Download Studies of Nanostructured Layers with UV-VIS Spectroscopic Ellipsometry

Project work done at
Linköping University
Studies of Nanostructured Layers with
UV-VIS Spectroscopic Ellipsometry
by Tijs Mocking
May 4th 2008
Supervisor: Hans Arwin
Examiner: Kenneth Järrendahl
Laboratory of Applied Optics, Linköping University, Sweden
Abstract
In this report a model analysis is presented for three different nanostructured layers:
silicon nanotips (SiNTs), gold nanosandwiches and Split Ring Resonators (SRR).
The last two materials are metamaterials and both may show a negative refraction
index. Experimental data are obtained for every sample using a variable angle
spectroscopic ellipsometer. For the gold nanosandwiches, also an infrared
ellipsometric measurement is done. For complex layers like these, advanced
modeling is necessary. A recently developed analysis program including options for
both anisotropic permittivity and permeability is used. A realistic model is presented
in this report for the gold nanosandwiches, which also includes the magnetic activity
in the layer itself. The results for the nanosandwiches are reasonable, and a
magnetic oscillation is found in the horizontal plane at around 260 nm although it was
not expected to have a resonance that far in the UV-range. For the SiNTs and the
SRR it was not possible to create an acceptable model.
2
Preface
This report is written as a part of my internship for the Laboratory of Applied Optics
on the University of Linköping, Sweden, which lasted three months. The internship is
part of my study, Applied Physics on the University of Twente in The Netherlands.
Doing this internship in a foreign country helped me to acquire some general
knowledge in doing research and making a report and a presentation. Also it helped
me to get some insights in scientific research and the people that are involved with
the research. Furthermore during this internship I got in contact with many people
from different countries and that was a very nice experience.
The most important person who made this possible is Hans Arwin, from Linköping
University who was my guide and supervisor during the internship. I want to thank
him for making the internship possible and also for getting me into the floorball group.
I also want to thank Kenneth Järrendahl who was my examiner and with whom I had
some nice conversations. From the University of Twente I like to thank Herbert
Wormeester. Without his relations in foreign countries it would not have been
possible to do this internship. I also like to thank the Erasmus organization in The
Netherlands who granted me with a Erasmus Internship Scholarship to help me
finance this internship. I want to conclude with thanking the JA Woollam Co. Inc. for
supplying me with a new version of their WVASE program to test and use during my
internship.
3
Table of contents
ABSTRACT .............................................................................................................................. 2
PREFACE ................................................................................................................................. 3
TABLE OF CONTENTS ......................................................................................................... 4
LIST OF SYMBOLS ................................................................................................................ 5
1. INTRODUCTION ................................................................................................................ 6
2. THEORETICAL ASPECTS ............................................................................................... 7
2.1 ELLIPSOMETRY ................................................................................................................. 7
2.1.1 Standard Ellipsometry ........................................................................................... 7
2.1.2 Generalized Ellipsometry ..................................................................................... 8
2.2 FIT PROCEDURE............................................................................................................... 9
2.3 OPTICAL MODELS ............................................................................................................ 9
2.3.1 Cauchy Model ........................................................................................................ 9
2.3.2 Lorentz Model....................................................................................................... 10
2.3.3 Effective Medium Approximation (EMA) .......................................................... 10
2.3.4 Biaxial Layer ......................................................................................................... 12
2.4 META-MATERIALS ........................................................................................................... 13
2.4.1 Negative Refractive Index .................................................................................. 14
2.4.2 The Split Ring Resonator (SRR) ....................................................................... 14
2.4.3 Gold Sandwiches ................................................................................................. 15
2.5 THE HARMONIC OSCILLATOR ........................................................................................ 16
3. EXPERIMENTAL ASPECTS .......................................................................................... 19
3.1 VASE ELLIPSOMETER ................................................................................................... 19
3.2 IR VASE ........................................................................................................................ 20
3.3 THE DIFFERENT SAMPLES ............................................................................................. 20
3.3.1 Silicon Nanotips ................................................................................................... 20
3.3.2 Gold Sandwiches ................................................................................................. 20
3.3.3 SRR ....................................................................................................................... 21
4. EXPERIMENTAL RESULTS .......................................................................................... 22
4.1 MEASUREMENTS ON SILICON NANOTIPS ...................................................................... 22
4.1.1 Substrate Characterization................................................................................. 22
4.1.1 Silicon Nanotips ................................................................................................... 23
4.2 MEASUREMENTS ON AU-SIO2-AU SANDWICHES .......................................................... 24
4.2.1 Substrate Characterization................................................................................. 24
4.2.2 Au-SiO2-Au Sandwiches Structures................................................................. 25
4.2.3 IR VASE Measurement on Nanosandwiches ................................................. 28
4.3 MEASUREMENTS ON COPPER NANOSTRUCTURES (SRR) ........................................... 30
5. CONCLUSIONS AND DISCUSSION ............................................................................. 32
6. REFERENCES ................................................................................................................... 33
7. APPENDICES .................................................................................................................... 34
APPENDIX A .......................................................................................................................... 34
APPENDIX B .......................................................................................................................... 36
4
List of Symbols
b
c
e
f
k
me
n
r
w
x0
z0
A, B, C
An
C
E
En
K
L
M
N
Rp, Rs
i
[kg/s2]
[m/s]
[C]
[]
[]
[kg]
[]
[m]
[m]
[m]
[m]
[]
[eV]
[F]
[V/m]
[eV]
[Kg/s]
[H]
[]
[]
[]
[º]
[]
[m]
[]
[]
[º]
[s]
[º]
[]
[s-1]
[º]
[eV]
[º]
Damping constant
Speed of light (3*108)
Elementary charge (1.6*10-19)
Fraction of a material
Extinction coefficient
Mass of an electron (9.1*10-31)
Refractive index
Radius
Width of a gold nano sandwich
Initial position of an electron in x-direction
Initial position of an electron in z-direction
Cauchy constants
Amplitude of resonance
Capacitance
Electric field
Position of resonance
Spring constant
Self inductance
Number of variable parameters in the model
Number of (Ψ, ∆) pairs
Reflection coefficient for p or s polarized light
Phase of the reflected wave
Electric permittivity
Wavelength
Magnetic permeability
Ratio of the reflection coefficients
Standard deviations on the experimental data points
Decay time
Angle of incidence
Ratio of the photon energies of the s-polarized light and
the p-polarized light
Angular frequency
Ellipsometric parameter (phase difference)
Broadening of a resonance
Ellipsometric parameter (amplitude ratio)
5
1. Introduction
Ellipsometry has become a very important tool for research nowadays. In the search
for a negative refractive index in so called meta-materials, materials that have
properties more due to their structure than their composition, ellipsometry is a
suitable technique to use.
Negative refractive index has been shown for micro wavelengths and 2D structures,
but for optical frequencies and 3D structures it is still a challenge to find a negative
refractive index. This is why a number of research groups is working on this subject
using spectroscopic ellipsometry [1].
The properties of nanostructures are becoming more interesting for different usages.
Trying to reduce the reflectivity of a surface E. Schubert [2] grows a SiO2
nanostructured layer on quartz while monitoring the reflectivity with ellipsometry. The
same is done by S. Y. Bae et al. [3] using silicon nanotips, for use on a micro sun
sensor for Mars rovers. The results presented by these researchers are very recent,
but still there is a need for further methodological development in monitoring
magnetic and electric activity in nanostructures.
The objectives of this project are:
- Exploring the possibilities to extract nanostructural parameters from
ellipsometric data, and
- Exploring possibilities to perform ellipsometric modeling of
metamaterials with potential negative refractive index.
6
2. Theoretical aspects
2.1 Ellipsometry
2.1.1 Standard Ellipsometry
Ellipsometry is an optical technique with which it is possible to measure the optical
properties and also different other properties of thin films, for example their thickness.
As an optical method, ellipsometry is nondestructive and contactless. Spectroscopic
ellipsometry is a very sensitive technique and finds applications in many areas of
research and industry.
Ellipsometry works as follows: monochromatic light comes from a light source, and is
polarized. A chopper can be placed just after the light source, which improves the
signal-noise ratio by modulating the signal. After being polarized, the monochromatic
light can pass an optional compensator (for example a quarter wave plate for giving a
phase delay). Then the light hits the sample and the reflected light can pass an
optional compensator again. Finally the light is measured and analyzed for changed
polarization (see Fig.1) using an analyzer and a detector. There are many different
compositions of an ellipsometric system, which each can have a specialized target.
Fig.1 - Schematic view of an ellipsometer where being the angle of incidence.
The incident light and the normal together forming the plane of incidence.
With ellipsometry it is possible to find the thicknesses of layers that are thinner than
the wavelength of the probing light. It is sensitive to a single layer of atoms and even
less. Also multilayer systems with different kind of layers can be examined with this
technique. Optical inhomogeneous, isotropic and anisotropic layers can be
measured.
The polarization state of the light incident upon the sample can be described by its s
and p-components. The s component is oscillating parallel to the sample surface and
7
is perpendicular to the plane of incidence, which is formed by the surface normal and
the incident light. The p-component oscillates perpendicular to the s-component and
the incoming beam. The complex-valued amplitudes of the s- and p-reflection
coefficients are named Rs and Rp, respectively.
1)
R p = Rp e
iδ p
and R s = Rs e iδ s
there |Rp| and |Rs| are the amplitudes and p and s are the phases of the reflection
coefficients. The ratio of Rp and Rs is measured with standard ellipsometry.
2)
ρ=
Rp
Rs
= tan(Ψ )e i∆
Here tan( )=|Rp|/|Rs| and = p- s are the amplitude ratio upon reflection and the
phase shift, respectively. Optical parameters like layer thicknesses and refractive
indices can normally not be calculated directly from measured
and
values.
However the opposite, i.e. to calculate the and from an optical model with values
on the thicknesses and refractive indices, is possible.
This means that for gathering information about the sample it is necessary do some
model analysis. With a layer model the results of the measurements are fitted, using
a mean square minimization procedure, so parameters like for example thickness
and optical constants can be determined. This model analysis has in this project
been done in the program WVASE32tm from J.A. Woollam Co. Inc.
2.1.2 Generalized Ellipsometry
Standard ellipsometry (or just short 'ellipsometry') is applied, when no s-polarized
light is converted into p-polarized light or vice versa. When a sample is anisotropic,
generalized ellipsometry can be applied. This includes measurements of at least
three polarization changes at three different polarizations of the probe beam.
Anisotropic samples are generally described by a non-diagonal Jones matrix
3)
Rr =
R pp
R ps
Rsp
Rss
At least three values of at three different i=Epi/Esi (ratio of the ingoing electric fields
of the p- and s-polarized light) are required for an ellipsometric characterization. Now
three pairs of
and are defined. Three complex-valued ellipsometric parameters
are measured: pp, ps and sp which are defined as:
4a)
ρ pp =
4b)
ρ ps =
4c)
ρ sp =
R pp
Rss
R ps
R pp
Rsp
Rss
= tan(Ψ pp )e
= tan(Ψ ps )e
= tan(Ψsp )e
i∆ pp
i∆ ps
i∆ sp
8
Now the Jones matrix can be rewritten as:
5)
R r = Rss
ρ pp
ρ ps ρ pp
ρ sp
1
Finding the values pp, ps and sp is the objective in generalized ellipsometry. To
improve the precision of the measurement one can measure at more then three i’s.
2.2 Fit Procedure
The program for analysis (WVASE32tm) has a procedure to determine the best fit that
a model can give for measured data. This procedure works so that the mean square
error (MSE) is calculated for a set of data and a model, and the fit for which the MSE
is smallest is the best fit. WVASE32tm uses the following expression for calculating
the MSE [5]:
6)
1
MSE =
2N − M
N
i =1
ψ imod − ψ iexp
σ ψexp,i
2
+
∆mod
− ∆exp
i
i
2
σ ∆exp,i
where N is the number of (Ψ, ∆) pairs, M is the number of variable parameters in the
model, σ ψexp,i is the measurement uncertainty for , and σ ∆exp
,i is the measurement
uncertainty for
, ψ imod is the evaluated (fitted) point of
and ψ iexp is the measured
point of , ∆mod
is the evaluated (fitted) point of and ∆exp
.
i
i is the measured point of
The MSE is weighted by the uncertainties on each measured data point.
2.3 Optical Models
For many materials the optical properties are known, but for many materials, like
organic materials, or combinations of different materials, the optical properties are
still unknown, or not available in the analysis program. There are models to describe
some of the materials mentioned, and in this section a few different models will be
described.
2.3.1 Cauchy Model
The Cauchy model is a relation between the refractive index and the wavelength of
the light for a material [5]. The Cauchy model can be used on transparent materials
and the equation is given by:
7)
n (λ ) = A +
B
λ
2
+
C
λ4
Where n is the refractive index, is the wavelength and A, B and C are coefficients
which can be fitted using an analysis program. This model is very good for
wavelengths in the visible region if a material is transparent, but quite poor in the
9
infrared region. A different model is made by Sellmeier, who invented the Sellmeier
equation which works very good in the UV, visible and infrared region. The Cauchy
model will be used to model a glass substrate of which the material is unknown.
2.3.2 Lorentz Model
The Lorentz oscillator model is very useful for metals. For analyzing a thin layer of
metal on an opaque substrate the Lorentz model is the best choice to get reasonable
results for the film thickness and optical constants. Also Lorentz oscillators are very
useful for describing resonant absorption peaks.
The Lorentz oscillator model is based on the assumption that electrons in a material
react on an external electric field like a harmonically driven mass on a spring reacts
on an external force [5]. In this model the mass is the electron and the spring is the
force that holds the electron and the nucleus together. The Lorentz model is
formulated as follows:
8)
ε~( E ) = ε 1 (∞) +
N
i =1
Ai
E − E 2 − iΓi E
2
i
Where ε~ ( E ) is the dimensionless dielectric function as a function of the photon
energy E. N is the total number of oscillators and ε 1 (∞) is the real part of the
dielectric function at very large photon energies. Each oscillator is described by three
parameters; Ai, i and Ei. Ai is the amplitude, i is the broadening and Ei is the center
energy of the ith oscillator.
2.3.3 Effective Medium Approximation (EMA)
A heterogeneous material (multiple components) can be modeled with an Effective
Medium Approximation (EMA). An EMA provides an expression for and average
(effective) value on which is called EMA, This EMA can be expressed in the dielectric
functions and amounts of the different structures in the heterogeneous material. To
give an example: let us consider two ideal two-component structures with composite
materials A and B as shown in Fig.2.
Fig.2 - Two different two-component structures with in a) the material boundaries
parallel to the applied electric field, and in b) the boundaries perpendicular to the
field. Here A and B are the permittivity and fA and fB the volume fraction of
material A and B, respectively [10].
10
In Fig.2a) the boundaries of the two components in the heterogeneous material are
parallel to the applied electric field. The dielectric function EMA= || is then given by:
9)
ε || = f A ε A + f B ε B
where f A and f B are the volume fractions of material A and B respectively and ε A
and ε B are the dielectric functions. In Fig.2b) the boundaries are perpendicular to the
applied electric field and then the dielectric function EMA= ε ⊥ is given by
10)
ε⊥ =
1
fA
εA
+
fB
εB
If the values for the volume fractions and the dielectric functions are set equal to each
other for the two microstructures, it results in a different value for EMA.
These two examples show a phenomenon that is called screening. In the parallel
situation the two components contribute according to their volume fraction. By
definition there is no screening in this situation. In the other case where the
boundaries are perpendicular to the electric field there is maximum screening. The
material with the lowest has the most influence on the ε ⊥ . The material with the
highest will be “screened”.
In this project some materials will be modeled using an EMA and an example of such
a material is given in Fig.3. With an EMA or with a few different layers of EMA’s the
optical properties of nanostructure are examined. The thickness of the EMA layer d
and also the volume fraction of the material can be found.
Fig.3 - Schematic side view of a silicon substrate with a layer of silicon
nanotips on top with height ’d’.
In the analysis of the experiments done, especially the Bruggeman EMA will be used.
It is also called the Bruggeman Effective Medium Theory. The aggregate structure
shown in Fig.4 can be approximated with a unit cell also shown in Fig.4. For the
dielectric function , Bruggeman derived the following equation:
11
11)
fA
εA −ε
ε −ε
+ (1 − f A ) B
=0
ε A + 2ε
ε B + 2ε
This equation can be used when there are two different materials in the structure.
However this equation can simply be altered to make it applicable on a structure with
more then two materials. It then changes to:
12)
fA
ε −ε
εA −ε
ε −ε
+ fB B
+ fC C
+ ... = 0
ε A + 2ε
ε B + 2ε
ε C + 2ε
The Bruggeman theory is widely used in optical analysis, and it can be used for all
values of fA.
Fig.4 - The random unit cell in the Bruggeman theory. The unit cell
has probability fA to be material A and fB=1-fA to be material B [10].
2.3.4 Biaxial Layer
All of the samples that are examined are anisotropic in at least two dimensions.
When the optical axis is perpendicular to the surface the optical constants in the xand y-directions (parallel to the surface) are isotropic, but for the z-direction they are
different. Such a material is uniaxial, so x = y
z . When the optical constants also
differ for the x- and y-direction the material is biaxial, x y
z. This information is
very important when one wants to make a model of the sample. A biaxial layer can
be included in the model. There one can choose the optical constants per direction in
the uni- or biaxial layer.
When magnetic behavior of a material is included in the model the ‘meta6 layer’ is
used. Here a similar thing happens. The permeability is not the same for different
directions in the material. In the Meta6 layer the permittivity and the permeability will
be calculated using the following equations:
13a)
D = ε 0 εE
13b)
B = µ 0 µH
D is the electric displacement, and B is the magnetic flux density. In vacuum the
permittivity and permeability are 0 and 0
12
2.4 Meta-materials
The term meta-material came to use quite recently. An artificial material that gains its
optical properties from its structure more than from its composition is a meta-material.
Also a meta-material is usually a material with “unusual” properties. Their three
dimensional structure is periodic in two dimensions in most structures developed so
far. The dimensions of the periodic structure are normally smaller than the
wavelength of he electromagnetic waves it interacts with for these unusual properties
to exist. See Fig.5 for an example of such a periodic structure.
Fig.5 – An example of a metamaterial layer. These are small Split
Ring Resonators which can exhibit a negative refractive index [13].
An example of these “unusual” properties is a possible negative refractive index.
Such a meta-material will bend electromagnetic waves – such as microwaves, radio
waves and visible light – the “wrong” way. The electromagnetic waves will not be
refracted away from the source they are coming from, but they will refract back in the
opposite direction towards the source as shown in Fig.6. This is consistent with
Snell’s law if the refractive index is negative.
These materials are also called “left handed
meta-materials”. The existence of a negative
refractive index has been proven already for
the microwave frequency range [8]. The
possibility of finding a negative refractive
index is the reason why many research
groups investigate meta-materials, since this
property is not found in any existing natural
material.
There
are
already
some
applications of these materials introduced
and suggested for medicine, lenses and
military application. A few examples are: the Fig.6 – The behavior of electromagnetic
“super lens” and the “stealth suit” [14], [15]. waves entering a material with a negative
There are some different kinds of meta- refractive index.
materials. The sizes of these materials
depends on if the researchers want to find a negative refraction for a larger
wavelength, or a shorter wavelength.
13
Some examples of different kinds of meta-materials:
- Single or double Split Ring Resonators
- Area’s with ellipsoidal voids in a metal sheet
- Wire pairs meta-material
- Plate pairs
- Metal sandwiches
2.4.1 Negative Refractive Index
For describing the behavior of light in a medium the refractive index N=n+ik is very
important. In this definition n is the real part of the refractive index, and k is the
imaginary part, also the extinction coefficient. The real part of the refractive index n is
positive for all naturally existing materials, but the existence of n < 0 is possible.
The refractive index can also be calculated with N = ± εµ where ε = ε '+iε " is the
permittivity and µ = µ '+iµ " is the permeability of the material with a real and an
imaginary part for both and .
Some metals (for example silver and gold) already have a negative permittivity for
optical frequencies, however, to have a negative refractive index, also the
permeability should be smaller then zero:
< 0 and
< 0 [6]. Of course the
multiplication of and will be positive if both values are smaller than zero, and in
this situation the negative value of the root in N = ± εµ should be taken. The
behavior of electromagnetic waves in a medium with a negative refractive index is
shown in Fig.6. That both the permittivity and the permeability are smaller than zero
is a sufficient but not a necessary condition for negative refraction. The necessary
condition to have a negative refractive index is ε " µ '+ µ " ε ' < 0 , however the materials
with just one of the - and - values negative will not by useful for application,
because such a material will have a large loss [6].
2.4.2 The Split Ring Resonator (SRR)
Two concentric rings with a gap in
both rings as shown in Fig.7 are
known as a Split Ring Resonator
(SRR). These SRR’s are the basis of
most of the meta-materials with
negative
magnetic
permeability
nowadays. It works as follows.
Because of the magnetic field in the
electromagnetic radiation charges are
moved in the rings and will accumulate
near the gaps. The gaps in the rings
prevent the charges to follow the ring,
which means that some charges cross
the distance w to the other ring and in
this way completing the L-C circuit,
which results in a negative magnetic
permeability. The capacitance across
the rings causes this SRR to be
Fig.7 – Schematic top view of a SRR with definitions of
the sizes.
14
resonant [9]. Resonance occurs at the resonance frequency: ω m =
1
and at this
LC
frequency there is a large absorbance of the external magnetic field. In Fig.8 the
resulting permeability is shown. The real part of is shown in the left graph and the
imaginary part is shown in the graph in Fig.8.
Fig.8 – In the left graph the real part of the permeability is shown for different resistances of the
material. To the right the imaginary part of the permeability is shown for r2=1.5 mm and w=0.2
mm [9].
If the layer of SRR’s now also has a negative permittivity for these frequencies, the
material can exhibit a negative refractive index. As said before this has already been
demonstrated in the microwave range, but for optical frequencies it is still a challenge
to get a negative refractive index.
2.4.3 Gold Sandwiches
Another meta-material that is expected to show some negative refractive index is
shown in Fig.9. On a glass surface there are a number of gold/SiO2/gold sandwiches.
The figure is just a 2D version with cylindrical structures which shows one cylindrical
gold sandwich, but the structures are in fact three dimensional. The thicknesses of
the Au-layers are 20 nm and the thickness of the SiO2 is 20 nm for one type of
samples and 40 nm in another type.
Fig.9 – Schematic side view of a gold sandwich with a
layer of SiO2 between gold layers on a glass substrate [6].
15
In this structure an effective permeability was predicted because of localized
plasmonic resonances [4], [5]. In these structures there are two different resonances
possible: a symmetric and an asymmetric resonance, which result in an effective
permittivity and an effective permeability respectively. These resonances correspond
to an electric and a magnetic resonance.
In Fig.10 the magnetic field and the electric field are shown for a different kind of
nano sandwiches, with silver instead of gold. The 2D structure of Fig.9 is elongated
for infinite length in the y-direction for this example (directions as in Fig.9).
Fig.10 – Field maps in the sandwich structure during a spectroscopic scan for 1720 nm (left)
and 910 nm (right). The arrows represent the electric displacement and the colormap represents
the magnetic field [6].
The arrows are the electric displacement and the colors represent the value of the
magnetic field. In the figure it is clearly visible where the permittivity and the
permeability come from. For a measurement with a wavelength of 1720 nm the
electric displacement vectors are opposite to each other in the different nanostrips.
This gives a strong magnetic field in between the nanostrips and that gives rise to a
magnetic permeability. For 910 nm the electric displacement vectors are parallel,
which give a symmetric resonance and thus an effective permittivity. The
wavelengths for which this phenomenon should appear depend largely on the width
of the nanostrips and also on the aspect ratio. The resonances of the permittivity and
the permeability ideally should overlap, because then there is a possibility that both
are negative, and then it might be possible to find a negative refractive index.
As said before the measurements for this project are done on gold sandwiches, not
on nanostrips. However, it might be possible to find this negative refractive index in
these gold nanosandwiches.
2.5 The Harmonic Oscillator
Due to an applied electromagnetic field an electron in a symmetric structure moves.
Magnetic resonances are expected, but it will be interesting to find more information
about the fundamental origin of these resonances. Do these resonances origin from
the ordinary movement of an electron in an atomic structure, or do they come from
the microscopic structure?
The movement of an electron in an atomic structure can be approximated with the
Harmonic Oscillator model. Due to the electric field the charge starts moving, and
16
because of the magnetic field the electron starts moving in a perpendicular direction
as shown in Fig.11.
The following equations of motions describe the situation shown in Fig.11
14a)
14b)
dx
d 2x
Kx = −eE + Kx 0 − b − me 2
dt
dt
dx E
dz
d 2z
Kz = −
e + Kz 0 − b − me 2
dt c0
dt
dt
Where b=me/ is the damping constant with the mass of an electron me and a decay
time . The spring constant K is given by K= 2*me with angular frequency . The
initial position of the electron is given by x0 and z0. e is the constant elementary
charge and c0 is the speed of light in vacuum. E is the applied electric field.
With this information an approximation of the movement of an electron in an
electromagnetic field can be made using the Simulink function in MATLAB. In this
way it can be shown if there is an electric and/or a magnetic resonance. For the
Simulink model, see Appendix A.
Fig.11 – An electron with mass me, oscillating in a symmetric
atomic structure. The E-field is pointing in the x-direction, while
the B-field points perpendicular to the E-field. The electron is
moving in the x- and z-direction.
According to this model the electron would move in its structure as shown in Fig.12
The displacement in x-direction is much larger than the displacement in the zdirection. This means that the electric field has a much larger influence on the
electrons in a structure than the magnetic field. So if a resonance in the magnetic
permeability is found during any measurement, one can conclude that this is not due
to the movement of the electrons, but it is more due to the microscopic structure of
the layer that is examined. This is the property of a metamaterial.
17
Fig.12 – The movement of an electron in an xz-plane due to the
electromagnetic field from a plane wave propagating in the zdirection. Notice that there is a 20 order difference in the scales.
18
3. Experimental aspects
3.1 VASE Ellipsometer
VASE stands for Variable Angle Spectroscopic Ellipsometer. For the experiments a
VASE from J. A. Woollam Co., Inc. was used. A schematic picture of the VASE is
given in Fig.13. The light source is a xenon arc lamp with a monochromator.
The instrument can measure with wavelengths between 188 nm and 1700 nm. The
angle of incidence can be varied between 20 and 90 degrees. One should perform
the following steps to perform a measurement. The first action is to mount a sample
on the sample holder. This sample stays on the holder because of a vacuum pump.
Then the alignment process is performed. The alignment detector can be removed
after aligning, but this is not necessary if the measurements are done below 5 eV.
These steps should be done on a silicon calibration sample first. After a calibration,
real measurements on samples under investigation can be done.
Typical experimental parameter settings are:
- Wavelength range: 200 – 1700 nm
- Angle of incidence range: 40 – 80 degrees
- Time of measurements: 30 to 200 minutes.
Some pictures of the experimental setup are shown in Appendix B.
Fig.13 – The VASE ellipsometer with angle of incidence
analyzer VASE is shown.
with the normal. Here a rotating
19
3.2 IR VASE
Measurements in the near and far infrared range were performed using an InfraRed
Variable Angle Spectroscopic Ellipsometer (IR VASE) from J.A. Woollam, Co., Inc.
The principle of the IR VASE measurement is similar to the VASE experiment for the
visible range. This IR VASE has a spectral range from 2 to 30 m, and the angle of
incidence can be varied from 30° to 90° with a precision better than 0,005°. A special
feature of the IR VASE is the Rotating Compensator Ellipsometer (RCE)
configuration, which provides very accurate “delta” data from 0 to 360 as well as
advanced measurements like anisotropy, Mueller matrix and depolarization
measurements.
3.3 The Different Samples
The following samples have been used for measurements:
• Silicon nanotips (SiNT) on silicon
• Sandwiched gold/SiO2/gold nanodots on glass
• Split Ring Resonators (SRR)
3.3.1 Silicon Nanotips
Two different samples with SiNTs
were available: ECR 1006 and ECR
1008. An example of a layer with
nanotips is shown in Fig.14. The
properties of the different structures
are:
ECR 1006 - SiNTs
Length = 950 ~ 900 nm
Diameter = 100 ~ 140 nm
Spacing = ~ 200 nm
ECR 1008 - SiNTs
Length = ~1350 nm
Diameter = 100 ~ 130 nm
Spacing = ~ 200 nm
Fig.14 – SEM image of the ECR 1008 SiNT. The tips
have a height of about 1350 nm.
The percentage of void in the SiNT
(Silicon NanoTips) is about 85%. This percentage changes for different depths in this
SiNT-layer.
3.3.2 Gold Sandwiches
A schematic example of the layer layout is shown in Fig.15. The sandwiches have
three layers and they are placed on a glass substrate. The gold layers are 20 nm
each, but the SiO2-layer varies between 20 nm and 40 nm for two different samples.
20
The diameter of such a sandwich is approximately 170 nm. It is predicted that in
these sandwiches there will be electric resonances but also magnetic resonances,
which should make a negative refractive index at optical frequencies possible.
Fig.15 – Schematic side view of cylindrical gold sandwiches on a glass substrate.
The thickness of the SiO2 layer d is 20 nm or 40 nm.
3.3.3 SRR
The third substrate that is examined is the SRR. An SRR is shown in Fig.16. All the
different sizes of the structures are given in the figure. These structures are copper
structures embedded in a silicon surface. The height of the copper structure is about
800 nm.
Fig.16 – SEM micrograph of the SRR pattern on the silicon
substrate. [9]
21
4. Experimental results
4.1 Measurements on Silicon NanoTips
The substrates for the samples are measured first to make sure that a good
approximation of the silicon substrate including the oxide layer can be made. After
that the real SiNTs structures are examined.
4.1.1 Substrate Characterization
The substrates consist of silicon with a top layer of silicon dioxide, because the
oxygen in the air reacts with the silicon. The model used is shown in Fig.17. An
intermix layer is included to model the interface and this gives a better result for the
fit. The mean square error is: MSE=0.438. The experimental and fit data are shown
in Fig.18.
2 sio2
1 Intermix
0 si
2.361 nm
1.021 nm
1 mm
Fig.17 – Layer structure of the substrate on which
the silicon nanotips had been grown.
Fig.18 – Experimental data and fit data of the substrate on
which the SiNTs had been grown.
22
4.1.1 Silicon Nanotips
18 biaxial5
17 ema11 (si)/98% void
16 ema10 (si)/98% void
15 biaxial4
14 ema9 (si)/97% void
13 ema8 (si)/97% void
12 biaxial3
11 ema7 (si)/95% void
10 ema6 (si)/95% void
9 biaxial2
8 ema5 (si)/90% void
7 ema4 (si)/90% void
6 biaxial
5 ema3 (si)/85% void
4 ema2 (si)/85% void
3 sio2
2 ema (si)/19% sio2
1 sio2
0 si
Fig.19 – Optical model of the SiNTs
sample. The nanotip layer is assumed to be
anisotropic. In the horizontal plane the
layer is symmetric, but not in the vertical
plane. The EMA model is used to
approximate the SiNTs layer.
4.914 nm
0.000 nm
0.000 nm
10.239 nm
0.000 nm
0.000 nm
51.787 nm
0.000 nm
0.000 nm
382.172 nm
0.000 nm
0.000 nm
124.776 nm
0.000 nm
0.000 nm
2.200 nm
1.870 nm
2.361 nm
1.004 mm
Because it is expected that the widths
of the nanotips changes for different
heights above the surface, a model as
shown in Fig.19 has been made for a
measurement on a ECR 1006 sample.
The nanotips are expected to have a
void fraction of about 85%, but the
higher one gets above the surface,
the more narrow the SiNTs will
become, so there is more void at that
height. The different layers of the
nanotips are modeled as an effective
media layer. It should be possible to
approximate the optical constants with
a mixture of silicon and void. The result of this modeling is shown in Fig.20. It is clear
that the result is not as good as would be expected. The multilayer EMA model is not
sufficient for modeling this complicated nanostructure and another way of modeling
this structure should be found. The optical constants that describe the optical
behavior of the SiNT layer are shown in Fig.21. There are big differences in these
values when changing the wavelength of the light or the angle of incidence.
Si Nanotips
60
Model Fit
Exp E 50°
Exp E 60°
Exp E 70°
Exp E 80°
50
40
1.4
1.2
<n>
30
0.60
Exp < n>-E 50°
Exp < n>-E 70°
Exp < k>-E 50°
Exp < k>-E 70°
20
0
400
620
840
1060
Wavelength (nm)
1280
Fig.20 – Experimental and fit data of the
measurement on the silicon nanotips.
1500
0.40
0.30
1.0
0.20
0.8
10
0.50
0.10
0.6
400
600
800
1000
Wavelength (nm)
1200
0.00
1400
Fig.21 – Measured graphs of the refractive
index and the extinction coefficient for two
different angles of incidence.
23
<k>
Ψ in d e g re e s
Optical constants Si Nanotips
1.6
4.2 Measurements on Au-SiO2-Au sandwiches
First a measurement on a bare glass surface without the gold sandwiches was done.
This is the characterization of the substrate. If this is done, better results for the real
measurements can be obtained. Subsequently these experimental results will be
used for fitting, using a model that does not include the magnetic resonances in the
material. Also a model will be made that does include these resonances.
4.2.1 Substrate Characterization
The experimental results are fitted using a Cauchy model. The Cauchy parameters
are as follows: An=1.494, Bn=6.412*10-3 and Cn=-8.604*10-5. The experimental
results and the fit are shown in Fig.22. The error of the fit is: MSE=0.536. The optical
constants of the Cauchy layer are shown in Fig.23 and it is clear that this is the result
for a glass substrate, because the refractive index n is around 1.50 which is the nvalue for glass. The refractive index gets higher in the UV-range. The extinction
coefficient stays zero for all wavelengths.
Substrate Characterization
35
Model Fit
Exp E 40°
Exp E 50°
Exp E 60°
Exp E 70°
Exp E 80°
Ψ in degrees
30
25
20
15
10
5
0
400
600
800
1000
Wavelength (nm)
1200
1400
Fig.22 – Experimental and fit data of the substrate on which
the gold sandwiches are grown.
cauchy Optical Constants
0.10
n
k
1.540
0.08
1.530
0.06
1.520
0.04
1.510
0.02
1.500
1.490
200
400
600
800
1000
Wavelength (nm)
1200
Extinction Coefficient ' k'
Index of refraction ' n'
1.550
0.00
1400
Fig.23– Experimental and fit data of the real part and the
imaginary part of the refractive index of the substrate.
24
4.2.2 Au-SiO2-Au Sandwiches Structures
For modeling the actual Au-SiO2-Au sample the Cauchy parameters in section 4.2.1
are used in the model as the substrate of the sample. In Fig.24 the model is shown
that is used for fitting the experimental results without including magnetic effects.
Because the material is asymmetric (the x and y directions are similar and form the
horizontal plane, but the vertical z-direction is different), a biaxial layer is used. Two
different Lorentzian oscillators are used to make an accurate approximation of the
gold sandwich layer.
3
2
1
0
biaxial
lorenz2
lorentz
substrate nanosandwich
59.072 nm
0.000 nm
0.000 nm
1 mm
Fig.24 –optical model of the gold nano sandwiches sample, without
including magnetic effects. The Lorentz layer simulates the electric
resonances in the x- and y- directions and the Lorentz2 layer
simulates the response in z-direction.
In Fig.25 the experimental data and the fit resulting from the model shown in Fig.24
are shown. The model gives an MSE of 1.482, so this fit is quite accurate.
Gold sandwiches 20-20-20
25
Model Fit
Exp E 56°
Exp E 58°
Exp E 60°
Exp E 62°
Exp E 64°
Exp E 66°
Exp E 68°
Exp E 70°
20
Ψ in degrees
In
Fig
.26
the
mo
del
is
sh
ow
n
for
mo
del
ing
the
ex
15
10
5
0
200
400
600
800
1000
Wavelength (nm)
1200
1400
Fig.25 – Experimental and fit data for gold nanosandwiches on a surface. This measurement was
done on a sample with gold nanosandwiches that had layers of 20 nm Au, 20 nm SiO2 and 20 nm Au.
perimental results when also magnetic behavior in the material is included. The
meta6 layer is the layer where one can decide if the magnetic resonances in the
material should be included or not. For approximating the permeability in the layer a
Lorentzian-oscillator is used. In references [11] and [12] it is explained why the
Lorentzian-oscillator is a good approximation for the behavior of the permeability.
25
7
6
5
4
3
2
1
0
meta6
biaxial2
lorentz4
lorentz3
biaxial
lorenz2
lorentz
substrate nanosandwich
61.423 nm
0.000 nm
0.000 nm
0.000 nm
0.000 nm
0.000 nm
0.000 nm
1 mm
Fig.26 – Layer model of the gold nano
sandwiches sample, when concerning the
magnetic response of the material. The biaxial
layer simulates the electric response, and the
biaxial2 layer the magnetic response, which is
also uniaxial. These two come together in the
Meta6 layer.
The experimental data and the resulting fit are shown in Fig.27. The mean square
error is MSE=1.151. The fit is quite good, and using magnetic resonances in the
model for the gold sandwiches gives a better result than when the magnetic behavior
is neglected. There is one magnetic resonance, and it is observed in the Lorentz3
sub layer, which means that there is a magnetic resonance in the horizontal plane.
The permeability is shown in Fig.28. The magnetic resonance is situated around 260
nm, in the UV-range. The permeability in Lorentz4 has a constant value of =1.251.
Gold sandwiches 20-20-20
25
Model Fit
Exp E 56°
Exp E 58°
Exp E 60°
Exp E 62°
Exp E 64°
Exp E 66°
Exp E 68°
Exp E 70°
Ψ in degrees
20
15
10
5
0
0
300
600
900
1200
Wavelength (nm)
1500
1800
Fig.27 – Experimental and fit data for gold nanosandwiches on a surface. Again all
three layers of the sandwiches are 20 nm thick. While fitting, also the magnetic response
of the material was taken into account.
lorentz3 Optical Constants
1.140
0.030
ε1
ε2
1.130
0.025
0.020
1.125
0.015
1.120
0.010
1.115
1.110
0
300
600
900
1200
Wavelength (nm)
1500
Imag permeability
Real permeability
1.135
0.005
1800
Fig.28 – The magnetic response of the gold nanosandwiches. The magnetic resonance is
situated around 260 nm and it is observed in the horizontal plane.
26
Table.1 Oscillator parameters per layer
Layer
Number of oscillator
Lorentz
1
Lorentz
2
Lorentz2
1
Lorentz2
2
Lorentz3
Lorentz4
1
1
1.237
1.237
0.981
0.981
Am
0.477
1.120
0.188
0.042
(eV)
0.140
0.322
0.172
0.300
En (eV)
1.475
1.861
1.540
2.451
1.091
1.251
0.904
0
7.648
0
5.434
0
There are multiple oscillators in the dielectric functions of different layers. In Table.1
the properties of the oscillators in the different layers are shown. Here
is the
permittivity of the layer and
the permeability for low or values of the wavelength.
Am is the amplitude and is the broadening of the resonance. En is then the position
of the resonance in the energy spectrum. Lorentz2 and Lorentz4 are the vertical
directions, and Lorentz and Lorentz3 are the horizontal directions. So there is a
resonance in the permeability for the horizontal direction, and the permeability is
constant for the vertical direction. There are two dielectric resonances for both the xand y-direction and the z-direction.
The dielectric functions of the Lorentz and Lorentz2 layer are shown in Fig.29, and
the optical constants of the Meta6 layer are shown in Fig.30. Most optical variation is
seen between 500 nm and 1000 nm.
Lorentz and Lorentz2 Optical Constants
2.5
1.5
1.0
1.0
0.5
0.5
0
300
600
900
1200
Wavelength (nm)
1500
Fig.29 – Dielectric functions (real
and imaginary parts) of the gold
nanosandwich layer.
0.0
1800
2.0
0.60
0.50
0.40
1.8
0.30
1.6
0.20
1.4
0.10
1.2
300
600
900
1200
Wavelength (nm)
1500
<k>
1.5
Model Fit
Exp < n>-E 56°
Model Fit
Exp < k>-E 56°
2.2
2.0
2.0
Optical constants Meta6 layer
2.4
<n>
ε1x
ε1z
ε2x
ε2z
2.5
Im a g D ie le c tr ic C o n s ta n t, ε 2
R e a l D ie le c tr ic C o n s ta n t, ε 1
3.0
0.00
1800
Fig.30 – Measured graph of the refractive
index and the extinction coefficient for
two different angles of incidence.
27
4.2.3 IR VASE Measurement on Nanosandwiches
T. Driscoll et al. [16] have found that in a structure like Split Ring Resonator a
resonance in the permeability could be found at long wavelengths in the infrared
range. This could also count for the gold nanosandwiches. However, a resonance in
the IR region has not been found yet using the VASE for the visible range, so by
using the IR VASE maybe a resonance can be found. A measurement on only the
glass substrate can be seen in Fig.31.
The optical constants of the Cauchy layer of the substrate in chapter 4.2.1. are used
as the initial values and then the optical constants n and k of the layer are fitted to the
experimental data. The resulting fit and the experimental data are shown in Fig.31.
The resulting value for the MSE is MSE = 1,16. The optical constants n and k for the
glass substrate are shown in Fig.32.
Substrate Characterization
50
Ψ in degrees
40
30
Model Fit
Exp E 50°Visible range
Exp E 60°Visible range
Exp E 70°Visible range
Exp E 50°
Exp E 60°
Exp E 70°
20
10
0
0
2000
4000
6000
8000
Wavelength (nm)
10000
12000
Fig.31 – Experimental and fit data of the glass substrate in the IR range.
Also the data measured in the visible range are plotted in the graph. The
gap at 1400-1800 nm arises from the fact that both the VIS and IR VASE
are not accurate enough for these wavelengths.
Optical constants IR
2.4
2.1
1.2
0.9
1.5
0.6
1.2
0.3
0.9
-0.0
0.6
0
3000
6000
9000
Wavelength (nm)
12000
<k>
<n>
1.8
1.5
Model Fit
Exp < n>-E 50°
Exp < n>-E 70°
Model Fit
Exp < k>-E 50°
Exp < k>-E 70°
-0.3
15000
Fig.32 – Optical constants of the substrate. Both the experimental and fit
data are plotted.
28
The experimental data of the actual measurement on the nanosandwiches is shown
in Fig.33. Making a fit that was accurate for both the visible range and the IR range
did not work, but these experimental data give some important information. The result
is almost the same for the measurement on the glass substrate. This means that the
layer with sandwiches has no major effect on the light reflecting on the surface. If
there would be a resonance in the permeability in the IR range, some activity in the
would be expected. However, the fact that this is not observed in the results must
mean that there is no resonance in the permeability in the IR range, also according to
the experimental data for that are shown in Fig.34. The action starts around 7000
nm, which is the same for the
data and this is typical for a glass sample and it
looks like there is no magnetic action in the IR region.
50
Ψ in degrees
40
Gold nanosandwiches IR measurement
Exp E 60°Visible range
Exp E 70°Visible range
Exp E 50°
Exp E 60°
Exp E 70°
30
20
10
0
0
2000
4000
6000
8000
Wavelength (nm)
10000
12000
Fig.33 – Experimental data of the nanosandwich sample. Both the visible
and IR range measurements are plotted. The result is very similar to the
result of only the glass substrate.
300
Gold nanosandwich IR measurement
Exp E 60°
Exp E 70°
Exp E 50°
Exp E 60°
Exp E 70°
∆ in degrees
200
100
0
-100
0
2000
4000
6000
8000
Wavelength (nm)
10000
12000
Fig.34 – Experimental data of the nanosandwich sample. Optical
activity starts at 7000 nm and higher.
29
4.3 Measurements on Copper Nanostructures (SRR)
Because it was not possible to measure on the substrate on which the SRRs were
grown, the approximation is made that the substrate consists of silicon with a layer of
2 nm silicon dioxide on it. The fabrication process of the SRR’s is explained in
reference [9]. The layer model is shown in Fig.35. Also a generalized ellipsometry
measurement is done to find out if there is some anisotropy in the structure.
4
3
2
1
0
biaxial
lorenz2
lorentz
sio2
si
819.890 nm
0.000 nm
0.000 nm
2.000 nm
1 mm
Fig.35 – Optical model simulating the Cu nanostructure
on the substrate.
The experimental and fitting data for this metamaterial are shown in Fig.36. The
mean square error is: MSE=47.76. The resulting fit is not very good. More research
should be done to find the right model for this sample. Then it will be possible to find
information about the optical constants of this metamaterial.
Split Ring Resonator
80
Model Fit
Exp AnE -50°
Exp AnE -40°
Exp AnE -30°
Exp Aps -50°
Exp Aps -40°
Exp Aps -30°
Exp Asp -50°
Exp Asp -40°
Exp Asp -30°
Ψ in degrees
60
40
20
0
0
400
800
1200
1600
Wavelength (nm)
2000
2400
Fig.36 – Experimental and fit data for the copper Split Ring Resonators. The fit is not
accurate, and there is some anisotropy measured for wavelengths higher than 900 nm.
The Aps and Asp data hold the information about the anisotropy. The AnE data are used
to find the optical parameters of the sample.
30
The optical constants of the biaxial layer are obtained from the experimental data and
they are shown in Fig.37. The graphs of these optical constants are quite special,
because the refractive index is smaller than one for wavelengths smaller than 1500
nm. But the refractive index of a vacuum is only one. This could mean that the
complexity of the SRR-layer and -structure have influenced the refractive index to
behave like this. The optical model should be optimized to be able to make some
reasonable conclusion on this particular behavior. This near zero refractive index
happens for both the angles of incidence, so with a larger sample this also could be
tested for larger (more standard) angles of incidence.
biaxial Optical Constants
0.15
nx
nz
kx
kz
0.99
0.12
0.96
0.09
0.93
0.06
0.90
0.03
0.87
0
300
600
900
1200
Wavelength (nm)
1500
Extinction Coefficient ' k'
Index of refraction ' n'
1.02
0.00
1800
Fig.37 – Refractive index and extinction coefficient for two different angles of
incidence. The refractive index reaches very high values, while the imaginary part of the
refractive index gets below zero for some values of the wavelength.
31
5. Conclusions and Discussion
There were three different kinds of samples on which measurements were
performed. For two of these samples, SiNTs and SRRs, there was not enough time
to get acceptable models, but still the experimental results gave some interesting
information. One sample however, the gold nanosandwiches, gave less problems in
modeling, and a model including the magnetic behavior of the nanosandwich layer
has been made.
SiNTs
A model of the substrate was easily made with a very small error. The model for the
SiNTs layer however was not very good. Mathematically the model seemed
appropriate, but is physically not correct. The model was not sufficient for a
complicated structure like the SiNTs.
SRR
Measurements on just the substrate were not possible for this sample. An
assumption has been made to model the substrate. A model has been made for this
sample, but also for this sample the model failed to give an accurate fit to the
experimental data. It gives some global approximation, but it is not good enough to
make conclusions from. The experimental data however give some special
information about the refractive index that which is smaller than one for wavelengths
up to 1500 nm.
Gold Nanosandwiches
The glass substrate is modeled using a Cauchy layer, which gave a quite accurate fit.
The gold nanosandwich layer is modeled in the spectral range up to 1700 nm using
Lorentzian oscillators for a uniaxial layer. In the infrared a point to point fit was done.
The fit of the gold nanosandwiches was acceptable, but when the magnetic behavior
of the complex structure was included in the model using a new version of the
WVASE32tm program the fit was even more accurate, and a magnetic oscillation is
found in the horizontal plane in the UV-range. Still one cannot say that it is a fact that
this magnetic oscillation is exactly there. The fit should still be better if one wants to
conclude that with more certainty. Oscillations were expected in the IR-range, that is
why an IR measurement has been done, but the result was different than was
expected. There was no magnetic activity in the IR-range, the measurement on only
the substrate looked almost identical to the measurement on the sample including the
gold nanosandwiches.
In future experiments different approaches or improvements for modeling the SiNTs
and the SRR structures should be developed. The SiNTs and the SRR hold very
interesting information on how the microstructure of a layer has more influence on the
optical behavior of the layer than the composition of this layer.
The version of the WVASE32tm program in which magnetic behavior could be
included was only a tryout that seemed to work acceptable. This program should be
tested more so the value of the program will be known when one wants to model
different complex metamaterials.
32
6. References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
N. Liu and H. Guo, “Three-dimensional photonic metamaterials at optical
frequencies,” Nature Mat., 7, pp. 31-37, 2008
E. Schubert, “Sub-Wavelength Antireflection Coatings From Nanostructure
Sculptured Thin Films,” Phys., 47, pp. 545–550, 2007
S. Y. Bae, C. Lee, S. Mobasser and H. Manohara, “Silicon Nanotips
Antireflection Surface for Micro Sun Sensor,” Nanotechn., 2, pp. 527–530,
2006
N. Feth, C. Enkrich and M. Wegener, “Large-area magnetic metamaterials via
compact interference lithography,” Opt. Express, 15, pp. 501-507, 2007
J. A. Woollam Co., Inc., “Guide to using WVASE32TM,” WexTech Systems,
Inc., New York, 1995
U. K. Chettiar, A. V. Kildishev, T. A. Klar† and V. M. Shalaev, “Negative index
meta-material combining magnetic resonators with metal films,” Opt. Express,
14, pp. 7872-7877, 2006
M. W. McCall, A. Lakhtakia and W. S. Weiglhofer, “The negative index of
refraction demystified,” Eur. J. Phys., 23, pp. 353–359, 2002
R. A. Shelby, D. R. Smith and S. Schultz, “Experimental Verification of a
Negative Index of Refraction,” Science, 292, pp. 77-79, 2001
P. Han and W. Ni, “Optical Studies of Cu-based Meta-materials,” Thesis
report, 2007
H. Arwin, “Thin Films Optics and Polarized Light,” Lecture notes Linköping
University, 2007
T. Driscoll, D. N. Basov, W. J. Padilla, J. J. Mock and D. R. Smith
“Electromagnetic characterization of planar metamaterials by oblique angle
spectroscopic measurements,” Phys. Rev. B75, 115114, 2007
A. Ishimaru, S. Lee, Y. Kuga and V. Jandhyala, “Generalized Constitutive
Relations for Metamaterials Based on the Quasi-Static Lorentz Theory,” IEEE
trans. antennas propagate. 51, pp. 2550-2557, 2003
2006
The
Spokesman-Review,
“Building
for
the
future,”
http://www.spokesmanreview.com/cover.htm
A. Cho, “High-tech metamaterials could render objects invisible,” Science, 312,
pp.1120, 2006
F. Mesa, M. J. Freire, R. Marqués, and J. D. Baena, “Three-dimensional
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33
7. Appendices
Appendix A
Fig.A1 – Simulink model for finding a solution for the coupled differential equations which give
the movement of an electron in an electromagnetic field. The upper half of the model calculates
the x-movement (in reaction to an electric field in the x-direction), while the lower half
calculates the z-movement (in reaction to the magnetic field in the y-direction). The differential
equations are coupled by the first derivate of the x-displacement that is multiplied with the
electric field (given as a sine wave) and the e/c0 constant. The result is plotted separately and
also in a XY graph.
34
Fig.A2 – Displacement of the electron in the horizontal
direction. On the vertical axis the position in meters is
plotted against the time on the horizontal axis.
Fig.A3 – Displacement of the electron in the vertical
direction. Very interesting is the amount of displacement
in this direction. It is a negligible value.
Fig.A4 – The driving force: the electrical field. On the
vertical axis the electric field in V/m is plotted.
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Appendix B
Fig.B1 – The VASE Ellipsometer and the lamp plus the monochromator.
Fig.B2 – Front view of the VASE Ellipsometer.
36