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

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

Jun Seok Lee

2011

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 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.

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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.

Dedication

This document is dedicated to my family.

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!

Vita

August, 2004 ...... B.S. Electrical Eng., Kyungpook National University, Daegu, South Korea

June 2004 to June 2005 ...... Assistant Engineer, Samsung Electronics, Tangjung, South Korea

December, 2006 ...... 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

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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 ...... 11

2.1. Introduction ...... 11 2.2. Review of Conventional On-Wafer Measurement Methods for Dielectric Materials ...... 13 2.2.1. Overview of Non-Resonant Method ...... 15 2.2.1.1. Transmission Line Method - Theory ...... 15 2.2.1.2. Transmission Line Method - Experiments ...... 20 2.2.2. Overview of Resonant Method ...... 26 2.2.2.1. T-Resonator Method - Theory ...... 29 2.2.2.2. T-Resonator Method - Experiments...... 34 2.3. Review of Conventional On-Wafer Measurement Methods for Magnetic- Dielectric Materials ...... 38 2.3.1. Transmission Line Method (Theory) ...... 39

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Chapter 3. An Improved T-Resonator Method for the Dielectric Material On-Wafer Characterization ...... 45

3.1. Introduction ...... 45 3.2. Method of Analysis ...... 46 3.2.1. T-Resonator Matrix Model ...... 47 3.2.2. Consideration of Loss Measurements ...... 51 3.3. T-Resonator Measurement Results ...... 53 3.4. Summary ...... 65

Chapter 4. Novel Electromagnetic On-Wafer Characterization Method for Magnetic- Dielectric Materials ...... 66

4.1. Introduction ...... 66 4.2. Method of Analysis - System Matrix Model ...... 67 4.3. Method of Analysis - Transmission Line Models ...... 69 4.4. Simulated Results with Sensitivity Test ...... 74 4.5. Error Analysis ...... 80 4.6. Measurement Results ...... 87 4.7. Summary ...... 90

Chapter 5. New On-Wafer Characterization Method for Magnetic-Dielectric Materials Using T-Resonators ...... 92

5.1. Introduction ...... 92 5.2. Method of Analysis ...... 93 5.3. Simulated Results...... 96 5.4. Consideration of the Effective T-Stub Length ...... 100 5.5. Measurement Results ...... 103 5.6. Summary ...... 107

Chapter 6. On-Wafer Electromagnetic Characterization Method for Anisotropic Materials ...... 109

6.1. Introduction ...... 109 6.2. Method of Analysis – Uniaxial and Biaxial Anisotropic Materials ...... 110 ix

6.3. Method of Analysis – General Biaxial Anisotropic Materials ...... 115 6.4. Simulation and Measurement Results ...... 121 6.5. Summary ...... 127

Chapter 7. Conclusion ...... 129

7.1. Summary and Conclusion ...... 129 7.2. Future Works ...... 133

Appendix A. Crystal System (Bravais Lattice)...... 135

Appendix B. Conformal Mapping of a Microstrip Line with Duality Relation ...... 137

Appendix C. The Permittivity Tensor in the Measurement Coordinate System ...... 141

References ...... 143

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 ...... 100

Table 5.2. The simulated results using the effective T-stub length ...... 103

Table 5.3. The measured results for ε'r and μ'r using two T-resonators ...... 106

Table 5.4. The measured results for ε"r and tanδ. (The nominal value of tanδ is 0.005) ...... 107

Table A.1. Classification of tensor forms by crystal system ...... 135

List of Figures

Figure 1.1. Simple illustrations for (a) permittivity and (b) permeability measurements including their equivalent circuit models ...... 3

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 εr using transmission line method (ε'r of Pyrex 7740 is 4.6): (a) coplanar waveguide transmission line and (b) microstrip transmission line ...... 23

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Figure 2.10. De-embedded S11 of the Thru standard. From the de-embedded S11 result of the Thru standard, calibration is valid from3.7GHz to 14.5GHz ...... 24

Figure 2.11. 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 2.12. Three different types of microstrip resonators: (a) ring resonator, (b) T- resonator, and (c) straight-ribbon resonators ...... 27

Figure 2.13. T-resonator models: (a) Microstrip T-resonator and (b) Coplanar waveguide T-resonator with air-bridge ...... 30

Figure 2.14. T-resonators on the Pyrex 7740 wafers: (a) coplanar waveguide T-resonators and (b) microstrip T-resonators ...... 35

Figure 2.15. S21(dB) measurement results for T-resonators: (a) coplanar waveguide T- resonators and (b) microstrip T-resonators ...... 36

Figure 2.16. 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) S11 and (b) S21 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 R11 and (b) phase angle of R11 for short-stub coplanar waveguide T-resonator. The green dashed lines in the plots indicate the S11 resonant points ...... 56

Figure 3.5. Measured open-stub microstrip T-resonator (a) S11 and (b) S21 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 R11 and (b) phase angle of R11 for open-stub microstrip T-resonator. The green dashed lines in the plots indicate the S11 resonant points ...... 60 xiii

Figure 3.7. Error analysis with ±95 confidence limits of εr 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. [Ra], [Rb], [RD1], and [RD2] 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 (le) and DUT (L) are 500μm and 5mm, respectively ...... 75

Figure 4.3. Simulated results of εr and μr extraction for lossless case (εr=3 and μr=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 rw=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 4.10. 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

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Figure 4.11. 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 4.12. 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 4.13. Extracted results of the real parts of εr and μr of the Pyrex 7740 wafer: (a) used proposed method and (b) used conventional method (The nominal values of real parts of εr and μr of the Pyrex 7740 wafer are 4.6 and 1, respectively) ...... 89

Figure 4.14. Extracted result of the imaginary parts of εr 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) S21 (dB) in overall frequency range and (b) S21 (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 open- end effect and the T-junction discontinuity effect ...... 101

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 ...... 104

Figure 5.6. Comparison of measured |S21| for two T-resonators. Top figure is S21 comparison for the overall frequency range and bottom 4 figures are detailed S21 at the resonant frequency points ...... 105

Figure 6.1. Cross section of (a) microstrip on anisotropic substrate and (b) equivalent microstrip on isotropic substrate ...... 112

Figure 6.3. The simulated results for the anisotropic material characterizations: (a) uniaxial and (b) biaxial anisotropic substrates ...... 114

Figure 6.4. The principal axes of the permittivity tensor (x´y´z´ system) and the measurement coordinate system (xyz system) ...... 116

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

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

Figure 6.7. The simulated results of the R-plane sapphire wafer characterizations: (a) diagonal elements (b) off-diagonal elements ...... 124

Figure 6.8. (a) Layout design for the sapphire wafer measurement and (b) the fabricated sapphire wafer sample...... 125

Figure 6.9. C-plane sapphire measurement results for εx and εz. The nominal values of εx and εz are 9.4 and 11.6, respectively, up to 1GHz ...... 126

Figure 6.10. R-plane sapphire measurement results for diagonalized matrix elements of εx, εy, and εz. The nominal values of εx, εy, and εz are 9.4, 9.4, and 11.6, respectively, up to 1GHz ...... 127

Figure B.1. Conformal mapping of a microstrip in z-plane into a two parallel plates in w- plane ...... 138

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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 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:

DE H (1.1)

BH P (1.2) where, E, H, D, and B are the electric field, magnetic field, and electric and magnetic flux densities, respectively. In addition, ε = ε0εr and μ = μ0μr are complex permittivity

-12 -7 and permeability, respectively; ε0 (8.854×10 ) and μ0 (4π×10 ) are the free space permittivity and permeability, respectively; and εr = ε'r - jε"r and μr = μ'r - jμ"r are the relative complex permittivity and permeability, respectively. The real and imaginary parts of εr and μr are related to the energy storage terms and the loss terms, respectively. The real and imaginary parts of εr can be described as the capacitance (C) and conductance (G) in the capacitor, respectively, while the real and imaginary parts of μr 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 εr are tC/ε0A and tG/ωε0A, respectively, where t is the thickness of the sample being tested and ω is the angular frequency. In addition, the real and imaginary

Electrode (Area = A)

LR CG

(a) (b)

Figure 1.1. Simple illustrations for (a) permittivity and (b) permeability measurements including their equivalent circuit models

2 2 parts of μr are lLeff/μ0N AC and l(Reff - Rw)/μ0ωN AC, respectively, where l, Leff, N, AC, Reff, and Rw 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).

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 Rectangular Rectangular dielectric Sample dielectric Free space waveguide waveguide

n1 n2 n1 εr1 εr2 Sa mple termina ted by metal plate d1 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 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 ( H and P ); 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 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 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 on-

wafer 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

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.

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 magnetic-

dielectric 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.

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.

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

P1 P2 S21

S11 S22

S12

Coaxial to coplanar transition To network To network analyzer Probes analyzer

ε μ r r Coplanar cell Conductive strips

Figure 2.1. Typical configuration of the on-wafer measurement using probe station 11

(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 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 non- resonant and resonant methods. Furthermore, we will discuss conventional characterization methods for both isotropic magnetic-dielectric and anisotropic dielectric materials.

2.2. 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 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 (εr) 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

εr=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 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.

2.2.1. 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, on- wafer 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.

2.2.1.1. 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]:

JD jjk E H (2.1) 0 eff 15

(a) (b)

Figure 2.4. Electric field distribution of (a) microstrip and (b) coplanar waveguide structures

where εeff is the effective dielectric constant of either microstrip transmission line or coplanar waveguide transmission line, expressed as εeff = 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].

HHcc111 HcMS rr (2.2) eff 22 1 12(hW / )

Hcc1 Kk Kk HcCPW 1 r 5 (2.3) eff 2 Kkc Kk 5 where K(k) is the complete elliptic integral of the first kind. The moduli k, kˊ, k5, and kˊ5 are given by [41]:

cb22 a k 22 (2.4) bc a

ac22 b c 2 kk 1 22 (2.5) bc a

sinh SSSch / 2 sinh22 bh / 2 sinh ah / 2 k (2.6) 5 sinhSSSbh / 2 sinh22 ch / 2 sinh ah / 2

sinh SSSah / 2 sinh22 ch / 2 sinh bh / 2 kkc 12 (2.7) 55sinhSSSbh / 2 sinh22 ch / 2 sinh ah / 2

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, εr, 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

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:

DD cd D (2.8) where αc and αd 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, αc, is given by [41]:

RR12 Dc (2.9) 2Z0 where R1 and R2 are the normalized series-distributed resistances for the signal metal line and ground plane, respectively. Equations for R1 and R2 of both the microstrip line and coplanar waveguide line are given by [40, 41]:

MS RLRS u §·11§· 4SW R1 ¨¸2 ln ¨¸ (2.10) WT©¹SS ©¹

R Wh/ RMS S u (2.11) 2 WWh/ 5.8 0.03 hW /

ªº CPW RkS §·8Sa §·1 0 Rk10 «»S ln¨¸ ln ¨¸ (2.12) 22 Tk1 81akKk 00 ¬¼©¹ ©¹ 0

ªº CPW kR00S §·81Sb §·1 k R2 «»S ln¨¸ ln ¨¸ (2.13) 22 Tk1 k 81akKk 00 ¬¼©¹00©¹

1/2 where RS=(ωμ/2σ) is the surface resistivity of the conductor, K(k0) is the complete elliptic integrals of the first kind, and k0 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 d 0.5 ° h LR ® 2 (2.14) ° WW§· W 0.94 0.132 0.0062¨¸ for 0.5 d 10 ¯° hh©¹ h

The dielectric loss tangent can be determined from the attenuation constant, αd, namely, [40, 41]:

2D tanG d (2.15) qk0 Heffc

-1 -1 where q=(1-(εˊeff) )/(1-(εrˊ) ) 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 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., εr and tanδ) can be determined from the measured complex propagation constant using the transmission line method.

2.2.1.2. 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 0.005 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 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)

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 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 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 εr 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

6 6

5.8 5.8

5.6 5.6

5.4 5.4

5.2 5.2 r r

H 5 H 5

4.8 4.8

4.6 4.6

4.4 4.4

4.2 4.2

4 4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 4 5 6 7 8 9 10 11 12 13 14 Frequency (Hz) 10 Frequency (Hz) 9 x 10 x 10

(a) (b)

Figure 2.9. Extraction results of εr using transmission line method (ε'r of Pyrex 7740 is 4.6): (a) coplanar waveguide transmission line and (b) microstrip transmission line

23 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.

-10 Valid Region

-20

-30 S11 (dB) S11

-40

-50

-60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 x 10

Figure 2.10. De-embedded S11 of the Thru standard. From the de-embedded S11 result of the Thru standard, calibration is valid from3.7GHz to 14.5GHz. 24

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 0.034 to 0.011 over the frequency range of 5GHz to 20GHz while the extracted loss tangent from the microstrip line measurement vary from 0.012 to 0.004 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

0.05 0.015 0.014 0.045 0.013 0.04 0.012 0.011 0.035 0.01 0.03 0.009 0.008 G 0.025 G

tan tan 0.007 0.02 0.006 0.005 0.015 0.004 0.01 0.003 0.002 0.005 0.001 0 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 4 5 6 7 8 9 10 11 12 13 14 Frequency (Hz) 10 Frequency (Hz) 9 x 10 x 10 (a) (b)

Figure 2.11. 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 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.

2.2.2. 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.

r Gapl1 Gap

Lstub Gap l2 Gap

W (a) (b) (c)

Figure 2.12. 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]:

2 §·nc Heff ¨¸ for n 1,2,3, (2.16) ©¹2Srfn

th where r is the mean radius of the ring, fn is the n 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 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]:

2 ªºcnf 12nn nf 21 Heff «» (2.17) ¬¼«»2 ffnn122 l l 1 where the subscript 1 and 2 refer the straight-ribbon resonator 1 and 2, respectively. In

th addition, l is the ribbon length, fn is the n 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 open- end 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.

2.2.2.1. 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 non- resonant 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 short- stub 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.

(a) (b)

Figure 2.13. 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]:

nc L (2.18) stub 4 f H neff

th where Lstub is the T-stub length, fn is the n 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:

2 §·nc Heff ¨¸ (2.19) 4Lf ©¹stub n

Once εeff 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 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, Lstub, in (2.19) needs to be considered as an effective T-stub length, Leff, 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

LLeff stub'' l end l junction (2.20) where Lstub is the physical length of the T-stub measured from the center of the feed to the end of the T-stub. In addition, Δlend and Δljunction in (2.20) are the correction factors for the open-end effect and T-junction effect, respectively. The correction factor Δlend for the microstrip line can be taken into account as follows [59]:

[[[123 'lhend u (2.21) [4 where

0.81 0.8544 Heff 0.26 Wh/ 0.236 [1 0.434907 u0.81 u0.8544 (2.22) Heff 0.189 Wh/0.87

0.5274tan1 ªº 0.084Wh / 1.9413/[5 ¬¼ [2 1 0.9236 (2.23) Heff

7.5 Wh / [3 1 0.218e (2.24)

1.456 [ 1 0.0377 tan1 ªº 0.067Wh / uªº 6 5 e0.036 1Hr (2.25) 4 ¬¼ ¬¼

Wh/ 0.371 [5 1 (2.26) 2.358Hr 1 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 result s for determining the correction factor due to the open-end effect for the range of normalized widths 0.01 ≤

W/h ≤ 100 and εr ≤ 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]:

2 'l ªº§·f junction 0.5«» 0.05 0.7e1.6 0.25 (2.27) ¨¸ Wf«»©¹p1 ¬¼ where fp1[GHz] = 0.4×Z0/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, QL, is given by:

f QL (2.28) BW3dB

The loaded quality factor, QL, 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, Q0, which is given by [61]:

QL Q0 (2.29) 1210u LA /10 where LA is the insertion loss at the resonant frequency. In addition, the unloaded quality factor, Q0, can be written as:

1111 (2.30) QQQQ0 dcr where Qd, Qc, and Qr 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, Qc, can be calculated; (2.31) shows the equation for Qc [54].

20 S Qc (2.31) ln10 DOcg where λg is the guided wavelength in the microstrip line and αc is the attenuation constant due to the conductor losses given in (2.9). The quality factor due to the radiation losses,

Qr, in (2.30) is given by [54]:

nZ0 Qr 2 (2.32) 480SO hF / Heff where F(εeff) 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 junction radiations are given by [62, 63]:

2 Heff 1 HHeff11 eff Fopen log (2.33) Heff 21HHeff eff H eff

2 §· 31HHeff eff 1HH HH eff 21 eff 1 F logeff log ¨¸ eff (2.34) T 8214HH3/2 HHH121 ¨¸ H effeff eff©¹ eff eff eff

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, Qd; the loss tangent of the dielectric material can then be determined using the following relationship [54].

HH 1 tanG eff r (2.35) QdrHH eff1

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.

2.2.2.2. 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 2.2.1.2. For the metal deposition, coplanar waveguide T-resonators used aluminum while microstrip T-resonators used gold for both the top test fixtures and 34 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 wire- bondings 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 2.2.1.2. As in the previous experiments described in section 2.2.1.2, 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 2.14.

Figure 2.15 shows examples of T-resonator measurements for both microstrip and

(a) (b)

Figure 2.14. T-resonators on the Pyrex 7740 wafers: (a) coplanar waveguide T-resonators and (b) microstrip T-resonators

35 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 εr extraction are accurate compared to the nominal values in [44].

According to the results, the minimum and maximum relative error of εr extraction results

0 0

-2

-5 -4

-6 -10

-8 S21 (dB) S21 (dB) -15 -10

-12 -20

-14 Before TRL Before TRL After TRL After TRL -16 -25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 Frequency (Hz) 10 x 10 x 10

(a) (b)

Figure 2.15. S21(dB) measurement results for T-resonators: (a) coplanar waveguide T- resonators and (b) microstrip T-resonators 36 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 εr. According to the extraction results of εr, 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 εr 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

εr tanδ f (GHz) Value Error (%) Value CPW 8.904 4.688 1.913 0.027 T-resonator 18.023 4.553 1.022 0.014 2.714 4.784 4.0 0.026 Microstrip 8.192 4.70 2.174 0.011 T-resonator 13.574 4.734 2.913 0.007 18.926 4.74 3.043 0.008 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 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 2.2.1.2.

2.3. 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 on- wafer characterization of magnetic-dielectric materials, because it is necessary to determine both the propagation constant and the characteristic impedance simultaneously.

Thus, an in-depth review of the non-resonant method for the magnetic-dielectric materials is presented in this section.

2.3.1. Transmission Line Method (Theory)

The transmission mission line method is the well-known non-resonant method for on- wafer 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:

JHP jk0 eff eff (2.36)

Peff ZZ00 c (2.37) Heff

39 where Z'0 is the characteristic impedance when εr = μr = 1. Note that the effective permittivity (εeff) and permeability (μeff) are complex numbers, and they are given by:

HHeff effcccj H eff (2.38)

PPeff effcccj P eff (2.39)

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 εeff and μeff can be calculated easily by dividing and multiplying of γ and Z0. Since the substrate has both magnetic and dielectric losses, the relative permittivity (εr = ε'r - jε"r) and permeability (μr

= 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:

1 ªº 1/PPcc 1 1/ 1 1 PcMS «»rr (2.40) eff 22 ¬¼«»1 12(hW / )

1 CPW ªº 1/Prcc 1 Kk Kk 5 Peffc «»1 (2.41) 2 Kkc Kk ¬¼«» 5 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

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., qd,loss and qm,loss, which are referred to as “filling factors.” The filling factors for the dielectric and magnetic losses, i.e., qd,loss and qm,loss, respectively, are given by [43]:

1 1Heffc qdloss, 1 (2.42) 1H rc

1 Peffc qmloss, (2.43) 1 Prc

Now, consider the effective dielectric and magnetic loss tangents, i.e., tanδd,eff and tanδm,eff, respectively. It is shown in [19, 43] that these effective loss tangents can be expressed in terms of the filling factor introduced above:

ccc1 Heffcc 1HHeff r tanGG q tan (2.44) deff,,c dloss d 1 Heff 1HHrrcc

ccc Peffcc 1 PPeff r tanGGmeff,, q mloss tan m (2.45) PPPeffccc 1 r r where tanδd = εʺr/εʹr and tanδm = μʺr/μʹr. Since the complex numbers εeff and μeff 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:

§·Hc 1 HHcc ¨¸r cc (2.46) reff¨¸c ©¹Heff 1

PPrrcc 1 PPreffcc cc (2.47) PPeffcc 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

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]:

ªºXX 1 §*ªº111YZ ª · º 11 12 * 1 r (2.48) «»2 ¨¸«» « » ¬¼XX21 22 1* ©¹¬¼* 1112 ¬ ¼ where Xij 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]:

ªºªRR11 121 SSSSS 12 21 11 22 11 º «»« » (2.49) ¬¼¬RR21 22S21 S 22 1 ¼ where Rij and Sij 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]:

2 ZZ XX * 0 r 12 21 (2.50) ZZ 2 0 r 4 XX12 21

43 where Z0 is the characteristic impedance of the error box, and Zr is the reference impedance of the probe tip, typically 50 Ω. Since X12 and X21 already have been determined from the TRL calibration and Zr also has been determined from the SOLT (or

LRM) calibration, it is possible to determine Z0 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].

Zr:Z0

Probe tip Interface between probe- Pad capacitances Impedance transformer tip and contact pad

Figure 2.16. Probe tip/contact pad model and its equivalent circuit model

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

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.

3.2. 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 role in material characterization using the T-resonator. In this chapter, we will discuss a new way to use the T-resonator method.

3.2.1. 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:

ªºeJ l 0 R TLine «»J l (3.1) ¬¼0 e

ªºYZ YZ 100 «»22 R «» (3.2) Y YZ YZ «»001 ¬¼«»22 where γ is the propagation constant of the transmission line and Z0 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.

ªº2J l feed §·YZ00 YZ «»e ¨¸1 ©¹22 >@R «»(3.3) Tres «» YZ002J l feed §· YZ «»e ¨¸1 ¬¼22©¹

lfeed

lfeed lfeed

Z0 Y Z0 Lstub

Feed line T-stub with Feed line

at port 1 length Lstub at port 2

(a) (b)

where Y is the input admittance of the stub, given by:

eeJJLLstub stub Yopen for open-stub (3.4) ZeJJLLstub e stub 0

eeJJLLstub stub Yshort for short-stub (3.5) ZeJJLLstub e stub 0

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:

ªºªSS11 121 RRRRR 12 11 22 12 21 º «»« » (3.4) ¬¼¬SS21 22R22 1 R 21 ¼

Thus, the S-matrix of the T-resonator is given by:

ªºYZ e2J l feed 2e2J l feed «»0 «»22YZ00 YZ >@STres (3.5) «»2e2J l feed YZ e2J l feed «»0 «»22YZ YZ ¬¼00

Let’s consider the resonances in the T-resonator for both open-stub and short-stub cases. Resonances in T-resonators occur when |S21| goes the minimum; it is possible to express |S21| using (3.5) for both open-stub and short-stub T-resonators. Thus, |S21| of the open-stub and short-stub T-resonators for the lossless case is given by:

2cos E Lstub S21 for open-stub (3.6) 4cos22EELL sin stub stub

2sin E Lstub S21 for short-stub (3.7) 4sin22EELL cos stub stub

Equations (3.6) and (3.7) clearly demonstrate that the |S21| minimum occurs when

|cos(βLstub)| and |sin(βLstub)| are zero for the open-stub and short-stub T-resonators, respectively. In other words, the |S21| minimum occurs when βLstub 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:

nc n 1,3,5 for opopen-stube fr where ® (3.8) 4L H n 2,4,6 for short-stubsho stub eff ¯

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 S11 are considered in the conventional T-resonator method. Based on (3.5), it is possible to determine |S11| of the

T-resonator model for both open-stub and short-stub cases.

2sin E Lstub S11 for open-stub (3.9) 4cos22EELL sin stub stub

2cos E Lstub S11 for short-stub (3.10) 4sin22EELL cos stub stub

According to (3.6), (3.7), (3.9), and (3.10), resonances in S11 and S21 for both open- stub and short-stub T-resonators only depend on the T-stub length, not the length of feed lines. However, we noticed that YZ0/2 in (3.3) goes to zero at the S11 resonances for

jl2E feed lossless cases. As a result, R11 in (3.3) is e at the S11 resonant frequency. Thus, β at the S11 resonant frequency is given by:

j E ln R (3.11) 2l 11 feed

The effective permittivity can be found from the determined β at the S11 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 S11 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 S21 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 proposed method, on the other hand, uses the resonant effect in S11 (which makes YZ0/2 in R11 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.

3.2.2. 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 R11. The R11 of open-stub T-resonator in (3.3) for the lossy material is given by:

ªº 2J l feed sinh J Lstub Re11 «»1 (3.12) 2cosh J L ¬¼ stub

As previously stated, we are interested in R11 at S11 resonant frequency points, and R11 will be e jl2J feed for low loss materials. Thus, it is possible to determine the attenuation constant at S11 resonant frequency points. The attenuation constant and the phase constant at the S11 resonance points are given by:

1 D ln R (3.13) 2l 11 feed

j E R (3.14) 2l 11 feed

The measured attenuation constant, α, can be broken down into different components, with the total attenuation constant given by:

DD D D (3.15) cdr where αc is the attenuation constant due to the conductor losses, αd is the attenuation constant due to the dielectric losses, and αr 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 αc and αd.

We discussed how to calculate αc, αd, and αr 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,

αc, and the attenuation constant due to the radiation losses, αr, from the equations in

Chapter 2. Thus, it is possible to determine the attenuation constant due to dielectric losses, αd, 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.

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 εr of 4.6 and tanδ of

0.005 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

S11 and S21. 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 S21 are not changed by performing TRL calibration. In other words, the resonant frequencies in S21 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

S11 are changed by TRL calibration. However, the resonant frequencies in S11 after TRL

(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 |S11| in our matrix model in (3.10).

First, we consider the conventional coplanar waveguide T-resonator analysis and determine εr of the Pyrex wafer from the measured S21 resonant frequencies. The measured resonant frequencies in S21 are 8.904GHz and 18.023GHz. Thus, the extracted

εr are 4.688 and 4.553 at the first and second resonant frequencies in S21, respectively. All the parasitic effects are considered for the extraction of εr 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 R11 and the phase angle of R11, which are related to α and β, respectively.

Figure 3.4 shows both the magnitude and phase angle of the measured R11; both behave

0 0

-2

-5 -4

-6 -10

-8 S21 (dB) S11 (dB) S11 -15 -10

-12 -20 -14 Before TRL Before TRL After TRL After TRL -25 -16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 Frequency (Hz) 10 x 10 x 10

(a) (b)

well at S11 resonant frequency points, which are 4.411GHz and 13.722GHz. The extracted εr can be found using (3.14). Thus, the extracted εr are 4.633 and 4.590 at the first and second resonances in S11, respectively. Table 3.1 shows the extracted εr 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 εr from both methods have very small relative error with respect to the nominal value, which is εr 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

1.3

1.2

1.1

Mag(R11) 0.9

0.8

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 x 10 (a)

150

100

-50 Ang(R11) (in degree)

-100

-150

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 x 10 (b)

Figure 3.4. Measured (a) magnitude of R11 and (b) phase angle of R11 for short-stub coplanar waveguide T-resonator. The green dashed lines in the plots indicate the S11 resonant points 56

f (GHz) εr Relative Error (%)

Proposed 4.411 4.633 0.717 Method 13.722 4.590 0.217

Conventional 8.904 4.688 1.913 Method 18.023 4.553 1.022 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 S11 make the R11 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 εr 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 0.027 and 0.014 at the frequencies of

8.906GHz and 18.023GHz, respectively. These values are much higher than the nominal value of the Pyrex 7740 wafer, which is tanδ of 0.005. On the other hand, the determined tanδ using our proposed method are 0.0030 and 0.0013 at the frequencies of 4.411GHz and 13.722GHz, respectively. These determined tanδ are also different from the nominal

57 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 S11 and S21 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 S21 of the microstrip T-resonator are 2.714GHz, 8.192GHz, 13.574GHz, and 18.926GHz. In addition, the measured resonant frequencies in S11 of the microstrip

T-resonator are 5.488GHz, 10.874GHz, and 16.086GHz. Unlike the previous short-stub coplanar waveguide T-resonator, the microstrip T-resonator in this measurement has an

0 0

-5 -5

-10

-10 -15

S11 (dB) S11 -20 (dB) S21 -15

-25

-20 -30 Before TRL Before TRL After TRL After TRL -35 -25 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 Frequency (Hz) 10 x 10 x 10

(a) (b)

Figure 3.5. Measured open-stub microstrip T-resonator (a) S11 and (b) S21 in dB. Blue and red lines indicate the measured S-parameters with and without performing TRL calibrations, respectively.

58 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 εr are 4.784, 4.700, 4.734, and 4.74 for each of the resonant frequency points in

S21. 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 εr 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 R11 data to (3.13) and (3.14) to extract the material properties. First of all, we need to determine the resonant frequencies in S11. According to Figure 3.5, the resonant frequencies in S11 are 5.488GHz, 10.74GHz, and 16.086GHz. Then, we need to apply the measured R11 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 R11 for the microstrip T-resonator; both demonstrate good behavior at the S11 resonant frequency points, which are marked on

Figure 3.6 with green dashed lines. The extracted εr using our proposed method are 4.596,

4.579, and 4.630 for each of the resonant frequencies in S11. The relative errors of the extracted value of εr 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

5.5

4.5

3.5

Mag(R11) 2.5

1.5

0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 x 10 (a)

200

150

100

-50 Ang(R11) (in degree)

-100

-150

-200 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 x 10 (b)

Figure 3.6. Measured (a) magnitude of R11 and (b) phase angle of R11 for open-stub microstrip T-resonator. The green dashed lines in the plots indicate the S11 resonant points 60 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

f (GHz) ε'r Relative Error (%) 5.488 4.596 0.094 Proposed 10.874 4.579 0.457 Method 16.086 4.630 0.657 2.714 4.784 4.0

Conventional 8.192 4.70 2.174 Method 13.574 4.734 2.913 18.926 4.74 3.043 Table 3.2. The measurement results comparison for microstrip T-resonator 61 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 0.008 at each of the resonant frequency points in S21. For the proposed T- resonator method, the measured dielectric loss tangents are 0.019, 0.012, and 0.004 at each of the resonant frequency points in S11. The determined dielectric loss tangents for both methods have large relative errors with respect to the nominal value of 0.005 compared to the relative errors in the determination of εr. 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 εr from the measurements of T-resonators on the different wafers The standard error, SE, is V / n , where n is the size of the sample and σ is the sample standard deviation. The sample standard deviation, σ, is given by

¦()/xx 2 n, where x is the sample mean average. Figure 3.7 shows the standard error analysis for the extraction of εr 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 xSEru( 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

(a)

(b)

Figure 3.7. Error analysis with ±95 confidence limits of εr 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 εr 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 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 εr 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 εr extraction for the conventional and proposed T- resonator methods.

Avg. f (GHz) εr Relative Error (%) 5.521 4.535 – 4.584 0.338 – 1.405 Proposed 11.080 4.623 – 4.692 0.504 – 2.001 Method 16.301 4.617 – 4.675 0.366 – 1.632 2.751 4.751 – 4.784 3.277 – 4.001

Conventional 8.284 4.698 – 4.713 2.127 – 2.448 Method 13.722 4.731 – 4.752 2.845 – 3.308 19.161 4.726 – 4.746 2.740 – 3.175 Table 3.3. The error analyses comparison for microstrip T-resonator measurements 64

3.4. Summary

In this chapter, we discussed a new and improved on-wafer characterization for thin- film 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.

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 εr and μr from the measured characteristic impedance and propagation constant of transmission lines printed on this class of materials. We already discussed how to determine both εr and μr 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 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 magnetic- dielectric 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, εr and μr 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.

4.2. 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, Z0 [27].

Thus, the de-embedded scattering parameters of the DUT are relative to Z0. 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, Z01, while DUT2 has a different characteristic impedance, Z02. 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 Z01 and Z02. 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 [Rm1] and [Rm2], equations (4.1) and (4.2) show the system matrices of test sets (a) and (b), respectively.

RRRR (4.1) >@>@>@>@maDb11

>@>@>@>@>@>@>@>@>@RRRRRRRRR (4.2) m22 a D b a mis 12'2 D mis b

Error box A DUT1 Error box B

(a) Z01 Z01 Z01 Z01 Z01 Z01 Z01 [Ra] [RD1] [Rb]

Error box A DUT2 Error box B

(b) Z01 Z01 Z02 Z02 Z02 Z01 Z01 [Ra] [RD2] [Rb]

Reference planes

Figure 4.1. Block diagram of two sets of DUT’s with same error boxes. [Ra], [Rb], [RD1], and [RD2] are the wave cascade matrices of Error box A, Error box B, DUT1, and DUT2, respectively 68 where [Ra], [Rb], [RD1], and [RD2] are the wave cascade matrices of Error box A, Error box B, DUT1, and DUT2, respectively. Note that [RD2'] in (4.2) is the wave cascade matrix without the impedance mismatch. In addition, [Rmis1] and [Rmis2] are the wave cascade matrices representing the impedance mismatch between DUT2 and the error boxes A and B. [Rmis1] and [Rmis2] can be expressed in terms of Z01 and Z02—namely:

1 ªºZZZZ01 02 02 01 >@Rmis1 «»(4.3) 2 ZZ ZZZZ 01 02 ¬¼02 01 01 02

1 ªºZZZZ01 02 01 02 >@Rmis2 «»(4.4) 2 ZZ ZZZZ 01 02 ¬¼01 02 01 02

It is impossible to determine Z01 and Z02 directly from (4.3) and (4.4) without first knowing either Z01 or Z02. 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.

4.3. 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 Z01, can be written as:

ªºeJ1L 0 >@RD1 «» (4.5) 0 eJ1L ¬¼

However, the wave cascade matrix of DUT2 (the second transmission line with the characteristic impedance of Z02) includes the impedance mismatch matrices. Its wave cascade matrix can be found from (4.2) to (4.4).

22 ª JJ22LL 1 eZZ 01 02 eZZ 01 02 >@RD2 « 4ZZ «eZZeZZJJ22LL22 22 01 02 ¬ 02 01 01 02 (4.6) JJ22LL22 22º eZZeZZ 01 02 02 01 » 22 eZZJJ22LL eZZ» 01 02 01 02 ¼

The propagation constant in (4.5), γ1, which is the propagation constant of DUT1, can be found easily through TRL calibration [25]. The propagation constant in (4.6), γ2, 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.

DD22 1 1 §·RR11 22 J 2 cosh ¨¸ (4.7) L 2 ©¹

D2 where Rij is a matrix element in [RD2].

Thus, two unknowns, Z01 and Z02, are left in (4.6); however, Z01 and Z02 cannot be determined directly. Therefore, we must consider the characteristic impedance ratio, r =

Z01/Z02, plugging it into (4.6). The following wave cascade matrix for DUT2 is obtained in terms of r:

JJLL22JJ22LL22 1 ªer22 11 er er 11 e rº R « » (4.8) >@D2 JJLL22 22 4r «erer22 11 erJJ22LL11 er » ¬ ¼

From (4.8), the characteristic impedance ratio of r can be found after several steps of

70 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:

Z RRRRDDDD2222 r 01 21 12 22 11 (4.9) ZL2sinh J 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]:

RjL Z Z0 (4.10) GjC Z 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]:

Z ªº22 RhdSedS «»PHeffcc t effcc z (4.11) i 2 ³³S 0 ¬¼S

Z ªº22 GedShdS «»HPeffcc t effcc z (4.12) v 2 ³³S 0 ¬¼S

1 ªº22 CedShdS «»HPeffcc t eff z (4.13) v 2 ³³S 0 ¬¼S

1 ªº22 LhdSedS «»PHeffcc t eff z (4.14) i 2 ³³S 0 ¬¼S where v0 and i0 are the normalization constants for the waveguide voltage and waveguide

±γz ±γz current, which are v(z) = v0e and i(z) = i0e , respectively. The effective permittivity and permeability are given by εeff = εʹeff - jεʺeff and μeff = μʹeff - jμʺeff, respectively.

Equations (4.11) through (4.14) do not include metal conductivity as an explicit term in

εeff, but it is absorbed in εʺeff [27]. 71

From (4.10) to (4.14), it is easy to find the characteristic impedance in terms of L, C, ε, and μ—namely:

1 HPc2 P ZL eff eff eff (4.15) 0 C HPHc2 eff eff eff

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, εeff, and μeff. Note that C and L are defined as C =

Caε'eff and L = μ'eff / Ca, where Ca is the capacitance of the transmission line when it is air- filled; therefore, it only depends on geometry [40]. In addition, the propagation constant

1/2 and the index of refraction are related by neff = (εeffμeff) = jγ/k0. After several simple algebraic steps, the characteristic impedance ratio, r can be expressed as:

C H J r a2 eff 2 1 (4.16) C HJ aeff112

C P J r a2 eff 1 2 (4.17) C PJ aeff121 where r, γ1, and γ2 in (4.16) and (4.17) were found using (4.9), (4.5), and (4.8), respectively. In addition, the air-filled capacitances Ca1 and Ca2 can be found if we know the geometry of the transmission line. The air-filled capacitance per unit length, Ca, of the microstrip line is shown in (4.18) [40]: 72

2SH CWh o for / 1 a §·8hW ln ¨¸ ©¹Wh4 (4.18) ªºWW§· CWhao H «»1.393 0.667ln¨¸ 1.444 for / ! 1 ¬¼hh©¹ where W is the microstrip line width and h is the substrate thickness. Thus, the only unknowns in (4.16) and (4.17) are εeff2/εeff1 and μeff1/μeff2. Finally, from εeff2/εeff1 and

μeff1/μeff2, it is possible to extract the actual εr and μr, because εeff and μeff depend on εr, μr, 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 εr and μr 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:

DD dmc D D (4.19) where αc, αd, and αm, are the attenuation constants due to the conductor losses, dielectric losses, and magnetic losses, respectively.

We already described αc in (2.9). In addition, the summation of αd and αm is given by [43]:

k0 HP''eff eff DDd m tan G d,, eff tan G m eff (4.20) 2 where tanδd,eff and tanδm,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:

ªº§·1 kn 1 Heffc 1,2 §·1 Pc DD 01,2eff«»¨¸ Hcc eff 1,2 P cc AB H cc P cc (4.21) dm1,2 «» r¨¸2 r1,2 r 1,2 21¨¸HPPrrrccc©¹ ¬¼©¹

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:

BBDD AA DD HPcc 21 12 and cc 21 12 (4.22) rrAB AB AB AB 12 21 21 12

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.

4.4. 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 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 εr=3 and μr=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 3.064 to 3.109 over the frequency range of 1GHz to 10GHz. These results demonstrate very good agreement with the actual value of 3. The minimum and the

le L le

Transmission Z W1 Z Z Line 1 01 01 01

Transmission Z W2 Z Z Line 2 01 02 01

Error Error DUT Box1 Box2

3.2

2.8

2.6

2.4

2.2 Relative permittivity and permeability permittivity and Relative 2 Relative permittivity Relative permeability 1.8 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) 9 x 10

Figure 4.3. Simulated results of εr and μr extraction for lossless case (εr=3 and μr=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 1.981 to

2.058 over the frequency range of 1GHz to 10GHz. The minimum error of the extracted

μr 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 εr and μr 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 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 rw = 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 rw=1 because our proposed method does not work for rw=1. According to Table 4.1, when rw 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 rw 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 0.005. Thus, ε"r and μ"r are 0.015 and 0.01,

εʹ μʹ r r r w Min. (%) Max. (%) Min. (%) Max. (%) 1.05 9.716 17.114 8.738 15.384 1.1 3.184 3.904 1.308 1.349 1.2 2.133 3.633 0.950 2.900 1.3 0.635 3.463 0.738 5.119 1.4 0.175 3.744 1.128 5.920 1.5 1.945 3.845 1.267 9.262 1.6 3.069 4.567 2.251 11.151 1.7 4.373 5.236 3.145 13.469 1.8 5.478 6.199 4.405 15.487 1.9 6.626 7.153 5.620 17.645 2 7.823 8.276 7.012 20.023 Table 4.1. Minimum and Maximum Relative Error of the Extraction Results for the Frequency Range of 1GHz to 10GHz

77 respectively. Figure 4.4 shows the simulated results for the extracted ε'r and μ'r. The extracted ε'r varies from 3.064 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 2.054 to 2.074. 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 0.0115 to 0.0195 whereas μ"r varies from 0.0092 to 0.0157. 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, |0.015-0.0115|=0.0035 and |0.01-0.0195|=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 lines with two different widths play a very important role, making it necessary to present an error analysis given such uncertainties.

3.2

2.8

2.6

2.4

2.2

2 Real part of relative permittivity (H ')

Real part of relative permittivity and permeability permittivity and relative of part Real r Real part of relative permeability ( ') Pr 1.8 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) 9 x 10

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)

0.025

0.02

0.015

0.01

Imaginary part of relative permittivity ( ") Hr

Imaginary part of relative permittivity and permeability permittivity and relative of part Imaginary Imaginary part of relative permeability ( ") Pr 0.005 1 2 3 4 5 6 7 8 9 10 Frequency (Hz) 9 x 10

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)

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 standard error, SE, isV / n , where n is the size of the sample and σ is the sample standard deviation. The sample standard deviation, σ, is given by ¦()/xx 2 n, where 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 xSEru( 1.96) . The maximum and minimum variations of ε'r for the upper and lower 95% limits are 0.113 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

3.2 Avg. Upper 95% limit Lower 95% limit 3.18 3.16 3.14

) 3.12 r ε 3.1

Re( 3.08 3.06 3.04 3.02 3 12345678910 Frequency (GHz) 2.18 2.16 Avg. Upper 95% limit Lower 95% limit 2.14 2.12 2.1 )

r 2.08 μ 2.06

Re( 2.04 2.02 2 1.98 1.96 1.94 12345678910 Frequency (GHz)

81 minimum 95% variations for μ'r are 0.120 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 0.016 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 0.042 and 0.041, respectively. This corresponds to the

3.09 Avg. Upper 95% limit Lower 95% limit 3.085 3.08 3.075 ) r ε 3.07 Re( 3.065 3.06 3.055 3.05 12345678910 Frequency (GHz) 2.08 Avg. Upper 95% limit Lower 95% limit 2.07 2.06 2.05

) 2.04 r μ 2.03

Re( 2.02 2.01 2 1.99 1.98 12345678910 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

3.09 Avg. Upper 95% limit Lower 95% limit 3.085 3.08 3.075

) 3.07 r ε 3.065

Re( 3.06 3.055 3.05 3.045 3.04 12345678910 Frequency (GHz) 2.08 Avg. Upper 95% limit Lower 95% limit 2.07 2.06 2.05 ) r 2.04 μ 2.03 Re( 2.02 2.01 2 1.99 12345678910 Frequency (GHz)

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 0.150 and 0.080, respectively; for μ'r, these variations are 0.151 and 0.075, respectively. These variations result in errors of 4.985% 84

3.18 Avg. Upper 95% limit Lower 95% limit 3.16 3.14 3.12

) 3.1 r ε 3.08

Re( 3.06 3.04 3.02 3 2.98 12345678910 Frequency (GHz) 2.25 Avg. Upper 95% limit Lower 95% limit 2.22 2.19 2.16 2.13 ) r

μ 2.1 2.07 Re( 2.04 2.01 1.98 1.95 1.92 12345678910 Frequency (GHz)

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 4.10. The maximum and minimum variations of ε'r within the 95% confidence limits are 0.183 and

0.141, respectively, while the maximum and minimum variations of μ'r are 0.179 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

3.3 Avg. Upper 95% limit Lower 95% limit 3.25 3.2 )

r 3.15 ε

Re( 3.1 3.05 3 2.95 12345678910 Frequency (GHz) 2.25 Avg. Upper 95% limit Lower 95% limit 2.2 2.15 2.1 ) r μ 2.05 Re( 2 1.95 1.9 1.85 12345678910 Frequency (GHz)

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 0.0016 and 0.0015, respectively. The nominal values of ε"r and μ"r used in the simulation are 0.015 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

0.02

0.019

) 0.018 r ε

Im( 0.017

0.016 Avg. Upper 95% limit Lower 95% limit 0.015 12345678910 Frequency (GHz) 0.013 Avg. Upper 95% limit Lower 95% limit

0.012

) 0.011 r μ

Im( 0.01

0.009

0.008 12345678910 Frequency (GHz)

the results show similar behaviors to the results shown in Figure 4.11.

4.6. Measurement Results

The test fixture of microstip lines was fabricated on a Pyrex 7740 wafer, which has εr of 4.6 and μr 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 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 εr and μr of the Pyrex 7740 wafer are shown in Figure 4.13 (a). The nominal values of the real parts of εr and μr 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 εr 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 εr are 10.45% and 13.07%, respectively.

The extracted real part of μr 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 4.12. 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

8 7 Re( ) Re( ) Hr Hr Re(P ) Re(P ) 7 r 6 r

6 5 ) ) r r )

) 5 P r r P μ μ 4 4 ) Re( and ) and Re( ) r r ) and Re( ) and H ) and Re( ) and H 3 r r ε ε 3 Re( Re( Re( Re( 2 2

1 1

0 0 4 5 6 7 8 9 10 11 12 13 14 4 5 6 7 8 9 10 11 12 13 14 Frequency (GHz) 9 Frequency (GHz) Frequency (GHz) x 10 (a) (b)

results are 13.8% and 17.0%, respectively. The relative error of the extracted results of μr seems higher than the extracted results of εr because the nominal value of the real part of

μr is a small number. In addition, Figure 4.13 (b) depicts the extracted results for both real parts of εr and μr 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 0.005 [44]. The extracted value of tanδd 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.

0.018

0.016

0.014

0.012

G 0.01 tan

0.008

0.006

0.004

0.002 4 5 6 7 8 9 10 11 12 13 14 Frequency (GHz)

Figure 4.14. Extracted result of the imaginary parts of εr 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 0.003 to 0.013 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.

4.7. Summary

In this chapter, we proposed a new method to measure εr and μr 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 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 εr and μr, 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 εr and μr 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 waveguide- to-microstrip transitions were used for the on-wafer material characterization.

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, HPeff 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 εr and μr for the characterization of magnetic-dielectric materials.

In this chapter, we propose a new method for the characterization of magnetic- dielectric 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 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 εr and μr 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 on- wafer material, but most previous studies have focused on dielectric materials (εr 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 εr and μr [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

l lfeed feed

Z01 Z02

Z01 Lstub Z01 Lstub

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).

ªº2J1l feed §·YZstub01 YZ stub 01 «»e ¨¸1 ©¹22 >@R «»(5.1) T1 «» YZstub012J1l feed §· YZ stub 01 «»e ¨¸1 ¬¼22©¹

ªº2J 2l feed §·YZstub02 YZ stub 02 «»e ¨¸1 ©¹22 >@R «»(5.2) T 2 «» YZstub022J 2l feed §· YZ stub 02 «»e ¨¸1 ¬¼22©¹ where γ1 and γ2 are the propagation constants in the feed lines of T-resonator 1 and 2, respectively. Ystub 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),

γ1, 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

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 Z01 and Z02. Thus, it is possible to determine Z01/Z02, which is the ratio of the two different characteristic impedances and the characteristic impedance ratio, r, is given by:

Z RT1 r 01 12 (5.3) ZRT 2 02 12

T1 T 2 where R12 and R12 in (5.3) indicate the wave cascade matrix elements of T-resonator 1 and 2, respectively. We already discussed the expression of r in terms of εeff and μeff in

Chapter 4. The characteristic impedance ratio, r, also can be expressed as [70]

C H J r a2 eff 2 1 (5.4) C HJ aeff112

C P J r a2 eff 1 2 (5.5) C PJ aeff121 where Ca 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 Z01 and Z02, respectively. The propagation constant,

γ1, 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, γ2, in the feed line can be found easily through the TRL calibration [25]. Thus, the unknowns in equations (5.4) and (5.5) are εeff2/εeff1 and μeff1/μeff2, respectively. It is possible to extract the actual εr and μr, because εeff and μeff depend on εr, μr, as well as the geometry of the transmission line. The procedure for evaluating εr and μr was discussed in Chapter 4. 95

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, γ1 and γ2, at the resonant frequency points. Therefore, we can find the attenuation constants, α1 and α2, 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, αc, must be considered, and these losses can be determined by equation (2.9). Therefore, when αc is subtracted from α, the attenuation constants due to the dielectric and magnetic losses, αd and αm, respectively, are left. Thus, the summation of αd and αm in terms εʺr and μʺr is given in equation (4.21). Also, εʺr and μʺr can be calculated using equation (4.22).

5.3. 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 10.15 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 εr and μr 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 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 S12 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 3.656 GHz and 3.650

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

0 -20

-10 -30

-20 -40

-30 -50 S12 (dB)

-40 S12(dB) Comparison -60

-50 -70

T-resonator #1 T-resonator #1 T-resonator #2 T-resonator #2 -60 -80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 3.6 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.7 Frequency (Hz) 10 Frequency (Hz) 9 x 10 x 10

(a) (b)

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 R12 values of T-resonators 1 and 2 can be approximated as shown in equations (5.6) and (5.7).

1.8

1.6 r

1.4

1.2

0.8

0.6

Characteristic Impedance Ratio, 0.4

0.2 Original value of r Regularized value of r 0 3.6 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.7 Frequency (Hz) 9 x 10

Figure 5.3. Simulated results of the characteristic impedance ratio, r (red solid line) and its value obtained by regularization (blue dot line)

a 2 RaaxxaxxT1 |0 (5.6) 12xx 1 2 0 3 0 0

b 2 RbbxxbxxT 2 |0 (5.7) 12xx 1 2 0 3 0 0

Thus, using equations (5.6) and (5.7), the characteristic impedance ratio, r, can be expressed as shown in equation (5.8).

23 RT 2 b bxx bxx bxx r 12 01 0 2 0 3 0 (5.8) RT1 a axx axx23 axx 12 01 0 2 0 3 0

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 εr and μr 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 εr and μr at the first resonant frequency point were 3.178 and 1.905, respectively. The relative errors of the determined εr and μr at the first resonant frequency point were 5.947% and 4.755%, respectively. The εr and μr 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.

Relative Error Relative Error f (GHz) ε μ r (%) r (%) 3.653 3.178 5.947 1.905 4.755 10.903 3.257 8.563 1.869 6.575 18.046 3.401 13.363 1.794 10.325 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 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.

LLLL (5.9) eff x stub end where Lx is the unknown length in the feed line due to the T-junction discontinuity, Lstub is the physical length measured from the bottom of the feed line to the end of the T-stub, and Lend is the unknown length due to the open-end effect. Let’s consider two different T- resonators that have different Lstub values in equation (5.9) but the same width. This implies that the lengths Lx and Lend for both resonators are the same. In addition, the effective T-stub length can be expressed as βLeff = nπ/2. Thus, the effective T-stub lengths of two different T-resonators can be written as

nc fLnx11 L stubend L (5.10) neff

Feed line center

Leff Lstub

Open-end effect L end Figure 5.4. The effective T-stub length in the T-resonator model which includes the open- end effect and the T-junction discontinuity effect 101

nc fLLnx22 stubend L (5.11) neff where fn1 and fn2 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 Lx and Lend, can be determined.

fL fL LL nstubnstub22 11 (5.12) xend ff nn12

Note that Lx and Lend cannot be determined separately. Although this method does not provide each value of Lx and Lend, 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 10.25 mm and 10.20 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 stub lengths for T-resonator 1 and 2, and, at the first resonant frequency point, they were

10.156 mm and 10.139 mm, respectively. Using these effective T-stub lengths, it is possible to determine more accurate values of εr and μr, 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 εr and μr 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.

Relative Error Relative Error f (GHz) ε μ r (%) r (%) 3.653 3.113 3.760 2.027 1.355 10.903 3.093 3.087 2.090 4.505 18.046 2.986 0.467 2.238 11.880 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

(a) (b)

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 S21 comparison of the two T- resonators with the detailed S21 comparisons at each resonant frequency points. From the measured data from two T-resonators, we can determine the real parts of εr and μr 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

-2

-4

-6

-8

-10 S21 (dB) -12

-14

-16

-18 T-Resonator #1 T-Resonator #2 -20 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Frequency (Hz) 10 x 10

0 0 (a) 1st resonant frequency (b) 3rd resonant frequency -2 -2

-4 T1 : 2.768 GHz -4 T1 : 8.324 GHz -6 T2 : 2.768 GHz -6 T2 : 8.318 GHz -8 -8

-10 -10 S21 (dB) S21 (dB) -12 -12

-14 -14

-16 -16

-18 T-Resonator #1 -18 T-Resonator #1 T-Resonator #2 T-Resonator #2 -20 -20 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 7.8 7.9 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Frequency (Hz) 9 Frequency (Hz) 9 x 10 x 10 0 0 th th -2 (c) 5 resonant frequency -2 (d) 7 resonant frequency

-4 T1 : 13.786 GHz -4 T1 : 19.278 GHz -6 T2 : 13.778 GHz -6 T2 : 19.218 GHz -8 -8

-10 -10 S21 (dB) S21 (dB) S21 -12 -12

-14 -14

-16 -16

-18 T-Resonator #1 -18 T-Resonator #1 T-Resonator #2 T-Resonator #2 -20 -20 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.88 1.89 1.9 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 Frequency (Hz) 10 Frequency (Hz) 10 x 10 x 10

Figure 5.6. Comparison of measured |S21| for two T-resonators. Top figure is S21 comparison for the overall frequency range and bottom 4 figures are detailed S21 at the resonant frequency points. 105

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 μr was 0. Therefore, we had difficulty

Relative Error Relative Error f (GHz) ε' μ' r (%) r (%) 2.768 4.656 1.213 1.009 0.930 8.321 4.630 0.661 0.996 0.440 13.782 4.609 0.200 0.976 2.440 19.248 4.657 1.237 0.962 3.810

Table 5.3. The measured results for ε'r and μ'r using two T-resonators 106 for this measurement, and the imaginary part of μr was 0. Therefore, we had difficulty determining the μ"r, because the μr = 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 0.005. The extracted dielectric loss tangents in Table 5.4 are higher than the nominal value of 0.005. 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) 2.768 8.321 13.782 19.248

ε"r 0.0853 0.0353 0.0272 0.0428 tanδ 0.0183 0.0076 0.0059 0.0092

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 εr and μr 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 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

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 (H ) and permeability

(P ). 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 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.

6.2. 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 can be expressed as a scalar constant of “isotropy-ized” permittivity, εg. The physical height of the material in the anisotropic region, however, transforms into the effective height, He, in the isotropic region. Thus, a microstrip line in an anisotropic region with the permittivity tensor, H , and substrate thickness, H, can be transformed into a microstrip line in the isotropic region with a permittivity of εg and a substrate thickness of

He [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, H , 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 (x- y plane) optical axis. In other words, when the optical and measurement coordinate systems are the same and the matrix becomes diagonal, namely,

ªºH x 00 HH «»00 (6.1) «»y «»00H ¬¼z

First, we consider a uniaxial anisotropic substrate (εx = εy ≠ εz) with the thickness of H.

For a transformed microstrip line in the isotropic region, the “isotropy-ized” permittivity,

εg, is HHxz , and the effective height, He, is H HHxz/ for the propagation along the y- axis [74]. The effective permittivity of an anisotropic substrate is given by [40]:

111

W W

z u ε H g He x v

Z-plane W-plane

(a) (b)

Figure 6.1. Cross section of (a) microstrip on anisotropic substrate and (b) equivalent microstrip on isotropic substrate.

ªº1/2 HHgg11§·He CHae Heff «»¨¸112 (6.2) 22©¹WCH ¬¼«»a where Ca is the capacitance for the air-filled micrsotrip line [40]. The Ca for the micrsotrip line is given in (4.18).

According to equation (6.2), there are two unknowns, i.e., εg and He, 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 εg and He from the two effective permittivity equations. Finally,

εx and εz can be found easily from the definitions of εg and He.

Now, we can consider a biaxial anisotropic material (sometimes called Type III anisotropic material) that has εx ≠ εy ≠ εz. 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 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:

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 εg,x

of HHyz and He,x of H HHyz/ , and it is possible to determine εy and εz. Similarly, we

y x Optical axes of biaxial Optical axes of biaxial anisotropic x anisotropic material 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 can consider εg,y of HHxz and He,y of H HHxz/ for the propagation along the y-axis, and it is possible to determine εx and εz.

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 εx = εy = 3 and εz = 9. Also, for the biaxial anisotropic simulation, the permittivity elements are εx = 3, εy = 6, and εz = 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 εx and εz with respect to the nominal values were approximately 2% over the frequency range of 1 to 10 GHz. Similar to the simulation of

10 10 9 9 8 8

7 z 7 ε z

ε 6 6 5 5 , and y and and x 4 ε 4 , ε x

3 ε 3 2 2 1 ε ε 1 x z εx εy εz 0 0 12345678910 12345678910 Frequency (GHz) Frequency (GHz)

(a) (b)

Figure 6.3. The simulated results for the anisotropic material characterizations: (a) uniaxial and (b) biaxial anisotropic substrates 114 the characterization of biaxial anisotropic materials, the maximum relative errors for εx, εy, and εz 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.

6.3. 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]:

ªºªºªcosTT sin 0 1 0 0 cos TTITI sin cos sin sin º «»«»« » U «»«»«sinTT cos 0 0 cos I sin I sin TTITI cos cos cos sin »(6.5) «»«»«0010sincos0II sincos I I » ¬¼¬¼¬ ¼

115

z I z'

θ x' x

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, H c , (see Appendix C), is given by:

ªºHHHxx xy xz «» T (6.6) HH' UU «» HHHyx yy yz «»HHH ¬¼zx zy zz where

22222 HHxx xcos THT y sin cos IHT z sin sin I 22 HHTTHTTxy xsin cos y sin cos cos IHTT z sin cos sin I

Hxz H ysin TIIHTII sin cos z sin sin cos

HHyx xy 22222 HHyy xsin THT y cos cos IHT z cos sin I (6.7)

Hyz H ycos TIIHTII sin cos z cos sin cos

HHzx xz

HHzy yz HH sin22 IHI cos zz y z 116

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, εg, and the effective height, He, for different propagation directions can be determined from the measured effective permittivity and can be expressed with the permittivity tensor elements.

z y' y z'

x' x' H θ H I H x x

(a) (b)

117

Equations (6.8) and (6.9) provide εg and He for x-axis propagation and y-axis propagation, respectively.

HH H2 HHHH 2 and HH yy zz yz (6.8) g,, x yy zz yz e x H 2 zz

HH H2 HHHH 2 and HH xx zz xz (6.9) gy,, xxzz xz ey H 2 zz

From equations (6.8) and (6.9), only εzz can be found. However, since both εg and He have off-diagonal elements, i.e., εyz and εxz, it is impossible to solve the permittivity tensor elements, εxx, εyy, εxz, and εyz. 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, H D , is given by:

DDD ªºHHHxx xy xz «» aTc DDD (6.10) HH UUDD «» HHHyx yy yz «»HHHDDD ¬¼zx zy zz where

ªºcosDD sin 0 U «»sinDD cos 0 (6.11) D «» «»001 ¬¼

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 H D also can be expressed in terms of εx, εy, εz, θ, ϕ, and α. 118

D 22 HHxx xxcos DH yy sin DH 2 xy sin DD cos D 22 HHHDDHDxy xx yysin cos xy sin cos D D HHxz xzcos DHD yz sin DD HHyx xy D 22 HHyy xxsin DH yy cos DH 2 xy sin DD cos (6.12) D HHyz xzsin DHD yz cos DD HHzx xz DD HHzy yz HHD zz zz

Let us consider microstrip lines with different propagation directions, one for the direction of α and another for the direction of α + 90°. The “isotropy-ized” permittivity, εg, and the effective height, He, for these two different propagation directions can be determined. Equations (6.13) and (6.14) are εg and He for the propagation along the α direction and the propagation along the α + 90° direction, respectively.

2 HHDD H D DD D2 yy zz yz HHHHgyyzzyze,,DD and HH (6.13) D 2 H zz

2 HHDD H D DD D2 xx zz xz HHHH HHxx zz Hx xz and HHH H (6.14) ge,90DD ,90, D 2 H zz

We could find the “isotropy-ized” permittivity, εg, and the effective height, He, 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 H , such as εx, εy, and εz, and the misalignment angles, such as θ and

119

ϕ, is better than finding matrix elements in H c . First, we can simplify the relationships of the different εg 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

HH22 HHHH (6.15) gx,, gy x zz y z

HH22 cos 2 THHHH (6.16) gy,, gx x zz y z

HH22 sin 2 THHHH (6.17) gg,,90DD xzzyz

From equations (6.16) and (6.17), it is possible to determine the in-plane misalignment angle, θ.

In addition, εx can be determined using equations (6.15), (6.16), and (6.18), and it is given by:

Again, εzz has already been determined using equation (6.8). The other unknowns in equations (6.15), (6.16), and (6.17) are εy and εz. It is impossible to determine εy and εz using equations (6.15) to (6.19), but we can determine εyεz, which is given by:

HH H22 H HH (6.20) yz gx,, gy xzz

So far, we have determined εx, εzz, θ, and εyεz. It is possible to determine εy and εz if we know (εy+εz), which is shown in equation (6.21).

120

HHHT22 cos HH gx, y z H (6.21) yzHTsin2 zz x

Thus, we can find εy and εz from equations (6.20) and (6.21). The last unknown is the misalignment angle of ϕ and it can be easily determined from εzz in equation (6.7).

Finally, we can determine all the unknowns, i.e., εx, εy, εz, θ, and ϕ. It is also possible to express these unknowns in terms of εxx, εyy, εzz, εxy, εxz, and εyz by using the values determined above. This method for the measurement of anisotropic thin-film materials is verified and discussed in the following section.

6.4. 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 Al2O3. 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

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:

ªº9.4 0 0 H «»0 9.4 0 (6.23) C «» «»0 0 11.6 ¬¼

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 HC . 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 C-plane

57.6° y z′ 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 gives the permittivity tensor of an R-plane sapphire wafer.

ªº9.4 0 0 H «»0 10.97 0.99 (6.24) R «» «»0 0.99 10.03 ¬¼

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:

ªº9.6801 0.6007 0.4206 «» H R 0.6007 10.6882 0.9021 (6.25) T 25 «»« ¬¼«»« 0.4206 0.9021 10.0316

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

εxx, εyy, and εzz are 8.917%, 6.994%, and 2.131%, respectively. Since the non-zero, off-

123

11 3

10 2

zz 1 ε

9 yz ε 0

, and 8 yy , and

ε -1 xz , ε xx 7 , ε

xy -2 ε 6 -3 εxx εyy εzz εxy εxz εyz 5 -4 12345678910 12345678910 Frequency (GHz) Frequency (GHz)

(a) (b)

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 εxy, εxz, and εyz are 0.081, 0.396, and 0.9615. According to Figure 6.7, the off-diagonal element values increase as frequency increases, and this trend results in large absolute errors in the high- frequency 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

(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 εx and εz. In this case, we used our proposed method for uniaxial anisotropic materials, so we assumed that εx and εy were the same. The measured results showed that the extraction results for both εx and εz 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

15 14 13

z 12 ε 11 and x ε 10 9 8 εx εz 7 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Frequency (GHz)

Figure 6.9. C-plane sapphire measurement results for εx and εz. The nominal values of εx and εz are 9.4 and 11.6, respectively, up to 1GHz.

this measurement, we also measured the sample wafer over the frequency range of 3 to16

GHz, and Figure 6.10 shows the extraction results for the diagonalized matrix elements,

εx, εy, and εz. Since we didn’t know the orientation of the in-plane optical axes of the R- plane sapphire wafer, it was impossible to build our test structures on the wafer so that they were perfectly aligned with the in-plane optical axes. Thus, we had to determine the in-plane misalignment angle and then express the diagonalized permittivity tensor using the misalignment angles. The in-plane misalignment angle was determined by our proposed characterization method, and the in-plane misalignment angle was found to be approximately -7.5º. According to the results measured for the R-plane sapphire wafer, εx had a relative error of around 5%, and εy and εz have relative errors of approximately 10% with respect to the nominal values over the frequency range of 3 to 16 GHz. In a comparison of C-plane and R-plane sapphire wafer measurements, the R-plane results were more stable than the C-plane results. The main reason for the difference in the 126

12 z ε 11 and y

ε 10 x, ε 9

8 εx εy εz 7 3456789 10111213141516 Frequency (GHz)

Figure 6.10. R-plane sapphire measurement results for diagonalized matrix elements of εx, εy, and εz. The nominal values of εx, εy, and εz are 9.4, 9.4, and 11.6, respectively, up to 1GHz.

measured results could be the fabrication quality. Since our proposed method uses two different microstrip lines to extract the permittivity tensor elements, better fabrication quality of the test structures will provide better extraction results.

6.5. Summary

In this chapter, we proposed a new method of on-wafer characterization of the permittivity tensor for anisotropic thin-film materials. The main idea of this method is the use of two different microstrip lines with different widths to determine two different effective permittivity values. We also showed a full mathematical derivation of the extraction procedure of our proposed method. We discussed the characterizations of both uniaxial and biaxial, anisotropic, thin-film material using microstrip lines. Furthermore, we showed how to characterize the more general case of biaxial, anisotropic, thin-film 127 material for which the optical axes are unknown.

We also verified our proposed method for measuring the characteristics of anisotropic wafers using C-plane and R-plane sapphire wafers. According to the measured results, the extraction results for the permittivity tensor elements had relative errors of approximately

10% with respect to the nominal values of the tensor elements. In addition, our on-wafer measurements included TRL calibration so the parasitic effects between probe tips and contact pads could be eliminated. Thus, the measurement errors caused by these parasitic effects were reduced by our on-wafer measurement technique.

128

Chapter 7

CONCLUSION

7.1. Summary and Conclusion

The main purpose of this study is to develop new on-wafer characterization methods that overcome the limitations of the conventional on-wafer characterization methods. In this dissertation, we presented four different newly developed on-wafer characterization methods suitable for different types of materials. To realize on-wafer measurements, the test fixture should be implemented with planar structures. Therefore, all the theoretical derivations for each method were focused on planar structures, such as the coplanar waveguide and microstrip lines. In addition, we clearly stated the limitations for the conventional on-wafer characterization methods and reasons for the developments of new on-wafer characterization methods at the beginning of each chapter. In each chapter, we provided not only the theoretical derivations of the newly developed characterization methods, but also both simulated and experimental results in this dissertation.

In Chapter 2, the conventional on-wafer characterization methods were reviewed.

Both transmission line method and T-resonator method—the most widely used methods for the on-wafer characterization as non-resonant and resonant methods, respectively— were fully reviewed. In addition, we presented and compared the experimental results for

129 both conventional methods. The experimental results clearly showed the advantage and disadvantage of each method. For example, the T-resonator method provided more accurate results than the transmission line method; however, the transmission line methods showed continuous results of the material extraction whereas the T-resonator method only provided the extracted results at a discrete number of frequency points.

In Chapter 3, we discussed a new on-wafer characterization method using T- resonators. Although the conventional T-resonator method included the parasitic effects, such as open-ended and T-junction effects, the problems in the determination of the effective T-stub length still existed. A newly developed T-resonator method discussed in this chapter eliminates these parasitic effects almost completely so that the measurement results using our newly developed method provided better accuracy than the conventional

T-resonator method. The measurements were performed over the frequency range of

1GHz to 20GHz, and the measured results using the new method with both coplanar waveguide and microstrip T-resonators achieved less than 1% of the relative errors for the extracted results of permittivity while the conventional method had relative errors of

1% to 4% over the frequency range of 1GHz to 20GHz. The main problem in the conventional T-resonator method includes all the parasitic effects to the permittivity extraction procedure; however, our newly developed method excluded these parasitic effects in the permittivity extraction procedure. This is the main reason why the newly developed method could achieve a very high accuracy in the measurements.

Chapter 4 discussed a new transmission line method for the magnetic-dielectric material characterization. For the on-wafer measurement with the microstrip structures, it

130 is necessary to include coplanar waveguide-to-microstrip transitions. Therefore, the conventional transmission line method for the magnetic-dielectric material characterization using microstrip lines might have problems determining the characteristic impedance of the microstrip line. Consequently, the extraction results of the material properties for the magnetic-dielectric material might not be correct. The newly developed transmission line method in Chapter 4 used two different microstrip transmission lines to determine the characteristic impedance ratio. Therefore, εr and μr of the magnetic-dielectric material could be determined from the measured propagation constants and the characteristic impedance ratio. This chapter also presented both simulated and measured results. The measured results showed that the maximum relative errors of the εr and μr extractions were 13.07% and 17.0%, respectively, over the 4GHz to 14GHz frequency range. The measured results had larger relative errors for both εr and

μr extractions. However, it would have been better had the sample wafer being tested had a μr value of more than 1 because the equations used in this method determined the ratio of the effective permeability for two different microstrip lines. However, the effective values of permeability ratio are 1 when μr is 1, which may increase the uncertainties in the calculation procedure. Another accomplishment in this chapter is error analysis. We also presented error analyses due to the uncertainties of the structure geometry in Chapter

4. Although the error analyses in this chapter used only simulated results, the results of error analyses clearly showed which geometrical parameters play the important role in this method.

131

A new on-wafer characterization method for magnetic-dielectric materials using T- resonators was discussed in Chapter 5. Similar to the method in Chapter 4, we used two different T-resonators and determined the characteristic impedance ratio. The material properties of the magnetic-dielectric materials were determined using the measured effective refractive index and the characteristic impedance ratio at the resonant frequency points. In addition, we presented a new and easy way to determine the effective T-stub length, which was similar to the modified straight-ribbon resonator method; the measured results, including the consideration of the effective T-stub length, indicated that the maximum relative errors for εr and μr extractions were 1.24% and 3.81%, respectively, across the 1GHz to 20GHz frequency range. The results clearly demonstrated that the T- resonator method provided better accuracy of the measurement than the transmission line method.

The final chapter of the main part of this dissertation offered a new on-wafer characterization method for anisotropic materials. Unlike isotropic material characterization, anisotropic characterization needed to consider the permittivity as a tensor form. We used a mapping technique to transform the anisotropic region into an isotropic region. For the special cases, which considered the optical axes of the anisotropic material and the measurement axes to be perfectly matched cases, the characterization using microstrip lines were not complicated. However, for general cases, we needed to consider the misalignment between the optical axes and the measurement axes, which produced non-zero off-diagonal elements in the permittivity tensor.

Therefore, the extraction procedures were more complicated than the special cases. We

132 provided full theoretical derivations for the general case of anisotropic material measurements. In addition, the measured results of sapphire wafers using microstrip lines were discussed in this chapter. Although the maximum relative error for diagonal permittivity element extractions was approximately 10% with respect to the nominal values, we could also determine the in-plane misalignment angle between the optical and measurement axes, and the determined in-plane misalignment angle was around -7.5º over the 3GHz to 16GHz frequency range. Therefore, it is possible to obtain a full matrix of the permittivity tensor with non-zero elements.

7.2. Future Work

We discussed newly developed on-wafer characterization methods for different types of materials. However, the methods we discussed in this dissertation need further improvements to apply to more different types of materials. In addition, further improvements are needed to reduce measurement errors using the methods described in this dissertation.

First of all, we developed both a transmission line method and T-resonator method for the on-wafer characterization method for isotropic materials. However, the transmission line method was only used for anisotropic material on-wafer characterization in this dissertation. According to the measured results for both the transmission line method and

T-resonator method, the measured results had better accuracy than the results using the transmission line method. Therefore, applying the T-resonator method to anisotropic material characterization will provide better accuracy in the measurements. The nature of

133 the anisotropic materials have different permittivity in different directions of optical axes;

T-resonators with different directions of T-stub on anisotropic material will not result in resonances at the same frequency point even if the T-resonators have the same .physical length. According to recent research on the measurement of the liquid crystal using the patch resonator, resonant frequencies shifted by changing the alignment of the liquid crystals which means changing the dielectric constants [80]. Therefore, it is difficult to determine the permittivity tensor elements at the same frequency points. Averaging the same order of the resonant frequencies may be one solution for anisotropic material characterization using T-resonators if the difference of the resonant frequencies is not large.

Furthermore, it is necessary to extend our proposed anisotropic material characterization method to the characterization for both H and P . In this study, only the dielectric anisotropic material characterization was considered. To approach the anisotropic material characterization of H and P , we can start with the same analysis method for the permittivity tensor, which is a mapping technique of the anisotropic region into the isotropic region. The permeability tensor analysis may result in a duality relationship. Whenever the permeability characterization method is available, we can apply characterization algorithms for both permittivity and permeability tensors to our newly developed magnetic-dielectric thin-film characterization methods, as discussed in

Chapter 4. Therefore, it may be possible to determine both permittivity and permeability tensors for anisotropic substrates that present both permittivity and permeability as tensors. 134

Appendix A

CRYSTAL SYSTEM (BRAVAIS LATTICE)

Number of Classification System Bravais Lattice independent Tensor form coefficient

ªºH 00 Isotropic «» Cubic 1 00H (Anaxial) a «» «» a ¬¼00H a

Tetragonal c

a a

ªºH1 00 c «» Uniaxial 2 00H1 Hexagonal «» ¬¼«»00H3

a α=β=γ≠90° γ Rhombohedral a β a α a Continued

Table A.1. Classification of tensor forms by crystal system [13]

135

Table A. 1. continued

a≠b≠c

ªºH1 00 «»00H Orthorhombic 3 «»2 c «»00H ¬¼3 a b γ HH 0 Biaxial ªº11 12 «»HH 0 Monoclinic 4 «»12 22 β α ¬¼«»00H33 α≠90°, β=γ=90° γ ªºHHH11 12 13 «»HHH Triclinic 6 «»12 22 23 β α ¬¼«»HHH13 23 33 α≠β≠γ≠90°

136

Appendix B

CONFORMAL MAPPING OF A MICROSTRIP LINE WITH DUALITY RELATION

The conformal mapping for microstrip analysis is the most widely used technique.

This technique uses a conformal transformation induced by introducing a dielectric constant that is effective for the equivalent capacitance of the microstrip [38, 39]. The transformation for the wide microstrip is [52]:

§·w zj S dtanh¨¸ w (B.1) ©¹2 where z is the microstrip plane, and w is the plane in which the microstrip is mapped into a parallel plate. The parameter d is approximately g´ in w-plane. Figure B.1 shows a microstrip configuration in the z-plane and its mapping in the w-plane. The dielectric-air boundary of the microstrip substrate in the z-plane is mapped into an arc (ba' curve in

Figure B.1 (b)) in the w-plane. We can approximate that the dielectric-air boundary curve in (b) to a rectangle in (c). Thus, the area over the dielectric-air boundary curve in (b) is

πs' and is the same as the area over the rectangle in (c), which is a sum of the parallel area of πs" and a series area of π(s'-s"). Furthermore, these parallel and series areas in (c) are the same as the parallel area in (d).

137

Im(z) Im(w) ↑ w-plane ᬅ πs' z-plane b=π ᬅ,ᬋ ᬊ

ᬆ ᬇ

ᬈ Re(z) ᬉ

ᬉ ᬋ→ ᬆ ᬇ ᬈ Re(w) ᬊ 0 a´ g´ (a) (b) Im(w) Im(w) π(s'-s'') πs'' w-plane w-plane s b=π b=π

Re(w) Re(w) 0 a´ g´ 0 a´ g´ (c) (d)

Figure B.1. Conformal mapping of a microstrip in z-plane into a two parallel plates in w- plane

ssccc ss cc (B.2) H r

Thus, the effective filling factor, which is defined as the ratio of dielectric area over the total area in the rectangle of the mapping field, is given by:

gas'' q (B.3) g '

The effective capacitance, Ceff, is the sum of C1, C2, and C3 in (d) and given by:

CCCCeff 123 (B.4)

Equation (B.4) can be expressed in terms of the effective and relative dielectric constants and the parallel plate areas.

138

HH00effgass'' H HH 00 r HH r ga '' (B.5)

Then, it is possible to find g' in terms of εr and εeff.

H 1 gas'' r (B.6) HHreff

Also, g'-(a'-s) can be found easily from (B.5) by rearranging of (B.5), given by:

H 1 gas'' eff as ' (B.7) HHreff

From (B.7), the effective filling factor, q, can be expressed in terms of the effective and relative dielectric constants.

gas'' Heff 1 q (B.8) g '1Hr

Thus, the effective dielectric constant is:

HHeff 11q r (B.9)

Now, consider the effective permeability, μeff, of the microstrip. Similar to the analysis of the microstrip on a dielectric substrate, we can consider the effective inductance for the microstrip on a magnetic substrate, and the effective inductance is given by:

1111 (B.10) LLLLeff 123

Equation (B.10) can be also expressed in terms of the effective and relative permeabilities and the parallel area parameters.

gassga'' '' (B.11) PP0000eff P PP r PP r

From (B.11), g' and g'-(a'-s) can be written as: 139

1/P 1 gas'' r (B.12) 1/PPreff 1/

1/Peff 1 gas'' as ' (B.13) 1/PPreff 1/

Thus, the effective filling factor, q, can be expressed in terms of the effective and relative permeabilities.

gas'' 1/Peff 1 q (B.14) g '1/1 Pr

Finally, the effective permeability is given by:

140

Appendix C

THE PERMITTIVITY TENSOR IN THE MEASUREMENT COORDIATE SYSTEM

The measurement is performed in the xyz system, but the permittivity tensor is in x´y´z´ system. Figure 6.3 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 I is the rotation angle of the y-axis. Then, the rotation transformation matrix U is given by [71,

72]:

ªºªºªcosTT sin 0 cos I 0 sin I cos TIT cos sin cos TI sin º «»«»« » U «»«»«sinTT cos 0 0 1 0 sin TIT cos cos sin TI sin » (D.1) ¬¼¬¼¬«»«»«001sin0cossin0II I cos I ¼»

Let us assume that H is the permittivity tensor of biaxial anisotropic material; then the transformed permittivity tensor is:

HH' UUT cosTI cos sin T cos TIH sin 0 0 cos TI cos sin TI cos sin I ªºªºªºx (D.2) «»«»«»sinTI cos cos T sin TI sin 0 H 0 sin T cos T 0 «»«»«»y «»«»«»¬¼¬¼¬¼sinIIHTITII 0 cos 0 0z cos sin sin sin cos

First, we consider the in-plane (xy-plane) misalignment angle to be θ only. In this case, the rotation angle I is zero, and the H ' is given by:

141

ªº22 HTHTHHTTxycos sin xy sin cos 0 «»ªºHHxx xy 0 «»22«» HHHTTHTHT' x y sin cos x sin y cos 0 HH yx yy 0 (D.3) «»«» «»00H ¬¼«»00H zz ¬¼z

Similarly, we can consider the misalignment angle, I , only.

22 ªºHIHIxcos z sin 0 HHIIH x z sin cosªº xx 0 H xz «» HH'0 0 «» 00 H (D.4) «»yyy«» «»22«» ¬¼ HHxzsin I cos I 0 H x sin IH z cos I¬¼ H zx 0 H zz

According to (D.3) and (D.4), off-diagonal elements exist if there are differences between the angles of the principal axes of the permittivity tensor and the measurement axes.

142

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