On-Wafer Characterization of Electromagnetic Properties of Thin-Film RF Materials

Transcription

1 On-Wafer Characterization of Electromagnetic Properties of Thin-Film RF Materials Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Jun Seok Lee, B. S., M. S. Graduate Program in Electrical and Computer Engineering The Ohio State University 2011 Dissertation Committee Professor Roberto G. Rojas, Adviser Professor Patrick Roblin Professor Fernando L. Teixeira

2 Copyright by Jun Seok Lee 2011

3 ABSTRACT At the present time, newly developed, engineered thin-film materials, which have unique properties, are used in RF applications. Thus, it is important to analyze these materials and to characterize their properties, such as permittivity and permeability. Unfortunately, conventional methods used to characterize materials are not capable of characterizing thin-film materials. Therefore, on-wafer characterization methods using planar structures must be used for thin-film materials. Furthermore, most new, engineered materials are usually wafers consisting of thin films on a thick substrate. Therefore, it is important to develop measurement techniques for on-wafer films that involve the use of a probe station. The first step of this study was the development of a novel, on-wafer characterization method for isotropic dielectric materials using the T-resonator method. Material characterization using a T-resonator provides more accurate extraction results than the non-resonant method. Although the T-resonator method provides highly accurate measurement results, there is still a problem in determining the effective T-stub length, which is due to the parasitic effects, such as the open-end effect and the T-junction effect. Our newly developed method uses both the resonant effects and the feed-line length of the T-resonator. In addition, performing the TRL calibration provides the exact length of ii

4 the feed line, thereby minimizing the uncertainty in the measurements. As a result, our newly developed method showed more accurate measurement results than the conventional T-resonator method, which only uses the T-stub length of the T-resonator. The second step of our study was the development of a new on-wafer characterization method for isotropic, magnetic-dielectric, thin-film materials. The on-wafer measurement approach that we developed uses two microstrip transmission lines with different characteristic impedances, which allow the determination of the characteristic impedance ratio. Therefore, permittivity and permeability can be determined from the characteristic impedance ratio and the measured propagation constants. In addition, this method involves Thru-Reflect-Line (TRL) calibration, which is the most fundamental calibration technique for on-wafer measurement, and it eliminates the parasitic effects between probe tips and contact pads. Therefore, this novel characterization method provides an accurate way to determine relative permittivity and permeability. The third step of this study was the development of an on-wafer characterization method for magnetic-dielectric material using T-resonators. Similar to our second proposed method, this method uses two different T-resonators that have the same T-stub lengths and widths but different widths of feed lines. This method allows the determination of the ratio of the characteristic impedance to the effective refractive index of the magnetic-dielectric materials at the resonant frequency points. Therefore, permittivity and permeability can be determined. Although this method does not provide continuous extractions of material properties, it provides more accurate experimental results than the transmission line methods. iii

5 The last step of this research was the evaluation and assessment of an anisotropic, thin-film material. Many of the new materials being developed are anisotropic, and previous techniques developed to characterize isotropic materials cannot be used. In this step, we used microstrip line structures with a mapping technique to characterize anisotropic materials, which allowed the transfer of the anisotropic region into the isotropic region. In this study, we considered both uniaxial and biaxial anisotropic material characterization methods. Furthermore, in this step, we considered a characterization method for biaxial anisotropic material that has misalignments between the optical axes and the measurement axes. Thus, our newly developed anisotropic material characterization method can be used to determine the diagonal elements in the permittivity tensor as well as the misalignment angles between the optical axes and the measurement axes. iv

6 Dedication This document is dedicated to my family. v

7 Acknowledgments First and foremost, it is a pleasure to thank my advisor, Prof. Roberto G. Rojas, for his guidance and efforts made this dissertation possible. He has always encouraged me to pursue a career in the electrical engineering. He has enlightened me through his wide knowledge of Electrical Engineering and his deep intuitions about where it should go and what is necessary to get there. I am also very grateful to my dissertation committee members, Prof. Fernando L. Teixeira and Prof. Patrick Roblin. Their academic guidance and input and personal cheering are greatly appreciated. I would like to thank my fellow graduate students at ElectroScience Laboratory (ESL) Keum-su Song, Bryan Raines, Idahosa Osaretin, Brandan T Strojny, and Renaud Moussounda. It has been a great experience to work with them past four years. I also want to thank to other Korean graduate students at ESL - Gil Young Lee, James Park, Chun-Sik Chae, Haksu Moon, Jae Woong Jeong, and Woon-Gi Yeo. Finally, I would like to thank all my family members, specially my parents and parents-in-law, for their unconditional love, encouragement, and support over the years. Last but not least, I would like to express the deepest gratitude to my wife, Hyun-su Kim, for being with me through all of this. Without her, it would be much harder to finish this work. Thank you and I love you! vi

8 Vita August, B.S. Electrical Eng., Kyungpook National University, Daegu, South Korea June 2004 to June Assistant Engineer, Samsung Electronics, Tangjung, South Korea December, M.S. Electrical and Computer Eng. University of Rochester, Rochester, NY, USA September 2007 to present...graduate Research Associate, ElectroScience Laboratory, The Ohio State University, Columbus, OH, USA Fields of Study Major Field: Electrical and Computer Engineering vii

9 Table of Contents Abstract... ii Dedication...v Acknowledgments... vi Vita... vii List of Tables... xi List of Figures... xii Chapter 1. Introduction...1 Chapter 2. Review of Conventional On-Wafer Measurement Methods Introduction Review of Conventional On-Wafer Measurement Methods for Dielectric Materials Overview of Non-Resonant Method Transmission Line Method - Theory Transmission Line Method - Experiments Overview of Resonant Method T-Resonator Method - Theory T-Resonator Method - Experiments Review of Conventional On-Wafer Measurement Methods for Magnetic- Dielectric Materials Transmission Line Method (Theory)...39 viii

10 Chapter 3. An Improved T-Resonator Method for the Dielectric Material On-Wafer Characterization Introduction Method of Analysis T-Resonator Matrix Model Consideration of Loss Measurements T-Resonator Measurement Results Summary...65 Chapter 4. Novel Electromagnetic On-Wafer Characterization Method for Magnetic- Dielectric Materials Introduction Method of Analysis - System Matrix Model Method of Analysis - Transmission Line Models Simulated Results with Sensitivity Test Error Analysis Measurement Results Summary...90 Chapter 5. New On-Wafer Characterization Method for Magnetic-Dielectric Materials Using T-Resonators Introduction Method of Analysis Simulated Results Consideration of the Effective T-Stub Length Measurement Results Summary Chapter 6. On-Wafer Electromagnetic Characterization Method for Anisotropic Materials Introduction Method of Analysis Uniaxial and Biaxial Anisotropic Materials ix

11 6.3. Method of Analysis General Biaxial Anisotropic Materials Simulation and Measurement Results Summary Chapter 7. Conclusion Summary and Conclusion Future Works Appendix A. Crystal System (Bravais Lattice) Appendix B. Conformal Mapping of a Microstrip Line with Duality Relation Appendix C. The Permittivity Tensor in the Measurement Coordinate System References x

12 List of Tables Table 2.1. Pyrex 7740 wafer measurement results using different types of T-resonators (ε' r and tanδ of Pyrex 7740 are 4.6 and 0.005, respectively)...37 Table 3.1. The measurement results comparison for coplanar waveguide T-resonator...57 Table 3.2. The measurement results comparison for microstrip T-resonator...61 Table 3.3. The error analyses comparison for microstrip T-resonator measurements...64 Table 4.1. Minimum and Maximum Relative Error of the Extraction Results for the Frequency Range of 1GHz to 10GHz...77 Table 5.1. The simulated results for using two T-resonators Table 5.2. The simulated results using the effective T-stub length Table 5.3. The measured results for ε' r and μ' r using two T-resonators Table 5.4. The measured results for ε" r and tanδ. (The nominal value of tanδ is 0.005) Table A.1. Classification of tensor forms by crystal system xi

13 List of Figures Figure 1.1. Simple illustrations for (a) permittivity and (b) permeability measurements including their equivalent circuit models...3 Figure 1.2. Examples of conventional material characterization method configurations (a) Reflection method with open-ended coaxial probe, (b) Free-space bistatic reflection method, and (c) Rectangular dielectric waveguide method...4 Figure 2.1. Typical configuration of the on-wafer measurement using probe station...11 Figure 2.2. (a) Probe station measurement configuration. (b) Contact between GSG (Ground-Signal-Ground) probe tip and contact pad (upper) and probes on the wafer sample (lower )...12 Figure 2.3. (a) Microstrip transmission line and (b) coplanar waveguide transmission line...14 Figure 2.4. Electric field distribution of (a) microstrip and (b) coplanar waveguide structures...16 Figure 2.5. Equivalent circuit model of the transmission line...18 Figure 2.6. Fabricated test structures on Pyrex 7740 wafer (a) coplanar waveguide test structures and (b) microstrip test structures. Both microstrip and coplanar waveguide transmission lines and TRL calibration kits are fabricated on Pyrex 7740 wafers...21 Figure 2.7. The test fixture consists of a microstrip line as a DUT and coplanar waveguide-to-microstrip transitions as error boxes...21 Figure 2.8. E-field distributions at (a) A-A' plane and (b) B-B' plane. The upper and lower figures represent the magnitude and the vector of E-fields, respectively...22 Figure 2.9. Extraction results of ε r using transmission line method (ε' r of Pyrex 7740 is 4.6): (a) coplanar waveguide transmission line and (b) microstrip transmission line...23 xii

14 Figure De-embedded S 11 of the Thru standard. From the de-embedded S 11 result of the Thru standard, calibration is valid from3.7ghz to 14.5GHz...24 Figure Dielectric loss tangent (tanδ of Pyrex 7740 is 0.005) extraction results for using (a) coplanar waveguide transmission line and (b) microstrip transmission line...25 Figure Three different types of microstrip resonators: (a) ring resonator, (b) T- resonator, and (c) straight-ribbon resonators...27 Figure T-resonator models: (a) Microstrip T-resonator and (b) Coplanar waveguide T-resonator with air-bridge...30 Figure T-resonators on the Pyrex 7740 wafers: (a) coplanar waveguide T-resonators and (b) microstrip T-resonators...35 Figure S 21 (db) measurement results for T-resonators: (a) coplanar waveguide T- resonators and (b) microstrip T-resonators...36 Figure Probe tip/contact pad model and its equivalent circuit model...44 Figure 3.1. (a) A typical T-resonator configuration and (b) its equivalent circuit model for T-resonator. Each section in the equivalent circuit model can be expressed with a wave cascade matrix model...48 Figure 3.2. Fabricated test structures on Pyrex 7740 wafers which have diameter of 100mm. (a) Coplanar waveguide structures and (b) microstrip test structures...54 Figure 3.3. Measured short-stub coplanar waveguide T-resonator (a) S 11 and (b) S 21 in db. Blue and red lines indicate the measured S-parameters with and without performing TRL calibrations, respectively...55 Figure 3.4. Measured (a) magnitude of R 11 and (b) phase angle of R 11 for short-stub coplanar waveguide T-resonator. The green dashed lines in the plots indicate the S 11 resonant points...56 Figure 3.5. Measured open-stub microstrip T-resonator (a) S 11 and (b) S 21 in db. Blue and red lines indicate the measured S-parameters with and without performing TRL calibrations, respectively...58 Figure 3.6. Measured (a) magnitude of R 11 and (b) phase angle of R 11 for open-stub microstrip T-resonator. The green dashed lines in the plots indicate the S 11 resonant points...60 xiii

15 Figure 3.7. Error analysis with ±95 confidence limits of ε r extraction using (a) conventional T-resonator method and (b) proposed T-resonator method...63 Figure 4.1. Block diagram of two sets of DUT s with same error boxes. [R a ], [R b ], [R D1 ], and [R D2 ] are the wave cascade matrices of Error box A, Error box B, DUT1, and DUT2, respectively...68 Figure 4.2. The actual simulated microstrip transmission lines. DUT1 is the top figure while DUT2 is the bottom figure. In the simulation, W1 and W2 are 500μm and 600μm, respectively. The length of error box (l e ) and DUT (L) are 500μm and 5mm, respectively...75 Figure 4.3. Simulated results of ε r and μ r extraction for lossless case (ε r =3 and μ r =2 are the exact values)...76 Figure 4.4. Simulated results of ε' r and μ' r extraction for lossy case (ε' r =3 and μ' r =2 are the exact values)...79 Figure 4.5. Simulated results of ε" r and μ" r extraction for lossy case (ε" r =0.015 and μ" r =0.01 are the exact values)...79 Figure 4.6. Simulated error analysis results for variation in 600μm line width.. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values)...81 Figure 4.7. Simulated error anlaysis results for variations in the error boxes connected to 600μm microstrip line. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values)...83 Figure 4.8. Simulated error analysis results (for r w =1.2) for variation in 500μm line width. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values)...84 Figure 4.9. Simulated error analysis results for variations in TRL calibration kits. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values)...85 Figure Simulated error analysis results for uncertainties in both DUT s and TRL calibration kits. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values)...86 xiv

16 Figure Simulated error analysis results for uncertainties in both DUT s and TRL calibration kits. Imaginary parts of permittivity (top) and permeability (bottom) with standard error analysis (ε" r =0.015 and μ" r =0.01 are the exact values)...87 Figure The test fixtures ofmicrostrip transmission lines for the measurements. The widths of DUT1 and DUT2 are 500μm and 600μm, respectively, and both DUT s are the line length of 5mm...88 Figure Extracted results of the real parts of ε r and μ r of the Pyrex 7740 wafer: (a) used proposed method and (b) used conventional method (The nominal values of real parts of ε r and μ r of the Pyrex 7740 wafer are 4.6 and 1, respectively)...89 Figure Extracted result of the imaginary parts of ε r of the Pyrex 7740 wafer (The nominal value of the dielectric loss tangent of the Pyrex 7740 wafer is 0.005)...90 Figure 5.1. Two T-resonator models with same characteristic impedance at the T-stub, but different characteristic impedances at the feed lines...94 Figure 5.2. Simulated results of two T-resonators which have same T-stub length and width, but different feed line widths. (a) S 21 (db) in overall frequency range and (b) S 21 (db) for region near the first resonant frequency...97 Figure 5.3. Simulated results of the characteristic impedance ratio, r (red solid line) and its value obtained by regularization (blue dot line)...98 Figure 5.4. The effective T-stub length in the T-resonator model which includes the openend effect and the T-junction discontinuity effect Figure 5.5. Two different microstrip T-resonators for the measurements. T-resonator (a) and (b) have T-stub length of 15mm and width of 500μm while the feed line widths are 500μm and 400μm for T-resonator (a) and (b), respectively Figure 5.6. Comparison of measured S 21 for two T-resonators. Top figure is S 21 comparison for the overall frequency range and bottom 4 figures are detailed S 21 at the resonant frequency points Figure 6.1. Cross section of (a) microstrip on anisotropic substrate and (b) equivalent microstrip on isotropic substrate Figure 6.2. Schematic diagrams of the microstrip lines on a biaxial anisotropic material with different propagation directions: Microstrip lines along the x-axis (left) and y-axis (right) xv

17 Figure 6.3. The simulated results for the anisotropic material characterizations: (a) uniaxial and (b) biaxial anisotropic substrates Figure 6.4. The principal axes of the permittivity tensor (x y z system) and the measurement coordinate system (xyz system) Figure 6.5. (a) Top-view of microstrip transmission line with misalignment angle θ between in-plane optical axis and propagation direction, (b) cross sectional view of microstrip line with misalignment angle ϕ between the principal axis and x-y plane (x, y, and z are the geometrical axes of microstrip lines; and x, y, and z are the optical axes of anisotropic thin-film substrate Figure 6.6. Orientation of C-plane and R-plane in the conventional unit cell of a single crystal sapphire (a, b, and c are the optical axes of sapphire crystal) Figure 6.7. The simulated results of the R-plane sapphire wafer characterizations: (a) diagonal elements (b) off-diagonal elements Figure 6.8. (a) Layout design for the sapphire wafer measurement and (b) the fabricated sapphire wafer sample Figure 6.9. C-plane sapphire measurement results for ε x and ε z. The nominal values of ε x and ε z are 9.4 and 11.6, respectively, up to 1GHz Figure R-plane sapphire measurement results for diagonalized matrix elements of ε x, ε y, and ε z. The nominal values of ε x, ε y, and ε z are 9.4, 9.4, and 11.6, respectively, up to 1GHz Figure B.1. Conformal mapping of a microstrip in z-plane into a two parallel plates in w- plane xvi

18 Chapter 1 INTRODUCTION In microwave engineering, there are numerous methods for determining material properties, such as permittivity and permeability, for both bulk media and thin-film materials [1]. The characterization of thin-film materials is currently important as the use of new and complex materials in the fabrication of electric circuits increases continuously. Recent progress in engineered materials has provided new materials with unique electromagnetic behaviors; thus, the accurate measurement of their electromagnetic material properties is crucial for assessing whether they can be used in a variety of applications. Therefore, the study of electromagnetic material characterization can be used to determine the electromagnetic properties of the materials by demonstrating that the material properties allow for the designing of appropriate microwave applications, such as 50Ω matched microwave devices. In addition, electromagnetic characterization can often be used in the measurement of the complex permittivity of biological tissue for medical applications [2, 3]. Several different types of microwave sensors, such as resonator sensors, transmission sensors, and reflection sensors, are used in industrial areas [4]. Therefore, accurate measurements of the electromagnetic material characterization are very important for many fields of engineering in order to achieve 1

19 more accurate measurement results, which is highly desired and the main motivation of this study. In electromagnetic material characterization, complex permittivity and permeability are typically determined. Both permittivity and permeability are described as the interactions between the electric and magnetic fields. Therefore, complex permittivity and permeability can be defined based on the constitutive relations: D E (1.1) B H (1.2) where, E, H, D, and B are the electric field, magnetic field, and electric and magnetic flux densities, respectively. In addition, ε = ε 0 ε r and μ = μ 0 μ r are complex permittivity and permeability, respectively; ε 0 ( ) and μ 0 (4π 10-7 ) are the free space permittivity and permeability, respectively; and ε r = ε' r - jε" r and μ r = μ' r - jμ" r are the relative complex permittivity and permeability, respectively. The real and imaginary parts of ε r and μ r are related to the energy storage terms and the loss terms, respectively. The real and imaginary parts of ε r can be described as the capacitance (C) and conductance (G) in the capacitor, respectively, while the real and imaginary parts of μ r can be described as inductance (L) and resistance (R) [5]. Therefore, the permittivity and permeability can be measured using commercial LCR meters by measuring the capacitance and inductance, respectively [6]. Figure 1.1 depicts simple illustrations for measuring capacitance and inductance as well as their equivalent circuit models. In Figure 1.1, the real and imaginary parts of ε r are tc/ε 0 A and tg/ωε 0 A, respectively, where t is the thickness of the sample being tested and ω is the angular frequency. In addition, the real and imaginary 2

20 Electrode (Area = A) C G L R (a) (b) Figure 1.1. Simple illustrations for (a) permittivity and (b) permeability measurements including their equivalent circuit models parts of μ r are ll eff /μ 0 N 2 A C and l(r eff - R w )/μ 0 ωn 2 A C, respectively, where l, L eff, N, A C, R eff, and R w are average magnetic path length of toroidal core, inductance of toroidal coil, number of turns, cross-sectional area of toroidal core, equivalent resistance of magnetic core loss including wire resistance, and resistance of wire only, respectively [6]. The problem in the permittivity measurement using the LCR meter is the air-gap between the electrodes and the sample being tested due to the surface roughness of the sample; these air-gaps produce uncertainties in the measurements. In addition, the permeability measurement using the LCR meter cannot provide accurate results when the sample material has high permittivity because the capacitance being produced between the sample and test fixture should not be neglected if the sample s permittivity is high. Furthermore, in conventional material characterization methods, reflection methods and transmission/reflection methods are commonly used. In the reflection method, material properties can be determined from the reflection, which is caused by the impedance mismatch between a transmission line and the sample. One example of the reflection method is the use of an open-ended coaxial probe, as shown in Figure 1.2 (a). 3

21 Although the open-ended coaxial probe reflection method allows for operations in broadband measurements despite the relatively small sensing area, the coaxial probe should contact the sample material directly; however, due to imperfections, an air gap is created between the probe and sample [7]. A free-space bistatic reflection technique is another example of the reflection method. Unlike most reflection methods, this method uses two antennas to transmit and receive signals; the configuration is shown in Figure 1.2 (b). This method measures different reflections at different incident angles in order to minimize errors stemming from multiple reflections. However, this measurement requires special calibrations [8]. Meanwhile, in the transmission/reflection methods, material properties are determined from the reflection and transmission coefficients. A rectangular dielectric waveguide method one example of the transmission/reflection method can determine the permittivity of test samples with various thicknesses and cross-sections; its measurement configuration is shown in Figure 1.2 (c) [9]. However, this method cannot Transmit antenna Coaxial dielectric probe Free space Rectangular dielectric waveguide Sample Rectangular dielectric waveguide ε r1 d 1 ε r2 Sample terminated by metal plate n 1 n 2 n 1 z=0 z=d Receive antenna (a) (b) (c) Figure 1.2. Examples of conventional material characterization method configurations (a) Reflection method with open-ended coaxial probe, (b) Free-space bistatic reflection method, and (c) Rectangular dielectric waveguide method 4

22 provide an accurate measurement of the loss tangent due to the open discontinuity problem between the rectangular dielectric waveguide and sample These examples of conventional material characterization methods are not considered in on-wafer measurements. Typically, on-wafer measurements use planar circuits, such as a microstrip and coplanar waveguide structures in conjunction with a probe station. The main advantage of these types of structures is that no air gap presents between the metallic structures and the sample being tested. Thus, on-wafer measurement methods can minimize measurement errors due to an air gap. In addition, the on-wafer measurement method can be used directly in the development of the planar circuits on the sample being tested, thereby allowing in-situ measurements. For the on-wafer measurements, resonant and non-resonant methods are commonly used; we will present an in-depth review for both resonant and non-resonant methods in the following chapter. In this study, we realized the need to develop accurate on-wafer measurement methods not only for isotropic thin-film materials, but also for anisotropic thin-film materials. Anisotropic materials present the permittivity and permeability as tensors ( and ); the accurate characterization of the electromagnetic properties of new, on-wafer thin films is crucial for accessing their potential use in the design of microwave devices, antennas, and a variety of sensors. Furthermore, many of the new materials being developed are anisotropic, and previous on-wafer techniques that have been developed to characterize isotropic materials cannot be used. Several methods for determining the permittivity and permeability tensors of the anisotropic materials exist, such as the free space method [10], waveguide method [11], and the transmission/reflection method [12]. The main 5

23 ideas of these measurement methods are similar to isotropic material measurement methods, except that they consider different directions of the electric field. However, these measurement methods are not performed as on-wafer measurements for thin-film materials. Therefore, it is necessary to develop a suitable on-wafer characterization for the anisotropic thin-film materials. In addition, the permittivity tensors of anisotropic materials have different forms depending on the crystal system of the materials (see Appendix A) [13]. Thus, it is necessary to develop on-wafer characterization methods for the most general case of anisotropic materials. Another important aspect of our research goal in material measurement is error analysis. The sources of errors in measurements can be measurement set-up-related errors (e.g., gaps between the sample and sample holder, uncertainty in sample length, and connector mismatch) and calibration-related errors (e.g., uncertainty in reference plane position and imperfection of calibration) [14-16]. Air-gap errors have previously been studied [14, 17]; however, the on-wafer measurement method does not present air gaps between metallic structures and test samples. Thus, calibration-related errors and geometrical uncertainty in the test structures can be considered as the dominant source of on-wafer measurement errors. Analyses of calibration errors have been conducted [18], and a modified Thru-Reflect-Line (TRL) calibration technique has been proposed to reduce calibration errors due to the imperfections of calibration standards. This modified calibration method uses redundant measurements of the calibration standards to eliminate random errors in the calibration standards. Previous research of error analysis due to uncertainties in test structures is also available [15]. This error analysis sets the error 6

24 boundaries that can be predicted from the actual scattering parameters and imperfect scattering parameters, which are the calculated scattering parameters with ideal calibration standards and imperfect calibration standards, respectively. Therefore, we will also consider adopting an error analysis of the on-wafer measurements and discuss the measurement errors due to geometrical uncertainties of the test structures, including calibration standards, in this study. Therefore, here is a summary of the main reasons the development of new on-wafer characterization methods are needed: 1. Newly developed engineered materials are usually formed as wafers in the configuration of layered structures on a thick substrate. Therefore, appropriate onwafer characterization methods are essential for analyzing the electromagnetic properties of those kinds of materials in the microwave frequency region. 2. Although several different types of on-wafer characterization methods are already available, these conventional methods still have significant limitations. In addition, the conventional methods are not capable of characterizing newly developed thin-film materials that have unique properties (e.g., anisotropy in the material properties), since the conventional on-wafer characterization methods are focused mainly on the characterization of the permittivity of isotropic materials. 3. Another limitation of the conventional methods is that the measurement results are not sufficiently accurate, which is the most essential problem with their measurements. Although the conventional methods take into account all the 7

25 possible uncertainties in the measurements, improvement of the measurement accuracy is still needed, and achieving this is a highly desirable goal. As previously stated, the main goal of this study is to develop more accurate on-wafer material characterization methods for different types of materials. Furthermore, it is important to study not only the measurement method itself, but also the data analysis for the measured data for the on-wafer material characterization. Therefore, developing and modifying the data analysis method for the on-wafer characterization is another goal of this study. In this dissertation, we will discuss newly developed on-wafer characterization methods for different types of materials and will also discuss the data analyses of these measurements. First of all, we will provide in-depth reviews for the conventional on-wafer characterization for both non-resonant and resonant methods in the following chapter. We will also show the measurement results using conventional methods in Chapter 2. In Chapter 3, we will discuss a newly developed on-wafer characterization method using the T-resonator for dielectric materials. We will present full mathematical derivations and measurement results in Chapter 3. The on-wafer characterization methods using both non-resonant and resonant methods for the magnetic-dielectric materials will be discussed in Chapters 4 and 5, respectively. A newly developed transmission line method for the magnetic-dielectric materials will also be presented in Chapter 4. In addition, we will provide not only the measurement results, but also conduct an error analysis based on the geometrical uncertainties in Chapter 4. Chapter 5 will include a discussion of a newly developed T-resonator method for the magnetic-dielectric material characterization. 8

26 We will also present an easy way to determine the effective T-stub length and show the measurement results in Chapter 5. In Chapter 6, we will discuss how to characterize anisotropic material using on-wafer measurement methods. In this chapter, we will discuss the transformation of the permittivity tensor due to a misalignment between the optical axes and the measurement axes. Therefore, different on-wafer characterization methods for different permittivity tensor forms will be discussed in Chapter 6. We will also present the measurement results of a sapphire wafer, which is a well-known anisotropic material, in Chapter 6. The last chapter in this dissertation will conclude our presented studies on this dissertation and the discussion of future research topics. Here are the key contributions of this study through the main chapters. 1. The development of a new T-resonator method for the on-wafer characterization of dielectric material: The main achievement of this newly developed method is that it provides much more accurate measurements than the conventional T-resonator methods. This is possible because the new method eliminates parasitic effects due to open-end and T-junction effects of the T-stub. Therefore, the method is capable of achieving a relative error of extraction for permittivity values below 1% with respect to the nominal value of the sample wafer up to the frequency range of 16 GHz. 2. Development of a new on-wafer characterization method for magneticdielectric materials using microstrip transmission lines: The main achievement of this method is that it overcomes the limitation of the conventional transmission line method for the on-wafer characterization of magnetic-dielectric materials. 9

27 Therefore, compared to the conventional methods, this method allows the use of a greater variety of test structures for on-wafer characterization. In addition, this method provides measurements with relative errors of approximately 10% for both permittivity and permeability extractions over the frequency range of 4 GHz to 14 GHz. 3. Development of a new T-resonator method for the on-wafer characterization of magnetic-dielectric materials: This is the first time the T-resonator method has been used for the on-wafer characterization of magnetic-dielectric materials. The main achievement of this method is that it improves the accuracy of the extractions for both permittivity and permeability. Therefore, it is capable of achieving approximately 1% and 3% relative errors for the extracted results of permittivity and permeability, respectively, up to a frequency of 19 GHz. 4. Development of a new on-wafer characterization method for anisotropic materials using microstrip transmission lines: The main achievement of this method is the determination of the full range of matrix elements of biaxial anisotropic materials with misalignment between the optical axes and the measurement axes of the anisotropic material. We demonstrated this method using R-plane sapphire wafers, and the measured results showed relative errors of approximately 5% to 10% for the extraction of the matrix elements over the frequency range of 3 GHz to 16 GHz. In addition, this method allows the determination of the misalignment angle between the optical axes and the measurement axes. 10

28 Chapter 2 REVIEW OF CONVENTIONAL ON-WAFER MEASUREMENT METHODS 2.1. Introduction Typically, on-wafer measurements use planar circuits, such as a microstrip and coplanar waveguide structures in conjunction with a probe station. Figure 2.1 shows a schematic diagram for a typical configuration of the two port on-wafer measurement system using a probe station [19]. Meanwhile, Figure 2.2 depicts the actual configuration of the probe station measurement. Two well-known electromagnetic on-wafer material characterization techniques exist namely: resonant and non-resonant methods [1]. This chapter will review the theoretical background of both non-resonant and resonant P 1 P 2 S21 S 11 S 22 S 12 To network analyzer Coaxial to coplanar transition Probes To network analyzer Conductive strips ε r μ r Coplanar cell Figure 2.1. Typical configuration of the on-wafer measurement using probe station 11

29 (a) (b) Figure 2.2. (a) Probe station measurement configuration. (b) Contact between GSG (Ground-Signal-Ground) probe tip and contact pad (upper) and probes on the wafer sample (lower) methods. The chapter will also demonstrate how to determine the relative permittivity of dielectric materials using both non-resonant and resonant methods for on-wafer measurements. For the on-wafer measurements, it is critical to remove parasitic effects between the probes and contact pads to achieve accurate measured results. Several different calibration methods can be used for on-wafer measurements, such as Short-Open-Load- Thru (SOLT), Line-Reflect-Match (LRM), and Thru-Reflect-Line (TRL) [20-24]. However, the TRL calibration method is the most fundamental calibration technique for 12

30 on-wafer measurement [25, 26] as this method is crucial for removing the parasitic effects [27]. By performing TRL calibration, the reference planes are moved close to the DUT, and the de-embedded scattering parameters of the DUT are the scattering parameters with respect to the characteristic impedance at the center of the Thru standard [27]. Unlike other calibration methods, TRL calibration uses on-wafer calibration standards without requiring matched resistance standards. Thus, the TRL calibration method is very useful for on-wafer material characterization. Therefore, all the measurements in this dissertation are performed using TRL calibration. This chapter will discuss the conventional non-resonant and resonant methods in depth. Since all the studies in this dissertation are based on the on-wafer measurements, it is important to incorporate some parts of these conventional methods in order to apply newly developed on-wafer characterization methods in this study. Thus, full mathematical derivations are discussed in this section. We will also show the measurement results for dielectric material on-wafer characterization using both nonresonant and resonant methods. Furthermore, we will discuss conventional characterization methods for both isotropic magnetic-dielectric and anisotropic dielectric materials Review of Conventional On-Wafer Measurement Methods for Dielectric Materials Numerous studies on the on-wafer electromagnetic material characterizations for dielectric materials have been conducted [28-32]. Both resonant and non-resonant on- 13

31 wafer material characterization methods are commonly used. A resonant method, such as using a T or some other type of resonator, provides accurate results for material properties; however, it provides material properties at a discrete number of equally spaced frequencies [33, 34]. On the other hand, a non-resonant method using transmission lines the so-called transmission line method can provide material properties over a finite frequency band from the measured propagation constant or characteristic impedance of a transmission line [35, 36]. These methods focus primarily on dielectric properties of electromagnetic materials, making it possible to determine the relative permittivity (ε r ) by measuring either the characteristic impedance or the propagation constant of the transmission line. For the on-wafer measurements of both resonant and non-resonant methods, planar waveguide structures are commonly used. Figure 2.3 shows typical examples of planar waveguide structures which are micrsotrip and coplanar waveguide structures. General microstrip and coplanar waveguide transmission line structures on a substrate of thickness h, with relative permittivity of ε r =ε' r -jε" r, are shown in Figure 2.3. Note that the imaginary part of the relative (a) (b) Figure 2.3. (a) Microstrip transmission line and (b) coplanar waveguide transmission line 14

32 permittivity relates to the dielectric loss of the substrate. The following section will discuss how to determine material properties using either microstrip or coplanar waveguide structures for on-wafer characterization methods Overview of Non-Resonant Method The transmission line method is a widely used method for on-wafer measurements. In this method, planar waveguide structures (e.g., a microstrip and coplanar waveguide structure) are typically used. The main advantage of using these types of structures is that no air gap presents between the metallic structures and the sample being tested. Thus, onwafer measurement methods can minimize measurement errors due to air gaps. Another advantage of this method is that it provides continuous values of the material properties over a given frequency range. In addition, the on-wafer measurement method can be used directly in the development of the planar circuits on the sample being tested, thereby allowing in-situ measurements. We will review this well-known material characterization method in the following section Transmission Line Method Theory The transmission line method assumes that the dominant propagation mode in the transmission line is a quasi-tem mode; Figure 2.4 depicts the electric field distributions of both microstrip and coplanar waveguide structures. Thus, it is possible to calculate material properties from the measured propagation constant, which is given by [37]: j jk 0 eff (2.1) 15

33 (a) (b) Figure 2.4. Electric field distribution of (a) microstrip and (b) coplanar waveguide structures where ε eff is the effective dielectric constant of either microstrip transmission line or coplanar waveguide transmission line, expressed as ε eff = ε' eff - jε" eff [27]. The effective dielectric constants of these planar types of transmission lines can be considered as the equivalent dielectric constants of a homogeneous medium in which the transmission lines are embedded. The effective dielectric constants that replace the air and dielectric substrate regions can be obtained using conformal mapping techniques [38, 39]. The real part of the effective dielectric constants for both microstrip and coplanar waveguide transmission lines are shown in (2.2) and (2.3), respectively [40, 41]. MS r 1 r 1 1 eff ( h/ W) (2.2) CPW eff 1 r 1 Kk 2 Kk K k K k 5 5 (2.3) where K(k) is the complete elliptic integral of the first kind. The moduli k, kˊ, k 5, and kˊ5 are given by [41]: 16

34 c b a k b c a (2.4) k k k 2 1k a c b b c a sinh c/ 2h sinh b/ 2h sinh a/ 2h sinh b/ 2h sinh c/ 2h sinh a/ 2h sinh / 2 sinh / 2 sinh / 2 5 1k5 2 2 a h c h b h sinh b/ 2h sinh c/ 2h sinh a/ 2h (2.5) (2.6) (2.7) Note that the effective dielectric constants of both microstrip and coplanar transmission lines are functions of the relative dielectric constant, the substrate thickness, and the geometry of the transmission lines. As a result, the material property, ε r, can be found when the effective dielectric constant of the transmission line is determined; the effective dielectric constant is easily found from the measured propagation constant. This method is a very well-known transmission line method for on-wafer material characterization [1, 35]. Figure 2.5 shows an equivalent circuit model of the transmission line; the circuit parameters L, C, R, and G are the inductance, capacitance, resistance, and conductance per unit length of transmission line, respectively [27]. Loss measurements are also important to consider. The attenuation constant, α, is related to the losses in the measurement. The total attenuation stems from the finite conductivity of the conductors, the dielectric loss of the substrate, and radiation losses (if applicable). The attenuation due to finite conductivity of the conductors accounts for the series resistance, R, and 17

35 L R C G Figure 2.5. Equivalent circuit model of the transmission line dielectric losses account for the shunt conductance, G, in the equivalent circuit model of the transmission line [27]. Therefore, the total attenuation constant is given by: (2.8) c d where α c and α d are the attenuation constants due to conductor losses and dielectric losses, respectively. To determine the dielectric loss tangent of the material, it is necessary to first determine the conductor loss due to the finite conductivity of the metals. The attenuation constants due to conductor losses for both microstrip and coplanar waveguide lines are related to the series-distributed resistances of signal metal lines and ground planes [40, 42]. Thus, the attenuation constant, α c, is given by [41]: R R 1 2 c (2.9) 2Z0 where R 1 and R 2 are the normalized series-distributed resistances for the signal metal line and ground plane, respectively. Equations for R 1 and R 2 of both the microstrip line and coplanar waveguide line are given by [40, 41]: R R LR 1 1 4W ln W T MS S 1 2 (2.10) 18

36 R MS 2 RS W / h W W / h h/ W (2.11) R R 8 a 1 k ln k ln CPW S a 1 k K k T k0 R CPW 0 S a k K k T k0 k0 kr 8 b 1 1 k ln ln (2.12) (2.13) where R S =(ωμ/2σ) 1/2 is the surface resistivity of the conductor, K(k 0 ) is the complete elliptic integrals of the first kind, and k 0 is a/b [40, 41]. Note that the superscripts MS and CPW refer to the microstrip and coplanar waveguide structures, respectively. In addition, LR is the loss ratio in the microstrip line, given by [40]: W 1 for 0.5 h LR 2 W W W for h h h (2.14) The dielectric loss tangent can be determined from the attenuation constant, α d, namely, [40, 41]: 2 d tan qk 0 eff (2.15) where q=(1-(εˊeff ) -1 )/(1-(ε rˊ) -1 ) is the filling factor due to the dielectric loss [41, 43]. The main advantage of the transmission line method is that it provides continuous values of the measured material properties over the finite frequency bandwidth while the resonant method only provides material properties with a discrete number of equally spaced frequencies. In addition, the characterization of material properties is relatively simple since this method only needs to measure the complex propagation constant of the 19

37 transmission line. However, the accuracy of the extracted results is relatively lower than the resonant method. The material characterization method needs to measure the complex propagation constant from the S-parameters, which is a voltage ratio, whereas the resonant method only needs to determine resonant frequencies of the resonator, thus providing a more robust measurement result. In summary, the on-wafer electromagnetic material characterization for isotropic dielectric material uses the transmission line method as a non-resonant method where the material properties (e.g., ε r and tanδ) can be determined from the measured complex propagation constant using the transmission line method Transmission Line Method Experiments This section shows the isotropic-dielectric wafer measurement results using the transmission line method. We fabricated both microstrip and coplanar waveguide test structures on a Pyrex 7740 wafer; Figure 2.6 shows the fabricated Pyrex 7740 wafers with a thickness of 500μm. The given material properties of Pyrex 7740 are a relative dielectric constant of 4.6 and the loss tangent of at 1MHz frequency [44]. We used a lift-off process to deposit the metal on Pyrex 7740 wafers; aluminum and gold were used to deposit the top metal layers for coplanar waveguide and microstrip test structures, respectively. We also deposited gold on the back side of the wafer as a ground plane for the microstrip test structures. In addition, TRL calibration kits were embedded into the Pyrex 7740 to perform TRL calibration for the measurements. Because our measurements 20

38 are based on the on-wafer technique, using a probe station and TRL calibration is fundamental to achieve good accuracy [45]. Unlike coplanar waveguide test structures, microstrip test structures require coplanar waveguide-to-microstrip transitions to implement on-wafer measurements using the probe station [46]. Thus, the test fixtures consist of microstrip transmission lines as DUTs and coplanar waveguide-to-microstrip transitions as error boxes. Figure 2.7 shows the (a) (b) Figure 2.6. Fabricated test structures on Pyrex 7740 wafer (a) coplanar waveguide test structures and (b) microstrip test structures. Both microstrip and coplanar waveguide transmission lines and TRL calibration kits are fabricated on Pyrex 7740 wafers A' B' A B Figure 2.7. The test fixture consists of a microstrip line as a DUT and coplanar waveguide-to-microstrip transitions as error boxes 21

39 microstrip test fixture including the coplanar waveguide-to-microstrip transition. Several different types of vialess coplanar waveguide-to-microstrip transition models exist [47-51]; our vialess coplanar waveguide-to-microstrip transition is based on [48]. Unlike the coplanar waveguide-to-microstrip transition model in [48], our transition model also has a ground plane on the back side of the probe pads since there is no problem maintaining a proper coplanar waveguide mode at the beginning of the transition. Because the gap between the signal line of the coplanar waveguide and the top ground plane is much smaller than the thickness of the wafer [52], it can reduce additional fabrication processes for the ground plane on the back side. Figure 2.8 depicts the E-fields at the A-A' and B-B' planes using a full-wave electromagnetic solver. Figure 2.8 clearly shows that the (a) (b) Figure 2.8. E-field distributions at (a) A-A' plane and (b) B-B' plane. The upper and lower figures represent the magnitude and the vector of E-fields, respectively. 22

40 coplanar waveguide mode is dominant at the A-A' plane and the microstrip mode is dominant at the B-B' plane. The extracted results for the real part of ε r using both microstrip and coplanar waveguide transmission lines are shown in Figure 2.9. According to the extraction results of the relative permittivity in Figure 2.9, the maximum relative errors compared to the nominal value of 4.6 using coplanar waveguide and microstrip transmission lines are approximately 11% and 6%, respectively. According to Figure 2.9, the extracted results using microstrip transmission line show better accuracy than using coplanar waveguide transmission line. Typically, microstrip transmission line provides better electric field concentration to the substrate than coplanar waveguide transmission line. Therefore, microstrip transmission line provides better accuracy for the extraction of the material properties than coplanar waveguide transmission line. Note that the extraction results r 5 r Frequency (Hz) x Frequency (Hz) x 10 9 (a) (b) Figure 2.9. Extraction results of ε r using transmission line method (ε' r of Pyrex 7740 is 4.6): (a) coplanar waveguide transmission line and (b) microstrip transmission line 23

41 using both coplanar waveguide and microstrip transmission lines in Figure 2.9 show in the frequency ranges of 5GHz to 20GHz and 4GHz to 14GHz, respectively. Because of the TRL calibration criteria, which states that the phase angle of the Line standard should be within 20 to 160 [25], the extracted results are valid in those frequency regions. In addition, the microstrip transmission lines in this measurement include the transitions, making it necessary to determine the frequency range where the transitions are valid. It is possible to determine the valid region from the de-embedded return loss of the Thru standard. Figure 2.10 shows the return loss of the de-embedded Thru standard; the region where the magnitude of the de-embedded return loss is lower than -35dB is valid [51]. According to Figure 2.10, the valid calibration region of the frequency range is approximately 3.7GHz to 14.5GHz. In other words, the measured results in the frequencies below 3.7GHz and above 14.5GHz may not be correct Valid Region -20 S11 (db) Frequency (Hz) x Figure De-embedded S 11 of the Thru standard. From the de-embedded S 11 result of the Thru standard, calibration is valid from3.7ghz to 14.5GHz. 24

42 Figure 2.11 shows the extracted results for the dielectric loss tangent using both coplanar waveguide and microstrip transmission lines. As previously stated, coplanar waveguide and microstrip structures use different types of metal deposition on the Pyrex 7740 wafers. According to Figure 2.11, the extracted loss tangents from the coplanar waveguide line measurement vary from to over the frequency range of 5GHz to 20GHz while the extracted loss tangent from the microstrip line measurement vary from to over the frequency range of 4GHz to 14GHz range. Although these extracted results have larger relative errors than the extracted results for the relative permittivity, the absolute errors of the extracted results for the dielectric loss tangent using coplanar waveguide and microstrip lines are small enough to use in the dielectric material characterizations. In general, the transmission line method, which is one of the non-resonant methods, provides less accuracy in the extraction results than the resonant methods. The tan Frequency (Hz) x (a) tan Frequency (Hz) x 10 9 (b) Figure Dielectric loss tangent (tanδ of Pyrex 7740 is 0.005) extraction results for using (a) coplanar waveguide transmission line and (b) microstrip transmission line 25

43 experimental results for the extraction of the material properties in this section show good agreement with the nominal values of the material properties. We will also show and compare the experimental results using the resonant method in a later section Overview of Resonant Method The main advantage of using a resonator method is that it provides accurate results of the material property extraction based on the simple measurement of the resonant frequencies, since the resonant frequencies depend on the effective permittivity and the resonator geometry. In other words, resonance frequencies of the resonators are independent from other factors besides the effective permittivity and the resonator geometry. Although resonant methods provide accurate results in the material characterization, the extracted material parameters can only be determined at the resonant frequencies while non-resonant methods provide continuous values of the material properties over a certain frequency range. On-wafer material characterization requires planar circuit structures (e.g., microstrip and coplanar waveguide) as resonators while the substrates are the material under test. Several different types of resonators are used, including ring resonator [34], T-resonator [33], and straight-ribbon resonator [53], for the on-wafer material characterization. Figure 2.12 shows the different types of resonators in microstrip structures. A ring resonator, depicted in Figure 2.12 (a), has resonances when the mean circumference is equal to the multiple of a guided wavelength. Thus, it provides the effective permittivity of the substrate being tested by measuring resonant frequencies. 26

44 r Gap l 1 Gap L stub Gap l 2 Gap W (a) (b) (c) Figure Three different types of microstrip resonators: (a) ring resonator, (b) T- resonator, and (c) straight-ribbon resonators The relationship between the effective permittivity and the resonant frequency is given by [34]: eff nc 2 rf n 2 for n 1,2,3, (2.16) where r is the mean radius of the ring, f n is the n th resonant frequency, c is the speed of light, and n is the mode number. The resonant frequencies of the ring resonator can be measured directly while the effective dielectric constant of the substrate can be determined using (2.16) and the structure geometries. In addition, the loss tangent of the substrate can be determined from the measured quality factor [54]. Unlike other types of resonators, there is no open end, making it possible to minimize the radiation losses, which is the main advantage of the ring resonator [34, 55]. The main issue with using a ring resonator in the material characterization is the need to determine a suitable coupling gap separating the feed line from the ring, which will ensure that the ring resonator can have selective frequencies. A 27

45 large coupling gap, for example, does not affect the resonant frequencies of the ring resonator whereas a small gap creates a deviation of resonant frequencies [34, 56]. Another type of microstrip resonator is the straight-ribbon resonator method, shown in Figure 2.12 (c). Similar to the ring resonator, the straight-ribbon resonator method provides material properties by measuring the resonant frequencies related to the length of the ribbon [53]. However, it is necessary to consider the ribbon length in determining the effective length due to the coupling gaps, which create incremental changes in the effective ribbon length. A modified straight-ribbon resonator method was proposed by [53]. According to [53], the open-end effects of the coupling gaps can be eliminated by using two or more series resonators. The relationship between the effective permittivity and resonant frequency is given by [53]: eff cnf nf 2 f f l l 1 n2 2 n1 n1 n (2.17) where the subscript 1 and 2 refer the straight-ribbon resonator 1 and 2, respectively. In addition, l is the ribbon length, f n is the n th resonant frequency, c is the speed of light, and n is the mode number. The material loss tangent can be determined from the measured quality factor at the resonant frequency point [54]. Although this modified method includes consideration of the coupling gap effects, it is not completely free of the openend effects. In addition, the straight-ribbon resonator method usually has a lower quality factor than the ring resonator method [1]. The T-resonator is one of the most popular type of resonator for on-wafer material characterization. This method will be discussed in more detail in the following section. 28

46 T-Resonator Method Theory The T-resonator method is widely used for on-wafer material characterization as a resonant method. Unlike the transmission line method, which is commonly used as a nonresonant method, the T-resonator method provides accurate material properties for a discrete number of equally spaced frequencies [33, 57]. These resonant frequencies depend on the material properties of the substrate and the geometry of the resonators, such as the T-stub length in the T-resonator. This method uses a simple T-pattern consisting of feed lines and a T-stub. The T-stub is a quarter-wave stub that provides approximately odd (even) integer multiples of its quarter-wavelength frequency for the open-stub (shot-stub). Figure 2.13 shows a microstrip and coplanar waveguide implementation of a T-resonator. To avoid unwanted modes for the coplanar waveguide T-resonator, it is necessary to include an air-bridge depicted in Figure 2.13 (b) where air-bridges have been added at the junction area. The main reason for using an air-bridge in the coplanar T-resonator is to suppress the parasitic-coupled slotline mode at the T-junction as discontinuities at the T- junction produce mode conversion, which can create excessive losses in the measurement [58]. In addition, air-bridges in the coplanar T-resonator help maintain the even mode the desired mode in the coplanar waveguide structure by suppressing the odd mode (i.e., the undesired mode) [57]. Another advantage of using the T-resonator in the coplanar waveguide structure is the ease of implementing a short-stub T-resonator. Using a shortstub T-resonator removes the open-end effect, which is the main reason for the uncertainties of T-stub length in the open-stub T-resonator. 29

47 (a) (b) Figure T-resonator models: (a) Microstrip T-resonator and (b) Coplanar waveguide T-resonator with air-bridge Regarding the microstrip type T-resonator, as previously stated, a t-stub in the T- resonator is a quarter-wave stub; the basic equation for a quarter-wave stub resonator is given by [33]: L stub nc (2.18) 4 f n eff where L stub is the T-stub length, f n is the n th resonant frequency, c is the speed of light, and n is odd integers for open stub and even integers for short stub. Thus, the effective dielectric constant can be easily determined through the resonant frequencies using (2.18), namely: eff nc 4Lstub f n 2 (2.19) Once ε eff is known, the relative permittivity can be determined from the effective permittivity using conformal mapping of the planar waveguide structure [40, 41]. According to (2.19), the effective permittivity depends only on the T-stub length and the 30

48 resonant frequencies for the conventional T-resonator method. In addition, it is necessary to consider the open-end effect at the end of the T-stub for the open-stub T-resonators. Using a short-stub T-resonator can minimize the open-end effect of the T-resonator. However, it is also necessary to consider the T-junction effects of both the open-stub and short-stub T-resonators. Thus, the T-stub length, L stub, in (2.19) needs to be considered as an effective T-stub length, L eff, to include both the open-end and T-junction effects of T- resonators. The open-end effect in the T-resonator model will increase the electrical length of the T-stub [33]. The T-junction reference plane will shift downward due to the T-junction effect in the T-resonator model [33]. As a result, the effective T-sub length can be considered as Leff Lstub lend ljunction (2.20) where L stub is the physical length of the T-stub measured from the center of the feed to the end of the T-stub. In addition, Δl end and Δl junction in (2.20) are the correction factors for the open-end effect and T-junction effect, respectively. The correction factor Δl end for the microstrip line can be taken into account as follows [59]: end (2.21) 4 l h where eff W h eff W h 0.26 / / 0.87 (2.22) eff / 5 W h tan / 1 (2.23) 31

49 W/ h e (2.24) r tan W / h 6 5e W / h r (2.25) (2.26) where W is the microstrip line width of the top conductor and h is the substrate thickness. The expressions from (2.21) to (2.26) provide accurate results for determining the correction factor due to the open-end effect for the range of normalized widths 0.01 W/h 100 and ε r 128 [59]. When using the short-stub T-resonator, one can ignore this open-end effect; thus, only the T-junction effect has to be considered to determine the effective T-stub length. However, it is difficult to use the short-stub T-resonator with the microstrip line, because it is necessary to use via holes to implement short-stub T- resonators. However, for the coplanar waveguide T-resonator, it is much easier to implement the short-stub T-resonator for the on-wafer measurements. The correction factor due to the T-junction effect for the microstrip line can be taken into account as follows [60]: l junction W f e 0.25 f p1 (2.27) where f p1 [GHz] = 0.4 Z 0 /h[mm] is the first higher-order mode cutoff frequency [60]. It is also imperative to determine material losses. Similar to other resonator methods, material losses can be determined from the measured quality factors in the T-resonator method. The loaded quality factor, Q L, is given by: 32

50 f Q (2.28) L BW 3dB The loaded quality factor, Q L, in (2.28) contains both the quality factor of the T- resonator and the external loading due to the measurement system. Thus, it is necessary to determine the unloaded quality factor, Q 0, which is given by [61]: Q 0 Q L L 1210 A /10 (2.29) where L A is the insertion loss at the resonant frequency. In addition, the unloaded quality factor, Q 0, can be written as: (2.30) Q Q Q Q 0 d c r where Q d, Q c, and Q r are the quality factors due to the dielectric losses, the conductor losses, and the radiation losses, respectively. The quality factor due to the conductor losses, Q c, can be calculated; (2.31) shows the equation for Q c [54]. Q c 20 (2.31) ln10 c g where λ g is the guided wavelength in the microstrip line and α c is the attenuation constant due to the conductor losses given in (2.9). The quality factor due to the radiation losses, Q r, in (2.30) is given by [54]: Q r nz 0 (2.32) 2 h F eff 480 / where F(ε eff ) is a radiation form factor and is the sum of the open-end and the T-junction form factors. The expressions for the radiation form factors due to the open-end and T- 33

51 junction radiations are given by [62, 63]: F open 2 eff eff eff log 2 1 eff eff eff eff (2.33) F T 2 3eff 1 eff 1 eff 2eff 1 eff eff 1 log log 3/2 8eff 1 2eff eff eff eff eff (2.34) Thus, one unknown is left in (2.30): the quality factor due to the dielectric losses. From the measured and calculated quality factors, it is possible to determine the quality factor due to the dielectric losses, Q d ; the loss tangent of the dielectric material can then be determined using the following relationship [54]. tan 1 eff r Q d r eff 1 (2.35) Based on equations from (2.19) to (2.35), the material properties the relative permittivity and the dielectric loss tangent can be determined from the measured T- resonators. In the following section, we will show experimental results for the on-wafer material characterization using T-resonators T-Resonator Method Experiment In this section, we will provide the experimental results of the on-wafer characterization using T-resonators. Both microstrip and coplanar waveguide test structures were fabricated on the Pyrex 7740 wafer; its electrical properties were described in section For the metal deposition, coplanar waveguide T-resonators used aluminum while microstrip T-resonators used gold for both the top test fixtures and 34

52 bottom ground plane. Figure 2.14 shows the fabricated T-resonator test structures on Pyrex 7740 wafers. The coplanar waveguide T-resonators shown in Figure 2.14 (a) have both open-stub and short-stub T-resonators since it is easy to implement the short-stub T- resonator in the coplanar waveguide structures. As previously stated, air-bridges are required to suppress the parasitic coupled mode at the T-junction; thus, we used wirebondings as air-bridges. The microstrip T-resonators shown in Figure 2.14 (b) have the coplanar waveguide-to-microstrip transitions at each end of the feed line, and the transitions used here are the same transition model as discussed in section As in the previous experiments described in section , the experimental measurements in this section are also based on the on-wafer measurements, making it necessary to perform TRL calibrations to remove the parasitic effects from the interface between the probe tip and contact pads. Note that all the TRL calibration kits are also fabricated on the same wafers, although the TRL calibration kits are not shown in Figure Figure 2.15 shows examples of T-resonator measurements for both microstrip and (a) (b) Figure T-resonators on the Pyrex 7740 wafers: (a) coplanar waveguide T-resonators and (b) microstrip T-resonators 35

53 coplanar waveguide structures. The T-resonators used in Figure 2.15 have a T-stub length of 10mm for the coplanar waveguide structure and 15.25mm for the microstrip structure. In addition, the coplanar waveguide T-resonator has a short-stub while the microstrip T- resonator has an open-stub. Unlike the open-stub T-resonator, the short-stub T-resonator does not require compensation due to the open-end effect. Based on the measured resonant frequencies of T-resonators, the material properties (e.g., relative permittivity and dielectric loss tangent) can be determined using equations from (2.19) to (2.35). The extracted material properties of the Pyrex 7740 substrate using both coplanar and microstrip T-resonators are summarized in Table 2.1. Although the T- resonator method provides material properties for only the resonant frequency points, the results of the ε r extraction are accurate compared to the nominal values in [44]. According to the results, the minimum and maximum relative error of ε r extraction results S21 (db) S21 (db) Before TRL After TRL Frequency (Hz) x Before TRL After TRL Frequency (Hz) x (a) (b) Figure S 21 (db) measurement results for T-resonators: (a) coplanar waveguide T- resonators and (b) microstrip T-resonators 36

54 for the coplanar waveguide T-resonator are 1.022% and 1.913%, respectively. The microstrip T-resonator also has a minimum relative error of 2.174% and maximum relative error of 4.0% for the extraction results of ε r. According to the extraction results of ε r, the coplanar waveguide short-stub T-resonator provides better accuracy then the microstrip open-stub T-resonator. It is most likely due to the open-end effect at the microstrip T-stub. Although all the parasitic effects are taken into the effective stub length calculation, the parasitic effects cannot be removed completely for open-stub T- resonator. Because the equations used for the effective T-stub length calculation still contain uncertainties. Although the extraction results of the loss tangent are not good compared to the ε r extraction results in regard to the relative error comparison, the extraction results of the loss tangents in Table 2.1 are closed to the loss tangent measurement results in the previous section which are used transmission line methods. Since loss tangent calculations deal with very small numbers compared to the relative dielectric constant calculations, the relative error in the loss tangent could be high. In addition, the dielectric CPW T-resonator Microstrip T-resonator f (GHz) Table 2.1. Pyrex 7740 wafer measurement results using different types of T-resonators (ε' r and tanδ of Pyrex 7740 are 4.6 and 0.005, respectively) 37 ε r tanδ Value Error (%) Value

55 loss tangent measurement might be more affected than the relative permittivity measurement by fabrication quality, losses in the metallic conductor, wire-bonding quality, and/or other effects. Although the T-resonator method provides material properties at a discrete number of selective frequencies, the extraction results for both the relative permittivity and dielectric loss tangent have better accuracy than using the transmission line method discussed in section Review of Conventional On-Wafer Measurement Methods for Magnetic-Dielectric Materials On-wafer characterization methods for dielectric materials were discussed in the previous section. The material properties that we want to determine for a dielectric material are its relative permittivity and dielectric loss tangent. However, for on-wafer characterization of magnetic-dielectric materials, additional properties must be determined, including relative permittivity, relative permeability, dielectric loss tangent, and magnetic loss tangent. Similar to the on-wafer characterization of dielectric materials, mainly microstrip and coplanar waveguide structures are used for the on-wafer characterization of magnetic-dielectric materials, and the analyses are based on quasi- TEM mode propagation. In addition, non-resonant methods are used mostly for the onwafer characterization of magnetic-dielectric materials, because it is necessary to determine both the propagation constant and the characteristic impedance simultaneously. 38

56 Thus, an in-depth review of the non-resonant method for the magnetic-dielectric materials is presented in this section Transmission Line Method (Theory) The transmission mission line method is the well-known non-resonant method for onwafer electromagnetic characterizations, and it has been used in many research efforts to determine both the relative permittivity and permeability [64, 65]. Just as the transmission line method is used for dielectric materials, microstrip and coplanar waveguide transmission lines are used in this method. In the previous section, we discussed how to determine the relative permittivity of the dielectric wafer using the transmission line method, and the relative permittivity was determined by measuring the propagation constant of a transmission line. However, for magnetic-dielectric materials, it is impossible to determine the relative permittivity and permeability without accurately measuring both the characteristic impedance and the propagation constant of the transmission line. In other words, relative permittivity and permeability can be found easily from the measured propagation constant and characteristic impedance when the transmission line has the quasi-tem dominant mode. A simple expression for the propagation constant and the characteristic impedance is shown below: jk 0 eff eff (2.36) Z eff 0 Z0 (2.37) eff 39

57 where Z' 0 is the characteristic impedance when ε r = μ r = 1. Note that the effective permittivity (ε eff ) and permeability (μ eff ) are complex numbers, and they are given by: j (2.38) eff eff eff j (2.39) eff eff eff Thus, both the propagation constant and the characteristic impedance in equations (2.36) and (2.37) are complex numbers as well. The complex numbers of ε eff and μ eff can be calculated easily by dividing and multiplying of γ and Z 0. Since the substrate has both magnetic and dielectric losses, the relative permittivity (ε r = ε' r - jε" r ) and permeability (μ r = μ' r - jμ" r ) are also complex numbers. It is possible to determine ε' r using either equation (2.2) or equation (2.3) for microstrip or coplanar waveguide transmission lines. For the permeability calculation, the duality relationship is used. The analytical equations for the dielectric case in equations (2.2) and (2.3), i.e., the microstrip and coplanar waveguide transmission line equations, respectively, can be used for the magnetic case by replacing ε with 1/μ [43] (see Appendix B). Thus, equations (2.2) and (2.3) can be rewritten for the expression of μ. The effective permeability for microstrip and coplanar waveguide transmission lines is given by: MS 1/ r 1 1/ r 1 1 eff ( h/ W) CPW eff 1 1/ r 1 Kk 2 Kk K k K k (2.40) (2.41) where K(k) is the complete elliptic integral of the first kind and the modulus (k) are defined in equations (2.4) through (2.7). 40

58 Now, let s discuss both dielectric and magnetic losses. The imaginary parts of the permittivity (ε" r ) and permeability (μ" r ) are related to substrate losses, and it is possible to express ε" r and μ" r in terms of two functions, i.e., q d,loss and q m,loss, which are referred to as filling factors. The filling factors for the dielectric and magnetic losses, i.e., q d,loss and q m,loss, respectively, are given by [43]: q 1 1 eff dloss, 1 1 r (2.42) q mloss, 1 eff 1 r (2.43) Now, consider the effective dielectric and magnetic loss tangents, i.e., tanδ d,eff and tanδ m,eff, respectively. It is shown in [19, 43] that these effective loss tangents can be expressed in terms of the filling factor introduced above: tan tan 1 eff eff r deff, qdloss, tand 1 eff 1 r r 1 1 eff eff r meff, qmloss, tanm eff 1 r r (2.44) (2.45) where tanδ d = εʺr/εʹr and tanδ m = μʺr/μʹr. Since the complex numbers ε eff and μ eff already have been determined from the measured propagation constant and the characteristic impedance, the only unknowns in equations (2.44) and (2.45) are ε" r and μ" r, respectively. Thus, the imaginary parts of the relative permittivity and permeability can be written as: r 1 r eff eff 1 (2.46) 41

59 r 1 r r eff (2.47) 1 eff Therefore, both dielectric and magnetic losses can be calculated using equations (2.46) and (2.47). The extraction procedure in the transmission line method for the characterization of dielectric-magnetic materials is simple if the propagation constant and the characteristic impedance are known. The measurement of the complex propagation constant is not a problem because it can be easily determined from the measured scattering parameters of the transmission line. For on-wafer measurement, however, it is impossible to determine the characteristic impedance from the de-embedded scattering parameters if only the TRL calibration technique is used in the measurement. As mentioned earlier, the TRL calibration technique is the most fundamental calibration technique for on-wafer measurements to de-embed the parasitic effects between the probe tip and the contact pad. Unlike other calibration methods, such as SOLT or LRM, the TRL calibration method does not have a matched load standard (50 Ω). Thus, after performing the TRL calibration, the de-embedded scattering parameters of the DUT are the scattering parameters with respect to the characteristic impedance at the center of the Thru standard [27]. This means that the characteristic impedance of the DUT cannot be determined by the de-embedded scattering parameters of DUT. However, the calibration comparison method provides a way to measure the characteristic impedance using the TRL calibration [45, 67]. Basically, this method compares a planar transmission line under test and the reference impedance at the probe tip. Thus, this method involves two calibration methods, i.e., the so-called two-tier calibration, such as TRL calibration and eff 42

60 SOLT (or LRM) calibration. The first calibration (first tier), i.e., the SOLT or LRM calibration, is performed with the reference impedance at the probe tip set to 50 Ω. The second calibration (second tier) is the TRL calibration, which is conducted with the characteristic impedance of the transmission line being tested set to the characteristic impedance of the error boxes. Figure 2.16 shows an equivalent circuit model that includes an impedance transformer between the probe tip and the error box [67]. From the equivalent model in Figure 2.16, it is possible to express the wave cascading matrix of the error box as [67]: X11 X12 1 YZ r X X (2.48) where X ij represents the matrix elements of the wave cascade matrix of the error box, which can be determined from the TRL calibration. The wave cascade matrix can be defined in terms of the scattering parameters, and equation (2.49) gives the relationship between the scattering parameters and the wave cascade matrix [27]: R11 R12 1 S12S21 S11S22 S11 R R S S (2.49) where R ij and S ij are the matrix elements of the wave cascade matrix and the scattering matrix, respectively. Also, the reflection coefficient Γ in equation (2.48) can be expressed as [67]: X Z0 Zr 12 X21 Z Z 4 X X 2 0 r (2.50) 43

61 where Z 0 is the characteristic impedance of the error box, and Z r is the reference impedance of the probe tip, typically 50 Ω. Since X 12 and X 21 already have been determined from the TRL calibration and Z r also has been determined from the SOLT (or LRM) calibration, it is possible to determine Z 0 from equation (2.50). The results extracted from the measurements of characteristic impedance showed good agreement with results reported in prior research related to the calibration comparison method [67, 68]. Z r :Z 0 Y Probe tip Interface between probetip and contact pad Pad capacitances Impedance transformer Figure Probe tip/contact pad model and its equivalent circuit model 44

62 Chapter 3 AN IMPROVED T-RESONATOR METHOD OF THE DIELECTRIC MATERAL ON-WAFER CHARACTERIZATION 3.1. Introduction In this chapter, we will introduce a new and improved on-wafer characterization method using T-resonators. The conventional T-resonator method only uses the T-stub length of T-resonator; however, a problem occurs in the determination of the effective T- stub length for the conventional T-resonator method. The open-stub T-resonator results in an open-end effect, making it difficult to determine the effective length of the T-stub [33, 59] as previously discussed in Chapter 2. For the short-stub T-resonator, it is possible to reduce the open-end effect; however, there still exists an uncertainty in the determination of the T-stub length, including uncertainties in defining the beginning and end points of the T-stub. This uncertainty can produce an error in the measurement result. In this chapter, we will approach the T-resonator analysis in a different manner. The conventional T-resonator analysis only uses the length of the T-stub to determine material properties at the resonant frequencies; however, our proposed method in this chapter will use both the resonant effects due to the T-stub of the T-resonator and the feed line length of the T-resonator. Since our measurement is based on the on-wafer measurement, the 45

63 TRL calibration method, the most fundamental calibration technique for on-wafer measurement will be used [25, 27]. By performing TRL calibration, we can set the measurement reference planes, which will provide the exact feed line length of the T- resonator. Thus, it is possible to minimize the uncertainty in determining the length of the T-resonator. Consequently, the measurement results will have less error than the results from the conventional method. We will discuss our proposed method analysis in the following section. We will also show our measurement results of the T-resonator using both the conventional method and our proposed method Method of Analysis The T-resonator method is commonly used for material characterization; as a resonant method, it provides accurate results for material properties at a discrete number of equally spaced frequencies. This method uses a simple T-pattern, which consists of feed lines and the T-stub. The T-stub is a quarter-wave stub that provides approximately odd (even) integer multiples of its quarter-wavelength frequency for the open stub (shot stub). The basic equation for the effective dielectric constant of a quarter-wave stub resonator is given in (2.19). The relative permittivity can then be determined from the effective permittivity using conformal mapping of the planar waveguide structures [41, 52]. According to (2.19), the effective permittivity only depends on the T-stub length and the resonant frequencies, not the feed lines of the T-resonator. In other words, the information of the feed lines for the T-resonator is not needed to determine material properties. However, we believe that the feed lines of the T-resonator play an important 46

64 role in material characterization using the T-resonator. In this chapter, we will discuss a new way to use the T-resonator method T-Resonator Matrix Model First of all, we consider the T-resonator as an equivalent circuit model, as shown in Figure 3.1. Each section in the equivalent circuit model can be considered as a transmission line model, single stub model, and transmission line model, respectively. In addition, each sectional model can be expressed with a wave cascade model [27]. The wave cascade matrices of the transmission line model with length l and the shunt resistance (Y) model are given by: R e 0 l 0 TLine l e (3.1) R Y YZ0 YZ YZ0 YZ (3.2) where γ is the propagation constant of the transmission line and Z 0 is the characteristic impedance at the ports of the shunt resistance model. From (3.1) and (3.2), it is possible to express the equivalent circuit model as a series of wave cascade matrix models; (3.3) gives the wave cascade matrix for the T-resonator. R Tres e YZ YZ YZ YZ e l feed l feed 0 (3.3) 47

65 l feed l feed l feed L stub Z 0 Y Z 0 Feed line at port 1 T-stub with length L stub Feed line at port 2 (a) (b) Figure 3.1. (a) A typical T-resonator configuration and (b) its equivalent circuit model for T-resonator. Each section in the equivalent circuit model can be expressed with a wave cascade matrix model. where Y is the input admittance of the stub, given by: Y Y e Lstub e Lstub open Lstub Lstub Z0 e e e Lstub e Lstub short Lstub Lstub Z0 e e for open-stub for short-stub (3.4) (3.5) Thus, the S-matrix of the T-resonator can easily be found from the wave cascade matrix of the T-resonator in (3.3) and the conversion from the wave cascade matrix to the S-matrix given by: S11 S12 1 R12 R11R22 R12R21 S S R 1 R (3.4) Thus, the S-matrix of the T-resonator is given by: 48

66 S YZ e 2YZ 2YZ 2 l feed 2 l feed 0 2e 0 0 Tres 2 l feed 2 l feed 2e YZ0e 2YZ 2YZ 0 0 (3.5) Let s consider the resonances in the T-resonator for both open-stub and short-stub cases. Resonances in T-resonators occur when S 21 goes the minimum; it is possible to express S 21 using (3.5) for both open-stub and short-stub T-resonators. Thus, S 21 of the open-stub and short-stub T-resonators for the lossless case is given by: S 21 4cos 2cosLstub L sin L 2 2 stub stub for open-stub (3.6) S 21 4sin 2sinLstub L cos L 2 2 stub stub for short-stub (3.7) Equations (3.6) and (3.7) clearly demonstrate that the S 21 minimum occurs when cos(βl stub ) and sin(βl stub ) are zero for the open-stub and short-stub T-resonators, respectively. In other words, the S 21 minimum occurs when βl stub is equal to nπ/2 with odd integers for the open-stub T-resonator and even integers for the short-stub T- resonators. Thus, the resonant frequency of the T-resonator is given by: f r nc 4L stub eff where n 1,3,5 for open-stub n 2, 4, 6 for short-stub (3.8) The resonant frequency in (3.8) for both open-stub and short-stub T-resonators is exactly same as the conventional resonant frequency formulas. However, none of the resonances of T-resonators in S 11 are considered in the conventional T-resonator method. Based on (3.5), it is possible to determine S 11 of the 49

67 T-resonator model for both open-stub and short-stub cases. S 11 4cos 2sinLstub L sin L 2 2 stub stub for open-stub (3.9) S 11 4sin 2cosLstub L cos L 2 2 stub stub for short-stub (3.10) According to (3.6), (3.7), (3.9), and (3.10), resonances in S 11 and S 21 for both openstub and short-stub T-resonators only depend on the T-stub length, not the length of feed lines. However, we noticed that YZ 0 /2 in (3.3) goes to zero at the S 11 resonances for lossless cases. As a result, R 11 in (3.3) is the S 11 resonant frequency is given by: e j2lfeed at the S 11 resonant frequency. Thus, β at j ln R11 (3.11) l 2 feed The effective permittivity can be found from the determined β at the S 11 resonant frequency. Since, in this study, we use T-resonators implemented with planar structures, such as the microstrip line structure or coplanar waveguide structure, we can determine the relative permittivity at the frequency of the S 11 resonance using conformal mapping techniques [41, 52]. It is important to discuss the difference between the conventional T-resonator method and our proposed method. The conventional method uses the resonant frequency in S 21 to characterize material properties where the resonance frequency only depends on the length of the T-stub. However, uncertainty exists when determining the exact T-stub length, such as the open-end and T-junction effects discussed in the previous chapter. Our 50

68 proposed method, on the other hand, uses the resonant effect in S 11 (which makes YZ 0 /2 in R 11 equal to zero) and the feed line length of the T-resonator. Using the feed line of the T-resonator can minimize the uncertainty when determining the exact feed line length of the T-resonator, which is an advantage of our proposed method over the conventional method. In this study, we use the TRL calibration method the most fundamental calibration method for on-wafer measurement setting up the reference planes where we want to measure using the TRL calibration method [27]. In other words, it is possible to minimize the uncertainty in measuring the feed line length by measuring the distance between two reference planes. We will show and compare the T-resonator measurement results using both the conventional method and our proposed method later Consideration of Loss Measurements The loss calculations for the conventional T-resonator method were discussed in Chapter 2. Now we will consider material loss determination using our proposed method. Our proposed method can determine material loss using the measured R 11. The R 11 of open-stub T-resonator in (3.3) for the lossy material is given by: R 11 2 l sinh L feed stub e 1 2cosh Lstub (3.12) As previously stated, we are interested in R 11 at S 11 resonant frequency points, and R 11 will be e j2 l feed for low loss materials. Thus, it is possible to determine the attenuation constant at S 11 resonant frequency points. The attenuation constant and the phase constant at the S 11 resonance points are given by: 51

69 1 ln R11 2l (3.13) feed j R (3.14) 2l feed 11 The measured attenuation constant, α, can be broken down into different components, with the total attenuation constant given by: c d r (3.15) where α c is the attenuation constant due to the conductor losses, α d is the attenuation constant due to the dielectric losses, and α r is the attenuation constant due to the radiation losses. If we use open-stub T-resonators, we need to consider all loss terms in (3.15). However, if we use short-stub T-resonators, the radiation losses can be neglected. In that case, the total attenuation constant can be considered as a sum of α c and α d. We discussed how to calculate α c, α d, and α r for both microstrip and coplanar waveguide structures in Chapter 2. From (3.13), we can determine the total attenuation constant, α. We can also determine the attenuation constant due to the conductor losses, α c, and the attenuation constant due to the radiation losses, α r, from the equations in Chapter 2. Thus, it is possible to determine the attenuation constant due to dielectric losses, α d, as well as the dielectric loss tangent using (2.15). We will show and compare the measurement results using both the conventional and our proposed methods in the following section. We will also show and verify that our proposed method will not be affected by the effective T-stub length using both short- and open-stub T-resonators. 52

70 3.3. T-Resonator Measurement Results We built and measured T-resonators to verify our proposed method. As in the previous chapter, we fabricated both coplanar waveguide and microstrip T-resonators on a 500μm Pyrex 7740 wafer, whose nominal electrical properties are ε r of 4.6 and tanδ of at the 1MHz frequency [44]. We deposited aluminum and gold on top of the Pyrex wafer as coplanar waveguide and microstrip test structures, respectively. We also generated TRL calibration kits on the same wafer to perform TRL calibration for each of the T-resonator measurements. In addition, the coplanar waveguide-to-microstrip transitions are included in the microstrip T-resonator models; we discussed these transition models in Chapter 2. Figure 3.2 depicts the fabricated test sample structures on a Pyrex 7740 wafer with a diameter of 100mm. Our measurements were performed on the probe station (Cascade Microtech) using a vector network analyzer (Agilent); our frequency range of the measurement was 1GHz to 20GHz, and the measurement configuration was the same as in Figure 2.2 in Chapter 2. Figure 3.3 shows the measured coplanar waveguide T-resonator S-parameters for both S 11 and S 21. The measured T-resonator had 10mm of shorted T-stub length and 2.425mm of feed line length after moving the reference plane from the probe tip to the beginning of the DUT. According to Figure 3.3, the resonant frequencies in S 21 are not changed by performing TRL calibration. In other words, the resonant frequencies in S 21 of the T- resonator depend only on the length of the T-stub. This is the main advantage of using the T-resonator in the material characterization. On the other hand, resonant frequencies in S 11 are changed by TRL calibration. However, the resonant frequencies in S 11 after TRL 53

71 (a) (b) Figure 3.2. Fabricated test structures on Pyrex 7740 wafers which have diameter of 100mm. (a) Coplanar waveguide structures and (b) microstrip test structures calibration are very close to the resonant frequencies of S 11 in our matrix model in (3.10). First, we consider the conventional coplanar waveguide T-resonator analysis and determine ε r of the Pyrex wafer from the measured S 21 resonant frequencies. The measured resonant frequencies in S 21 are 8.904GHz and GHz. Thus, the extracted ε r are and at the first and second resonant frequencies in S 21, respectively. All the parasitic effects are considered for the extraction of ε r which are discussed in the previous chapter. The measurement results using our proposed method show similar, albeit more accurate, results. According to (3.13) and (3.14), our proposed method uses the magnitude of R 11 and the phase angle of R 11, which are related to α and β, respectively. Figure 3.4 shows both the magnitude and phase angle of the measured R 11 ; both behave 54

72 S11 (db) -15 S21 (db) Before TRL After TRL Frequency (Hz) x Before TRL After TRL Frequency (Hz) x (a) (b) Figure 3.3. Measured short-stub coplanar waveguide T-resonator (a) S 11 and (b) S 21 in db. Blue and red lines indicate the measured S-parameters with and without performing TRL calibrations, respectively. well at S 11 resonant frequency points, which are 4.411GHz and GHz. The extracted ε r can be found using (3.14). Thus, the extracted ε r are and at the first and second resonances in S 11, respectively. Table 3.1 shows the extracted ε r comparison between the conventional method and our proposed method for coplanar waveguide T-resonators. According to the extracted results, both methods provide very accurate results. In other words, the extracted results for ε r from both methods have very small relative error with respect to the nominal value, which is ε r of 4.6 for the Pyrex 7740 wafer. Yet by comparing both methods, it becomes clear that our proposed method provides more accurate extracted results than the conventional method. The conventional method has approximately 2% of the maximum relative error whereas our proposed method has less than 1% of the maximum relative error with respect to the nominal value 55

73 Mag(R11) Frequency (Hz) x (a) Ang(R11) (in degree) Frequency (Hz) x (b) Figure 3.4. Measured (a) magnitude of R 11 and (b) phase angle of R 11 for short-stub coplanar waveguide T-resonator. The green dashed lines in the plots indicate the S 11 resonant points 56

74 Proposed Method Conventional Method f (GHz) ε r Relative Error (%) Table 3.1. The measurement results comparison for coplanar waveguide T-resonator of 4.6. As previously discussed, our proposed method uses both resonant effects due to T- stub and the feed line of the T-resonator. Resonances in S 11 make the R 11 in the wave cascade matrix depend only on the feed line length of the T-resonator. In addition, it is possible to set the measurement reference planes by performing TRL calibration, which provides the exact feed line length of the T-resonator. As a result, the uncertainty in the measurement of the feed line can be minimized, and the extracted results of ε r will have fewer relative errors. In regard to the loss measurement of the T-resonator, we discussed how to determine tanδ of the sample being tested using the T-resonator for both the conventional and proposed method in the previous section. For the conventional method, as discussed in Chapter 2, tanδ of the Pyrex 7740 wafer are and at the frequencies of 8.906GHz and GHz, respectively. These values are much higher than the nominal value of the Pyrex 7740 wafer, which is tanδ of On the other hand, the determined tanδ using our proposed method are and at the frequencies of 4.411GHz and GHz, respectively. These determined tanδ are also different from the nominal 57

75 value of the Pyrex 7740 wafer; however, these values are much closer to the nominal value than those determined using the conventional method. Furthermore, the method of analysis for the microstrip T-resonator is the same as the coplanar waveguide T-resonator. However, the microstrip T-resonators used in this measurement are open-stub T-resonators. Figure 3.5 shows the measured S 11 and S 21 of the microstrip T-resonator. The microstrip T-resonator used in Figure 3.5 has an open T- stub with a stub length of 15.25mm and 2.5mm of feed line. The measured resonant frequencies in S 21 of the microstrip T-resonator are 2.714GHz, 8.192GHz, GHz, and GHz. In addition, the measured resonant frequencies in S 11 of the microstrip T-resonator are 5.488GHz, GHz, and GHz. Unlike the previous short-stub coplanar waveguide T-resonator, the microstrip T-resonator in this measurement has an S11 (db) S21 (db) Before TRL After TRL Frequency (Hz) x Before TRL After TRL Frequency (Hz) x (a) (b) Figure 3.5. Measured open-stub microstrip T-resonator (a) S 11 and (b) S 21 in db. Blue and red lines indicate the measured S-parameters with and without performing TRL calibrations, respectively. 58

76 open T-stub. Therefore, it is necessary to consider the open-end effect and T-junction effect when determining the effective T-stub length. We already discussed how to accounts for the open-end effect in the effective T-stub length in Chapter 2. Using the conventional T-resonator method including the open-end effect and T-junction effect, the extracted ε r are 4.784, 4.700, 4.734, and 4.74 for each of the resonant frequency points in S 21. The minimum and maximum relative errors with respect to the nominal value of 4.6 are 2.174% and 4.0%, respectively. The extracted results of ε r using the conventional T- resonator method demonstrate good agreement with the nominal value of the Pyrex 7740 wafer. Meanwhile, as previously stated, our proposed method does not need to consider both the open-end effect and T-junction effect. Therefore, we just apply the measured R 11 data to (3.13) and (3.14) to extract the material properties. First of all, we need to determine the resonant frequencies in S 11. According to Figure 3.5, the resonant frequencies in S 11 are 5.488GHz, 10.74GHz, and GHz. Then, we need to apply the measured R 11 data to (3.13) and (3.14) to determine the material properties. Figure 3.6 shows both the magnitude and phase angle of the measured R 11 for the microstrip T-resonator; both demonstrate good behavior at the S 11 resonant frequency points, which are marked on Figure 3.6 with green dashed lines. The extracted ε r using our proposed method are 4.596, 4.579, and for each of the resonant frequencies in S 11. The relative errors of the extracted value of ε r with respect to the nominal value of 4.6 are 0.094%, 0,457%, and 0.657%. Thus, our proposed method gives a maximum relative error of less than 1%. This means that our proposed method has much better accuracy compared to the 59

77 Mag(R11) Frequency (Hz) x (a) Ang(R11) (in degree) Frequency (Hz) x (b) Figure 3.6. Measured (a) magnitude of R 11 and (b) phase angle of R 11 for open-stub microstrip T-resonator. The green dashed lines in the plots indicate the S 11 resonant points 60

78 conventional T-resonator method. The main reason for this high accuracy is that our proposed method is not affected by the open-end effect and T-junction effect, which is the most advantageous part of our proposed method. Table 3.2 summarizes a comparison between the conventional and proposed T-resonator methods. According to Tables 3.1 and 3.2, our proposed T-resonator method provides more stable results than the conventional T-resonator method for both open-stub and short-stub T-resonators. The relative errors for our proposed T-resonator method stay below 1%, while the relative errors for the conventional T-resonator method vary from 1% to 4%. The fluctuation in the relative errors for the conventional T-resonator method reflects that the accurate determination of the effective stub length is a crucial part of the conventional T-resonator method. Moreover, the conventional T-resonator method still has uncertainty problems with the open-end effect and T-junction effect, although these parasitic effects can be managed in this method. Regarding the loss measurements of the microstrip T-resonator, we already discussed Proposed Method Conventional Method f (GHz) ε' r Relative Error (%) Table 3.2. The measurement results comparison for microstrip T-resonator 61

79 how to determine the dielectric loss tangent using the T-resonator in Chapter 2. Unlike the dielectric loss tangent calculation for the coplanar waveguide T-resonator, it is necessary to consider the open-end effect to achieve accurate results. We also discussed measurement losses due to the open-end effect in Chapter 2. For the conventional T- resonator method, the measured dielectric loss tangents of the material are 0.026, 0,011, 0.007, and at each of the resonant frequency points in S 21. For the proposed T- resonator method, the measured dielectric loss tangents are 0.019, 0.012, and at each of the resonant frequency points in S 11. The determined dielectric loss tangents for both methods have large relative errors with respect to the nominal value of compared to the relative errors in the determination of ε r. However, the determined dielectric loss tangents based on the proposed method are much closer to the nominal value than the determined dielectric loss tangents using the conventional method. Another observation regarding the microstrip T-resonator measurements comparison stems from error analysis comparison. The error analysis used in this chapter is the standard error analysis for the extraction of ε r from the measurements of T-resonators on the different wafers The standard error, SE, is / n, where n is the size of the sample and σ is the sample standard deviation. The sample standard deviation, σ, is given by x x n, where x is the sample mean average. Figure 3.7 shows the standard 2 ( ) / error analysis for the extraction of ε r using both conventional and proposed T-resonator methods. Figure 3.7 also includes upper and lower 95% confidence error bars, which can be determined from SE and are given by x( SE 1.96). Note that we used 24 samples, which provide about 20% of the margin of error in 95% of confidence limits, in this error 62

80 (a) (b) Figure 3.7. Error analysis with ±95 confidence limits of ε r extraction using (a) conventional T-resonator method and (b) proposed T-resonator method analysis for each method. Therefore, 24 samples are not enough to provide an accurate error analysis; however, it is possible to see the error behavior in the extraction of ε r for each method. Each of the resonant frequency points in Figure 3.7 are the average resonant frequency points of the samples, and the deviation of the resonant frequencies at each point is very small. According to Figure 3.7, the maximum variations in the ±95% 63

81 confidence limits for the conventional and proposed T-resonator methods are ±0.017 and ±0.034, respectively. In addition, the minimum variations in the ±95% confidence limits for the conventional and proposed T-resonator methods are ±0.007 and ±0.025, respectively. The conventional T-resonator method has lower maximum and minimum variations in the ±95% confidence limits than the proposed T-resonator method. Although our proposed method has larger variations in the ±95% confidence limits, the absolute values of the variation are still sufficiently small. In addition, our proposed T- resonator method has smaller relative errors for ε r in the ±95% confidence limits than the relative errors for the conventional T-resonator method. The minimum and maximum relative errors in the ±95% confidence limits for our proposed method are 0.338% and 3.299%, respectively, while the conventional method s minimum and maximum relative errors in the ±95% confidence limits are 2.127% and 4.001%, respectively. Table 3.3 summarizes the error analyses of ε r extraction for the conventional and proposed T- resonator methods. Proposed Method Conventional Method Avg. f (GHz) ε r Relative Error (%) Table 3.3. The error analyses comparison for microstrip T-resonator measurements 64

82 3.4. Summary In this chapter, we discussed a new and improved on-wafer characterization for thinfilm materials using T-resonators. Unlike the conventional T-resonator method, our proposed method uses the resonant effects in the feed line of the T-resonator instead of the resonator itself. The main advantage of our proposed method is that it can minimize the uncertainty in determining the length of the T-resonator. Thus, our proposed method can increase accuracy in the measurement results. In this chapter, we also showed and compared on-wafer measurement results of both coplanar waveguide and microstrip T- resonators using both the conventional method and our proposed method. The measurement results clearly indicated that our proposed method provides better results than the conventional method. In addition, we verified that our proposed method is not affected by the open-end effect or T-junction effect even if the open-stub T-resonator is used in the measurement. 65

83 Chapter 4 NOVEL ELECTROMAGNETIC ON-WAFER CHARACTERIZATION METHOD FOR MAGNETIC-DIELECTRIC MATERIALS 4.1. Introduction In this chapter, we will introduce a new on-wafer characterization method for magnetic-dielectric materials. Unlike nonmagnetic-dielectric materials, it is necessary to determine both ε r and μ r from the measured characteristic impedance and propagation constant of transmission lines printed on this class of materials. We already discussed how to determine both ε r and μ r in Chapter 2. We also discussed the TRL calibration, which is a very fundamental calibration technique for the on-wafer measurements, in Chapter 2. However, after performing TRL calibration, the de-embedded scattering parameters of DUT are the scattering parameters with respect to the characteristic impedance at the center of the Thru standard [27]. This means that the characteristic impedance of the DUT cannot be determined by the de-embedded S-parameters of the DUT. Therefore, a two-tier calibration method is conducted to determine the characteristic impedance of the DUT; this method is called the calibration comparison method [45]. Although the calibration comparison method can accurately determine the characteristic impedance, this method determines the characteristic impedance of the 66

84 error box. Thus, this method can be used if the characteristic impedance of the DUT is the same as the characteristic impedance of the error box. However, sometimes on-wafer measurements require the DUT to have a different characteristic impedance from its error box, such as a microstrip line with a coplanar waveguide-to-microstrip transition without via holes [47-51]. Since microstip structures allow for a better concentration of the field into the substrate, microstrip structures are more suitable for the electromagnetic material characterization. Therefore, a coplanar waveguide-to-microstrip transition is needed to use the microstrip structure in the on-wafer electromagnetic material characterization. In this case, the discussed method for determining the characteristic impedance may not be appropriate. In this chapter, we will discuss a new on-wafer characterization method for magneticdielectric materials. This method uses two transmission lines that have the same line length, but different line widths to determine the characteristic impedance ratio of these two transmission lines on a homogeneous and isotropic substrate material. Then, ε r and μ r can be determined from the measured propagation constants and the characteristic impedance ratio. We will present the theoretical derivation for this method in the following section Method of Analysis - System Matrix Model TRL calibration is a well-known and the most fundamental on-wafer calibration method. One property of TRL calibration is that the reference impedance of a DUT is set as being equal to the characteristic impedance at the center of the Thru standard, Z 0 [27]. 67

85 Thus, the de-embedded scattering parameters of the DUT are relative to Z 0. Let s consider that two DUTs have different characteristic impedances; namely, DUT1 has the same characteristic impedance as the characteristic impedance at the center of the Thru standard, Z 01, while DUT2 has a different characteristic impedance, Z 02. In addition, DUT2 has the same error boxes as DUT1. Figure 4.1 shows block diagrams of these two test structures. Error boxes A and B can be removed after TRL calibration; however, the de-embedded scattering parameters of DUT2 will include the impedance mismatch between Z 01 and Z 02. Thus, it is possible to express two measurement sets with wave cascade matrices that can be written in terms of the scattering parameters using (2.45). Regarding the measured wave cascade matrices of DUT1 and DUT2, including the error boxes [R m1 ] and [R m2 ], equations (4.1) and (4.2) show the system matrices of test sets (a) and (b), respectively. R R R R (4.1) m1 a D1 b R R R R R R R R R (4.2) m2 a D2 b a mis1 D2' mis2 b (a) Error box A Z 01 [R a ] DUT1 Z 01 Z 01 Z 01 Z 01 Z 01 [R D1 ] Error box B Z 01 [R b ] (b) Error box A Z 01 [R a ] DUT2 Z 01 Z 02 Z 02 Z 02 Z 01 [R D2 ] Error box B Z 01 [R b ] Reference planes Figure 4.1. Block diagram of two sets of DUT s with same error boxes. [R a ], [R b ], [R D1 ], and [R D2 ] are the wave cascade matrices of Error box A, Error box B, DUT1, and DUT2, respectively 68

86 where [R a ], [R b ], [R D1 ], and [R D2 ] are the wave cascade matrices of Error box A, Error box B, DUT1, and DUT2, respectively. Note that [R D2' ] in (4.2) is the wave cascade matrix without the impedance mismatch. In addition, [R mis1 ] and [R mis2 ] are the wave cascade matrices representing the impedance mismatch between DUT2 and the error boxes A and B. [R mis1 ] and [R mis2 ] can be expressed in terms of Z 01 and Z 02 namely: R R mis1 mis2 1 Z Z Z Z Z01Z Z Z01 Z01 Z 02 1 Z Z Z Z Z01Z Z Z02 Z01 Z 02 (4.3) (4.4) It is impossible to determine Z 01 and Z 02 directly from (4.3) and (4.4) without first knowing either Z 01 or Z 02. However, it is possible to find the ratio of the characteristic impedance. A more specific derivation of the transmission line case will be examined on the following section Method of Analysis - Transmission Line Models Let s consider two different transmission lines which have same length, L, but different line widths. Thus, these two transmission lines have different characteristic impedances. In addition, the wave cascade matrix of a transmission line can be expressed with the propagation constant and line length. Thus, DUT1, which is the first transmission line with the characteristic impedance Z 01, can be written as: R D1 e 0 1L e 0 1L (4.5) 69

87 However, the wave cascade matrix of DUT2 (the second transmission line with the characteristic impedance of Z 02 ) includes the impedance mismatch matrices. Its wave cascade matrix can be found from (4.2) to (4.4). R 1 L e Z Z e Z Z 2 2 2L L 2 2 2L 2 2 e Z01 Z02 e Z02 Z01 D2 2L 2L 4Z01Z02 e Z02 Z01 e Z01 Z02 2L 2 2 2L e Z01 Z02 e Z01 Z02 (4.6) The propagation constant in (4.5), γ 1, which is the propagation constant of DUT1, can be found easily through TRL calibration [25]. The propagation constant in (4.6), γ 2, which is the propagation constant of DUT2 but excluding the impedance mismatch matrices, can be found from (4.6) through several steps of derivation. Equation (4.7) is the propagation constant of DUT2. D2 D cosh R R L 2 (4.7) where D2 R ij is a matrix element in [R D2 ]. Thus, two unknowns, Z 01 and Z 02, are left in (4.6); however, Z 01 and Z 02 cannot be determined directly. Therefore, we must consider the characteristic impedance ratio, r = Z 01 /Z 02, plugging it into (4.6). The following wave cascade matrix for DUT2 is obtained in terms of r: R L 1 1 L 1 1 2L L e r e r e r e r 4r e 1r e r 1 e r1 e r1 D2 2L 2 2L L 2L (4.8) From (4.8), the characteristic impedance ratio of r can be found after several steps of 70

88 derivation. An expression for r can be obtained in terms of the propagation constant and the matrix elements of DUT2, which are all known parameters namely: D2 D2 D2 D2 Z01 R21 R12 R22 R11 r (4.9) Z 2sinh L 02 2 In addition, the characteristic impedance of the transmission line model, whose equivalent circuit models is shown in Figure 2.5, is given by [27]: Z 0 R G jl jc (4.10) where R, G, C, and L are the resistance, conductance, capacitance, and inductance per unit length of conventional transmission line theory, respectively; and are defined by [27]: R h ds e ds 2 i S eff t eff z (4.11) S G e ds h ds 2 v S eff t eff z (4.12) S 1 C e ds h ds 2 v S eff t eff z (4.13) S 1 L h ds e ds 2 i S eff t eff z (4.14) S where v 0 and i 0 are the normalization constants for the waveguide voltage and waveguide current, which are v(z) = v 0 e ±γz and i(z) = i 0 e ±γz, respectively. The effective permittivity and permeability are given by ε eff = εʹeff - jεʺeff and μ eff = μʹeff - jμʺeff, respectively. Equations (4.11) through (4.14) do not include metal conductivity as an explicit term in ε eff, but it is absorbed in εʺeff [27]. 71

89 From (4.10) to (4.14), it is easy to find the characteristic impedance in terms of L, C, ε, and μ namely: Z 2 1 eff eff eff 0 L 2 C eff eff eff (4.15) The effective values of the permittivity and permeability in a microstrip line can be considered to be the equivalent permittivity and permeability of a homogeneous medium in which the transmission line is embedded. These effective values, which replace the air and magnetic-dielectric substrate regions, can be obtained using conformal mapping techniques [52]. Next, (4.15) can be used in (4.9), resulting in an expression for the characteristic impedance ratio, r, in terms of L, C, ε eff, and μ eff. Note that C and L are defined as C = C a ε' eff and L = μ' eff / C a, where C a is the capacitance of the transmission line when it is airfilled; therefore, it only depends on geometry [40]. In addition, the propagation constant and the index of refraction are related by n eff = (ε eff μ eff ) 1/2 = jγ/k 0. After several simple algebraic steps, the characteristic impedance ratio, r can be expressed as: r C C a2 eff 2 1 (4.16) a1 eff1 2 r C C a2 eff 1 2 (4.17) a1 eff 2 1 where r, γ 1, and γ 2 in (4.16) and (4.17) were found using (4.9), (4.5), and (4.8), respectively. In addition, the air-filled capacitances C a1 and C a2 can be found if we know the geometry of the transmission line. The air-filled capacitance per unit length, C a, of the microstrip line is shown in (4.18) [40]: 72

90 2 o Ca for W / h1 8h W ln W 4h W W Ca o ln for W / h1 h h (4.18) where W is the microstrip line width and h is the substrate thickness. Thus, the only unknowns in (4.16) and (4.17) are ε eff2 /ε eff1 and μ eff1 /μ eff2. Finally, from ε eff2 /ε eff1 and μ eff1 /μ eff2, it is possible to extract the actual ε r and μ r, because ε eff and μ eff depend on ε r, μ r, and the geometry of the transmission line. For a microstrip line, the effective permittivity is given in (2.2). Furthermore, the analytical equations for the effective permeability of the microstrip line are obtained based on a duality relationship. Thus, the effective permeability of the microstrip line is shown in (2.36). Therefore, it is possible to determine ε r and μ r by plugging the effective permittivity and permeability equations into (4.16) and (4.17), respectively. In the following section, we will verify this method by showing several simulated results; however, first we need to consider both the dielectric and magnetic losses of the thin-film substrate. Similar to the loss calculation in Chapter 2, we use the total attenuation constants of two different transmission lines in this analysis. The total attenuation constant, α, can be broken down into different components, with the total attenuation constant given by: d m c (4.19) where α c, α d, and α m, are the attenuation constants due to the conductor losses, dielectric losses, and magnetic losses, respectively. We already described α c in (2.9). In addition, the summation of α d and α m is given by [43]: 73

91 k0 ' eff ' eff d m tand, eff tanm, eff (4.20) 2 where tanδ d,eff and tanδ m,eff are the effective dielectric and magnetic loss tangents, as shown in (2.40) and (2.41), respectively. Thus, it is possible to express (4.20) as a function of εʺr and μʺr using (2.40) and (2.41). Therefore, (4.20) of DUT1 and DUT2 can be expressed as: 1 eff 1,2 kn 1 0 eff 1,2 1 eff 1,2 d m 1,2 r 2 r A1,2 r B1,2 2 r 1 r r (4.21) We already determined εʹr and μʹr, meaning that εʹeff and μʹeff can be easily found using conformal mapping techniques since we know the geometry of DUT1 and DUT2. Thus, there are two unknowns in (4.21): εʺr and μʺr. Having two unknowns and two equations, it is possible to solve them for εʺr and μʺr using: B B r and A A AB AB AB AB r (4.22) We have demonstrated all the theoretical derivations of our proposed on-wafer characterization method for the magnetic-dielectric materials in this section. Using our proposed method, it is possible to determine all the material parameters, such as εʹr, μʹr,εʺr, and μʺr, for the magnetic-dielectric material using two different transmission lines Simulated Results with Sensitivity Test Using a full-wave electromagnetic solver, we accurately simulated all the steps of the measurement procedure, including calibration, to access the accuracy of this proposed 74

92 method. Although we could have used a number of planar transmission lines, we used microstrip transmission lines because they are very common for wafer-based measurements. We initially used a lossless substrate with ε r =3 and μ r =2 and a thickness of 100μm. Figure 4.2 shows the actual test structure geometries used in the simulations. DUT1 is a microstrip transmission line with a length of 5mm and a width of W1=500μm. DUT2 has the same geometry except for its width, which is W2=600μm. Meanwhile, both test structures have the same error boxes at each end. In addition, as previously mentioned, all TRL calibration procedures were performed in the simulations, and the TRL calibration kits (Thru, Reflect, and Line) were based on the error box structures in Figure 4.2. Figure 4.3 shows the simulated results of the extracted relative permittivity and permeability values. The simulated results indicate that the relative permittivity varies from to over the frequency range of 1GHz to 10GHz. These results demonstrate very good agreement with the actual value of 3. The minimum and the l e L l e Transmission Line 1 Z 01 W1 Z 01 Z 01 Transmission Line 2 Z 01 W2 Z 02 Z 01 Error Box1 DUT Error Box2 Figure 4.2. The actual simulated microstrip transmission lines. DUT1 is the top figure while DUT2 is the bottom figure. In the simulation, W1 and W2 are 500μm and 600μm, respectively. The length of error box (l e ) and DUT (L) are 500μm and 5mm, respectively 75

93 3.2 3 Relative permittivity and permeability Relative permittivity Relative permeability Frequency (Hz) x 10 9 Figure 4.3. Simulated results of ε r and μ r extraction for lossless case (ε r =3 and μ r =2 are the exact values) maximum relative errors of the extracted relative permittivity are 2.13% and 3.63%, respectively. The simulated result for the relative permeability also shows good agreement with the actual value of 2. The extracted permeability varies from to over the frequency range of 1GHz to 10GHz. The minimum error of the extracted μ r is 0.95% while the maximum relative error is 2.9% As shown in Figure 4.2, this simulation uses two different microstrip lines with the same error boxes. Thus, step discontinuities exist at the interfaces between the DUT2 and the error boxes. Although the simulated results do account for these discontinuities, the model used to extract ε r and μ r does not at this time, however, it can be easily added to the model. Therefore, the proposed method may not work as well for cases where the difference in width between the two microstrip lines is large. In addition, when the 76

94 difference in the width between the two microstrip lines is too small, the method loses sensitivity. Thus, it is necessary to determine a range of appropriate ratios for the two microstrip line widths, which is referred to as r w = W2/W1. Table 4.1 summarizes the minimum and maximum relative errors of the extracted results for εʹr and μʹr. Table 4.1 does not include the case when r w =1 because our proposed method does not work for r w =1. According to Table 4.1, when r w is close in value to 1.1 or 1.2, this proposed method yields more accurate extracted values for both εʹr and μʹr than other cases. In addition, the results in Table 4.1 clearly show that the effect of the step discontinuity becomes more important as r w increases. This implies that we cannot neglect the step discontinuity effects if the two microstrip lines have large differences in width. Next, we consider a lossy substrate. In this case, we used the same configuration as the previous lossless case, except both dielectric and magnetic losses are included. We set both dielectric and magnetic loss tangents to Thus, ε" r and μ" r are and 0.01, r w εʹr μʹr Min. (%) Max. (%) Min. (%) Max. (%) Table 4.1. Minimum and Maximum Relative Error of the Extraction Results for the Frequency Range of 1GHz to 10GHz 77

95 respectively. Figure 4.4 shows the simulated results for the extracted ε' r and μ' r. The extracted ε' r varies from to 3.085; these values are similar to the previous lossless case. The minimum and maximum relative errors are 2.13% and 2.83%, respectively. The extracted value of μ' r varies from to This result is slightly worse than lossless case, although it still shows very good agreement with the actual value of 2. The minimum and maximum relative errors of extraction for μ' r are 2.5% and 3.7%, respectively. Thus, simulated results for both lossless and lossy cases show that this method provides very accurate values for the real part of the material properties. Once the real parts have been determined, the next step is to extract both dielectric and magnetic losses. Figure 4.5 shows the extracted values of ε" r and μ" r. The extracted value of ε" r varies from to whereas μ" r varies from to Since the nominal values of the imaginary part of the permittivity and permeability are small numbers (0.015 and 0.01, respectively), the absolute errors of ε" r and μ" r are small namely, = and =0.0095, respectively. Note that relative error is not a good measure when dealing with small numbers and therefore is not used to assess the accuracy of the imaginary parts. This newly developed method for on-wafer measurements requires test fixtures consisting of planar transmissions (microstrip), pads for the probes, coplanar waveguide transmission lines, a fixture to transition from a coplanar waveguide to a microstrip line, and various calibration fixtures. However, the generated fixtures will have fabrication errors due to imperfections in the fabrication process. In this proposed method, microstrip 78

96 lines with two different widths play a very important role, making it necessary to present an error analysis given such uncertainties. 3.2 Real part of relative permittivity and permeability Real part of relative permittivity ( r ') Real part of relative permeability ( r ') Frequency (Hz) x 10 9 Figure 4.4. Simulated results of ε' r and μ' r extraction for lossy case (ε' r =3 and μ' r =2 are the exact values) Imaginary part of relative permittivity and permeability Imaginary part of relative permittivity ( r ") Imaginary part of relative permeability ( r ") Frequency (Hz) x 10 9 Figure 4.5. Simulated results of ε" r and μ" r extraction for lossy case (ε" r =0.015 and μ" r =0.01 are the exact values) 79

97 4.5. Error Analysis Although this chapter does not present error analyses due to uncertainties in the substrate thickness and the transmission line lengths, errors due to uncertainties in the width of the transmission lines are discussed here since they have the largest impact on the accuracy of the proposed method. The error due to uncertainties in substrate thickness can be considered minor in this proposed method because the electromagnetic characteristics of the guided waves are more sensitive to the transmission line width than the substrate thickness. As a result, errors due to uncertainties in the substrate thickness can be neglected. To simulate uncertainties in the transmission line width, we generated various sets of random numbers for the transmission line widths of 500μm and 600μm. These random number sets were used to generate transmission line test sets; each set included ±1σ (σ is a standard deviation) deviations from the nominal values of 500μm and 600μm, respectively. This corresponds to a maximum deviation of ±0.5% of the nominal values. Note that each of the transmission line sets consisted of 100 samples, which provides a margin of error of less than 10% for the ±95% confidence limit. In this error analysis, we initially considered one error at a time (one random variable); we then considered all of them together. We will first consider errors due to uncertainties in the width of the 600μm microstrip line. In this initial error analysis, only the width of the 600μm microstrip line is allowed to vary. In other words, the 500μm microstrip line width and the widths of the TRL calibration kits are fixed. Figure 4.6 shows the standard error analysis for ε' r and μ' r. The 80

98 standard error, SE, is / n, where n is the size of the sample and σ is the sample standard deviation. The sample standard deviation, σ, is given by x x n, where 2 ( ) / x is the sample mean average. Figure 4.6 also includes upper and lower 95% confidence error bars, which can be determined from SE and are given by x( SE 1.96). The maximum and minimum variations of ε' r for the upper and lower 95% limits are and 0.110, respectively. In addition, the relative error of the maximum and minimum variations for the upper and lower 95% confidence limits relative to the real part permittivity of 3 are 5.565% and 1.662%, respectively. Similarly, the maximum and Re(ε r ) Re(μ r ) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Figure 4.6. Simulated error analysis results for variation in 600μm line width.. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values) 81

99 minimum 95% variations for μ' r are and 0.119, respectively. The relative error for the maximum and minimum 95% confidence limits relative to the real part permeability of 2 is 6.512% and 1.724%, respectively. Thus, based on this analysis, we can expect the extracted values of εʹr and μʹr within the upper and lower 95% limits to have the relative errors of less than 6% and 7%, respectively, despite the existence of uncertainties in the microstrip line width of 600μm with a ±0.5% error. We also considered the effect of uncertainties in the error boxes connected to transmission line 2, as shown in Figure 4.2. Our proposed method uses the TRL calibration method, which removes errors due to the errors boxes in the test structures by moving reference planes. Thus, we can expect the extraction errors for both ε' r and μ' r due to the uncertainties in the width of the error boxes to be small. Figure 4.7 shows the standard error analysis for ε' r and μ' r. According to Figure 4.7, the maximum variations for ε' r and μ' r within the 95% confidence limits are and 0.015, respectively. Thus, the maximum relative errors for the extraction of ε' r and μ' r are 2.626% and 3.057%, respectively. Compared to the previous error analysis, this analysis shows that uncertainties in the error boxes connected to DUT2 generate small errors in the extraction of ε' r and μ' r. In other words, the width of transmission line 2 plays a more important role than the width of error boxes connected to line 2. Regarding errors due to uncertainties in the line width of the 500μm microstrip line only (with other parameters held constant), Figure 4.8 shows the standard error analysis of the extracted values of ε' r and μ' r where the maximum variations for ε' r and μ' r within the 95% confidence limits are and 0.041, respectively. This corresponds to the 82

100 Re(ε r ) Re(μ r ) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Figure 4.7. Simulated error analysis results for variations in the error boxes connected to 600μm microstrip line. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values) maximum extraction errors of 2.869% and 3.498% for ε' r and μ' r, respectively. This error analysis indicates that the uncertainty in the line width of the 500μm microstrip line generates fewer extraction errors than the 600μm line. Similar to the results in Figure 4.7, this error analysis result shows relatively small standard errors. However, this latter error analysis identified a different behavior than previous results. Note that our proposed method uses propagation constants of both DUT1 and DUT2. Keeping in mind that uncertainties in width of the 500μm microstrip line produce uncertainties in the propagation constant of DUT1, the standard errors in Figure 4.8 are due to these uncertainties in the propagation constant of DUT1. Also, note that in each set of curves in 83

101 Re(ε r ) Re(μ r ) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Figure 4.8. Simulated error analysis results (for r w =1.2) for variation in 500μm line width. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values) Figure 4.8, all curves intersect at 6.8 GHz. This behavior needs further investigation to determine why the curves intersect. Next, we consider width variations of lines in TRL calibration kits only (i.e., all other line widths are fixed). As previously discussed, TRL calibration is a crucial step in our method. The TRL calibration kits (Thru, Reflect, and Line) are designed based on the error boxes of the test structures shown in Figure 4.2. Figure 4.9 shows simulated results for both ε' r and μ' r. According to these results, the maximum and minimum variation of ε' r within the 95% confidence limits are and 0.080, respectively; for μ' r, these variations are and 0.075, respectively. These variations result in errors of 4.985% 84

102 Re(ε r ) Re(μ r ) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Figure 4.9. Simulated error analysis results for variations in TRL calibration kits. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values) and 9.541% for ε' r and μ' r, respectively. Compared to the previous results, the maximum variations within the 95% confidence limits for both ε' r and μ' r are larger. These results demonstrate the importance of TRL calibration in our proposed method. Finally, we need to consider all possible variations in both DUTs and TRL calibration kits. The simulations and the standard error analysis results are shown in Figure The maximum and minimum variations of ε' r within the 95% confidence limits are and 0.141, respectively, while the maximum and minimum variations of μ' r are and 0.135, respectively. As expected, this overall error analysis yields larger variations for both ε' r and μ' r than previously discussed results. According to Figure 4.10, the maximum 85

103 Re(ε r ) Re(μ r ) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Figure Simulated error analysis results for uncertainties in both DUT s and TRL calibration kits. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε' r =3 and μ' r =2 are the exact values) relative errors for the extracted values of ε' r and μ' r within the 95% confidence limits are 6.980% and 9.488%, respectively. Another consideration is the standard error analysis of loss. Figure 4.11 shows the simulated results with standard error analysis for both ε" r and μ" r. The results shown in Figure 4.11 include width uncertainty for both DUTs and TRL calibration kits. The maximum and minimum variations within the 95% confidence limits are very small. The maximum variations for ε" r and μ" r are and , respectively. The nominal values of ε" r and μ" r used in the simulation are and 0.01, respectively. In this chapter, we do not include error analyses for ε" r and μ" r for the first four cases; however, 86

104 Im(ε r ) Avg. Upper 95% limit Lower 95% limit Frequency (GHz) Avg. Upper 95% limit Lower 95% limit Im(μ r ) Frequency (GHz) Figure Simulated error analysis results for uncertainties in both DUT s and TRL calibration kits. Imaginary parts of permittivity (top) and permeability (bottom) with standard error analysis (ε" r =0.015 and μ" r =0.01 are the exact values) the results show similar behaviors to the results shown in Figure Measurement Results The test fixture of microstip lines was fabricated on a Pyrex 7740 wafer, which has ε r of 4.6 and μ r of 1 while its thickness is 500μm [44]. Since suitable magnetic-dielectric wafers are hard to find, we used well-known dielectric wafers. We deposited gold on top of a Pyrex 7740 wafer as a test structure using a lift-off process; we also deposited gold on the back side of the wafer as a ground plane. The test fixtures, shown in Figure 4.12, 87

105 consist of microstrip lines as DUTs and coplanar waveguide-to-microstrip transitions as error boxes. This measurement is based on on-wafer measurement, meaning it is required a transition from the coplanar waveguide probe pads to the microstrip line. This vialess coplanar waveguide-to-microstrip transition is based on [48]. We discussed this transition model in Chapter 2. The extracted values of the real parts of ε r and μ r of the Pyrex 7740 wafer are shown in Figure 4.13 (a). The nominal values of the real parts of ε r and μ r of the Pyrex 7740 wafer are 4.6 and 1, respectively. According to Figure 4.13 (a), the minimum and maximum extracted values of the real part of ε r are 4.12 and 5.20, respectively, over the frequency range of 4GHz to 14GHz. Thus, the relative errors of the minimum and maximum extracted values of the real part of ε r are 10.45% and 13.07%, respectively. The extracted real part of μ r varies from 0.86 to 1.17 over the frequency range of 4GHz to 14GHz, and the relative errors of the minimum and maximum values of the extracted (a) (b) Error Box A DUT Error Box B Figure The test fixtures ofmicrostrip transmission lines for the measurements. The widths of DUT1 and DUT2 are 500μm and 600μm, respectively, and both DUT s are the line length of 5mm. 88

106 8 Re( r ) 7 Re( r ) 7 Re( r ) 6 Re( r ) 6 5 Re(ε Re( r ) and Re( r ) r ) and Re(μ r ) Re(ε Re( ) and r Re( ) r ) and Re(μ r ) r Frequency (GHz) (a) Frequency (GHz) x 10 9 (b) Figure Extracted results of the real parts of ε r and μ r of the Pyrex 7740 wafer: (a) used proposed method and (b) used conventional method (The nominal values of real parts of ε r and μ r of the Pyrex 7740 wafer are 4.6 and 1, respectively) results are 13.8% and 17.0%, respectively. The relative error of the extracted results of μ r seems higher than the extracted results of ε r because the nominal value of the real part of μ r is a small number. In addition, Figure 4.13 (b) depicts the extracted results for both real parts of ε r and μ r using conventional transmission line method with the calibration comparison method discussed in Chapter 2. The extracted results in Figure 4.13 (b) clearly show that the conventional transmission line method with calibration comparison method cannot be used for on-wafer material characterization using microstrip with coplanar waveguide-to-microstrip transitions. Regarding the dielectric and magnetic losses of the Pyrex 7740 wafer, the given value of the dielectric loss tangent is [44]. The extracted value of tanδ d is shown in Figure 4.14, since it is difficult to use (4.21) and (4.22) when μ' r is 1 and μʺr is 0. Because (4.21) and (4.22) obtain singularities when μ' r is close to 1 and μʺr is close to 0. 89

107 tan Frequency (GHz) Figure Extracted result of the imaginary parts of ε r of the Pyrex 7740 wafer (The nominal value of the dielectric loss tangent of the Pyrex 7740 wafer is 0.005) Therefore, we assumed μ' r of 1 and μʺr of 0 in the loss calculation. Figure 4.14 indicates that the dielectric loss tangent varies from to over the frequency range of 4GHz to 14GHz. The measurement results for the dielectric loss tangent are not good enough to compare the measurement results of ε' r and μ' r. This means that the loss measurements are very difficult in the material characterization measurements Summary In this chapter, we proposed a new method to measure ε r and μ r of on-wafer magnetic dielectric materials using two transmission lines of different widths. In addition, this method can be used in more general cases of on-wafer characterization. A complete mathematical derivation of this new method was presented, including simulation, error 90

108 analysis, and measurement. As this method also includes TRL calibration, the parasitic effects between the probes of a probe station and contact pads can be removed. As a result, the novel method proposed in this chapter provides accurate results for the extraction of relative permittivity and permeability. Moreover, we verified this method through computer simulations for both lossless and lossy cases; the results demonstrated very good agreement with exact values. Furthermore, we performed standard error analyses with random variables using an electromagnetic simulation tool. According to these analyses, the real parts of the relative permittivity and permeability can be extracted with a maximum error of less than 10% within the 95% confidence limits. We also built microstrip transmission line models on the Pyrex 7740 wafer and discussed the measurement results for both ε r and μ r, and the relative errors for the extracted results were approximately 10% with respect to the nominal value. In addition, we showed the extracted results of ε r and μ r using convention transmission line method with calibration comparison method in this chapter and the conventional transmission line method didn t provide correct extracted results when microstrip transmission with coplanar waveguideto-microstrip transitions were used for the on-wafer material characterization. 91

109 Chapter 5 NEW ON-WAFER CHARACTERIZATION METHOD FOR MAGNETIC-DIELECTRIC MATERIALS USING T-RESONATORS 5.1 Introduction As we discussed in Chapter 2, the T-resonator method is commonly used for the characterization of dielectric materials [33, 57]. The main advantage of the T-resonator method is that it provides very accurate results for material properties based on the measurement of resonant frequencies. For magnetic-dielectric materials, only the effective refractive index, eff eff, can be determined by measuring resonant frequencies. This is the main reason that non-resonant methods, such as the transmission-line method, are mainly used for the characterization of magnetic-dielectric material. However, with non-resonant methods, it is necessary to determine both the characteristic impedance and the effective refractive index to find the relative values of ε r and μ r for the characterization of magnetic-dielectric materials. In this chapter, we propose a new method for the characterization of magneticdielectric materials using T-resonators. The proposed method is capable of determining both the characteristic impedance ratio and the effective refractive index at the resonant frequency points. To determine the characteristic impedance ratio, we used a concept that 92

110 is similar to that described in Chapter 4. Then, it was possible to use the values obtained for the characteristic impedance ratio and the effective refractive index to determine the relative values of ε r and μ r at the resonant frequency points. Furthermore, we introduce a new way to determine the effective T-stub length accurately, which is crucial in the T- resonator measurement because an open-end effect exists and produces uncertainty in the measurement result [33]. Our proposed method allows the effective T-stub length to be determined accurately, thereby enhancing the accuracy of the measurement. We show simulated and measured results in the following sections to verify the accuracy of our proposed method. 5.2 Method of Analysis The T-resonator method is very commonly used to characterize the properties of onwafer material, but most previous studies have focused on dielectric materials (ε r and tanδ). The method that we propose in this chapter is based on the T-resonator method and can be used to characterize magnetic-dielectric, thin-film materials. In Chapter 4, we used two different transmission lines to characterize magnetic-dielectric, thin-film materials, which provided the ratio of two different characteristic impedances to determine both ε r and μ r [70]. In this study, similar to our previous study, we used two different T- resonators that had the same T-stub length and the same characteristic impedance at the T-stub but had different characteristic impedances at the feed lines. Figure 5.1 shows two different T-resonator models. Each T-resonator model can be written as a wave cascade matrix using equation (3.3), and the wave cascade matrices of the two T-resonators in 93

111 l feed l feed Z 01 Z 02 Z 01 L stub Z 01 L stub Figure 5.1. Two T-resonator models with same characteristic impedance at the T-stub, but different characteristic impedances at the feed lines. Figure 5.1 are shown in equations (5.1) and (5.2). R R T1 T 2 e e Y Z Y Z YstubZ Y Z e l feed stub 01 stub l feed stub 01 Y Z Y Z YstubZ Y Z e l feed stub 02 stub l feed stub 02 (5.1) (5.2) where γ 1 and γ 2 are the propagation constants in the feed lines of T-resonator 1 and 2, respectively. Y stub in equations (5.1) and (5.2) for the open-stub and short-stub T- resonators are given in equations (3.4) and (3.5), respectively. In equations (3.4) and (3.5), γ 1, the propagation constant in the T-stub, is equal to the propagation constant in the feed line of T-resonator 1, since the widths of the feed line and the T-stub are the same. 94

112 The wave cascade matrix of T-resonator 1 (equation (5.1)) is a regular T-resonator wave cascade matrix in equation (3.3), but the wave cascade matrix of T-resonator 2 contains both Z 01 and Z 02. Thus, it is possible to determine Z 01 /Z 02, which is the ratio of the two different characteristic impedances and the characteristic impedance ratio, r, is given by: Z R r Z (5.3) T T 2 02 R12 where R and T1 12 R in (5.3) indicate the wave cascade matrix elements of T-resonator 1 T 2 12 and 2, respectively. We already discussed the expression of r in terms of ε eff and μ eff in Chapter 4. The characteristic impedance ratio, r, also can be expressed as [70] r C C a2 eff 2 1 (5.4) a1 eff1 2 r C C a2 eff 1 2 (5.5) a1 eff 2 1 where C a is the capacitance of the transmission line when it is air-filled, therefore it only depends on the geometry [40]. Note that subscripts 1 and 2 indicate the transmission lines with the characteristic impedance of Z 01 and Z 02, respectively. The propagation constant, γ 1, in the T-stub can be determined from the effective refractive index, which can be used to determine the measured resonant frequencies. Also, the propagation constant, γ 2, in the feed line can be found easily through the TRL calibration [25]. Thus, the unknowns in equations (5.4) and (5.5) are ε eff2 /ε eff1 and μ eff1 /μ eff2, respectively. It is possible to extract the actual ε r and μ r, because ε eff and μ eff depend on ε r, μ r, as well as the geometry of the transmission line. The procedure for evaluating ε r and μ r was discussed in Chapter 4. 95

113 Now, it is important to consider the loss calculations since both dielectric and magnetic losses can be determined using this method. The main idea for the loss calculations is basically the same as it was for the loss calculations in Chapter 4. The loss calculations in Chapter 4 used the attenuation constants of two different transmission lines. In this chapter, we determined the complex propagation constants, γ 1 and γ 2, at the resonant frequency points. Therefore, we can find the attenuation constants, α 1 and α 2, at the resonant frequency points. Since the metal used in the sample was not a perfect electric conductor, the attenuation due to the conductor losses, α c, must be considered, and these losses can be determined by equation (2.9). Therefore, when α c is subtracted from α, the attenuation constants due to the dielectric and magnetic losses, α d and α m, respectively, are left. Thus, the summation of α d and α m in terms εʺr and μʺr is given in equation (4.21). Also, εʺr and μʺr can be calculated using equation (4.22) Simulated Results We simulated T-resonators with the same T-stub width and different feed-line widths. Both T-resonator 1 and 2 have a T-stub length of mm and a width of 500 μm; however, the feed-line widths are 400 μm and 500 μm, respectively. The substrate that was used in the simulations had a thickness of 100 μm, and ε r and μ r were 3 and 2, respectively. Note that we used a lossless substrate and a perfect electrical conductor in the simulations. However, we simulated all the TRL calibration kits as well, and the extraction procedures used in the simulations were exactly the same as those used in the actual measurements. The simulated results are shown in Figure 5.2, which shows that 96

114 the resonant frequencies of the two T-resonators are almost the same. Since the T-stub lengths and widths are the same, the resonant frequencies should be the same. However, the feed line widths of the two T-resonators were different, and this difference resulted in different T-junction effects. Therefore, the resonant frequencies of the two T-resonators were slightly different even though the two T-resonators had the same T-stub lengths. Now, let s consider only the first resonant frequency. Note that Figure 5.2 (b) shows the detailed S 12 of T-resonators 1 and 2 in the region around the first resonant frequency. The exact first resonant frequencies for T-resonators 1 and 2 were GHz and GHz, respectively, and the difference between the two resonant frequencies was 6 MHz, which can be considered as a small difference and its relative error is about 0.164% with respect to the first resonant frequency of T-resonator 1. The resonant frequency difference is very small at the first resonant frequency, and, even though the differences S12 (db) S12(dB) Comparison T-resonator #1 T-resonator # Frequency (Hz) x T-resonator #1 T-resonator # Frequency (Hz) x 10 9 (a) (b) Figure 5.2. Simulated results of two T-resonators which have same T-stub length and width, but different feed line widths. (a) S 21 (db) in overall frequency range and (b) S 21 (db) for region near the first resonant frequency 97

115 became larger for higher orders, they were still small enough to use our method. Now, we need to determine the characteristic impedance ratio, r, using equation (5.3). Since the first resonant frequencies of T-resonators 1 and 2 are slightly different, the average of the two resonant frequencies was used. Figure 5.3 shows the characteristic impedance ratio, r, near the first resonant frequency, which is shown by the solid red line. According to Figure 5.3, the characteristic impedance ratio, r, at the resonant frequency contains a singularity, since the T-resonators are not ideal T-resonators. Thus, the value of r at the resonant frequency shows a very sharp peak. To determine the characteristic impedance ratio, r, we must eliminate the singularity in r at the resonant point using regularization. The R 12 values of T-resonators 1 and 2 can be approximated as shown in equations (5.6) and (5.7). Characteristic Impedance Ratio, r Original value of r Regularized value of r Frequency (Hz) x 10 9 Figure 5.3. Simulated results of the characteristic impedance ratio, r (red solid line) and its value obtained by regularization (blue dot line) 98

116 a R a a xx a xx 2 T x x0 b R b b xx b xx 2 T x x0 (5.6) (5.7) Thus, using equations (5.6) and (5.7), the characteristic impedance ratio, r, can be expressed as shown in equation (5.8) R b b xx b xx b xx r R a a x x a x x a x x T T (5.8) The regularized value of the characteristic impedance ratio, r, is shown in Figure 5.3 by the dashed blue line. Equation (5.8) can be used to eliminate the singularity near the first resonant frequency. The value of r at the first resonant frequency is 1.161, and this value is very close to the theoretical value. Now, we can determine both ε r and μ r at the first resonant frequency point from the determined values, which are the value of r, the resonant frequency, the propagation constants, and the information of structure geometry. The determined ε r and μ r at the first resonant frequency point were and 1.905, respectively. The relative errors of the determined ε r and μ r at the first resonant frequency point were 5.947% and 4.755%, respectively. The ε r and μ r at the higher resonant frequency points also can be determined by the method described above, and the results are summarized in Table 5.1. The results in the Table 5.1 were obtained without considering the effective T-stub length, which is discussed in more detail in the following section. 99

117 f (GHz) ε r Relative Error Relative Error μ (%) r (%) Table 5.1. The simulated results for using two T-resonators 5.4. Consideration of the Effective T-Stub Length In this study, we used microstip line open-stub T-resonators. Unlike shorted-stub T- resonators, open-stub T-resonators contain an open-end effect, and it is difficult to determine the effective length of the T-stub exactly. The effective length of the T-stub is a function of the physical length as well as the open-end effect and T-junction discontinuity [33]. Thus, it is very important to determine the effective T-stub length accurately during the characterization of the material using a T-resonator. There are empirical studies on the open-end effect and T-junction effect for microstrip lines [59, 60]. However, in this study, we introduced an easy way to determine the effective T-stub length accurately in the T-resonator measurements. The method that is described in this section is similar to the method used in the straight-ribbon resonator method discussed in Chapter 2. In the previous section, we used two different T-resonators that had the same stub width and length but different feed-line widths. We assumed that the open-end effects of the two T-resonators were the same because they had the same T-stub lengths and widths. However, according to the simulated results shown in Figure 5.2, the two resonators had different resonant frequencies even though they had same T-stub length. This means that 100

118 the T-junction discontinuity effect also affected the T-resonator measurements of the effective length of the T-stub. The effective T-stub length is given by equation (5.9), and the results are depicted in Figure 5.4. Leff Lx Lstub Lend (5.9) where L x is the unknown length in the feed line due to the T-junction discontinuity, L stub is the physical length measured from the bottom of the feed line to the end of the T-stub, and L end is the unknown length due to the open-end effect. Let s consider two different T- resonators that have different L stub values in equation (5.9) but the same width. This implies that the lengths L x and L end for both resonators are the same. In addition, the effective T-stub length can be expressed as βl eff = nπ/2. Thus, the effective T-stub lengths of two different T-resonators can be written as nc fn 1Lx Lstub 1Lend (5.10) n eff Feed line center L x L eff L stub Open-end effect L end Figure 5.4. The effective T-stub length in the T-resonator model which includes the openend effect and the T-junction discontinuity effect 101

119 nc fn2lx Lstub2 Lend (5.11) n where f n1 and f n2 are the resonant frequencies of the two T-resonators. Also, we assumed that the effective values of the refractive indices of the two T-resonators were the same, since the two T-resonators had the same T-stub widths. Note the similarity between equations (5.10) and (5.11) and their similarity to the equations used for the modified, straight-ribbon resonator discussed in Chapter 2. After several simple algebraic steps, the unknown values, such as L x and L end, can be determined. eff L x L end f L f f L f n2 stub2 n1 stub1 n1 n2 (5.12) Note that L x and L end cannot be determined separately. Although this method does not provide each value of L x and L end, an accurate effective T-stub length of the T-resonator can be determined. In the previous section, we used two different T-resonators that had the T-stub length of 10 mm and the same T-stub width of 500 μm. However, T-resonators 1 and 2 had different feed-line widths of 500 μm and 400 μm, respectively. To apply the method that we explained in this section, we simulated two additional T-resonators for T-resonators 1 and 2. We call these additional structures as T-resonators 1ʹ and 2ʹ, and these are the same structures as T-resonators 1 and 2, except that they have different T-stub lengths. T- resonators 1ʹ and 2ʹ had T-stub lengths of mm and mm, respectively, and these T-stub lengths were measured from the bottom edge of the feed line to the end of the T-stub. By applying equation (5.12) to each of the two T-resonator sets, i.e., T- resonators 1 and 1ʹ and T-resonators 2 and 2ʹ, it is possible to determine the effective T- 102

120 stub lengths for T-resonator 1 and 2, and, at the first resonant frequency point, they were mm and mm, respectively. Using these effective T-stub lengths, it is possible to determine more accurate values of ε r and μ r, which are shown in Table 5.2. Note that the frequencies shown in Table 5.2 are used as the average value of the resonant frequencies of T-resonator 1 and 2. The ε r and μ r values in Table 5.2 have smaller relative errors than the values Table 5.1. This means that the effective T-stub length has a significant effect on T-resonator measurements. As a result, the accurate effective T-stub lengths that were determined in this study using the T-resonator method provided better accuracy in the characterization of magnetic-dielectric materials. In the following section, we verified the method proposed in this chapter by comparing its results to actual, experimental results. f (GHz) ε r Relative Error Relative Error μ (%) r (%) Table 5.2. The simulated results using the effective T-stub length 5.5. Measurement Results We had the same problem with the measurements described in the Chapter 4 that no magnetic-dielectric wafers were available. Therefore, we used Pyrex 7740 wafers for this measurement. The electrical properties of the Pyrex 7740 wafers are discussed in previous chapters. Figure 5.5 shows the microstrip T-resonator test structures. 103

121 (a) (b) Figure 5.5. Two different microstrip T-resonators for the measurements. T-resonator (a) and (b) have T-stub length of 15mm and width of 500μm while the feed line widths are 500μm and 400μm for T-resonator (a) and (b), respectively. The T-resonator structures had the same T-stub length of 15 mm, but the feed-line widths of T-resonators 1 and 2 were 500 μm and 400 μm, respectively. Note that the coplanar waveguide-to-microstrip transitions for each T-resonator were different because the feed lines for the two T-resonators were different. Therefore, it is necessary to build different sets of TRL calibration kits for the different T-resonators on the same wafer. The coplanar waveguide-to-microstrip transitions used in this measurement were discussed in Chapter 2. Figure 5.6 shows the measured S 21 comparison of the two T- resonators with the detailed S 21 comparisons at each resonant frequency points. From the measured data from two T-resonators, we can determine the real parts of ε r and μ r using the equations in the previous section. We also considered the effective T-stub length, which was discussed in the previous section. We used two T-resonators that had different 104

122 S21 (db) T-Resonator #1 T-Resonator # Frequency (Hz) x S21 (db) (a) 1 st resonant frequency T1 : GHz T2 : GHz S21 (db) (b) 3 rd resonant frequency T1 : GHz T2 : GHz S21 (db) -18 T-Resonator #1 T-Resonator # Frequency (Hz) 0 x (c) 5 th resonant frequency T1 : GHz T2 : GHz S21 (db) -18 T-Resonator #1 T-Resonator # Frequency (Hz) 0 x (d) 7 th resonant frequency T1 : GHz T2 : GHz T-Resonator #1 T-Resonator # Frequency (Hz) x T-Resonator #1 T-Resonator # Frequency (Hz) x Figure 5.6. Comparison of measured S 21 for two T-resonators. Top figure is S 21 comparison for the overall frequency range and bottom 4 figures are detailed S 21 at the resonant frequency points. 105

123 T-stub lengths. The different T-sub lengths used in this measurement were 15mm and 15.25mm. Therefore, we were able to include the effects due to both open-end and T- junction in the extraction procedure. The extracted results for ε' r and μ' r are shown in Table 5.3. Note that the frequencies in Table 5.3 are the averaged frequencies for the resonant frequencies of the two T-resonators. According to Table 5.3, the extracted results are very accurate for both ε' r and μ' r, since the relative error is smaller than 4% for all cases shown in Table 5.3. The extraction results of ε' r are slightly worse than the extraction results in Table 3.2 in Chapter 3. The measurements in this chapter use two T- resonators rather than using one T-resonator as was done in Chapter 3, so the measurement error should be larger than the measurement in Chapter 3. Compared to the extracted results for both ε' r and μ' r in Chapter 4, however, the measurement results for both ε' r and μ' r in this chapter were much better than the results in Chapter 4. Although the measured results show only at the resonant frequency points, the measured results were very accurate compared to the non-resonant method, and this is the main advantage of the T-resonator method. Now, let s consider the loss measurements. As stated above, the measured losses can be determined from the measured attenuation constants. We used non-magnetic wafers for this measurement, and the imaginary part of μ r was 0. Therefore, we had difficulty f (GHz) ε' r Relative Error Relative Error μ' (%) r (%) Table 5.3. The measured results for ε' r and μ' r using two T-resonators 106

124 for this measurement, and the imaginary part of μ r was 0. Therefore, we had difficulty determining the μ" r, because the μ r = 1-j0 creates singularities in the equations for the loss calculation. In addition, these singularities produce huge uncertainties in the loss calculations, and these uncertainties also affect the determination of ε" r. Therefore, we were able to consider only the dielectric loss in this measurement. The measurement results of the dielectric loss tangent are shown in Table 5.4. The nominal value of the dielectric loss tangent for Pyrex 7740 wafers was The extracted dielectric loss tangents in Table 5.4 are higher than the nominal value of Also, the extracted results in Table 5.4 show that the dielectric loss tangents at the third and fifth resonant frequency points are closer to the nominal value than the dielectric loss tangents at the first and seventh resonant frequency points. This pattern is similar to the extracted results of ε' r in Table 5.3. However, both ε' r and tanδ measurement results show good agreement with the nominal values over all of the resonant frequency points. f (GHz) ε" r tanδ Table 5.4. The measured results for ε" r and tanδ. (The nominal value of tanδ is 0.005) 5.6. Summary In this chapter, we discussed how to determine ε r and μ r of magnetic-dielectric material using the T-resonator on-wafer characterization method. We combined the concepts of the T-resonator method and our proposed magnetic-dielectric material 107

125 characterization method, which was discussed in Chapter 4. Similar to the method in Chapter 4, we used two different T-resonators with the same T-stubs, but different feed lines, in the T-resonators. Therefore, it was possible to determine the characteristic impedance ratio, r, at the resonant frequency points. From the measured effective refractive index of T-resonator and the characteristic impedance ratio, r, it was possible to determine both ε' r and μ' r at the resonant frequency points. In addition, we applied a new way to determine the effective T-stub length in this measurement. As a result, the measured ε' r and μ' r values using our proposed method showed very good agreement with the nominal values of Pyrex 7740 wafers. In addition, the measured results showed much better accuracy than the non-resonant method used for the magnetic-dielectric material on-wafer characterization. 108

126 Chapter 6 ON-WAFER ELECTROMAGNETIC CHARACTERIZATION METHOD FOR ANISOTROPIC MATERIALS 6.1. Introduction Recent progress in engineered materials is providing new materials that have unique electromagnetic behaviors, such as anisotropies in the permittivity ( ) and permeability ( ). The accurate measurement of the electromagnetic properties of these new materials is crucial to access whether they can be used in a variety of applications. Furthermore, on-wafer characterization of thin-film materials is important since new electronic circuits use new and complex materials in the form of thin-film materials on wafers at the present time. Thus, accurate on-wafer characterization of anisotropic material properties is very important. There are several different methods to characterize anisotropic materials, and those that are commonly used include the free space method, the waveguide method, and the transmission/reflection method [10-12]. These conventional methods, however, are not suitable for characterizing anisotropic thin-film materials because they are too thin (typically, micron range of thickness) to measure in a certain direction. In addition, it is difficult to measure small areas using the conventional measurement methods. Thus, on- 109

127 wafer measurement methods must be used to characterize these thin-film anisotropic materials. Typically, planar structures are used for the on-wafer measurements. In this chapter, we ll discuss how to characterize anisotropic thin-film materials using microstrip lines. In the following section, we discuss characterization methods for uniaxial anisotropic materials that have the same permittivity values in the in-plane direction, but different permittivity values in the normal direction [71]. In addition, we expand our proposed method to biaxial anisotropic materials that have different permittivity values in different axes [71]. Furthermore, we ll consider the more general case of biaxial anisotropic material characterizations, which include misalignments between the optical axes of the anisotropic material and the measurement axes [71, 72]. In the last section, we show measurement results for our proposed anisotropic wafer characterization method. We designed and fabricated our test structures on anisotropic sapphire wafers. Our measurement results using sapphire wafers showed good agreement with nominal values of the sapphire permittivity tensor Method of Analysis Uniaxial and Biaxial Anisotropic Materials Let s discuss how to characterize uniaxial anisotropic materials (sometimes called Type II anisotropic materials) using microstrip lines. The method that was used in this study is based on the mapping of two-dimensional anisotropic regions [73]. This mapping theory allows us to map an anisotropic region in the Z-plane into an isotropic region in the W-plane [73]. In addition, the relative permittivity tensor of the anisotropic material 110

128 can be expressed as a scalar constant of isotropy-ized permittivity, ε g. The physical height of the material in the anisotropic region, however, transforms into the effective height, H e, in the isotropic region. Thus, a microstrip line in an anisotropic region with the permittivity tensor,, and substrate thickness, H, can be transformed into a microstrip line in the isotropic region with a permittivity of ε g and a substrate thickness of H e [74]. Consider a microstrip line on an anisotropic thin-film material; Figure 6.1 shows a cross section of the microstrip in the Z-plane and the W-plane. Thus, a transformed microstrip line in the isotropic region can be managed as a well-known microstrip line on isotropic substrate analysis [74]. The permittivity tensor,, of the anisotropic substrate is given (6.1). Initially, to test our methodology, we will assume that we know the optical axes of the material so that we can build the test fixtures (planar waveguides) in the same direction as the in-plane (xy plane) optical axis. In other words, when the optical and measurement coordinate systems are the same and the matrix becomes diagonal, namely, x y z (6.1) First, we consider a uniaxial anisotropic substrate (ε x = ε y ε z ) with the thickness of H. For a transformed microstrip line in the isotropic region, the isotropy-ized permittivity, ε g, is, and the effective height, H e, is H / for the propagation along the y- x z axis [74]. The effective permittivity of an anisotropic substrate is given by [40]: x z 111

129 W W z x H u v ε g H e Z-plane (a) W-plane (b) Figure 6.1. Cross section of (a) microstrip on anisotropic substrate and (b) equivalent microstrip on isotropic substrate. eff W Ca H 1/2 g g H C e a He (6.2) where C a is the capacitance for the air-filled micrsotrip line [40]. The C a for the micrsotrip line is given in (4.18). According to equation (6.2), there are two unknowns, i.e., ε g and H e, if we know the effective permittivity and structure geometry. Thus, we need two equations to determine the two unknowns. Let us consider two microstrip transmission lines with different line widths. It is possible to have two different effective permittivity values from the measurements of the two microstrip transmission lines, and each effective permittivity also has two unknowns. As a result, there are two equations and two unknowns. Thus, it is possible to determine ε g and H e from the two effective permittivity equations. Finally, ε x and ε z can be found easily from the definitions of ε g and H e. Now, we can consider a biaxial anisotropic material (sometimes called Type III anisotropic material) that has ε x ε y ε z. In this case, we also assume that the optical axes of the material are known and they are the same as the measurement axes. We will 112

130 consider two different propagation directions along the in-plane (x-y plane) optical axes. One is the propagation along the x-axis, and the other is the propagation along the y-axis. Each propagation direction can be considered as a microstrip line on a uniaxial anisotropic substrate problem, and we need two different microstrip lines for each propagation direction. Figure 6.2 shows microstrip lines on a biaxial anisotropic material along x-axis and y-axis. The effective dielectric constants for the microstrip lines with the x-axis and y-axis propagations are given by: eff, x eff, y 1/2 gx, 1 gx, 1 H C 1 12 ex, a H 2 2 W Ca H 1/2 gy, 1 gy, 1 H C ey, a H W Ca H e, x e, y (6.3) (6.4) Equations (6.3) and (6.4) are the same as the effective permittivity of the uniaxial anisotropic material. Therefore, for the propagation along the x-axis, we can consider ε g,x of and H e,x of H /, and it is possible to determine ε y and ε z. Similarly, we y z y z y y x Optical axes of biaxial anisotropic material x Optical axes of biaxial anisotropic material Figure 6.2. Schematic diagrams of the microstrip lines on a biaxial anisotropic material with different propagation directions: Microstrip lines along the x-axis (left) and y-axis (right) 113

131 can consider ε g,y of and H e,y of H / for the propagation along the y-axis, and x z x it is possible to determine ε x and ε z. We also tested our proposed characterization methods for both uniaxial and biaxial anisotropic materials using a full-wave electromagnetic solver. In the simulation, substrates with thicknesses of 100 μm were used for both uniaxial and biaxial anisotropic simulations. For the uniaxial anisotropic simulation, the permittivity elements of the substrate were ε x = ε y = 3 and ε z = 9. Also, for the biaxial anisotropic simulation, the permittivity elements are ε x = 3, ε y = 6, and ε z = 9. Note that we considered the lossless case in the simulation. The microstrip lines used in the simulations have lengths of 10 mm and widths of 300 μm and 500 μm. Figure 6.3 shows the simulated results for the characterization of both uniaxial and biaxial anisotropic materials using microstrip lines. The simulated results the characterization of uniaxial anisotropic material show that the maximum relative errors for ε x and ε z with respect to the nominal values were approximately 2% over the frequency range of 1 to 10 GHz. Similar to the simulation of z ε x and ε z Frequency (GHz) (a) ε x, ε y, and ε z ε x ε z Frequency (GHz) (b) ε x ε y ε z Figure 6.3. The simulated results for the anisotropic material characterizations: (a) uniaxial and (b) biaxial anisotropic substrates 114

132 the characterization of biaxial anisotropic materials, the maximum relative errors for ε x, ε y, and ε z with respect to the nominal values were approximately 4% over the frequency range of 1 to 10 GHz. The simulation results for the characterization of both uniaxial and biaxial anisotropic materials showed very good agreement with the actual values. We will extend our proposed method in this section to the more general case of biaxial anisotropic materials in which the optical axes are not known a priori Method of Analysis General Biaxial Anisotropic Materials In the previous part, we discussed the special case of microstrip lines on anisotropic thin-film materials for which the optical axes were known and therefore the measurement axes can be chosen to coincide with these axes. In general, however, the optical axes of anisotropic materials are unknown a priori. In this case, the measurement axes are not aligned with the optical axes of the anisotropic thin-film material. This results in misalignment angles between those two coordinate systems, and the permittivity tensor is no longer diagonal [71, 72]. The measurement is performed in the xyz system, but the permittivity tensor is in the x y z system. Figure 6.4 shows the angle differences between the xyz and the x y z systems. Let us assume that θ is the rotation angle along the z-axis and that ϕ is the rotation angle of the x-axis. Then, the rotation transformation matrix U is given by [71, 72]: U cos sin cos sincos sinsin sin cos 0 0 cos sin sin coscos cossin sin cos 0 sin cos (6.5) 115

133 z z' y θ x' x y' Figure 6.4. The principal axes of the permittivity tensor (x y z system) and the measurement coordinate system (xyz system) The permittivity tensor in (6.1), which can be transformed with the transformation matrix U and the transformed permittivity tensor,, (see Appendix C), is given by: xx xy xz T ' UU yx yy yz zx zy zz (6.6) where cos sin cos sin sin xx x y z sincos sincoscos sincossin 2 2 xy x y z sinsincos sinsincos xz y z yx xy sin cos cos cos sin yy x y z cossincos cossincos yz y z zx zy xz yz sin cos 2 2 zz y z (6.7) 116

134 If the misalignment angles, which are θ and ϕ, are not zero, then the permittivity tensor has non-zero, off-diagonal elements. Thus, it is necessary to determine either all the elements in the permittivity tensor or diagonal elements with misalignment angles. Let us consider a microstrip line on a biaxial, anisotropic, thin-film material where the measurement axes do not match the optical axes of the anisotropic material. In other words, misalignment angles exist between the measurement axes and the optical axes. Figure 6.5 shows top and cross sectional views of the microstrip line with misalignment angles of θ and ϕ. Similar to the previous analysis, we can consider two different propagation directions, i.e., along the x-axis and the y-axis. The isotropy-ized permittivity, ε g, and the effective height, H e, for different propagation directions can be determined from the measured effective permittivity and can be expressed with the permittivity tensor elements. y' y z' z θ x' x x' x H (a) (b) Figure 6.5. (a) Top-view of microstrip transmission line with misalignment angle θ between in-plane optical axis and propagation direction, (b) cross sectional view of microstrip line with misalignment angle ϕ between the principal axis and x-y plane (x, y, and z are the geometrical axes of microstrip lines; and x, y, and z are the optical axes of anisotropic thin-film substrate 117

135 Equations (6.8) and (6.9) provide ε g and H e for x-axis propagation and y-axis propagation, respectively. (6.8) 2 g, x yy zz 2 yy zz yz yz and He, x H 2 zz (6.9) 2 gy, xx zz 2 xx zz xz xz and Hey, H 2 zz From equations (6.8) and (6.9), only ε zz can be found. However, since both ε g and H e have off-diagonal elements, i.e., ε yz and ε xz, it is impossible to solve the permittivity tensor elements, ε xx, ε yy, ε xz, and ε yz. Thus, we need more equations to solve the permittivity tensor elements. Let us consider a microstrip line that has a known angle of α from the x- axis. The permittivity tensor will be transformed by rotation of the microstrip lines, and the transformed permittivity tensor,, is given by: xx xy xz a T UU yx yy yz zx zy zz (6.10) where cos sin 0 U sin cos (6.11) Therefore, the matrix element in (6.10) can be expressed in terms of the matrix elements in (6.6) and the known angle of α from the x-axis. Equation (6.12) are the matrix elements in (6.10) and each of the permittivity tensor elements in also can be expressed in terms of ε x, ε y, ε z, θ, ϕ, and α. 118

136 2 2 xx xx cos yy sin 2 xy sin cos 2 2 sin cos sin cos xy xx yy xy cos sin xz xz yz yx xy 2 2 yy xx sin yy cos 2 xy sin cos sin cos yz xz yz zx zy zz xz yz zz (6.12) Let us consider microstrip lines with different propagation directions, one for the direction of α and another for the direction of α The isotropy-ized permittivity, ε g, and the effective height, H e, for these two different propagation directions can be determined. Equations (6.13) and (6.14) are ε g and H e for the propagation along the α direction and the propagation along the α + 90 direction, respectively. (6.13) 2 yy zz yz g, yy zz yz and He, H 2 zz (6.14) 2 xx zz xz, 90 zz xz g xx and H H e, 90 2 zz We could find the isotropy-ized permittivity, ε g, and the effective height, H e, for several different directions; however, it is impossible to determine the permittivity tensor elements from equations (6.8), (6.9), (6.13), and (6.14) directly. Thus, several steps of mathematical derivations are required to solve the unknowns. In addition, finding diagonal elements in, such as ε x, ε y, and ε z, and the misalignment angles, such as θ and

137 ϕ, is better than finding matrix elements in. First, we can simplify the relationships of the different ε g values, and equations (6.15), (6.16), and (6.17) show the simplified relationships. Using a rotation angle α of 45 in this analysis, we obtain (6.15) 2 2 gx, gy, x zz y z 2 2 gy gx x zz y z,, cos 2 (6.16) 2 2 g, g, 90 sin 2 xzz yz (6.17) From equations (6.16) and (6.17), it is possible to determine the in-plane misalignment angle, θ tan 1 g, g, gy, gx, (6.18) In addition, ε x can be determined using equations (6.15), (6.16), and (6.18), and it is given by: gy, gx, 2 2 x g, x g, y 2 zz cos (6.19) Again, ε zz has already been determined using equation (6.8). The other unknowns in equations (6.15), (6.16), and (6.17) are ε y and ε z. It is impossible to determine ε y and ε z using equations (6.15) to (6.19), but we can determine ε y ε z, which is given by: (6.20) 2 2 y z g, x g, y x zz So far, we have determined ε x, ε zz, θ, and ε y ε z. It is possible to determine ε y and ε z if we know (ε y +ε z ), which is shown in equation (6.21). 120

138 cos (6.21) 2 2 gx, y z y z 2 zz x sin Thus, we can find ε y and ε z from equations (6.20) and (6.21). The last unknown is the misalignment angle of ϕ and it can be easily determined from ε zz in equation (6.7). cos z y 1 zz y (6.22) Finally, we can determine all the unknowns, i.e., ε x, ε y, ε z, θ, and ϕ. It is also possible to express these unknowns in terms of ε xx, ε yy, ε zz, ε xy, ε xz, and ε yz by using the values determined above. This method for the measurement of anisotropic thin-film materials is verified and discussed in the following section Simulation and Measurement Results The methodology for characterizing anisotropic, thin-film materials using microstrip lines was described in the previous section. In this section, the on-wafer characterization measurements of anisotropic thin-film material are discussed. We chose sapphire wafers to verify our proposed characterization method. Sapphire wafers are a good example of anisotropic material, and they have the rhombohedral crystal structure of Al 2 O 3. Several schemes for the measurements of the dielectric constants of the sapphire have been proposed [75-77]. However, those methods were focused on the bulk sapphire materials [75, 76]. Although a study of sapphire substrate characterization using microstrip line has been proposed, this study only determined the effective dielectric constant of the sapphire substrate [77]. 121

139 The given dielectric constants of sapphire are 11.6 for the parallel to the c-axis and 9.4 for the perpendicular to the c-axis and Figure 6.6 shows a conventional unit cell of a single sapphire crystal with the orientation of C-plane and R-plane [78, 79]. According to Figure 6.6, the permittivity tensor of the C-plane sapphire wafer is: C (6.23) The permittivity tensor of the C-plane sapphire wafer has the same form as the uniaxial anisotropic permittivity tensor. The permittivity tensor of an R-plane sapphire wafer can be calculated by the rotation of C. The angle between the c-axis and the normal to the R-plane is equal to 57.6, as shown in Figure 6.6 [79]. Thus, the permittivity tensor of an R-plane sapphire wafer can be calculated easily. Equation (6.24) C-plane c a b 57.6 z y y θ x x Figure 6.6. Orientation of C-plane and R-plane in the conventional unit cell of a single crystal sapphire (a, b, and c are the optical axes of sapphire crystal) 122

140 gives the permittivity tensor of an R-plane sapphire wafer. R (6.24) Although, we know the permittivity tensor of an R-plane sapphire theoretically, it is impossible to build test structures on the wafer that are perfectly aligned with the optical axes and the measurement axes. Thus, an in-plane misalignment angle exists between the optical axes and the measurement axes. As a result, the permittivity tensor of an R-plan sapphire wafer will be a full matrix with non-zero off-diagonal elements. However, all the values can be determined with our anisotropic characterization method. Before we discuss the sapphire wafer measurements, we will show the results of the R-plan sapphire wafer simulation. In the simulation, we assigned the in-plane misalignment angle, θ, to be 25. Therefore, the permittivity tensor of the R-plane sapphire can be considered as a full matrix with non-zero, off-diagonal elements and equation (6.24) can be expressed as: R (6.25) In the simulation, we used microstrip lines with the same geometries as in the previous simulation; however, we needed microstrip lines with different propagation directions. Figure 6.7 shows the simulated results for the characterization of the R-plane sapphire wafer with an in-plane misalignment angle of θ = 25. The maximum relative errors for ε xx, ε yy, and ε zz are 8.917%, 6.994%, and 2.131%, respectively. Since the non-zero, off- 123

141 ε xx, ε yy, and ε zz ε xx ε yy ε zz Frequency (GHz) (a) ε xy, ε xz, and ε yz (b) ε xy ε xz ε yz Frequency (GHz) Figure 6.7. The simulated results of the R-plane sapphire wafer characterizations: (a) diagonal elements and (b) off-diagonal elements diagonal elements are small numbers, using the absolute error rather than the relative error would be better for data analysis. The maximum absolute errors for ε xy, ε xz, and ε yz are 0.081, 0.396, and According to Figure 6.7, the off-diagonal element values increase as frequency increases, and this trend results in large absolute errors in the highfrequency region Now, let s discuss our sapphire wafer measurements. First of all, we designed several microstrip lines with different propagation directions. Our layout design and the fabricated sapphire wafer sample are shown in Figure 6.8. For the on-wafer measurement, we need to use a probe station and probes that have Ground-Signal-Ground (GSG) tips. Thus, we need a transition from a coplanar waveguide to the microstrip. In Figure 6.8, all the test structures of the microstrip lines include coplanar waveguide-to-microstrip transitions at each port [48]. In addition, it is also very important to remove any parasitic effects between the probes and contact pads to achieve accurate on-wafer measurements. Therefore, we used the TRL calibration technique in our measurement, and the TRL 124

142 (a) (b) Figure 6.8. (a) Layout design for the sapphire wafer measurement and (b) the fabricated sapphire wafer sample calibration kits are included in our layout design, as shown in Figure 6.8. The test structures were fabricated on a 500μm, C-plane sapphire wafer and a 330μm, R-plane sapphire wafer. We deposited Au on top of the sample wafers as test structures. Also, both test sample wafers had Au ground planes at the back of the wafers. Our first measurement was conducted for the C-plane sapphire wafer. We measured the sapphire wafer over the frequency range of 3 to 16 GHz, and Figure 6.9 shows the measured results for ε x and ε z. In this case, we used our proposed method for uniaxial anisotropic materials, so we assumed that ε x and ε y were the same. The measured results showed that the extraction results for both ε x and ε z had the maximum relative error of around 15% with respect to the given values. Another measurement that we conducted was the R-plane sapphire measurement. In 125