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

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Jun Seok Lee, B. S., M. S.

Graduate Program in Electrical and Computer Engineering

The Ohio State University

2011

Dissertation Committee

Professor Roberto G. Rojas, Adviser

Professor Patrick Roblin

Professor Fernando L. Teixeira

Copyright by

Jun Seok Lee

2011

ii

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

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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.

v

Dedication

This document is dedicated to my family.

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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!

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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

viii

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

ix

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

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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

xi

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

xii

List of Figures

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

Figure 1.2. Examples of conventional material characterization method configurations (a) Reflection method with open-ended coaxial probe, (b) Free-space bistatic reflection method, and (c) Rectangular dielectric waveguide method .................................................4

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

Figure 2.2. (a) Probe station measurement configuration. (b) Contact between GSG (Ground-Signal-Ground) probe tip and contact pad (upper) and probes on the wafer sample (lower ) ..................................................................................................................12

Figure 2.3. (a) Microstrip transmission line and (b) coplanar waveguide transmission line............................................................................................................................................14

Figure 2.4. Electric field distribution of (a) microstrip and (b) coplanar waveguide structures ............................................................................................................................16

Figure 2.5. Equivalent circuit model of the transmission line ...........................................18

Figure 2.6. Fabricated test structures on Pyrex 7740 wafer (a) coplanar waveguide test structures and (b) microstrip test structures. Both microstrip and coplanar waveguide transmission lines and TRL calibration kits are fabricated on Pyrex 7740 wafers ............21

Figure 2.7. The test fixture consists of a microstrip line as a DUT and coplanar waveguide-to-microstrip transitions as error boxes ...........................................................21

Figure 2.8. E-field distributions at (a) A-A' plane and (b) B-B' plane. The upper and lower figures represent the magnitude and the vector of E-fields, respectively ................22

Figure 2.9. Extraction results of 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

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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

xv

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 6.2. 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

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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

1

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

2

more accurate measurement results, which is highly desired and the main motivation of

this study.

In electromagnetic material characterization, complex permittivity and permeability

are typically determined. Both permittivity and permeability are described as the

interactions between the electric and magnetic fields. Therefore, complex permittivity

and permeability can be defined based on the constitutive relations:

D E (1.1)

B H (1.2)

where, E, H, D, and B are the electric field, magnetic field, and electric and magnetic

flux densities, respectively. In addition, ε = ε0εr and μ = μ0μr are complex permittivity

and permeability, respectively; ε0 (8.854×10-12) and μ0 (4π×10-7) 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

3

(a) (b)

Figure 1.1. Simple illustrations for (a) permittivity and (b) permeability measurements including their equivalent circuit models parts of μr are lLeff/μ0N2AC and l(Reff - Rw)/μ0ωN2AC, 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).

C G

Electrode (Area = A)

L R

4

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

(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

Coaxial dielectric probe

εr1 εr2

Free space

d1

Sample terminated by metal plate

Transmit antenna

Receive antenna

Rectangular dielectric waveguide

Rectangular dielectric waveguide

Sample

n1 n2 n1

z=0 z=d

5

provide an accurate measurement of the loss tangent due to the open discontinuity

problem between the rectangular dielectric waveguide and sample

These examples of conventional material characterization methods are not considered

in on-wafer measurements. Typically, on-wafer measurements use planar circuits, such as

a microstrip and coplanar waveguide structures in conjunction with a probe station. The

main advantage of these types of structures is that no air gap presents between the

metallic structures and the sample being tested. Thus, on-wafer measurement methods

can minimize measurement errors due to an air gap. In addition, the on-wafer

measurement method can be used directly in the development of the planar circuits on the

sample being tested, thereby allowing in-situ measurements. For the on-wafer

measurements, resonant and non-resonant methods are commonly used; we will present

an in-depth review for both resonant and non-resonant methods in the following chapter.

In this study, we realized the need to develop accurate on-wafer measurement methods

not only for isotropic thin-film materials, but also for anisotropic thin-film materials.

Anisotropic materials present the permittivity and permeability as tensors ( and ); the

accurate characterization of the electromagnetic properties of new, on-wafer thin films is

crucial for accessing their potential use in the design of microwave devices, antennas, and

a variety of sensors. Furthermore, many of the new materials being developed are

anisotropic, and previous on-wafer techniques that have been developed to characterize

isotropic materials cannot be used. Several methods for determining the permittivity and

permeability tensors of the anisotropic materials exist, such as the free space method

[10], waveguide method [11], and the transmission/reflection method [12]. The main

6

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

7

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

8

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.

9

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.

10

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.

11

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

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

εr μr

Probes

P1 P2 S21

S12

To network analyzer

To network analyzer

S11 S22

Coaxial to coplanar transition

Coplanar cell Conductive strips

12

(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

13

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-

14

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

15

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]:

0 effj jk

(2.1)

16

(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].

1 1 12 2 1 12( / )

MS r reff h W

(2.2)

5

5

11

2rCPW

eff

K k K kK k K k

(2.3)

where K(k) is the complete elliptic integral of the first kind. The moduli k, kˊ, k5, and kˊ5

are given by [41]:

17

2 2

2 2

c b akb c a

(2.4)

2 22

2 21 a c bk kb c a

(2.5)

2 2

5 2 2

sinh / 2 sinh / 2 sinh / 2sinh / 2 sinh / 2 sinh / 2

c h b h a hk

b h c h a h (2.6)

2 22

5 5 2 2

sinh / 2 sinh / 2 sinh / 21

sinh / 2 sinh / 2 sinh / 2a h c h b h

k kb h c h a h

(2.7)

Note that the effective dielectric constants of both microstrip and coplanar

transmission lines are functions of the relative dielectric constant, the substrate thickness,

and the geometry of the transmission lines. As a result, the material property, 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

18

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:

c 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]:

1 2

02cR R

Z (2.9)

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]:

1 2

1 1 4lnMS SR LR WRW T

(2.10)

L R

C G

19

2/

/ 5.8 0.03 /MS SR W hR

W W h h W (2.11)

01 02 2

00 0

18ln ln18 1

CPW SR kaR kT ka k K k

(2.12)

0 02 2 2

0 00 0

18 1ln ln18 1

CPW Sk R kbRT k ka k K k

(2.13)

where RS=(ωμ/2σ)1/2 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]:

2

1 for 0.5

0.94 0.132 0.0062 for 0.5 10

Wh

LRW W Wh h h

(2.14)

The dielectric loss tangent can be determined from the attenuation constant, αd,

namely, [40, 41]:

0

2tan d

effqk (2.15)

where q=(1-(εˊeff)-1)/(1-(εrˊ)-1) is the filling factor due to the dielectric loss [41, 43].

The main advantage of the transmission line method is that it provides continuous

values of the measured material properties over the finite frequency bandwidth while the

resonant method only provides material properties with a discrete number of equally

spaced frequencies. In addition, the characterization of material properties is relatively

simple since this method only needs to measure the complex propagation constant of the

20

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

21

are based on the on-wafer technique, using a probe station and TRL calibration is

fundamental to achieve good accuracy [45].

Unlike coplanar waveguide test structures, microstrip test structures require coplanar

waveguide-to-microstrip transitions to implement on-wafer measurements using the

probe station [46]. Thus, the test fixtures consist of microstrip transmission lines as DUTs

and coplanar waveguide-to-microstrip transitions as error boxes. Figure 2.7 shows the

(a) (b)

Figure 2.6. Fabricated test structures on Pyrex 7740 wafer (a) coplanar waveguide test structures and (b) microstrip test structures. Both microstrip and coplanar waveguide transmission lines and TRL calibration kits are fabricated on Pyrex 7740 wafers

Figure 2.7. The test fixture consists of a microstrip line as a DUT and coplanar waveguide-to-microstrip transitions as error boxes

A B

A' B'

22

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.

23

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

(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

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

x 1010

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

Frequency (Hz)

r

4 5 6 7 8 9 10 11 12 13 14

x 109

4

4.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

Frequency (Hz)

r

24

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.

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.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-60

-50

-40

-30

-20

-10

0

Frequency (Hz)

S11

(dB

)

Valid Region

25

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

(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

4 5 6 7 8 9 10 11 12 13 14

x 109

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

Frequency (Hz)

tan

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

x 1010

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Frequency (Hz)

tan

26

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.

27

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

for 1,2,3,2eff

n

nc nrf

(2.16)

where r is the mean radius of the ring, fn is the nth 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

r

W

Lstub

Gap Gap

Gap Gap

l1

l2

(a) (b) (c)

28

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

1 2 2 1

1 2 2 12n n

effn n

c n f n ff f l l

(2.17)

where the subscript 1 and 2 refer the straight-ribbon resonator 1 and 2, respectively. In

addition, l is the ribbon length, fn is the nth 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.

29

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.

30

(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]:

4stubn eff

ncLf

(2.18)

where Lstub is the T-stub length, fn is the nth 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

4effstub n

ncL f

(2.19)

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

31

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

eff stub end junctionL L l l (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]:

1 2 3

4endl h (2.21)

where

0.85440.81

1 0.85440.81

0.26 / 0.2360.434907

0.189 / 0.87eff

eff

W hW h

(2.22)

51.9413/1

2 0.9236

0.5274tan 0.084 /1

eff

W h (2.23)

32

7.5 /3 1 0.218 W he (2.24)

1.456 0.036 114 1 0.0377tan 0.067 / 6 5 rW h e (2.25)

0.371

5

/1

2.358 1r

W h

(2.26)

where W is the microstrip line width of the top conductor and h is the substrate thickness.

The expressions from (2.21) to (2.26) provide accurate results for determining the

correction factor due to the open-end effect for the range of normalized widths 0.01 ≤

W/h ≤ 100 and 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

1.6

1

0.5 0.05 0.7 0.25junction

p

l feW f

(2.27)

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:

33

3dBBWLfQ (2.28)

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]:

0 /101 2 10 A

L

L

QQ (2.29)

where LA is the insertion loss at the resonant frequency. In addition, the unloaded quality

factor, Q0, can be written as:

0

1 1 1 1

d c rQ Q Q Q (2.30)

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].

20ln10c

c g

Q (2.31)

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]:

02480 /

reff

nZQh F

(2.32)

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-

34

junction radiations are given by [62, 63]:

21 11

log2 1

eff effeffopen

eff eff eff eff

F (2.33)

2

3/2

3 1 1 2 1 1log log

8 2 1 41 2 1eff eff eff effeff eff

Teff eff effeff eff eff

F (2.34)

Thus, one unknown is left in (2.30): the quality factor due to the dielectric losses.

From the measured and calculated quality factors, it is possible to determine the quality

factor due to the dielectric losses, Qd; the loss tangent of the dielectric material can then

be determined using the following relationship [54].

1tan

1eff r

d r effQ (2.35)

Based on equations from (2.19) to (2.35), the material properties—the relative

permittivity and the dielectric loss tangent—can be determined from the measured T-

resonators. In the following section, we will show experimental results for the on-wafer

material characterization using T-resonators.

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

35

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

36

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

(a) (b)

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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

Before TRLAfter TRL

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-25

-20

-15

-10

-5

0

Frequency (Hz)

S21

(dB

)

Before TRLAfter TRL

37

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

f (GHz)

εr tanδ Value Error (%) Value

CPW T-resonator

8.904 4.688 1.913 0.027 18.023 4.553 1.022 0.014

Microstrip T-resonator

2.714 4.784 4.0 0.026 8.192 4.70 2.174 0.011 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)

38

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.

39

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:

0 eff effjk (2.36)

0 0eff

eff

Z Z (2.37)

40

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:

eff eff effj (2.38)

eff eff effj (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:

11 / 1 1 / 1 1

2 2 1 12( / )r rMS

eff h W (2.40)

1

5

5

1 / 11

2rCPW

eff

K k K kK k K k

(2.41)

where K(k) is the complete elliptic integral of the first kind and the modulus (k) are

defined in equations (2.4) through (2.7).

41

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

, 1

11

effd loss

r

q (2.42)

,

11

effm loss

r

q (2.43)

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:

1

, , 1

1tan tan

1eff reff

d eff d loss deff r r

q (2.44)

, ,

1tan tan

1eff reff

m eff m loss meff r r

q (2.45)

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:

11

rr eff

eff

(2.46)

42

11

r rr eff

eff eff

(2.47)

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

43

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]:

11 12

221 22

1 1 11 11 1 121

rX X YZX X

(2.48)

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]:

11 12 12 21 11 22 11

21 22 2221

11

R R S S S S SR R SS

(2.49)

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]:

212 210

20 12 214

r

r

X XZ ZZ Z X X

(2.50)

44

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].

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

Zr:Z0

Y

Pad capacitances Impedance transformerProbe tip Interface between probe-

tip and contact pad

45

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

46

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

47

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:

00

l

T Line l

eR

e (3.1)

0 0

0 0

12 2

12 2

Y

YZ YZ

RYZ YZ

(3.2)

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.

2 0 0

20 0

12 2

12 2

feed

feed

l

T resl

YZ YZeR

YZ YZe

(3.3)

48

(a) (b)

Figure 3.1. (a) A typical T-resonator configuration and (b) its equivalent circuit model for T-resonator. Each section in the equivalent circuit model can be expressed with a wave cascade matrix model. where Y is the input admittance of the stub, given by:

0

for open-stubstub stub

stub stub

L L

open L L

e eYZ e e

(3.4)

0

for short-stubstub stub

stub stub

L L

short L L

e eYZ e e

(3.5)

Thus, the S-matrix of the T-resonator can easily be found from the wave cascade

matrix of the T-resonator in (3.3) and the conversion from the wave cascade matrix to the

S-matrix given by:

11 12 12 11 22 12 21

21 22 2122

11

S S R R R R RS S RR

(3.4)

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

YZ0 Z0

lfeed lfeed

Feed line at port 1

Feed line at port 2

T-stub with length Lstub

Lstub

lfeed

49

2 20

0 022

0

0 0

22 2

22 2

feed feed

feedfeed

l l

T res ll

YZ e eYZ YZ

SYZ ee

YZ YZ

(3.5)

Let’s consider the resonances in the T-resonator for both open-stub and short-stub

cases. Resonances in T-resonators occur when |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:

21 2 2

2 cos for open-stub

4cos sinstub

stub stub

LS

L L (3.6)

21 2 2

2 sin for short-stub

4sin cosstub

stub stub

LS

L L (3.7)

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:

1,3,5 for open-stub where

2,4,6 for short-stub4rstub eff

nncfnL

for ope for sho

(3.8)

The resonant frequency in (3.8) for both open-stub and short-stub T-resonators is exactly

same as the conventional resonant frequency formulas.

However, none of the resonances of T-resonators in S11 are considered in the

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

50

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

11 2 2

2 sin for open-stub

4cos sinstub

stub stub

LS

L L (3.9)

11 2 2

2 cos for short-stub

4sin cosstub

stub stub

LS

L L (3.10)

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

lossless cases. As a result, R11 in (3.3) is 2 feedj le at the S11 resonant frequency. Thus, β at

the S11 resonant frequency is given by:

11ln2 feed

j Rl

(3.11)

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

51

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:

211

sinh1

2coshfeedl stub

stub

LR e

L (3.12)

As previously stated, we are interested in R11 at S11 resonant frequency points, and R11

will be 2 feedj le 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:

52

111 ln

2 feed

Rl

(3.13)

112 feed

j Rl

(3.14)

The measured attenuation constant, α, can be broken down into different components,

with the total attenuation constant given by:

c d r (3.15)

where 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.

53

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

54

(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

55

(a) (b)

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. 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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-25

-20

-15

-10

-5

0

Frequency (Hz)

S11

(dB

)

Before TRLAfter TRL

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

Before TRLAfter TRL

56

(a)

(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

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

0.7

0.8

0.9

1

1.1

1.2

1.3

Frequency (Hz)

Mag

(R11

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-150

-100

-50

0

50

100

150

Frequency (Hz)

Ang

(R11

) (in

deg

ree)

57

f (GHz) εr Relative Error (%)

Proposed Method

4.411 4.633 0.717

13.722 4.590 0.217

Conventional Method

8.904 4.688 1.913

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

58

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

(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.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-35

-30

-25

-20

-15

-10

-5

0

Frequency (Hz)

S11

(dB

)

Before TRLAfter TRL

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-25

-20

-15

-10

-5

0

Frequency (Hz)

S21

(dB

)

Before TRLAfter TRL

59

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

60

(a)

(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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Frequency (Hz)

Mag

(R11

)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-200

-150

-100

-50

0

50

100

150

200

Frequency (Hz)

Ang

(R11

) (in

deg

ree)

61

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 (%)

Proposed Method

5.488 4.596 0.094

10.874 4.579 0.457

16.086 4.630 0.657

Conventional Method

2.714 4.784 4.0

8.192 4.70 2.174

13.574 4.734 2.913

18.926 4.74 3.043 Table 3.2. The measurement results comparison for microstrip T-resonator

62

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 / n , where n is the size of the sample

and σ is the sample standard deviation. The sample standard deviation, σ, is given by

2( ) /x x 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 ( 1.96)x SE . Note that we used 24 samples,

which provide about 20% of the margin of error in 95% of confidence limits, in this error

63

(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%

64

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 (%)

Proposed Method

5.521 4.535 – 4.584 0.338 – 1.405

11.080 4.623 – 4.692 0.504 – 2.001

16.301 4.617 – 4.675 0.366 – 1.632

Conventional Method

2.751 4.751 – 4.784 3.277 – 4.001

8.284 4.698 – 4.713 2.127 – 2.448

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

65

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.

66

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

67

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].

68

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.

1 1m a D bR R R R (4.1)

2 2 1 2 ' 2m a D b a mis D mis bR R R R R R R R R

(4.2)

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

Error box AZ01

[Ra]

DUT1Z01

[RD1]

Error box BZ01[Rb]

Z01 Z01 Z01 Z01

Error box AZ01

[Ra]

DUT2Z02

[RD2]

Error box BZ01

[Rb]Z01 Z02 Z02 Z01

(a)

(b)

Reference planes

69

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:

01 02 02 011

02 01 01 0201 02

12mis

Z Z Z ZR

Z Z Z ZZ Z (4.3)

01 02 01 022

01 02 01 0201 02

12mis

Z Z Z ZR

Z Z Z ZZ Z (4.4)

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:

1

110

0

L

D L

eR

e (4.5)

70

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).

2 2

2 2

2 2

2 2

2 201 02 01 02

2 2 2 2 201 02 02 01 01 02

2 2 2 201 02 02 01

2 201 02 01 02

14

L L

D L L

L L

L L

e Z Z e Z ZR

Z Z e Z Z e Z Z

e Z Z e Z Z

e Z Z e Z Z

(4.6)

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.

2 21 11 22

21 cosh

2

D DR RL

(4.7)

where 2DijR 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:

2 22 2

2 2 2 2

2 2 2 2

2 2 2 2 2

1 11 114 1 1 1 1

L LL L

D L L L L

e r e re r e rR

r e r e r e r e r

(4.8)

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

71

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:

2 2 2 201 21 12 22 11

02 22sinh

D D D DZ R R R RrZ L

(4.9)

In addition, the characteristic impedance of the transmission line model, whose

equivalent circuit models is shown in Figure 2.5, is given by [27]:

0R j LZG j C

(4.10)

where R, G, C, and L are the resistance, conductance, capacitance, and inductance per

unit length of conventional transmission line theory, respectively; and are defined by [27]:

2 22

0eff t eff zS

S

R h dS e dSi

(4.11)

2 22

0eff t eff zS

S

G e dS h dSv

(4.12)

2 22

0

1eff t eff zS

S

C e dS h dSv

(4.13)

2 22

0

1eff t eff zS

S

L h dS e dSi

(4.14)

where v0 and i0 are the normalization constants for the waveguide voltage and waveguide

current, which are v(z) = v0e±γz and i(z) = i0e±γz, 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].

72

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

and μ—namely:

2

0 2

1 eff eff eff

eff eff eff

Z LC

(4.15)

The effective values of the permittivity and permeability in a microstrip line can be

considered to be the equivalent permittivity and permeability of a homogeneous medium

in which the transmission line is embedded. These effective values, which replace the air

and magnetic-dielectric substrate regions, can be obtained using conformal mapping

techniques [52].

Next, (4.15) can be used in (4.9), resulting in an expression for the characteristic

impedance ratio, r, in terms of L, C, 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

and the index of refraction are related by neff = (εeffμeff)1/2 = jγ/k0. After several simple

algebraic steps, the characteristic impedance ratio, r can be expressed as:

22 1

1 1 2

effa

a eff

CrC

(4.16)

12 2

1 2 1

effa

a eff

CrC

(4.17)

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]:

73

2 for / 18ln

4

1.393 0.667 ln 1.444 for / 1

oa

a o

C W hh W

W hW WC W hh h

(4.18)

where W is the microstrip line width and h is the substrate thickness. Thus, the only

unknowns in (4.16) and (4.17) are 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:

d m c (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]:

74

0, ,

' 'tan tan

2eff eff

d m d eff m eff

k (4.20)

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:

1

1,20 1,2 1,21,2 1,221,2

1 12 1

effeff effd m r r r

r r r

k nA B (4.21)

We already determined εʹr and μʹr, meaning that εʹeff and μʹeff can be easily found using

conformal mapping techniques since we know the geometry of DUT1 and DUT2. Thus,

there are two unknowns in (4.21): εʺr and μʺr. Having two unknowns and two equations,

it is possible to solve them for εʺr and μʺr using:

2 1 1 2 2 1 1 2

1 2 2 1 2 1 1 2

and r rB B A AA B A B A B A B

(4.22)

We have demonstrated all the theoretical derivations of our proposed on-wafer

characterization method for the magnetic-dielectric materials in this section. Using our

proposed method, it is possible to determine all the material parameters, such as εʹr, μʹr,εʺr,

and μʺr, for the magnetic-dielectric material using two different transmission lines.

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

75

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

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

DUTError Box1

Error Box2

W1

W2

le leL

Z01 Z01 Z01

Z01 Z02 Z01

Transmission Line 1

Transmission Line 2

76

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

1 2 3 4 5 6 7 8 9 10

x 109

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Frequency (Hz)

Rel

ativ

e pe

rmitt

ivity

and

per

mea

bilit

y

Relative permittivityRelative permeability

77

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,

rw εʹr μʹr 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

78

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

79

lines with two different widths play a very important role, making it necessary to present

an error analysis given such uncertainties.

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)

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)

1 2 3 4 5 6 7 8 9 10

x 109

1.8

2

2.2

2.4

2.6

2.8

3

3.2

Frequency (Hz)

Rea

l par

t of r

elat

ive

perm

ittiv

ity a

nd p

erm

eabi

lity

Real part of relative permittivity ( r')

Real part of relative permeability ( r')

1 2 3 4 5 6 7 8 9 10

x 109

0.005

0.01

0.015

0.02

0.025

Frequency (Hz)

Imag

inar

y pa

rt of

rela

tive

perm

ittiv

ity a

nd p

erm

eabi

lity

Imaginary part of relative permittivity ( r")

Imaginary part of relative permeability ( r")

80

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

81

standard error, SE, is / n , where n is the size of the sample and σ is the sample

standard deviation. The sample standard deviation, σ, is given by 2( ) /x x 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 ( 1.96)x SE . 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

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)

33.023.043.063.083.1

3.123.143.163.18

3.2

1 2 3 4 5 6 7 8 9 10

Re(ε r

)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

1.941.961.98

22.022.042.062.08

2.12.122.142.162.18

1 2 3 4 5 6 7 8 9 10

Re(μ r

)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

82

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

83

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

3.05

3.055

3.06

3.065

3.07

3.075

3.08

3.085

3.09

1 2 3 4 5 6 7 8 9 10R

e(ε r

)Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

1.981.99

22.012.022.032.042.052.062.072.08

1 2 3 4 5 6 7 8 9 10

Re(μ r

)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

84

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) 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%

3.043.045

3.053.055

3.063.065

3.073.075

3.083.085

3.09

1 2 3 4 5 6 7 8 9 10R

e(ε r

)Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

1.992

2.012.022.032.042.052.062.072.08

1 2 3 4 5 6 7 8 9 10

Re(μ r

)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

85

Figure 4.9. Simulated error analysis results for variations in TRL calibration kits. Real parts of permittivity (top) and permeability (bottom) with standard error analysis (ε'r =3 and μ'r=2 are the exact values) and 9.541% for ε'r and μ'r, respectively. Compared to the previous results, the maximum

variations within the 95% confidence limits for both ε'r and μ'r are larger. These results

demonstrate the importance of TRL calibration in our proposed method.

Finally, we need to consider all possible variations in both DUTs and TRL calibration

kits. The simulations and the standard error analysis results are shown in Figure 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

2.983

3.023.043.063.08

3.13.123.143.163.18

1 2 3 4 5 6 7 8 9 10R

e(ε r

)Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

1.921.951.982.012.042.07

2.12.132.162.192.222.25

1 2 3 4 5 6 7 8 9 10

Re(μ r

)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

86

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) 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,

2.95

3

3.05

3.1

3.15

3.2

3.25

3.3

1 2 3 4 5 6 7 8 9 10R

e(ε r

)Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

1 2 3 4 5 6 7 8 9 10

Re(μ r

)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

87

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) 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,

0.015

0.016

0.017

0.018

0.019

0.02

1 2 3 4 5 6 7 8 9 10Im

(εr)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

0.008

0.009

0.01

0.011

0.012

0.013

1 2 3 4 5 6 7 8 9 10

Im(μ

r)

Frequency (GHz)

Avg. Upper 95% limit Lower 95% limit

88

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

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.

(a)

(b)

Error Box A Error Box BDUT

89

(a) (b)

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) 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.

4 5 6 7 8 9 10 11 12 13 140

1

2

3

4

5

6

7

8

Frequency (GHz)

Re(

r) an

d Re

(r)

Re( r)

Re( r)

4 5 6 7 8 9 10 11 12 13 14

x 109

0

1

2

3

4

5

6

7

Frequency (GHz)

Re(

r) and

Re(

r)

Re( r)

Re( r)

Re(ε r)

and

Re(μ r

)

Re(ε r)

and

Re(μ r

)

Frequency (GHz) Frequency (GHz)

90

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

4 5 6 7 8 9 10 11 12 13 140.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Frequency (GHz)

tan

91

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.

92

Chapter 5

NEW ON-WAFER CHARACTERIZATION METHOD FOR MAGNETIC-DIELECTRIC MATERIALS USING T-RESONATORS

5.1 Introduction As we discussed in Chapter 2, the T-resonator method is commonly used for the

characterization of dielectric materials [33, 57]. The main advantage of the T-resonator

method is that it provides very accurate results for material properties based on the

measurement of resonant frequencies. For magnetic-dielectric materials, only the

effective refractive index, eff eff , can be determined by measuring resonant frequencies.

This is the main reason that non-resonant methods, such as the transmission-line method,

are mainly used for the characterization of magnetic-dielectric material. However, with

non-resonant methods, it is necessary to determine both the characteristic impedance and

the effective refractive index to find the relative values of 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

93

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

94

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).

1

1

2 01 01

1201 01

12 2

12 2

feed

feed

l stub stub

Tlstub stub

Y Z Y ZeR

Y Z Y Ze

(5.1)

2

2

2 02 02

2202 02

12 2

12 2

feed

feed

l stub stub

Tlstub stub

Y Z Y ZeR

Y Z Y Ze

(5.2)

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.

Z01

Z01 Lstub

lfeed

Z02

Z01 Lstub

lfeed

95

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:

101 12

202 12

T

T

Z RrZ R

(5.3)

where 112TR and 2

12TR 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]

22 1

1 1 2

effa

a eff

CrC

(5.4)

12 2

1 2 1

effa

a eff

CrC

(5.5)

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.

96

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

97

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

(a) (b)

Figure 5.2. Simulated results of two T-resonators which have same T-stub length and width, but different feed line widths. (a) S21 (dB) in overall frequency range and (b) S21 (dB) for region near the first resonant frequency

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-60

-50

-40

-30

-20

-10

0

Frequency (Hz)

S12

(dB

)

T-resonator #1T-resonator #2

3.6 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.7

x 109

-80

-70

-60

-50

-40

-30

-20

Frequency (Hz)

S12

(dB

) Com

paris

on

T-resonator #1T-resonator #2

98

became larger for higher orders, they were still small enough to use our method.

Now, we need to determine the characteristic impedance ratio, r, using equation (5.3).

Since the first resonant frequencies of T-resonators 1 and 2 are slightly different, the

average of the two resonant frequencies was used. Figure 5.3 shows the characteristic

impedance ratio, r, near the first resonant frequency, which is shown by the solid red line.

According to Figure 5.3, the characteristic impedance ratio, r, at the resonant frequency

contains a singularity, since the T-resonators are not ideal T-resonators. Thus, the value

of r at the resonant frequency shows a very sharp peak. To determine the characteristic

impedance ratio, r, we must eliminate the singularity in r at the resonant point using

regularization. The R12 values of T-resonators 1 and 2 can be approximated as shown in

equations (5.6) and (5.7).

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

3.6 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.7

x 109

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency (Hz)

Cha

ract

eris

tic Im

peda

nce

Rat

io, r

Original value of rRegularized value of r

99

21 012 1 2 0 3 0

0

T aR a a x x a x xx x

(5.6)

22 012 1 2 0 3 0

0

T bR b b x x b x xx x

(5.7)

Thus, using equations (5.6) and (5.7), the characteristic impedance ratio, r, can be

expressed as shown in equation (5.8).

2 320 1 0 2 0 3 012

2 3112 0 1 0 2 0 3 0

T

T

b b x x b x x b x xRrR a a x x a x x a x x

(5.8)

The regularized value of the characteristic impedance ratio, r, is shown in Figure 5.3

by the dashed blue line. Equation (5.8) can be used to eliminate the singularity near the

first resonant frequency. The value of r at the first resonant frequency is 1.161, and this

value is very close to the theoretical value. Now, we can determine both 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.

100

f (GHz) εr Relative Error

(%) μr Relative Error

(%) 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

101

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.

eff x stub endL L L L (5.9)

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

1 1n x stub endeff

ncf L L Ln

(5.10)

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

Lstub

Lend

Lx

Feed line center

Leff

Open-end effect

102

2 2n x stub endeff

ncf L L Ln

(5.11)

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.

2 2 1 1

1 2

n stub n stubx end

n n

f L f LL Lf f

(5.12)

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-

103

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.

f (GHz) εr Relative Error

(%) μr Relative Error

(%) 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.

104

(a) (b)

Figure 5.5. Two different microstrip T-resonators for the measurements. T-resonator (a) and (b) have T-stub length of 15mm and width of 500μm while the feed line widths are 500μm and 400μm for T-resonator (a) and (b), respectively. The T-resonator structures had the same T-stub length of 15 mm, but the feed-line

widths of T-resonators 1 and 2 were 500 μm and 400 μm, respectively. Note that the

coplanar waveguide-to-microstrip transitions for each T-resonator were different because

the feed lines for the two T-resonators were different. Therefore, it is necessary to build

different sets of TRL calibration kits for the different T-resonators on the same wafer.

The coplanar waveguide-to-microstrip transitions used in this measurement were

discussed in Chapter 2. Figure 5.6 shows the measured 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

105

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.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1010

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

T-Resonator #1T-Resonator #2

7.8 7.9 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

x 109

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

T-Resonator #1T-Resonator #2

T1 : 8.324 GHzT2 : 8.318 GHz

1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43

x 1010

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

T-Resonator #1T-Resonator #2

T1 : 13.786 GHzT2 : 13.778 GHz

1.88 1.89 1.9 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98

x 1010

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

T-Resonator #1T-Resonator #2

T1 : 19.278 GHzT2 : 19.218 GHz

2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3

x 109

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Frequency (Hz)

S21

(dB

)

T-Resonator #1T-Resonator #2

T1 : 2.768 GHzT2 : 2.768 GHz

(a) 1st resonant frequency (b) 3rd resonant frequency

(c) 5th resonant frequency (d) 7th resonant frequency

106

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

f (GHz) ε'r Relative Error

(%) μ'r Relative Error

(%) 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

107

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

108

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.

109

Chapter 6

ON-WAFER ELECTROMAGNETIC CHARACTERIZATION METHOD FOR ANISOTROPIC MATERIALS

6.1. Introduction Recent progress in engineered materials is providing new materials that have unique

electromagnetic behaviors, such as anisotropies in the permittivity ( ) and permeability

( ). The accurate measurement of the electromagnetic properties of these new materials

is crucial to access whether they can be used in a variety of applications. Furthermore,

on-wafer characterization of thin-film materials is important since new electronic circuits

use new and complex materials in the form of thin-film materials on wafers at the present

time. Thus, accurate on-wafer characterization of anisotropic material properties is very

important.

There are several different methods to characterize anisotropic materials, and those

that are commonly used include the free space method, the waveguide method, and the

transmission/reflection method [10-12]. These conventional methods, however, are not

suitable for characterizing anisotropic thin-film materials because they are too thin

(typically, micron range of thickness) to measure in a certain direction. In addition, it is

difficult to measure small areas using the conventional measurement methods. Thus, on-

110

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

111

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, , 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, , 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,

0 00 00 0

x

y

z

(6.1)

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 , and the effective height, He, is for the propagation along the y-

axis [74]. The effective permittivity of an anisotropic substrate is given by [40]:

x z /x zH

112

(a) (b)

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

1/21 11 12

2 2g g a ee

effa

C HHW C H

(6.2)

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

z

x

W

Hu

v

W

Heεg

Z-plane W-plane

113

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:

1/2,, , ,

,

1 11 12

2 2a e xg x g x e x

eff xa

C HHW C H

(6.3)

1/2,, , ,

,

1 11 12

2 2a e yg y g y e y

eff ya

C HHW C H

(6.4)

Equations (6.3) and (6.4) are the same as the effective permittivity of the uniaxial

anisotropic material. Therefore, for the propagation along the x-axis, we can consider εg,x

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

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)

y z /y zH

x

y

Optical axes of biaxial anisotropic material

x

y

Optical axes of biaxial anisotropic material

114

can consider εg,y of and He,y of 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

(a) (b)

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

x z /x zH

012345678910

1 2 3 4 5 6 7 8 9 10

ε x an

d ε z

Frequency (GHz)

εx εz012345678910

1 2 3 4 5 6 7 8 9 10

ε x, ε

y, an

d ε z

Frequency (GHz)

εx εy εz

115

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]:

cos sin 0 1 0 0 cos sin cos sin sinsin cos 0 0 cos sin sin cos cos cos sin

0 0 1 0 sin cos 0 sin cosU

(6.5)

116

Figure 6.4. The principal axes of the permittivity tensor (x´y´z´ system) and the measurement coordinate system (xyz system) The permittivity tensor in (6.1), which can be transformed with the transformation

matrix U and the transformed permittivity tensor, , (see Appendix C), is given by:

'xx xy xz

Tyx yy yz

zx zy zz

U U

(6.6)

where

2 2 2 2 2

2 2

2 2 2 2 2

cos sin cos sin sin

sin cos sin cos cos sin cos sin

sin sin cos sin sin cos

sin cos cos cos sin

cos sin cos cos sin cos

xx x y z

xy x y z

xz y z

yx xy

yy x y z

yz y z

zx x

2 2sin cos

z

zy yz

zz y z

(6.7)

x

y

z

θ

x'

y'

z'

117

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.

(a) (b) Figure 6.5. (a) Top-view of microstrip transmission line with misalignment angle θ between in-plane optical axis and propagation direction, (b) cross sectional view of microstrip line with misalignment angle ϕ between the principal axis and x-y plane (x, y, and z are the geometrical axes of microstrip lines; and x´, y´, and z´ are the optical axes of anisotropic thin-film substrate

x

x'

yy'

θx

z

H

xxx

x'

z'

118

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

respectively.

22

, , 2 and yy zz yzg x yy zz yz e x

zz

H H

(6.8)

22

, , 2 and xx zz xzg y xx zz xz e y

zz

H H

(6.9)

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, , is given by:

xx xy xza T

yx yy yz

zx zy zz

U U

(6.10)

where

cos sin 0sin cos 0

0 0 1U

(6.11)

Therefore, the matrix element in (6.10) can be expressed in terms of the matrix elements

in (6.6) and the known angle of α from the x-axis. Equation (6.12) are the matrix

elements in (6.10) and each of the permittivity tensor elements in also can be

expressed in terms of εx, εy, εz, θ, ϕ, and α.

119

2 2

2 2

2 2

cos sin 2 sin cos

sin cos sin cos

cos sin

sin cos 2 sin cos

sin cos

xx xx yy xy

xy xx yy xy

xz xz yz

yx xy

yy xx yy xy

yz xz yz

zx xz

zy yz

zz zz

(6.12)

Let us consider microstrip lines with different propagation directions, one for the

direction of α and another for the direction of α + 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.

22

, , 2 and yy zz yzg yy zz yz e

zz

H H

(6.13)

22

2, 90 , 90 and xx zz xz

xx zz xzg ezz

H H 2, 90 and x H Hxx ,,and H

(6.14)

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 , such as εx, εy, and εz, and the misalignment angles, such as θ and

120

ϕ, is better than finding matrix elements in . 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

2 2, ,g x g y x zz y z (6.15)

2 2, , cos2g y g x x zz y z

(6.16)

2 2, , 90 sin 2g g x zz y z

(6.17)

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

angle, θ.

2 2, , 9012 2

, ,

1 tan2

g g

g y g x (6.18)

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

given by:

2 2, , 2 2

, ,1

2 cosg y g x

x g x g yzz

(6.19)

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:

2 2, ,y z g x g y x zz (6.20)

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).

121

2 2,

2

cossin

g x y zy z zz

x (6.21)

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).

1cos zz y

z y (6.22)

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].

122

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:

9.4 0 00 9.4 00 0 11.6

C

(6.23)

The permittivity tensor of the C-plane sapphire wafer has the same form as the

uniaxial anisotropic permittivity tensor. The permittivity tensor of an R-plane sapphire

wafer can be calculated by the rotation of C . The angle between the c-axis and the

normal to the R-plane is equal to 57.6°, as shown in Figure 6.6 [79]. Thus, the

permittivity tensor of an R-plane sapphire wafer can be calculated easily. Equation (6.24)

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)

c

a

b

C-plane

x′

y′ z′ 57.6°

x

y

θ

123

gives the permittivity tensor of an R-plane sapphire wafer.

9.4 0 00 10.97 0.990 0.99 10.03

R

(6.24)

Although, we know the permittivity tensor of an R-plane sapphire theoretically, it is

impossible to build test structures on the wafer that are perfectly aligned with the optical

axes and the measurement axes. Thus, an in-plane misalignment angle exists between the

optical axes and the measurement axes. As a result, the permittivity tensor of an R-plan

sapphire wafer will be a full matrix with non-zero off-diagonal elements. However, all

the values can be determined with our anisotropic characterization method.

Before we discuss the sapphire wafer measurements, we will show the results of the

R-plan sapphire wafer simulation. In the simulation, we assigned the in-plane

misalignment angle, θ, to be 25°. Therefore, the permittivity tensor of the R-plane

sapphire can be considered as a full matrix with non-zero, off-diagonal elements and

equation (6.24) can be expressed as:

25

9.6801 0.6007 0.42060.6007 10.6882 0.9021

0.4206 0.9021 10.0316R (6.25)

In the simulation, we used microstrip lines with the same geometries as in the previous

simulation; however, we needed microstrip lines with different propagation directions.

Figure 6.7 shows the simulated results for the characterization of the R-plane sapphire

wafer with an in-plane misalignment angle of θ = 25°. The maximum relative errors for

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

124

(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

5

6

7

8

9

10

11

1 2 3 4 5 6 7 8 9 10

ε xx, ε y

y, an

d ε z

z

Frequency (GHz)

εxx εyy εzz-4

-3

-2

-1

0

1

2

3

1 2 3 4 5 6 7 8 9 10

ε xy, ε x

z, an

d ε y

z

Frequency (GHz)

εxy εxz εyz

125

(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

126

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

7

8

9

10

11

12

13

14

15

3 4 5 6 7 8 9 10 11 12 13 14 15 16

ε xan

dε z

Frequency (GHz)

εx εz

127

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

7

8

9

10

11

12

13

14

3 4 5 6 7 8 9 10 11 12 13 14 15 16

ε x, ε

y an

dε z

Frequency (GHz)

εx εy εz

128

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.

129

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

130

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

131

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.

132

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

133

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

134

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 and . In this study, only the

dielectric anisotropic material characterization was considered. To approach the

anisotropic material characterization of and , 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.

135

Appendix A

CRYSTAL SYSTEM (BRAVAIS LATTICE)

Classification System Bravais Lattice Number of

independent coefficient

Tensor form

Isotropic (Anaxial) Cubic

1 0 0

0 00 0

Uniaxial

Tetragonal

2 1

1

3

0 00 00 0

Hexagonal

Rhombohedral

Continued

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

a a

a

a a

c

a

c

a a

a α

β

γ

α=β=γ≠90°

136

Table A. 1. continued

Biaxial

Orthorhombic

3

1

2

3

0 00 00 0

Monoclinic

4 11 12

12 22

33

00

0 0

Triclinic

6 11 12 13

12 22 23

13 23 33

a b

c

a≠b≠c

α β

γ

α≠90°, β=γ=90°

α β

γ

α≠β≠γ≠90°

137

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]:

tanh2wz j d w (B.1)

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).

138

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

r

s ss s (B.2)

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:

' ''

g a sq

g (B.3)

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

1 2 3effC C C C (B.4)

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

and the parallel plate areas.

Im(z)

Re(z)

z-plane

↑

→

Im(w)

Re(w)

w-plane

,

b=π

0 a´ g´

πs'

Im(w)

Re(w)

w-plane

b=π

0 a´ g´

s

(a) (b)

(c) (d)

Im(w)

Re(w)

w-plane

b=π

0 a´ g´

π(s'-s'') πs''

139

0 0 0 0' ' ' 'eff r rg a s s g a (B.5)

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

1' 'r

r eff

g a s (B.6)

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

1

' ' 'eff

r eff

g a s a s (B.7)

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

relative dielectric constants.

1' '' 1

eff

r

g a sq

g (B.8)

Thus, the effective dielectric constant is:

1 1eff rq (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:

1 2 3

1 1 1 1

effL L L L (B.10)

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

and the parallel area parameters.

0 0 0 0

' ' ' '

eff r r

g a s s g a (B.11)

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

140

1/ 1' '

1/ 1/r

r eff

g a s (B.12)

1/ 1' ' '

1/ 1/eff

r eff

g a s a s (B.13)

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

permeabilities.

1/ 1' '' 1/ 1

eff

r

g a sq

g (B.14)

Finally, the effective permeability is given by:

1 11 1eff r

q (B.15)

141

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 is the

rotation angle of the y-axis. Then, the rotation transformation matrix U is given by [71,

72]:

cos sin 0 cos 0 sin cos cos sin cos sinsin cos 0 0 1 0 sin cos cos sin sin

0 0 1 sin 0 cos sin 0 cosU (D.1)

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

transformed permittivity tensor is:

'cos cos sin cos sin 0 0 cos cos sin cos sinsin cos cos sin sin 0 0 sin cos 0

sin 0 cos 0 0 cos sin sin sin cos

T

x

y

z

U U

(D.2)

First, we consider the in-plane (xy-plane) misalignment angle to be θ only. In this case,

the rotation angle is zero, and the is given by: '

142

2 2

2 2

cos sin sin cos 0 0' sin cos sin cos 0 0

0 00 0

x y x yxx xy

x y x y yx yy

zzz

(D.3)

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

2 2

2 2

cos sin 0 sin cos 0' 0 0 0 0

sin cos 0 sin cos 0

x z x z xx xz

y yy

x z x z zx zz

(D.4)

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.

143

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