Material Characterization by Millimeter

Material Characterization by Millimeter
Masters’s Thesis
Material Characterization by
Millimeter-Wave Techniques
Leonard Andersson
Department of Electrical and Information Technology,
Faculty of Engineering, LTH, Lund University, 2016.
Material Characterization by
Millimeter-Wave Techniques
Department of Electrical and Information TechnologyFaculty of Engineering, LTH, Lund University
SE-221 00, Lund, Sweden
Author:
Leonard Andersson
Supervisors:
Lars-Erik Wernersson
Sebastian Heunisch
Examiner:
Erik Lind
March 2016
Abstract
Abstract
This master thesis investigates material characterization by reflection and
transmission of electromagnetic waves in the 40-60 GHz band (millimeter-wave
spectrum) for different materials. The free-space measurement method is a fast,
efficient and non-destructive way of examining a material and is being
researched by both academics and industries.
The theory of how electromagnetic waves interact with different materials such
as dielectrics and conductors is reviewed as well as how the reflection and
transmission from such materials can be computed theoretically. This theory is
partially derived from Maxwell’s equations. From this theory, simulations are
performed to get signal levels of reflection and transmission for different
materials and varying material parameters. From the simulations it is shown that
certain materials are better examined in either transmission or reflection.
Measurements were performed in time domain (with a wavelet generator and an
oscilloscope) and in frequency domain (with a network analyzer). Both reflection
and transmission were measured for all samples. Four samples were investigated
thoroughly: two PMMA (Poly(Methyl MethAcrylate)) samples, one silicon sample
and a thin gold film sample.
Before the measured data can be compared to the simulated, it is necessary to
apply signal processing to both the measured and the simulated data. This is
done to make sure the comparison of the two data sets works and it consists of
removing multiple reflections and other unwanted noise from the signal. The
material characterization could then be performed, by extracting a specific
material parameter, such as permittivity or conductivity. This is done by
comparing simulated data iteratively to measured data. The best fit should then,
in theory, correspond to the actual material parameter.
The material characterization worked, although sometimes differences in time
and frequency domain were found. Permittivity values were extracted for the
PMMA samples and conductivity values for the silicon and thin gold film samples.
The values extracted compared well with reference values for the PMMA samples
and the thin gold film sample.
i
Contents
Contents
ABSTRACT ....................................................................................................................... I
CONTENTS ...................................................................................................................... II
1 INTRODUCTION ............................................................................................................1
1.1 BACKGROUND & MOTIVATION........................................................................................... 1
1.2 PROJECT DESCRIPTION ...................................................................................................... 2
1.3 METHODOLOGY .............................................................................................................. 2
1.4 OUTLINE OF THE REPORT ................................................................................................... 2
2 THEORETICAL BACKGROUND ........................................................................................3
2.1 MAXWELL’S EQUATIONS, PERMITTIVITY AND PERMEABILITY...................................................... 3
2.1.1 Dielectrics ............................................................................................................ 7
2.1.2 Conductors .......................................................................................................... 8
2.1.3 Semiconductors ................................................................................................... 9
2.2 UNIFORM ELECTROMAGNETIC PLANE WAVES ....................................................................... 10
2.2.1 Reflection and Transmission ............................................................................. 10
2.2.2 One-layer structure ........................................................................................... 13
2.2.3 Multilayer structure .......................................................................................... 15
2.3 SIMULATION PRE-STUDY & DESIGN PLOTS .......................................................................... 17
2.3.1 One-layer structure ........................................................................................... 17
2.3.2 Multilayer structure .......................................................................................... 21
3 METHODOLOGY .......................................................................................................... 25
3.1 MEASUREMENTS ........................................................................................................... 25
3.3.1 Frequency domain ............................................................................................. 27
3.3.2 Time domain ..................................................................................................... 29
3.2 SIGNAL PROCESSING ...................................................................................................... 30
3.3 PARAMETER EXTRACTION ................................................................................................ 33
4 RESULTS AND DISCUSSION ......................................................................................... 36
4.1 PMMA SHEET .............................................................................................................. 36
4.2 PMMA CONTAINER ....................................................................................................... 38
4.3 SILICON WAFER ............................................................................................................. 40
4.4 THIN GOLD FILM ON SILICON ............................................................................................ 42
5 CONCLUSIONS ............................................................................................................ 45
5.1 DISCUSSION.................................................................................................................. 45
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5.2 FUTURE WORK .............................................................................................................. 46
6 ACKNOWLEDGEMENTS ............................................................................................... 47
7 REFERENCES ............................................................................................................... 48
8 APPENDICES ............................................................................................................... 51
8.1 SOURCE CODE ............................................................................................................... 51
Design plots – one layer ............................................................................................. 51
Design plots – two layer ............................................................................................. 53
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1 Introduction
CHAPTER
1
1 Introduction
This thesis was a part of the final project examination for the degree Master of
Science in Engineering Nanoscience from Lund University, Lund, Sweden. The
work presented here was performed with the Nanoelectronics Group at the
Department of Electrical and Information Technology, Lund University.
1.1 Background & Motivation
The recent advancement of material characterization by time- or frequency
domain measurements has received a lot of attention lately because of its nondestructive, fast and efficient way of probing a material. Research in this area
confirms that free-space probing by transmission and reflection measurements
makes it possible to calculate the complex permittivity of a material and thereby
decide its composition, without being in contact with the material.
It is known that the dielectric properties of a material corresponds to different
material characteristics and recent research also shows that this relation can be
used to determine not only electrical conductivity, permittivity and permeability
but also properties such as chemical concentration, moisture content, bulk
density, bio-content and stress-strain relationship. This is not only interesting for
the scientific community but also for the industrial world. This technology could
be useful in food science, medicine, agriculture, chemistry, defense industry,
electrical devices, biology and civil engineering [1] [2].
One of the main challenges today is to accurately measure the material
properties of thin samples (100 to 1000 nm range). Many methods exist for
measuring these properties, such as parallel plate capacitors, transmissionline/waveguides methods or free space methods. For higher frequencies, where
the wavelength is roughly a millimeter, free space methods are favored. This is
because if the wavelength is smaller than the sample, one can neglect effects
such as diffraction and charge buildup at the edges of the sample [3] [5].
1
1 Introduction
1.2 Project Description
The aim with this Master’s thesis was to study reflection and transmission of
millimeter-wave wavelets propagated through dielectrics and absorbing
materials with conductivity and/or permittivity. The materials investigated were,
among others, silicon-wafers and dielectrics. Further work could look at materials
such as organic solar cells and thin film dielectrics with semiconducting
nanostructures. The thesis includes reference measurements on known material
as well as measurements on new materials. The work also contains modeling and
simulations of material properties and their effect on transmission and reflection.
1.3 Methodology
The methodology used can be divided into two steps. First, models were chosen
to describe how electromagnetic waves interact with materials and simulations
of these models were implemented. Second, measurements in time and
frequency domain were performed and data collected. This data was examined
and its consistency with the simulations were evaluated.
1.4 Outline of the report
This master’s thesis starts with a thorough review of the theory behind reflection
and transmission from electromagnetic waves impinging upon different kind of
materials, as well as material models for conductors and dielectrics in Chapter 2.
Chapter 3 explains the methodology used, as well as how and why the
simulations were done. In Chapter 4 the results are discussed and analyzed.
Finally, chapter 5 draws conclusions and summarizes the whole report.
2
2 Theoretical background
CHAPTER
2
2 Theoretical background
The theoretical background of this report is electromagnetic wave propagation
and the electrodynamics of solids, which is a vital cornerstone for understanding
the simulations, measurements and conclusions.
The theory and statements in this chapter are mainly based on concepts covered
in Orfanidis [4], Dressel and Grunner [5] and Bishop [6].
2.1 Maxwell’s equations, permittivity and permeability
Between 1861 and 1862 James Clerk Maxwell published the famous Maxwell’s
equations (which in turn are based on Ampere’s circuit law, Faradays law of
induction, Gauss’s law for magnetism and Gauss’s flux theorem).
These integral equations, in combination with the law of the Lorentz force,
formed the groundwork for classical electromagnetic theory and optics. In (1),
Maxwell’s equations are stated in SI-units.
(1)
The following derivation of equations follows that of reference [4]. The four
different vectors on the left-hand side in (1), the electric and magnetic field
intensities
and the flux densities
are related to each other by
constitutive relations.
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2 Theoretical background
Equation (2) shows these constitutive relations in their most simple form (for
vacuum):
(2)
The two constants combining these quantities, and , are the permittivity and
permeability of vacuum. With the permittivity and permeability of vacuum we
can define the speed of light in vacuum and the characteristic impedance of
vacuum:
⁄√
and
√
⁄ .
The same constitutive relations can be written for materials. For a homogeneous
isotropic material they are:
(3)
Where and is the materials absolute permittivity and permeability. By dividing
the materials permittivity and permeability by its respective vacuum equivalent,
one can define the relative permittivity and permeability:
(4)
The refractive index of a material (5) is defined as the square root of the relative
permittivity multiplied with the relative permeability:
√
(5)
By using the values of a material’s permittivity and permeability, one can
calculate the speed by which light travels in a material and the characteristic
impedance of a material the same way as for vacuum. By using this fact together
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2 Theoretical background
with (4) and (5) one can obtain the useful relations in (6) and (7) with some
substitutions and extrapolations:
(6)
(7)
This master’s thesis only examines non-magnetic materials, which means
⁄ .
or
and (6) simplifies to
In general, the absolute permittivity
expressed as
,
is a complex quantity which can be
.
(8)
where
is the angular frequency and
the frequency. This can be
understood by considering a dipole exposed to an alternating electric field. The
dipole exposed to the field will rotate and try to align itself with it. After a certain
time, the electric field reverses direction and the dipole must re-align with the
field (to remain parallel to the correct polarity). As this oscillation occurs, the
friction experienced by the dipole through the acceleration and deceleration of
the rotation causes it to lose energy through heat generation. (This is also how
microwave ovens work, the dipole in that case is polar water molecules [7]) The
imaginary part of the permittivity is a measure of the degree to which the dipole
is out of phase with the electric field. The resulting losses through heat therefore
determine how large the imaginary permittivity is. A larger imaginary part
indicates a lot of energy is being dissipated through heat. The imaginary part
therefore directly reflects the loss in the material. [6]
A dipole is created when an imbalance of charge is present. For molecules this
takes form in polar covalent bonds, where one or more atoms of the molecule
have a higher electronegativity than the rest. This makes the electron cloud
unevenly distributed across the molecule and creates a nonzero dipole moment.
These molecules are known as polar molecules. Non-polar molecules (such as O2)
can, however, temporarily become dipoles under electric fields and therefore
dissipate energy from the field. This happens because of the mass imbalance
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2 Theoretical background
between the positive nucleus and negative electron cloud. The lighter electron
cloud will align much faster with the field and therefore a dipole is momentarily
created [6] [8].
One possible way to model these properties is by the following equation:
̈
̇
(9)
⁄ ,
where ̇
is the electron rest mass and the electric field is only
present in the x-direction. This equation describes the force a bound or unbound
electron experiences when exposed to an electromagnetic field. The
factor
comes from the desire of the negative electron to go back to its original state
near the positive nucleus and the
̇ factor from any friction force,
proportional to the electron-velocity. From Hooke’s law [9] we know the constant
is linked to the resonance frequency,
, of the (atomic)-spring by
√ ⁄ . Rewriting (9) together with this fact gives:
̈
̇
(10)
Some interesting characteristics can immediately be seen in (10). When
there is no force pulling the electrons to stay near the nuclei, therefore this case
describes free moving electrons i.e. a conductor. The term ̇ appears from
collisions that, on average, slow the electron down. The
parameter can
therefore be interpreted as the rate of collisions per unit time, implying that
⁄ is the mean-time between collisions.
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2 Theoretical background
2.1.1 Dielectrics
In dielectrics,
and
, and if the electromagnetic field can be
described as a sine wave,
with angular frequency , then the
solution to (10) will be on the form of
. Inserting this and replacing
the time derivatives by their frequency part
gives:
(11)
Rearranging the terms gives the final solution:
(12)
Following this and further mathematical steps, “Orfanidis” [4] defines the
effective permittivity as:
(13)
and by defining a material parameter called the plasma frequency,
⁄
, one can rewrite (13) in a more convenient form as:
(14)
Materials that can be described by (14) are known as “Lorentz dielectrics”. As
stated above, the real and imaginary parts of the effective permittivity
corresponds to two different phenomena. The real part is responsible for
refractive properties and the imaginary part for absorptive properties. Following
the convention set up in (8) one obtains:
(15)
for the real and imaginary part respectively.
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2 Theoretical background
2.1.2 Conductors
The conducting characteristics of a material are described by Ohm’s law:
(16)
This law can be derived from the “Lorentz dielectrics” model above. In
“Orfanidis” [4] this derivation can be examined in detail. From this derivation,
the conductivity can be expressed as:
(17)
The similarity to (14) is obvious and a new way to state a materials effective
permittivity is:
(18)
As mentioned above, a metal have unbound conduction charges and therefore
. Simplifying (17) with this fact gives:
(19)
for a metal. This is known as the “Drude model”.
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2 Theoretical background
2.1.3 Semiconductors
A material that exhibits both dielectric and conductivity properties (a
semiconductor) can therefore be described by the sum of two terms; the first
term describing bound charges and the second unbound charges [4]. Assigning
different parameters
for each term, the total permittivity becomes:
(20)
Merging the first two terms as
and the third as
⁄
one obtains
(21)
which is the total effective permittivity of a material with both dielectric and
conductive properties.
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2.2 Uniform electromagnetic plane waves
One way of describing an electromagnetic wave is to assume a uniform plane sine
wave, propagating along a fixed direction. At one frequency and in a lossless
material such waves can be described by:
(22)
From superposition and the theory of electromagnetic fields we know that the
electric- and magnetic field vectors can be described in terms of a forward and
backwards moving field :
[
(23)
]
By following the derivations from “Orfanidis” [4], one obtains the general
solution for a single-frequency wave expressed as the superposition of forward
and backward components
(24)
[
]
The reflection and transmission from uniform plane waves, with normal
incidence, is discussed next.
2.2.1 Reflection and Transmission
Consider the uniform plane wave propagating in the z-direction in an isotropic
and lossless material. If the field is linearly polarized in the x-direction, one gets
[
]
10
[
]
(25)
2 Theoretical background
following the convention of (24). One can also express the backward and forward
moving fields
in terms of
and
by extrapolating in (25):
[
]
[
]
(26)
From this, two valuable quantities can be defined: the wave impedance and the
reflection coefficient.
(27)
(28)
The most simple form of reflection and transmission comes from a planar
interface dividing two dielectric and/or conducting material with characteristic
impedances
, as in Figure 1.
𝜂
𝜂
𝐸
𝐸
𝐸
𝐸
𝜌𝜏
𝜌 𝜏
Figure 1: A field propagating forward and backward through an interface.
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The elementary reflection ( ) and transmission ( ) coefficients (also known as
Fresnel coefficients) are defined as:
(29)
(30)
where the last equal sign is derived from (6). Since the field is normally incident
the electric and magnetic field on one side of the interface must equal the
electric and magnetic field on the other side.
(31)
The equations in (31) can be rewritten in matrix form as:
[
]
[
][
Using (28) together with (32) and the fact that
]
(32)
yields:
(33)
for the reflection coefficient from a single interface. One could also look at the
total transmission, defining it as what comes out from the interface divided by
what goes in:
(34)
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2 Theoretical background
In our measurements, there will (theoretically) only be an incident wave from the
⁄
left in Figure 1, so that
. This simplifies (33) and (34) as
to:
(35)
(36)
As we can see, for a single interface the reflection and transmission is easily
calculated.
2.2.2 One-layer structure
The second most simple kind of interface is that of a single dielectric slab, see
Figure 2. In this figure we see a two-interface slab with characteristic impedance
separated by the two mediums and .
𝑡
𝜂𝑥
𝜂 𝑘
𝜂𝑦
𝐸
𝐸
𝐸
𝐸
𝐸
𝐸
𝜌 𝜏
𝐸
𝜌 𝜏
𝑍
𝛤 𝛤
𝑍
𝛤 𝛤
Figure 2: A single dielectric one-layer slab.
⁄ , where is the
The thickness of the slab is and the wavenumber is
speed of light in the material. The electromagnetic wave is set to be incident
from the left and therefore there is only a forward wave in the medium .
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Using the same matrix notation as in the previous section (32) gives:
[
]
[
][
]
[
][
][
]
(37)
[
][
]
[
][
]
Performing the matrix multiplications yields
(38)
(39)
for the incoming and outgoing fields. With this information we can calculate the
reflection and transmission signal, as in the previous section:
(40)
(41)
Since there is no field going to the left in media , the reflection from the
second interface is
, as derived in previous section. The transmission has a
delay factor of
, which represents the direct-way travel time
delay through the slab. The reflection is instead composed of two parts, one from
the first interface and another (two-way travel delayed) from the second.
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2 Theoretical background
2.2.3 Multilayer structure
Moving on to the more general case of an M-layer structure, as seen in Figure 3,
we finalize our theory for reflection and transmission.
𝑖
𝑖
𝜂𝑎
𝜂𝑖
𝑡𝑖
𝑘𝑖
𝜂
𝑡
𝑘
𝐸
𝐸
𝜌
𝑀
𝑀
𝐸𝑖
𝜌
𝐸𝑖
𝜌𝑖
𝛤
𝛤
𝜂𝑀
𝑡𝑀
𝑘𝑀
𝐸𝑀
𝜌𝑖
𝛤𝑖
𝐸𝑀
𝐸𝑀
𝜌𝑀
𝛤𝑀
𝛤𝑖
𝜂𝑏
𝜌𝑀
𝛤𝑀
Figure 3: A multilayer structure
In this figure there are slabs and
interfaces, and together with the left
⁄
and right media,
dielectrics. The entire reflection response
is obtained recursively by matrix multiplications (as done in the previous section).
The reflection coefficients
are defined as before, in terms of characteristic
impedances or refractive indices:
(42)
In (43) the forward/backwards fields at the left of interface are related to those
at the right.
[
]
[
][
15
]
(43)
2 Theoretical background
From this, the reflection and transmission response can be calculated:
(44)
(45)
These equations can be implemented in simulations to handle any M-layer
structure with each layer having different thicknesses, permittivity and
conductivity.
One phenomenon that can occur for one-layer structures or higher is destructive
and constructive interference of the reflected or transmitted wave. In (44) and
(45) this would correspond to the numerator approaching zero for destructive
interference and the denominator approaching zero for constructive
interference. For reflection the destructive interference occurs when the wave
reflected from the first interface is out of phase from the wave reflected at the
second interface. For the one-layer case in Figure 2 this happens when the
thickness is half a wavelength or multiples thereof, ⁄ .
Let’s assume the thickness is
and we want to calculate the
permittivity the material must have to correspond to the ⁄ situation. From (5)
and (7) we get:
√
from which the permittivity can be extracted.
16
(
)
(46)
2 Theoretical background
2.3 Simulation Pre-study & Design plots
This subsection uses the theory derived above to simulate transmission and
reflection for different material parameters. This gave a good foundation for
deciding what materials would be interesting to measure upon (where one would
get the most data from) as well as deciding what thickness the material should be
and at what frequency the measurement should be performed for optimal data
extraction. All of the graphs in this section are simulated at 60 GHz operating
frequency, unless mentioned otherwise.
2.3.1 One-layer structure
Starting with a single-layer structure, the reflection and transmission was
simulated. It was simulated by letting an electromagnetic wave travel through
vacuum toward a substrate, as in Figure 4.
Figure 4: A uniform planar electromagnetic wave impinges upon a one-layer substrate.
The following figures display expected transmission and reflection when
sweeping over different material parameters, such as conductivity, permittivity
and thickness of the substrate. By looking at each variable separately, the action
of each material parameter on its own will be clarified. In Figure 5 three graphs
are plotted: the reflection when varying a material’s conductivity, the reflection
vs. the thickness of the substrate and finally the reflection vs. permittivity.
In Figure 5a and Figure 5c the thickness of the substrate is 300 μm. In Figure 5b
the conductivity is
which is equivalent to gold’s conductivity at
room temperature [10]. The real part of the permittivity is held constant at 1 in
both Figure 5a and Figure 5b. For Figure 5c the conductivity is set to zero.
As (40) and (21) predicts, the reflection in Figure 5a increases linearly in the dBscale as the conductivity increases, until it starts to approach 0 dB asymptotically.
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Figure 5: Plot a) The reflection of a 300 μm one-layer slab over varying conductivities. The real part of
the permittivity is 1, representing a metal.
Plot b) The reflection from a one-layer slab with varying thickness. The conductivity used in this
simulation corresponds approximately to that of gold at room temperature. The thickness range is
from 0.1 nm to 10 μm. The real part of the permittivity is 1.
Plot c) The reflection vs. permittivity of a 300 μm slab. The occurrence of the repetitive destructive
half-wavelength interference is visible. The conductivity is set to zero.
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From Figure 5b one observes a similar appearance; the reflection increases
asymptotically towards 0 dB as the thickness of the substrate is increased. This is
also consistent with (40). If the conductivity of the substrate is high, the reflection
will approach full reflection at a certain thickness. This is related to the skin depth
of a material, expressed as
, which is a figure of merit for the power
√ ⁄
attenuation a field experiences traveling through a material. If the conductivity,
permeability or frequency increases, the skin depth would decrease, resulting in a
higher reflection (lower transmission) [11].
In Figure 5c a difference in appearance is noticed, as the reflection drastically
decreases at certain permittivity values. This corresponds to the half-wavelength
destructive interference. It is also worth noting that “metals” of such low
conductivities as simulated here do not exist to any known extent. The materials
with such low conductivity would instead have a real permittivity higher than 1,
which would increase the reflection from the material [12].
By combining the variation of conductivity and thickness into a colormap showing
the reflection and the transmission, one acquires Figure 6. The color scale is the
reflection in dB and the real part of the permittivity is 1. The region where the
reflection magnitude has a significant gradient is the most promising area to
measure upon, since one would see a distinct difference in reflection when
varying conductivity or thickness of the substrate.
Figure 6 also plots the corresponding colormap for transmission. Its gradient is
located approximately 2 orders of magnitude higher in conductivity, pointing in
favor for measuring materials with higher conductivities in transmission.
Preferable one would like to obtain both reflection and transmission data since
this maximizes the parameter range where material characterization is possible.
Figure 7 maps the reflection and transmission when sweeping over the material’s
real permittivity and conductivity. The thickness is set to 300 μm, since this was
approximately the same thickness as our silicon wafers had during measurement.
It appears that above a certain conductivity (
) the effect of the
permittivity no longer alters the reflection, but for low conductivities the
periodicity from the permittivity that was seen in Figure 5c is present. A line for
possible silicon (Si) substrate conductivities is included, showing what kind of
reflection and transmission one would expect from this substrate [13].
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Figure 6: Reflection and Transmission (respectively) vs. thickness and conductivity for a single-layer
slab. The real part of the permittivity is 1.
Figure 7: Reflection and Transmission (respectively) vs. real permittivity and conductivity for a
single-layer slab. The thickness of the slab is set to 300 μm.
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2.3.2 Multilayer structure
Figure 8 depicts a two-layer structure, representing a thin film on a thicker
substrate. The abbreviation PEDOT:PSS used in this section stands for poly(3,4ethylenedioxythiophene) polystyrene sulfonate and is a conductive polymer
which can be used for thin films.
Figure 8: An electromagnetic wave impinges upon a two-layer slab.
In these simulations the substrate is silicon (Si) and the thin film’s parameters are
swept. The silicon is modeled to have a conductivity of
and a
[ ]
real permittivity of
[10] [13]. Figure 9 maps the reflection and
transmission as the conductivity and the thickness of the thin film varies. The
change in reflection is not as profound (mind the scale of the colorbar) as in the
figures above, since the silicon substrate’s variables are constant. The
consequence of this is that the structure will always have a certain minimum
reflection, which will be much higher than in the one-layer case.
A very low transmission is observed when the thin film is highly conductive
⁄ ), for example gold (Au), titanium (Ti) and possible PEDOT:PSS thin
(
films are marked in the Figure 9 [10] [14] [15]. The gold shows a very high
reflection for most thicknesses and it’s not until very very thin (1-10 nm) films are
applied that one starts to see some measurable transmission. The titanium
should be very interesting to measure upon since it spans the gradient part of the
colormap very well. Unfortunately it seems that the PEDOT:PSS possibilities are a
bit too low in conductivities to really gain much information from such a sample.
In Figure 10 the permittivity and conductivity of the thin film are swept. The
dependence of the permittivity is not as apparent as in Figure 7. This is mostly
because the thin film is considerable thinner than the one-layer slab case.
In Figure 11 the substrate thickness vs. the thin film conductivity is swept. One
can observe that the two-layer slab will give a minimum reflection at certain
substrate thicknesses; this corresponds to the ⁄ thickness explained in (46).
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For real measurements it becomes impractical having to fit the substrate
thickness to minimize its reflection. Luckily when one does measurements, the
frequency is usually not fixed to a specific value. Therefore, as the frequency is
swept during measurements, one will hopefully cross the minimum values of
substrate reflection and be able to maximize the relative response from the thin
film sample.
Figure 9: Reflection and transmission vs. thickness and conductivity of thin film. The substrate is 300
μm and the thin film has a real permittivity of 1.
Figure 10: Reflection and transmission vs. real permittivity and conductivity of thin film. The
substrate is 300 μm and the thin film is 300 nm.
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Figure 11: Reflection and transmission vs. substrate thickness and conductivity of thin film. The thin
film is 300 nm and its real permittivity is 1.
Table 1 on the next page shows the 4 samples chosen to be measured and
simulated. The two dielectrics were made from PMMA. PMMA stands for
Poly(Methyl MethAcrylate) and is a thermoplastic polymer, also known under
one of its trade names; Plexiglas. They were chosen because reference
measurements on these samples from the institution existed. The two
conductors chosen were: a single-layer slab of silicon and a two-layer slab of thin
gold film on silicon. Table 1 also discloses all the constant material parameters
used for the simulations. Figure 12 shows a picture of the samples.
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Sample
PMMA sheet
Assumed value
[ ]
[16]
PMMA container
[ ]
Si wafer
[ ]
11.68 [13]
Gold film on Si
[ ]
1
[ ]
11.68
Table 1: Assumed values for the different samples.
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3 Methodology
CHAPTER
3
3 Methodology
The two different measurement setups are examined and how the
measurements were performed is explained. The necessary processing steps for
matching simulated to measured data is covered and the method used for
extracting parameters is shown.
3.1 Measurements
Figure 12 shows a picture of the measurement samples focused on in this report.
From left to right: A PMMA sheet, a PMMA container (small air gap between two
PMMA sheets) and wafers, such as low resistivity silicon and a thin gold film on
silicon.
Figure 12: From the left to the right: PMMA sheet, PMMA container and different wafers.
25
3 Methodology
Measurements were performed in both time- and frequency domain. A VNA
(Vector Network Analyzer, Agilent Technologies E8361A) was used for the
frequency-domain while a sampling oscilloscope (Agilent Technologies 86100D)
together with a sampling module (Agilent Technologies 86118A) and an in-house
wavelet generator was used for the time-domain. Figure 13 and Figure 14 depicts
a crude schematic for the transmission and reflection measurement setups. In
time domain measurements the signal is created by a short pulse, which is
equivalent to a broadband signal. Therefore many frequencies are present at the
same time, enabling time resolved measurements to be possible. In frequency
domain the VNA measures the reflected and transmitted scattering parameters
in a narrowband frequency signal, the frequency is then changed and the process
repeated.
Electromagnetic wave
MUT – Material Under
Test
Horn antenna
Connecting cables
VNA or Wavelet
generator + Oscilloscope
Figure 13: A schematic of transmission measurements.
26
3 Methodology
VNA or Wavelet
generator + Oscilloscope
Figure 14: A schematic of reflection measurements.
3.3.1 Frequency domain
Frequency domain measurements were done for both reflection and
transmission, using lens and horn antennas. The antennas were connected to
cables via waveguide adapters and free space measurements were performed.
The reference for the reflection setup was a copper-plate and in later
measurements a thick gold film on silicon, as this material theoretically should
have the highest possible reflection (of what can be produced in-house in our
labs), as we can see in Figure 9. One problem that later arose when examining
some data was that the reference object had not been placed at exactly the same
distance as the samples being investigated.
27
3 Methodology
A reflection measurement setup can be seen in Figure 15. In this figure the thick
gold film on silicon is seen. In these measurements the aperture for holding the
wafers consisted of a hole in an absorber sheet, whereupon the wafer was
placed.
Gold reference
Absorber
holder
Figure 15: Reflection setup from the final round of measurements. A thick gold film on an Siwafer is seen in the picture, this served as the normalization reference.
28
3 Methodology
3.3.2 Time domain
Both reflection and transmission measurements were performed in time domain
as well. A transmission measurement setup can be seen in Figure 16. The lens
antennas were used in this case.
MUT wafer
Absorber
Lens antenna
PMMA Holder
Figure 16: Transmission setup in time-domain.
In this setup another kind of holder aperture for the wafers was used. It consisted
of a PMMA sheet and an absorber with a hole. The wafer was placed on the back
of the PMMA sheet. The problem with this method is that when you remove the
wafer and place another it is hard to make certain that it arrives in the exact
same spot. It also introduces more things to simulate and measure since the
PMMA sheet has to be incorporated into the simulations.
29
3 Methodology
3.2 Signal Processing
To be able to compare the measured data to the simulated, post measurement
processing was necessary. Since no calibration kit for the waveguide adapter
existed (we could only calibrate to the cable tip), we used a method where the
data was normalized to a reference sample. For reflection this means that the
reference object would be something with a very high reflection at all
frequencies, for example a thick metal sheet. For transmission this reference
would correspond to open transmission without anything in the pathway.
After this was done, one had to remove multiple reflections from the data to get
the wanted signal. Figure 17 shows a signal processing flow-chart, representing
the necessary steps which had to be performed before the simulated data could
be compared to the measured. If the data came from time domain (with the inhouse wavelet generator) one had to transform to frequency domain before
normalization was performed. If the data was obtained in frequency domain,
normalization could be performed immediately. Therefore it does not matter
whether the data was obtained from frequency or time domain.
Figure 17: Signal processing flow-chart.
30
3 Methodology
The six steps outlined in the figure are:
1, Normalization of the sample to reference. This was used to overcome the fact
that we did not have calibration kit for the antenna. Any misalignment or angle
error in both the sample and reference would also be managed with this method.
2, Gate the frequency span of interest. In our case this corresponded to 40-60
GHz.
3, Padding. A mathematical trick used to fill the data vector with equally stepped
zeros from 40 GHz down to 0 GHz to mitigate gating artifacts.
4, Transform to time domain.
5, Gate in time-domain. This was done to remove multiple reflections from the
measured data and, in some cases, internal antenna reflections.
6, Transform to frequency domain. The processing steps completed. The data
could now be compared to simulated data, which had undergone the same
processing steps.
Figure 18 and Figure 19 shows two plots in step 5, where gating is performed in
time domain. Figure 18 is from a reflection measurement of a PMMA container
and Figure 19 from a transmission measurement of a gold film on a silicon
substrate.
31
3 Methodology
Figure 18: Time domain signal of reflection from a PMMA container, notice the
internal antenna reflection at the start of the signal, as well as the multiple
reflections after the first reflection.
Figure 19: Time domain signal of transmission from a thin gold film on silicon.
Notice the multiple reflections.
32
3 Methodology
3.3 Parameter extraction
By simulating the measurement setup as close to reality as possible, the
simulated data can be compared to the measured data and a parameter
extraction is possible.
Figure 20 shows the simulated reflection with added thermal noise of a two-layer
slab: a thin gold film on a silicon wafer. The gold layer was 5 nm thick and had a
conductivity of
[10]. The silicon substrate was 300 µm thick
[ ]
and had a permittivity of
and conductivity
[10] [13].
The complex noise added in Figure 20 is Gaussian noise, which amplitude is
calculated according to √
, where is Boltzmann’s constant, is
√
the absolute temperature in kelvins and
the bandwidth of the measurement
⁄
(in this case 10 kHz) [17]. The noise is added as
. The air
gap in the simulation is added to mirror an actual setup as much as possible.
Figure 20: Simulated reflection from a two-layer sample (thin gold
film on silicon) in dB with added thermal noise.
33
3 Methodology
This “layer” will theoretically not change the amplitude of the wave in any way (in
measurements it would however, due to spherical radiation properties of the
antennas) but will alter its phase. That’s why it’s important to measure things
exactly before executing real measurements.
The simulation with noise can then, for the sake of illustrating the method, be
imagined to represent a real life measurement,
. By comparing the
measurement to many simulations with a varying material parameter, an error
|
| . In Figure 21 such an
plot can be constructed:
√
error plot is shown when varying the thin film conductivity, assuming that all
other parameters are known. Similar plots can be made to extract the
permittivity or thickness of the material.
Figure 21: Error vs. Conductivity of the thin film. It finds the right value of
𝝈 𝟒 𝟏 𝟏𝟎𝟕 𝑺 𝒎.
In some measurements a constant phase discrepancy was apparent, making the
matching with simulations more difficult. This was induced because the distance
from the antenna to the sample and the antenna to the reference sample
differed slightly. When the normalization of the sample to the reference was
34
3 Methodology
performed it therefore introduced a constant phase error. To be able to fit
measurement and simulation in these cases, the gradient of the group delay
⁄ , where is the phase) and the
(group delay being calculated as
magnitude of the signal were compared separately:
||
|
||
|
|
|
|
||
|
To make sure that the error from the magnitude and the error from the phase
were at the same order of magnitude, a weighting factor
was added to the
phase error. The size of this weighting factor was decided manually (to
) by
looking at the magnitude error and phase error, making sure they arrived at the
same order of magnitude. A normalization vector was also applied to the phase
error. This was done because at reflection minima of the signal, the phase turned
into large spikes which made the phase error vector useless. The normalization
vector therefore consisted of disregarding the small areas of reflection minima.
Finally the phase error and magnitude error were added together to get a pooled
error and an error plot could be constructed.
One can also sweep more than one parameter at the same time, but this
increases the amount of calculations exponentially. Preferably one should have
very good guesses for most parameters and only one parameter being totally
unknown. If this is not the case the calculations may take days and it might also
find the “wrong” material parameters since if all parameters are chosen
arbitrarily more than one possible solution exists. This can be seen by examining
(44) and (45). The reflection and transmission coefficient in these two equations
depend on the refractive index of the material, which in turn depend on the
permittivity and permeability of the material (5). From (21) we know that by
changing the real part of the permittivity or conductivity we change the effective
permittivity. If one sweeps, for example, both thickness and conductivity over
large spans the result could therefore be that it finds more than one match. For
example, high conductivity thin film vs. lower conductivity thicker film.
35
4 Results and discussion
CHAPTER
4
4 Results and discussion
4.1 PMMA sheet
The PMMA sheet on the left in Figure 12 was examined first. Figure 22 shows the
fitting from reflection setups in frequency and time domain. The group-delay
method, chapter 3, was used for this matching.
Looking at the legends in Figure 22, the measurements are the data obtained
from the VNA or the sampling oscilloscope together with the in-house wavelet
generator. The simulation is the matched signal that has undergone the same
processing steps as the measured data. The ideal simulation is how the matched
signal would look like without performing any signal processing. The material
parameter fitted was the real part of the permittivity. In Figure 22 and Figure 23
one can see that the time and frequency domain signal is shifted compared to
each other. This most likely happened because of an angle difference between
the two setups (approximately 9°). The material parameter extracted for time
domain was
[
]
and for frequency domain
[
]
.
Figure 23 shows the results from transmission data in frequency and time
domain. The magnitude match does not seem to be as good in this case; but this
is most likely because the scale of the graph is different. It does, however, not
appear to have the same angle problem as for the reflection setup since it finds
almost the same value for both time and frequency domain matching. For time
domain it extrapolates
This is
[
]
and for frequency domain
lower than the expected value of
36
[
]
[18].
[
]
.
4 Results and discussion
Figure 22: Time and frequency domain measurements of reflection from the
PMMA sheet. The matched signal is shown in both ideal and processed state.
The group-delay method of finding the material parameter was used for this
case.
Figure 23: Time and frequency domain measurements of transmission from
the PMMA sheet. The matched signal is shown in both ideal and processed
state.
37
4 Results and discussion
4.2 PMMA container
The PMMA container sample can be seen in the middle of Figure 12. Figure 24
shows the results from the reflection setups in frequency and time domain. The
group-delay method was used in this matching. The simulated signals from both
time and frequency domain looks very similar and this is also confirmed since it
found very similar values for the real part of the permittivity. It also appears that
an angle difference is still present, looking at the phase of the two measurement
signals. The real part of the permittivity found was [ ]
for time domain
and
[ ]
for frequency domain. This is only
lower than the
expected value.
Figure 24: Time and frequency domain measurements of reflection from
PMMA container. The matched signal is shown in both ideal and processed
state. The group-delay method of finding the material parameter was used for
this case.
38
4 Results and discussion
Figure 25 shows the transmission data from frequency and time domain. The
angle difference is present here as well (approximately 7°), looking at the phase.
It also seems like the measured transmission is a bit lower than the simulated,
even though the imaginary part of the permittivity is borrowed from an earlier
PhD student’s work with similar PMMA sheets [16]. The found values were
[
]
for time domain and
[
]
for frequency domain. This is
off from the expected value.
Figure 25: Time and frequency domain measurements of transmission from
PMMA container. The matched signal is shown in both ideal and processed
state.
The calculated percentage value off from the expected value of the permittivity is
for those measurements only and there is no certainty that in repeating the
measurements the same error would be found reliably.
39
4 Results and discussion
4.3 Silicon wafer
The silicon wafer reflection results from frequency and time domain can be seen
in Figure 26. The parameter fitted was the resistivity and in this case we had
specifications from the manufacturer of what the resistivity of the silicon should
be:
. That’s why there are two additional lines in this plot that
correspond to the expected appearance of the signal. The time domain lines are
quite different from the frequency domain lines. This is because two different
holders for attaching the wafers were used. In time domain a PMMA sheet was
used to hold the wafer, this caused the pattern of large dips and tops that can be
seen. For frequency domain a holder made from an absorbing aperture was used.
For the time domain measurement the expected signal seems to be a better fit
than the simulation, but shifted in a similar way as the reflection case for the
PMMA sheet, Figure 22. One possible explanation for the discrepancy of the
fitted value and expected could therefore be that an angle difference was
present between the sample and the reference. For the frequency domain
measurement the expected value does not seem to fit since it has a higher
Figure 26: Time and frequency domain measurements of reflection from Siwafer. The matched signal is shown in both ideal and processed state. The
group-delay method of finding the material parameter was used.
40
4 Results and discussion
reflection than the measured. The resistivity fitted was
for time
domain and
for frequency domain, about 2 orders of magnitude
off from the expected value.
Figure 27 shows the transmission result for the frequency and time domain. For
the time domain it seems the expected value is a better fit once again, if
accounted for an angle. The difference in frequency domain is however very
small. The extrapolated values from time and frequency domain were still off by
2 orders of magnitude;
frequency domain.
for time domain and
Figure 27: Time and frequency domain measurements of transmission from Siwafer. The matched signal is shown in both ideal and processed state.
41
for
4 Results and discussion
4.4 Thin gold film on silicon
The final sample was a thin gold film on a silicon substrate. Figure 28 shows the
reflection results from frequency domain. Only frequency domain measurements
were performed. The conductivity of the thin film was the material parameter
fitted. As one can see from the figure, a very high reflection from the gold surface
is present. The simulated fit found a value,
, approximately half
of what it should be according to the expected value for bulk gold:
[10]. This can be explained by the fact that thin sputtered films can
have lower conductivities than in bulk form.
Figure 29 shows the transmission results from frequency domain. The phase
match was originally not as good as shown here, what is shown here is instead
what it would look like with an angle of 18° degrees away from normal incidence.
The path traveled in the sample is therefore 20 µm longer than it should be. The
value extrapolated,
reflection case.
, is on the same order of magnitude as in the
Figure 28: Frequency domain measurements of reflection from thin gold film on
silicon substrate. The matched signal is shown in both ideal and processed state.
42
4 Results and discussion
Figure 29: Frequency domain measurements of transmission from Si-wafer. The
matched signal is shown in both ideal and processed state.
Table 2 on the next page summarizes the extrapolated and expected values for
the different samples and the two measurement setups. In the appendix the
source code for the design plots (section 2.3) can be seen in their generalized
form.
43
4 Results and discussion
Sample
PMMA sheet
Reflection
Transmission
Expected
PMMA container
Reflection
Transmission
Expected
Time domain
[ ]
[ ]
[ ]
[18]
[ ]
[ ]
[ ]
Frequency domain
[ ]
[ ]
[ ]
[ ]
[ ]
[ ]
Si wafer
Reflection
Transmission
Expected
Gold film on Si
Reflection
Transmission
Expected (Bulk)
-
Table 2: Fitted values from time and frequency domain, for both reflection and
transmission measurements. Expected values for the wafers are included.
44
[10]
5 Conclusions
CHAPTER
5
5 Conclusions
5.1 Discussion
The thesis set out to characterize material parameters through reflection and
transmission. The materials investigated thoroughly were: a one-layer PMMA
sheet, a PMMA container, a silicon wafer and a thin gold film.
The material characterization was performed in two steps. First, the material to
be measured upon was chosen and all relevant data, such as thickness of sample,
distance from sample to antenna, reference sample etc. that could be obtained
was collected. The second step consisted of simulating the measurement with
one free parameter. These simulations were compared to measurements and
material parameters, for example conductivity or permittivity, were extracted.
From the simulation pre-study section it became apparent that certain materials
are easier to measure and extract material parameters from in either reflection
or transmission. For example; the silicon in Figure 7 would be better to measure
in transmission since there is a bigger difference in received magnitude over
varying conductivities. In reflection it would be harder to discern a low
conductivity from high conductivity silicon.
In a few cases there existed an angle difference between the measurements from
time domain compared to frequency domain. This was the cause of the bad
overlap between the time and frequency domain for the magnitude of the
reflection in Figure 22 and the phase of the transmission in Figure 24. Special care
therefore needs to be taken about positioning and angle accuracy of the sample.
For the silicon-wafer in Figure 26 and Figure 27 the extrapolated value was quite
off compared to the expected, see Table 2. This has to do with the fact that
silicon gives approximately the same reflection or transmission over a certain
conductivity span, see Figure 7. The expected resistivity value of
45
5 Conclusions
corresponds to a conductivity of
conductivity range.
which is within this
In the end it does not matter whether you perform the material characterization
by using measurements from time domain or frequency domain, since once the
data has been gathered it’s just a matter of Fourier transform between them.
Overall the method of material characterization by free space measurements
seems to be a very promising area. Its possible applications within industry and
the scientific community are wide and there is still more to explore. It is a young
and hot topic within academics and some companies are already working on
complete solutions, trying to turn them into profitable products.
5.2 Future work
The predicament with the sample not being perfectly perpendicular to the
incoming electromagnetic wave is something that could be integrated into the
fitting procedure. By sweeping the angle simultaneously as the material
parameter of interest, a better match is achieved if an angle is present. This
would however require a complete revision of the theory used since oblique
incidences alter the reflection and transmission from an interface depending on
the value of the angle. This kind of multi-variable optimization is the next logical
step in further works. In this thesis only one parameter at a time was extracted. If
one would instead sweep two or more variables, for example thickness,
permittivity and angle, the fit might become even better. However, this poses the
problem of exponentially increasing the computing time required. As the amount
of variables increase, the number of iterations the simulation has to run through
increases exponentially. This creates an upper limit on how many variables can
be swept at the same time. The best way to combat this would be to keep the
sweeping span very narrow for most parameters. Another difficulty of having
many parameters is that it creates a global optimization problem; the program
might find a very good fit but the correct one is outside the parameter sweep
span.
Further materials that could be interesting to measure and apply the method on
are solar cells and structured materials.
46
6 Acknowledgements
CHAPTER
6
6 Acknowledgements
I would like to thank my supervisors Professor Lars-Erik Wernersson and PhD
student Sebastian Heunisch for providing help and guidance during the master’s
thesis. I would also like to thank Dr. Lars Ohlsson for letting me use his signal
processing scripts and for proofreading/commenting on the master’s thesis work.
47
7 References
CHAPTER
7
7 References
[1] Tzu-Yang Yu, Jose Alberto Ortega Oral Büyüköztürk, "A methodology for
determining complex permittivity of construction materials based on
transmission-only coherent, wide-bandwidth free-space measurements.,"
Cement & Concrete Composites, ScienceDirect, p. 11, 2006.
[2] Jerzy Krupka, "Frequency domain complex permittivity measurements at
microwave frequencies," IOPScience, 2006.
[3] M.V.Jacob, J.Krupka C.D.Easton, "Non-destructive complex permittivity
measurement of low permittivity thin film materials," IOPScience,
Measurement Science and Technology, 2007.
[4] Sophocles J. Orfanidis, Electromagnetic Waves and Antennas.: Rutgers
University, 2014.
[5] Martin Dressel and George Gruner, Electrodynamics of Solids, Optical
properties of Electrons in Matter. Cambridge: Cambridge University Press,
2003.
[6] Chris Bishop, "The Relationship Between Loss, Conductivity, and Dielectric
Constant," 2001.
[7] J. G. Chafee, "Method and apparatus for heating dielectric materials,"
2,147,689.
[8] Loretta Jones Peter Atkins, Chemical Principles; The quest for insight, 5th ed.
New York, United States of America: W. H. Freeman and Company, 2010.
[9] Göran Jönsson, Våglära och Optik. Lund, Sweden: Media-Tryck, 2010, pp. 12-
48
7 References
16.
[10] Raymond A. Serway, Principles of Physics (2nd ed.). Fort Worth, Texas, 1998,
p. 602.
[11] Mats Gustafsson Daniel Sjöberg, Kretsteori, ellära och elektronik. Lund,
Sverige: KFS AB, 2013.
[12] Crish Bishop, "The Relationship Between Loss, Conductivity, and Dielectric
Constant," 2001.
[13] Virginia Semiconductor, "The General Properties of Si, Ge, SiGe, SiO2 and
Si3N4," Virginia Semiconductor , Fredericksburg, 2002.
[14] Sean C. Andrews, Steve Park, Julia Reinspach, Nan Liu, Michael F. Toney,
Stefan C. B. Mannsfeld and Zhenan Bao Brian J. Worfolk, "Ultrahigh electrical
conductivity in solution-sheared polymeric transparent films," Proceedings of
the National Academy of Sciences of the United States of America, no. 112,
2015.
[15] Liangqi. Musumeci, Chiara. Jafari, Mohammad Javad. Ederth, Thomas.
Inganäs, Olle Ouyang, "Imaging the Phase Separation Between PEDOT and
Polyelectrolytes During Processing of Highly Conductive PEDOT:PSS Films,"
ACS Applied Materials and Interfaces, 2015.
[16] Lars Ohlsson, Lars-Erik Wernersson and Mats Gustafsson Iman Vakili, "TimeDomain System for Millimetre-Wave Material Characterization," Department
of Electrical and Information Technology, Faculty of Engineering, LTH, Lund,
2015.
[17] Paul H. Young, Electronic Communication Techniques, fifth edition. Ohio:
Pearson, 2004.
[18] W. Culshaw and M. Anderson, "Measurement of permittivity and dielectric
loss with a millimetre-wave Fabry-Perot interferometer," IEE Proc. Electron.
Commun. Eng. pt. B, vol. 109, no. 23, pp. 820-826, 1962.
49
7 References
[19] Milo W. Hyde IV and Michael J. Havrilla, "Measurement of Complex
Permittivity and Permeability Using Two Flanged Rectangular Waveguides,"
Air Force Institute of Technology, 2007.
50
8 Appendices
CHAPTER
8
8 Appendices
8.1 Source code
Design plots – one layer
clc
clear all
c0 = 3e8;
e0 = 8.854*10^-12;
f1 = 60*10^9;
lambda1 = c0./f1;
omega = 2*pi.*f1;
k = 2*pi./lambda1;
d = 300e-6;
% Conductivity vs permittivity
e1 = linspace(1.01,80,1000);
cond = logspace(-2,8,1000);
e2 = cond./(omega*e0);
[E1, C] = meshgrid(e1, cond);
[Gamma1, Tau1] = mesh_one_layer(d,k,e1,e2,cond);
figure(1);
FigHandle = figure(1);
set(FigHandle, 'Position', [50, 100, 1350, 470]);
subplot(1,2,1)
mesh(log10(C),E1, 20*log10(abs(Gamma1)));
ylabel('Re[\epsilon]', 'FontSize', 14); xlabel('logarithmic
cond (log(\sigma/(S/m)))', 'FontSize', 14);
zlabel('Reflectivity \Gamma (dB)', 'FontSize', 16)
title('Reflection', 'FontSize', 14);
colorbar; view([0 90]); caxis([-50 0]);
hold on
axis([min(log10(cond)) max(log10(cond)) e1(1) e1(end)])
subplot(1,2,2)
limit = 1e-10;
Tau1(abs(Tau1)<limit)=limit;
mesh(log10(C),E1, 20*log10(abs(Tau1)));
ylabel('Re[\epsilon]', 'FontSize', 14); xlabel('logarithmic
cond (log(\sigma/(S/m)))', 'FontSize', 14);
zlabel('Transmission \tau (dB)')
title('Transmission', 'FontSize', 14);
51
8 Appendices
colorbar; view([0 90]); caxis([-50 0]);
axis([min(log10(cond)) max(log10(cond)) e1(1) e1(end)])
% Conductivity vs depth
e1 = linspace(1,1,1000);
d2 = logspace(-9,-3,1000);
[D2, C] = meshgrid(d2, cond);
[Gamma2, Tau2] = mesh_one_layer(D2,k,e1,e2,cond);
figure(2)
FigHandle = figure(2);
set(FigHandle, 'Position', [50, 100, 1350, 470]);
subplot(1,2,1)
mesh(log10(C),log10(D2),20*log10(abs(Gamma2)));
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('logarithmic Thickness (log(m/m))', 'FontSize',
14)
title('Reflection','FontSize', 14);
colorbar; view([0 90]); caxis([-50 0]);
subplot(1,2,2)
limit = 1e-10;
Tau2(abs(Tau2)<limit)=limit;
mesh(log10(C),log10(D2),20*log10(abs(Tau2)));
xlabel('cond (log(\sigma/(S/m)))', 'FontSize', 14);
ylabel('Thickness (log(m/m))', 'FontSize', 14)
title('Transmission','FontSize', 14);
colorbar; view([0 90]); caxis([-50 0]);
function [gamma, tau] = mesh_one_layer(d,k, e1, e2, cond)
n0 = ones(1000,1000);
[E11, E22] = meshgrid(e1, e2);
N = sqrt(E11 - 1i*E22);
p1 = (n0 - N)./(n0 + N);
p2 = (N - n0)./(N + n0);
t1 = 1 + p1;
t2 = 1 + p2;
z1 = exp(-2*1i*(N.*k).*d);
z2 = exp(-1i*(N.*k).*d);
tau = (t1.*t2.*z2)./(1 + p1.*p2.*z1);
gamma = (p1 + p2.*z1)./(1 + p1.*p2.*z1);
end
52
8 Appendices
Design plots – two layer
clc
clear all
c0 = 3*10^8;
e0 = 8.854*10^-12;
f1 = 60*10^9;
lambda1 = c0./f1;
omega = 2*pi.*f1;
k = 2*pi./lambda1;
d = 300e-9;
d_Si = 300e-6;
e1 = linspace(1.01,80,1000);
cond = logspace(-2,8,1000);
e2 = cond./(omega*e0);
e_Si = 11.86 - 1i*((1/(6.4*10^2)))./(omega*e0);
% Meshplot
% Real permittivity vs real conductivity
[E1, C] = meshgrid(e1, cond);
[Gamma1, Tau1] = mesh_two_layers(d,d_Si,k,e1,e2,e_Si);
figure(1);
FigHandle = figure(1);
set(FigHandle, 'Position', [50, 100, 1350, 470]);
subplot(1,2,1)
mesh(log10(C),E1,20*log10(abs(Gamma1)));
axis([log10(min(cond)) log10(max(cond)) min(e1) max(e1)])
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('Re[\epsilon]', 'FontSize', 14)
title('Reflection','FontSize', 14)
colorbar; view([0 90]);
subplot(1,2,2)
mesh(log10(C),E1,20*log10(abs(Tau1)));
axis([log10(min(cond)) log10(max(cond)) min(e1) max(e1)])
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('Re[\epsilon]', 'FontSize', 14)
title('Transmission','FontSize', 14)
caxis([-50 0]); colorbar; view([0 90]);
e1 = linspace(10,10,1000);
d2 = logspace(-9,-6,1000);
[D2, C] = meshgrid(d2, cond);
[Gamma2, Tau2] = mesh_two_layers(D2,d_Si,k,e1,e2,e_Si);
figure(2)
FigHandle = figure(2);
set(FigHandle, 'Position', [50, 100, 1350, 470]);
subplot(1,2,1)
mesh(log10(C),log10(D2),20*log10(abs(Gamma2)));
53
8 Appendices
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('logarithmic Thin-film thickness (log(m/m))',
'FontSize', 14)
title('Reflection','FontSize', 14)
view([0 90]);
colorbar;
subplot(1,2,2)
mesh(log10(C),log10(D2),20*log10(abs(Tau2)));
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('logarithmic Thin-film thickness (log(m/m))',
'FontSize', 14)
title('Transmission','FontSize', 14)
view([0 90]); caxis([-50 0]); colorbar;
% Real part vs depth (substrate)
d3 = logspace(-7,-3,1000);
[D3, cond] = meshgrid(d3, cond);
[Gamma3, Tau3] = mesh_two_layers(d,D3,k,e1,e2,e_Si);
figure(3)
FigHandle = figure(3);
set(FigHandle, 'Position', [50, 100, 1350, 470]);
subplot(1,2,1)
mesh(log10(C),D3./(10^-6),20*log10(abs(Gamma3)));
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('Substrate depth \mum', 'FontSize', 14)
title('Reflection','FontSize', 14)
caxis([-50 0]); view([0 90]); colorbar;
subplot(1,2,2)
mesh(log10(C),D3./(10^-6),20*log10(abs(Tau3)));
xlabel('logarithmic cond (log(\sigma/(S/m)))', 'FontSize',
14); ylabel('Substrate depth \mum', 'FontSize', 14)
title('Transmission','FontSize', 14)
caxis([-50 0]); view([0 90]); colorbar;
function [gamma, tau] =
mesh_two_layers(d_film,d_sub,k,e1,e2,e_sub)
[E11, E22] = meshgrid(e1, e2);
N0 = ones(1000,1000);
N1 = sqrt(E11 - 1i*E22);
N2 = sqrt(e_sub);
N2 = ones(1000,1000).*N2;
p1 = (N0 - N1)./(N0 + N1);
p2 = (N1 - N2)./(N1 + N2);
p3 = (N2 - N0)./(N0 + N2);
t1 = 1 + p1;
t2 = 1 + p2;
t3 = 1 + p3;
54
8 Appendices
z1p = exp(1i*(N1.*k).*d_film);
z2p = exp(1i*(N2.*k).*d_sub);
z1n = exp(-1i*(N1.*k).*d_film);
z2n = exp(-1i*(N2.*k).*d_sub);
z1 = exp(-2*1i*(N1.*k).*d_film);
z2 = exp(-2*1i*(N2.*k).*d_sub);
gamma = (p1 + p2.*z1 + p1.*p2.*p3.*z2 + p3.*z1.*z2)./(1 +
p1.*p2.*z1 + p2.*p3.*z2 + p1.*p3.*z1.*z2);
tau = (t1.*t2.*t3)./(z1p.*z2p + p1.*p2.*z1n.*z2p +
p2.*p3.*z1p.*z2n + p1.*p3.*z1n.*z2n);
end
55
Series of Master’s theses
Department of Electrical and Information Technology
LU/LTH-EIT 2016-488
http://www.eit.lth.se
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