Propagation of Electromagnetic
Waves in Graphene Waveguides
Bachelorarbeit
von
Christoph Helbig
vorgelegt
am 20. Oktober 2011
Lehrstuhl für Theoretische Physik II
Universität Augsburg
Propagation of Electromagnetic
Waves in Graphene Waveguides
Bachelorarbeit
von
Christoph Helbig
vorgelegt
am 20. Oktober 2011
Name:
Christoph Helbig
Matrikelnummer: 1071660
Studiengang:
Bachelor Physik
Erstprüfer:
Zweitprüfer:
Prof. Dr. Ulrich Eckern
Prof. Dr. Arno Kampf
5
Contents
1 Motivation and Introduction
7
2 Electromagnetic Waves and Graphene Properties
2.1 Electromagnetic Waves . . . . . . . . . . . . . .
2.2 Graphene Lattice . . . . . . . . . . . . . . . . .
2.3 Electronic Configuration . . . . . . . . . . . . .
2.4 Charge Carrier Density . . . . . . . . . . . . . .
2.5 Gate Voltage . . . . . . . . . . . . . . . . . . .
2.6 Conductivity . . . . . . . . . . . . . . . . . . .
3 General Parallel Plate Waveguides (PPWG)
3.1 Maxwell Equations . . . . . . . . . . . .
3.2 Single Layer . . . . . . . . . . . . . . . .
3.3 Solution of Differential Equation . . . . .
3.4 Equal Conductivities . . . . . . . . . . .
3.4.1 Optical Plasmon . . . . . . . . .
3.4.2 Acoustical Plasmon . . . . . . . .
3.5 Arbitrary Conductivities . . . . . . . . .
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34
4 Graphene Waveguides
4.1 Dimensionless Units . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Single Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Equal Conductivities . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Analytical Solution for Small Frequencies with Scattering .
4.3.2 Numerical Solution for All Frequencies Without Scattering
4.4 Arbitrary Conductivities . . . . . . . . . . . . . . . . . . . . . . .
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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 numerical 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:
i(qx−ωt)
~ r, t) = E(z)e
~
E(~
,
(2.1)
i(qx−ωt)
~ r, t) = B(z)e
~
B(~
.
(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.


Ex (z)


~
E(z)
= 0  ,
Ez (z)

0


~
B(z)
= By (z) .
0
(2.3)

(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 0 + iq 00 . Then if q 00 > 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 0 (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 Bravais 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
√
~ 1 = 2πa−1 (1, 1/ 3)
basis vectors of the reciprocal lattice then can be written as G
√
~ 2 = 2πa−1 (1, −1/ 3). This means that every second corner of the hexagon
and G
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 0 . K and K 0 are equivalent concerning charge carrier
density and density of states. Figure 2.3 shows the first Brillouin zone of graphene.
14
Chapter 2 Electromagnetic Waves and Graphene Properties
Figure 2.2: The honeycomb lattice of graphene. Two Bravais sublattices can be identified. All points
of 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 every
atom in a honeycomb lattice has three in total. Dashed lines show the boundaries of the
elementary cell. a is the lattice constant. Picture taken from Tudorovskiy [11].
2.2 Graphene Lattice
15
~ 1 and
Figure 2.3: The Brillouin zone of graphene. The basis vectors of the reciprocal lattice are G
~
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, since
they 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 0 . Picture taken from
Tudorovskiy [11].
16
Chapter 2 Electromagnetic Waves and Graphene Properties
2.3 Electronic Configuration
Now with a tight-binding model we can derive the electronic band structure of graphene. The result will be the relation for the energy eigenvalues in dependence of the
wave vector [4, 12].
2~V ~ ~
~ ~
~ ~
E(~k) = ± √ |eik·b1 + eik·b2 + eik·b3 | .
(2.6)
a 3
Here b~1 , b~2 and b~3 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 of
1/a. Valence band and conduction band are symmetrical around zero-energy and the
band gap is zero. Touching points are the six corners K and K 0 of the first Brillouin
zone. For µ = 0 and T = 0 all negative energy states are filled, all positive energy states
are 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
~0 = K
~ 2 . If we define a relative wave vector ~q by ~k = K
~ + ~k̃ for corners K
close to K
~ 0 + ~k̃ for corners K 0 both with k̃ K respectively k̃ K 0
and respectively ~k = K
then we get in the calculation of the band energies:
~0 ~ ~
~0 ~ ~
~0 ~ ~
ei((K +k̃)·b1 ) + ei((K +k̃)·b2 ) + ei((K +k̃)·b3 ) ≈
~0 ~
~0 ~
~0 ~
= (1 + i~k̃ · b~1 )eiK ·b1 + (1 + i~k̃ · b~2 )eiK ·b2 + (1 + i~k̃ · b~3 )eiK ·b3 =
(2.7)
√
3
=a
(k̃x − ik̃y ) .
2
p
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
~ =K
~ 5 and we get [14]:
vectors close to K
√
e
~ ~k̃)·b~1 )
i((K+
~ ~k̃)·b~2 )
i((K+
+e
~ ~k̃)·b~3 )
i((K+
+e
≈a
3
(−k̃x − ik̃y ) .
2
(2.8)
The total energy relation near Brillouin corner points gets:
E± (~k̃) = ±~V k̃ − O((k̃/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 0 [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.
18
Chapter 2 Electromagnetic Waves and Graphene Properties
Figure 2.5: Energy band structure of graphene around Dirac point K 0 = K2 . k is given in units of
1/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 dependencies of charge carrier density.
Z
Z
1X ~
gs gv
ns =
f0 (k) =
dkx dky f0 (~k) .
S
(2π)2
(2.10)
~k,σ,v
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]:
4
ns =
2π
(2π)2
Zµ
dE
0
⇔ ns =
1 E
,
~V ~V
µ2
.
π~2 V 2
(2.11)
(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 potential 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:
SU
.
(2.14)
ens S =
4πd
20
Chapter 2 Electromagnetic Waves and Graphene Properties
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
µ2
= ns =
.
4πed
π~2 V 2
(2.16)
This gives us the relation between chemical potential and the voltage applied.
r
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~
~ =0.
(2.18)
P~ + + eE
τ
In this equation, τ is the relaxation time of charge carriers in the material. With
non-relativistic impulse we get:
m~v˙ +
m~v
~ =0.
− eE
τ
(2.19)
~ with frequency ω we can not assume
Note that with an alternating electric field E
~
that ~v˙ = 0. Instead since we have a periodic behaviour with E(t)
= E~0 e−iωt , so for
charge carrier velocity applies ~v (t) = v~0 e−iωt . This leads to a reduced amplitude of
current [16]:
m~v
~ .
= eE
(2.20)
m(−iω~v ) +
τ
With ~j = −ens~v this gets:
2
~ .
~j = e ns 1 E
(2.21)
m τ1 − iω
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~0 e−iωt ] ,
E
(2.23)
~j = <[σ(ω)E~0 e−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) =
ins e2
.
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 =
e2 gs gv 4i
ins e2 V 2
=
.
(ω + i0)µ
16~ 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
e2 gs gv
=
16~
i 2 + Ωµ Θ(|Ωµ | − 2) − ln .
π
2 − Ωµ (2.27)
Here dimensionless frequency Ωµ = ~ω/µ has been introduced. ns = gs gv µ2 /4π~2 V 2
is the density of electrons at T = 0 as was already shown in chapter 2.4 [17]. Then
22
Chapter 2 Electromagnetic Waves and Graphene Properties
total frequency depending intra- and interband conductivity of graphene with real
and imaginary part can then be written as:
σtot
e2 gs gv
=
16~
i
Θ(|Ωµ | − 2) +
π
2 + Ωµ 4
− ln .
Ωµ
2 − Ωµ (2.28)
Figure 2.6 shows an overview over the change of real and imaginary part of conductivity for values of Ωµ from 0 to 4 [17]:
Conductivity
3
2
1
1
2
3
4
WΜ
-1
-2
Figure 2.6: Dimensionless conductivity of graphene (real and imaginary part) in dependence of dimensionless frequency Ωµ = ~ω/µ. Conductivity is displayed in units of e2 gs gv /16~.
Imaginary intraband (blue, dashed), imaginary interband (blue, dotted), imaginary total
(blue, solid) and real interband/total (red, solid). Imaginary part diverges at Ωµ = 0
and Ωµ = 2; For Ωµ 1 interband conductivity can be neglected. The real interband
conductivity has a jump point, because as soon as the photon energy is ~ω > 2µ = 2EF
there 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 the
algebraic 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 ) =
e2 gs gv 4i
e2 gs gv i
=
.
16~ πΩµ
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 intraband conductivity counts, we get the important part of the equation:
σ(ω) =
ie2 ns V 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:
r
ie2 V
U
.
(2.32)
σ(ω) =
2π~(ω + iγ) ed
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:
~
~ = − 1 ∂B .
∇×E
c ∂t
(3.1)
The second important Maxwell equation in our problem is Ampère’s circuital law:
~
~ = 4π ~j + 1 ∂ E .
∇×B
c
c ∂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
~ r, t) = E(~
~ r)e−iωt and
Ampère’s law (3.2). Concerning the time dependency we use E(~
~ r, t) = B(~
~ r)e−iωt (see equations (2.1) and (2.2)). Faraday’s law then simplifies to:
B(~
∂Ex ∂Ez
1 ∂By
iω
−
=−
= By .
∂z
∂x
c ∂t
c
(3.3)
26
Chapter 3 General Parallel Plate Waveguides (PPWG)
When considering the simplification of Ampère’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.
iω
∂By
= − Ez ,
∂x
c
(3.4)
∂By
iω
4π
= Ex −
[σ1 (ω)Ex (z1 )δ(z − z1 ) + σ2 (ω)Ex (z2 )δ(z − z2 )] .
(3.5)
∂z
c
c
Now we can differentiate the simplified Faraday’s law (3.3). Concerning the differentiation with respect to location for electromagnetic waves propagating in x-direction
iqx
~ r) = E(z)e
~
in our waveguide we have to keep in mind that E(~
. 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:
∂Ex
ω2
+ q 2 Ez = 2 Ez .
(3.6)
iq
∂z
c
Now this gives us an expression for Ez which we differentiate once again with respect
to z:
iq
∂ 2 Ex
∂Ez
=− 2
.
(3.7)
∂z
q − ω 2 /c2 ∂z 2
Now let’s look at the differentiation of (3.3) with respect to z: This gives us:
∂ 2 Ex
∂Ez ω 2
4πiω
−
iq
+ 2 Ex = − 2 [σ1 Ex (z1 )δ(z − z1 ) + σ2 Ex (z2 )δ(z − z2 )] .
2
∂z
∂z
c
c
(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 = q 2 − ω 2 /c2 and we receive the differential equation for Ex which we will analyse
further:
∂ 2 Ex
4πiκ2
2
−
κ
E
=
[σ1 Ex (z1 )δ(z − z1 ) + σ2 Ex (z2 )δ(z − z2 )] .
x
∂z 2
ω
(3.9)
3.2 Single Layer
Before we go into details of the two-layer problem, a short consideration about plasmons 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) = E1 eκz , z < 0
= E2 e−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:
ε
4πiκ2
∂Ex (z) σ(ω)E .
(3.12)
=
∂z ε
ω
This leads with ε → 0 to the dispersion equation of for a plasmon propagating along
a single graphene monolayer:
1+
2πiκ
σ(ω) = 0 .
ω
Or the same equation with re-substituting κ by
q
2πi q 2 −
1+
ω
ω2
c2
p
(3.13)
q 2 − ω 2 /c2 :
σ(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) = E1 eκ(z+d/2) , z < −d/2
= A sinh κz + B cosh κz, < d/2 < z < d/2
= E2 e−κ(z−d/2) , z > d/2 .
(3.15)
28
Chapter 3 General Parallel Plate Waveguides (PPWG)
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 sinh
κd
κd
+ B cosh
= Ex (−d/2 + 0) .
2
2
κd
κd
+ B cosh
= Ex (+d/2 + 0) .
2
2
Therefore we can express A and B in matters of E1 and E2 :
Ex (d/2 − 0) = E2 = A sinh
(3.16)
(3.17)
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) = E1 eκ(z+d/2) , z < −d/2
(E2 − E1 ) sinh κz (E2 + E1 ) cosh κz
+
, −d/2 < z < d/2
=
2 sinh κd/2
2 cosh κd/2
= E2 e−κ(z−d/2) , z > d/2 .
(3.20)
We can also list the solution Ez (z):
q ∂Ex
κ2 ∂z
q
= −i E1 eκ(z+d/2) , z < −d/2
κ
q E2 − E1 cosh κz
E2 + E1 cosh κz
= −i
+
, −d/2 < z < d/2
κ
2
sinh κd/2
2
sinh κd/2
q
= i E2 e−κ(z−d/2) , z > d/2 .
(3.21)
κ
Ez (z) = −i
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
Z
∂ 2 Ex
−
dz
∂z 2
−d/2+ε
Z
−d/2+ε
Z
2
4πiκ2
[σ1 Ex (z1 )δ(z−z1 )+σ2 Ex (z2 )δ(z−z2 )] ,
dz
ω
dz κ Ex =
−d/2−ε
−d/2−ε
d/2+ε
Z
d/2+ε
Z
−d/2−ε
(3.22)
dz
∂ 2 Ex
−
∂z 2
d/2−ε
d/2−ε
dz κ2 Ex =
d/2+ε
Z
dz
4πiκ2
[σ1 Ex (z1 )δ(z − z1 ) + σ2 Ex (z2 )δ(z − z2 )] .
ω
d/2−ε
(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 κ2 Ex vanishes. Meanwhile the integration over the
second differentiation of Ex gives difference between the border terms of the first
differentiation for both integration intervals:
R
−d/2+ε
4πiκ2
∂Ex =
σ1 (ω)E1 ,
∂z −d/2−ε
ω
(3.24)
d/2+ε
∂Ex 4πiκ2
=
σ2 (ω)E2 .
∂z d/2−ε
ω
(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
E2 + E1
4πiκ
coth κd/2 −
tanh κd/2 =
σ1 (ω)E1 ,
2
2
ω
(3.26)
E2 − E1
E2 + E1
4πiκ
coth κd/2 −
tanh κd/2 =
σ2 (ω)E2 .
(3.27)
2
2
ω
This system of equations can be written in a compact notation using matrices:
−E2 −
!
!
4πiκ
[1 + coth(κd/2) + 4πiκ
σ
(ω)]
−[1
+
coth(κd/2)
+
σ
(ω)]
E
1
1
2
ω
ω
=0 .
4πiκ
4πiκ
[1 + tanh(κd/2) + ω σ1 (ω)]
[1 + tanh(κd/2) + ω σ2 (ω)]
E2
(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
30
Chapter 3 General Parallel Plate Waveguides (PPWG)
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:
4πiκ
σ(ω) (E1 − E2 ) = 0 ,
(3.29)
1 + coth(κd/2) +
ω
4πiκ
1 + tanh(κd/2) +
σ(ω) (E1 + E2 ) = 0 .
(3.30)
ω
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
σ(ω) = 0 .
ω
(3.32)
We can rewrite this equation with an exponential function rather than a hyperbolic
function:
2πiκ
1+
(3.33)
1 + e−κd σ(ω) = 0 .
ω
p
Or the same equation with re-substituting κ by q 2 − ω 2 /c2 :
q
2πi q 2 −
1+
ω
ω2
c2
q
2
− q 2 − ω2 d
1+e
c
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 cosh κz
, −d/2 < z < d/2
=
cosh(κd/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 .
Ex
1.0
0.8
0.6
0.4
0.2
-3
-2
-1
1
2
3
Z
Figure 3.1: Electric field in propagation direction versus location (optical mode) in dimensionless
units. K = 1 and D = 1. Nondimensionalized with K = κ/q0 , D = dq0 = 1 and
Z = zq0 .
And this way we can also write down Ez (z):
Ez (z) = −
iq ∂Ex
iq
= − Eeκ(z+d/2) , z < −d/2
2
κ ∂z
κ
iq E sinh κz
= −
, −d/2 < z < d/2
κ cosh(κd/2)
iq −κ(z−d/2)
=
Ee
, z > d/2 .
κ
(3.37)
32
Chapter 3 General Parallel Plate Waveguides (PPWG)
Figure 3.2 shows Ez for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1
and Z = zq0 .
Ez
1.0
0.5
-3
-2
1
-1
2
3
Z
-0.5
-1.0
Figure 3.2: Electric field perpendicular to propagation direction versus location (optical mode) in
dimensionless 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
σ(ω) = 0 .
ω
(3.39)
Like in the optical case, transform into equation with exponential function rather
than hyperbolic function:
1+
2πiκ
1 − e−κd σ(ω) = 0 .
ω
(3.40)
3.4 Equal Conductivities
33
Or the same equation with replacing κ by
q
2πi q 2 −
1+
ω2
c2
ω
p
q 2 − ω 2 /c2 :
1−e
q
2
− q 2 − ω2 d
c
σ(ω) = 0 .
(3.41)
In the acoustical case the solution looks as follows:
A=
E
,B = 0,
sinh(κd/2)
(3.42)
Ex (z) = −Eeκ(z+d/2) , z < −d/2
E sinh κz
, −d/2 < z < d/2
=
sinh(κd/2)
= Ee−κ(z−d/2) , z > d/2 .
(3.43)
(3.44)
(3.45)
Figure 3.3 shows Ex for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1
and Z = zq0 .
Ex
1.0
0.5
-3
-2
1
-1
2
3
Z
-0.5
-1.0
Figure 3.3: Electric field perpendicular to propagation direction versus location (acoustical mode) in
dimensionless units. K = 1 and D = 1. Nondimensionalized with K = κ/q0 , D = dq0 =
1 and Z = zq0 .
34
Chapter 3 General Parallel Plate Waveguides (PPWG)
Ez (z) = −
iq ∂Ex
iq κ(z+d/2)
=
Ee
, z < −d/2
2
κ ∂z
κ
iq E cosh κz
= −
, −d/2 < z < d/2
κ sinh(κd/2)
iq −κ(z−d/2)
Ee
, z > d/2 .
=
κ
(3.46)
Figure 3.4 shows Ez for nondimensionalized parameters K = κ/q0 = 1, D = dq0 = 1
and Z = zq0 .
Ez
2.0
1.5
1.0
0.5
-3
-2
-1
1
2
3
Z
Figure 3.4: Electric field perpendicular to propagation direction versus location (acoustical mode) in
dimensionless 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 +
[1 +
2πiκ
σ1 (ω)(1
ω
2πiκ
σ1 (ω)(1
ω
− e−κd )] −[1 +
+ e−κd )]
[1 +
2πiκ
σ2 (ω)(1
ω
2πiκ
σ2 (ω)(1
ω
Or the same equation with re-substituting κ by

[1 +





[1 +

q
2
2πi q 2 − ω2
c
ω
σ1 (ω)(1 − e
q
2
− q 2 − ω2 d
c
p
!
!
− e−κd )]
E1
=0 .
+ e−κd )]
E2
q 2 − ω 2 2/c2 :

)]


!
 E
c
)]
−[1 +
σ
(ω)(1
−
e

1
2
ω
q
=0.
q

2
ω2
2πi q 2 − ω2
2
 E2
− q − 2d
c
c

σ
(ω)(1
+
e
)]
1
ω
q

q
2
2
2πi q 2 − ω2
− q 2 − ω2 d
c
c
σ2 (ω)(1 + e
[1 +
)]
ω
q
2
2πi q 2 − ω2
c
(3.47)
q
2
− q 2 − ω2 d
(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:
2πiκσ2 (ω)
2πiκσ1 (ω)
(1 − e−κd )][1 +
(1 + e−κd )] +
ω
ω
2πiκσ1 (ω)
2πiκσ2 (ω)
[1 +
(1 + e−κd )][1 +
(1 − e−κd )] = 0 .
ω
ω
[1 +
(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) 2πiκσ2 (ω) 2πiκσ1 (ω) 2πiκσ2 (ω)
+
+
(1 − e−2κd ) = 0 .
ω
ω
ω
ω
(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 σ1 + σ2
2πiκ σ1 + σ2
(1 − e−κd )][1 +
(1 + e−κd )] −
ω
2
ω
2
2πiκ σ1 − σ2
2πiκ σ1 − σ2
(1 − e−κd )
(1 + e−κd ) = 0 .
ω
2
ω
2
(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 finding 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 independent 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 ins e2 V 2
=0.
ω
ωµ
(4.1)
This dispersion equation is solved by the following plasma frequency:
s
ω=
2πns e2 V 2
q.
µ
(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:
2πns e2 V 2
q = c2 q 2 .
µ
(4.3)
38
Chapter 4 Graphene Waveguides
Ω
Ω0
q
q0
Figure 4.1: Frequency versus wave number. Light line (dashed) and plasma frequency (solid). Crossing 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 =
2e2 V √
2πns e2 V 2
2e2 µ
=
.
πn
=
s
µc2
~c2
~2 c2
(4.4)
The corresponding frequency is then:
ω0 = cq0 =
2πns e2 V 2
2e2 V √
2e2 µ
=
πns = 2 .
µc
~c
~c
(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:
2e2 V d √
D = dq0 =
πns .
~c2
(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. Nevertheless 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 = ω0 Ω and so we
can replace every ω by ω0 Ω in the formula (2.28). The two dimensionless frequencies
are linked in the following way:
Ωµ =
~ω
~ω0 Ω
~ 2e2 µ
2e2
=
=
Ω
=
Ω = ηΩ .
µ
µ
µ ~2 c
~c
2
(4.9)
Here we introduced a constant factor η = ~ωµ0 = 2e
. η is dimensionless in Gaussian
~c
units and has the value 0.0146.
ηΩ + 2 e2
i
4
σ(Ω) =
Θ(|ηΩ| − 2) +
− ln .
(4.10)
4~
π ηΩ
ηΩ − 2 40
Chapter 4 Graphene Waveguides
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:
µ2
α=
.
(4.11)
µ1
In case of equal conductivities, α accordingly is 1. If the dimensionless frequency is
defined with the first layer (ω0 = 2e2 µ21 /~2 c), then by using α, the conductivity of the
second layer can be express with the following relation:
Ωµ =
2e2 µ1
ηΩ
~ω
=
Ω=
.
µ2
~c µ2
α
(4.12)
And the total expression for the conductivity changes to:
" ηΩ e2
i
σ2 (Ω) =
Θ( − 2) +
4~
α
π
!#
ηΩ + 2 4α
α
− ln ηΩ
.
α − 2
ηΩ
(4.13)
This way we have derived dimensionless expressions for the conductivity, the frequency, 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 waveguide 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
σ(ω) = 0 .
ω
(4.14)
With the now introduced dimensionless units and the full graphene conductivity with
scattering we can rewrite the dispersion equation:
πη
1+i
4
p
ηΩ + 2 Q2 − Ω2
i
4
Θ(|ηΩ| − 2) +
− ln =0.
Ω
π η(Ω + iΓ)
ηΩ − 2 (4.15)
4.2 Single Layer
41
Note that here the constant factor η has been used as in the previous chapter with
2
η = 2e
. This way we can find the solution for Q(Ω) analytically:
~c
s
ηΩ + 2 −2
16
i
4
Q = Ω 1 − 2 2 Θ(|ηΩ| − 2) +
− ln .
π η
π η(Ω + iΓ)
ηΩ − 2 (4.16)
Here we can search for the real part of Q = Q0 + iQ00 :

−2
ηΩ + 2 16
i
4
 . (4.17)
Q0 = Ω <  1 − 2 2 Θ(|ηΩ| − 2) +
− ln π η
π η(Ω + iΓ)
ηΩ − 2 s
While the real part Q0 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 (Q00 > 0).
s

−2
ηΩ + 2 16
i
4
 .
Q00 = Ω =  1 − 2 2 Θ(|ηΩ| − 2) +
− ln π η
π η(Ω + iΓ)
ηΩ − 2 (4.18)
The mode will be dampened with e
=e
. With Q00 = q 00 /q0 and Z = zq0 .
Figure 4.2 displays the real and the imaginary part of the wave number in dependency
of dimensionless frequency.
−q 00 z
−Q00 Z
For small values of Ω, the dispersion relation of single layer mode with Drude scattering 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 dimensionless 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:
p
1−
Q2 − Ω2
Ω
1
η ηΩ + 2 − ln
=0,
Ω 4 ηΩ − 2 (4.19)
42
Chapter 4 Graphene Waveguides
Q',Q''
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
1.2
W
Figure 4.2: Wave number versus frequency (single layer) in dimensionless units for analytical solution
with scattering, 0 < Ω < 1.2. Green: Γ = 0.1; Red: Γ = 1; Blue: Γ = 10. Real part
always solid, imaginary part always dashed. Light line dotted. For large scattering, the
real part can exceed the light line, but in that case damping is very high.
4.2 Single Layer
43
Q',Q''
10
8
6
4
2
0.5
1.0
1.5
2.0
2.5
3.0
W
Figure 4.3: Wave number versus frequency (single layer) in dimensionless units for analytical solution
with scattering, 0 < Ω < 3. Green: Γ = 0.1; Red: Γ = 1; Blue: Γ = 10. Real part
always solid, imaginary part always dashed. Light line dotted. For large scattering, the
real part can exceed the light line, but in that case damping is very high.
44
Chapter 4 Graphene Waveguides
Here as well we can write the solution of the dispersion equation as Q(Ω) with the
analytical solution:
s
Q=Ω 1+
−2
η ηΩ + 2 1
− ln
.
Ω 4 ηΩ − 2 (4.20)
So for low values of Ω this will be linear and transform into a parabolic profile afterwards. This is shown in figure 4.4.
Q
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
W
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 exceed
the light line. For low values of Ω ( 1), dispersion relation is linear; for higher values
the relation transforms into a parabolic profile.
ηΩ+2 When conductivity in case of Γ = 0 gets zero (which is the case when
ln ηΩ−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
1
− η4
Ω
4.3 Equal Conductivities
45
of Ω well below 100 we can assume the conductivity to be truly Drude-like. The exact
zero position can be calculated with:
η ηΩ + 2 1
− ln
=0.
Ω 4 ηΩ − 2 (4.21)
This equation can only be solved numerically and with η = 2e2 /~c = 0.0146 the root
is found at Ω = 114.186.
Q
5 ´ 107
4 ´ 107
3 ´ 107
2 ´ 107
1 ´ 107
113.8
114.0
114.2
114.4
W
Figure 4.5: Wave number versus frequency (single layer) in dimensionless units near the root of
imaginary part of conductivity. Wave number diverges at the root of real part of graphene
conductivity 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:
2πiκ
1+
σ(ω)(1 + e−κd ) = 0 .
(4.22)
ω
46
Chapter 4 Graphene Waveguides
With the same dimensionless units as above and the full graphene conductivity this
dispersion equation transforms into:
ηΩ + 2 i
4
πη K
−DK
(1 + e
) Θ(|ηΩ| − 2) +
− ln = 0 . (4.23)
1+i
4 Ω
π η(Ω + iΓ)
ηΩ − 2 2
Here η = 2e
was introduced as in the previous chapters for simplification reasons.
~c
For the acoustical mode we derived the following dispersion equation:
1+
2πiκ
σ(ω)(1 − e−κd ) = 0 .
ω
(4.24)
With dimensionless units and graphene conductivity this once again transforms into:
ηΩ + 2 πη K
i
4
−DK
1+i
(1 − e
) Θ(|ηΩ| − 2) +
− ln = 0 . (4.25)
4 Ω
π η(Ω + iΓ)
ηΩ − 2 2
. Now these dispersion equations can no longer be
Here again η is defined as η = 2e
~c
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:
πη
1+i
4
p
ηΩ + 2 Q2 − Ω2
i
4
2 Θ(|ηΩ| − 2) +
− ln =0.
Ω
π η(Ω + iΓ)
ηΩ − 2 (4.26)
Which leads to the analytical solution:
v
u
Q = Ωu
t1 −
π2η2
h
Θ(|ηΩ| − 2) +
i
π
4
4
η(Ω+iΓ)
i .
ηΩ+2 2
− ln ηΩ−2 (4.27)
4.3 Equal Conductivities
47
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].
W
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Q',Q''
Figure 4.6: Frequency versus wave number (optical case) in dimensionless units for analytical solution
with scattering, 0 < Ω < 1. Q0 always solid, Q00 always dashed. Blue: Γ = 1; Red:
Γ = 10; Green: Γ = 0 (Only Q0 , because Q00 = 0); Light line: dotted.
The dispersion relation for acoustical mode transforms into:
p
ηΩ + 2 Q2 − Ω2 p 2
4
i
D Q − Ω2 Θ(|ηΩ| − 2) +
− ln =0.
Ω
π η(Ω + iΓ)
ηΩ − 2 (4.28)
Which leads to the other analytical solution:
πη
1+i
4
v
u
u
Q = tΩ2 +
h
πηD Θ(|ηΩ| − 2) +
i4Ω
i
π
4
η(Ω+iΓ)
i .
− ln ηΩ+2
ηΩ−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 optical mode increases with increasing scattering rate, the propagation velocity of the
acoustical mode decreases.
48
Chapter 4 Graphene Waveguides
W
5
4
3
2
1
1
2
3
4
5
Q',Q''
Figure 4.7: Frequency versus wave number (optical case) in dimensionless units for analytical solution
with scattering, 0 < Ω < 5. Q0 always solid, Q00 always dashed. Blue: Γ = 1; Red:
Γ = 10; Green: Γ = 0 (Only Q0 , because Q00 = 0); Light line: dotted.
W
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Q',Q''
Figure 4.8: Frequency versus wave number (acoustical case) in dimensionless units for analytical
solution with scattering, 0 < Ω < 1. Plate spacing D = 1. Q0 always solid, Q00 always
dashed. Blue: Γ = 1; Red: Γ = 10, Green: Γ = 0 (only Q0 , because Q00 = 0); Light line:
dotted.
4.3 Equal Conductivities
49
W
5
4
3
2
1
1
2
3
4
5
Q',Q''
Figure 4.9: Frequency versus wave number (acoustical case) in dimensionless units for analytical
solution with scattering, 0 < Ω < 5. Plate spacing D = 1. Q0 always solid, Q00 always
dashed. Blue: Γ = 1; Red: Γ = 10, Green: Γ = 0 (only Q0 , because Q00 = 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:
p 2
√
Qop − Ω2
η ηΩ + 2 1
−D Q2op −Ω2
(1 + e
− ln
1−
)
=0.
Ω
Ω 4 ηΩ − 2 (4.30)
If we ignore all real parts of the conductivity in the dispersion relation of the acoustical mode we get:
p
√
Q2ac − Ω2
1
η ηΩ + 2 −D Q2ac −Ω2
(1 − e
)
− ln
=0.
1−
Ω
Ω 4 ηΩ − 2 (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.
50
Chapter 4 Graphene Waveguides
W
3.0
2.5
2.0
1.5
1.0
0.5
2
4
6
8
10
Q
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.
4.3 Equal Conductivities
51
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 frequencies 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.
W
1.0
0.8
0.6
0.4
0.2
0.5
1.0
1.5
2.0
Q
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,
52
Chapter 4 Graphene Waveguides
so the logarithmic term is negligible. We get simplified dispersion equations. For the
optical case applies a solution independent of plate spacing D:
r
Qac = Ω 1 +
Ω2
.
4
(4.32)
For the acoustical case applies a solution dependent of plate spacing D, but completely
linear:
r
1
Qac = Ω 1 +
.
(4.33)
D
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
Ω
ωq0
=
=
.
c
Q
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 =
Ω
v
=
=1.
c
Q
(4.35)
The speed of the acoustical waveguide mode however depends on the dimensionless
parameter D:
r
Ω
v
D
Cac = =
=
.
(4.36)
c
Q
D+1
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
4.3 Equal Conductivities
53
W
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Q
Figure 4.12: Frequency versus wave number. Comparison between analytical and numerical solution.
Plate spacing D = 1. Acoustical mode solid, optical mode dashed. Numerical solution
blue, analytical solution red. Analytical solution is approximately valid for small values
of Ω (Ω . 1 for optical mode, Ω . 0.5 for acoustical mode). Only then, the assumption
κd 1 is justified. Acoustical case is approximated by purely linear dispersion relation.
Optical case is approximated by linear dispersion relation for Ω 1 and parabolic
dispersion relation for higher dimensionless frequencies. Numerical analysis shows both
cases pass into approximately parabolic profile for higher values of Ω.
54
Chapter 4 Graphene Waveguides
dimensionless plate spacing, but its speed is also independent of the dimensionless
frequency. That behaviour is displayed in figure 4.13.
W
0.5
0.4
0.3
0.2
0.1
0.1
0.2
0.3
0.4
0.5
Q
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 small
frequencies in both optical and acoustical modes. Optical modes all propagate with
almost the velocity of light while propagation speed of acoustical modes highly depends
in 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
Ω
1
∝ 2 = .
Q
Ω
Ω
(4.37)
4.3 Equal Conductivities
55
C=vc
1.0
0.8
0.6
0.4
0.2
1011
1012
1013
ns
Figure 4.14: Propagation speed versus charge carrier density (acoustical mode), high plate spacing;
Solid: d = 1 µm ; Dotted: d = 10 µm ; Dashed: d = 100 µm ; Dotted-dashed: d = 1 mm.
C=vc
0.10
0.08
0.06
0.04
0.02
1011
1012
1013
ns
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.
56
Chapter 4 Graphene Waveguides
C=vc
1.00
0.99
0.98
0.97
0.96
1018
1019
ns
Figure 4.16: Propagation speed versus charge carrier density (acoustical mode), metal parameters;
Solid: d = 1 µm ; Dotted: d = 100 µm ; 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, respectively 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
p
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) = E1 eK(Z+D/2) , Z < −D/2
(E2 − E1 ) sinh KZ (E2 + E1 ) cosh KZ
+
, −D/2 < Z < D/2
=
2 sinh KD/2
2 cosh KD/2
= E2 e−K(Z−D/2) , Z > D/2 .
(4.39)
4.3 Equal Conductivities
57
W
7
6
5
4
3
2
1
10
20
30
40
50
Q
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 optical
and acoustical modes approach each other for high frequencies.
58
Chapter 4 Graphene Waveguides
For Ez we get:
Q
E1 eK(Z+D/2) , Z < −D/2
K
Q E2 − E1 cosh KZ
E2 + E1 sinh KZ
= −i
+
, −D/2 < Z < D/2
K
2
sinh KD/2
2
cosh KD/2
Q
= i E2 e−K(Z−D/2) , Z > D/2 .
(4.40)
K
p
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.
Ez (z) = −i
Figures 4.18 and 4.19 show Ex of the optical mode for the two cases κd 1 and
κd 1.
Ex
1.0
0.8
0.6
0.4
0.2
-1.0
-0.5
0.5
1.0
Z
Figure 4.18: Electric field in propagation direction versus location (optical mode) in dimensionless
units 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.
4.3 Equal Conductivities
59
Ex
1.0
0.8
0.6
0.4
0.2
-10
5
-5
10
Z
Figure 4.19: Electric field in propagation direction versus location (optical mode) in dimensionless
units for κd 1 . K = 10 and D = 10.
Ex
1.0
0.5
-1.0
0.5
-0.5
1.0
Z
-0.5
-1.0
Figure 4.20: Electric field in propagation direction versus location (acoustical mode) in dimensionless
units for κd 1. K = 0.1 and D = 0.1.
60
Chapter 4 Graphene Waveguides
Ex
1.0
0.5
-10
5
-5
10
Z
-0.5
-1.0
Figure 4.21: Electric field in propagation direction versus location (acoustical mode) in dimensionless
units 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 +
[1 +
2πiκ
σ1 (ω)(1
ω
2πiκ
σ1 (ω)(1
ω
− e−κd )] −[1 +
+ e−κd )]
[1 +
2πiκ
σ2 (ω)(1
ω
2πiκ
σ2 (ω)(1
ω
!
!
− e−κd )]
E1
=0.
+ e−κd )]
E2
(4.41)
With dimensionless units Ω = ω/ω0 and Q = q/q0 as well as the full graphene
conductivity this dispersion relation transforms into:
i
h

ηΩ+2 i
K
4
−DK
1 + i πη
(1
−
e
)
Θ(|ηΩ|
−
2)
+
−
ln
ηΩ−2 4 Ω
π η(Ω+iΓ)
ηΩ i
h

!
α +2 

ηΩ
K
i
4α
−DK
(1
−
e
)
Θ(|
|
−
2)
+
−
ln
−1 − i πη
ηΩ −2  E1

4 Ω
α
π η(Ω+iΓ)

i
α

h

ηΩ+2  E =0.
πη K
i
4
−DK
) Θ(|ηΩ| − 2) + π η(Ω+iΓ) − ln ηΩ−2 2
1 + i 4 Ω (1 + e


ηΩ i
h
α +2 K
4α
1 + i πη
(1 + e−DK ) Θ(| ηΩ
| − 2) + πi η(Ω+iΓ)
− ln ηΩ
4 Ω
α
−2

α
(4.42)
This dispersion relation can no longer be solved analytically. Since no clearly separable 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 Q0 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:
√


√
Q2 −Ω2
ηΩ+2 η
1
−D Q2 −Ω2
(1
−
e
−
ln
)
1−
Ω
Ω
4
ηΩ−2

√
ηΩ 
√

!
Q2 −Ω2
2
2
α +2 
η
α
Q
−Ω
−D

−1 +
) Ω − 4 ln ηΩ
(1 − e

E
Ω


1
−2
α
√
=0.
√


 E2
1 − Q2 −Ω2 (1 + e−D Q2 −Ω2 ) 1 − η ln ηΩ+2 

Ω
Ω
4
ηΩ−2
√

ηΩ 
√
2
2
Q −Ω
2
2
α +2 (1 + e−D Q −Ω ) Ωα − η4 ln ηΩ
1−
Ω
−2
α
(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.
W
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Q
Figure 4.22: Frequency versus wave number (arbitrary conductivities) in dimensionless units for different values of α; light line (dashed); Upper lines are optical solution, lower lines are
acoustical 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
62
Chapter 4 Graphene Waveguides
E2 , so we will call them upper and lower mode, referring to the propagation velocity
in the parallel plate waveguide.
W
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Q
Figure 4.23: Frequency versus wave number (arbitrary conductivities) in dimensionless units for different values of α; light line (dashed); Upper lines are optical solution, lower lines are
acoustical 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 conductivity 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 dramatically 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 coefficient 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
2.2
2.3
2.4
2.5
2.6
3.1
3.2
3.3
3.4
4.1
4.2
Geometry of the graphene waveguide layers and p-polarized mode . .
The honeycomb lattice of graphene [11] . . . . . . . . . . . . . . . . .
The Brillouin zone of graphene [11] . . . . . . . . . . . . . . . . . . .
Energy band structure of graphene for whole first Brillouin zone . . .
Energy band structure of graphene around Dirac point . . . . . . . .
Conductivity of graphene; intra-, interband and total, real and imaginary part, no scattering . . . . . . . . . . . . . . . . . . . . . . . . . .
Electric field in propagation direction versus location (optical mode)
in dimensionless units . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electric field perpendicular to propagation direction versus location
(optical mode) in dimensionless units . . . . . . . . . . . . . . . . . .
Electric field in propagation direction versus location (acoustical mode)
in dimensionless units . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electric field perpendicular to propagation direction versus location
(acoustical mode) in dimensionless units . . . . . . . . . . . . . . . .
Frequency versus wave number, definition of ω0 and q0 . . . . . . . .
Dispersion relation for single layer for full conductivity and different
scattering rates, 0 < Ω < 1.2 . . . . . . . . . . . . . . . . . . . . . . .
4.3 Dispersion relation for single layer for full conductivity and different
scattering rates, 0 < Ω < 3 . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Wave number versus frequency (single layer) in dimensionless units
without scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Wave number versus frequency (single layer) in dimensionless units
near the zero of imaginary part of conductivity . . . . . . . . . . . . .
4.6 Frequency versus wave number (optical case) in dimensionless units for
analytical solution with scattering, 0 < Ω < 1 . . . . . . . . . . . . .
4.7 Frequency versus wave number (optical case) in dimensionless units for
analytical solution with scattering, 0 < Ω < 5 . . . . . . . . . . . . .
4.8 Frequency versus wave number (acoustical case) in dimensionless units
for analytical solution with scattering, 0 < Ω < 1 . . . . . . . . . . .
4.9 Frequency versus wave number (acoustical case) in dimensionless units
for analytical solution with scattering, 0 < Ω < 5 . . . . . . . . . . .
4.10 Frequency versus wave number (equal conductivities) in dimensionless
units, 0 < Ω < 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
14
15
16
18
22
31
32
33
34
38
42
43
44
45
47
48
48
49
50
66
Chapter List of Figures
4.11 Frequency versus wave number (equal conductivities) in dimensionless
units, 0 < Ω < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.12 Comparison between analytical and numerical solution . . . . . . . .
4.13 Frequency versus wave number (equal conductivities) in dimensionless
units, 0 < Ω < 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.14 Propagation speed versus charge carrier density (acoustical mode), high
plate spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.15 Propagation speed versus charge carrier density (acoustical mode), low
plate spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.16 Propagation speed versus charge carrier density (acoustical mode),
metal waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.17 Frequency versus wave number (equal conductivities) in dimensionless
units, 0 < Ω < 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.18 Electric field in propagation direction versus location (optical mode)
in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . .
4.19 Electric field in propagation direction versus location (optical mode)
in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . .
4.20 Electric field in propagation direction versus location (acoustical mode)
in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . .
4.21 Electric field in propagation direction versus location (acoustical mode)
in dimensionless units for κd 1 . . . . . . . . . . . . . . . . . . . .
4.22 Frequency versus wave number (arbitrary conductivities) in dimensionless units for different values of α, D = 1 . . . . . . . . . . . . . . . .
4.23 Frequency versus wave number (arbitrary conductivities) in dimensionless units for different values of α, D = 0.1 . . . . . . . . . . . . . . .
51
53
54
55
55
56
57
58
59
59
60
61
62
67
Bibliography
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URL
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T. & Mikhailov,
S. A.
Intervalley
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69
Lists
List of Used Symbols
Symbol
a
α
~
B
c
C
C
d
D
δ
e
e
η
~
E
EF
g
γ
Γ
~
i
=
~j
K
κ
m
µ
n
O
ω
Ω
P
π
Units (Gaussian)
cm
1
G
cm/s
cm
1
cm
1
F
1
1
statV/cm
erg
1
1/s
1
erg s
1
Fr/s cm3
1
1/cm
g
erg
1/cm2
1/s
1
Dyn s
1
Description
Lattice constant of graphene
Relation between chemical potential: µ2 /µ1
Magnetic field
Speed of light in vacuum
Capacity
Dimensionless propagation velocity C = v/c
Plate spacing
Dimensionless plate spacing
Dirac delta function
Elementary charge
Euler’s number
Proportional factor: η = 2e2 /~c
Electric field
Fermi energy level
Degeneration coefficient
Scattering rate
Dimensionless scattering rate
Reduced Planck constant
Imaginary unit
Imaginary part
Current density
Dimensionless decay parameter in z-direction
Field decay parameter in z-direction
Charge carrier mass
Chemical potential
Charge carrier density
Terms of this order or higher order
Frequency of EM waves
Dimensionless frequency
Momentum
Circle number
70
Chapter Lists
Symbol
q
Q
Q
<
S
σ
T
Θ
τ
U
v
V
Z
Units (Gaussian)
1/cm
F
1
cm2
cm/s
K
s
statV
cm/s
cm/s
1
Description
Wave number in z-direction
Total charge
Dimensionless wave number
Real part
Surface area
Conductivity
Temperature
Heaviside step function
Relaxation time
Gate voltage
Propagation velocity of the wave
Fermi velocity in graphene
Dimensionless z-coordinate
List of Used Nature Constants
Constant Value (Gaussian)
c
3 · 1010 cm/s
e
4.8 · 10−10 Fr
1 · 10−27 erg s
~
η
0.0146
Description
Speed of light in vacuum
Elementary charge
Reduced Planck constant
η = 2e2 /~c
The mathematical constants e and π were used with their exact values as far as
Mathematica allows.
71
Acknowledgements
At this point I would like to thank all persons who made this work possible:
• Prof. Dr. Ulrich Eckern for being the first corrector and his support in the
deciding phase of the work,
• Prof. Dr. Arno Kampf for being willing to make the second correction of my
Bachelor’s thesis,
• Dr. Sergey Mikhailov for setting up the topic of this Bachelor’s thesis, for very
useful discussions and guidance during the whole process as well as proofreading,
• Priv.-Doz. Dr. Wolfgang Häusler for his lecture about the general electronic
properties of graphene at the Institute of Physics in Augsburg in the summer
semester 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 and
support.