conductivity, we will use the Drude model for transport of electrons

in materials. In the Drude model, motion of the charge carriers is determined by a

differential equation representing the impact of the forces on one charge carrier with

charge e:

~P +~P

τ+ e ~E = 0 . (2.18)

In this equation, τ is the relaxation time of charge carriers in the material. With

non-relativistic impulse we get:

m~v +m~v

τ− e ~E = 0 . (2.19)

Note that with an alternating electric field ~E with frequency ω we can not assume

that ~v = 0. Instead since we have a periodic behaviour with ~E(t) = ~E0e−iωt, so for

charge carrier velocity applies ~v(t) = ~v0e−iωt. This leads to a reduced amplitude of

current [16]:

m(−iω~v) +m~v

τ= e ~E . (2.20)

With ~j = −ens~v this gets:

~j =e2nsm

11τ− iω

~E . (2.21)

2.6 Conductivity 21

Now this way we get the AC conductivity σ(ω):

σ(ω) =σ0

1− iωτ. (2.22)

The explicit expressions for electric field and current are then:

~E = <[ ~E0e−iωt] , (2.23)

~j = <[σ(ω) ~E0e−iωt] . (2.24)

It is convenient to introduce the reciprocal value of the relaxation time τ ; γ = 1/τ

is called called the scattering rate. With this value introduced we get the following

expression for the frequency depending conductivity:

σ(ω) =inse

2

m(ω + iγ). (2.25)

Electrons and holes as charge carriers in graphene are considered not to have an

effective rest mass. We will therefore only work with the term m = µ/V 2 resulting

from mass-energy-equivalence.

Inter- and Intraband Conductivity of Graphene

In their work from 2007, Mikhailov and Ziegler have used the intraband and interband

conductivity of graphene at T/µ→ 0:

σintra =inse

2V 2

(ω + i0)µ=e2gsgv16~

4i

πΩµ

. (2.26)

This conductivity term is Drude-like. Here the scattering rate γ was assumed to be

negligible. Note that for low frequencies that might not be justified.

σinter =e2gsgv16~

(Θ(|Ωµ| − 2)− i

πln

∣∣∣∣2 + Ωµ

2− Ωµ

∣∣∣∣) . (2.27)

Here dimensionless frequency Ωµ = ~ω/µ has been introduced. ns = gsgvµ2/4π~2V 2

is the density of electrons at T = 0 as was already shown in chapter 2.4 [17]. Then


total frequency depending intra- and interband conductivity of graphene with real

and imaginary part can then be written as:

σtot =e2gsgv16~

(Θ(|Ωµ| − 2) +

i

π

(4

Ωµ

− ln

∣∣∣∣2 + Ωµ

2− Ωµ

∣∣∣∣)) . (2.28)

Figure 2.6 shows an overview over the change of real and imaginary part of conduct-

ivity for values of Ωµ from 0 to 4 [17]:

1 2 3 4WΜ

-2

-1

1

2

3Conductivity

Figure 2.6: Dimensionless conductivity of graphene (real and imaginary part) in dependence of di-mensionless frequency Ωµ = ~ω/µ. Conductivity is displayed in units of e2gsgv/16~.Imaginary intraband (blue, dashed), imaginary interband (blue, dotted), imaginary total(blue, solid) and real interband/total (red, solid). Imaginary part diverges at Ωµ = 0and Ωµ = 2; For Ωµ 1 interband conductivity can be neglected. The real interbandconductivity has a jump point, because as soon as the photon energy is ~ω > 2µ = 2EFthere is an unoccupied state in the conduction band (E(~k) = +~V k) for the same ~k as

for an occupied state in the valence band (E(~k) = −~V k). This is independent of thealgebraic sign of the voltage applied.

Now let’s analyse how intraband and interband conductivity matter in our case with

graphene waveguides. Concerning the waves we assume waves with f . 300 GHz. For

the charge carrier density it has become appropriate to assume ns ≈ 6 · 1012 1/cm2.

Then Ωµ 1 and thus:

σinter(Ωµ) ≈ 0 , (2.29)

σ(Ωµ) ≈ σintra(Ωµ) =e2gsgv16~

4i

πΩµ

=e2gsgvi

4π~Ωµ

. (2.30)

This way only the intraband conductivity is relevant for the later studied waveguides.

Note that this expression is still without scattering, or in other words: Here the

relaxation time was assumed to be infinite.

2.6 Conductivity 23

Now with the conductivity derived in chapter 3.3 and the assumption that only intra-

band conductivity counts, we get the important part of the equation:

σ(ω) =ie2nsV

2

µ(ω + iγ). (2.31)

We can replace the material components ns and µ with the relations from above and

thus get the gate voltage dependency of the frequency dependent electric conductivity

of graphene:

σ(ω) =ie2V

2π~(ω + iγ)

√U

ed. (2.32)

25

3 General Parallel Plate Waveguides

(PPWG)

This section is based on the paper “Waveguide modes in graphene” by Dr. Sergey

Mikhailov from January 2011 [18]. We will introduce the different waveguide modes

for parallel plate waveguides and also discuss the single layer case. No graphene

conductivity will be implemented yet. We assume a parallel plate waveguide (PPWG)

made of graphene. As explained in chapter 2.1, we will assume a waveguide with layer

seperation d, propagation direction x and only p-polarized modes.

3.1 Maxwell Equations

We will start with having a look at the Maxwell equations in Gaussian units for matter

with σ 6= 0. First important Maxwell equation is Faraday’s law of induction:

∇× ~E = −1

c

∂ ~B

∂t. (3.1)

The second important Maxwell equation in our problem is Ampere’s circuital law:

∇× ~B =4π

c~j +

1

c

∂ ~E

∂t. (3.2)

Since only p-polarized waves are considered some components of the electric and the

magnetic field are zero: Ey = Bx = Bz = 0. This automatically means we just have

to look at the y-component of Faraday’s law (3.1) and the x- and z-component of

Ampere’s law (3.2). Concerning the time dependency we use ~E(~r, t) = ~E(~r)e−iωt and~B(~r, t) = ~B(~r)e−iωt (see equations (2.1) and (2.2)). Faraday’s law then simplifies to:

∂Ex∂z− ∂Ez

∂x= −1

c

∂By

∂t=iω

cBy . (3.3)

26 Chapter 3 General Parallel Plate Waveguides (PPWG)

When considering the simplification of Ampere’s law we need to remember that the

whole conductivity of each of the graphene monolayers is concentrated in one infinitely

thin layer, resulting in a total of two delta functions.

∂By

∂x= −iω

cEz , (3.4)

∂By

∂z=iω

cEx −

4π

c[σ1(ω)Ex(z1)δ(z − z1) + σ2(ω)Ex(z2)δ(z − z2)] . (3.5)

Now we can differentiate the simplified Faraday’s law (3.3). Concerning the differen-

tiation with respect to location for electromagnetic waves propagating in x-direction

in our waveguide we have to keep in mind that ~E(~r) = ~E(z)eiqx. Especially we are

interested in finding Ex(z). Let’s first look at the differentiation of (3.3) with respect

to x. This gives us:

iq∂Ex∂z

+ q2Ez =ω2

c2Ez . (3.6)

Now this gives us an expression for Ez which we differentiate once again with respect

to z:∂Ez∂z

= − iq

q2 − ω2/c2

∂2Ex∂z2

. (3.7)

Now let’s look at the differentiation of (3.3) with respect to z: This gives us:

∂2Ex∂z2

− iq ∂Ez∂z

+ω2

c2Ex = −4πiω

c2[σ1Ex(z1)δ(z − z1) + σ2Ex(z2)δ(z − z2)] . (3.8)

To eliminate the unknown Ez, it comes in handy that we already derived equation

(3.7). We insert this expression into (3.8), simplify and introduce the value κ with

κ2 = q2 − ω2/c2 and we receive the differential equation for Ex which we will analyse

further:

∂2Ex∂z2

− κ2Ex =4πiκ2

ω[σ1Ex(z1)δ(z − z1) + σ2Ex(z2)δ(z − z2)] . (3.9)

3.2 Single Layer

Before we go into details of the two-layer problem, a short consideration about plas-

mons propagating along a single graphene monolayer should be made. We assume

that the layer lies at z = 0. As we are searching for the solution of a differential

3.3 Solution of Differential Equation 27

equation of order 2, we have two constant parameters for each of the areas. The

solution of the differential equation is then

Ex(z) = E1eκz, z < 0

= E2e−κz, z > 0 . (3.10)

In (3.10) it was already assumed that the field vanishes in infinity. The remaining

two parameters will be defined by the boundary condition, which is that we want the

electric field to be continuous:

Ex(−0) = Ex(+0)⇒ E1 = E2 . (3.11)

Now we will integrate the adapted differential equation of (3.9) over a small interval

around 0:∂Ex(z)

∂z

∣∣∣∣εε

=4πiκ2

ωσ(ω)E . (3.12)

This leads with ε→ 0 to the dispersion equation of for a plasmon propagating along

a single graphene monolayer:

1 +2πiκ

ωσ(ω) = 0 . (3.13)

Or the same equation with re-substituting κ by√q2 − ω2/c2:

1 +2πi√q2 − ω2

c2

ωσ(ω) = 0 . (3.14)

3.3 Solution of Differential Equation

To find the form of the solution for the given differential equation of second order (3.9)

in the waveguide case with two graphene layers, we divide space into three areas. The

solution for Ex(z) can then be written as:

Ex(z) = E1eκ(z+d/2), z < −d/2

= A sinhκz +B coshκz,< d/2 < z < d/2

= E2e−κ(z−d/2), z > d/2 . (3.15)


Again, we have two constant parameters for each of the areas, but as in the single

layer case, in (3.15) it was already assumed that the field vanishes in infinity. The

remaining four parameters will be defined by the boundary conditions. First, we

demand the electric field to be continuous. This gives us two boundary conditions:

Ex(−d/2− 0) = Ex(−d/2 + 0) and Ex(+d/2− 0) = Ex(+d/2 + 0). This way E1, E2,

A and B are no longer independent.

Ex(−d/2− 0) = E1 = −A sinhκd

2+B cosh

κd

2= Ex(−d/2 + 0) . (3.16)

Ex(d/2− 0) = E2 = A sinhκd

2+B cosh

κd

2= Ex(+d/2 + 0) . (3.17)

Therefore we can express A and B in matters of E1 and E2:

A =E2 − E1

2 sinh(κd/2). (3.18)

B =E2 + E1

2 cosh(κd/2). (3.19)

Now with inserting these expressions for A and B into the solution of the differential

equation we get a solution only depending on E1 and E2:

Ex(z) = E1eκ(z+d/2) , z < −d/2

=(E2 − E1) sinhκz

2 sinhκd/2+

(E2 + E1) coshκz

2 coshκd/2,−d/2 < z < d/2

= E2e−κ(z−d/2) , z > d/2 . (3.20)

We can also list the solution Ez(z):

Ez(z) = −i qκ2

∂Ex∂z

= −i qκE1e

κ(z+d/2) , z < −d/2

= −i qκ

(E2 − E1

2

coshκz

sinhκd/2+E2 + E1

2

coshκz

sinhκd/2

),−d/2 < z < d/2

= iq

κE2e

−κ(z−d/2) , z > d/2 . (3.21)

3.3 Solution of Differential Equation 29

In order to get the second pair of boundary conditions, we integrate the differential

equation (3.9) over small intervals around z = −d/2 respectively z = d/2:

−d/2+ε∫−d/2−ε

dz∂2Ex∂z2−−d/2+ε∫−d/2−ε

dz κ2Ex =

−d/2+ε∫−d/2−ε

dz4πiκ2

ω[σ1Ex(z1)δ(z−z1)+σ2Ex(z2)δ(z−z2)] ,

(3.22)d/2+ε∫d/2−ε

dz∂2Ex∂z2

−d/2+ε∫d/2−ε

dz κ2Ex =

d/2+ε∫d/2−ε

dz4πiκ2

ω[σ1Ex(z1)δ(z−z1)+σ2Ex(z2)δ(z−z2)] .

(3.23)

For integration over delta functions we use∫f(x)δ(x− x0) dx = f(x0). If we assume

ε → 0 then the integration over κ2Ex vanishes. Meanwhile the integration over the

second differentiation of Ex gives difference between the border terms of the first

differentiation for both integration intervals:

∂Ex∂z

∣∣∣∣−d/2+ε

−d/2−ε=

4πiκ2

ωσ1(ω)E1 , (3.24)

∂Ex∂z

∣∣∣∣d/2+ε

d/2−ε=

4πiκ2

ωσ2(ω)E2 . (3.25)

Now if we look at the differentiation of the solution of our differential equation given

in equation (3.20), we can insert the border values (z = −d/2 and z = d/2) and gain

two equations for the link between E1, E2 and σ1, σ2.

−E1 +E2 − E1

2cothκd/2− E2 + E1

2tanhκd/2 =

4πiκ

ωσ1(ω)E1 , (3.26)

−E2 −E2 − E1

2cothκd/2− E2 + E1

2tanhκd/2 =

4πiκ

ωσ2(ω)E2 . (3.27)

This system of equations can be written in a compact notation using matrices:([1 + coth(κd/2) + 4πiκ

ωσ1(ω)] −[1 + coth(κd/2) + 4πiκ

ωσ2(ω)]

[1 + tanh(κd/2) + 4πiκωσ1(ω)] [1 + tanh(κd/2) + 4πiκ

ωσ2(ω)]

)(E1

E2

)= 0 .

(3.28)

This is now the basic equation the following calculations of dispersion equations work

with. We can now discuss different kinds of solutions for different specifications of σ1


and σ2, as well as parameter conditions E1 and E2 given in each case. Note that up to

now none of the calculations are made especially for graphene except the monolayers

assumed as infinitely thin with finite conductivity.

3.4 Equal Conductivities

Let’s consider the special case when the two conductivities of the two monolayers are

equal (σ1 = σ2 = σ). Then the equation system in (3.28) simplifies to these two

equations: [1 + coth(κd/2) +

4πiκ

ωσ(ω)

](E1 − E2) = 0 , (3.29)[

1 + tanh(κd/2) +4πiκ

ωσ(ω)

](E1 + E2) = 0 . (3.30)

This leads to two different solutions which we will call optical and acoustical.

3.4.1 Optical Plasmon

The optical plasmon solution is defined by in-phase oscillation of charges in different

layers:

E1 = E2 = E . (3.31)

Then (3.29) is trivially fulfilled while (3.30) simplifies to:

1 + tanh(κd/2) +4πiκ

ωσ(ω) = 0 . (3.32)

We can rewrite this equation with an exponential function rather than a hyperbolic

function:

1 +2πiκ

ω

(1 + e−κd

)σ(ω) = 0 . (3.33)


1 +2πi√q2 − ω2

c2

ω

(1 + e−

√q2−ω2

c2d

)σ(ω) = 0 . (3.34)

3.4 Equal Conductivities 31

Now let’s have a look at the solution of our original differential equation with the

specifications of the optical special solution:

A = 0, B =E

cosh(κd/2). (3.35)

With these, the solution looks like this:

Ex(z) = Eeκ(z+d/2) , z < −d/2

=E coshκz

cosh(κd/2),−d/2 < z < d/2

= Ee−κ(z−d/2) , z > d/2 . (3.36)

Figure 3.1 shows Ex for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1

and Z = zq0.

-3 -2 -1 1 2 3 Z

0.2

0.4

0.6

0.8

1.0

Ex

Figure 3.1: Electric field in propagation direction versus location (optical mode) in dimensionlessunits. K = 1 and D = 1. Nondimensionalized with K = κ/q0, D = dq0 = 1 andZ = zq0.

And this way we can also write down Ez(z):

Ez(z) = − iqκ2

∂Ex∂z

= −iqκEeκ(z+d/2) , z < −d/2

= −iqκ

E sinhκz

cosh(κd/2),−d/2 < z < d/2

=iq

κEe−κ(z−d/2) , z > d/2 . (3.37)


Figure 3.2 shows Ez for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1

and Z = zq0.

-3 -2 -1 1 2 3 Z

-1.0

-0.5

0.5

1.0

Ez

Figure 3.2: Electric field perpendicular to propagation direction versus location (optical mode) indimensionless units. K = 1 and D = 1. Nondimensionalized with K = κ/q0, D = dq0 =1 and Z = zq0.

3.4.2 Acoustical Plasmon

In a second special case of equal conductivities we will now consider out-of-phase

oscillation of charges in different layers:

E1 = −E2 = E . (3.38)

The action is done in the same way as in the optical case. In the acoustical case (3.30)

is trivially fulfilled while (3.29) simplifies to:

1 + coth(κd/2) +4πiκ

ωσ(ω) = 0 . (3.39)

Like in the optical case, transform into equation with exponential function rather

than hyperbolic function:

1 +2πiκ

ω

(1− e−κd

)σ(ω) = 0 . (3.40)


Or the same equation with replacing κ by√q2 − ω2/c2:

1 +2πi√q2 − ω2

c2

ω

(1− e−

√q2−ω2

c2d

)σ(ω) = 0 . (3.41)

In the acoustical case the solution looks as follows:

A =E

sinh(κd/2), B = 0 , (3.42)

Ex(z) = −Eeκ(z+d/2) , z < −d/2 (3.43)

=E sinhκz

sinh(κd/2),−d/2 < z < d/2 (3.44)

= Ee−κ(z−d/2) , z > d/2 . (3.45)

Figure 3.3 shows Ex for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1

and Z = zq0.

-3 -2 -1 1 2 3 Z

-1.0

-0.5

0.5

1.0Ex

Figure 3.3: Electric field perpendicular to propagation direction versus location (acoustical mode) indimensionless units. K = 1 and D = 1. Nondimensionalized with K = κ/q0, D = dq0 =1 and Z = zq0.


Ez(z) = − iqκ2

∂Ex∂z

=iq

κEeκ(z+d/2) , z < −d/2

= −iqκ

E coshκz

sinh(κd/2),−d/2 < z < d/2

=iq

κEe−κ(z−d/2) , z > d/2 . (3.46)

Figure 3.4 shows Ez for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1

and Z = zq0.

-3 -2 -1 1 2 3 Z

0.5

1.0

1.5

2.0

Ez

Figure 3.4: Electric field perpendicular to propagation direction versus location (acoustical mode) indimensionless units. K = 1 and D = 1. Nondimensionalized with K = κ/q0, D = dq0 =1 and Z = zq0.

3.5 Arbitrary Conductivities

In equation (3.28) we expressed the compact form of the equation system leading to

dispersion relations with a matrix. We can also analyse modes in the case σ1 6= σ2.

Then there will also be two modes, but they won’t be completely optical and acoustical

Like in the derivation of the dispersion relation for the optical and the acoustical case

3.5 Arbitrary Conductivities 35

the analysis gets easier if we rewrite equation (3.28) with exponential functions rather

than hyperbolic functions.:([1 + 2πiκ

ωσ1(ω)(1− e−κd)] −[1 + 2πiκ

ωσ2(ω)(1− e−κd)]

[1 + 2πiκωσ1(ω)(1 + e−κd)] [1 + 2πiκ

ωσ2(ω)(1 + e−κd)]

)(E1

E2

)= 0 . (3.47)


[1 +2πi

√q2−ω2

c2

ωσ1(ω)(1− e−

√q2−ω2

c2d)]

−[1 +2πi

√q2−ω2

c2

ωσ2(ω)(1− e−

√q2−ω2

c2d)]

[1 +2πi

√q2−ω2

c2

ωσ1(ω)(1 + e−

√q2−ω2

c2d)]

[1 +2πi

√q2−ω2

c2

ωσ2(ω)(1 + e−

√q2−ω2

c2d)]

(E1

E2

)= 0 . (3.48)

This equation can only be fulfilled if the determinant of the matrix is 0 and has two

different solutions which have to be found. So the equation which has to be solved is:

[1 +2πiκσ1(ω)

ω(1− e−κd)][1 +

2πiκσ2(ω)

ω(1 + e−κd)] +

[1 +2πiκσ1(ω)

ω(1 + e−κd)][1 +

2πiκσ2(ω)

ω(1− e−κd)] = 0 . (3.49)

We will solve this equation for graphene conductivities numerically in the next chapter.

If we multiply the expressions out the equation gets equal to:

1 +2πiκσ1(ω)

ω+

2πiκσ2(ω)

ω+

2πiκσ1(ω)

ω

2πiκσ2(ω)

ω(1− e−2κd) = 0 . (3.50)

If we transform this into a term with expressions of (σ1 + σ2)/2 and (σ1 − σ2)/2 this

equation gets equal to:

[1 +2πiκ

ω

σ1 + σ2

2(1− e−κd)][1 +

2πiκ

ω

σ1 + σ2

2(1 + e−κd)] −

2πiκ

ω

σ1 − σ2

2(1− e−κd)2πiκ

ω

σ1 − σ2

2(1 + e−κd) = 0 . (3.51)

In this term (3.51) we see that if σ1 = σ2 the second term vanishes and the dispersion

equations (optical and acoustical case) of equal conductivities are recovered since

always only one factor has to become zero.

37

4 Graphene Waveguides

Now we want to analyse the properties of graphene waveguides. For this we will

both need the properties of graphene monolayers explained in chapter 2 as well as the

waveguide modes introduced in chapter 3.

4.1 Dimensionless Units

To analyse the dispersion relations with graphene specific conductivities, it will be

important to introduce dimensionless units. There are plenty possibilities for find-

ing dimensionless units for wave number and frequency. All will result in different

parameters which can be changed according to different boundary conditions for the

parallel plate waveguide. In this work we will use dimensionless units which are in-

dependent of the plate spacing d. This way we can use the same dimensionless units

for both single layer plasmons as well as the waveguide modes.

Let’s start with a dispersion relation for the single-layer case with Drude conductivity

without scattering and the assumption of c→∞ which gives the identity κ = q.

1 +2πiq

ω

inse2V 2

ωµ= 0 . (4.1)

This dispersion equation is solved by the following plasma frequency:

ω =

√2πnse2V 2

µq . (4.2)

Let’s compare this with the light line ω = cq and see where both relations intersect.

Figure 4.1 shows both the plasma frequency relation and the light line. We define the

intersection point as (q0, ω0) by the following equation:

2πnse2V 2

µq = c2q2 . (4.3)

38 Chapter 4 Graphene Waveguides

q0

q

Ω0

Ω

Figure 4.1: Frequency versus wave number. Light line (dashed) and plasma frequency (solid). Cross-ing point defines the dimensionless units and is located at (q0, ω0).

This gives a wave number only dependent of nature constants and the charge carrier

density and/or the chemical potential in graphene:

q0 =2πnse

2V 2

µc2=

2e2V

~c2

√πns =

2e2µ

~2c2. (4.4)

The corresponding frequency is then:

ω0 = cq0 =2πnse

2V 2

µc=

2e2V

~c√πns =

2e2µ

~2c. (4.5)

This way we can define both the dimensionless wave number Q = q/q0 and the

dimensionless frequency Ω = ω/ω0. It is important to know which values these

dimensionless units can take in our problem. For this we will also compare our

graphene problem with metals. At a frequency of f = 100 GHz and with a charge

carrier density of ns = 1018 cm−2 for metals as well as a charge carrier density of

ns = 1011 cm−2 − 1012 cm−2 for graphene, we get:

Ωmetal ≈ 10−5,Ωgraphene ≈ 10−1 . (4.6)

4.1 Dimensionless Units 39

So, normally the dimensionless frequency in metallic case will be much smaller than

1 while the frequency in graphene can be up to 1.

Dimensionless Parameters

For rewriting the dispersion equations of different cases in dimensionless units we also

introduce three dimensionless parameters: D, Γ and α:

D = dq0 =2e2V d

~c2

√πns . (4.7)

D displays the dimensionless plate spacing. The real plate spacing d can vary from

1 nm to 1 mm, while the charge carrier density basically ns can vary from 0 to

1013 cm−2. For graphs we will consider the minimal density to be 1010 cm−2. With

these extrema the parameter D varies from 9 · 10−7 to 27.

Γ =γ

ω0

. (4.8)

Γ displays the dimensionless scattering rate. This parameter lies in our hands. Nev-

ertheless this parameter will be assumed to be negligible at many points in this work.

The dimensionless frequency introduced here is different to the one used in chapter 2

for the total frequency of graphene. We need to transform the full conductivity into

a form dependent of the new dimensionless frequency Ω = ω/ω0. In chapter 2 the

dimensionless frequency was defined by Ωµ = ~ω/µ. Now we use ω = ω0Ω and so we

can replace every ω by ω0Ω in the formula (2.28). The two dimensionless frequencies

are linked in the following way:

Ωµ =~ωµ

=~ω0Ω

µ=

~µ

2e2µ

~2cΩ =

2e2

~cΩ = ηΩ . (4.9)

Here we introduced a constant factor η = ~ω0

µ= 2e2

~c . η is dimensionless in Gaussian

units and has the value 0.0146.

σ(Ω) =e2

4~

[Θ(|ηΩ| − 2) +

i

π

(4

ηΩ− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)] . (4.10)


Note that if the two graphene layers have different charge carrier densities, still only

one dimensionless frequency can be defined. Then we define the ratio between both

chemical potential as α:

α =µ2

µ1

. (4.11)

In case of equal conductivities, α accordingly is 1. If the dimensionless frequency is

defined with the first layer (ω0 = 2e2µ21/~2c), then by using α, the conductivity of the

second layer can be express with the following relation:

Ωµ =~ωµ2

=2e2

~cµ1

µ2

Ω =ηΩ

α. (4.12)

And the total expression for the conductivity changes to:

σ2(Ω) =e2

4~

[Θ(

∣∣∣∣ηΩ

α

∣∣∣∣− 2) +i

π

(4α

ηΩ− ln

∣∣∣∣∣ ηΩα

+ 2ηΩα− 2

∣∣∣∣∣)]

. (4.13)

This way we have derived dimensionless expressions for the conductivity, the fre-

quency, the wave number and the plate spacing which we can use instead of the

dimensional quantities in the dispersion relations of single-layer plasmons and wave-

guide modes.

4.2 Single Layer

Let’s first begin with solving the dispersion equation of the single layer plasmon for

the full conductivity of graphene. The dispersion equation was derived in chapter 3.2:

1 +2πiκ

ωσ(ω) = 0 . (4.14)

With the now introduced dimensionless units and the full graphene conductivity with

scattering we can rewrite the dispersion equation:

1 + iπη

4

√Q2 − Ω2

Ω

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)] = 0 . (4.15)

4.2 Single Layer 41

Note that here the constant factor η has been used as in the previous chapter with

η = 2e2

~c . This way we can find the solution for Q(Ω) analytically:

Q = Ω

√1− 16

π2η2

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)]−2

. (4.16)

Here we can search for the real part of Q = Q′ + iQ′′:

Q′ = Ω<

√1− 16

π2η2

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)]−2 . (4.17)

While the real part Q′ defines the wave length and the propagation velocity, the

scattering rate also leads to damping of the mode. For this the imaginary part is

important (Q′′ > 0).

Q′′ = Ω=

√1− 16

π2η2

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)]−2 .

(4.18)

The mode will be dampened with e−q′′z = e−Q

′′Z . With Q′′ = q′′/q0 and Z = zq0.

Figure 4.2 displays the real and the imaginary part of the wave number in dependency

of dimensionless frequency.

For small values of Ω, the dispersion relation of single layer mode with Drude scatter-

ing can exceed the light line. In this case the mode will be relaxational or the packet

will contract as it propagates along the layer [19]. Note that both the Heaviside term

and the logarithmic term in the dispersion equation can be neglected for dimension-

less frequencies Ω well below 100. Only the Drude scattering is important for typical

dimensionless frequencies applied for the graphene waveguide.

Figure 4.3 displays the same relation of real wave number versus frequency, but in a

larger section. Since scattering rate is assumed to be a fixed value independent of the

wave frequency, its influence falls with increasing frequency.

At γ ω < 2µ we can ignore any real parts of the conductivity and get a more

simple dispersion relation for single layer case:

1−√Q2 − Ω2

Ω

(1

Ω− η

4ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣) = 0 , (4.19)


0.2 0.4 0.6 0.8 1.0 1.2 W

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Q',Q''

Figure 4.2: Wave number versus frequency (single layer) in dimensionless units for analytical solutionwith scattering, 0 < Ω < 1.2. Green: Γ = 0.1; Red: Γ = 1; Blue: Γ = 10. Real partalways solid, imaginary part always dashed. Light line dotted. For large scattering, thereal part can exceed the light line, but in that case damping is very high.

4.2 Single Layer 43

0.5 1.0 1.5 2.0 2.5 3.0 W

2

4

6

8

10

Q',Q''

Figure 4.3: Wave number versus frequency (single layer) in dimensionless units for analytical solutionwith scattering, 0 < Ω < 3. Green: Γ = 0.1; Red: Γ = 1; Blue: Γ = 10. Real partalways solid, imaginary part always dashed. Light line dotted. For large scattering, thereal part can exceed the light line, but in that case damping is very high.


Here as well we can write the solution of the dispersion equation as Q(Ω) with the

analytical solution:

Q = Ω

√1 +

(1

Ω− η

4ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)−2

. (4.20)

So for low values of Ω this will be linear and transform into a parabolic profile after-

wards. This is shown in figure 4.4.

0.2 0.4 0.6 0.8 1.0 W

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Q

Figure 4.4: Wave number versus frequency (single layer) in dimensionless units (solid); Light line(dashed). Without scattering being considered, the dispersion relation will not exceedthe light line. For low values of Ω ( 1), dispersion relation is linear; for higher valuesthe relation transforms into a parabolic profile.

When conductivity in case of Γ = 0 gets zero (which is the case when 1Ω− η

4ln∣∣∣ηΩ+2ηΩ−2

∣∣∣ =

0 is fulfilled), Q diverges. Figure 4.5 shows this divergence. Since that is the case at

about Ω = 114.2 and we explained in chapter 4.1 that typical values of Ω in graphene

can be up to 1, this graphene specific intraband conductivity normally doesn’t matter

for graphene waveguides. One would need frequencies of over 10 THz to reach these

values of Ω with standard graphene charge carrier densities. So we see that for values


of Ω well below 100 we can assume the conductivity to be truly Drude-like. The exact

zero position can be calculated with:

1

Ω− η

4ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣ = 0 . (4.21)

This equation can only be solved numerically and with η = 2e2/~c = 0.0146 the root

is found at Ω = 114.186.

113.8 114.0 114.2 114.4 W

1´107

2´107

3´107

4´107

5´107

Q

Figure 4.5: Wave number versus frequency (single layer) in dimensionless units near the root ofimaginary part of conductivity. Wave number diverges at the root of real part of grapheneconductivity at Ω = 114.186.

4.3 Equal Conductivities

Now let us consider waveguide modes of parallel plate waveguides. For the case of

equal conductivities on both layers we had derived two dispersion equations: One for

the optical mode and one for the acoustical mode. The equation for the optical mode

was:

1 +2πiκ

ωσ(ω)(1 + e−κd) = 0 . (4.22)


With the same dimensionless units as above and the full graphene conductivity this

dispersion equation transforms into:

1 + iπη

4

K

Ω(1 + e−DK)

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)] = 0 . (4.23)

Here η = 2e2

~c was introduced as in the previous chapters for simplification reasons.

For the acoustical mode we derived the following dispersion equation:

1 +2πiκ

ωσ(ω)(1− e−κd) = 0 . (4.24)

With dimensionless units and graphene conductivity this once again transforms into:

1 + iπη

4

K

Ω(1− e−DK)

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)] = 0 . (4.25)

Here again η is defined as η = 2e2

~c . Now these dispersion equations can no longer be

solved analytically without any assumptions. So we will now pursue two strategies:

First, to make assumptions in order to further solve the problem analytically. Second,

to ignore all scattering effects in order to make the dispersion equation completely

real and solve the equation numerically.

4.3.1 Analytical Solution for Small Frequencies with Scattering

If we are in the regime κd 1, we can make the equations non-transcendent by

assuming that (1 + e−κd) ≈ 2 in optical case and (1 − e−κd) ≈ κd in acoustical case.

With these assumptions the dispersion relation for optical mode transform into:

1 + iπη

4

√Q2 − Ω2

Ω2

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)] = 0 . (4.26)

Which leads to the analytical solution:

Q = Ω

√√√√1− 4

π2η2[Θ(|ηΩ| − 2) + i

π

(4

η(Ω+iΓ)− ln

∣∣∣ηΩ+2ηΩ−2

∣∣∣)]2 . (4.27)


The result of the dispersion relation for the optical case is displayed in figures 4.6 and

4.7. Here again counts that if the dispersion relation exceeds the light line, the packet

either contracts during propagation or will become relaxational [19].

0.2 0.4 0.6 0.8 1.0 Q',Q''

0.2

0.4

0.6

0.8

1.0

W

Figure 4.6: Frequency versus wave number (optical case) in dimensionless units for analytical solutionwith scattering, 0 < Ω < 1. Q′ always solid, Q′′ always dashed. Blue: Γ = 1; Red:Γ = 10; Green: Γ = 0 (Only Q′, because Q′′ = 0); Light line: dotted.

The dispersion relation for acoustical mode transforms into:

1 + iπη

4

√Q2 − Ω2

ΩD√Q2 − Ω2

[Θ(|ηΩ| − 2) +

i

π

(4

η(Ω + iΓ)− ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣)] = 0 .

(4.28)

Which leads to the other analytical solution:

Q =

√√√√Ω2 +i4Ω

πηD[Θ(|ηΩ| − 2) + i

π

(4

η(Ω+iΓ)− ln

∣∣∣ηΩ+2ηΩ−2

∣∣∣)] . (4.29)

The dispersion relation for the acoustical case is displayed in figures 4.8 and 4.9.

Here we see that optical and acoustical mode, according to these calculations react

differently on the consideration of scattering. While the propagation velocity of op-

tical mode increases with increasing scattering rate, the propagation velocity of the

acoustical mode decreases.


1 2 3 4 5 Q',Q''

1

2

3

4

5

W

Figure 4.7: Frequency versus wave number (optical case) in dimensionless units for analytical solutionwith scattering, 0 < Ω < 5. Q′ always solid, Q′′ always dashed. Blue: Γ = 1; Red:Γ = 10; Green: Γ = 0 (Only Q′, because Q′′ = 0); Light line: dotted.

0.2 0.4 0.6 0.8 1.0 Q',Q''

0.2

0.4

0.6

0.8

1.0

W

Figure 4.8: Frequency versus wave number (acoustical case) in dimensionless units for analyticalsolution with scattering, 0 < Ω < 1. Plate spacing D = 1. Q′ always solid, Q′′ alwaysdashed. Blue: Γ = 1; Red: Γ = 10, Green: Γ = 0 (only Q′, because Q′′ = 0); Light line:dotted.


1 2 3 4 5 Q',Q''

1

2

3

4

5

W

Figure 4.9: Frequency versus wave number (acoustical case) in dimensionless units for analyticalsolution with scattering, 0 < Ω < 5. Plate spacing D = 1. Q′ always solid, Q′′ alwaysdashed. Blue: Γ = 1; Red: Γ = 10, Green: Γ = 0 (only Q′, because Q′′ = 0); Light line:dotted.

4.3.2 Numerical Solution for All Frequencies Without Scattering

We can also solve the dispersion equations for optical and acoustical mode without

assuming κd 1. Though, we then ignore Drude scattering and other real parts of

graphene conductivity. If we ignore all real parts of the conductivity in the dispersion

relation of the optical mode we get:

1−√Q2op − Ω2

Ω(1 + e−D

√Q2op−Ω2

)

(1

Ω− η

4ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣) = 0 . (4.30)

If we ignore all real parts of the conductivity in the dispersion relation of the acous-

tical mode we get:

1−√Q2ac − Ω2

Ω(1− e−D

√Q2ac−Ω2

)

(1

Ω− η

4ln

∣∣∣∣ηΩ + 2

ηΩ− 2

∣∣∣∣) = 0 . (4.31)

Figures 4.10, 4.11, 4.16 and 4.17 display the dispersion relations for both optical and

acoustical waveguide modes for different ranges of dimensionless frequency Ω. We can

identify different regimes in those graphs.


2 4 6 8 10 Q

0.5

1.0

1.5

2.0

2.5

3.0

W

Figure 4.10: Frequency versus wave number (equal conductivities) in dimensionless units, 0 < Ω < 3;light line (dotted). Optical modes are dashed, acoustical modes solid. Blue: D = 0.1;Green: D = 1; Red: D = 10.


Figure 4.10 shows for three different dimensionless plate spacings how optical and

acoustical dispersion relations are situated in the Q-Ω-space. Here going up to fre-

quencies of Ω = 3 already exceeds the usual frequencies of graphene waveguides

slightly. Only be exceeding the usual range, the typical transition from linear to

parabolic profile is made visible. For D = 1 and D = 10 we also already see that for

higher frequencies the difference between optical and acoustical mode vanishes and

both modes approach the single layer case.

0.5 1.0 1.5 2.0Q

0.2

0.4

0.6

0.8

1.0

W

Figure 4.11: Frequency versus wave number (equal conductivities) in dimensionless units, 0 < Ω < 1;light line (dotted). Optical modes are dashed, acoustical modes solid. Blue: D = 0.1;Green: D = 1; Red: D = 10.

While analysing the different regimes of those dispersion relations, we should compare

the numerical solution with the analytical solution which was derived for the regime

with κd 1 in chapter 4.3.1. For comparison we combine the two assumptions made

in analytical and numerical calculation and derive the solutions for small frequencies

as well as without scattering. We can then also assume that Ω is smaller than 100,


so the logarithmic term is negligible. We get simplified dispersion equations. For the

optical case applies a solution independent of plate spacing D:

Qac = Ω

√1 +

Ω2

4. (4.32)

For the acoustical case applies a solution dependent of plate spacing D, but completely

linear:

Qac = Ω

√1 +

1

D. (4.33)

If we compare these two analytical solutions for small values of Ω, we can display

graphically when they drift away from the numerical solutions valid for all frequencies.

This is shown in figure 4.12 where we can see the analytical solutions together with the

numerical solutions. We see that for small frequencies, the analytical approximation

is valid.

The dimensionless propagation speed of a wave mode is given by:

C =v

c=

Ω

Q=ωq0

qω0

. (4.34)

So we see that the propagation speed of optical waveguide modes at the assumptions

κd 1 and for low values of Ω is always the speed of light.

Cop =v

c=

Ω

Q= 1 . (4.35)

The speed of the acoustical waveguide mode however depends on the dimensionless

parameter D:

Cac =v

c=

Ω

Q=

√D

D + 1. (4.36)

Figures 4.14 and 4.15 display the change of propagation velocity with charge carrier

density for seven different plate spacings of graphene waveguides. This represents the

full spectrum of the dimensionless plate spacing parameter D.

For very small values of Ω, the optical modes all propagate approximately with the

speed of light, and are independent of the dimensionless frequency. The acoustical

mode propagates with a speed lower than the speed of light and dependent on the


0.2 0.4 0.6 0.8 1.0 1.2 1.4 Q

0.2

0.4

0.6

0.8

1.0

W

Figure 4.12: Frequency versus wave number. Comparison between analytical and numerical solution.Plate spacing D = 1. Acoustical mode solid, optical mode dashed. Numerical solutionblue, analytical solution red. Analytical solution is approximately valid for small valuesof Ω (Ω . 1 for optical mode, Ω . 0.5 for acoustical mode). Only then, the assumptionκd 1 is justified. Acoustical case is approximated by purely linear dispersion relation.Optical case is approximated by linear dispersion relation for Ω 1 and parabolicdispersion relation for higher dimensionless frequencies. Numerical analysis shows bothcases pass into approximately parabolic profile for higher values of Ω.


dimensionless plate spacing, but its speed is also independent of the dimensionless

frequency. That behaviour is displayed in figure 4.13.

0.1 0.2 0.3 0.4 0.5 Q

0.1

0.2

0.3

0.4

0.5

W

Figure 4.13: Frequency versus wave number (equal conductivities) in dimensionless units, 0 < Ω <0.5; light line (dotted). Optical modes are dashed, acoustical modes solid. Blue: D =0.1; Green: D = 1; Red: D = 10. Here we see the linear dispersion relations for smallfrequencies in both optical and acoustical modes. Optical modes all propagate withalmost the velocity of light while propagation speed of acoustical modes highly dependsin the dimensionless plate spacing D.

Figure 4.16 displays the same dependencies of propagation velocity with characteristic

parameters for metal case. It shows that the speed of the acoustical wave mode in

a metal waveguide can hardly be manipulated by change of charge carrier density or

different plate spacing. The wave propagates always almost with the velocity of light.

When κd 1 is valid, the propagation velocity of both modes in graphene waveguides

gets dependent of the charge carrier density for higher frequencies. Because of the low

charge carrier density, only in graphene dimensionless frequencies of Ω > 1 can occur

if high frequencies of about 10 THz are used. For frequencies over 1, the dispersion

relation of both modes is changed into an approximately parabolic profile. This way

the propagation velocity changes with dimensionless frequency:

Cκd1 =Ω

Q∝ Ω

Ω2=

1

Ω. (4.37)


1011

1012

1013 ns

0.2

0.4

0.6

0.8

1.0

C=vc

Figure 4.14: Propagation speed versus charge carrier density (acoustical mode), high plate spacing;Solid: d = 1µm ; Dotted: d = 10µm ; Dashed: d = 100µm ; Dotted-dashed: d = 1 mm.

1011 1012 1013 ns

0.02

0.04

0.06

0.08

0.10

C=vc

Figure 4.15: Propagation speed versus charge carrier density (acoustical mode), low plate spacing;Solid: d = 1 nm ; Dotted: d = 10 nm ; Dashed: d = 100 nm.


1018 1019 ns

0.96

0.97

0.98

0.99

1.00

C=vc

Figure 4.16: Propagation speed versus charge carrier density (acoustical mode), metal parameters;Solid: d = 1µm ; Dotted: d = 100µm ; Dashed: d = 1 mm.

Since the dimensionless frequency depends on the charge carrier density with Ω ∝1/√ns, the propagation velocity depends also on the charge carrier density:

Cκd1 ∝√ns . (4.38)

The dimensionless parameter D = dq0 also characterises for which frequencies the

difference between optical and acoustical mode persists. For high values of Ω, re-

spectively Q, the difference between optical and acoustical mode vanishes. The

reason gets clear if we have a look on the properties of κ and its consequences on

the solutions. Here Ex(z) exponentially decays with κ, which means for high values

of κ = q0

√Q2 − Ω2, the electric field is localized mostly directly near the graphene

layers. The overlap term of ±e−κd vanished for κd 1. This approaching of optical

and acoustical case is displayed in figure 4.17.

For making the solutions Ex(z) of the differential equation visible, we introduce a

dimensionless location Z with Z = zq0 just as D was defined by D = dq0. This way

the definition of the three space areas remains the same: z = d/2⇔ Z = D/2. Then

the general solutions look like this:

Ex(z) = E1eK(Z+D/2) , Z < −D/2

=(E2 − E1) sinhKZ

2 sinhKD/2+

(E2 + E1) coshKZ

2 coshKD/2,−D/2 < Z < D/2

= E2e−K(Z−D/2) , Z > D/2 . (4.39)


10 20 30 40 50 Q

1

2

3

4

5

6

7

W

Figure 4.17: Frequency versus wave number (equal conductivities) in dimensionless units, 0 < Ω < 7;light line (dotted). Optical modes are dashed, acoustical modes solid. Blue: D = 0.1;Green: D = 1; Red: D = 10. Here we clearly see that dispersion relations of opticaland acoustical modes approach each other for high frequencies.


For Ez we get:

Ez(z) = −iQKE1e

K(Z+D/2) , Z < −D/2

= −iQK

(E2 − E1

2

coshKZ

sinhKD/2+E2 + E1

2

sinhKZ

coshKD/2

),−D/2 < Z < D/2

= iQ

KE2e

−K(Z−D/2) , Z > D/2 . (4.40)

Here Q = q/q0 and K = κ/q0 =√Q2 − Ω2. The exact relation between Q and K can

be derived from the numerical solution of the dispersion equation. For low frequencies

though we can estimate K Q, so the z-component of the electric field will be much

higher than the x-component: Ez Ex. In this case the electric field will be almost

perpendicular to the graphene layers.

Figures 4.18 and 4.19 show Ex of the optical mode for the two cases κd 1 and

κd 1.

-1.0 -0.5 0.5 1.0 Z

0.2

0.4

0.6

0.8

1.0Ex

Figure 4.18: Electric field in propagation direction versus location (optical mode) in dimensionlessunits for κd 1. K = 0.1 and D = 0.1.

Figures 4.20 and 4.21 show Ex of the acoustical mode for the two cases κd 1 and

κd 1.


-10 -5 5 10 Z

0.2

0.4

0.6

0.8

1.0Ex

Figure 4.19: Electric field in propagation direction versus location (optical mode) in dimensionlessunits for κd 1 . K = 10 and D = 10.

-1.0 -0.5 0.5 1.0 Z

-1.0

-0.5

0.5

1.0Ex

Figure 4.20: Electric field in propagation direction versus location (acoustical mode) in dimensionlessunits for κd 1. K = 0.1 and D = 0.1.


-10 -5 5 10 Z

-1.0

-0.5

0.5

1.0Ex

Figure 4.21: Electric field in propagation direction versus location (acoustical mode) in dimensionlessunits for κd 1. K = 10 and D = 10.

4.4 Arbitrary Conductivities

Apart from the now discussed waveguide modes for graphene waveguides with equal

conductivities, in chapter 3.5 we also introduced waveguide dispersion relations for

arbitrary conductivities. This dispersion relation was written with matrices.([1 + 2πiκ

ωσ1(ω)(1− e−κd)] −[1 + 2πiκ

ωσ2(ω)(1− e−κd)]

[1 + 2πiκωσ1(ω)(1 + e−κd)] [1 + 2πiκ

ωσ2(ω)(1 + e−κd)]

)(E1

E2

)= 0 . (4.41)

With dimensionless units Ω = ω/ω0 and Q = q/q0 as well as the full graphene

conductivity this dispersion relation transforms into:1 + iπη

4KΩ

(1− e−DK)[Θ(|ηΩ| − 2) + i

π

(4

η(Ω+iΓ)− ln

∣∣∣ηΩ+2ηΩ−2

∣∣∣)]−1− iπη

4KΩ

(1− e−DK)[Θ(|ηΩ

α| − 2) + i

π

(4α

η(Ω+iΓ)− ln

∣∣∣ ηΩα

+2ηΩα−2

∣∣∣)]1 + iπη

4KΩ

(1 + e−DK)[Θ(|ηΩ| − 2) + i

π

(4

η(Ω+iΓ)− ln

∣∣∣ηΩ+2ηΩ−2

∣∣∣)]1 + iπη

4KΩ

(1 + e−DK)[Θ(|ηΩ

α| − 2) + i

π

(4α

η(Ω+iΓ)− ln

∣∣∣ ηΩα

+2ηΩα−2

∣∣∣)]

(E1

E2

)= 0 .

(4.42)

This dispersion relation can no longer be solved analytically. Since no clearly sep-

arable optical and acoustical solutions exist any more, even making the equation

non-transcendent for small frequencies by approximating the exponential functions

as we did in chapter 4.3.1 doesn’t really help. The result would be a cubic complex

implicit equation with three very complicated term as solutions, though we know that

4.4 Arbitrary Conductivities 61

a differential equation of second order can only have two independent solutions. So

the one of the three solutions with negative wave number Q′ would have to be sorted

out. We wont pursue that idea here. Instead we will ignore any real part of the

graphene conductivity and analyse the dispersion equation numerically. If we ignore

any real conductivity, we get the following dispersion equation:

1−√Q2−Ω2

Ω(1− e−D

√Q2−Ω2

)(

1Ω− η


∣∣∣)−1 +

√Q2−Ω2

Ω(1− e−D

√Q2−Ω2

)(αΩ− η

4ln∣∣∣ ηΩα

+2ηΩα−2

∣∣∣)1−√Q2−Ω2

Ω(1 + e−D

√Q2−Ω2

)(

1Ω− η


∣∣∣)1−√Q2−Ω2

Ω(1 + e−D

√Q2−Ω2

)(αΩ− η

4ln∣∣∣ ηΩα

+2ηΩα−2

∣∣∣)

(E1

E2

)= 0 .

(4.43)

The numerical solution of this dispersion equation is displayed in figure 4.22 and 4.23

for two different dimensionless plate spacings D.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 Q

0.2

0.4

0.6

0.8

1.0

W

Figure 4.22: Frequency versus wave number (arbitrary conductivities) in dimensionless units for dif-ferent values of α; light line (dashed); Upper lines are optical solution, lower lines areacoustical solutions. D = 1 for all lines. Blue: α = 1; Red: α = 2; Green: α = 10.

It is no longer justified to call these two modes resulting from the dispersion equation

optical and acoustical since they will not have changed behaviour concerning E1 and


E2, so we will call them upper and lower mode, referring to the propagation velocity

in the parallel plate waveguide.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 Q

0.2

0.4

0.6

0.8

1.0

W

Figure 4.23: Frequency versus wave number (arbitrary conductivities) in dimensionless units for dif-ferent values of α; light line (dashed); Upper lines are optical solution, lower lines areacoustical solutions. D = 0.1 for all lines. Blue: α = 1; Red: α = 2; Green: α = 10.

Here we see that the propagation velocity of the wave in the lower mode is also

dependent on the value of α. Higher α leads to higher propagation velocity, which

is not surprising since due to our definition of α, for example going from α = 1 to

α = 10 increases the average chemical potential to 5.5 times the original average

potential. And was shown in the modes of equal conductivities, an increased charge

carrier density which is the result of an increased chemical potential will lead to higher

propagation velocity in the acoustical mode. Here the inequality of chemical potential

will not change the definition of Ω in the dispersion equation, but the velocity of the

lower mode solution of the arbitrary conductivity case will be increased due to higher

charge carrier density on the second graphene layer.

63

5 Conclusion

In this thesis, I studied the propagation of p-polarized electromagnetic waves (TM

modes) in graphene waveguides, in particular, parallel plate waveguides (PPWG). The

graphene layers were assumed to be infinitely large. Additionally to the Drude con-

ductivity the imaginary part of the interband graphene conductivity was considered.

When the assumptions allowed analytical solutions, also scattering was included.

An important conclusion of my work is that the propagation speed of the acoustical

mode, in case of equal conductivities on both graphene layers, can be changed dra-

matically by changing the charge carrier density. The charge carrier density can be

manipulated by an applied gate voltage. For parameters leading to a dimensionless

frequency of Ω < 1, which will be the case i.e. for values up to a frequency of 400 GHz

with a charge carrier density of 1012 cm−2 (higher charge carrier density will allow

higher frequency), the propagation speed of the optical mode is at least 0.894 times

the velocity of light. In contrast, with varying plate spacings from d = 1 nm to

d = 1 mm a very large variety of propagation velocities for the acoustical mode can

be realised.

Changing the charge carrier density by one scale can change the propagation velocity

by a factor of 2 (see figures 4.14 and 4.15). For metal PPWGs, in comparison, neither

can such small plate spacings in the thickness of nm be achieved, nor can such low

and variable charge carrier density be realised. This leads to the fact that for metal

PPWGs, both optical and acoustical modes propagate with at least 0.95 times the

speed of light (see figure 4.16). For the more general case of arbitrary conductivities

on both layers, the same behaviour has been found: One (the higher) mode will

propagate with at least 0.9 times the speed of light while the other (the lower) can be

manipulated by changing the charge carrier density even only on one layer. Smaller

plate spacings will lead to slower lower modes.

Analytical analysis of scattering effects have shown though, that the damping coeffi-

cient can be of the same scale as the wave number. For higher than usual frequencies,

64 Chapter 5 Conclusion

acoustical as well as optical modes in graphene waveguides will have a propagation

velocity which is dependant on the charge carrier density. These kinds of solutions

can not be achieved normally by metal waveguides because of their high charge carrier

density.

Possible next steps in the analysis may be to consider a finite scattering rate γ.

Also the assumption of T = 0 could be removed which changes the relation between

chemical potential and charge carrier density. The propagation of s-polarised waves

(TE modes) could also be studied which should lead to different waveguide modes.

65

List of Figures

2.1 Geometry of the graphene waveguide layers and p-polarized mode . . 122.2 The honeycomb lattice of graphene [11] . . . . . . . . . . . . . . . . . 142.3 The Brillouin zone of graphene [11] . . . . . . . . . . . . . . . . . . . 152.4 Energy band structure of graphene for whole first Brillouin zone . . . 162.5 Energy band structure of graphene around Dirac point . . . . . . . . 182.6 Conductivity of graphene; intra-, interband and total, real and imagin-

ary part, no scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Electric field in propagation direction versus location (optical mode)in dimensionless units . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Electric field perpendicular to propagation direction versus location(optical mode) in dimensionless units . . . . . . . . . . . . . . . . . . 32

3.3 Electric field in propagation direction versus location (acoustical mode)in dimensionless units . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Electric field perpendicular to propagation direction versus location(acoustical mode) in dimensionless units . . . . . . . . . . . . . . . . 34

4.1 Frequency versus wave number, definition of ω0 and q0 . . . . . . . . 384.2 Dispersion relation for single layer for full conductivity and different

scattering rates, 0 < Ω < 1.2 . . . . . . . . . . . . . . . . . . . . . . . 424.3 Dispersion relation for single layer for full conductivity and different

scattering rates, 0 < Ω < 3 . . . . . . . . . . . . . . . . . . . . . . . . 434.4 Wave number versus frequency (single layer) in dimensionless units

without scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.5 Wave number versus frequency (single layer) in dimensionless units

near the zero of imaginary part of conductivity . . . . . . . . . . . . . 454.6 Frequency versus wave number (optical case) in dimensionless units for

analytical solution with scattering, 0 < Ω < 1 . . . . . . . . . . . . . 474.7 Frequency versus wave number (optical case) in dimensionless units for

analytical solution with scattering, 0 < Ω < 5 . . . . . . . . . . . . . 484.8 Frequency versus wave number (acoustical case) in dimensionless units

for analytical solution with scattering, 0 < Ω < 1 . . . . . . . . . . . 484.9 Frequency versus wave number (acoustical case) in dimensionless units

for analytical solution with scattering, 0 < Ω < 5 . . . . . . . . . . . 494.10 Frequency versus wave number (equal conductivities) in dimensionless

units, 0 < Ω < 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

66 Chapter List of Figures

4.11 Frequency versus wave number (equal conductivities) in dimensionlessunits, 0 < Ω < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.12 Comparison between analytical and numerical solution . . . . . . . . 534.13 Frequency versus wave number (equal conductivities) in dimensionless

units, 0 < Ω < 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.14 Propagation speed versus charge carrier density (acoustical mode), high

plate spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.15 Propagation speed versus charge carrier density (acoustical mode), low

plate spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.16 Propagation speed versus charge carrier density (acoustical mode),

metal waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.17 Frequency versus wave number (equal conductivities) in dimensionless

units, 0 < Ω < 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.18 Electric field in propagation direction versus location (optical mode)

in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . . 584.19 Electric field in propagation direction versus location (optical mode)

in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . . 594.20 Electric field in propagation direction versus location (acoustical mode)

in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . . 594.21 Electric field in propagation direction versus location (acoustical mode)

in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . . 604.22 Frequency versus wave number (arbitrary conductivities) in dimension-

less units for different values of α, D = 1 . . . . . . . . . . . . . . . . 614.23 Frequency versus wave number (arbitrary conductivities) in dimension-

less units for different values of α, D = 0.1 . . . . . . . . . . . . . . . 62

67

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Lists

List of Used Symbols

Symbol Units (Gaussian) Description

a cm Lattice constant of grapheneα 1 Relation between chemical potential: µ2/µ1

~B G Magnetic fieldc cm/s Speed of light in vacuumC cm CapacityC 1 Dimensionless propagation velocity C = v/cd cm Plate spacingD 1 Dimensionless plate spacingδ - Dirac delta functione F Elementary chargee 1 Euler’s numberη 1 Proportional factor: η = 2e2/~c~E statV/cm Electric fieldEF erg Fermi energy levelg 1 Degeneration coefficientγ 1/s Scattering rateΓ 1 Dimensionless scattering rate~ erg s Reduced Planck constanti 1 Imaginary unit= - Imaginary part~j Fr/s cm3 Current densityK 1 Dimensionless decay parameter in z-directionκ 1/cm Field decay parameter in z-directionm g Charge carrier massµ erg Chemical potentialn 1/cm2 Charge carrier densityO - Terms of this order or higher orderω 1/s Frequency of EM wavesΩ 1 Dimensionless frequencyP Dyn s Momentumπ 1 Circle number

70 Chapter Lists

Symbol Units (Gaussian) Description

q 1/cm Wave number in z-directionQ F Total chargeQ 1 Dimensionless wave number< - Real partS cm2 Surface areaσ cm/s ConductivityT K TemperatureΘ - Heaviside step functionτ s Relaxation timeU statV Gate voltagev cm/s Propagation velocity of the waveV cm/s Fermi velocity in grapheneZ 1 Dimensionless z-coordinate

List of Used Nature Constants

Constant Value (Gaussian) Description

c 3 · 1010 cm/s Speed of light in vacuume 4.8 · 10−10 Fr Elementary charge~ 1 · 10−27 erg s Reduced Planck constantη 0.0146 η = 2e2/~c

The mathematical constants e and π were used with their exact values as far asMathematica allows.

71

Acknowledgements

At this point I would like to thank all persons who made this work possible:

• Prof. Dr.UlrichEckern for being the first corrector and his support in thedeciding phase of the work,

• Prof. Dr.ArnoKampf for being willing to make the second correction of myBachelor’s thesis,

• Dr. SergeyMikhailov for setting up the topic of this Bachelor’s thesis, for veryuseful discussions and guidance during the whole process as well as proofreading,

• Priv.-Doz.Dr.WolfgangHausler for his lecture about the general electronicproperties of graphene at the Institute of Physics in Augsburg in the summersemester 2011,

• Prof. Dr.Gert-Ludwig Ingold for helpful discussions,

• the Max-Weber-Programm Bayern for supporting my studies.

Additionally I want to thank my family and my friends for their understanding andsupport.

propagation of electromagnetic waves in graphene waveguides · propagation of electromagnetic waves...

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