MScThesis_JSap_20-05

MScThesis_JSap_20-05
Modelling optical properties of layers
for thin-film silicon solar cells
MSc. Thesis
by
Jeroen Alexander Sap
Born on the 6th of September 1985 in Rotterdam
Supervisors:
Prof. Dr. M. Zeman
O. Isabella, MSc.
K. Jäger, MSc.
Photovoltaic Materials and Devices
Delft University of Technology
May 20th 2010
Master of Science Thesis, 2010
.
Picture on coverpage is the
Robert O. Schulz Solar Farm
built by Conergy Americas.
http://www.conergy.us/
Preface
This thesis report is written to conclude the master phase of my studies at
Delft University of Technology. The project is carried out in the Photovoltaic Materials and Devices group that is part of the Electric Sustainable
Energy department at the faculty of Electrical Engineering, Mathematics
and Computer Sciences. The project deals with the characterization of
different materials using optical measurements of reflectance and transmittance. The background of this project is thin-film silicon solar cell research
such that only materials that find application in these cells are characterized
in this thesis. In this project the measured spectra are also simulated with
a software package and a fitting procedure of the simulated spectra on the
measurements yields valuable information.
The thesis report is divided in 6 chapters. The first chapter is the introduction. After that a thorough overview is given of the models and
theory that is applied in the experimental part of this work. In chapter 3
the equipment is briefly introduced. For readers that are merely interested
in the results these chapters can be skipped and in chapter 4 the detailed
results can be found of the experimental work for different materials. The
conclusions and recommendations are given in chapters 5 and 6 respectively.
This report concludes my master graduation project and is the crown on
seven years of hard work at Delft University of Technology. Looking back
at all the ups and downs I start to realize that there are many people who
have helped and supported me to make it to this point. Not the least of
which are my parents who gave me full support and confidence for which I’d
like to thank them.
Furthermore I would like to thank prof. dr. Miro Zeman for inspiring me
to specialize on solar cells and for giving me the opportunity to be part of the
Photovoltaic Materials and Devices group. In this group I did an internship
and consequently the graduation project that resulted in this final thesis. I
would also like to thank him for giving me the opportunity to present my
results to the public on the SAFE workshop in Veldhoven. It was a good
experience and interesting way to meet other researchers in the field.
Thanks also to my two daily supervisors, Olindo Isabella, Msc. and
Klaus Jäger, MSc. who helped me in this project by supplying everything
necessary to complete my work. Besides the discussions they also helped
i
ii
PREFACE
me to use all the equipment and did a good job in tracking my progress and
reviewing of all the deliverables for which I owe them special thanks.
Furthermore I’d like to thank the people who did the depositions of all
the required samples. Those are Olindo Isabella and Benjamin Bolman for
the AZO and ITO depositions, Martijn Tijssen and Stefaan Heirman for the
a-Si:H and µc-Si:H depositions and Rudi Santbergen and Tristan Temple for
the nanoparticle depositions. Thanks also goes out to Jan Gilot from the
Technical University of Eindhoven for supplying the polymer samples and
Ruben Abellon for assisting me in the etching of FTO samples.
Special thanks goes out to dr. René van Swaaij and dr. ir. Tom Savenije
for reviewing my thesis and accepting to be in the examination board. Furthermore I’d like to thank all the people of the PVMD group with whom it
was a pleasure to cooperate. In particular dr.ir Rudi Santbergen for working together on the section of silver nanoparticles and last but certainly not
least my office mates, Benjamin, Chare and Gerald with whom I shared the
experience and had a great time.
Delft, May 20th 2010
Jeroen Sap
Abstract
With the growing concern about climate change and depleting fossil fuels,
the need for sustainable energy alternatives is high. Solar energy is a promising alternative because it directly converts sunlight into electricity without
moving parts and noise. In the PhotoVoltaic Materials and Devices group
at Delft University of Technology research concentrates on thin-film silicon
solar cells. This type of cell has no toxic components and can be deposited
al lower temperatures compared to its crystalline counterpart.
In the research on solar cells, accurate characterization techniques are of
great importance. In this work a specific technique is studied that makes use
of transmitted and reflected light from a thin layer. Some material properties
can be extracted by fitting a mathematical model on these measured spectra.
This model consists of sub-models that describe the physical properties of
the layer such as the bandgap of the material or the free carrier absorption. A
close fit of the model on the measurements then reveals all these parameters.
The modelling is done with the aid of a software package called SCOUT. In
this software all the sub-models are available as ‘building blocks’ and one
can compose the right interface for a certain material.
In this work an interface is created for different materials which is capable
of simultaneously fitting 17 spectra. These spectra are obtained at different angles of incidence and polarizations. The materials are Transparent
Conductive Oxides (AZO, ITO and FTO), amorphous and microcrystalline
silicon, layers of silver nanoparticles and polymer layers. These materials all
find application in solar cell devices. An interface is also created for rough
TCO layers. This required the implementation of a model in SCOUT that
includes light scattering behaviour.
This method of characterization turns out to be a highly accurate way to
obtain material properties. The fitting results of the model on the measured
spectra are accurate for all studied materials and an error analysis shows that
a unique solution is found for all the parameters. The obtained properties
are comparable to values found in literature and results obtained with state
of the art characterization techniques.
The creation of a specific interface in SCOUT for each material has
provided the PVMD group with a powerful tool for optical characterization
on which further research on material optimization can be based.
iii
iv
ABSTRACT
Contents
Preface
i
Abstract
1 Introduction
1.1 Background and motivation
1.2 History of solar cells . . . .
1.3 Thin-film silicon solar cells .
1.4 Polymer solar cells . . . . .
1.5 Outline of the thesis . . . .
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2 Theoretical background
2.1 Optics . . . . . . . . . . . . . . . .
2.1.1 Electromagnetic radiation .
2.1.2 Refraction . . . . . . . . . .
2.1.3 Interference . . . . . . . . .
2.2 Modelling of dielectric functions . .
2.2.1 Simulation of R/T spectra
2.2.2 Dielectric models . . . . . .
2.2.3 Effective medium theory . .
2.2.4 SCOUT software package .
2.3 Scattering from rough surfaces . .
2.3.1 Wave equations . . . . . . .
2.3.2 Scalar wave equation . . . .
2.3.3 Specular component of R/T
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rough surfaces
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3 Equipment
3.1 Deposition . . . . . . . . . . . . . . . . . .
3.1.1 Magnetron sputtering . . . . . . .
3.1.2 Chemical Vapor Deposition (CVD)
3.1.3 Thermal evaporation . . . . . . . .
3.2 Characterization . . . . . . . . . . . . . .
3.2.1 Automated R/T Analyser (ARTA)
3.2.2 Total integrating sphere . . . . . .
3.2.3 Mini-RT Setup . . . . . . . . . . .
3.2.4 Atomic Force Microscopy (AFM) .
3.2.5 Hall Setup . . . . . . . . . . . . . .
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vi
CONTENTS
3.2.6
Spectroscopic ellipsometry . . . . . . . . . . . . . . . .
4 Modelling results
4.1 Glass substrates . . . . . . . . . . . . . . .
4.2 Transparent Conductive Oxides (TCOs) . .
4.2.1 Modelling . . . . . . . . . . . . . . .
4.2.2 Aluminium-doped Zinc Oxide (AZO)
4.2.3 Tin-doped Indium Oxide (ITO) . . .
4.2.4 Fluorine-doped Tin Oxide (FTO) . .
4.2.5 Error Analysis . . . . . . . . . . . .
4.2.6 Verification . . . . . . . . . . . . . .
4.3 Silicon Layers . . . . . . . . . . . . . . . . .
4.3.1 Modelling . . . . . . . . . . . . . . .
4.3.2 Amorphous silicon (a-Si:H) . . . . .
4.3.3 Microcrystalline silicon (µc-Si:H) . .
4.3.4 Conclusion . . . . . . . . . . . . . .
4.4 Silver nanoparticles . . . . . . . . . . . . . .
4.4.1 Modelling . . . . . . . . . . . . . . .
4.4.2 Layer deposition . . . . . . . . . . .
4.4.3 Annealing . . . . . . . . . . . . . . .
4.4.4 Particle size series . . . . . . . . . .
4.4.5 Conclusion . . . . . . . . . . . . . .
4.5 Polymer samples . . . . . . . . . . . . . . .
4.5.1 Modelling . . . . . . . . . . . . . . .
4.5.2 Fitting . . . . . . . . . . . . . . . . .
4.5.3 Conclusion . . . . . . . . . . . . . .
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5 Conclusions
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6 Recommendations
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A Thickness from interference pattern
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B Derivation of the wave equations
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C Mini-RT models for layer thickness
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D SCOUT interfaces
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Bibliography
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List of Figures
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Nomenclature
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Index
112
Chapter 1
Introduction
1.1
Background and motivation
The growing world population and related energy consumption pose a threat
to the current standard of living. The world energy consumption will be
growing significantly in the coming decades according to predictions made
by the World Energy Council (WEC), shown in figure 1.1. Three different scenarios were evaluated where A is the most pessimistic scenario with
high economic growth rates and C is the ecologically driven and optimistic
counterpart.
Figure 1.1: Predictions of world energy consumption according to different
scenarios. [60]
Satisfying this rapid growth in energy consumption with fossil fuels can
have a significant impact on the climate. The CO2 concentration in the atmosphere is due to the industrial revolution already higher than ever before
as illustrated in figure 1.2.
1
2
CHAPTER 1. INTRODUCTION
Figure 1.2: Concentration of greenhouse gasses over the last 2000 years. [39]
Although there is a debate whether greenhouse gasses really cause a climate change one cannot deny the fact that fossil fuels are rapidly depleting
with the current consumption rates. When taking the growth in energy consumption into account the estimated oil and gas reserves last only for another
50 years [65]. Needless to say that there is a strong demand for renewable
alternatives including: wind, solar, biomass, hydropower and geothermal
energy. This thesis report focusses on solar energy.
Solar cells convert sunlight directly into electricity. This photovoltaic
(PV) effect requires no moving parts which makes the solar cells an elegant and noise free alternative for energy production. Furthermore there
are no more greenhouse gas emissions after installing the system. In the
Photovoltaic Materials and Devices (PVMD) group at Delft University of
Technology research concentrates on thin-film silicon solar cells. The advantage of this type of cells is that the layers can be deposited at a much
lower temperature which improves the energy efficiency for production and
allows roll to roll processing on flexible substrates. Besides that, silicon is
an abundant and non-toxic material which makes it perfectly suitable from
a practical and environmental point of view.
The main drawback of thin-film silicon solar cell technology compared to
the more common crystalline cells is that the efficiency is lower. Research is
done at the PVMD group to find ways to improve the efficiency of thin-film
silicon solar cells by means of material improvement, light management and
novel absorber materials such as nanoparticles and photonic crystals. Over
the entire range of research there is the need for accurate characterization
methods. This thesis aims at the accurate determination of material properties by fitting simulations on optical measurements and thereby to assist
in the research on thin-film silicon solar cell technology.
1.2. HISTORY OF SOLAR CELLS
1.2
3
History of solar cells
The photovoltaic effect was discovered by the French physicist Becquerel in
1839 who observed that a voltage developed when light fell upon a solid
electrode. It took almost half a century for the first solar cell to be built
by Charles Fritts around 1883 [29]. Fritts coated a selenium semiconductor
with a thin layer of gold to form junctions and obtained an efficiency of 1%.
The more commonly known silicon solar cells originated in the 1950ies when
Bell Laboratories experimented with silicon and observed that doped silicon
was very sensitive to light. From that moment on the silicon solar cells were
developed further leading to current efficiencies exceeding 25% [33].
The main drawback for successful commercialisation of these crystalline
silicon (c-Si) solar cells are the high production costs and hence the large
initial investment that is required for installing such a system. That is why
a new line of research started that investigates the possibilities of amorphous
silicon (a-Si). The first experimental a-Si solar cell was reported by Carlson
and Wronski in 1976. This cell had a conversion efficiency of 2.4% in AM-1
sunlight (i.e. zenith angle of 0◦ ) [2]. The main advantage of using amorphous
silicon are the reduced production costs. However the efficiency is up to this
day significantly lower than c-Si counterparts. Solar devices formed by thin
films of absorber materials, such as a-Si, cadmium-telluride (CdTe) or other
quaternary compounds (CIGS) are usually known as second generation solar
cells [75, 10, 42]. Organic and dye-synthesized solar cells are also part of this
family [32, 20]. Another line of research referred to as the third generation
solar cells aims at combining low production costs with higher efficieny by
means of advanced concepts such as tandem cells or quantum dots. Figure
1.3 gives an overview of the three generations of solar cells with regard to
efficiency and costs.
Figure 1.3: Three generations in solar cell research. [21]
4
1.3
CHAPTER 1. INTRODUCTION
Thin-film silicon solar cells
Silicon Thin-film silicon solar cells can be based on amorphous (a-Si:H)
or microcrystalline silicon (c-Si:H). In a-Si:H the atoms have a less ordered
structure compared to crystalline silicon, as shown in figure 1.4. Because
of this less structured allocation it occurs that some silicon atoms are not
able to form four covalent bonds with neighboring atoms causing dangling
bonds as indicated in figure 1.4. Because these loose bonds are a trap for free
carriers, they are passivated with hydrogen. This hydrogenated amorphous
silicon is therefore abbreviated as a-Si:H.
Figure 1.4: Structure of amorphous silicon. [28]
In defect free crystalline silicon the valence and conduction bands are
separated by a clear energy bandgap as shown in figure 1.5(a). In a-Si:H
there is an amount of disorder in the material causing the energy states
to spread into the bandgap [see figure 1.5(b)]. There is however a big advantage related to the disorder in the system being that the absorption in
the visible part of the spectrum is significantly larger for a-Si:H compared
to crystalline silicon (see figure 1.6). This improved absorption also implies
that the thickness of the layer can be further reduced leading to less material
usage and lower production costs.
Figure 1.5: Bandgap of (a) crystalline silicon and (b) amorphous silicon.
[79]
1.3. THIN-FILM SILICON SOLAR CELLS
5
Figure 1.6: Absorption coefficient of crystalline and amorphous silicon. [79]
The structure of an a-Si:H solar cell is shown in figure 1.7(a). The
cell consists of thin p and n-type layers and a relatively thick undoped or
intrinsic film, forming a so-called p-i-n junction. This silicon structure is
covered by a TCO layer at the front of the cell to function as front electrode
and a TCO layer at the back. The back contact is an aluminium layer and
is sometimes replaced by a silver layer to prevent high absorption losses
in the aluminium and to enhance the reflectance at the back side. Unlike
crystalline silicon solar cells an a-Si:H cell is a drift device because diffusion
is not an option due to the large amount of defects in the material. An
electric field occurs in the intrinsic layer, as shown in the band diagram of
figure 1.7(b), which separates the generated charges.
Figure 1.7: (a) Structure and (b) band diagram of an a-Si:H solar cell. [79]
6
CHAPTER 1. INTRODUCTION
Transparent Conductive Oxides (TCO) A film of transparent Conductive Oxide is used in a variety of applications among which thin-film solar
cells. The main function of this layer is to effectively conduct the charges to
the external circuit while maintaining high transparency and low absorption
losses. For this purpose different materials are proposed being: aluminiumdoped zinc oxide (ZnO:Al or AZO) , tin-doped indium oxide (In2 O3 :SnO2
or ITO) and fluorine-doped tin oxide (SnO2 :F or FTO) . The layers are
doped to increase the conductivity. However an increased amount of impurities leads also to higher absorption losses. A similar trade-off between
resistance and transmittance is found when it comes to the thickness of the
TCO layer. Optimizing a TCO layer is therefore a delicate job and accurate
characterization is a crucial step in the process of optimization. The AZO
and ITO layers used in this work are deposited in the PVMD laboratory at
Delft University of Technology while the FTO is obtained from Asahi Glass
company [4].
TCOs are also used to scatter the incoming light. This scattering of
incoming solar radiation is achieved by making the surface of the TCO
layer rough. Various techniques are used for creating textures on TCO
surfaces suitable for light scattering: variation of deposition parameters [46],
wet-etching in chemical baths and plasma etching. Figure 1.8(a) shows a
typical surface roughness of AZO after wet-chemical etching in hydrochloric
acid (HCl). The effect of surface roughness is illustrated in figure 1.8(b)
where the transmittance of the incident radiation is shown. The transmitted
radiation consist of a specular component and diffuse light. The diffusely
scattered light has a prolonged optical path through the cell leading to
a higher probability of absorption and carrier generation. The target for
research on TCOs is therefore to scatter light into large angles while still
maintaining high transparency and conductivity.
Figure 1.8: (a) Typical surface texture of AZO after etching and (b) effect
of surface roughness on light transmission.
Silver nanoparticles The application of silver nanoparticles in a solar cell device is a novel concept to enhance the light absorption. Figure 1.9 schematically shows how the incident radiation is scattered by the
1.3. THIN-FILM SILICON SOLAR CELLS
7
metal nanoparticles and is contained within the semiconductor layer. When
nanoparticles are placed at the interface of two media, the light will scatter
preferentially in the material with the highest permittivity [5].
Figure 1.9: Scattering from nanoparticles. [5]
The size and geometry of the particles have a strong influence on the
coupling of the light into the solar cell. Catchpole and Polman [19] found
that smaller particles with a dipole moment closer to the semiconductor layer
couple a larger fraction of the light into the semiconductor. This is shown in
figure 1.10(a). The coupling is 96% in case of a point dipole at 10 nm from
the layer, which demonstrates the potential of this scattering technique.
Figure 1.10(b) illustrates the maximum path-length enhancement for the
same particles geometries at a wavelength of 800 nm.
Figure 1.10: Effect of nanoparticle geometry on (a) light coupling into the
solar cell and (b) optical path enhancement. [19]
At the PVMD group research also focusses on optimizing the plasmonic
effect of silver nanoparticles in thin film silicon solar cells. For this purpose
the optical properties of the layers are of importance. The possibilities of
modelling these optical properties are therefore also studied in this work.
8
1.4
CHAPTER 1. INTRODUCTION
Polymer solar cells
Another line of research focusses on potentially less expensive types of solar
cells based on polymers or plastics. The cost reduction is mainly due to the
lower processing costs. The operation of a polymer solar cell is different from
the silicon solar cells described above. In polymer solar cells the radiation
does not directly create free charge carrier but excitons, i.e. tightly bound
and neutral electron-hole pairs. For charge separation an acceptor polymer
is used with a higher electron affinity. Because of this higher electron affinity,
the electron moves into the acceptor polymer once the exciton reaches the
interface of the two polymers. The separated holes and electrons then diffuse
to the electrodes. Because of the necessity of interfaces for charge separation,
these cells are often made out of blends of the two polymers as illustrated in
figure 1.11(a). In this way there is a large contact surface area between the
polymers. One drawback of this method is that small isolated island may
form which have no connection to the electrodes. This reduces the efficiency.
Figure 1.11(b) shows the band diagram of the polymers and how the charges
are separated.
Figure 1.11: (a) Layer stack of a polymer solar cell and (b) band diagram.
[79]
In this work polymer samples are obtained from the Technical University
of Eindhoven where the focus was on PCPDTBT as donor and PCBM as
acceptor polymer. The group at Eindhoven had problems in determining the
refractive index of these samples and therefore a study is also done in this
work on the optical characterization of polymer samples through modelling.
1.5
Outline of the thesis
The main objective of this thesis is to provide an accurate method to optically characterize different materials that are used in thin-film silicon solar
cells. This is done by simulating reflectance and transmittance spectra that
are fitted on spectroscopic measurements. A close fit on the measurements
reveals the material properties that are afterwards analysed and verified
1.5. OUTLINE OF THE THESIS
9
with literature and state of the art measurement techniques. The main focus is on TCO layers. Models are created for AZO, ITO and FTO layers
taking also into account an eventual roughness of the surface. After that
also the main absorber layer of the solar cells, silicon, is characterized using
a similar approach. Both amorphous and microcrystalline silicon layers are
analysed. Consequently the characterization of silver nanoparticle layers is
studied. The last topic is about polymer layers for organic solar cells.
10
CHAPTER 1. INTRODUCTION
Chapter 2
Theoretical background
In this chapter an overview of the applied theory is given. This overview
consists of a short review on optics (section 2.1), the concept of dielectric
modelling (section 2.2) and the scalar scattering theory (2.3).
2.1
Optics
In this section relevant fundamental laws of optics are described. These
laws are applied to obtain reflectance R and transmittance T spectra from
modelled dielectric functions. This includes the basics of electromagnetic radiation in subsection 2.1.1 and the theory of refraction in subsection 2.1.2.
In the last subsection a short explanation is given about the origin of interference fringes on measured R/T spectra.
2.1.1
Electromagnetic radiation
Plane harmonic waves
Maxwell formulated the electromagnetic theory in a set of four partial differential equations that describe the properties of the electric and magnetic
field. These equations are given by [55, 64]
∇×H=
4π
1 ∂D
Jf +
c
c ∂t
∇×E=−
1 ∂B
c ∂t
(2.1)
(2.2)
∇ · D = 4πρ
(2.3)
∇·B=0
(2.4)
11
12
CHAPTER 2. THEORETICAL BACKGROUND
where D is the electric displacement, E the electric field, B the magnetic induction and H is the magnetic field. Jf , ρ and c are the free current density,
total charge density and speed of light respectively. When assuming a material that is isotropic, non-dispersive and uniform the electric displacement
and magnetic field are given by the relations [55]
D = εE
(2.5)
1
H= ·B
(2.6)
µ
where ε is the permittivity or dielectric constant and µ is the permeability
of the medium. From these equations one can derive the wave equations as
shown in appendix B. For the electric field this equation is:
εµ
Ë = 0.
(2.7)
c2
The propagation of electromagnetic radiation along the z-axis of a Cartesian coordinate system is often described as a plane harmonic wave satisfying
Maxwell’s equations with [64]
∇2 E −
Ex = E0 cos(νt − kz), Ey = 0, Ez = 0,
(2.8)
E0
· cos(νt − kz), Bz = 0,
(2.9)
c
in which k is the wavenumber and ω the angular frequency. Since it is more
common to express a wave in terms of wavelength, λ, and frequency, ν, the
following two relations are used for conversion [64]:
Bx = 0, By =
ν = 2πω
(2.10)
2π
.
λ
(2.11)
k=
Wave-particle duality
Under some circumstances, electromagnetic radiation behaves like discrete
particles called photons. These photons have particle-like properties and the
energy of a photon is given by Planck’s equation [64]:
hc
= hω.
(2.12)
λ
In this equation h represents Planck’s constant. This equation shows a
relation between electromagnetic radiation as a particle and as a wave. In
this work mainly the wave notation is used but sometimes radiation as a
particle is used for a better understanding when dealing with for example
bandgap energy. In this case the given relation is used for conversion.
E=
2.1. OPTICS
13
Electromagnetic spectrum
Solar cells are often said to convert light into electricity. According to the
dictionary light is defined as [1]:
“Electromagnetic radiation to which the organs of sight react, ranging in
wavelength from about 400 to 700 nm and propagated at a speed of 186.282
mi./s (299.972 km/s), considered variously as a wave, corpuscular, or quantum phenomenon.”
Light is therefore only a term based on the human eye to describe a
relatively small part of the total spectrum of electromagnetic radiation as
can be seen in figure 2.1. The region with shorter wavelength than the
visible part is ultraviolet (UV) and the region with larger wavelength is
the infrared (IR) part of the spectrum. In this work the wavelength of
investigation ranges from 300 to 1500 nm covering the UV/Visible/NIR
part of the electromagnetic spectrum.
Figure 2.1: Spectrum of electromagnetic radiation. [2]
2.1.2
Refraction
Snell’s law
Light is reflected or transmitted when it encounters an interface separating
two media. While the angle of reflection is equal to the angle of the incident
light, this is not true for transmitted light. Snell’s law describes the relation
between the angle of the incident light θi and the angle of the transmitted
light θt with [64]
n1 · sin(θi ) = n2 · sin(θt )
(2.13)
where n1 and n2 denote the refractive indices of the two media. In case of
absorbing layers the refractive index is made complex consisting of a real
refractive index n and an imaginary extinction coefficient k that relates to
14
CHAPTER 2. THEORETICAL BACKGROUND
the amount of absorption loss in the material. Complex refractive indices
are denoted by ñ and are related to the complex permittivity by:
n
e = n + ik (=
√
ε1 + iε2 ).
(2.14)
Fresnel equations
The angles of refraction are schematically shown in figure 2.2. This figure
also distinguishes two different polarization states that describe the orientation of the electric field vector. When the vector is parallel to the plane
of incidence it is called p-polarization and when the vector is perpendicular
to the plane of incidence it is s-polarization as indicated in figure 2.2 by the
superscripts (p) and (s).
Figure 2.2: Reflection and transmission at the boundary of two media.
The Fresnel equations provide a method to relate the intensity of the
transmitted and reflected parts of the electromagnetic waves to the refractive indices of the media. These equations assume that the magnetic permeability of both materials is the same. In applying the boundary conditions
in the derivation of the Fresnel equations, a distinction is made between pand s-polarized light. For s-polarization the Fresnel equations are [55, 64]
t⊥ =
2n1 cos(θi )
n1 cos(θi ) + n2 cos(θt )
and r⊥ =
n1 cos(θi ) − n2 cos(θt )
n2 cos(θi ) + n1 cos(θt )
(2.15)
while for p polarization the Fresnel equations are written as
tk =
2n1 cos(θi )
n1 cos(θt ) + n2 cos(θi )
and rk =
n2 cos(θi ) − n1 cos(θt )
.
n2 cos(θi ) + n1 cos(θt )
(2.16)
In these equations t and r represent the amplitude transmission and reflection coefficients. The intensities, T and R, are obtained by taking the square
of these coefficients. Equations 2.15 and 2.16 are also valid for absorbing
2.1. OPTICS
15
layers by replacing the real refractive indices n1 and n2 by the complex
refractive index n
e1 and n
e2 .
When working with known materials the only unknown to solve the
Fresnel equations is the angle of the transmitted light. This angle can be
obtained using Snell’s Law. Equations 2.13 through 2.16 therefore allow
the calculation of the transmittance and reflectance with as only input parameters the (complex) refractive indices of the two media and the angle of
incidence of the incoming light.
2.1.3
Interference
Transmittance and reflectance spectra of samples with a finite thickness
show so-called interference patterns [see figure 2.3(a)]. These oscillations are
due to multiple reflections (R1 , T1 etc) from the layer as shown schematically
in figure 2.3(b). This causes the intensity of the transmitted or reflected light
to be enhanced or decreased depending on the phase difference between the
waves.
Figure 2.3: Internal reflections causing interference.
Whether or not the two waves are in phase depends on the wavelength
of the incoming light and the thickness of the layer. These two parameters
are related through the interference equation [71, 49]:
wλ = 2nd, w = 1, 1.5, 2, 2.5...
(2.17)
where n is the real part of the refractive index of the layer, d is the thickness
of the layer and w is the order number. The order number is an integer
for maxima and a half-integer for minima in the oscillation. This physically
means that when w is an integer the waves are exactly in phase and thereby
enhancing each other to the maximum intensity and when w is a half-integer
the waves are out of phase leading to the minimum intensity. From this
equation, Manifacier [49] derived an expression for the layer thickness as
16
CHAPTER 2. THEORETICAL BACKGROUND
function of the position of the extrema and the corresponding refractive
index at that wavelength:
d=
wλ1 λ2
.
2 · [n(λ1 )λ2 − n(λ2 )λ1 ]
(2.18)
The subscripts 1 and 2 denote the first and second extrema respectively.
The thickness can be calculated for any pair of extrema so in some cases it
is possible to calculate the thickness for up to a hundred different combinations for one spectrum. The average then provides a reasonable estimate
of the thickness. It is however known that this method is not very accurate and strongly depends on the correctness of the refractive index. In
appendix A the thickness of an AZO layer is determined with this method
for demonstration.
2.2
Modelling of dielectric functions
The intensity of reflected and transmitted light can be modelled by combining several ‘dielectric models’ that describe some of the characteristic
phenomena that occur when light travels through a medium. These models
can be fitted on measurements to yield a large variety of optical properties
of the layer(s). The procedure for modelling and fitting R/T spectra is explained in subsection 2.2.1. Afterwards some of the applied dielectric models
are presented in subsection 2.2.2. Subsection 2.2.3 contains an overview of
effective medium theory for modelling inhomogeneous or mixed layers. The
section ends with an introduction to the SCOUT software package that is
used in this work for the modelling.
2.2.1
Simulation of R/T spectra
The model for simulating spectra consists of different sub-models. These
sub-models all provide wavelength-dependent susceptibility equations that
model specific features such as the bandgap and free carrier absorption.
The complex dielectric function of the layer is then given by summing these
susceptibilities, χn [72];
ε = εre + iεim = 1 +
X
χn .
(2.19)
n
The complex refractive index is calculated by taking the square root of
the dielectric function (eq. 2.14). Through these equations, the complex refractive index is computed from all the parameters present in the sub-models
of the interface. Calculation of R and T from this refractive index according
to the Fresnel equations described in section 2.1.2 provides a feedback loop
to enable a fitting procedure of the models on actual R/T measurements.
This procedure is illustrated in figure 2.4.
2.2. MODELLING OF DIELECTRIC FUNCTIONS
17
Figure 2.4: Feedback loop for fitting R/T spectra.
Because most of the equations in the sub-models contain the wavelength
of the light as parameter, the dielectric function and R and T show a wavelength dependency as observed in measurements.
A characteristic shape of R/T spectra was depicted in figure 2.3. It
shows high transmittance in the visible part of the spectrum (350-750 nm).
In the UV and IR part the transmittance decreases rapidly due to bandgap
and free carrier absorption respectively. These are probably the most important phenomena that have to be included in the dielectric model for
TCO samples and are modelled with the OJL-interband and Drude models
respectively. Another model called a Brendel oscillator is implemented to
model the normal mode vibrations in the material. These models will be
presented in the next section.
2.2.2
Dielectric models
Extended Drude model
The Drude model, proposed in 1900 [26, 27], describes the transport properties of electrons in materials. One of the main results of Drude’s theory is
the equation of motion that relates the average momentum p of the charged
particle to the electric field E, charge q and mean free time between ionic
collisions or relaxation time τ .
d
p(t)
p(t) = qE −
.
dt
τ
(2.20)
The particle has an effective mass m eff meaning that the average momentum
of the particle is given by
p(t) = meff ·
dx
.
dt
(2.21)
Substitution of equation 2.21 into 2.20 yields a differential equation for the
distance x along the path of the charged particle or electron:
18
CHAPTER 2. THEORETICAL BACKGROUND
meff ·
d2 x
d2 x
dx
dx
e
=
−Γ
·
m
·
+
eE
⇒
+Γ
=
· E.
eff
dt2
dt
dt2
dt
meff
(2.22)
where Γ = 1/τ is the damping factor and e is the elementary charge. In
the derivation of the Drude model E and x are assumed to have sinusoidal
waveforms of a single frequency and are hence defined as
E(t) = E · e−iωt
(2.23)
x(t) = x · e−iωt .
(2.24)
Substitution of these two equations into the equation of motion, eq. 2.22,
provides a solution for x. Note that the exponential terms can be crossed
out during the substitution.
(−ω 2 − Γωi)x =
eE
1
eE
⇒x=
·
.
meff
meff −ω 2 − γωi
(2.25)
Knowing that the susceptibility is given by P/E and that P=n · e · x, the
following relation is obtained for the susceptibility and resulting dielectric
function:
2
ωpl
ne2
1
χ=
·
⇒ ε=1+
meff −ω 2 − Γωi
−ω 2 − Γωi
2
=
with ωpl
Ne e2
ε0 meff .
(2.26)
(2.27)
where Ne is the density of electrons and e0 is the permittivity of free space.
ωpl is referred to as the plasma frequency, i.e. the frequency at which the
charged particle resonates with the alternating electric field.
The classical Drude model is extended for improving the fits for doped
layers in the NIR part of the transmittance spectra. Hamberg et al.[34]
mentioned a deviation of the classical Drude model in case of doped layers
because of ionized impurity scattering. A better fit is obtained in case of
doped layers when the damping factor is defined as
Γτ,low − Γτ,high
π
−1 ω − ωτ,crossover
Γτ (ω) = Γτ,low −
tan
+
. (2.28)
π
ωτ,width
2
This damping factor adds four new fit parameters being: low and high
frequency damping, crossover frequency and width of the transition region.
The damping goes from a constant value at low frequency to a constant level
at high frequency with a transition region in between. Note that setting the
2.2. MODELLING OF DIELECTRIC FUNCTIONS
19
low and high frequency damping equal to each other eliminates the second
term in equation 2.28 and yields a constant damping again as in case of the
classical Drude model.
OJL interband model
Bandgap absorption is observed in the UV/Visible part of the spectrum.
The strong absorption is due to the fact that the energy of the photons is
higher than (or equal to) the bandgap of the material. The photons can
then be absorbed to excite an electron from the valence into the conduction
band. Multiple models are proposed for this phenomenon [72] but for the
case of TCOs the O’Leary-Johnson-Lim (OJL) interband transitions model
[58] was found to give best fitting results in this work.
The OJL model is an empirical model that describes the density of states
(DOS) functions of the valence and conduction band. The shape of the bands
is parabolic with tail states that exponentially decay into the bandgap. Tail
states are the regions that spread into the bandgap and are a measure of
the amount of disorder in the material which makes this model suitable for
crystalline as well as amorphous structures. Figure 2.5 shows a schematic
drawing of the energy bands.
Figure 2.5: Valence and conduction band in the OJL interband model. [72]
The DOS function of the conduction band is then given by
√
Nc (E) =
∗3/2
2mc
2 3
π h̄
 √
, E ≥ Ec +
 rE − Ec
·
E−E
c
γc
− 12

· e γc
, E < Ec +
2 ·e
γc
2
γc
2
(2.29)
where m∗c represents the effective mass associated with the conduction band,
E c the disorderless bandedge and γ c the breadth of the conduction band tail
or Urbach energy. Similarly the DOS function for the valence band is given
by
20
CHAPTER 2. THEORETICAL BACKGROUND
√
Nv (E) =
∗3/2
2mv
2 3
π h̄
·
 r

γ
 √
− 12
·e ·e
Ev − E
v
2
Ev −E
γv
, E ≥ Ev +
, E < Ev +
γv
2
γv
2 .
(2.30)
From these expressions of the density of states it is possible to calculate
the imaginary part of the susceptibility. The real part is obtained through
a Kramers-Kronig transformation [25]. The fitting parameters associated
with this interband model are the energy bandgap E bg , Urbach energy, γv ,
strength of the transition and a decay parameter that is added to drag the
imaginary part down for high frequencies.
Brendel oscillator
The normal mode vibrations in the material are modelled with a Brendel
oscillator [13, 12] which is an extended version of a damped harmonic oscillator. In case of a damped harmonic oscillator the system can be seen as a
mass m connected to a spring with stiffness L and a damper with damping
coefficient a as shown in figure 2.6 [38]. A harmonic external force F is
applied with amplitude F 0 .
Figure 2.6: Schematic system of a damped harmonic oscillator.
The equation of motion of this system is obtained by summing the forces
in x-direction. The external force can also be written in its complex form:
mẍ + aẋ + Lx = F0 cos(ωt) = F0 · eiωt
(2.31)
where the dots denote the time derivatives. Dividing this equation by the
mass and setting a/m=Γ, L/m=ω 0 and F 0 /m=f 0 gives
ẍ + Γẋ + ω0 x = f0 · eiωt .
(2.32)
To find a solution for this differential equation it is assumed that the
particular solution has a similar form as the external force with X equal to
a complex valued constant:
xp (t) = X · eiωt .
Substitution in equation 2.32 yields for the complex valued constant:
(2.33)
2.2. MODELLING OF DIELECTRIC FUNCTIONS
X=
ω02
f0
.
− ω 2 + Γωi
21
(2.34)
In dielectric modelling the theory of a damped harmonic oscillator is
implemented in a similar way by defining a susceptibility function according
to equation 2.34. Since an atom in a crystal structure is able to move in
more than one direction, there is a harmonic oscillator for each degree of
freedom. The susceptibility function then yields
!
m
2
X
ωpl
χ=
(2.35)
ω02 − ω 2 + Γωi
i=1
where the sum is taken over the amount of vibration modes. This harmonic
oscillator model is therefore suitable for modelling materials in which the
vibrations are restrained to a certain amount of vibration modes as in case
of crystal structures. In amorphous structures however the degree of freedom can be infinite. This requires the extension of the harmonic oscillator
into a Brendel oscillator in which the integral is taken from -∞ to ∞ with
the assumption of a Gaussian distribution of resonance frequencies. The
Gaussian distribution has a probability density function given by [68]
1
2
2
f (x; µ, σs2 ) = p
· e−(x−µ) /(2σs )
2πσs2
(2.36)
in which σs is the standard deviation and µ is the expected value. Substitution of the integrated damped harmonic oscillator, eq. 2.35, in this equation
yields for the Brendel oscillator:
χ(ω) = p
1
2πσs2
2
ωpl
−(x − ω0 )2
exp
· 2
dx.
2σs2
x − ω 2 + Γωi
−∞
Z
∞
(2.37)
This susceptibility equation will provide better fitting results in case of amorphous layers in proximity of the bandgap (200-500nm). This model will
therefore be used in this work.
2.2.3
Effective medium theory
In some cases the layer that has to be modelled is inhomogeneous and consists of a mixture of two different materials. In effective medium theory, this
classical problem is studied resulting in a number of models that are widely
used. Development of these models goes back to 1904 when the MaxwellGarnett Theory [50] was published. Later on Bruggeman [16] and Bergman
[31, 69] published more models. This section describes the basic theory of
effective medium approximation and presents the characteristic features of
the different models mentioned earlier.
22
CHAPTER 2. THEORETICAL BACKGROUND
Figure 2.7: Definition of an ‘effective medium’.
Recursive system
The effective medium theory assumes that an effective dielectric function
εeff can be composed of the dielectric functions of particles εp with a certain
predefined geometry and the host material εh in which they are embedded
(see figure 2.7). The mixed layer is thus modelled as a homogeneous layer in
order to simplify the characterization process. All effective medium theories
have as prerequisite that the particles are small in comparison with the
wavelength of the incident light.
The derivation of an effective dielectric function is an iterative process
starting with a pure host material containing randomly placed particles with
a volume fraction f [31]. The dielectric function of the composite is then a
function of the two dielectric functions and the volume fraction,
(1)
εeff = M (εp , εh , f ),
(2.38)
in which the function M is defined by the geometry and/or the spacing of
the particles. Equation 2.38 completes the first step of the iteration process.
For the second step the dielectric function of the host medium is replaced by
the newly computed one. After the nth step the effective dielectric function
is then given by [31]
(n)
εeff = M (εp , εn−1
eff , f ).
(2.39)
The key to solving this problem lies within the form of the function M . As
mentioned before there are different models proposed for different conditions.
The remainder of this chapter explains some of the most commonly used
models.
Bergman representation
The most flexible model describes the iterative procedure with the Bergman
spectral representation. The M function for the first iteration is then given
by the following equation [31, 69].
2.2. MODELLING OF DIELECTRIC FUNCTIONS
(1)
εeff
Z
= M (εp , εh , f ) = εh 1 − f
0
1
G(s)
ds
t−s
23
with
t=
1
ε . (2.40)
1 − εph
In this equation, G(s) is the spectral density that contains information about
the particle microgeometry and is independent of εp and εh . The spectral
density is a real non-negative function normalized in the range [0,1].
When high volume fractions are expected it is common to include percolation in this model. Percolation means interaction of the particles when
they are for example connected as shown in figure 2.7. In case of percolation
the spectral density can be split into a Dirac delta distribution at s=0 and
a continuous rest [31, 72],
G(s) = G0 δr (s) + G(s).
(2.41)
With this in mind the M function becomes
Z 1
G0 f
G(s)
(1)
εeff = εh 1 −
−f
ds
t
0 t−s
with
t=
1
ε
1 − εph
(2.42)
where G0 is the so-called percolation strength that is defined between 0
and 1. Repeating equation 2.42 in the iterative process described in section
2.2.3 yields an effective dielectric function for the replacing homogeneous
layer. Implementation of this model requires parameter input of the percolation strength and the spectral density function as well as assignment of
materials/dielectric functions to the host and embedded particles.
Simplifications
The knowledge of the spectral density function is essential to solving equation 2.42 and obtaining an effective dielectric constant. In the past decades
some models have been derived that deal with certain assumptions on the
geometry and percolation of the encapsulated particles. The most commonly
used models are the Maxwell-Garnett Theory (MGT) and the Bruggeman
model. Later on it appeared that all these models are special cases of the
Bergman representation with a concrete definition of the spectral density
function and percolation strength. This section briefly explains these two
surface mix models.
Maxwell-Garnett Theory (MGT) This model was introduced in 1904
[50] and assumes spherical inclusions in a homogeneous host material without any interaction of neighboured particles. The effective dielectric function
depends on εp , εh and f and is given by
24
CHAPTER 2. THEORETICAL BACKGROUND
Figure 2.8: Spectral density function of the Maxwell-Garnett Theory. [72]
(1)
εeff − εh
(1)
εeff
=f·
+ 2εh
εp − ε h
.
εp + 2εh
(2.43)
Although this equation was derived in a different way using equations
of average electric fields and polarization in the particles [76]; Ghosh et
al. proved [31] that this equation is also a specific case of the Bergman
representation with the percolation strength and spectral density function
given by
G0 =
2
3−f
(2.44)
G(s) = A · δ(s − s0 )
(2.45)
with A=(1-f )/(3-f )=1-G0 and s0 =1-f /3. This model is therefore a simplification of the Bergman representation based on a spectral density function
that describes the geometry of spheres in a host material (Figure 2.8). This
limits the applicability of this model. Another limitation of the MGT model
is that it requires a low volume fraction and particles that are far away from
each other, i.e. no percolation [72].
Bruggeman Model This model published by Bruggeman in 1936 [16]
and often called Effective Medium Approximation (EMA), is probably the
most often applied effective medium model and proved successful in a vast
number of studies involving macroscopically inhomogeneous media. The
effective dielectric function is defines as
(1)
(1 − f ) ·
εh − εeff
(1)
εh + 2εeff
(1)
+f ·
εp − εeff
(1)
εp + 2εeff
= 0.
(2.46)
2.2. MODELLING OF DIELECTRIC FUNCTIONS
25
The Bruggeman model incorporates percolation making it not only useful
for low volume fractions but also for higher volume fractions. As with the
MGT the geometry consists of spheres. Also in this case Ghosh et.al. [31]
proved that this model is a specific case of the Bergman representation with
percolation strength and spectral density function for a three dimensional
layer.
For a volume fraction below 1/3 there is no percolation strength and
above 1/3 the percolation strength increases with volume fraction as shown
in figure 2.9(a). Figure 2.9(b) shows the spectral density function that is
again only a function of s and f . Looking at eq. 2.46, the volume fraction is
the only parameter that has to be defined to solve for the effective dielectric
function, besides of course the definition of the materials. This means that
there can be no external influence on the spectral density function and percolation function. This is limiting the model to this specific case and since
there is no specific reason why these function would behave like so, it might
in some cases not be accurate enough.
Figure 2.9: Spectral density function of the Bruggeman model. [72]
2.2.4
SCOUT software package
The iterative process of dielectric modelling that was presented in section
2.2.1 is carried out with the SCOUT software package [72]. In this program
one can define a layer stack composed of different materials. The dielectric
functions of these materials can be either taken from the SCOUT database
or be composed manually by selecting the appropriate sub-models as shown
in figure 2.10.
26
CHAPTER 2. THEORETICAL BACKGROUND
Figure 2.10: Material definitions with a manually composed master model
(bottom left) and the SCOUT database (right).
SCOUT is able to simulate the R/T spectra with all the equations presented in chapter 2 according to the predefined configuration of layers and
materials. A graphical user interface can be created in which the measurements are imported to start the fitting procedure. All the parameters that
are accessible in the equations of the sub-models are available as fitting parameters and hence all these parameters can function as output of the model.
Amongst the others, the most important output will be the refractive index
of the layer(s). In chapter 4 different materials used in thin-film silicon solar
cells are characterized and fitted with SCOUT to yield the material properties of the layers. This requires selection of the right sub-models and some
creativity with the SCOUT software. Other than understanding the physics,
the correct implementation of the models and equations into SCOUT has
therefore also become an important part of this work.
2.3
Scattering from rough surfaces
The dielectric models described in the previous section do not account for
surface roughness. When the roughness is small in comparison with the
wavelength the roughness can be modelled with an effective medium approach. However, for a higher roughness this is not sufficient. The difference with regard to flat layers is that more light is scattered away from the
specular direction. Scalar scattering theory provides a method to make predictions of the scattering properties of a rough layer to apply a correction
for surface roughness. The starting point are the wave equations that are
derived from Maxwell’s equations.
2.3. SCATTERING FROM ROUGH SURFACES
2.3.1
27
Wave equations
The propagation of electromagnetic radiation as waves was described according to Maxwell‘s equations in section 2.1.1. Starting from here, the
wave equations of the electromagnetic field can be derived (see Appendix
B). These wave equations for the electric and magnetic field are given by
∇2 E −
εµ
Ë + ∇ ln µ × (∇ × E) + ∇(E∇ ln ε) = 0
c2
(2.47)
εµ
Ḧ + ∇ ln ε × (∇ × H) + ∇(H∇ ln µ) = 0
(2.48)
c2
When the material is assumed to be homogeneous and non-magnetic
these equation can be simplified to their more familiar form:
∇2 H −
∇2 E −
2.3.2
εµ
Ë = 0,
c2
∇2 H −
εµ
Ḧ = 0.
c2
(2.49)
Scalar wave equation
Since the wave equations are difficult to solve it is convenient to simplify
them by means of some assumptions. When the relaxation time of the
material is assumed to be much shorter than the periodic time of vibration
of the wave, which is mostly the case for conducting materials, than the
second Maxwell equation, eq. 2.2, can be written as:
∇ · E = 0.
(2.50)
From this, the wave equation can be written as
µε
4πµσ
Ë + 2 Ė.
(2.51)
2
c
c
Here the assumption is made that the dielectric function of the material
is effectively constant over a length comparable to the wavelength of the
incoming radiation. For a monochromatic field the derivative with respect
to time is proportional to -iω. This yield for the first and second derivative
of the electric field E:
∇2 E =
Ė = −iωE
(2.52)
Ë = i2 ω 2 E = −ω 2 E.
(2.53)
Substitution of these two equations into equation 2.51 gives
∇2 E + k 2 E = 0
where:
(2.54)
28
CHAPTER 2. THEORETICAL BACKGROUND
k2 =
ω2µ
c2
ε+i
4πσ
ω
.
(2.55)
The main advantage of deriving this new function with these assumptions
is that there is no coupling between the Cartesian coordinates (x, y, z) any
more so that the equations can be solved separately. For a component U
this equation can therefore be written in the scalar form[40]:
∇2 U + k 2 U = 0.
(2.56)
From this theory, Bennett and Porteus and Carniglia derived the equations for the scattering properties of randomly textured surfaces. These
equations describe the drop in specular transmittance and reflectance as
will be explained in the next section.
2.3.3
Specular component of R/T for rough surfaces
Surface roughness causes the incoming radiation to be scattered away from
the specular component. Therefore a drop in the specular component of both
the reflectance and transmittance is observed. From scalar scattering theory
presented in this chapter it is possible to calculate the drop in intensity as
function of the angle of the incoming light and the roughness of the surface.
This theory is presented by Davies in 1954 [24] and developed for scattering of radar waves on rough water surfaces. Nevertheless it is also valid
for wavelengths in the UV/Vis/IR range of the electromagnetic spectrum.
In this model the RMS roughness is assumed to be small enough so that
no shadowing occurs, the surface is assumed to be perfectly conducting and
the height distribution is assumed to be Gaussian. Starting from the scalar
scattering theory, Bennett and Porteus derived the specular component of
the reflectance to decrease with roughness according to [9, 40]
Rspec
" #
4πnσR cos θi 2
= R0 · exp −
λ
(2.57)
in which, σR is the RMS roughness of the surface and R0 represents the
reflectance of a flat surface of a similar material. This exponential correction factor is shown in figure 2.11 for different roughnesses at zero angle of
incidence. This graph clearly shows that the correction is stronger in the
UV/Visible part than in the IR which implies that scattering from rough
surfaces is more effective at shorter wavelengths. This is also observed in
reality where UV or blue light is more easily scattered than IR longer wavelengths.
In a similar way a relation has been derived by Carniglia that describes
the specular component of the transmitted radiation [40, 18]:
2.3. SCATTERING FROM ROUGH SURFACES
29
Figure 2.11: Exponential correction on Rspec due to scattering.
(2πσR )2
2
Tspec = T0 · exp −
(n
cos
θ
−
n
cos
θ
)
1
i
2
t
λ2
(2.58)
where the angle of the refracted light θt depends on the refractive indices
at the interface according to Snell’s law (see section 2.1.2). This correction
shows a similar behaviour to the one seen for the reflectance in figure 2.11.
These two equations provide a way to calculate the drop in specular R/T
cause by the surface roughness and allows fitting of data obtained from
specular measurements with rough surfaces. Due to the assumptions the
accuracy is limited and especially for materials that do not show a Gaussian
height distribution.
30
CHAPTER 2. THEORETICAL BACKGROUND
Chapter 3
Equipment
3.1
3.1.1
Deposition
Magnetron sputtering
The AZO and ITO samples in this work are deposited with RF-magnetron
sputtering. Sputtering is a process that ejects atoms from a solid target due
to bombardment of the surface with ions. The ions are from an inert gas
such as argon. A voltage is applied to the target and the negative voltage
attracts the positive ions. When the energy transferred to the lattice is
greater than the binding energy of the target atoms, they can be ejected
from the target towards the substrate where they are deposited [61]. Figure
3.1(a) schematically shows this process and figure 3.1(b) is the setup at the
PVMD laboratory.
Figure 3.1: Magnetron sputtering; (a) schematic [54] and (b) setup.
3.1.2
Chemical Vapor Deposition (CVD)
The silicon layers in this work are deposited with a plasma enhanced CVD
process (PECVD). In CVD, the substrate is exposed to precursor gasses
31
32
CHAPTER 3. EQUIPMENT
that react and decompose on the substrate surface. Silane gas (SiH4 ) is
used as source for the silicon. The rates of the chemical reactions of the
precursors are enhanced with a plasma. This also allows lower deposition
temperatures [66]. For doped silicon layers other process gasses can be
added such as diborane (B2 H6 ) for doping with boron (p-type silicon) and
phosphine (PH3 ) for doping with phosphorus (n-type silicon). In this work
however only intrinsic silicon is characterized. Figure 3.2(a) schematically
shows the process and 3.2(b) is a picture of the setup.
Figure 3.2: Chemical vapor deposition; (a) schematic and (b) setup.
3.1.3
Thermal evaporation
The silver layers that are used for the creation of nanoparticles are deposited
using a thermal evaporation technique. The source material is heated and
evaporates. The evaporated atoms condense on the cold substrate to form
the thin layer. The process is carried out in vacuum to prevent interaction
with any other particles. Figure 3.3(a) schematically shows the process and
figure 3.3(b) shows the setup.
Figure 3.3: Thermal evaporation; (a) schematic and (b) setup.
3.2. CHARACTERIZATION
3.2
3.2.1
33
Characterization
Automated R/T Analyser (ARTA)
Most of the optical characterization in this work is done with Variable Angle
Spectroscopy (VAS) . The Perkin Elmer Lambda 950 is a spectrophotometer
that is capable of measuring Reflectance and Transmittance (R/T ) spectra
with high accuracy and reproducibility. The Lambda 950 spectrophotometer
has a wavelength range of 175–3300 nm which spans the UV/Visible/NIR
region of the spectrum (see figure 2.1). This broad range of wavelengths
makes it an excellent tool for characterization of TCO layers because other
than the UV/Visible part, where the bandgap and interference fringes play a
role, the NIR part is equally important because of the free carrier absorption.
Other than TCOs this technique can be used for any other material and will
in this work also be used for silicon, polymer and nanoparticle layers.
VAS will be done with the ARTA accessory that can be installed on the
Lambda 950. ARTA consists of a drum in which the detector and sample
holder have separate rotation stages. This allows automatic adjustment of
the angles during the measurements. The sample holder can be positioned
at any desirable angle while the detector is limited to 15-345 degrees to prevent blocking of the incoming light. ARTA uses a small integrating sphere
detector equipped with a photomultiplier for the UV/Visible spectrum region and a PbS cell for the NIR region. Figure 3.4 shows the Lambda/ARTA
configuration.
Figure 3.4: Lambda 950/ARTA configuration.
Measuring R/T spectra with ARTA Specular transmittance is obtained by fixing the detector behind the sample at 180 degrees because
according to Snell’s law, the angle of transmitted light is equal to the angle
of incidence when travelling from a layer with refractive index, n, through
a layer stack to another layer with an equal refractive index. This is illustrated in figure 3.5. There is however a small displacement, xd , depending
on the thickness of the stack, d.
34
CHAPTER 3. EQUIPMENT
Figure 3.5: Specular transmittance through a layer stack.
The specular reflectance measurements are done by positioning the detector at twice the angle of incidence because the angle of specular reflection
is equal to the angle of incidence on the sample. The angles in ARTA are
defined differently for the sample holder and the detector as shown in figure
3.6. The red shaded area is not reachable by the detector. Due to the red
region it is not possible to make specular reflection measurements at angles
smaller than 8 degrees.
Figure 3.6: Defined angles of (a) sample holder and (b) detector in ARTA.
The accuracy of the measurements can be improved by measuring the
R/T spectra at both positive and negative angles of incidence. This method
eliminates also any kind of misalignment in the sample holder because the
error is cancelled out. Therefore always the average is taken in R/T measurements in this work. The R/T measurements are performed at 01 , 15,
30, 45 and 60 degrees angle of incidence for both p- and s-polarization. That
sums up to a total of seventeen spectra when taking into account that there
is only one spectrum available at 0 degrees angle of incidence. This is due
to the fact that reflectance measurements are not possible and there is no
difference between the R/T spectra under p- and s-polarized light.
1
Only transmittance at 0 degrees.
3.2. CHARACTERIZATION
35
Measurement accuracy An accuracy test has been carried out to verify
the reproducibility of the R/T measurements and hence the accuracy of the
obtained data. This is done for a rough AZO sample (σ R = 40 nm) at 45
degrees angle of incidence and s-polarization. This sample is chosen because
it has average roughness compared to the entire batch of AZO samples.
The R/T measurements are done three times at a different location on the
sample with the TCO facing the incoming light. The resulting (averaged)
spectra are shown in figure 3.7.
Figure 3.7: Accuracy test for a rough AZO layer at 45 degrees.
For each wavelength the standard deviation is determined with respect
to the average value. The largest deviation was found to be within 1.5%.
This deviation is mainly due to measuring at different spots on the sample. The thickness is larger in the center of the deposition than on the
edges. Furthermore, etched AZO layers have a random surface texture and
a slightly different roughness can also affect the measurement. Nevertheless
in future work a sample will only be measured once so from this analysis
it is shown that the accuracy is limited to some degree when dealing with
rough surfaces. This has to be taken into account when assigning a rating
system to the acquired fit.
Sample placement The samples can be placed in the sample holder either
with the glass in front or with the layer in front. Although in the solar cell
the light first impedes on the glass it might in some cases be more useful to
measure the spectra with the layer in front as will be explained later. The
effect of changing the measurement configuration on the measured spectra
is shown in figure 3.8. This plot shows the R/T spectra for a flat AZO
layer and an AZO layer with an RMS roughness of 40nm. The spectra are
measured at 45 degrees angle of incidence and s-polarization.
36
CHAPTER 3. EQUIPMENT
Figure 3.8: R/T spectra for different sample placement configurations.
The transmittance hardly changes because the transmitted light in both
cases goes through the entire stack and encounters the same layer interfaces.
The specular reflectance however changes because the reflected light only
encounters the first interface and some of the light is now scattered away
from the specular direction. Besides that also the intensity changes because
of the different refractive indices of AZO and glass according to Fresnel’s
equations (eq. 2.15-2.16).
3.2.2
Total integrating sphere
Another accessory for the Lambda 950 spectrophotometer is the total integrating sphere. This accessory is capable of measuring the total and diffuse
transmittance and reflectance at normal incidence. This accessory is useful
for determining the Haze, i.e. portion of the light that is diffusely scattered.
The accessory is equipped with an integrating sphere with a diameter of
150 mm. The detector is thus a small portion of the total surface area and
hence the light is homogeneous along the surface of the sphere before it
reaches the detector. The total R/T is obtained by closing all the ports in
the sphere except where the light enters and where the sample is located.
For the diffuse part, a port can be opened to let the specular part escape.
For reflectance measurements an angle of incidence of 5 degrees is applied
to prevent interference with the incoming light. Figure 3.9 shows the setup.
3.2. CHARACTERIZATION
37
Figure 3.9: Lambda 950 equipped with the integrating sphere accessory.[80]
3.2.3
Mini-RT Setup
The mini-RT is a device that measures the reflectance and transmittance
of a sample with two CCD cameras. The difference with the Lambda spectrophotometer is that the wavelength range is much smaller (375–1060 nm)
and that there is no option to either polarize the light or measure under an
angle of incidence. This limits the user to measuring only two spectra per
sample. The setup from Eta-Optik is shown in figure 3.10.
Figure 3.10: Eta-Optik mini-RT Setup at Delft University of Technology.
The results when characterizing a layer with this setup are not as accurate as for the Lambda/ARTA configuration because of the lack of spectra
and limited wavelength range. However, the setup proved to be useful for
thickness determination of silicon and flat TCO layers. This is because the
thickness has a strong effect on the spectra and can be more easily fitted with
less data. More information about thickness determination with mini-RT is
given in appendix C.
38
3.2.4
CHAPTER 3. EQUIPMENT
Atomic Force Microscopy (AFM)
An atomic force microscope is used to map the surface topology of the samples. In AFM a cantilever with a tip scans over the surface. Any deflection
of the cantilever is measured with a laser beam that is reflected off the top
of the cantilever into a photodiode as shown in figure 3.11(a). In this work
only semi-contact mode AFM is used which means that the cantilever is
driven to oscillate near its resonance frequency. Due to van der Waals forces
the amplitude of the oscillation get smaller when the tips gets closer to the
surface [74]. This decrease is noticed by the sensor and fed back into the
system to control the sample position in vertical direction to restore the amplitude to its original value. This is schematically shown in figure 3.11(b).
The vertical correction together with the relative x and y-coordinates give
the user enough information for a three dimensional map of the surface.
Figure 3.11: AFM: (a) operation principle, (b) semi-contact mode. [74]
In this work scans are made over a 10x10 µm area with a 256x256 resolution. With post processing software the root mean square of the surface
roughness is determined according to
v
u
N
u1 X
σR = t
(zi − ẑ)2
(3.1)
N
i=1
with zi equal to the ith position and ẑ representing the average surface level.
The AFM setup is shown in figure 3.12.
3.2.5
Hall Setup
Hall measurements are conducted to verify the obtained electrical parameters. Hall measurements provide multiple parameters of which the free
carrier concentration and mobility are the most important for this work.
The geometry of the Hall effect on which the measurements are based
is shown schematically in figure 3.13. A current j is passed through the
material in x-direction. The electrons move with a velocity v in opposite
3.2. CHARACTERIZATION
39
Figure 3.12: AFM Setup at Delft University of Technology.
direction. Due to the magnetic field B there is a Lorentz force −ev × B
which tends to deflect the electrons. The Lorentz force is counterbalanced
by an induced electric field EH [36]. This electric field is given by the relation
EH = RH B × j,
(3.2)
Figure 3.13: Hall effect [36].
where RH is the Hall coefficient which is inversely related to the electron
concentration ne and describes the relation between conductivity σH and
mobility µH with [52]
1
σ.
(3.3)
Ne e
In this work the Hall setup shown in figure 3.14 is used. The samples
should have a size of approximately one square centimeter and be as square
as possible for reliable and accurate data. The sample is placed in the
sample holder with the four contacts exactly on the corners of the sample
after which the sample holder is placed on top of the device. During the
measurements the voltages between the points are monitored when a set of
magnets slide past the sample. In this way the electric field EH and hence
µ = |RH | σH =
40
CHAPTER 3. EQUIPMENT
the hall coefficient can be calculated. This is all done automatically and the
software provides, among others, values for the conductivity and mobility.
Figure 3.14: Hall Setup at Delft University of Technology.
3.2.6
Spectroscopic ellipsometry
Spectroscopic ellipsometry is a tool to determine the thickness of a layer
and its refractive index. When a linearly polarized beam is reflected from
a sample surface it can get elliptically polarized. Elliptical polarization
means that the electric field components parallel (p) and perpendicular (s)
to the plane of incidence differ in phase and amplitude as illustrated in figure
3.15(a). Figure 3.15(b) schematically shows the measurement configuration.
Figure 3.15: (a) Elliptical polarization and (b) ellipsometry configuration.
In spectroscopic elipsometry the change in phase and amplitude of these
electric field components is measured and expressed in two parameters, ∆
and Ψ. Ψ is the amplitude ratio and ∆ is the phase shift. These parameters
are related to the reflection coefficients as follows:
ρ=
rp
= tan(Ψ)ei∆ .
rs
(3.4)
The reflection coefficients, rp and rs , can be calculated from the Fresnel
equations (eq. 2.15 and 2.16). The refractive index in these equations is
modelled with the dielectric models described in section 2.2. In this way the
3.2. CHARACTERIZATION
41
Ψ and ∆ spectra can be simulated after which a fitting procedure reveals the
material properties. This makes ellipsometry a model-based characterization
technique.
Ellipsometry is used in this work to determine the optical properties of
layers of silver nanoparticles. The possibility to do in-situ measurements
during the formation process of the nanoparticles is in this case a big advantage over spectroscopy discussed in section 3.2.1.
42
CHAPTER 3. EQUIPMENT
Chapter 4
Modelling results
In this chapter the results of the research are presented. Different materials are characterized by applying the method of dielectric modelling to fit
models on specular R/T measurements. Section 4.1 contains the results
of the required glass calibrations. Sections 4.2 to 4.5 contain the details
about modelling and fitting of TCOs, a-Si:H/µc-Si:H, silver nanoparticles
and polymer samples respectively.
4.1
Glass substrates
The first step to obtain accurate fitting results is to customize the default
glass model from the SCOUT database to represent the actual substrates
that are used. In this work two different types of substrates are used, being
‘Corning Eagle 2000tm ’ for TCO and silicon samples and the glass that is
used by Asahi glass company for the FTO samples which will be referred to
as ‘Asahi glass’. This calibration method and results are briefly summarized
in this section.
Corning Eagle 2000 For the Corning Eagle 2000 substrates, transmittance and reflectance measurements are performed at normal incidence with
the Lambda equipped with the integrating sphere. The SCOUT glass model
consists of a number of Brendel oscillators which are then fitted on these
measurements to obtain the exact parameters of Corning Eagle 2000 substrates. Figure 4.1(a) shows the obtained fitting results and figure 4.1(b)
presents the obtained refractive index. This fitting was done earlier in an
internship project.
Asahi Glass For Asahi a similar approach is followed except that now
the ARTA is used to produce more spectra at different polarizations and
angles for improved accuracy. In order to get a clear glass substrate to
do the measurements the FTO layer had to be etched away. This is done
43
44
CHAPTER 4. MODELLING RESULTS
Figure 4.1: (a) Fitting results and (b) refractive index of Corning Eagle 2000
glass.
Figure 4.2: Asahi Glass; (a) fitting results at 30 degrees angle of incidence
and (b) refractive indices.
by wet chemical etching in diluted HCl (10% of J.T.Baker(37%)[7]) with
Zinc powder as a catalyst. The ARTA R/T measurements are performed
at 15, 30 and 45 degrees both for p- and s-polarization and at positive and
negative angles. The SCOUT default glass model is simultaneously fitted
on the averaged spectra as shown in figure 4.2(a) for 30 degrees angle of
incidence. The parameters obtained from this fitting are used in the FTO
model. The refractive index of the Asahi glass model is compared with the
Corning Eagle 2000 glass in figure 4.2(b).
4.2
Transparent Conductive Oxides (TCOs)
Different TCO materials are characterized using SCOUT. These are Aluminiumdoped Zinc Oxide (AZO), Tin-doped Indium Oxide (ITO) and Fluorinedoped Tin Oxide (FTO). For these materials a model is first made for flat
layers and after that a model is created that supports surface roughness
effects. Since the physical behaviour of the three different TCO materials
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
45
is comparable, the model is first presented after which the results are given
for the different materials specifically.
4.2.1
Modelling
Flat TCO layers
The bandgap of a TCO is modelled with the OJL interband model described
in section 2.2.2. The OJL model is available in SCOUT and has to be placed
inside a so-called KKR (Kramer-Kronig relation) susceptibility [72]. This
function automatically calculates the corresponding real part of the refractive index with the Kramer-Kronig relation after defining the imaginary
part with the available fitting parameters. The amount of data point in this
model must be a multiple of 2 to carry out this transformation in SCOUT.
Another important aspect is that of free carrier absorption. Free carrier
absorption is an oscillation of the free carrier density in the material. The
electrons screen the electric field of the light so that light with a frequency
below the plasma frequency is reflected [45]. This decrease in transmittance
and increase in reflectance can best be modelled with the Drude model for
free electrons that was presented in section 2.2.2. The ‘extended’ Drude
model is used because the layer is a doped material and impurity scattering
requires a non constant damping factor.
Normal mode vibrations of the atoms have a small impact on the R/T
spectra. A Brendel Oscillator is chosen because this provides is suitable
for both crystalline and amorphous materials and is a good choice when
the exact structure of the layers is not (yet) known. This specific oscillator
uses a Gaussian distribution of resonance frequencies as explained in section
2.2.2.
Optically flat layers still have a certain roughness. This small roughness
does not have a big impact on the spectra but the fits can be improved if they
are still taken into account. Because the features are small in comparison
with the wavelength an effective medium approach is sufficient to enhance
the fitting accuracy. In this case the Bruggeman effective medium model is
applied by assuming that there is a small roughness layer on top consisting
of TCO particles in air. For a larger surface roughness this approach is
however too rigorous and the following model is developed for that purpose.
Rough TCO layers
The surface roughness is merely a correction factor on the spectra for flat
layers as it will be shown later. The equations derived from the scalar
scattering theory, presented in section 2.3.3, are used to model the surface
roughness of the TCO layers. The main difficulty is the correct implementation in SCOUT. In SCOUT a ‘rough layer’ can be introduced in the stack
46
CHAPTER 4. MODELLING RESULTS
to multiply the reflection or transmission coefficient with a wavelength dependent function. In this wavelength dependent function no more than two
other fitting parameters can be introduced. The difficulty arises as SCOUT
does not allow the use of either the refractive index or the angle of incidence
as fitting parameter.
For reflectance spectra the refractive index can be left out of the equation when the samples are measured with the TCO in front. This is because
the first layer before encountering the air/TCO interface is then air with
a refractive index of 1.0. The angle of incidence problem can be solved by
copying the complete layer stack and by assigning a different layer stack to
each angle of incidence. In this way the angle of incidence can be implemented as a constant value as illustrated in figure 4.3. The layer thickness’s
of the different stacks are coupled to one ‘master parameter’ since these
should always be the same. These two tricks bypass the problems for reflectance spectra. The RMS roughness in the equations is now the only
parameter that is left and this parameter can be used for fitting.
Figure 4.3: Layer stack definitions for rough TCO samples.
Transmittance spectra are more difficult to implement since there are
two refractive indices present in the equations so one cannot avoid the use
of the refractive index of the TCO layer. Since there is so far only one
parameter used as fitting parameter (RMS roughness) there is still one parameter left. This open slot is used to approximate the refractive index as
a linear function of the wavelength. Calibration of the model results in a
linear approximation that can be used for a specific material. However because of this approximation, the accuracy of the fits on transmittance will
not be as high as for the reflectance.
Nevertheless the transmittance spectra do have an added value during
the fitting procedure when it comes to fitting the bandgap and free carrier
absorption since these phenomena are less clear in the reflectance spectra.
However because the modelled reflectance spectra are significantly more ac-
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
47
curate a weight factor of 0.1 has been applied to the transmittance fits.
This optimum number of 0.1 is determined experimentally. With this factor
the influence of the linear approximation for the refractive index in case of
transmittance spectra is reduced. The transmittance spectra now provide a
good guideline for fitting the bandgap and free carrier absorption yet they
don’t disturb the accurate (simultaneous) fits on the reflectance.
The final point of attention for implementing surface roughness is that
the ‘rough layer‘ in SCOUT puts a correction on the r or t coefficient.
Because the intensity is defined as the square of this coefficient one has
to multiply the coefficients with the square root of the exponential term
in equations 2.57 and 2.58. This is illustrated for the reflectance by the
equation below:
Rspec
2
p
4πnσR cos θi 2
= R0 · exp [a] = r0 · exp [a] with a = −
(4.1)
λ
Overview
The fitting parameters for fitting rough TCO layers with this model are
shown in table 4.1. The italic parameters are manually adjustable in the interface. The interface for fitting rough TCO layers can be found in appendix
D.
Submodel
OJL bandgap
Extended Drude model
Brendel oscillator
Other
Fitting parameter
Bandgap energy
Urbach energy
OJL strength
OJL decay
Plasma frequency
Low freq. damping
High freq. damping
Crossover frequency
Crossover width
Resonance frequency
Oscillator strength
Oscillator distrib. width
Dielectric background
AZO layer thickness
RMS roughness
Table 4.1: Rough TCO fitting parameters
units
[eV]
[meV]
[-]
[-]
[eV]
[cm−1 ]
[cm−1 ]
[cm−1 ]
[cm−1 ]
[eV]
[-]
[cm−1 ]
[-]
[nm]
[nm]
48
4.2.2
CHAPTER 4. MODELLING RESULTS
Aluminium-doped Zinc Oxide (AZO)
One of the big advantages of using AZO as a front contact in solar cells
is the natural abundance and low material costs. Its instability in acids
allows surface roughening by wet chemical etching in diluted hydrochloric
acid HCl (0,5%) [56]. This roughness scatters the light and although this
is beneficial to the light absorption in the cell, this roughness also makes
it more complex to optically characterize the layer. In this work a batch
of nine AZO samples is deposited on Corning Eagle 2000 glass substrates
with radio frequency magnetron sputtering (see section 3.1.1). The samples
are wet-chemically etched in diluted HCl with etching times ranging from 0
(optically flat surface) to 50 seconds. The previously discussed model and a
graphical interface are created in SCOUT that are able to fit specular R/T
spectra of these rough layers to yield an accurate refractive index.
Characterization
The setups used for the characterization are ARTA, AFM and the Hall setup.
The ARTA data are the most important and are used to fit the simulated
spectra. The AFM and Hall data are used for a better understanding of
the structural and electrical properties but more importantly to verify the
fitting results of the models.
AFM The surface of all AZO samples is analysed with AFM. The results of
this analysis are shown in figure 4.4 for three different samples. These scans
show a crater-like surface roughness and the features become larger with
etching time. The resulting root mean square of the roughness is summarized
in figure 4.5 for all samples and this graph shows that there is a mainly linear
relation between etching time and surface roughness with an etching rate of
approximately 1.5 nm/s. The uncertainty of the measurements is not known
because initially only one measurement was done for each sample. In later
attempts no reliable AFM data were obtained from the samples because the
cantilevers did not give a good resonance peak.
Figure 4.4: AFM scans of AZO samples with 10, 35 and 50 seconds of
etching.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
49
Figure 4.5: Roughness of etched AZO layers.
ARTA For all nine samples ARTA is used to measure the specular reflectance and transmittance for both p- and s-polarized light at the angles:
0, 15, 30, 45 and 60 degrees. The average is taken over positive and negative
angles for the reasons explained in section 3.2.1. Since this approach gives
a large amount of spectra for the entire batch, some selections had to be
made in this report to clearly illustrate the important aspects.
The effect of the surface roughness on the R/T spectra is shown in figure
4.6 by selecting the measured spectra at s-polarization and at 45 degrees
angle of incidence. The reported roughness is the one found with AFM.
Figure 4.6: The effect of surface roughness at 45 degrees angle of incidence
and s-polarized light.
50
CHAPTER 4. MODELLING RESULTS
From this graph it is observed that the specular transmittance and reflectance decrease in the visible part of the spectrum with increasing surface
roughness. An increasing part of the light is scattered away from the specular direction which however does not imply that the total transmittance
or reflectance is also lower. The transmittance in the NIR part of the spectrum seems to increase due to surface roughness however this it physically
not possible since the material properties do not change and light will be
scattered away from the specular direction (see also figure 2.11). The increase in transmittance is most likely due to the decrease in layer thickness
with etching time. Another important feature in these spectra is that the
interference fringes are diminishing with increasing roughness. This is due
to the fact that the light is scattered and hence the multiple reflections inside
the material are also scattered in other directions and interfere less with the
specular part.
Hall measurements Hall measurements are conducted on all nine samples. As explained in section 3.2.5 these measurements give the conductivity
and mobility of the layers. Figure 4.7 shows these two measured layer properties as a function of the etching time as well as the carrier concentration
that was calculated by dividing the conductivity, σH , by eµH . Each sample
is measured five times and with different placements. This means that the
sample is rotated such that the four contacts are not connected to the same
four corners of the sample each time1 . The resulting error bars represent the
standard deviation and show that the measurements were highly reproduceable with the exception of the flat sample and the sample with 35 seconds
of etching.
The material is equal for all nine samples yet the mobility is slightly
decreasing with etching time. Other authors also observed a decrease in
mobility that was related to a decrease in layer thickness [77][73]. The main
reason for this was mentioned to be the scattering from grain boundaries and
surface defects that becomes stronger when the layer gets thinner and with
a higher surface roughness. The conductivity and free carrier concentration
also show a slight decrease.
1
One configuration is measured twice since there are only four possible orientations
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
51
Figure 4.7: Mobility, conductivity and carrier density as function of etching
time for AZO.
Fitting results
Flat AZO layers The model for flat AZO layers is used to fit the ARTA
measurements of the sample that was not etched. From AFM measurements
the roughness was determined to be around 18 nm (see figure 4.5). The
fitting is done simultaneously on all seventeen spectra. The fitting results
are shown in figure 4.8 and 4.9 for p- and s-polarized light respectively. The
arrows indicate the correct y-axis. The fit on the transmittance at zero
degrees provided an equally good fit but is left out to make the graphs more
clear.
The simultaneous fit is close on all seventeen spectra which yields accurate optical properties of this layer. It also indicates that the sample is
indeed optically flat and the surface mix is sufficient to model the slight RMS
roughness of 18 nm. Figure 4.10 shows the most important parameters and
the obtained refractive index.
52
CHAPTER 4. MODELLING RESULTS
Figure 4.8: Fitting results for a flat AZO layer under p-polarized light.
Figure 4.9: Fitting results for a flat AZO layer under s-polarized light.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
53
Figure 4.10: Obtained optical properties of a flat AZO layer.
The shape and values of the refractive index are similar to those found
in literature [62, 17, 3] which, to some extent, confirms the correctness of
the model. Furthermore a literature review shows that the bandgap for
ZnO is approximately 3.3 eV [22]. Doping the material with aluminium fills
the states at the bottom of the conduction band and thereby increases the
bandgap to values in the range of 3.4-4.0 eV [62, 22, 63]. For the plasma
frequency of AZO, values ranging from 0.9-1.4 eV1 have been reported by
several authors[62, 53] whereby the value is closely linked to the level of
doping (i.e. free carrier concentration). The Urbach energy depends on the
amount of disorder in the material but is in the same order of magnitude as
the value of 160 meV as reported by Qiao[62]. The model therefore provides
plausible results when looking at earlier work.
Rough AZO layers For larger roughness the model for rough TCO layers
is applied (see section 4.2.1). In the prepared batch of AZO samples there are
eight etched samples that have non-negligible surface roughness. A simultaneous fit is performed on each sample. The fitting results for the sample
with 40 seconds of etching and a corresponding 68 nm of RMS roughness
are shown in figure 4.11 and 4.12 for p- and s-polarized light respectively.
These graphs clearly show that the fit on transmittance is not as good
as on the reflectance which was already expected from the discussion in
section 4.2.1 about correct implementation of the equations in SCOUT.
Nevertheless it also shows that with the introduced weight factor on the
transmittance fits, the model does indeed take the transmittance spectra
into account to fit the bandgap and free carrier absorption (1000–1500 nm)
but does not influence the good fitting on the reflectance spectra. In this
way the transmittance can still provide better accuracy despite the worse
fit. Also for this layer the obtained properties are shown in figure 4.13.
1
the plasma frequency will be expressed in eV instead of Hz to get values close to 1
which are easier to handle and compare.
54
CHAPTER 4. MODELLING RESULTS
Figure 4.11: Fitting results for a rough AZO layer under p-polarized light.
Figure 4.12: Fitting results for a rough AZO layer under s-polarized light.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
55
In the n,k-graph the blue line represents the refractive index of the flat
layer for reference.
Figure 4.13: Obtained optical properties for a chemically wet-etched AZO
layer.
These results are quite similar to the ones obtained for the flat sample
and since the refractive index and optical properties should not change due
to this roughness, the model for rough AZO samples also provides a good
accuracy. This can also be seen in the value for the RMS roughness that
was found for the model (70.4 nm) compared with the actual roughness
from AFM (68 nm). Note also that the layer thickness is much lower for the
rough sample. This is due to the etching during which not only the surface
morphology is changed but also a large portion of the bulk AZO is etched
away. The other eight samples were successfully fitted with equal accuracy.
4.2.3
Tin-doped Indium Oxide (ITO)
ITO is used in a variety of application ranging from LCD and flat panel
displays [41, 23] to toys [6]. Indium is a relatively rare metal (see figure
4.14) and is mainly used for the production of ITO (about 85% [70]).
Despite the disadvantage of low abundance, ITO films have high transparency in visible light and low resistivity [43]. Better electrical properties
can be obtained by a thermal annealing process which increases the crystallinity and therefore also the conductivity, mobility and carrier concentration [8]. However also for ITO films there is the trade off between electrical
and optical properties. The increased carrier concentration results in more
free carrier absorption in the infra red region and hence a lower transmittance [51]. Optimization of the ITO layers requires accurate characterization
methods among which the accurate determination of the refractive index.
For this purpose an ITO sample is deposited by rf-magnetron sputtering
(see section 3.1.1) and fitted with the TCO SCOUT model.
56
CHAPTER 4. MODELLING RESULTS
Figure 4.14: Abundance of chemical elements. [48]
Characterization
AFM The roughness of the ITO surface after deposition is small in comparison with the wavelength. AFM is not necessary because the ITO sample
is not post treated and is therefore expected to be optically flat.
ARTA The ITO sample is measured with the TCO in front at 0, 15, 30,
45 and 60 degrees angle of incidence with p- and s-polarized light over a
wavelength range of 250-1500 nm. The wavelength range is extended a little
further into the UV part of the spectrum to get a clear view on the bandgap
that is expected to be higher than for AZO. The measurements are done
at both positive and negative angles and the average spectra are used to fit
the models. The measured spectra have a shape similar to those for AZO
as can be seen in figure 4.15 for p-polarization.
Figure 4.15: Measured spectra of an ITO sample at p-polarization and different angles of incidence.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
57
The bandgap lies further in the UV as expected and another difference
compared to the AZO spectra in section 4.2.2 is the high plasma frequency
that is accompanied by a rapid drop in transmittance and high reflectance
in the IR part.
Hall measurements Two small pieces of one square centimeter are extracted from the ITO sample to do Hall measurements. Both samples are
measured five times and the results are averaged. The conductivity of the
ITO layer was measured at 4.468·103 Ω−1 cm−1 which is significantly larger
than that of AZO ( 0.669·103 Ω−1 cm−1 ). The mobility was measured at 42.9
cm2 /Vs and from this the carrier concentration is calculated at 6.47·1020
cm−3 which is roughly a factor two higher than for AZO. This higher carrier
concentration leads to higher optical losses in the infrared. The improvement
in electrical performance therefore comes at the cost of optical performance.
Fitting Results
Flat ITO layers The TCO model for flat layers was applied to the ARTA
measurements of a flat ITO layer. The averaged spectra over positive and
negative angles function as input for the model. The fitting results for pand s-polarized light are presented in figures 4.16 and 4.17 respectively.
Figure 4.16: Fitting results for a flat ITO layer under p-polarized light.
58
CHAPTER 4. MODELLING RESULTS
Figure 4.17: Fitting results for a flat ITO layer under s-polarized light.
These graphs are showing close fits to the measurements yielding the
refractive index and material properties that are given in figure 4.18.
Figure 4.18: Obtained optical properties of a flat ITO layer.
The bandgap of ITO is typically in the range of 3.5-4.3 eV [44, 35]. The
bandgap for the deposited layer is on the high side of this range which is
probably due to the relatively high carrier concentration compared to for
example the AZO sample in section 4.2.2. The plasma frequency is highly
dependent on the tin doping concentration and values are reported ranging
from 1.1 up to even 6 eV [14, 15]. Again the value obtained from the model
lies well within this range so also for ITO the results are comparable to
literature.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
4.2.4
59
Fluorine-doped Tin Oxide (FTO)
FTO is a widely used commercial TCO that was developed and first produced in the 1940ies. The primary use of this TCO was for anti fogging
coatings of cockpits of World War II aircraft [30]. The main advantage with
respect to optics is that FTO has a larger bandgap compared to AZO and
ITO and hence lower absorption losses in the UV. Furthermore the transmittance is significantly higher in the NIR part of the spectrum as will be
shown in this section. The FTO samples that were used in this thesis are
standard Asahi U-Type samples that were provided by Asahi glass company [4]. These samples have a non-negligible roughness of approximately
40 nm RMS. There are no flat FTO samples available so the practical use
of the roughness model will be tested thoroughly because no information is
available from earlier fittings on flat layers.
Characterization
AFM The roughness of the surface is analysed with AFM. The samples
are supplied by Asahi company and since wet chemical etching is no option
for FTO layers, there is only one roughness available. It is therefore sufficient
to analyse the surface of one small piece of FTO that is extracted from the
sample. Figure 4.19 shows the obtained 5×5 µm AFM scan. The roughness
of the layer was found to be 36.6 nm.
Figure 4.19: Surface roughness of an FTO layer.
ARTA The measurement conditions are the same as for AZO and ITO
being 0, 15, 30, 45 and 60 degrees both positive and negative, p- and spolarization and the TCO layer in front. The wavelength range is kept at
250–1500 nm to get a clear view on the bandgap. The measured (averaged)
spectra are shown in figure 4.20 for p-polarized light.
Besides the high bandgap located at approximately 300 nm, the FTO
sample also shows lower absorption at longer wavelengths compared to AZO
60
CHAPTER 4. MODELLING RESULTS
Figure 4.20: Measured spectra of an FTO sample at p-polarization and
different angles of incidence.
and ITO. Another interesting observation is the weak interference pattern
and smooth decrease in transmittance when approaching the bandgap as
seen with the rough AZO samples in section 4.2.2. This indicates that the
surface has indeed a non-negligible roughness that scatters a portion of the
light away from the specular directions.
Hall measurements The measurements are done on two substrates of one
square centimeter that were extracted from the same FTO sample. Both
are measured five times in the Hall Setup and the results are averaged.
The conductivity is found to be 1.02·103 Ω−1 cm−1 and the mobility is 39.0
cm2 /Vs. From these two values the carrier concentration in the FTO layer
was found to be 1.63·1020 cm−3 . Like ITO the conductivity is higher than
for AZO ( 0.669·103 Ω−1 cm−1 ) however in the case of FTO this is also
accompanied by a lower carrier concentration. This implies that a better
conductivity is achieved with less free carrier absorption in the considered
wavelength range.
Fitting Results
Rough FTO layers The FTO sample has an RMS roughness of 37 nm.
This is too high to be modelled as an effective medium so the rough TCO
model is used for the FTO samples. Also in this case a linear approximation
is used in the equation for specular transmittance (eq. 2.58) for the reasons
explained in section 4.2.1. The rough TCO model together with the calibrated glass model provides the fits that are shown in figure 4.21 and 4.22
for p- and s-polarization respectively.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
61
Figure 4.21: Fitting results for a rough FTO layer under p-polarized light.
Figure 4.22: Fitting results for a rough FTO layer under s-polarized light.
The graphs show fitting results that are comparable to those of the other
TCO materials. Also in this case the reflectance spectra have good fits and
the transmittance spectra were able to guide the bandgap and free carrier
absorption parameters. The resulting refractive index of the FTO samples
is given in figure 4.23 together with some other material properties.
62
CHAPTER 4. MODELLING RESULTS
Figure 4.23: Obtained optical properties of a flat FTO layer.
4.2.5
Error Analysis
The accuracy of the TCO models was up to now assumed to be reasonable
due to the large amount of simultaneously fitted spectra. In this section this
accuracy is quantified by performing a series of tests on the models. In these
tests the spectra are loaded for the flat and rough (σR =68 nm) AZO samples,
the flat ITO sample and the FTO sample (σR =37 nm). The main parameters
are randomly chosen such that there is no close fit and the automated fitting
procedure is started. A threshold was set on the fitting deviation below
which the fitting was stopped and the parameters were gathered. For each
sample this test is repeated 10 times at 10 different starting positions. The
standard deviations of the most important parameters are given in figure
4.24 with respect to their average values. The bars represent the rough
samples and the dots within represent the flat samples.
Figure 4.24: Standard deviation of the fitting parameters after 10 automated
fittings.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
63
The standard deviations show that most of the parameters are within a
5% accuracy. The only parameter that can have a larger deviation is the Urbach energy which slightly interferes with the parameters of the OJL model
and the Brendel oscillator. This error can however be reduced with human
intervention. As mentioned the fitting was started automatically from an
off position. In practice it is however relatively easy to fit the bandgap parameter manually after which the Urbach energy will automatically adjust
to its correct value. This indicates that it is always better to start from
a manually obtained close fit,although the model is able to find a fit after
some time with good reproducability.
The refractive index is wavelength dependent and therefore the standard
deviation can also be presented as a function of wavelength as in figure 4.25.
Note that this graph only shows the deviation of the ‘rough TCO model’
since this model was found to be in general slightly less accurate. Since no
rough sample was available for ITO it is left out of this graph. Nevertheless
a similar accuracy is expected also for ITO.
The refractive index is within 2% accuracy in the wavelength range of
600–1500 nm. At values closer to the bandgap the accuracy becomes somewhat lower. This is probably due to the increased standard deviation of the
Urbach energy but is also related to the approximations made for correct
implementation of the scattering model for transmittance spectra as mentioned in section 4.2.1. Besides this, the extinction coefficient is close to zero
meaning that the larger deviation might also be due to the low signal/noise
ratio. Nevertheless the models do indeed provide a unique solution with
acceptable accuracy.
Figure 4.25: Standard deviation of the refractive index from rough samples.
64
4.2.6
CHAPTER 4. MODELLING RESULTS
Verification
In this section an analysis is presented with the aim of verifying the unique
solutions from the models. For this purpose the obtained results from the
models are compared with other characterization techniques. This includes
morphology from AFM and electrical properties from Hall measurements.
Morphology
Roughness The model for rough TCO layers used the RMS roughness
as fitting parameter in the scalar scattering equations. The RMS roughness
obtained from the models is compared with the AFM measurements in figure
4.26 for the batch of AZO samples.
Figure 4.26: Comparison of RMS roughness from AFM and the model.
Most of the samples in this graph are near the dotted line indicating that
the roughness found by modelling is close to the value found with AFM. The
samples with 20 and 50 seconds of etching are clear outliers. This is more
likely due to the AFM measurements since they also appear as outliers in
the AFM analysis in figure 4.5. It was however mentioned before that there
is no information about the accuracy of these measurement. Overall it can
be said that the model is able to find representative RMS roughness’s but
more important this confirms the validity of the scalar scattering model.
Electrical properties
Hall measurements are conducted to verify the electrical properties that
resulted from the model. The extended Drude model contains a non constant
damping term as explained in section 2.2.2. Based on the obtained fitting
parameters the damping term is plotted according to equation 2.28 for each
TCO material in figure 4.27. The circles represent the damping at the
plasma frequency that was calculated from the parameters in the table.
4.2. TRANSPARENT CONDUCTIVE OXIDES (TCOS)
65
For further calculations the damping at this plasma frequency is taken as
suggested by Mergel and Qiao [52] and will be called Drude damping, ΓDr .
Figure 4.27: Wavelength dependent damping term for TCO materials.
The Drude damping is used to obtain the average collision time and
afterwards the mobility of the free carriers according to [52]:
hτ i =
1
,
ΓDr
(4.2)
MDr =
e hτ i
.
meff
(4.3)
In these equations, meff represents the effective mass. In literature this
effective electron mass for a TCO varies from 0.25-1.02me [78, 52, 62, 11].
Because of this large range an approximation is made by plotting the squared
plasma frequency in figure 4.28 against the carrier concentration ne that was
obtained from the Hall measurements. According to the equation for the
plasma frequency (eq. 2.26) there is a linear relation between these parameters. The slope of the line is a measure for the effective mass. The plot
also shows a comparable analysis performed on AZO by Qiao and Brehme
[62, 11].
66
CHAPTER 4. MODELLING RESULTS
Figure 4.28: Determination of the effective mass.
Figure 4.28 reveals that although the materials are different the samples
still show a linear trend. From this observation the assumption is made
that the effective electron mass is equal for all samples with meff =0.37me
which is related to the slope of the line. With this effective mass the Drude
mobilities from the model are calculated from equation 4.3 and compared
with the measured mobilities in figure 4.29.
Figure 4.29: Verification of the mobility.
The plot reveals that there is no clear relation between the measured
and modelled mobility. It was mentioned by Mergel [52] that the carrier
concentration obtained from modelling is in most cases higher than the con-
4.3. SILICON LAYERS
67
centration obtained from Hall measurements which is due to optically active
carriers that do not contribute to the direct current of the Hall setup. Optically determined parameters are therefore generally larger than electrically
determined parameters so this method of optical characterization is not suitable for accurate determination of electrical properties of the layer.
4.3
Silicon Layers
Since silicon is the main absorber material for thin-film silicon solar cells.
A model is also made for the determination of the optical properties of
silicon layers. As mentioned in section 1.3, the structure in thin-film silicon
solar cells can be amorphous (a-Si:H) or microcrystalline (µc-Si:H). One way
to slightly control the microstructure of the deposited layer is through the
substrate temperature during deposition. To analyse the effects of substrate
temperature on the refractive index a temperature series is deposited with
RF-PECVD (see section 3.1.2) both for amorphous and microcrystalline
intrinsic silicon over a range of 100–300◦ C with steps of 50◦ C. The refractive
indices will be obtained in a similar way as for the TCO layers by fitting a
SCOUT model on R/T measurements.
4.3.1
Modelling
The bandgap of the silicon is modelled with the OJL interband transitions
model. This model fits best because of the amorphous structure with tail
states that decay into the bandgap. This model is suitable for amorphous
silicon but also for microcrystalline silicon. The difference is that for microcrystalline the disorder in the material is different. This is reflected by the
shape of the tail states that can be modified with the fitting parameters of
the model. Furthermore a Brendel oscillator will be required for modelling
the vibrations in the material. The model includes also the surface mix
which is a Bruggeman effective medium model for modelling a small surface
roughness.
Unlike for TCOs a Drude model will not be necessary for the silicon
model. The films in this case are intrinsic meaning that the free carrier
concentration is significantly lower than for doped materials and hence the
free carrier absorption is not observed in the considered wavelength range.
The SCOUT model that is prepared is equal in size to the TCO model
meaning that also seventeen spectra a required for a simultaneous fitting
procedure. The model is expected to produce accurate results over a wavelength range of 300–1500 nm. In this work however the model is reduced to
fitting only two spectra because of a hardware failure of the ARTA accessory. As alternative the mini-RT setup is used for measuring the reflectance
and transmittance at zero degrees angle of incidence. This setup has a limited wavelength range of 400–1000 nm. Fortunately the reduced wavelength
68
CHAPTER 4. MODELLING RESULTS
range does not affect the fitting procedure itself because the bandgap is still
clearly visible and free carrier absorption can be neglected as mentioned
earlier so no important information will be missed. The reduced amount of
spectra however can slightly reduce the accuracy of the obtained results.
Overview
The fitting parameters for the silicon model are shown in table 4.2 and the
manually adjustable parameters are written in italic. The interface that is
created around the model is shown in appendix D.
Submodel
OJL bandgap
Brendel oscillator
Surface mix model
Other
Fitting parameter
Bandgap energy
Urbach energy
OJL strength
OJL decay
Resonance frequency
Oscillator strength
Oscillator distrib. width
Surface mix layer thickness
Volume fraction
Dielectric background
Silicon layer thickness
units
[eV]
[meV]
[-]
[-]
[eV]
[-]
[cm−1 ]
[nm]
[-]
[-]
[nm]
Table 4.2: Silicon model fitting parameters
4.3.2
Amorphous silicon (a-Si:H)
Before the hardware failure, the samples deposited at 100, 150 and 200◦ C
were successfully characterized with ARTA. The larger ARTA model is therefore properly tested on these samples but it is not enough to clearly reveal
the effects of substrate temperature. The characterization is therefore entirely redone using only the mini-RT. This section is split in a part that
describes the results of the mini-RT analysis and a part that compares these
results with the ARTA model that was used for the first three samples to see
whether the accuracy is significantly compromised by lowering the amount
of spectra.
Mini-RT results
The reduced model uses only the reflectance and transmittance at zero degrees angle of incidence. For each sample the model is simultaneously fitted
on these two spectra and the fitting results are shown in figure 4.30 for the
4.3. SILICON LAYERS
69
Figure 4.30: Fitting results for a-Si:H at T=100◦ C and 300◦ C.
samples with a substrate temperature of 100 and 300◦ C. The other samples
showed equally close fits.
The fitting was repeated five times for each sample and from this it
appeared that there is a reasonable accuracy (within 5%) on the obtained
results. Figure 4.31 shows the trends observed in the bandgap and Urbach
energy as the substrate temperature increases. The accuracy of the bandgap
was well within 1% meaning that the error bars in figure 4.31 are not visible.
Figure 4.31: Bandgap and Urbach energy as function of substrate temperature for a-Si:H.
The bandgap is decreasing with increasing temperature and this was
also observed by other authors [47, 59]. Platz et.al [59] measured a rate of
approximately -55 meV/100K which is comparable to these results. This
decrease in bandgap was mentioned to be correlated to lower hydrogen concentrations at elevated substrate temperatures. The Urbach energy in figure
4.31 is slightly increasing with temperature and this increased amount of
disorder might be due to this lower hydrogen concentration and hence the
increased defect density in the materials. This can however not be proven
with this data and more research would be needed to confirm this.
70
CHAPTER 4. MODELLING RESULTS
The effects of substrate temperature on the refractive index are shown
in figure 4.32. The extinction coefficient shifts to the right with increasing
temperatures and this is related to the decrease in bandgap. The real part
of the refractive index shows an upward trend.
Figure 4.32: Refractive index of a-Si:H for different substrate temperatures.
Mini-RT vs. ARTA model
The samples deposited at 100, 150 and 200◦ C were measured with ARTA
before the hardware failure. These measurement are used to test the large
ARTA-model for silicon samples. The observed fits are equally close as for
the mini-RT model. The bandgap and Urbach energy obtained with this
ARTA model are compared to the previously presented mini-RT results in
figure 4.33. The error bars are the result of repeating the fitting process five
times from different random starting positions.
Figure 4.33: Mini-RT vs. ARTA model for a-Si:H samples.
This analysis shows that the bandgap obtained with the ARTA model is
slightly lower than that of the mini-RT model but judging from the slope of
4.3. SILICON LAYERS
71
the line, the decreasing rate with substrate temperature is comparable. The
same is observed with the Urbach energy. It is interesting in this case that
the Urbach energy obtained with the ARTA model has a higher accuracy as
can be seen from the error bars. The differences in parameters are translated
into the refractive index which is compared in figure 4.34.
Figure 4.34: n,k comparison of a-Si:H from the mini-RT and ARTA model.
The smaller bandgap of the ARTA model compared to the mini-RT is
also observed in the refractive index by a right shifted extinction coefficient.
The real part of the refractive index also shows an offset when comparing the
results. The mini-RT model thus shows similar trends as the ARTA model
but with slightly less accuracy. This is due to the reduced amount of spectra.
For a more thorough research the ARTA model is therefore recommended.
4.3.3
Microcrystalline silicon (µc-Si:H)
The mini-RT model was also used to fit the µc-Si:H samples. The results
obtained in this fitting procedure are presented in figure 4.35 for 100 and
300◦ C. Also in this case the other samples showed equally close fits.
Figure 4.35: Fitting results for µc-Si:H at T=100◦ C and 300◦ C.
72
CHAPTER 4. MODELLING RESULTS
The effect of substrate temperature for µc-Si:H is presented in figure
4.36, where the trend of the bandgap and Urbach energy is shown. Also in
this case, the bandgap decreases with temperature but for µc-Si:H the Urbach energy tends to decrease with substrate temperature indicating lower
disorder in the material. This is probably due to a higher amount of crystallisation at higher temperatures.
Figure 4.36: Bandgap and Urbach energy as function of substrate temperature for µc-Si:H.
The refractive index of the µc-Si:H samples is shown in figure 4.37. For
microcrystalline silicon the same trend is observed in the refractive index
as for a-Si:H. The extinction coefficient shifts to the right which confirms
the lower bandgap for higher substrate temperatures. The real part of the
refractive index also increases with temperature but not as much as for
a-Si:H.
Figure 4.37: Refractive index of µc-Si:H for different substrate temperatures.
4.4. SILVER NANOPARTICLES
73
Also in this case a more thorough research is required to analyse the
exact cause of the observed trends but nevertheless the created models, and
especially the ARTA model, are a perfect tool for characterizing the layers
in such research.
4.3.4
Conclusion
Although the ARTA model is providing more accurate results, the miniRT analysis provided insight in the effects of substrate temperature on the
properties of the silicon layers. Table 4.3 summarizes the resulting properties
for amorphous and microcrystalline silicon deposited at 200◦ C.
parameter
Thickness [nm]
Optical bandgap [eV]
Urbach energy [meV]
a-Si:H
1029
1.780
79.10
µc-Si:H
986.7
2.250
223.3
Table 4.3: Properties for silicon samples with deposition temperature of
200◦ C.
4.4
Silver nanoparticles
The introduction of nanoparticles in a solar cell is a novel concept to enhance
the light absorption. Incident light excites surface plasmons in the silver
nanoparticles which can make the particles strongly absorb and scatter the
light into the layer. Research is still carried out to optimize the plasmonic
effect in solar cell applications [5]. The objective in this work is to investigate
whether a layer of silver nanoparticles can be represented by an effective
medium. This would make SCOUT useful for analysing the optical effects
of annealing on the growth of nanoparticles and the optical properties of
nanoparticles with different sizes.
4.4.1
Modelling
Silver nanoparticles can be seen as an inhomogeneous layer consisting of
silver particles in air. Figure 4.38 gives an indication of the microgeometry
of such layers. Since the particles are significantly smaller than the wavelength of the incoming light it should be valid to use an effective medium
approach. The most flexible and accurate effective medium model is the
Bergman representation that was introduced in section 2.2.3. The model
for silver nanoparticles samples consists of the substrate, a calibrated glass
layer or crystalline silicon layer, and this effective layer. The refractive index
of bulk silver is taken from the SCOUT database to function as input for
the Bergman representation.
74
CHAPTER 4. MODELLING RESULTS
Figure 4.38: Silver nanoparticles, left: AFM scan and right: SEM image.
4.4.2
Layer deposition
Silver is deposited by thermal evaporation (see section 3.1.3). The deposition of silver layers with a thickness in the order of several nanometers
results in inhomogeneous layers, i.e. consisting of isolated nanoparticles
[57]. When more silver is deposited, these particles grow and form a continuous/homogeneous layer. Ellipsometry is used to characterize layers with a
mass thickness ranging from 2.5 to 25 nm deposited on silicon substrates.
The mass thickness is the thickness of an equivalent flat and homogeneous
layer with equal mass. Each sample is measured at 55, 60, 65, 70 and 75
degrees angle of incidence. Figure 4.39 shows the trend of the ellipsometry
data measured at 60 degrees angle of incidence for layers with a different
mass thickness.
Figure 4.39: Ellipsometry measurements of deposited silver layers with a
different mass thickness for a 60 degrees angle of incidence.
4.4. SILVER NANOPARTICLES
75
This graph already shows that the optical properties of the layers are
different. SCOUT is also able to simulate ellipsometry data and to do a
simultaneous fit on the different angles of incidence. The interface that
is created in SCOUT can be found in appendix D. Figure 4.40 gives an
indication of the accuracy obtained when fitting the measurements with the
Bergman representation. The resulting refractive indices after the fitting
procedure are shown in figure 4.41.
Figure 4.40: Fits on ellipsometry data for a silver layer at 55 and 70 degrees
angle of incidence.
Figure 4.41: Refractive index of deposited silver layers with a different mass
thickness.
This figure clearly shows that for thin inhomogeneous layers the optical properties are far from the bulk properties. When the layer thickness
increases, the silver layer approaches the properties of bulk silver which indicates that a more homogeneous layer is formed. Figure 4.42 gives the
corresponding volume fraction and percolation obtained from the SCOUT
model. Both are increasing steadily which means that larger islands are
formed with an increased level of interaction as the layer grows.
76
CHAPTER 4. MODELLING RESULTS
Figure 4.42: Volume fraction and percolation as function of deposited silver
mass thickness.
4.4.3
Annealing
The structure of a thin silver layer may change significantly when the layer is
annealed. This effect is analysed by annealing the samples, discussed above,
at 400◦ C. Ellipsometry measurements are performed on the annealed layers
and the refractive indices that resulted from the fitting with SCOUT are
shown in figure 4.43.
Figure 4.43: Refractive index of deposited silver layers with a different thickness after annealing at 400◦ C.
When these refractive indices are compared to those before annealing
(see figure 4.41) it is observed that the annealing process has shifted the
refractive index further away from the bulk properties for every initial mass
thickness. This indicates that the annealing process disintegrates the silver
layer.
The effect of annealing is further analysed for a new sample deposited on
a silicon substrate. The sample is preheated to 200◦ C and in-situ ellipsometry is done to monitor the optical properties of the layer. Figure 4.44 shows
4.4. SILVER NANOPARTICLES
77
the refractive indices at different stages of the process. From this graph it is
observed that the starting position at zero minutes is already away from the
bulk properties which can be due to a combination of the inhomogeneity of
the layer after deposition and the pre heating. During the annealing process
the refractive index steadily moves away from the bulk line which indicates
that nanoparticles are being formed. The transition is quick in the first ten
minutes after which it saturates to a stabilized value.
Figure 4.44: Evolution of the refractive index of a silver nanoparticle layer
during the formation process.
4.4.4
Particle size series
Up to now the fitting on ellipsometry data demonstrated that annealing
changes the microstructure and optical properties of a thin silver layer. To
better understand the relation between optical properties and layer thickness or particle size, a series of silver nanoparticle layers with thickness’s of
3, 6, 9, 12, 15 and 18 nm is deposited on glass substrates. The particles are
deposited on glass to be able to also measure the reflectance and transmittance which are interesting properties for such layers. The choice for glass
substrates requires a new SCOUT model that can fit the R/T spectra.
Figure 4.45 shows the effect of nanoparticle size on the specular transmittance and reflectance. Besides that, it also shows the results of fitting
the SCOUT model on these measurements. The measured spectra were
obtained with the Lambda 950 spectrophotometer equipped with the integrating sphere detector (see section 3.2.2). The transmittance drops as a
function of particle size and the reflectance increases. This can be explained
by the fact that the volume fraction increases and the layer is behaving
more like bulk silver which is highly reflective due to its high plasma frequency. This trend is also observed when looking at the resulting refractive
78
CHAPTER 4. MODELLING RESULTS
indices from the SCOUT model in figure 4.46. As particle size increases the
properties of the layer are approaching those of bulk silver.
Figure 4.45: Fits of the model on R/T for particles sizes of 6 and 18 nm.
Figure 4.46: Effect of nanoparticles size on the refractive index of the layer.
4.4.5
Conclusion
The main conclusion is that SCOUT and in particular the Bergman representation is able to model also layers of nanoparticles as an effective medium.
This makes SCOUT a valuable tool for optimising the plasmonic effect in
solar cell applications. The fitting can be done on ellipsometry data or R/T
spectra with the same underlying model. Furthermore, it is demonstrated
that the size of nanoparticles has a strong influence on the optical properties
of the layer.
4.5. POLYMER SAMPLES
4.5
79
Polymer samples
The previous sections have shown that with some experience SCOUT is a
valuable tool to model and characterize all sorts of materials. It is also
possible to model organic materials as will be demonstrated in this section.
The Technical University of Eindhoven is researching organic solar cells and
asked whether it is possible to obtain the refractive index for their polymer
layers. They provided a PCPDTBT polymer on a glass substrate as well
as a PCPDTBT:PCBM blend as it is used in organic solar cells. The layer
thickness is 100 nm for both samples.
4.5.1
Modelling
PCPDTBT
The modelling starts by looking at the molecular structure of the polymers.
These structures are shown in figure 4.47.
Figure 4.47: Molecular structures of PCPDTBT and PCPDTBT:PCBM.
[67]
The most important phenomena present in the R/T spectra of such
polymers are the strong absorption peaks. These are due to electronic transitions that can be assigned to different compounds in the polymer [37]. It
is proposed by Hoppe et.al. [37] to model these absorption peaks with Kim
oscillators which are quite similar to a Brendel oscillator but with less computational effort. Since there is, other than time, no significant advantage
of choosing a Kim over a Brendel oscillator the choice is made to use the
familiar Brendel oscillators. Four oscillators are used in this case which was
the minimum amount to obtain acceptable fits as will be explained later.
The bandgap is modelled using an OJL interband model. A dielectric background is added to the model because the real part of the dielectric function
will level off to a constant value in the IR part of the spectrum far from the
strong electronic interband transitions observed in the UV part [72].
80
CHAPTER 4. MODELLING RESULTS
PCPDTBT:PCBM
The molecular structure of PCBM reveals that it consist of a C60 fullerene
molecule with a small string attached to it. The SCOUT database contains
a dielectric model for C60 molecules and since this covers the largest part
of the PCBM polymer this model is used as starting point. The exact parameters are determined with the automatic fitting procedure to approach
the dielectric function of the real PCBM structure. The blend is modelled
with a Bruggeman effective medium model. The almost spherical structure
of the PCBM molecules encapsulated in a PCPDTBT medium fits the assumptions made in the Bruggeman model, i.e. spherical inclusions that are
significantly smaller than the wavelength of the incoming light. The total
model for the blend thus consists of a glass substrate with a Bruggeman
blend of C60 inside the previously modelled PCPDTBT. The implementation in SCOUT is shown in figure 4.48. The left window is the layer stack
and the bottom right window is the Bruggeman effective layer of the blend.
This blend consist of the PCPDTBT and C60 materials from the materials
list in the top right window.
Figure 4.48:
SCOUT.
4.5.2
Implementation of the PCPDTBT:PCBM layer stack in
Fitting
PCPDTBT
For this polymer the same seventeen spectra are measured with ARTA as
for the other materials, i.e. 0, 15, 30, 45 and 60 degrees with p- and spolarization. The average is taken over positive and negative angles of incidence. Figure 4.49 shows the spectra for 0, 30 and 60 degrees angle of
incidence to reveal the characteristic shape of the PCPDTBT spectra.
4.5. POLYMER SAMPLES
81
Figure 4.49: Specular R/T spectra of PCPDTBT at different angles and
polarizations.
The transmittance spectra show one big and one small absorption peak
in the visible part of the spectrum. Absorption for green light is less which
explains the green colour of the samples. By examining the large absorption
peak it is observed that it consists of two separate peaks because of the small
bump at the bottom of the transmittance spectra. Also the peak observed
in the transmittance spectra at 500 nm is not symmetrical indicating that
another oscillator is required there to flatten the right side of the peak. A
minimum of four oscillators is therefore required to model this shape.
With four Brendel oscillators the fits are as shown in figure 4.50. Only
the spectra at 30 degrees angle of incidence are presented here but the other
spectra of the simultaneous fit show similar fitting accuracy.
Figure 4.50: Fits on the measured spectra of the PCPDTBT sample.
82
CHAPTER 4. MODELLING RESULTS
The fits show that the small bump at the bottom of the transmittance
spectra is slightly overestimated while the peak in reflectance is underestimated. However, all seventeen spectra show an acceptable fit and the accuracy is already satisfactory. The fits can however be improved by adding
more oscillators and fine-tuning the parameters. The refractive index of the
polymers is presented in figure 4.53.
PCPDTBT:PCBM
The measured spectra of the blend are presented in figure 4.51. The main
difference with the plain PCPDTBT polymer is that there is more absorption
near the band gap which lowers the first peak in transmittance.
Figure 4.51: Specular R/T spectra of the PCPDTBT:PCBM blend at different angles and polarizations.
The proposed effective medium model gave the following fitting results
for the polymer blend. Also in this case the fit is acceptable and provides
an accurate refractive index. Figure 4.53 presents the refractive index of the
blend.
4.5. POLYMER SAMPLES
83
Figure 4.52: Fits on the measured spectra of the PCPDTBT:PCBM blend.
Figure 4.53: Refractive indices of the PCPDTBT and PCPDTBT:PCBM
samples.
4.5.3
Conclusion
With a set of Brendel Oscillators it is possible to characterize also polymer
layers and blends of two different polymers. The fits were already close
but can be further improved by fine tuning the parameters and amount of
oscillators which can be time consuming. In the outline of this thesis, the
polymer samples are merely a demonstration of the possibilities of SCOUT
and no further improvement or analysis is performed.
84
CHAPTER 4. MODELLING RESULTS
Chapter 5
Conclusions
Optical characterization with variable angle spectroscopy and SCOUT software proved to be an accurate method for determining the optical properties
of different layers of a thin-film silicon solar cell. For TCOs, a model composed of a Brendel oscillator, extended Drude model and OJL interband
model was sufficient for obtaining close simultaneous fitting results on specular reflectance and transmittance spectra measured at different polarizations and angles of incidence. With this model it is possible to characterize
AZO, ITO and FTO. The reproducibility of the results is within 5 percent
for most of the fitting parameters except for the Urbach energy that showed
a slightly larger deviation. The refractive index and optical bandgap are
comparable to expected values from literature and are a reliable output of
this model. The electrical parameters such as the mobility that can be calculated from the Drude parameters is however less reliable when comparing
the results with Hall measurements. This difference is mainly due to the
fact that for optical measurements only a small spot is analysed whereas for
Hall measurements the current passes through the entire layer and is more
influenced by grain boundaries and defects.
With the implementation of the equations for rough surfaces, based on
the scalar scattering theory, the model proved to be also useful for characterizing rough layers. In this case the fitting results on the reflectance are
highly accurate whereas the transmittance shows a larger deviation. This
is due to the model itself that is less accurate but also due to the fact that
SCOUT does not allow the refractive index as input. With these limitations
of SCOUT a compromise had to be made which approximates a linear refractive index in the considered wavelength range. By introducing a weight
factor of 0.1 on these transmittance spectra it was found that these less
accurate fits do not have a big influence on the final outcome.
Silicon layers were also characterized using a similar model as for TCOs.
The Drude model was taken out because the material is intrinsic and free
carrier absorption is not observed in the considered wavelength range. With
85
86
CHAPTER 5. CONCLUSIONS
this interface, substrate temperature series of amorphous and microcrystalline silicon were characterized. The interface provided accurate results
also in this case. The main influence of temperature on the optical properties is the decrease in optical bandgap with substrate temperature that
might be due to a lower hydrogen concentration at elevated temperatures.
This effect was seen both for amorphous and for microcrystalline silicon. In
both cases also the Urbach energy was changing with temperature indicating that the microstructure of the layers slightly differs with temperature.
A more thorough research is however required to analyse these trends which
is beyond the scope of this thesis.
Besides these most important layers of thin-film silicon solar cells a model
was also created for nanoparticle layers. Nanoparticles are a concept for improving the light absorption. For these layers the Bergman effective medium
approximation was used. This model approximates the layer as a homogeneous layer with a refractive index composed of those of the two different
materials. With the Bergman model it is possible to follow the changes of
the reflectance and transmittance spectra with particle size.
As a final topic also polymer samples were characterized. These samples
were supplied by the Technical University of Eindhoven. For the PCPDTBT
layers a set of Brendel oscillators against a dielectric background was sufficient for simulating the R/T spectra. After that the Bruggeman effective
medium approximation was applied to model the mixture of PCPDTBT
with PCBM.
As final conclusion of this research it can be said that SCOUT is a valuable and powerful tool for characterizing layers of many different materials.
For the most commonly used materials in the PVMD laboratory, interfaces
are created that allow quick and easy characterization of layers. The input
can be from a variety of measurement equipment being the ARTA accessory,
integrating sphere or mini-RT.
Chapter 6
Recommendations
The first and most important recommendation is to use the large ARTA
model for the most accurate results. Although the measurement time for
one sample is in the order of three hours per sample, the fitting results will be
significantly better compared to using the mini-RT setup and model. This
is especially the case for rough layers where bandgap, Brendel oscillator and
roughness model strongly interfere with each other in the bandgap region.
This region is for TCOs out of range for the mini-RT that can lead to strange
outcomes. For a quick analysis of the thickness of flat layers the mini-RT
or integrating sphere will be sufficient since the thickness has a strong effect
on the spectra and can be easily fitted with less parameters and spectra.
With regard to the modelling itself it is recommended to find a suitable way to implement the roughness equation for transmittance spectra.
With the linear approximation of the refractive index and lack of proper
input of wavelength dependent correction functions the fitting accuracy on
transmittance spectra is reduced to a large extent and weight factors are
required.
87
88
CHAPTER 6. RECOMMENDATIONS
Appendix A
Thickness from interference
pattern
In section 2.1.3 it was mentioned that the thickness of a layer can be estimated using the expression
d=
wλ1 λ2
2 · [n(λ1 )λ2 − n(λ2 )λ1 ]
(A.1)
where the refractive index n and locations of the maxima and minima of
the oscillation are used as input. A regular transmittance spectrum shows
multiple peaks. This means that for a single layer, multiple combinations of
peaks can be used to calculate the thickness. A script has been written to
calculate the thickness for every peak combination. An example is shown
in figure A.1 for an AZO layer of which the refractive index and thickness
(585 nm) were determined with the ARTA SCOUT model.
Figure A.1: Obtained thickness for an AZO layer with the interference
model.
89
90
APPENDIX A. THICKNESS FROM INTERFERENCE PATTERN
The obtained thickness’s have a Gaussian-like distribution. The box
plot indicates that the average is located around 600 nm. The blue box
represents the area in which the ‘middle’ 50% of the data can be found and
the whiskers point out the furthest datapoint within plus or minus 1.5 times
the box length. The rest of the data is indentified as potential outlier and
is represented by a red cross. There is still a large spread of data points
and also a large amount of outliers. Although the average value comes close
to the thickness determined with the ARTA model the accuracy of this
method is not sufficient. Furthermore this method is only applicable for flat
layers with a thickness in the order of hundreds of nanometers of which the
refractive index is known, which further limits its usability.
Appendix B
Derivation of the wave
equations
In this appendix the wave equations are derived staring from Maxwell’s
equations presented in section 2.1.1. The second Maxwell equation with a
substitution of B from the material equation 2.6 yields
1
∇ × E = − µḢ.
c
Dividing this equation by µ and applying the operator curl gives
i
1
1h
∇×
(∇ × E) +
∇ × Ḣ = 0.
µ
c
(B.1)
(B.2)
The time derivative of the first Maxwell equation (eq. 2.1), with D given
by the material equation 2.5, results in a function of ∇ × Ḣ:
1
ε
∇ × Ḣ = D̈ = 2 Ë
c
c
which can be substituted in eq. B.2 to give the following relation:
ε
1
∇×
(∇ × E) + 2 Ë = 0.
µ
c
(B.3)
(B.4)
From mathematics, the terms in this equation can be rewritten according
to the identities
∇ × uv = u(∇ × v) + ∇u × v
(B.5)
∇ × (∇ × v) = ∇(∇ · v) − ∇2 v
(B.6)
which allows equation B.4 to be rewritten in the form:
∇2 E −
εµ
Ë + ∇ ln µ × (∇ × E) − ∇(∇ · E) = 0.
c2
91
(B.7)
92
APPENDIX B. DERIVATION OF THE WAVE EQUATIONS
The divergence of E given in the last term of this equation is obtained by
taking the third Maxwell equation with again the material equation 2.5. this
yields for the divergence:
∇ · E = −E∇ ln ε.
(B.8)
Substitution into equation B.7 gives the wave equation
εµ
Ë + ∇ ln µ × (∇ × E) + ∇(E∇ ln ε) = 0
(B.9)
c2
and in a similar way the wave equation is obtained for the magnetic field
vector H
∇2 E −
εµ
Ḧ + ∇ ln ε × (∇ × H) + ∇(H∇ ln µ) = 0.
(B.10)
c2
When the material is assumed to be homogeneous, ∇ ln ε and ∇ ln µ
are zero and these equation can be simplified to the more familiar wave
equations:
∇2 H −
∇2 E −
εµ
Ë = 0,
c2
∇2 H −
εµ
Ḧ = 0.
c2
(B.11)
Appendix C
Mini-RT models for layer
thickness
The thickness of a layer has a strong impact on the R/T spectra because
of the interference pattern. The thickness can therefore be more easily determined with high accuracy than other parameters. A SCOUT interface is
prepared for determining the thickness with the mini-RT setup. This setup
is ideal for this purpose because it is quick and provides enough information
to do a fitting of the thickness. The model is a downgrade of the ARTA
model. The first difference is that the amount of spectra is brought down
from seventeen to only two spectra. These are reflectance and transmittance
at normal incidence over a wavelength range of 400–1000 nm. Because of
this smaller wavelength range the bandgap of TCOs is not visible in these
spectra. The parameters of the OJL bandgap model will therefore run away
to high (or low) and unrealistic values. The bandgap model is therefore
taken out. The Brendel oscillator, extended Drude model and Bruggeman
effective medium for the surface mixture are kept and the parameters in
table C.1 are used for the fitting.
Submodel
Extended Drude model
Brendel oscillator
Other
Fitting parameter
Plasma frequency
Low freq. damping
Crossover frequency
Crossover width
Resonance frequency
Oscillator distrib. width
Dielectric background
AZO layer thickness
units
[eV]
[cm−1 ]
[cm−1 ]
[cm−1 ]
[eV]
[cm−1 ]
[-]
[nm]
Table C.1: mini-RT flat TCO fitting parameters.
93
94
APPENDIX C. MINI-RT MODELS FOR LAYER THICKNESS
Figure C.1: mini-RT model fit on measurements of a flat ITO layer.
The flat ITO that was fitted with the ARTA interface in section 4.2.3 is
also characterized with this mini-RT model to determine the thickness. The
fitting results are presented in figure C.1. The thickness was found to be
506 nm which is comparable to the thickness found with the ARTA model
(485.8 nm). The difference is mainly due to the spot where the measurement
is done because the deposition is not uniform. The fitting is repeated five
times from different starting positions and the deviation of the thickness was
within 2 nm.
The mini-RT model can only be used for flat layers since the roughness
model contains to much parameters to provide a reliable fit. The layer stack
in SCOUT is positioned with the glass in front to prevent damaging the
layer when doing the mini-RT measurements. The interface of this model
can be found in appendix D.
Appendix D
SCOUT interfaces
TCOs with ARTA The interface for fitting rough TCO layers is presented in figure D.1. The bottom graphs show the simulated (blue) and
measured (red) spectra. In top right graph the resulting refractive index of
the TCO layer can be found. The interface also shows the fitting parameters
of which the five selected main parameters are manually adjustable with the
slider bars in the top left corner. The ‘view’ button in the toolbar toggles
between different pages containing the other spectra. The interface looks
similar for the different TCO materials except for the colour of the interface
for a better distinction.
Figure D.1: ARTA interface for rough TCO layers.
95
96
APPENDIX D. SCOUT INTERFACES
TCO‘s with mini-RT The interface for determining the layer thickness
of flat TCO samples with mini-RT is quite similar to the ARTA model
(see figure D.2). The layout is the same and the main difference is that
the amount of spectra is reduced to only the two spectra at the bottom.
Furthermore the amount of fitting parameters is reduced as explained in
appendix C. The slider bars now only control the layer thickness, plasma
frequency and oscillator resonance frequency.
Figure D.2: Mini-RT interface for thickness determination of TCO layers.
Silicon layers with mini-RT and ARTA In combination with ARTA
the interface in figure D.3 is used for the fitting procedure. The amount
of spectra is exactly the same as for TCO layers and the only difference is
that the extended Drude model is taken out because there is no significant
free carrier absorption in the considered wavelength range. This results in
less fitting parameters and the slider bar for plasma frequency is no longer
necessary.
In case of the mini-RT, the standard interface can be used that is delivered with the SCOUT software (see figure D.4). This interface functions
in a similar way as all the other interfaces but the visual appearance is a
little different. The difference with the ARTA model is the reduction of
spectra from seventeen to only the two spectra that can be measured with
the mini-RT. Furthermore the wavelength range is reduced.
97
Figure D.3: ARTA interface for silicon layers layers.
Figure D.4: Mini-RT interface for silicon layers.
98
APPENDIX D. SCOUT INTERFACES
Nanoparticles with ellipsometry The interface for fitting ellipsometry
data looks different from the TCO and silicon interfaces. There are only
three slider bars that control the thickness, volume fraction and percolation
which are the main inputs for the Bergman effective medium model. The
spectral density function cannot be added in the main interface and has to
be controlled internally. The graphs represent the ellipsometry parameters
measured at different angles of incidence so a simultaneous fit can be done
with up to five angles.
Figure D.5: Ellipsometry interface for nanoparticles layers.
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List of Figures
1.1
Predictions of world energy consumption according to different scenarios. [60] . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Concentration of greenhouse gasses over the last 2000 years.
[39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Three generations in solar cell research. [21] . . . . . . . . . .
1.4 Structure of amorphous silicon. [28] . . . . . . . . . . . . . .
1.5 Bandgap of (a) crystalline silicon and (b) amorphous silicon.
[79] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Absorption coefficient of crystalline and amorphous silicon. [79]
1.7 (a) Structure and (b) band diagram of an a-Si:H solar cell. [79]
1.8 (a) Typical surface texture of AZO after etching and (b) effect
of surface roughness on light transmission. . . . . . . . . . . .
1.9 Scattering from nanoparticles. [5] . . . . . . . . . . . . . . . .
1.10 Effect of nanoparticle geometry on (a) light coupling into the
solar cell and (b) optical path enhancement. [19] . . . . . . .
1.11 (a) Layer stack of a polymer solar cell and (b) band diagram.
[79] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
1
2
3
4
4
5
5
6
7
7
8
Spectrum of electromagnetic radiation. [2] . . . . . . . . . . .
Reflection and transmission at the boundary of two media. .
Internal reflections causing interference. . . . . . . . . . . . .
Feedback loop for fitting R/T spectra. . . . . . . . . . . . . .
Valence and conduction band in the OJL interband model. [72]
Schematic system of a damped harmonic oscillator. . . . . . .
Definition of an ‘effective medium’. . . . . . . . . . . . . . . .
Spectral density function of the Maxwell-Garnett Theory. [72]
Spectral density function of the Bruggeman model. [72] . . .
Material definitions with a manually composed master model
(bottom left) and the SCOUT database (right). . . . . . . . .
2.11 Exponential correction on Rspec due to scattering. . . . . . .
13
14
15
17
19
20
22
24
25
3.1
3.2
31
32
Magnetron sputtering; (a) schematic [54] and (b) setup. . . .
Chemical vapor deposition; (a) schematic and (b) setup. . . .
105
26
29
106
LIST OF FIGURES
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
Thermal evaporation; (a) schematic and (b) setup. . . . . . .
Lambda 950/ARTA configuration. . . . . . . . . . . . . . . .
Specular transmittance through a layer stack. . . . . . . . . .
Defined angles of (a) sample holder and (b) detector in ARTA.
Accuracy test for a rough AZO layer at 45 degrees. . . . . . .
R/T spectra for different sample placement configurations. . .
Lambda 950 equipped with the integrating sphere accessory.[80]
Eta-Optik mini-RT Setup at Delft University of Technology. .
AFM: (a) operation principle, (b) semi-contact mode. [74] . .
AFM Setup at Delft University of Technology. . . . . . . . . .
Hall effect [36]. . . . . . . . . . . . . . . . . . . . . . . . . . .
Hall Setup at Delft University of Technology. . . . . . . . . .
(a) Elliptical polarization and (b) ellipsometry configuration.
4.1
(a) Fitting results and (b) refractive index of Corning Eagle
2000 glass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asahi Glass; (a) fitting results at 30 degrees angle of incidence
and (b) refractive indices. . . . . . . . . . . . . . . . . . . . .
Layer stack definitions for rough TCO samples. . . . . . . . .
AFM scans of AZO samples with 10, 35 and 50 seconds of
etching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Roughness of etched AZO layers. . . . . . . . . . . . . . . . .
The effect of surface roughness at 45 degrees angle of incidence and s-polarized light. . . . . . . . . . . . . . . . . . . .
Mobility, conductivity and carrier density as function of etching time for AZO. . . . . . . . . . . . . . . . . . . . . . . . .
Fitting results for a flat AZO layer under p-polarized light. .
Fitting results for a flat AZO layer under s-polarized light. . .
Obtained optical properties of a flat AZO layer. . . . . . . . .
Fitting results for a rough AZO layer under p-polarized light.
Fitting results for a rough AZO layer under s-polarized light.
Obtained optical properties for a chemically wet-etched AZO
layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abundance of chemical elements. [48] . . . . . . . . . . . . .
Measured spectra of an ITO sample at p-polarization and
different angles of incidence. . . . . . . . . . . . . . . . . . . .
Fitting results for a flat ITO layer under p-polarized light. . .
Fitting results for a flat ITO layer under s-polarized light. . .
Obtained optical properties of a flat ITO layer. . . . . . . . .
Surface roughness of an FTO layer. . . . . . . . . . . . . . . .
Measured spectra of an FTO sample at p-polarization and
different angles of incidence. . . . . . . . . . . . . . . . . . . .
Fitting results for a rough FTO layer under p-polarized light.
Fitting results for a rough FTO layer under s-polarized light.
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
32
33
34
34
35
36
37
37
38
39
39
40
40
44
44
46
48
49
49
51
52
52
53
54
54
55
56
56
57
58
58
59
60
61
61
LIST OF FIGURES
107
4.23 Obtained optical properties of a flat FTO layer. . . . . . . . . 62
4.24 Standard deviation of the fitting parameters after 10 automated fittings. . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.25 Standard deviation of the refractive index from rough samples. 63
4.26 Comparison of RMS roughness from AFM and the model. . . 64
4.27 Wavelength dependent damping term for TCO materials. . . 65
4.28 Determination of the effective mass. . . . . . . . . . . . . . . 66
4.29 Verification of the mobility. . . . . . . . . . . . . . . . . . . . 66
4.30 Fitting results for a-Si:H at T=100◦ C and 300◦ C. . . . . . . . 69
4.31 Bandgap and Urbach energy as function of substrate temperature for a-Si:H. . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.32 Refractive index of a-Si:H for different substrate temperatures. 70
4.33 Mini-RT vs. ARTA model for a-Si:H samples. . . . . . . . . . 70
4.34 n,k comparison of a-Si:H from the mini-RT and ARTA model. 71
4.35 Fitting results for µc-Si:H at T=100◦ C and 300◦ C. . . . . . . 72
4.36 Bandgap and Urbach energy as function of substrate temperature for µc-Si:H. . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.37 Refractive index of µc-Si:H for different substrate temperatures. 73
4.38 Silver nanoparticles, left: AFM scan and right: SEM image. . 74
4.39 Ellipsometry measurements of deposited silver layers with a
different mass thickness for a 60 degrees angle of incidence. . 75
4.40 Fits on ellipsometry data for a silver layer at 55 and 70 degrees
angle of incidence. . . . . . . . . . . . . . . . . . . . . . . . . 76
4.41 Refractive index of deposited silver layers with a different
mass thickness. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.42 Volume fraction and percolation as function of deposited silver mass thickness. . . . . . . . . . . . . . . . . . . . . . . . . 77
4.43 Refractive index of deposited silver layers with a different
thickness after annealing at 400◦ C. . . . . . . . . . . . . . . . 77
4.44 Evolution of the refractive index of a silver nanoparticle layer
during the formation process. . . . . . . . . . . . . . . . . . . 78
4.45 Fits of the model on R/T for particles sizes of 6 and 18 nm. . 79
4.46 Effect of nanoparticles size on the refractive index of the layer. 79
4.47 Molecular structures of PCPDTBT and PCPDTBT:PCBM.
[67] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.48 Implementation of the PCPDTBT:PCBM layer stack in SCOUT. 81
4.49 Specular R/T spectra of PCPDTBT at different angles and
polarizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.50 Fits on the measured spectra of the PCPDTBT sample. . . . 82
4.51 Specular R/T spectra of the PCPDTBT:PCBM blend at different angles and polarizations. . . . . . . . . . . . . . . . . . 83
4.52 Fits on the measured spectra of the PCPDTBT:PCBM blend. 84
4.53 Refractive indices of the PCPDTBT and PCPDTBT:PCBM
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
108
LIST OF FIGURES
A.1 Obtained thickness for an AZO layer with the interference
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
C.1 mini-RT model fit on measurements of a flat ITO layer. . . .
94
D.1
D.2
D.3
D.4
D.5
95
96
97
97
98
ARTA interface for rough TCO layers. . . . . . . . . . . . . .
Mini-RT interface for thickness determination of TCO layers.
ARTA interface for silicon layers layers. . . . . . . . . . . . .
Mini-RT interface for silicon layers. . . . . . . . . . . . . . . .
Ellipsometry interface for nanoparticles layers. . . . . . . . .
Nomenclature
Parameters
B
Magnetic induction
[T]
D
Electric displacement field
E
Electric field
[V/m]
H
Magnetic field
[A/m]
j
Current
Jf
Free current density
p
Momentum
v
Velocity
a
Spring damping coefficient
c
Speed of light
d
Thickness
e
Elementary charge
Ebg
Bandgap energy
F
Force
f
Volume fraction
[-]
G
Spectral Density
[-]
G0
Percolation strength
[-]
h
Planck‘s constant
k
Extinction coefficient
kn
Wavenumber
[C/m2 ]
[A]
[A/m2 ]
[Ns]
[m/s]
[A]
[m/s]
[nm]
[C]
[eV]
[N]
[Js]
[-]
[cm−1 ]
109
110
LIST OF FIGURES
L
Spring stiffness
[N/m]
m
Mass
MDr
Drude mobility
n
Refractive index
[-]
Ne
Electron density
[m−3 ]
q
Charge
R
Reflectance
[-]
r
Reflection coefficient
[-]
RH
Hall coefficient
[-]
T
Transmittance
[-]
t
Transmission coefficient
[-]
w
Order number
[-]
x
Distance
z
Height of the surface
[kg]
[cm2 /Vs]
[C]
[m]
[nm]
Greek symbols
χ
Susceptibility
[-]
∆
Ellipsometry parameter [Phase shift]
[rad]
Γ
Damping constant
[s−1 ]
γ
Breadth of the Urbach tail
[eV]
λ
Wavelength
[nm]
µ
Permeability
[-]
µH
Hall mobility
[cm2 /Vs]
ν
Angular frequency
ω
Frequency
[s−1 ]
ωpl
Plasma frequency
[eV]
Ψ
Ellipsometry parameter [Amplitude ratio]
[rad]
ρ
Total charge density
[rad/s]
[C/m3 ]
LIST OF FIGURES
111
[cm−1 Ω−1 ]
σH
Hall conductivity
σR
Root mean square roughness
σs
Standard deviation
[-]
τ
Relaxation time
[s]
θ
Angle
ε
Permittivity/Dielectric function
Subscripts
c
Conduction band
Dr
Drude
d
Displacement
eff
Effective
H
Hall
h
Host
Im
Imaginary part
part
Particular
pl
Plasma
p
Particle
Re
Real part
v
Valence band
k
Parallel
⊥
Perpendicular
Abbreviations
µc-Si:H Hydrogenated microcrystalline silicon
a-Si:H Hydrogenated amorphous silicon
AFM Atomic force microscopy
AM
Air mass
ARTA Angular reflectance/transmittance analyzer
[nm]
[deg]
[-]
112
LIST OF FIGURES
AZO Aluminum-doped zinc xxide
CCD Charge-coupled device
CVD Chemical Vapor Deposition
DOS
Density of states
EMA Effective medium approximation
FTO Fluorine-doped tin oxide
HCl
Hydrochloric acid
IR
Infrared
ITO
Tin-doped indium oxide
KKR Kramer-Kroning relations
MGT Maxwell-Garnett theory
OJL
O’Leary, Johnson, Lim
PbS
Lead sulfide
PCBM [6,6]-phenyl C61-butyric acid methyl ester
PCPDTBT poly[2,6-(4,4-bis-(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b’]dithiophene)alt4,7,(2,1,3-benzothiadiazole)]
PECVD Plasma enhanced chemical vapor deposition
rf
Radio frequency
RMS Root mean square
TCO Transparent conductive oxide
TIS
Total Integrating Sphere
UV
Ultraviolet
VAS
Variable angle spectroscopy
Vis
Visible
ZnO
Zinc oxide
Index
Absorption, 82
Amorphous, 4, 21, 68
Angle of incidence, 13, 34, 46
Annealing, 77
ARTA, 49, 56, 59, 70
Asahi, 6
Atomic force microscopy, 38, 48, 64
AZO, 6, 48
AZO properties, 51
Bandgap, 4, 19, 20, 53, 69, 71
Bergman representation, 22, 74
Brendel oscillator, 45, 67, 80
Bruggeman model, 24, 67, 81
Etching time, 48
Evaporation, (thermal), 32
Extinction coefficient, 13
Fit parameters, 26, 47, 68
Fitting, 16, 26, 51, 57, 69, 71, 82
Free carrier absorption, 45
Fresnel equations, 14
FTO, 6, 59
FTO properties, 61
Fullerene, 81
Hall coefficient, 39
Hall measurements, 38, 50, 57, 60, 64
Infrared (IR), 13
Calibration, 46
Carrier concentration, 38, 50, 57, 60, Interference, 15, 89
Intrinsic, 5
65
ITO, 6, 55
Chemical vapor deposition, 31
ITO properties, 58
Conductivity, 39, 50, 57, 60
Damping, (Drude), 18, 65
Dangling bonds, 4
Density of states, 19
Dielectric background, 80
Dielectric function, 16
Diffuse light, 6
Doping, 53
Drude model, (Extended), 17, 45
Effective mass, 17, 19, 65
Effective medium, 21, 74
Electromagnetic radiation, 11
Electron density, 18
Ellipsometry, 40
Error, 62, 71
Etching, 48
Lambda 950, 33
Light, 13
Maxwell equations, 11
Maxwell-Garnett theory, 23
Microcrystalline, 71
Microgeometry, (Particles), 23
Mini-RT, 67, 68, 71, 93
Mobility, 38, 50, 57, 60, 65
Modelling, 16, 45, 67, 74
Nanoparticles, 6, 73
OJL model, 19, 45, 67
Oscillator, Brendel, 20
Particles, 22
113
114
PCBM, 81
PCPDTBT, 80
Percolation, 23–25
Photon, 12
Planck, 12
Plasma frequency, 18, 45, 64
Plasmonic effect, 74
Polarization, 14
Polymers, 8, 80
Reflectance, 17, 34
Refraction, 13
Refractive index, 16, 53, 55, 58, 61,
70, 71, 83
Relaxation time, 17, 27
Roughness, 6, 28, 38, 45, 48, 49, 53,
60
Scalar scattering theory, 26
Scattering, 6, 26, 28
SCOUT, 25, 80
Silicon, 4, 67
Silver, 74
Simulation, 16, 26
Snell’s law, 13, 29
Spectral density, 23–25
Spectroscopy, 33
Spectrum, 13
Specular, 6, 28, 33
Sputtering, 31
Standard deviation, 62
Surface, 38
Susceptibility, 16, 18, 20, 21
Tail states, 19
TCO, 6, 44, 47
Thickness, 15
Thin film, 4
Transitions, 19
Transmittance, 17, 33
Ultraviolet (UV), 13
Urbach energy, 19, 63, 69, 71
Vibrations, 20
INDEX
Volume fraction, 22
Wave equations, 27, 92
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