DESIGNING OPTICAL METAMATERIALS WITH HYPERBOLIC DISPERSION BASED ON AL/DIELECTRIC NANOLAYERED STRUCTURES. A Thesis. Presented to the.

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1 DESIGNING OPTICAL METAMATERIALS WITH HYPERBOLIC DISPERSION BASED ON AL/DIELECTRIC NANOLAYERED STRUCTURES A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Computational Science by Priscilla Noreen Kelly Spring 2016

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3 iii Copyright c 2016 by Priscilla Noreen Kelly

4 iv DEDICATION I dedicate this work to my grandmother, Linda Morales, whose sacrifice allowed my family and I to grow up in America. She was always there when we needed her and, at the end of her life, asked for nothing more than God s blessing and a good red lipstick.

5 v If you want to build a ship, don t drum up the men to gather wood, divide the work and give orders. Instead, teach them to yearn for the vast and endless sea. Antonie de Saint-Exupery

6 vi ABSTRACT OF THE THESIS DESIGNING OPTICAL METAMATERIALS WITH HYPERBOLIC DISPERSION BASED ON AL/DIELECTRIC NANOLAYERED STRUCTURES by Priscilla Noreen Kelly Master of Science in Computational Science San Diego State University, 2016 In the past decade, great leaps have been made in developing metamaterials man-made materials tailored to display behaviors not found in nature. This research looks away from conventional metals, such as Au and Ag, and towards Al which has been shown to have lower losses in the optical region and a higher plasma frequency, which is related to carrier concentration. Paired with nano layers of SiO 2 or ZnO, the overall structure can create hyperbolic dispersion. This dispersion occurs when the parallel and perpendicular permittivites have opposite signs leading to enhanced stimulated emission and decreased rates. Using spectroscopic ellipsometry, the deposition of Al with both SiO 2 and ZnO was investigated. This work utilizes the Transfer-Matrix formalism with the effective medium approximation to predict ellipsometry data, and the Levenberg-Marquardt algorithm is used to fit it. These techniques allow for conclusions to be made about material properties. Results show that at room temperature, RF sputtering of Al with SiO 2 does not deposit pure Al. Instead, a layer Al 3 O 2 was found to best-fit the data. Modeling efforts show that the Al content increased by 13% within the Al 3 O 2 layer when the temperature during deposition increased. The presence of Al 3 O 2 inhibits hyperbolic effects since it is a dielectric, so new techniques will be considered to fabricate Al/SiO 2 in further research. When Al is combined with ZnO using atomic layer deposition (ALD), it exists as a Al-doped ZnO layer, also called AZO. An initial investigation for the ellipsometry data used the Drude model and found that a nanolayered structure, made using ALD, had an order-of-magnitude lower losses than silver and other AZO/ZnO growing methods. Future work will look at developing a more complex model for AZO and experimental means of verifying these findings.

7 vii TABLE OF CONTENTS ABSTRACT... LIST OF TABLES... LIST OF FIGURES... ACKNOWLEDGMENTS... CHAPTER PAGE 1 INTRODUCTION Motivation Objectives MODELS FOR METALS AND DIELECTRICS Metals Dielectrics Silicon and Silicon Dioxide Zinc Oxide Aluminum Oxide ELLIPSOMETRY: MODELING AND FITTING Principles of Ellipsometry Modeling a Uniaxial, Anisotropic Material System Effective Medium Approximation Matrix Solutions for a Anistropic Material Transfer-Matrix Levenberg-Marquardt Algorithm FABRICATION METHODS: SPUTTERING AND ATOMIC LAYER DEPOSITION Sputtering: Aluminum and Silicon Dioxide Atomic Layer Deposition: Aluminum and Zinc Oxide MODELING FOR KNOWN STRUCTURES Proof of Concept with ZnO, Si, and SiO ALUMINUM AND SILICON DIOXIDE vi ix x xii

8 viii 6.1 Sample Specifications Fitting Results ALUMINUM-DOPED AND ZINC DIOXIDE Sample Specifications Fitting Results CONCLUSIONS AND FUTURE DIRECTIONS BIBLIOGRAPHY APPENDIX SUPPLEMENTARY CODE... 41

9 ix LIST OF TABLES PAGE Table 2.1. Al parameters for Drude-Critical Point model Table 2.2. Anisotropic Al 3 O 2 parameters for the Cauchy model in ev Table 3.1. Eigenvalue order for each iteration in the β summations Table 6.1. Growing parameters for Al/SiO 2 samples Table 6.2. Final Fitting heights [nm] for the Al/SiO 2 samples Table 7.1. Growing parameters, surface roughness, and EDX values for AZO/ZnO samples Table 7.2. Final modeling parameters for AZO/ZnO samples... 31

10 x LIST OF FIGURES PAGE Figure 1.1. Light propagation in a right-handed and left-handed material. Light is incident from the left, and the transmitted light in a right-handed material is shown in the solid arrow, while the dashed arrow represents light transmitted in a left-handed material Figure 1.2. Schematic of a nano-layered structure of a metal, in blue, and dielectric, in beige Figure 1.3. The iso-frequency surface shift from a closed to an open surface at the optical topological transition point Figure 2.1. Aluminum permittivity, real and imaginary, using the Drude-Critical Point model Figure 2.2. Si permittivity, real and imaginary, using the Sellmeier model Figure 2.3. SiO 2 permittivity, real and imaginary, using the Sellmeier model Figure 2.4. ZnO permittivity, real and imaginary, using the Adachi Model Figure 2.5. Al 3 O 2 permittivity, real and imaginary, using the Cauchy Model. In the Re(ε) lines, the top line is the parallel permittivity and the bottom is perpendicular Figure 3.1. s- and p-polarized light irradiated onto a sample. Ψ is the amplitude ratio between the rotated light and the incident light, and is the change in phase between the reflected s- and p-polarized light Figure 3.2. Model of s- and p- polarized light incident on an ambient/righthanded anisotropic film/isotropic substrate structure Figure 5.1. SEM image of ZnO/SiO2/Si using 5kV. Height measurements using SEM software predict the ZnO layer height to be 157 nm Figure 5.2. ZnO/SiO 2 /Si layers modeled with the Transfer-Matrix in red with measured ellipsometry data in black at an incident angle of 45 o Figure 6.1. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:1 with a model, in red, of 7% of 10 nm as Al at an incident angle of 45 o Figure 6.2. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:1 with 250 nm of SiO 2, in red, at an incident angle of 45 o Figure 6.3. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:1 with a layer of Al 2 O 3, in red, at an incident angle of 45 o Figure 6.4. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:2 with a layer of Al 2 O 3, in red, at an incident angle of 45 o

11 Figure 6.5. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:3 with a model of 7 nm of Al 2 O 3 and 4.9 nm of SiO 2, in red, at an incident angle of 45 o Figure 6.6. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:3 with a layer of SiO 2 and 13% Al in Al 2 O 3 and using the Maxwell-Garnett formula, in red, at an incident angle of 45 o Figure 7.1. SEM image of sample AZ:1 using 5kV. Height measurements using SEM software predict the AZO/ZnO layer height to be 302 nm Figure 7.2. SEM image of sample AZ:2 using 5kV. Height measurements using SEM software predict the AZO/ZnO layer height to be 313 nm Figure 7.3. Results of analysis of the generalized ellipsometry data with an incident angle of 70 o for AZ:1 which had a 1:9 ALD cycle ratio: AnE (a) Ψ and (b), Aps (c) Ψ and (d), and Asp (e) Ψ and (f) Figure 7.4. Results of analysis of the standard ellipsometry data (AnE) for AZ:1 which had a 1:9 ALD cycle ratio: 70 o (a) Ψ and (b), 60 o (c) Ψ and (d), and 45 o (e) Ψ and (f) Figure 7.5. Results of analysis of the generalized ellipsometry data with an incident angle of 70 o for AZ:2 which had a 1:49 ALD cycle ratio: AnE (a) Ψ and (b), Aps (c) Ψ and (d), and Asp (e) Ψ and (f) Figure 7.6. Fitted real (a), and imaginary (b) permittivity for AZ:1 and fitted real (c), and imaginary (d) permittivity for AZ:2 using the parameters in Table xi

12 xii ACKNOWLEDGMENTS In choosing a school to attend, I thought the best quality SDSU had was its close proximity to my family. However, I soon learned it was the professors and administrators that really held the worth. I d first like to thank Dr. Jose Castillo, the director of the Computational Science Research Center, and Dr. Satchi Venkataraman for mentoring me through the master s program. Without their support, I would not have been introduced to my advisor Dr. Lyuba Kuznetsova, received the G-STEM Scholarship, and the 2015 NSF Graduate Fellowship to continue my research. My deepest gratitude is given to Dr. K for her wisdom and enthusiasm about optics and the process of research. Although I started off dreading clean room days, being able to develop and characterize those materials to answer our research questions overshadowed that. This work is just the beginning of developing exciting, new materials, and I m looking forward to working with Dr. K on them for the next few years. I would also like to acknowledge Dr. Peter Blomgren. His practical teaching and encouragement solidified computational science as the career path for me. Lastly, I would like to thank my family and friends: my labmate and friend, Carla Bacco, for all her humor and motivation in the clean room and during rock climbing; my parents, Clifford and Patricia, and my sister, Elizabeth, for listening to my talks and praying for me each step of the way; my best friend, Susan, for always believing in me and agreeing to read through rough drafts late at night.

13 1 CHAPTER 1 INTRODUCTION The 20th century is marked by the advent of electronics. Research spanning the century has pushed the size of electronics down to the nanometer scale we know today [24]. While this is a great feat, the rate for increased speed has slowed as size reduction reaches its limits [24]. Transistors are made with more connectors each year, thereby increasing their output potential, but transferring that information out to other portions of the chip has become a problem. Even though the next transistor is perhaps a few centimeters away, traditional wires cannot transmit the load we now use [24]. New methods of transferring information across a device must be considered. One theoretical solution has been proposed in the use of light to transmit information in the field of photonics. Optical fibers have been shown to carry over one-thousand times more digital data than traditional transmitting media and operate at faster speeds [24]. The only problem is they are also one-thousand times larger than today s electronic devices. Unifying photonic and electric devices at the nanometer scale is an impossible task until the diffraction limit can be broken. Many researchers today think that negative-refractive materials with hyperbolic dispersion may be able to solve this [26]. 1.1 MOTIVATION A material s refractive index is a ratio of the speed of light within the material, ν compared to a vaccum, c. It can also be described by how the electromagnetic wave is affected by the material. Permittivity, ε, for the electric field, and permeabilty, µ for the magnetic. n = c ν = εµ (1.1) n is a complex value, like ε and µ. The real part represents the energy stored in the material and the imaginary is related to the dissapation of energy, or losses, in the material. In 1967, a Russian physicist named Viktor Veselago showed that it was theoretically possible to have a left-handed material arise from negative values in both permittivity and permeability [37]. Veselago noted that this relationship cannot distinguish whether or not ε or µ are simultaneously negative or positive. He then looked at where ε and µ appear separately in Maxwell s equations to see what behaviors could arise. Most notably, Veselago found that, for a plane monochromatic wave, its propagation behaves as a left-handed set instead of a right-handed set. This means that an incident light entering a left-handed material would transmit with a negative angle compared to the normal.

14 2 This is opposite of what happens in right-handed materials, which are ubiquitous in nature. Light transmitting in both left- and right-handed materials is shown in Figure 1.1. Figure 1.1. Light propagation in a right-handed and left-handed material. Light is incident from the left, and the transmitted light in a right-handed material is shown in the solid arrow, while the dashed arrow represents light transmitted in a left-handed material. Veselago also found that light moving in a left-handed material would also reverse the sign of the Poynting vector. This vector, calculated by S = 1 E B, (1.2) µ 0 measures the energy flux carried by the wave. In a right-handed material, this energy would travel down into the material. In a left-handed material, it is reversed as seen in Figure 1.1. Veselago s work remained purely a theoretical exercise until 2001, when Richard Shelby and colleagues at UCSD developed a left-handed material using unit cells of copper strips and split-ring resonators [35]. Concurrently, John Pendry developed a mathematical description for the transformation of optical waves through left-handed media [25]. These two events sparked the study of man-made materials, or metamaterials, with exotic properties never before seen. The focus of this research is the optical property of light dispersion within a material. Derived from the energy density of an electric field within a material, the dispersion relationship describes the behavior of a wave propagating in medium for a given frequency

15 3 Figure 1.2. Schematic of a nano-layered structure of a metal, in blue, and dielectric, in beige. [9]. For most materials, the dispersion relationship is not directionally dependent. For an anisotropic media, the dispersion is dependent on the material s optical constants. In 2005, Viktor Podolskiy and Evgenii Narimanov showed that a layered structure, such as the one in Figure 1.2, can exhibit negative refraction when transverse-magnetic (TM) modes travel through the material. In this case, no magnetic effects need to be considered. Particulary for optical and infrared light, this finding is essential as there are no naturally occuring materials with a negative magnetic response, µ < 0, as there are with the electric field, i.e. metals such as silver, gold, and aluminum. When considering the structure of a nanolayered material, as shown in Figure 1.2, the material s permittivity can be described as transversely-isotropic or uniaxial by optical scientists meaning optical properties are symmetrical about one axis. In contrast, biaxial materials exhibit behaviors along two optical axes. For the uniaxial case, waves traveling in the parallel direction, x-y axis, are called normal waves and those traveling along the perpendicular direction, z-axis, are called extraordinary waves. ε = ε xx ε yy 0 (1.3) 0 0 ε zz The equation governing the dispersion of a TM-polarized wave, which is the only mode that can exhibit negative refraction [26], in a uniaxial media is k 2 x + k 2 y ε + k2 z ε = ω2 c 2, (1.4) where the k-vector represents the wavevector in space [16]. Equation 1.4 visualizes the k-vector as characteristic surfaces that depends on the values of ε and ε.

16 4 Two types of k-vector surfaces can exist for a fixed frequency, or iso-frequency, when the signs of the parallel and perpendicular permittivities are considered. A closed, circular or elliptical surface occurs if ε and ε > 0 and they are not equal; this is shown in the left of Figure 1.3 (a). Figure 1.3 (b) shows the iso-frequency wavevector surface once ε becomes negative and ε remains positive. This shift is called the optical topological transition (OTT) [16]. Figure 1.3. The iso-frequency surface shift from a closed to an open surface at the optical topological transition point. Since the photonic density of states (PDOS) is related to the enclosed surface of the iso-frequency surface [3], there are theoretically an infinite number of states for photons to exist, and these wavevectors would include magnitudes that are larger than those allowed in a vacuum [16]. These qualities would result in the enhancement of spontaneous emission and decay rates which could potentially lead to ultra-fast and ultra-small LEDs. 1.2 OBJECTIVES The research presented here will focus on designing uniaxial nanolayered structures then identifying where hyperbolic dispersion occurs, i.e. when ε and ε are of opposite signs. This is done by first taking models for different materials in literature, explained more in Chapter 2, then comparing predicted data to measured ellipsometry data. The details of how model data will be fitted to ellipsometry will be expanded on Chapter 3. Two sets of samples were grown in collaboration with Brookhaven National Lab and UCSD s Nano3 clean room. Details on their growth will be covered in Chapter 4. The results of how known materials are fitted with ellipsometry data is given in Chapter 5. Where this research stands out is in modeling the properties of individual layers and how they relate to the overall sample s permittivity. The results of fitting sample data to a model are shown in Chapters 6 and 7.

17 5 CHAPTER 2 MODELS FOR METALS AND DIELECTRICS This chapter will review existing models for the permittivity of metals and dielectrics published in literature, particularly in the UV and infrared regions, 400 to 1700 nm. The Effective Medium Approximation (EMA) model is then used to extract the parallel and perpendicular permittivity of metal/dielectric layers. Equations for permittivity are either derived from physical theories of the material or empirically. They are given as functions of wavelength, λ, in nanometers, or frequency, ω, with units of inverse seconds. The relationship between these two is ω = 2πc/λ. 2.1 METALS The Drude model, Equation 2.1, is commonly used to predict permittivity in metals. The model considers oscillations of free electrons with a resonant or plasma frequency, ω D, and a damping frequency, γ. ε is the permittivity at infinite frequency. ε D (ω) = ε ω 2 D ω 2 + iγω This model is a relatively simple way to determine the permittivity of conventional metals. Lorentzian terms need to be added to consider long-range interactions. Aluminum (Al) was chosen as the metal component for the nanolayered material, (2.1) despite the rapid oxidation issues, because its ω D is in the ultraviolet region, approximately 82 nm [39]. This can potentially provide the bandwidth needed for an ultraviolet LED. Al is also CMOS-compatible, allowing for integration into current nano-fabrication methods [11]. In the range between 500 and 1000 nm, additional terms that model critical points in the band-gap must be added to the Drude model to accurately predict the permittivity of Al [38]. This modified model is called the Drude-Critical Point. Figure 2.1 shows the permittivity and the parameter values used for this research are given in Table 2.1. ε DCP (ω) = ε ω 2 D ω 2 + iγω + 2 ( A p Ω p p=1 e iφp Ω p ω iγ + ) e iφp Ω p + ω + iγ (2.2) 2.2 DIELECTRICS Two different dielectrics were chosen to pair with Aluminum: Silicon Dioxide (SiO 2 ) and Zinc Oxide (ZnO). SiO 2 is ubiquitous in current technology, so integrating it into a

18 6 Figure 2.1. Aluminum permittivity, real and imaginary, using the Drude-Critical Point model. Table 2.1. Al parameters for Drude-Critical Point model. Name [nm] A B 1.575e6 C 1e7 D E F G H hyperbolic metamaterial would allow for a straightforward incorporation by industry. ZnO was chosen because it has shown potential in ultraviolet lasing, photodetection, and infrared imaging applications [10], [30], [31] Silicon and Silicon Dioxide For the research shown here, data tables of permittivity were used for Si and SiO 2 [12]. Seven different samples of SiO 2 on an Si substrate were tested using spectroscopic ellipsometry to produce data over the wavelength range of 187 to 1770 nm. Using this data is preferred over the basic Sellmeier model because it considers the transition layer at the Si-SiO 2 interface [12]. Figure 2.2 shows the permittivity for Si and Figure 2.3 shows SiO 2.

19 7 Figure 2.2. Si permittivity, real and imaginary, using the Sellmeier model Zinc Oxide ZnO is weakly anisotropic, and it has a excitation peak at approximately 365 nm which makes modeling its permittivity difficult with conventional dielectric models. The Adachi model is used here because it incorporates parabolic conduction and valence bands, excitation states at the band gap, and the contribution of continuum excitons to accurately predict ZnO permittivity in its most important range of 250 to 833 nm [44]. Figure 2.4 shows the permittivity given by the Adachi model. However, due to the complexity of the model, it will not be discussed further here Aluminum Oxide Al is notorious for oxidizing when exposed to the atmosphere, and this is a problem for developing a hyperbolic metamaterial. Once Al becomes oxidized to Al 3 O 2, it loses its metallic properties as the permittivity is no longer negative. Therefore, for the purpose of developing a material with hyperbolic dispersion, limiting its growth is essential. There are two processes that can cause an oxide layer, Al 3 O 2, to grow. The first is natural oxidation which could occur when the structure is exposed to atmosphere. The second is in situ, or while the material is growing. Since the Al is grown in the same atmosphere with SiO 2, it is possible that the Al layer could either be completely oxidized or a mixture of Al and Al 3 O 2. The Cauchy model, used here for Al 3 O 2, is shown in Equations [42]. ñ = n + ik, (2.3) n(λ) = A + B λ + C λ + D 2 λ + E 3 λ, (2.4) 4

20 8 Figure 2.3. SiO 2 permittivity, real and imaginary, using the Sellmeier model. k(λ) = F 10 5 e [(λ G)/H]3 + I. (2.5) Table 2.2 gives both sets of parameters for Al 3 O 2, parallel and perpendicular and Figure 2.5 shows the permittivity. Table 2.2. Anisotropic Al 3 O 2 parameters for the Cauchy model in ev. Name A B C D E F 2.33e e-7 G H I

21 9 Figure 2.4. ZnO permittivity, real and imaginary, using the Adachi Model. Figure 2.5. Al 3 O 2 permittivity, real and imaginary, using the Cauchy Model. In the Re(ε) lines, the top line is the parallel permittivity and the bottom is perpendicular.

22 10 CHAPTER 3 ELLIPSOMETRY: MODELING AND FITTING This chapter will overview the principles and measurement methods of spectroscopic ellipsometry. Nanolayered Al/Dielectric material has optical anisotropy and strong absorption which calls for a more sophisticated analytic approach. In this research, the Transfer-Matrix formalisim is used to predict ellipsometry data with the models discussed in Chapter 2. The Levenberg-Marquardt algorithm was used to fit the model to the data and is briefly reviewed in this chapter. This is an inverted problem in that the models are used to predict ellipsometry data, then some parameters are allowed vary in order to fit measured ellipsometry data. 3.1 PRINCIPLES OF ELLIPSOMETRY In order to measure the permittivity of materials, the sample is systematically exposed to light and the optical response of the reflected light is recorded. This processes is done through a non-destructive method called ellipsometry. As seen in Figure 3.1, s- and p-polarized light is illuminated on a sample, and then the change in polarization due to reflection, or transmission, is measured [9]. The magnitude of the incident light, E is and E ip, is used with the reflected light, E rs and E rp, to calculate two parameters, Ψ and, using Equation 3.1 where r p and r s represent the reflection coefficients. In essence, Ψ represents the amplitude ratio and the phase difference between the polarized light. In an isotropic, homogeneous material, the refractive index, n, determines Ψ, and the extinction coefficient, k, is characterized by. ρ = tan(ψ)e i = r pp r ss = E rp E ip / E rs E is (3.1) The change in polarization that occurs can be described using the Jones and Mueller matrices. For this research, the Jones matrix is considered, since our samples do not depolarize light, which is when Mueller matrices are more appropriate [32]. The Jones matrix, shown in Equation 3.2, looks at the reflection coefficients for an isotropic system. ( E rp E rp ) = ( r pp 0 0 r ss ) ( Ellipsometry data that only has diagonal terms in the above Jones matrix with Equation 3.1 is called standard ellipsometry. E ip E ip ) (3.2)

23 11 Figure 3.1. s- and p-polarized light irradiated onto a sample. Ψ is the amplitude ratio between the rotated light and the incident light, and is the change in phase between the reflected s- and p-polarized light. For an anisotropic system, the diagonals no longer reduce to zero in the Jones matrix description. This type of data is called generalized ellipsometry, and the off-diagonal terms are named Aps and Asp depending on the polarization order. ( E rp E rp ) = ( r pp r ps r sp r ss ) ( There are many types of ellipsometers, each capable of rotating or modulating the E ip E ip ) (3.3) incident and reflected beam in order to pinpoint specific types of optical responses. Due to the importance of anisotropy in layered structures, the J. A. Woollam M-2000 ellipsometer was used in a wavelength range of 210 to 1692 nm to fill the Jones matrix elements. The full wavelength range was tested with an incident angle that varied from 45 to 70 in intervals of 5 o. The M-2000 is a rotating compensator ellipsometer with a high-speed CCD detector specifically built for thin film characterization [28]. For anisotropic samples, it is essential to have a compensator placed before and after the sample in order to measure all the off-diagonal components that are not present in an isotropic material [28]. The benefit of having a rotating compensator is that multiple frequencies of polarized light can be irradiated on the sample. This allows for a wide range of frequencies to be tested in a short amount of time [28]. The only limitation of this technique is the requirement of a model to fit calculated data. J. A. Woollam provides a software named W-VASE, which is generally used in literature to determine permittivity and sample thickness; however, the fitting process is hidden from the user. In regard to anisotropic samples with absorption, additional considerations for modeling,

24 such as the EMA with non-local corrections, need to be added in which is the focus of this work. The propagation of light is significantly different for an isotropic material than a uniaxial, nanolayered material (i.e., ε x = ε y ε x ). The z-axis defines the optical axis, and there is only a variation in propagation speed along the parallel direction (extraordinary waves) [9]. When varying the incident angle, the extraordinary waves will affect the overall path of light, whereas an isotropic material will not see that effect. We must consider the Transfer-Matrix approach because Snell s law and Fresnel s equations cannot characterize the light s behavior through a thin, anisotropic sample over any incident angle [9]. This approach allows us to determine the permittivity of an anisotropic material by normalizing Equation 3.3, dividing by r ss. There are three sets of parameters to characterize the anisotropic material. ( r pp r ps r sp r ss ) ( ) ( r pp /r ss r sp /r ss ρ pp = r ss = r ss r ps /r ss 1 ρ ps 1 The first diagonal term ρ pp is the standard ellipsometry set, called AnE. The off-diagonal terms are the additional Ψ and terms from the generalized data, called Aps and Asp. This gives rise to three separate equations and six independent parameters that can be used to deduce details about the nanolayered structure where only two would have been available with standard ellipsometry. ρ sp ) 12 (3.4) AnE = ρ pp = r pp /r ss = tan(ψ pp )e i pp (3.5) Aps = ρ ps = r ps /r ss = tan(ψ ps )e i ps (3.6) Asp = ρ sp = r sp /r ss = tan(ψ sp )e i sp (3.7) It is possible for even optically anisotropic materials to have zero off-diagonal elements in generalized ellipsometry with certain incident angles of light; this is why varying the angle of incidence is imperative to our data [9]. The basic system consists of three layers: infinite air, the nanolayered structure, and infinite silicon as seen in Figure 3.2. The silicon layer is considered to be infinite because the height of the substrate is on the order of 500 microns, while the heights of either the nanolayered composites are, at the largest, 320 nm. This is a difference in magnitude of 10 3.

25 13 Figure 3.2. Model of s- and p- polarized light incident on an ambient/right-handed anisotropic film/isotropic substrate structure. 3.2 MODELING A UNIAXIAL, ANISOTROPIC MATERIAL SYSTEM Effective Medium Approximation There is no general solution to modeling light as it travels through a multilayered material unless it s periodic. Generally, modeling how light is transmitted and reflected through a multilayered media requires a specialized matrix for each layer, M m for the metal and M d for the dielectric. How these matrices are calculated is not covered in this research, but it is well studied in conventional optics textbooks. If the multilayered material is periodic, and the number of layers is very large, the effective matrix, M eff, of the system saturates. M eff = M m M d M m M d... (M m M d ) n (3.8) For the metal/dielectric case, the entire multilayered composite can be treated like a photonic crystal so long as the heights of the layers are smaller than in incident light. This approximation is called the Effective Medium Approximation (EMA). The EMA takes models of permittivity for a metal and dielectric, Al and either SiO 2 or ZnO, and predicts the permittivity depending on the height of each layer. The permittivity can be estimated by averaging the permittivities of the composite materials based on their associated layer heights [17]. ε = (a 1 + a 2 )ε 1 ε 2 a 2 ε 1 + a 1 ε 2, ε = a 1ε 1 + a 2 ε 2 a 1 + a 2 (3.9)

26 where a 1 and a 2 are the heights of the metal and dielectric layer, and ε 1 and ε 2 are the respective permittivity values of the metal and dielectric, which can be found using the models shown in the previous Chapter. The EMA is computationally effective, since it can approximate the permittivity of a nanolayered material as a continuous medium. However, it only has reasonable agreement for physical nanolayered composites if there are a large number of layers and a 1,2 λ [8]. In 2007, J. Elser and colleagues showed that the reason EMA fails to describe the permittivity of nanolayered structures as the number of layers decreases is due to strong variations in the field along a single layer [8]. These variations exist at the subatomic scale and introduce long range interactions on the interface of the metal and dielectric layer that are considered non-local to the electron. To derive these corrections, the authors consider the interface as a two-layered system. By expanding the dispersion relation to the next non-vanishing Taylor series term, they correct the EMA for the non-local disturbance to the permittivity [8]. The EMA for nanolayered composites then becomes ε eff = ε 1 δ (k, ω), 14 ε eff = ε 1 δ (k, ω), (3.10) where δ = a2 1a 2 2(ε 1 ε 2 ) 2 ε 2 12(a 1 + a 2 ) 2 ε 2 1ε 2 2 δ = a2 1a 2 2(ε 1 ε 2 ) 2 12(a 1 + a 2 ) 2 ε ω 2 c 2, k x = 2π h, ( ) ω 2 ε c k2 x(ε 1 + ε 2 ) 2, 2 ε 2 (3.11) and h is the total height of the sample. Using the models for Al and SiO 2, it was found that these corrections approach the EMA once a multilayered sample approaches 30 layers. This work was published in SPIE The International Society for Optical Engineering 1. These corrections are used in the parameter fittings presented in Chapters 6, as the number of layers is low, but not in Chapter 7 as these samples are on the order of 200 layers. The EMA code used in this research is presented in A.1. 1 P. Kelly, D. White, and L. Kuznetsova, Theoretical design of nano-layered Al/SiO2 metamaterial with hyperbolic dispersion with minimum losses, Nanophotonic Materials XII:SPIE NanoScience + Engineering, San Diego, August 2015.

27 3.2.2 Matrix Solutions for a Anistropic Material The first-order solution to Maxwell equations for the wave function ψ, also known as Berreman s equation, is where B = ε yz ε zx ε xx ε xz ε zx ε zz 15 ψ z = iω c Bψ (3.12) ε k zx ε xx ε zz k zy xx ε zz 0 1 k2 xx ε zz ε zz ε yx kxx 2 ε ε yy + ε zy ɛ yz ε zz 0 k yz xx ε zz ε xy ε xz ε zy ε zz 0 k xx ε xz ε zz (3.13) Each ε corresponds to the elements of the dielectric tensor, Equation 1.3, which will be filled using the EMA presented in the previous section, after rotating the tensor using Euler s angles. The rotation that needs to be added in comes from the linearly polarized incident light at an orientation of 45 o relative to the E ip axis [9]. k xx is related to the propagation vector from the ambient layer and is given by k xx = n i sinθ i (3.14) where n i is the refractive index of the ambient layer and θ i is the angle of incidence. Since this solution is derived from Maxwell s equations, there are no imposed restrictions in the calculations, and multiple light reflections can be expressed explicitly from the equation [9]. Using Berreman s approach, the wave equation then has the form [ ψ( d) = exp i ω ] c B( d) ψ(0) (3.15) where d is the height of the layer, and ψ(d) and ψ(0) are the tangential components of the electric and magnetic field at z = d and 0, respectively [2] Transfer-Matrix Mathias Schubert introduced a three-level system using Berreman s work to calculate how light is transferred through each material [32]. He calculated the incident matrix, L i and exit matrix, L e, for the infinite layer of air and substrate. The matrix for the anisotropic material, T p ( d), is derived from solutions to the wave equation, Equation Therefore, the three-level system is T = L 1 i T p ( d)l t (3.16)

28 where and L 1 i = 0 1 1/(n i cos(θ i )) /(n i cos(θ i )) 0 1/cos(θ i ) 0 0 1/n i 1/cos(θ i ) 0 0 1/n i 0 0 cos(θ t ) 0 L t = n t cos(θ t ) n t 0 16 (3.17) (3.18) For both the ambient matrix, L 1 i, and substrate, L t, Snell s law stands. n i sinθ i = n t sinθ t (3.19) cosθ t = [1 (n i /n t ) 2 sin 2 θ i ] 1/2 (3.20) There are four solutions to Berreman s equation [ Ψ j (d) = exp i ω ] c q i( d) Ψ j (0), j = 1, 2, 3, 4 (3.21) where q represents the eigenvalues of the matrix B [2]. Using the four eigenvalues of B, T p ( d) can be approximated by a finite series [41]. [ T p ( d) = exp i ω ] c B( d) = β 0 I + β 1 B + β 2 2 B + β 3 3 B (3.22) where 4 exp [ i ω β 0 = q k q l q q c j( d) ] m (q j=1 j q k )(q j q l )(q j q m ) 4 exp [ i ω β 1 = (q k q l + q k q m + q l q m ) q c j( d) ] (q j=1 j q k )(q j q l )(q j q m ) 4 exp [ i ω β 2 = (q k + q l + q m ) q c j( d) ] (q j=1 j q k )(q j q l )(q j q m ) 4 exp [ i ω c β 3 = q j( d) ] (q j q k )(q j q l )(q j q m ) j=1 (3.23) (3.24) (3.25) (3.26) (3.27) For each summation, the order of eigenvalue terms used in each iteration, j, of the summation for β 0 3 is shown in Table 3.1.

29 17 Table 3.1. Eigenvalue order for each iteration in the β summations. j k l m Once the Transfer-Matrix, T in Equation 3.16, has been found using the matrices above, the Jones matrix parameters can be found using the equations below [43]. r pp = T 11T 43 T 13 T 41 T 11 T 33 T 13 T 31 (3.28) r sp = T 11T 23 T 13 T 21 T 11 T 33 T 13 T 31 (3.29) r ss = T 21T 33 T 23 T 31 T 11 T 33 T 13 T 31 (3.30) r ps = T 33T 41 T 31 T 43 T 11 T 33 T 13 T 31 (3.31) Using these reflection coefficients with Equations 3.5 through 3.7, the expected Ψ and values are calculated to fit ellipsometry data. The Transfer-Matrix code used for this research is presented in A LEVENBERG-MARQUARDT ALGORITHM To fit the EMA predictions to ellipsometry data, a non-linear, least-squares algorithm must be used. The most popular and effective algorithm used for fitting ellipsometry data is the Levenberg-Marquardt method. First proposed in 1963 by Donald W. Marquardt [20], the Levenberg-Marquardt algorithm emphasizes the strengths of the Gauss-Newton s method with a more computationally effective method. The Gauss-Newton method, a modified Newton s method for finding the roots of a function, minimizes a residual in order to solve a least-squares problem. A residual is generally defined as the difference between the model and data. The only significant change is the addition of a line search and that the Jacobian, J, which represents a first-order partial derivative of the function, can be used to approximate the Hessian, a second-order partial derivative matrix [22]. 2 f k J T k J k (3.32)

30 The line search component takes the direction calculated from the first and approximated second derivative towards negative gradient in order to find a local minimum. Line searches emphasize finding the step direction then calculating the step-size. At best, this method can have slow convergence rates since the step-size is directly proportional to the gradient, which goes to zero while approaching the minimum. This is nessecary for convergence, but calculating the step-size this way slows convergence. At worst, the step-size can end up too large due to a rank-deficient Jacobian, which can come from an insufficient amount of data, and it is no longer possible to prove global convergence [20], [22]. The Levenberg-Marquardt algorithm uses the strengths of the Gauss-Newton method, namely the approximation of the Hessian, and replaces line searches with trust-regions [22]. Trust-region methods define a region surrounding the current iterate where the model can be trusted to represent the function [22]. If the first calculated step is outside this region, the size of the next step and the trust-region is reduced. In contrast to line searches, the trust-region approach calculates the step-size limitation then the step direction. When the Jacobian becomes rank-deficient in the trust-region framework, the next step calculation can navigate around it [23]. In the best case scenario, when the Hessian is approximately second-order, both the Gauss-Newton and Levenberg-Marquart convergence is the same [23]. The Levenberg-Marquardt method works well in practice so long as there are reasonable starting parameters near a local minimum [27]. Therefore, it is important to understand how the parameters affect the model as discussed in Chapter 2. For any least-square fitting algorithm, the residual must be defined. For this case, the Mean-Square Error (MSE) between the ellipsometry data and the calculated model must be minimized. For the anisotropic case, there are three sets of data for every angle of incidence: AnE, Aps, and Asp. Each of these have their own Ψ and, as well as their respective errors. [ MSE = 1 N (Ψ ) exp Ψ model 2 ( ) exp model N δψ i i δ i ( ) 2 ( ) 2 Ψ exp ps Ψ model ps exp ps model ps + + δψ ps δ (3.33) ps i i ( ) 2 ( ) 2 Ψ exp sp exp sp Ψ model sp δψ sp i + model sp δ sp where N represents the number of data points used in the optimization. The MSE used here is modeled after the MSE used in [33]. Psuedo-code for the basic Levenberg-Marquardt algorithm used for this optimization was found in [18]. p represents the parameter vector, and x represents the current guess for i 18

31 19 the parameters. The user-defined parameters for the algorithm τ, ɛ 1, ɛ 2, ɛ 3, and k max are taken from [18]: 10 3, for all the ɛs, and 100, respectively. The Levenberg-Marquardt code used for this research is presented in A.3. Algorithm 1 : Levenberg-Marquardt Algorithm 1: k := 0; ν := 2; p := p 0 ; ρ := 100; 2: A := J T J; ɛ p := x f(p); g := J T ɛ p 3: stop:= ( g ɛ 1 ); µ := τ max i=1,..,m (A ii ); 4: while (not stop and k < k max ) do 5: k := k + 1; 6: while (ρ < 0 or stop) do 7: δ p := (A + µi) 1 g; 2 8: if ( δ p ɛ 2 p ) then 9: stop:= true; 10: else 11: p new := p + δ p ; 12: ρ := ( ɛ p 2 MSE)/(δ T p (µδ p + g)); 13: if ρ > 0 then 14: p := p new ; 15: A := J T J; ɛ p := x f(p); g := J T ɛ p ; 16: stop:= ( g ɛ 1 ) or ( ɛ p 2 ɛ 3 ); 17: µ := µ max( 1 3, 1 (2ρ 1)3 ); ν := 2; 18: else 19: µ := µ ν; ν := 2ν; 20: return p 2 The inversion is solved using QR Decomposition. This method does not add any additional error, as the error is usually related to the condition number of the Jacobian, J, not its square [22].

32 20 CHAPTER 4 FABRICATION METHODS: SPUTTERING AND ATOMIC LAYER DEPOSITION Here, the two methods of fabrication will be reviewed: Sputtering and Atomic Layer Deposition (ALD). All samples were grown on a silicon (100) substrate. 4.1 SPUTTERING: ALUMINUM AND SILICON DIOXIDE The Al and SiO 2 samples were grown using the Denton Discovery 18 vacuum sputter system at Nano3 Cleanroom located in the University of California, San Diego (UCSD). The set of samples presented here have between 10 and 20 layers, where a single layer is the joint stack of Al and SiO 2. Each layer was intended to be 10 nm in total, and the deposition time was calculated from the fill fraction ratio, f Al = d Al d di + d Al, (4.1) where the d Al is the height of Al in the layer and d di is the height of the dielectric once the daily growth rate was calculated. This technique is not automated, so there is human error attached to the deposition times. All the samples were grown with an applied power of 100 watts to the Al target and 400 watts to the SiO 2 target as suggested by the cleanroom technicians. Powering the material targets heats up the chamber by approximately 5 o C throughout the entire deposition time. 4.2 ATOMIC LAYER DEPOSITION: ALUMINUM AND ZINC OXIDE The Al and ZnO samples were fabricated using the Cambridge Nanotech Savannah S100 system at Brookhaven National Labs. ZnO was grown in DEZ/H 2 O and the Aluminum Oxide in TMA/H 2 O at a temperature 200 o C. A total of 2000 ALD cycles of ZnO and Aluminum Oxide were needed to generate a sufficient film thickness, approximately 300 nm. ALD doesn t deposit a solitary, nanometer thick layer of material like the sputtering technique does. The growth rate for the system is relatively low, and it provides a uniform coverage with atomic layer precision [6]. Each cycle of ALD is capable of doing this by self-limiting surface reactions with precursor chemicals, DEZ/H 2 O for ZnO, and TMA/H 2 O for Aluminum Oxide. These reactions control the film growth for each cycle. Once the cycle

33 21 has completed, excess precursors and by-products, such as Methane gas, are purged after every cycle. In comparison, the growth rate for ZnO grown with RF sputtering is nm/min and dual magnetron sputtering is 300 nm/min [6]. Brookhaven National Lab found that one ZnO cycle deposited approximately 1.57 angstroms, and Aluminum Oxide had approximately 1 angstrom per cycle. Therefore, with this atomic precison growth, an Al-doped ZnO (Al:ZnO) layer, commonly called AZO, is deposited instead of pure Al between ZnO layers. Many research groups look at bulk AZO [6], [7], [15], [19], [34], grown with ALD and a variety of other methods, but what makes this research different is that it looks to model how very thin films of AZO exist between ZnO layers. At the time of this work, only one other group looked at the permittivity of AZO within ZnO layers [21], their work was on the order of 10s of nanometers, while this research focuses on detailing individual layers which are on the order of angstroms using the EMA shown in Chapter 2.

34 22 CHAPTER 5 MODELING FOR KNOWN STRUCTURES This chapter serves as a verification for the Transfer-Matrix method presented in Chapter 3. Three layers are considered: ZnO grown using ALD, 200 nm of SiO 2, and a substrate of Si which were commercially grown. The permittivity model for these isotropic materials is taken from Chapter 2, and no part of the models is allowed to vary in the fitting. This way, we can test the Transfer-Matrix method without including the more complex responses of anisotropic materials. 5.1 PROOF OF CONCEPT WITH ZNO, SI, AND SIO 2 The only parameter that is allowed to vary for this Transfer-Matrix calculation is the height of ZnO. Figure 5.1 shows an image of the ZnO layer on SiO 2 using a scanning electron microscope (SEM). The SEM s build in software measured the ZnO layer to be 157 nm. Using the known models for all three dielectrics, the fitting algorithm should predict a height within the 2% error limits SEM height measurements have [36]. Figure 5.1. SEM image of ZnO/SiO2/Si using 5kV. Height measurements using SEM software predict the ZnO layer height to be 157 nm. When the ellipsometry data was fitted, shown in Figure 5.2, the Transfer-Matrix method found that a ZnO layer of 154 nm gave the best fit. This is within 2% of 157 nm, so this work is taken as a proof of concept for the Transfer-Matrix set up.

35 23 Figure 5.2. ZnO/SiO 2 /Si layers modeled with the Transfer-Matrix in red with measured ellipsometry data in black at an incident angle of 45 o. Figure 5.2 is a clear demonstration of how disordered bulk ZnO can be. In many growing techniques, ZnO has been shown to have vacancy defects. Oxygen vacancies being the most prominent, which lead to irregular grains in the film and produce a broad photoluminescence in the visible wavelengths [34]. These defects, while significant at a range of 150 nm, do not impact the validitiy of the data because ellipsometry measures a far-field response which is beyond the scale of defects.

36 24 CHAPTER 6 ALUMINUM AND SILICON DIOXIDE In this chapter, the fitting results for the Al/SiO 2 will be investigated using the Transfer-Matrix method and Levenberg-Marquardt algorithm shown in Chapter 3. Three samples were chosen to demonstrate the conclusions of this research. These results have been submitted to SPIE The International Society for Optical Engineering, and are awaiting approval. For these samples, only the AnE data is displayed here as the Asp and Aps data did not contribute as clearly to the final conclusions. 6.1 SAMPLE SPECIFICATIONS Initial fitting for the Al/SiO 2 samples proved to be difficult. Referencing the intended Al percentage and heights in Table 6.1, each sample was grown to have between 0.7 and 2.5 nm of Al in each 10 nm layer of Al/SiO 2. Table 6.1. Growing parameters for Al/SiO 2 samples. Sample Layers Intended Al Approx. Height Ambient Temperature Fabrication Date AS:1 20 7% 251 nm 25 o C Sep AS: % - nm 1 27 o C May 2015 AS: % 100 nm 65 o C Sep No papers on RF sputtering for layers of Al and SiO 2 were found in literature at the time of this research, so the first sample was aimed to test the system. RF sputtering cannot deposit as finely as ALD, so the initial choice was on the finest scale RF sputtering could provide. Sample AS:1 intended to have 7% Al out of a 10 nm layer, 0.7 nm of Al and 9.3 nm of SiO 2, which was calculated based on the growth rates of Al and SiO FITTING RESULTS When the ellipsometry data for this was compared with a model of pure Al layer in that height range, it became clear that very little pure Al was deposited on any of the samples. Fitting efforts found that reducing the Al layer to less than 0.01% best fit the data. Figure 6.1 shows what a pure sample of Al at 0.7 nm looks in comparison to the data and, Figure 6.2 shows the best-fit with no Al present. 1 Due to inconsistent measurements from both the SEM and Detak, no approximate height can be reported.

37 25 At first, the thought was the intended layer height of.7 nm Al was too thin, and the fill fraction was increased to 25% for sample AS:2. However, when ellipsometry data was fitted, a similar result was found: No reasonable height of pure Al fit the sample. Therefore, the conclusion of these experiments was that no Al was deposited. Figure 6.1. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:1 with a model, in red, of 7% of 10 nm as Al at an incident angle of 45 o. Figure 6.2. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:1 with 250 nm of SiO 2, in red, at an incident angle of 45 o. The next step was to test the data with a model of oxidized Al, Al 2 O 3. This proved to solve the mystery layer for sample AS:1 and AS:2, as shown in Figure 6.3 and 6.4. An interesting find arose with initial fitting efforts when the permittivity of Al 2 O 3 and SiO 2 were compared. Since both oxides permittivity values are similar over the measured range, 500 to 1690 nm, the Levenberg-Marquardt algorithm was able to reasonably fit the samples with pure SiO 2. This leads to the incorrect conclusion that no Al was deposited. Once Al 2 O 3 layers

38 were added instead of a pure slab of SiO 2 into the model, there was a 60% reduction in MSE for sample AS:1. 26 Figure 6.3. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:1 with a layer of Al 2 O 3, in red, at an incident angle of 45 o. Figure 6.4. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:2 with a layer of Al 2 O 3, in red, at an incident angle of 45 o. The fitting algorithm found that the Al 2 O 3 layer for both AS:1 and AS:2 proved to be much larger than expected, the intended Al height of nm, due to the additional oxygen content. The final fitting parameters are shown in Table 6.2. For sample AS:3, it was found that neither pure Al, pure Al 2 O 3, or separate layers of each were present in the layer. Figure 6.5 shows when a layer of Al 2 O 3 is used for the fitting, which did well overall, but did not fit as well as samples AS:1 and AS:2. What makes this sample different from the other two is that it was grown with an increased chamber temperature.

39 27 Sample AS:1 and AS:2 began their growth around 27 o C and ended near 33 o C. This occurred due to the energy expended by the sputtering machine to power both targets during deposition. When the oxidation results from sample AS:1 and AS:2 were presented to technicians at Nano3, they suggested increasing the chamber s ambient temperature. This would increase the deposition rate of Al and decrease interaction time with oxygen in the chamber s atmosphere. A literature search found that increasing the temperature to 57 o C helped with reducing Al 2 O 3 and smoothing the layer, although this method used a different sputtering system than was avaliable at Nano3 [29]. With these suggestions, sample AS:3 was grown at an increased temperature of 65 o C. Figure 6.5. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:3 with a model of 7 nm of Al 2 O 3 and 4.9 nm of SiO 2, in red, at an incident angle of 45 o. To model the mixed layer of Al and Al 2 O 3, the Maxwell-Garnett equation for evenly distributed atoms within a host material was used [4]. ε i ε h ε eff = ε h + 3fε h ε i + 2ε h f(ε i ε h ) (6.1) where ε h represents the host matrix, Al 2 O 3, and ε i represents the inclusions, Al. f is the fill fraction as shown in Equation 4.1. As seen in Figure 6.6, the Maxwell-Garnett equation did very well in fitting the peak on Ψ and the portion of the data. This was a 40% improvement in fitting even though most of Ψ could not be fit very well. It is possible the Al atoms are not evenly distributed, leading to poor fitting for Ψ. However, given the improvement in fitting, this research did find that increasing the temperature by 30 o C increased the Al content in the samples from 0 to 13%.

40 28 Figure 6.6. ANE ellipsometry data, in black, for Al/SiO 2 sample AS:3 with a layer of SiO 2 and 13% Al in Al 2 O 3 and using the Maxwell- Garnett formula, in red, at an incident angle of 45 o. It s important to note that this 13% Al content exists within the Al 2 O 3 layer, and there is not enough Al to move the permittivity to zero in one direction as needed for hyperbolic dispersion. Future work with Al and SiO 2 layers would need to employ more power and heat than currently avaliable at Nano3 s sputtering system, or a separate growing technique. Table 6.2. Final Fitting heights [nm] for the Al/SiO 2 samples. Sample Al 2 O 3 Al/Al 2 O 3 SiO 2 Sample Height AS: AS: AS:3 0 13% of

41 29 CHAPTER 7 ALUMINUM-DOPED AND ZINC DIOXIDE The fitting results for the AZO/ZnO samples will be investigated using the Transfer-Matrix method and Levenberg-Marquardt algorithm shown in Chapter 3. Two samples are shown here, and the results presented have been published in Applied Optics SAMPLE SPECIFICATIONS Given that atomic layer deposition (ALD) is capable of depositing very thin layers per deposition cycle, we anticipate an Al-doped ZnO, AZO, layer to exist between layers of ZnO, as described in Chapter 4. At this time, there are no parameters in literature that define thin layers of AZO, as they exist between ZnO layers, in the UV to near-infrared region. Therefore, this initial investigation will look at modeling AZO with the Drude model as shown in Equation 2.1 given its simplicity. To fully characterize our materials, additional measurements were taken of the sample s height, roughness, and mole concentration. The height, as with ZnO in Chapter 5, was measured with SEM images and is shown for both samples in Figures 7.1 and 7.2. These pictures demonstrate how increasing Al content, as present in Figure 7.1, significantly reduces the overall roughness of the material. This happens due to Al substitutions in ZnO sites and Zn vacancies, and has been seen before in literature [34], [45]. Another consequence of doping ZnO with Al is a shift in the band gap to higher energies due to increased carrier concentrations which is predicted by the Burstien-Moss shift, but was not investigated here [15]. The roughness, using atomic force microscopy (AFM), was measured and is consistent with the reported roughness of 2.5 nm in literature [7]. Lastly, the mole concentration of Al was measured using energy dispersive X-ray spectrosopy (EDX) at 20 kvolts and a takeoff angle of 35 o. All of these parameters are shown in Table P. Kelly, M. Liu, and L. Kuznetsova, Designing optical metamaterial with hyperbolic dispersion based on an Al:ZnO/ZnO nanostructure using the atomic layer deposition technique, Applied Optics, 55 (2016).

42 30 Table 7.1. Growing parameters, surface roughness, and EDX values for AZO/ZnO samples. Sample Cycle Ratio Num. of Periods Height AFM Roughness Mole Conc. AZ:1 1: nm 2.22 ± 0.28 nm 3.4 wt. % AZ:2 1: nm 2.89 ± 0.25 nm 0.88 wt. % Figure 7.1. SEM image of sample AZ:1 using 5kV. Height measurements using SEM software predict the AZO/ZnO layer height to be 302 nm. Figure 7.2. SEM image of sample AZ:2 using 5kV. Height measurements using SEM software predict the AZO/ZnO layer height to be 313 nm. 7.2 FITTING RESULTS To use the EMA, the number of layers must be provided. Therefore, for modeling purposes, it was assumed that the number of periods represents the number of layers within the sample.