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 . . . . iii . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3 4 8 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . rough surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 11 13 15 16 16 17 21 25 26 27 27 28 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 31 31 32 33 33 36 37 38 38 . . . . . . . . . . 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 . . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 43 43 44 45 48 55 59 62 64 67 67 68 71 73 73 74 74 77 78 79 80 80 81 84 5 Conclusions 85 6 Recommendations 87 A Thickness from interference pattern 89 B Derivation of the wave equations 91 C Mini-RT models for layer thickness 93 D SCOUT interfaces 95 Bibliography 99 List of Figures 103 Nomenclature 109 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. 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Ehancement of light trapping in thin film silicon solar cells. Master’s thesis, Delft University of Technology, 2009. 104 BIBLIOGRAPHY 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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