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The analysis of dielectric loss in co-planar waveguide structures using generalized transverse resonance

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The analysis of dielectric loss in co-planar waveguide structures using generalized transverse resonance
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Culver, James William
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Silicon
Permittivity
Numerical electromagnetic
Bcb
Package effects
Dissertations, Academic -- Electrical Engineering -- Doctoral -- USF   ( lcsh )
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Summary:
ABSTRACT: There are several methods for the full-wave characterization of waveguide discontinuities; Finite Element Method (FEM), Finite Difference Technique (FDT), and Method of Moments (MoM) are popular. However, these methods are not easily applied when studying the modal anatomy of a discontinuity. Other full-wave techniques are better suited. This dissertation discusses the formulation of a technique known as Generalized Transverse Resonance (GTR), which is a subset of Method of Moments. Generalized Transverse Resonance is a hybrid method combining the Transverse Resonance Method (TRM) with the Mode Matching Technique (MMT). The understanding of the generalized transverse resonance method starts with a discussion of Longitudinal Section Waves and from this derives the transverse resonance method for layered media para1lel to the wave propagation. It is shown that Maxwells equations can be represented as a mode function and voltage or current.This representation is used to reduce to the problem of merging the TRM and MMT into the GTR method by using network theory. The propagation constant is found by solving the wave equation, as an eigenvalue problem, subject to the boundary conditions. Discussed is the relative convergence phenomenon followed by the optimization strategy. Once the propagation constant is found, the cross sectional fields can be solved and from the fields the characteristic impedance is found. Theoretical data is compared to measure data to show the accuracy of the GTR method. Presented is an understanding of the propagation characteristics of a CPW transmission line in proximity with high and low loss silicon. This data will show the loss and propagation characteristics for four CPW structures using two separate silicon lids at six different heights above the transmission line. Two modes have been clearly identified and will be explained.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2005.
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Includes bibliographical references.
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by James William Culver.
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Document formatted into pages; contains 125 pages.
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Includes vita.

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ABSTRACT: There are several methods for the full-wave characterization of waveguide discontinuities; Finite Element Method (FEM), Finite Difference Technique (FDT), and Method of Moments (MoM) are popular. However, these methods are not easily applied when studying the modal anatomy of a discontinuity. Other full-wave techniques are better suited. This dissertation discusses the formulation of a technique known as Generalized Transverse Resonance (GTR), which is a subset of Method of Moments. Generalized Transverse Resonance is a hybrid method combining the Transverse Resonance Method (TRM) with the Mode Matching Technique (MMT). The understanding of the generalized transverse resonance method starts with a discussion of Longitudinal Section Waves and from this derives the transverse resonance method for layered media para1lel to the wave propagation. It is shown that Maxwells equations can be represented as a mode function and voltage or current.This representation is used to reduce to the problem of merging the TRM and MMT into the GTR method by using network theory. The propagation constant is found by solving the wave equation, as an eigenvalue problem, subject to the boundary conditions. Discussed is the relative convergence phenomenon followed by the optimization strategy. Once the propagation constant is found, the cross sectional fields can be solved and from the fields the characteristic impedance is found. Theoretical data is compared to measure data to show the accuracy of the GTR method. Presented is an understanding of the propagation characteristics of a CPW transmission line in proximity with high and low loss silicon. This data will show the loss and propagation characteristics for four CPW structures using two separate silicon lids at six different heights above the transmission line. Two modes have been clearly identified and will be explained.
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The Analysis of Dielectric Loss in Co-Plana r Waveguide Structures Using Generalized Transverse Resonance by James William Culver A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Thomas M. Weller, Ph.D. Glen Besterfield, Ph.D. Lawrence P. Dunleavy, Ph.D. Robert W. Flynn, Ph.D. Arthur D. Snider, Ph.D. Date of Approval: March 24, 2005 Keywords: Silicon, Permittivity, Numerical Electromagnetic, BCB, Package Effects Copyright 2005, James William Culver

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Dedication In loving memory of Dr. Clair H. Culver (1898 1983) SP4 Michael W. Culver (1964-1988)

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Acknowledgments Throughout this project there have been several people who have made this possible. I could not have star ted the pursuit of a Ph.D. wit hout the help of Ed Foreman, Tom Miles, Matt Smith and Miles Larson who supported me through the corporate bureaucracy. I want to thank them for their continued support. I want to express my appreciation to Dr. Tom Weller whose patience and guidance made this dissertation possible. In ad dition, I am grateful to Dr. A. Dave Snider and Dr. Larry Dunleavy for their comments a nd valuable suggestions. A special thanks goes to Balaji Lakshminarayanan and Sam Baylis who helped me with the fabrication and testing. It would be wrong not to mention the peopl e behind the scenes. I wish to thank Ed Grimes and Tim Erman for their support and encouragement. Finall y, I want to extend my deepest thanks and appreciation to Trac y, my wife. Her support, encouragement and friendship helped buffer the pressures of managing a full time job while earning this degree.

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Table of Contents List of Tables iii List of Figures iv Abstract viii Chapter One Introduction 1 Chapter Two Problem Formulation 9 2.1 Longitudinal Section Waves 9 2.2 Mode Functions 14 2.3 Transverse Resonance 18 2.4 Mode Matching 19 2.5 Generalized Transverse Resonance 22 2.6 Summary 31 Chapter Three Implementation of the GTR Algorithm 32 3.1 Relative Convergence 32 3.2 Numerical Technique For Solving the Transverse Resonance Condition 35 3.3 Field Calculation 38 3.4 Characteristic Impedance 41 3.5 Summary 43 Chapter Four Superstrate Layers on Coplanar Waveguides 45 4.1 Theory 47 i

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4.2 CPW and Lid Fabrication 48 4.3 Measurement Setup 49 4.4 Dielectric and Metal Losses 51 4.5 GTR Compared to HFSS 54 4.6 Measured Data Using the 2000 ohm-cm lid 57 4.7 Measured Data Using the 10 ohm-cm lid 63 4.8 Frequency Dependent Permittivity 69 4.9 Characteristic Impedance 76 4.10 Summary 79 Chapter Five Characteristics of MTIS Transmission Lines 81 5.1 Theory 83 5.2 MTIS CPW Fabrication 84 5.3 Measured Data Using the 25 ohm-cm and 10 ohm-cm Substrates 86 5.4 Measured Data Using the 0.4 ohm-cm Substrate 92 5.5 Summary 95 Chapter Six Summary and Recommendations 97 6.1 Data Summary 98 6.2 Recommendations For Further Work 99 References 102 Appendices 106 Appendix A: Metal Loss 107 Appendix B: GTR Compared to HFSS 109 About the Author End Page ii

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List of Tables Table 1. CPW line dimensions. 48 Table 2. MTIS CPW line dimensions. 85 Table 3. 2000 ohm-cm lid with no metal loss. 109 Table 4. 10 ohm-cm lid with no metal loss. 110 Table 5. 2000 ohm-cm lid with metal loss. 111 Table 6. 10 ohm-cm lid with metal loss. 112 iii

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List of Figures Figure 1. Network representation of GTR. 7 Figure 2. Layered dielectric structure normal mode set. 12 Figure 3. Layered dielectric structure LS mode set. 13 Figure 4. Transmission line representation for the layered dielectric structure. 18 Figure 5. Step discontinuity. 20 Figure 6. CPW cross section. 24 Figure 7. CPW example represented as a network. 25 Figure 8. Relative convergence of the odd mode for the CPW example (odd 34 mode is defined by the following E-field orientation in the slot ). Figure 9. Relative convergence of the even mode for the CPW example (even 35 mode is defined by the following E-field orientation in the slot ). Figure 10. Pictorial showing the behavior difference between using the 37 determinant and minimum singular value. Figure 11. Layout showing the CPW structures used in this study. 49 Figure 12. Diagram showing the fixture for suspending the lid; topand side-view. 50 Figure 13. Comparison showing the percent difference between the real part (phase) 54 of the propagation constant found using GTR and HFSS. Figure 14. Comparison showing the percent difference between the imaginary part 55 (loss) of the propagation constant found using GTR and HFSS. Figure 15. Comparison showing the measured loss data to simulated loss data for 56 CPW1 with a 2000 ohm-cm lid at 40 GHz. iv

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Figure 16. Comparison showing the measured loss data to simulated loss data for 56 CPW1 with a 10 ohm-lid at 40 GHz. Figure 17. Loss for CPW1 with 2000 ohm-cm lid. 58 Figure 18. Loss for CPW2 with 2000 ohm-cm lid. 58 Figure 19. Loss for CPW3 with 2000 ohm-cm lid. 59 Figure 20. Loss for CPW4 with 2000 ohm-cm lid. 59 Figure 21. Normalized phase constant for CPW1 with 2000 ohm-cm lid. 60 Figure 22. Normalized phase constant for CPW2 with 2000 ohm-cm lid. 61 Figure 23. Normalized phase constant for CPW3 with 2000 ohm-cm lid. 61 Figure 24. Normalized phase constant for CPW4 with 2000 ohm-cm lid. 62 Figure 25. Effective permittivity using the 2000 ohm-cm lid at a height of 5 um. 63 Figure 26. Effective permittivity using the 10 ohm-cm lid at a height of 5 um. 64 Figure 27. Loss for CPW1 with 10 ohm-cm lid. 65 Figure 28. Loss for CPW2 with 10 ohm-cm lid. 65 Figure 29. Loss for CPW3 with 10 ohm-cm lid. 66 Figure 30. Loss for CPW4 with 10 ohm-cm lid. 66 Figure 31. Normalized phase constant for CPW1 with 10 ohm-cm lid. 67 Figure 32. Normalized phase constant for CPW2 with 10 ohm-cm lid. 68 Figure 33. Normalized phase constant for CPW3 with 10 ohm-cm lid. 68 Figure 34. Normalized phase constant for CPW4 with 10 ohm-cm lid. 69 Figure 35. Real part of the lid permittivity for CPW3. 70 Figure 36. Imaginary part of the lid effective permittivity for CPW3. 71 Figure 37. Real part of the lid permittivity for CPW4. 71 v

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Figure 38. Imaginary part of the lid permittivity for CPW4. 72 Figure 39. Imaginary part of the lid permittivity for CPW1 and CPW2. 73 Figure 40. Corrected loss data for CPW3. 74 Figure 41. Corrected loss data for CPW4. 74 Figure 42. Corrected normalized phase for CPW3. 75 Figure 43. Corrected normalized phase for CPW4. 76 Figure 44. Characteristic impedance as a function of lid height at 4 GHz (Solid lines 77 2000 ohm-cm lid; dashed lines 10 ohm-cm lid). Figure 45. Characteristic impedance as a function of lid height at 40 GHz (Solid lines 78 2000 ohm-cm lid; dashed lines 10 ohm-cm lid). Figure 46. Difference in measured normalized phase between the 2000 and 10 ohm-cm 79 lid for the four CPW structures at 40 GHz. Figure 47. Mode regions with transition frequencies for 0.4, 10, and 25 ohm-cm silicon. 86 Figure 48. Real part of the substrate permittivity for MTIS1. 87 Figure 49. Imaginary part of the substrate permittivity for MTIS1. 88 Figure 50. Real part of the substrate permittivity for MTIS2. 88 Figure 51. Imaginary part of the substrate permittivity for MTIS2. 89 Figure 52. Loss data for MTIS1 on 25 ohm-cm silicon. 90 Figure 53. Loss data for MTIS2 on 25 ohm-cm silicon. 90 Figure 54. Loss data for MTIS1 on 10 ohm-cm silicon. 91 Figure 55. Loss data for MTIS2 on 10 ohm-cm silicon. 91 Figure 56. Pictorial of the skin depth mode. 92 Figure 57. One-half the magnetic field skin depth for a 0.4 ohm-cm substrate. 93 Figure 58. Loss data for MTIS1 on 0.4 ohm-cm silicon. 94 vi

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Figure 59. Loss data for MTIS2 on 0.4 ohm-cm silicon. 95 vii

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The Analysis of Dielectric Loss in Co-Planar Waveguide Structures Using Generalized Transverse Resonance James William Culver ABSTRACT There are several methods for the full-wave characterization of waveguide discontinuities; Finite Element Method (FEM), Finite Difference Technique (FDT), and Method of Moments (MoM) are popular. However, these methods are not easily applied when studying the modal anatomy of a discontinuity. Other full-wave techniques are better suited. This dissertation discusses the formulation of a technique known as Generalized Transverse Resonance (GTR), which is a subset of Method of Moments. Generalized Transverse Resonance is a hybrid method combining the Transverse Resonance Method (TRM) with the Mode Matching Technique (MMT). The understanding of the generalized transverse resonance method starts with a discussion of Longitudinal Section Waves and from this derives the transverse resonance method for layered media para1lel to the wave propagation. It is shown that Maxwells equations can be represented as a mode function and voltage or current. This representation is used to reduce to the problem of merging the TRM and MMT into the GTR method by using network theory. The propagation constant is found by solving the wave equation, as an eigenvalue problem, subject to the boundary conditions. Discussed viii

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is the relative convergence phenomenon followed by the optimization strategy. Once the propagation constant is found, the cross sectional fields can be solved and from the fields the characteristic impedance is found. Theoretical data is compared to measure data to show the accuracy of the GTR method. Presented is an understanding of the propagation characteristics of a CPW transmission line in proximity with high and low loss silicon. This data will show the loss and propagation characteristics for four CPW structures using two separate silicon lids at six different heights above the transmission line. Two modes have been clearly identified and will be explained. Also presented is a comparison between measured data and simulated data for two CPW structures fabricated on a layered BCB/silicon substrate. Three silicon resistivities were used which clearly show the two modes from the proximity experiment, in addition to a third mode. This third mode is identified and explained. ix

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Chapter One Introduction The trend in the RFIC communication industry is toward higher levels of integration, high frequency transceiver, analog IF, and digital circuits all on the same substrate. However, high frequency circuits require high resistivity substrates while digital CMOS circuits are typically fabricated on low resistivity substrates. For digital ICs a background dopant is typically added to the silicon to help with the purification. This background dopant lowers the resistivity of the silicon which can create a problem of high attenuation in RF transmission lines. Real world problems were the motivation for this research. The integration of high frequency circuits with digital circuits can be achieved in several ways. Flip-chip packaging is one example where the digital integrated circuit is flipped and suspended above a high frequency integrated circuit acting as a host substrate. The typical separation between the two substrates is 50 um and the close proximity of the lossy silicon affects the propagation characteristics of the transmission lines. Another integration technique is the use of a layer of low loss insulating dielectric between the transmission line and lossy substrate. The insulating dielectric allows low loss interconnect transmission lines on the same substrate as digital circuits. Two Co-Planar Waveguide (CPW) transmission line 1

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structures were studied using a full wave numerical electromagnetic program based on Generalized Transverse Resonance. The first study presents data showing the loss and propagation characteristics for four CPW structures when a silicon lid is present. Two separate resistivities for the silicon lid at six heights above the transmission line were looked at. This dissertation represents the first serious work on characterizing and understanding the proximity effects of silicon on the propagation characteristics of CPW transmission lines. Prior art does not attempt to understand or optimize the effect of stacked silicon layers at multiple heights or for different resistivities. The data presented here suggests the substrate resistivity and dielectric constant are dependent on frequency. This behavior occurs for transmission line structures with deep penetration of the fields into the substrate which is a function of line geometry. Possible causes for the frequency dependent material properties could be in the electromagnetic interaction with the material. It is possible that relaxations of the charges in the substrate are dependent on signal power, interaction with other charges, and/or tensor properties. This study extracted the material properties from the measured data and presents a frequency dependent complex dielectric constant whose imaginary part is a function of lid height; this lid height dependency is non-physical but necessary to accurately represent the measured data in the full-wave simulations, pointing to a need for further examination of the formulation governing the dielectric property behavior in these material systems. It can be theorized that the geometry of a lid over a CPW transmission line can be fabricated using 3-dimensional micromachining processes to give a target complex dielectric constant over a frequency band. This has applications 2

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in phase shifters, filters and chip scale antenna elements using 3-dimensional packaging such as flip-chip or bulk silicon micromachining. The second study further explored loss on low resistivity silicon substrates by looking at CPW transmission lines fabricated on metal-thick-insulating-semiconductor substrate. This dissertation advanced the understanding of using thick BCB insulators for low loss transmission lines on lossy silicon. The insertion loss data for the CPW transmission lines show that at frequencies below 5 GHz there was less attenuation of the signal as the silicon resistivity decreased from 10 ohm-cm to 0.4 ohm-cm. This is due to two different mode structures for each of the resistivities. For the 10 ohm-cm substrate a slow wave mode exists. The lossy silicon acts as a ground plane, shorting out the electric field while the magnetic field continues to penetrate the silicon. This mode is termed the slow-wave mode because boundary conditions force the signal to slow down to sustain TEM propagation. The 0.4 ohm-cm substrate exhibits the skin depth mode. The skin depth mode appears when the substrate conductivity is large and the frequency of the signal is low. The behavior is similar to the slow-wave mode in that the electric field is shorted at the silicon/BCB boundary. However, the magnetic field as it spreads into the silicon is bounded by the skin effect, effectively reducing the thickness of the silicon layer. This results in confining the field mostly to the low loss BCB layer. Decreasing the substrate resistivity to the point where the silicon exhibits the skin depth mode has application in creating low loss RF and high speed digital interconnects on CMOS grade silicon. 3

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Integral to both of these studies was the numerical simulation code developed specifically to study these types of CPW structures. There are several methods for the full-wave characterization of waveguide discontinuities; Finite Element Method (FEM), Finite Difference Technique (FDT), and Method of Moments (MoM) are popular. However, these methods are not easily applied when studying the modal anatomy of a discontinuity. Other full-wave techniques are better suited. This chapter introduces the formulation of a technique known as Generalized Transverse Resonance (GTR), which is a subset of Method of Moments. Generalized Transverse Resonance is a hybrid method combining the Transverse Resonance Method (TRM) with the Mode Matching Technique (MMT). Transverse Resonance was first used to design and characterize power flow in ridged waveguide [1]. The technique is used when the mode set for the boundary value problem is perpendicular to the actual power flow and normal to any discontinuities. In the transverse direction these discontinuities are seen as layered structures and the boundary conditions are satisfied such that the fields are uncoupled and each mode is independent. The individual modes are each characterized by a unique propagation constant and characteristic impedance and modeled using a one-dimensional wave equation. From a mathematical point of view, this is equivalent to saying the modes are orthogonal and the propagation constants are linearly dependent, knowing one propagation constant gives all because they are multiples of This orthogonality leads to examination of the one-dimensional wave equation as a transmission line [2]. The physical interpretation for the transmission line analogy is seen by looking at the 4

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distributed lumped element circuit approximation. In this approximation, it is implied that there is no mutual coupling between distributed segments. From a field theory point of view this is equivalent to having no E or H component in the direction of propagation, which is a TEM wave. Transverse resonance uses the fact that modes will not couple when the transverse field components are parallel to a discontinuity such as a dielectric boundary. The boundary conditions for the transverse fields of each mode are independently satisfied. Multi-mode analysis is quick for layered media because, once one mode is found all other modes are known. However, the modes are no longer orthogonal when the discontinuity is a metallic step and in order to satisfy the boundary conditions, mode coupling occurs across the discontinuity [3]. The step can be analyzed by either approximating the discontinuity using a reactive lumped element approximation or using a reaction matrix derived from a mode matching procedure. For discontinuities such as steps, the TRM can be used by representing the discontinuity as an equivalent lumped element circuit. The characteristic equation can be written using transmission line theory (representing a single mode) cascaded with the equivalent circuit [4]. This simple and fast analysis yields good approximations of the propagation constants. However, it has limitations; 1) high order mode interactions are not accounted for, 2) the method is only as good as the equivalent circuit model, and 3) no information about the modal spectrum is available. 5

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A rigorous representation of the discontinuity using the MMT will remove these limitations. Formulation of the MMT starts with factoring the transverse electric and magnetic fields into a mode function and a mode voltage or current respectively, equations 1 and 2. )(),(zVyxeE (1) )(),(zIyxhH (2) where, e(x,y) is the E-field mode function, V(z) is the mode voltage, h(x,y) is the H-field mode function, and I(z) is the mode current. The mode function, which is a form function, is only dependent on the coordinates transverse to the direction of propagation. The mode voltages and currents are amplitude functions, which depend on the coordinate in the direction of propagation [5]. The fields on each side of the discontinuity are expanded into a truncated series of N terms. Matching the tangential fields while enforcing the boundary conditions develops the behavior across the discontinuity [6]. The name Mode Matching comes from the fact that each term in the expansion represents a mode. Equations 3 and 4 represent the fundamental mode matching equations, which are a system of linear equations that completely describe the junction discontinuity. NNVeeVee221111 (3) 6

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NNIhhIhh222112 (4) The MMT gives an N-port matrix, which accounts for mode interactions while the TRM gives a transmission line representation for each mode. When combined this leads to a multiport network allowing multiple modes to be accounted for as shown in Figure 1 [7]. Using the MMT to improve the model for the discontinuity leads to the generalized transverse resonance method. The formulation of this code lends itself to using network theory to formulate the problem geometry. The code was written in a module way to take advantage of this and allow very complex problems to be formulated using a CAD approach. This problem formulation philosophy is unique to this formulation of the generalized transverse resonance method. Transverse ResonanceTransverse ResonanceMode Matching Transverse ResonanceTransverse ResonanceMode Matching Figure 1. Network representation of GTR. This dissertation is organized into two main parts. The first part discusses the formulation of the Generalized Transverse Resonance code in chapters 2 and 3. Chapter 4 is the study of propagation characteristics of four CPW transmission line structures in 7

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proximity with high and low resistivity silicon. Two propagation modes in the substrate were identified. Also discussed is a frequency and geometry dependent complex dielectric constant for transmission line in proximity to low resistivity silicon. Chapter 5 discusses CPW transmission lines fabricated on metal-thick-insulating-semiconductor substrates. These structures exhibited 3 propagation modes in the substrate which were dependent on the substrate resistivity. A frequency and geometry dependent complex dielectric constant was also identified for two of the modes. The mechanism for this dependency is left for further research, but could be due to shortcomings in either the models used for the material parameters and/or the assumptions in the electromagnetic interactions with the material. 8

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Chapter Two Problem Formulation This chapter will discuss the formulation of the Generalized Transverse Resonance (GTR) method in detail. The chapter starts with a discussion of Longitudinal Section Waves and from this derives the transverse resonance method for layered media para1lel to the wave propagation. It is shown that Maxwells equations can be represented as a mode function and voltage or current. This representation is used to reduce to the problem of merging the Transverse Resonance Method and Mode Matching Technique into the GTR method by using network theory. 2.1 Longitudinal Section Waves For wave propagation parallel to the dielectric boundaries the coordinate system must be rotated such that the boundary conditions for the transverse fields are independently satisfied. To do this longitudinal section waves are used. Time varying electromagnetic fields are governed by Maxwells equations which are coupled first order linear differential equations. To solve for the EM fields in a specific region, an appropriate coordinate system and boundary conditions are chosen. In a homogeneous and source free region both the electric and magnetic fields are 9

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solenoidal, allowing the fields to be described using a vector potential, which are solutions to the Helmholtz equation. Equations 5 and 6 are the vector Helmholtz equation 022AkA (5) 022FkF (6) and describe a plane wave in terms of the electric and magnetic vector potentials, respectively. Both these equations are solved as an eigenvalue problem, which leads to a discrete (and infinite) set of solutions or modes. Each mode is a solution to the wave equation as well as the superposition of these modes [8]. Maxwells equations can be factored into a set of equations that relate the potential functions to rectangular components of the electric and magnetic fields. A normal Eand H-field set can be constructed by assuming vector potentials in the direction of propagation. Define the propagation to be in the x-direction then assume the vector potentials to be and xaA 0 F This definition forms the TM to x field set. ]222[1''kxjxE (7a) yxjEy2''1 (7b) zxjEz2''1 (7c) 10

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0''xH (7d) zHy'' (7e) yHz'' (7f) Similarly if and 0A xaF then the TE to x field set can be derived. 0'xE (8a) zEy' (8b) yEz' (8c) yxjHy2'1 (8d) ][1222'kxjHx (8e) zxjHz2'1 (8f) Where the single prime indicates TE and the double prime indicates TM. This mode set when applied to the structure shown in figure 2, requires a complex series of higher order modes to satisfy the boundary conditions on the z-directed field component. 11

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zx zx Figure 2. Layered dielectric structure normal mode set. The vector potential is arbitrary so several representations of the field sets are possible. With this in mind, a choice for the vector potentials can be made that creates a set of equations that have no Eor H-field component in the transverse direction of the propagating wave. These sets of equations are known as Longitudinal Section (LS) waves [9]. There is no physical interpretation to describe an LS set of equations other than it is related to the normal set by a linear combination. The advantage of LS sets is that a mathematically simpler representation for the fields is created. For the layered structure shown in figure 3, the transverse fields are continuous across the common boundary and independent sets of modes are created which can be treated separately with no coupling to the other modes. 12

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zx zx Figure 3. Layered dielectric structure LS mode set. The LS mode set is derived by assuming zaA and 0 F Maxwells equations can be expanded into rectangular components leading to the TM to z mode set. zxjEx2''1 (9a) zyjEy2''1 (9b) ][1222''kzjEz (9c) yHx'' (9d) xHy'' (9e) 0''zH (9f) 13

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Similarly if and 0A zaF then the TE to z mode set can be derived. yEx' (10a) xEy' (10b) 0'zE (10c) zxjHx2'1 (10d) zyjHy2'1 (10e) ][1222'kzjHz (10f) Where the single prime indicates TE and the double prime indicates TM. 2.2 Mode Functions Solutions to the vector Helmholtz equation are found using the separation of variables technique [10]. The solution to equation 6 can be separated into rectangular coordinates. )()()(zZyYxX (11) 14

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At this point it is useful to define a two-dimensional del operator. zazt (12) and also represent the tangential fields as, )()(yYxXt (13) such that )(),(zZyxt (14) Using the TE to z mode set defined by equation 10, it is possible to represent the E and H fields as, )()('zZaEttzt (15) zzZjHttt)(1' (16) )(1222'kzjHy (17) 15

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For TM to z modes, )()(''zZaHttzt (18) zzZjEttt)(1'' (19) )(1222''kzjEy (20) For TE to z modes the above equations 15 through 17 can be factored in terms of mode functions and mode voltages and currents. '''VeEt (21a) '''IhHt (21b) where ttzae' (22a) tth' (22c) )('zZV (22b) zzZjI)(1' (22d) 16

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The TM to z modes can be written in terms of the mode functions and mode voltages and currents. (23a) '''''VeEt ''''''IhHt (23b) where tte'' (24a) ttzah'' (24c) zzZjV)(1'' (24b) )(''zZI (24d) The mode functions are form functions, which depend only on the cross-sectional coordinates. If the transmission media is uniform then the form function is identical for every cross section and the mode can be completely characterized by the amplitude. The variation of the amplitudes is given as a solution to the one-dimensional wave or transmission line equations [11]. 17

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2.3 Transverse Resonance Representing the fields across the common dielectric interface with LS modes creates a unique propagation constant and characteristic impedance for each mode and leads to a transmission line representation for each of the dielectric layers. To use the transverse resonance method the modes must be discrete, which is accomplished by enclosing the layered dielectric in a perfect electric conductor (PEC). The PEC will create shorted transmission lines for each of the modes which is represented in Figure 4. At the dielectric interface the impedance looking into each of the Zin1Zin2 Zin1Zin2 Figure 4. Transmission line representation for the layered dielectric structure. transmission lines must be equal giving the following transcendental equation. )tan()tan(222111lkZlkZzozo (25) Where Zo is the wave impedance, is the propagation constant for the z-directed field component, and is the dielectric thickness. With discrete modes the relationship for the zk l 18

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x-, y-, and z-directed field component propagation constant falls out from the separation of variables technique in terms of the free space constant and is given in equations 26a and 26b for each layer [12]. 11221212ozxkkbnk (26a) 22222222ozxkkbnk (26b) Equation 25 leads to the resonance condition, which is shown by equation 26c and has the general form of 0),( xkR 0)tan()tan(222111 lkZlkZzozo (26c) This equation leads to a singular matrix whose singular value is the propagation constant, which is a complex number. Setting the determinate of the characteristic equation to zero and using a search algorithm will find the propagation constant [13]. 2.4 Mode Matching Using mode functions along with the voltage and current, the behavior across a step discontinuity can be developed. The process starts with the transverse fields across the discontinuity boundary defined in Figure 5. 19

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Region 1Region 2S1S2yxz Region 1Region 2S1S2yxz Figure 5. Step discontinuity. The total fields can be represented by a truncated series as shown by equations 25 and 26. NyxNyxtVeeVeeE''''''''')()( (25) NyxNyxtIhhIhhH''''''''')()( (26) These equations written in matrix form are, NNyyxxNyxNyxtVeVVeeeeVeeVeeE'''''''''''''''''' (27) NNyyxxNyxNyxtIhIIhhhhIhhIhhH'''''''''''''''''' (28) where, e and are the mode functions represented as matrices and Vand h I are voltage and current vectors respectively. 20

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The series representation of the fields on both sides of the step discontinuity are set equal to one another as shown in equations 29 and 30. The voltage and current on one side of the step can be written in terms of the voltage and current on the other side using orthogonality. This relationship is shown by equations 31 and 32 where the NMVeVe2211 (29) NMIhIh2211 (30) normalized reaction matrix, e and h, transforms the voltage and current respectively. NMVeeeeV2211111 (31) NMIIhhhh2112122 (32) The reaction matrix is given by equations 33a and 33b and the normalization matrix is given by equations 34a and 34b. "2'2"2'2"1'1"1'121yyxxTNyyxxNeeeeeeeeee (33a) "1'1"1'1"2'2"2'212yyxxTNyyxxNhhhhhhhhhh (33b) 21

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"1'1"1'1"1'1"1'111yyxxTNyyxxNeeeeeeeeee (34a) "2'2"2'2"2'2"2'222yyxxTNyyxxNhhhhhhhhhh (34b) These two equations satisfy the following boundary conditions. 21ttEE on S 2 (33) 01tE on (S 1 -S 2 ) (34) 21ttHH on S 2 (35) This represents (M+N) linearly independent equations which completely describes the mode coupling across the junction discontinuity. Equations 31 and 32 form a description of the field interactions across the step discontinuity given in figure 5. 2.5 Generalized Transverse Resonance The GTR method combines the transverse resonance method and the mode matching technique. The combination of these two methods is accomplished using network theory since the fields are represented in a voltage and current form. The modes are combined as shown in Figure 1 and an equivalent matrix is created. This matrix leads to a homogeneous system of equations and unknowns. The condition for the nontrivial solution for this system of equations is the resonance condition [14]. The resonance 22

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condition is found by setting the determinant of this matrix to zero as described by equation 26c. To illustrate this process the characteristic equation for a simple coplanar waveguide is found. A general co-planar waveguide structure is shown in Figure 6 with the problem bounded using PEC boundaries so the potential functions form an infinite series and the eigenvalues are discrete. There are four regions. Region 1 is an air and substrate region, which is modeled using transmission line theory. Two transmission lines are cascaded together and the air side is terminated by the PEC. Region 4 is an air region, which is modeled using transmission line theory and one side terminated on the PEC. Region 2 and 3 are slot regions, which are modeled using transmission line theory (the use of transmission line theory allows metal thickness to be accounted for). 23

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1432 er b Hsub Hair t Hair Gnd Gnd S W W yxz 1432 er b Hsub Hair t Hair Gnd Gnd S W W yxz Figure 6. CPW cross section. The reaction matrix from the mode matching technique can completely represent the mode interaction across the metal step from the substrate to the slots and from the slots to the air. The PEC is far enough from the slot regions so that the fields do not interact with it. The slot regions are expanded and a two-port network representation is shown in figure 7. 24

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ABCDModeMatchingModeMatching Isub IairVa2Vb2Vd2Va3Vb3Vd3 IairIb2Ib2IairIb3Ib3Isub V1V4 Isub=YsubV1 Iair=YairV4Isub Vc2 Ic2Ic2 Vb3Ib3Ib3 ModeMatchingModeMatching ABCD ABCDModeMatchingModeMatching Isub IairVa2Vb2Vd2Va3Vb3Vd3 IairIb2Ib2IairIb3Ib3Isub V1V4 Isub=YsubV1 Iair=YairV4Isub Vc2 Ic2Ic2 Vb3Ib3Ib3 ModeMatchingModeMatching ABCD Figure 7. CPW example represented as a network. The GTR procedure starts by finding the driving point admittance looking into the shorted transmission lines for regions 1 and 4. Each region is characterized by multiple modes (multiple transmission lines) and an admittance matrix is formed. Since the modes are independent of one another the admittance matrix is diagonal. The admittance seen looking into region 4 and region 1 is represented as the matrix and respectively. 4Y 1Y 25

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111VYI (36a) 444VYI (36b) The slot regions are represented using ABCD matrices for each mode and are defined below. 22222222ccbbIVDCBAIV (37a) 33333333ccbbIVDCBAIV (37b) The next step is to model the metal edges (discontinuity) between the substrate and slots and the air and slots using equations 31 and 32. The following definitions are used. Region 2 to 1 (slot to substrate) 2212baVKV (38a) 2221baIIJ (38b) Region 3 to 1 (slot to substrate) 3313baVKV (39a) 26

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3331baIIJ (39b) Region 2 to 4 (slot to air) 2242cdVKV (40a) 2224cdIIJ (40b) Region 3 to 4 (slot to air) 3343cdVKV (41a) 3334cdIIJ (41b) Where K and J are the normalized reaction matrices. Equations 36 through 41 are combined using the equivalent network shown in figure 7 and subject to the following boundary conditions. 321aaVVV (42a) 321aaIII (42b) 324ddVVV (42c) 324ddIII (42d) 27

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The characteristic equation for the CPW example is given by equation 43. 022211211RRRR (43) Where the matrix elements are given by equations 44 and 45. tbtbbtYBYYBYAYYDCR2332322222222222211 (44a) tbbtbtYBYAYYBYYDR333233232322223212 (44a) tbtbbtYBYYBYAYYDR233332223223232321 (44a) tbbtbtYBYAYYBYYDCR3333333323232333322 (44a) 2442422KYJYt (45a) 3442423KYJYt (45b) 2443432KYJYt (45c) 3443433KYJYt (45d) 2112122KYJYb (45e) 3112123KYJYb (45f) 2113132KYJYb (45g) 3113133KYJYb (45h) 28

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To solve the characteristic equation the reaction matrices required by equations 38 through 41 are derived from the mode functions. This process starts with defining the potential functions for each region of the CPW. Region 1 and 4 zjkxjkzxeeybn)cos( (38) zjkxjkzxeeybn)sin( (39) Region 2 zjkxjkzxeeyGndWSWn))(cos( (40) zjkxjkzxeeyGndWSWn))(sin( (41) Region 3 zjkxjkzxeeyGndWn))(cos( (42) zjkxjkzxeeyGndWn))(sin( (43) 29

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The fields are found by substituting the potential functions into the mode set (equations 7 and 8) and the mode functions are found using the definitions given by equations 21 and 23. Region 1 and 4 TE to z nxjkxxeybnzZE]))[sin((' (44) nxjkyxeybnzZE]))[cos((' (45) nxjkxxxeybnjkzzZjH])cos([)(1' (46) ])sin([)(1'nxjkyxeybnbnzzZjH (47) TM to z nxjkxxxeybnjkzzZjE])sin([)(1'' (48) nxjkyxeybnbnzzZjE])cos([)(1'' (49) nxjkxxeybnbnzZH])cos()[('' (50) 30

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nxjkxyxeybnjkzZH])sin()[('' (51) Where Z(z) is the amplitude function and represents either voltage for the E-field or current for the H-field. These equations apply for regions 2 and 3 by making the following substitutions. Region 2: )(yGndWSy and Wb Region 3: and )(yGndy Wb 2.6 Summary This chapter discussed the development of generalized transverse resonance by combining the transverse resonance method with the mode matching technique. The understanding of the GTR method started with a discussion of longitudinal section waves and from this derived the transverse resonance method for layered dielectrics. It was shown that Maxwells equations can be represented as a mode function and mode voltage or current. This factored form of Maxwells equations was used to merge the TRM with MMT to form the GTR method using network theory. 31

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Chapter Three Implementation of the GTR Algorithm The two important parameters that define a transmission line from a circuit design point of view are the propagation constant and characteristic impedance. For practical transmission lines both these parameters are frequency dependent. The propagation constant is found by solving the wave equation, as an eigenvalue problem, subject to the boundary conditions. Once the propagation constant is found, the cross sectional fields can be solved and from the fields the characteristic impedance is found. This chapter describes the details of how the Generalized Transverse Resonance algorithm was implemented to find the propagation constant. Discussed first is the relative convergence phenomenon followed by the optimization strategy. The last two sections describe the solution for the fields and characteristic impedance. 3.1 Relative Convergence The numerical solution of the characteristic equation requires that the number of terms in the field expansions, equations 31 and 32, be truncated to finite values. The truncation of these terms affects the convergence of the algorithm and must be done in a way to avoid the relative convergence phenomenon [15]. 32

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The relative convergence phenomenon occurs in doubly infinite sets of equations such as the mode matching equations given in Chapter 2 equations 31 and 32. The term relative convergence comes from the fact that the solution converges to a different answer for different choices of the number of terms that represent the expansion equation, M and N. Referring to the mode matching equations given by equations 31 and 32 and subject to the boundary conditions given by equations 33 35 it can be shown that M must be greater than N [16]. This argument starts by referring to Figure 5. If the step is shorted such that the voltages in region 1 and 2 are zero then equation 31 can be written as equation 52. 2211110VeeeeNM (52) This implies that only the trivial solution is possible and the rank of this matrix is equal to N. Looking at the currents in equation 32, the boundary conditions show that the current in region 2 sees an open circuit while the current in region 1 sees a combination of open and short circuit, represented by equation 53. The current in region 1 cannot be zero, which implies that the rank of the matrix in equation 53 is less than M. 0112122IhhhhNM (53) This reasoning leads to the condition that M must be greater than N. The choice for the ratio M/N has been the subject of numerous studies and has been found to be linked to 33

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the ratio of the step size [17]. Convergence is achieved when the choice of M/N is 1.5 times the discontinuity step size [7]. The relative convergence behavior is clearly shown for a co-planar waveguide transmission line example [18]. The structure has a center conductor dimension of 0.2 mm and a slot dimension of 0.2 mm. The metal thickness is zero in this example. The substrate is 0.22 mm thick with a dielectric constant of 3.75. The structure was enclosed by PEC boundaries, 3.1 mm by 1.55 mm. Using the GTR technique the algorithm was used to find the odd and even mode propagation constant for different ratios of M/N. As seen in figures 8 and 9, convergence occurs when M/N is 12 which is consistent with the 1.451.501.551.601.651.7056789101112131415Ratio M/Nk/ko N = 2 N = 4 N = 6 N = 8 Figure 8. Relative convergence of the odd mode for the CPW example (odd mode is defined by the following E-field orientation in the slot ). 34

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criteria given above, 1.55 mm/0.2 mm times 1.5 gives M/N = 11.6. Final convergence for the odd mode is 1.475 and for the even mode 1.137. 1.111.121.131.141.151.1656789101112131415Ratio M/Nk/ko N = 2 N = 4 N = 6 N = 8 Figure 9. Relative convergence of the even mode for the CPW example (even mode is defined by the following E-field orientation in the slot ). 3.2 Numerical Technique For Solving the Transverse Resonance Condition The Generalized Transverse Resonance technique leads to the characteristic equation which is a homogeneous matrix equation in the form of 0),(xkRx where R is a complex matrix that is a function of frequency, and propagation constant, and is a column vector. The accepted method for solving the equation is to iteratively change the propagation constant until the determinant of the characteristic equation is zero, xk x 0R 35

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Numerically it is difficult to detect the zeros of the determinant because the characteristic equation is a rapidly changing function due to poles and zeros in close proximity causing the matrix to become singular or near singular. Complicating the problem further, the location of the poles and zeros cannot be expressed analytically. For this reason an iterative search was used to solve this equation based on singular value decomposition and looking for the minimum singular value [19]. In this technique the matrix R is decomposed into, whereis a diagonal matrix of singular values in decreasing order and Wand are the left and right singular vectors of WSVR S V R respectively. Instead of detecting zeros of the determinant, the technique detects the minima of the last element of the diagonal matrix S 0 m The advantage of using this technique over looking for the determinant zero crossing is robustness. Figure 10 is a pictorial example of the behavior showing the two techniques. The presence of a pole or steep gradient near a determinant zero crossing will cause the search algorithm to become unstable where in contrast, the minimum singular value technique is well behaved. An added benefit to this technique is the last column of the matrix automatically contains the solution for the vector, which is the slot voltage, Vc, as shown in figure 5. V x 36

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ObjectiveFunctionDeterminateMinSingValue ObjectiveFunctionDeterminateMinSingValue Figure 10. Pictorial showing the behavior difference between using the determinant and minimum singular value. A multidimensional optimizer was used as the search algorithm to find the real and imaginary parts of the propagation constant. This optimizer was a simplex method with the error function set up to search for the minimum singular value [20]. 37

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The propagation constant starting value was chosen so that it was equal to max0rk where maxr is the maximum relative dielectric constant in the structure. Built into the simplex optimizer is the ability to automatically adjust step size as the search algorithm converges to the correct solution. 3.3 Field Calculation Typically the quasi-TEM mode is the dominant mode in a transmission line and has no cut-off frequency. However, in practical transmission lines the signal is carried by a combination of the quasi-TEM mode and hybrid modes. These hybrid modes have cut-off frequencies, which are dependent on the order of the mode. The combination of all the modes defines the electric and magnetic fields. As discussed above, the last column of the matrix Vis a vector that contains the slot mode voltages. From this voltage vector all voltages and currents shown in figure 7 can be found. Starting with the slot voltage Vc the slot current can be found using equations 36 and 38-41 and the boundary conditions. Manipulation of these equations leads to equation 54. 323332232232ccttttccVVYYYYII (54) Voltage and currents above the CPW (in the air region) are found using equation 40 and 42c and d. 38

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3332234ccVKVKV (55) 444VYI (56) To find the voltage and currents in the substrate region, equations 36 42 are used. 22222222ccbbIVDCBAIV (57) 33333333ccbbIVDCBAIV (57) 3312211bbVKVKV (58) 111VYI (59) The x and y fields can be found using equations 44 through 46. The z field is found using Maxwells curl equation. The Hz component is found from the Ex and Ey components. Likewise, the Ez component can be found from the Hx and Hy components. yExEjHxyoz1 (60) yHxHjExyroz1 (61) 39

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From the potential functions and mode voltages and currents all the field components are defined. Shown below are region 1 and 4 fields; extension to the other regions is an argument change. TE modes nnxjkxzVeybnbnEx)(sin'' (62) nnxjkxyzVeybnjkEx)(cos'' (63) 0'zE (64) )(cos''zIeybnjkHnnxjkxxx (65) )(sin''zIeybnbnHnnxjkyx (66) nnxjkxozzVeybnbnkjHx)(cos1'22' (67) TM modes nnxjkxxzVeybnjkEx)(sin'''' (68) nnxjkyzVeybnbnEx)(cos'''' (69) 40

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nnxjkxrozzIeybnbnkjEx)(sin1''22'' (70) )(cos''''zIeybnbnHnnxjkxx (71) )(sin''''zIeybnjkHnnxjkxyx (72) 0''zH (73) These equations apply for regions 2 and 3 by making the following substitutions. Region 2: )(yGndWSy and Wb Region 3: and )(yGndy Wb The variation in the z direction is accounted for using transmission line theory. 3.4 Characteristic Impedance The voltage and current on a matched transmission line define the characteristic impedance Z o and for the TEM mode Z o is uniquely defined. However, the voltage and current are not uniquely defined for the hybrid modes and this ambiguity presents a problem in calculating a unique characteristic impedance. To resolve this ambiguity, a power definition is used [18]. Three common definitions exist for the characteristic impedance [21]: 41

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rrVIIVZ0 (74) 20rPIIPZ (75) PVZrPV20 (76) Where the subscript indicates the r th slot for voltage and the r th conductor for current. Each of these definitions will give different numerical results except at DC. The relationship between the three definitions is given below. PVPIVIZZZ000 (77) Equations 74 through 76 imply that transmission lines with several slots or several conductors can have multiple characteristic impedances [22]. The frequency dependent characteristic impedance is found using the fields. The voltage is found by integrating the field in the middle of the r th slot and the current is found by integrating the current density in the r th conductor. pathyryrdyEV (78) dsJIrr (79) 42

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The total power is found by integrating the transverse fields in each region and summing. dsaHEPsxzy (80) iitPP (81) The characteristic impedance for CPW transmission lines is simplified if the slots are symmetric [22]. For the dominant odd mode, a single mode propagation regime exists. However, for the parasitic even mode, a coupled slot regime is established and the propagation is characterized by an even and odd mode. From a circuit parameter point of view, the dominant mode characteristic impedance is found by finding the slot impedance and from symmetry dividing by two. The parasitic mode characteristic impedance is complicated in that a coupled slot transmission line is assumed and both an odd and even characteristic impedance is found. 3.5 Summary This chapter discussed the details of implementing the GTR method. The relative convergence phenomenon occurs in doubly infinite sets of equations such as the mode matching equations. The term relative convergence comes from the fact that the solution converges to a different answer for different choices of the number of terms that represent the expansion equation. A robust optimization strategy was used. Instead of solving for zeros of a determinant the matrix is decomposed using singular valued decomposition. A 43

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simplex direct search algorithm was used to minimize the singular value of the diagonal matrix. An added benefit of using singular valued decomposition was slot voltages were automatically found. These slot voltages were used to find the fields in the CPW cross section and from the fields the characteristic impedance was found. 44

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Chapter Four Superstrate Layers on Coplanar Waveguides The Co-Planar Waveguide (CPW) transmission line is a completely planar structure with the signal and ground conductors residing on top of the substrate. For this reason it is a popular choice for creating miniature RF circuits because it eliminates the uses of vias, simplifies the fabrication process, and allows straightforward integration of active and passive circuits. When adding a vertical dimension, these advantages allow miniaturization of complex systems leading to a multi-layered chip. These complete systems on a chip (SoC) can be very complex, comprising analog and digital circuitry, active and passive components, and antennas. Three-dimensional RF structures using silicon (Si) micromachining have been reported for applications through W-band [23] [24]. However, no attempt was made to understand or optimize the effect of stacked silicon layers at multiple heights or for different resistivities. This dissertation represents the first serious work on characterizing and understanding the proximity effects of silicon on the propagation characteristics of CPW transmission lines. The data presented here suggests the substrate resistivity and dielectric constant are dependent on frequency. This behavior occurs for transmission line structures with deep penetration of the fields into the substrate which is a function of line 45

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geometry. Possible causes for the frequency dependent material properties could be in the electromagnetic interaction with the material. It is possible that relaxations of the charges in the substrate are dependent on signal power, interaction with other charges, and/or tensor properties. This study extracted the material properties from the measured data and presents a frequency dependent complex dielectric constant whose imaginary part is a function of lid height; this lid height dependency is non-physical but necessary to accurately represent the measured data in the full-wave simulations, pointing to a need for further examination of the formulation governing the dielectric property behavior in these material systems. It can be theorized that the geometry of a lid over a CPW transmission line can be fabricated using 3-dimensional micromachining processes to give a target complex dielectric constant over a frequency band. This has applications in phase shifters, filters and chip scale antenna elements using 3-dimensional packaging such as flip-chip or bulk silicon micromachining. This chapter presents an understanding of the propagation characteristics of the CPW transmission line in proximity with high and low loss silicon. Presented will be data showing the loss and propagation characteristics for four CPW structures using two separate silicon lids at six heights above the transmission line. Two propagation modes in the substrate have been identified and will be discussed in sections 4.6 and 4.7. The data suggests that CPW transmission lines in proximity to a high conductivity silicon lid show a frequency dependent dielectric constant and substrate resistivity. This frequency dependence is discussed in section 4.8. 46

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4.1 Theory Substrate effects have a major impact on the performance of transmission lines over frequency. As the signal propagates at lower frequencies along a transmission line, any charges in the substrate are free to move with the period of the signal. The time it takes the charge to move from a high point to a low point in the period is the substrate relaxation time. A lossy substrate will act as a ground plane only to the point where the substrate relaxation time is less than the period of the signal. An interfacial polarization of the charge is created which will short the electric field. However, if the relaxation time is greater than the signal period the substrate begins to act more like a (lossy) dielectric because the charges cannot respond fast enough. In this regime the field will penetrate the substrate. Two propagation modes can be identified in the substrate with the transition from one to the other defined by the substrate relaxation frequency given by equation 82 [25], 21f (82) where is the substrate resistivity in ohm-cm and is the substrate permittivity. Below the relaxation frequency the substrate acts as a ground plane, shorting out the electric field. This mode is termed the slow-wave mode because boundary conditions force the signal to slow down to sustain TEM propagation. Above the relaxation frequency the 47

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substrate charges cannot respond and the substrate acts as a (lossy) dielectric. This mode is termed the dielectric quasi-TEM mode. 4.2 CPW and Lid Fabrication The dimensions of the four CPW lines used in this study are given in Table 1 and a layout of the lines is shown in Figure 11. Referring to Figure 6 in chapter 2, the center conductor width is represented in the GTR algorithm by the variable S and the slot width by the variable W. Ground conductor width for all the structures was 500 m. The total reticule size is 19 mm by 19 mm and contains two copies each of the CPW structures. To allow for wafer probing of the structures each of the transmission lines has the same input and output pitch. The slot width is 75 m and conductor width is 110 m so that a 150 m pitch probe can be used. The transmission lines were fabricated on high-resistivity (2000 ohm-cm) bare silicon, 400 m thick. The metal was evaporated chrome/gold (Cr/Au) and deposited to a total thickness of 1 m. The lines were formed using a lift-off process. Table 1. CPW line dimensions. Center Conductor Width (m) Slot Width (m) Nominal Impedance CPW1 54 101 59 CPW2 124 75 45 CPW3 250 150 49 CPW4 450 225 50 48

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CPW4CPW1CPW3CPW2CPW4CPW1CPW3CPW2 CPW4CPW1CPW3CPW2CPW4CPW1CPW3CPW2 Figure 11. Layout showing the CPW structures used in this study. 4.3 Measurement Setup Two bare 400 m thick pieces of silicon were cut 1.42 cm by 19 mm and used as lids to study the effects of high and low resistivity silicon in proximity to the CPW transmission lines; one 2000 ohm-cm and one 10 ohm-cm. Figure 12 shows a diagram of how the lid was suspended above the CPW host substrate; shown is a top and side view. Cantilevered from the micrometer assembly of the probe station was an aluminum arm with a threaded hole to accept a nylon screw. The lid was attached to the nylon screw using hard wax. A dielectric screw and wax for attachment was used to keep metal away from the silicon lid eliminating any effect on fringing fields. The distance from the aluminum cantilever and the lid was approximately one inch. The whole assembly could be adjusted up or down using the micrometer of the probe station. In order to take on49

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wafer measurements, probe access was achieved by extending the transmission lines beyond the lid. The lid was mechanically aligned using triangular markers printed on the reticule with the transmission lines as shown in Figure 11. LidHost substrate Micrometer isused to movelid fixture upand down Wafer probe accessWafer probe access Host substrateLid Nylon screw LidHost substrate Micrometer isused to movelid fixture upand down Wafer probe accessWafer probe access Host substrateLid Nylon screw Figure 12. Diagram showing the fixture for suspending the lid; topand side-view. The distance associated with each division marked on the micrometer was estimated using the microscope stage movement, which according to the factory was accurate to +/1 m. Moving the micrometer the same distance the stage moved calibrated the divisions on the micrometer to be +/-5 m. The lid needs to be moved to 50

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sub-micron accuracy and +/5 m is too much error. Using the stage to position the lid was not practical because it only had a 20 m maximum movement and would require the probe contacts to be adjusted for each lid movement. An alternate method was developed which is described below. Each of the transmission line structures were measured for six lid heights; 5 m, 10 m, 25 m, 50 m, 100 m, and no lid. The data was taken from 4 to 40 GHz, using a two-port vector network analyzer calibrated with a multi-line Thru-Reflect-Line calibration [26] [27] The calibration established reference planes so that the effective length of the transmission lines was 1.42 cm. Each of the lid heights were adjusted so that the measured insertion phase was +/1 degree of the simulated phase at the single frequency of 20 GHz. When the height was established a swept measurement was made using the network analyzer. The phase comparison method proved to be very accurate as shown in sections 4.5 and 4.6. The validity of the method can be seen when looking at the phase data in each of these sections. The lid height was established for one frequency but the dispersion matches across the entire frequency sweep. 4.4 Dielectric and Metal Losses The dielectric loss is accounted for in the GTR algorithm using a complex dielectric constant model given by equation 83, where is the conductivity of the substrate in S/cm. The loss tangent is derived from equation 83 and given in equation 84. 51

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oojj (83) otan (84) Equation 83 is a subset of and derived from the Debye relationship given by equations 85a and 85b. These equations represent the real and imaginary parts of the dielectric constants for a layered substrate. 2)(1 So (85a) oSo 2)(1)( (85b) Where: S is the dielectric constant at DC and is called the static dielectric constant. is the dielectric constant at high frequency, > 40 GHz, and is called the optical dielectric constant. is the Debye relaxation time constant. The optical dielectric constant is a constant with a value of 11.7 for silicon. The static dielectric constant is strongly dependent on the silicon doping concentration as shown in equation 86 [28]. The Debye relaxation time constant is a function of the dielectric relaxation frequency of the silicon and the relaxation frequency of the interfacial polarization. 52

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DDDSNNN211910172.1110635.1688.11)( (86) Examination of equation 86 shows the static dielectric constant does not change from 11.688 until the resistivity of the substrate is very low, < 0.1 ohm-cm. For a substrate resistivity of 10 ohm-cm the static dielectric constant is 11.688 which is approximately equal to the optical dielectric constant of 11.7. These values substituted into equation 85 gives the real part of the dielectric constant as 11.7 with the imaginary part equal to o which is equation 83. The GTR technique only accounts for the dielectric loss, which is the focus of this study. However, for a comparison to the measured data metal loss must be accounted for in some way. Closed form equations exist in the literature for predicting the metal loss [29] [30] [31]. However, these expressions do not compare well with one another when predicting metal loss at microwave frequencies. This lack of agreement is mostly due to the processing variations in fabricating metal transmission lines. Effects such as metal roughness, layering of different metals (typical fabrication requires adhesion layers), and low metal density are not accounted for in these studies. Metal loss was added using the conductor loss equation from [29]. A discussion of this equation is given in Appendix A. 53

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4.5 GTR Compared to HFSS The GTR algorithm was compared to a commercially available finite element electromagnetic field solver: Ansofts HFSS v9.1. Each of the CPW structures was modeled in HFSS using the same conditions as the GTR algorithm as described in Figure 6 of section 2.5. Figures 13 and 14 show this comparison for all lid heights of the CPW1 structure at 40 GHz. Figure 13 is the percent difference between the real part of the propagation constant for two cases, one accounting for metal loss and the other not accounting for metal loss. This data shows excellent correlation between the GTR technique and HFSS. -0.6-0.4-0.20.00.20.40.6050100150200250Lid Height (um)Difference (%) No metal loss Metal loss Figure 13. Comparison showing the percent difference between the real part (phase) of the propagation constant found using GTR and HFSS. 54

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-40-30-20-10010203040050100150200250Lid Height (um)Difference (%) No metal loss Metal loss Figure 14. Comparison showing the percent difference between the imaginary part (loss) of the propagation constant found using GTR and HFSS. Figure 14 is the percent difference between the imaginary part of the propagation constant for two cases, one accounting for metal loss and the other not accounting for metal loss. This data shows excellent correlation when no metal loss is accounted for. When metal loss is accounted for the comparison is off by as much as 30%. This difference is reasonable since different metal loss models are used for each technique. However, neither metal loss model accurately predicts the measured data. The comparison of measured loss data to GTR and HFSS is shown in Figures 15 and 16. Figure 15 and 16 shows the loss data as a function of lid height for CPW1 at 40 GHz with a 2000 ohm-cm and 10 ohm-cm lid respectively. 55

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-2.5-2.0-1.5-1.0-0.50.002040608010Lid Height (um)Loss (dB/cm) 0 Measured GTR HFSS Figure 15. Comparison showing the measured loss data to simulated loss data for CPW1 with a 2000 ohm-cm lid at 40 GHz. -12-10-8-6-4-20020406080100Lid Height (um)Loss (dB/cm) Measured GTR HFSS Figure 16. Comparison showing the measured loss data to simulated loss data for CPW1 with a 10 ohm-cm lid at 40 GHz. 56

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Figures 13 and 14 are a sample of the comparison between GTR and HFSS; Appendix B contains the complete comparison for all the CPW structures used in this study. 4.6 Measured Data Using the 2000 ohm-cm Lid Figures 17 through 20 are a comparison of the loss between the simulated and measured data for the CPW structures defined in Table 1, when using a 2000 ohm-cm lid. The data correspond to 6 lid heights, with the symbols representing the simulated data and the solid lines the measured data. The curves in the plot advance in sequence for each of the lid heights, from high loss (5 m lid height) to low loss (no lid). The slope of the simulated loss compares well with the measured data. This comparison is typical for insertion loss in that simulated data usually underestimates loss. An investigation into the difference shows that a compensation factor of about 2 applied to the metal loss would fit the simulated data to the measured data for all lid heights. The reason for the compensation factor can be explained by the inaccuracy in modeling the metal sheet resistivity in the metal loss equation. 57

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-2.5-2.0-1.5-1.0-0.50.0481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 17. Loss for CPW1 with 2000 ohm-cm lid. -2.0-1.8-1.6-1.4-1.2-1.0-0.8-0.6481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 18. Loss for CPW2 with 2000 ohm-cm lid. 58

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-1.8-1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 19. Loss for CPW3 with 2000 ohm-cm lid. -1.6-1.4-1.2-1.0-0.8-0.6-0.4-0.20.0481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 20. Loss for CPW4 with 2000 ohm-cm lid. 59

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The measured and simulated normalized phase constant are compared and shown for the four structures in Figures 21 through 24. As with the loss data, the data correspond to 6 lid heights, with the symbols representing the simulated data and the solid lines the measured data. The curves in the plot advance in sequence for each of the lid heights, from high phase constant (5 m lid height) to low phase constant (no lid). Figures 21 through 24 show excellent agreement between the measured and simulated data. 2.22.32.42.52.62.72.82.93.03.13.2481216202428323640Frequency (GHz)k/kono lid5 um Figure 21. Normalized phase constant for CPW1 with 2000 ohm-cm lid. 60

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2.22.32.42.52.62.72.82.93.03.13.2481216202428323640Frequency (GHz)k/kono lid5 um Figure 22. Normalized phase constant for CPW2 with 2000 ohm-cm lid. 2.22.32.42.52.62.72.82.93.03.13.2481216202428323640Frequency (GHz)k/kono lid5 um Figure 23. Normalized phase constant for CPW3 with 2000 ohm-cm lid. 61

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2.22.32.42.52.62.72.82.93.03.13.2481216202428323640Frequency (GHz)k/kono lid5 um Figure 24. Normalized phase constant for CPW4 with 2000 ohm-cm lid. The slow wave mode has the characteristic of a slow moving propagating signal, which relates to an increase in the effective permittivity below the transition frequency. Above the transition frequency the effective permittivity will be relatively constant with frequency. The transition frequency, defined by equation 82, is 77 MHz for the 2000 ohm-cm lid silicon. Shown in Figure 25 is the effective permittivity which was calculated from the measured phase data for a lid height of 5 m for each of the CPW structures. This lid height was chosen because it has the most effect on the CPW structures. Figure 25 shows that the high resistivity lid has a minor effect on the CPW transmission lines with respect to slowing the propagation signal. This suggests the signal propagates mostly as the dielectric quasi-TEM mode over the 4 to 40 GHz frequency band. 62

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8.68.78.88.99.09.19.29.39.49.59.6481216202428323640Frequency (GHz)Effective Permittivity CPW1CPW4CPW2CPW3 Figure 25. Effective permittivity using the 2000 ohm-cm lid at a height of 5 um. 4.7 Measured Data Using the 10 ohm-cm Lid The effective permittivity for the circuits measured using the 10 ohm-cm lid is shown in Figure 26 as a function of frequency with the transition frequency indicated as a horizontal line. As done for the 2000 ohm-cm case, the effective permittivity was calculated from the measured phase data for a lid height of 5 m for each of the CPW structures. Figure 26 shows that the low resistivity lid has a dramatic effect on the CPW transmission lines with respect to slowing the propagation signal. Two modes of propagation are clearly shown; the slow-wave mode below and the dielectric quasi-TEM mode above the transition frequency. 63

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89101112131415481216202428323640Frequency (GHz)Effective Permittivity CPW1CPW4 CPW2CPW3 Figure 26. Effective permittivity using the 10 ohm-cm lid at a height of 5 um. Figures 27 through 30 are a comparison of the loss between the simulated and measured data for the CPW structures using the 10 ohm-cm lid. The data correspond to 6 lid heights, with the symbols representing the simulated data and the solid lines the measured data. The curves in the plot advance in sequence for each of the lid heights, from high loss (5 m lid height) to low loss (no lid). The comparison between the measured and modeled data for the CPW1 and CPW2 structures is similar to the trend seen with the high resistivity lid. The simulated data underestimates the loss, which again can be explained by the model for the metal loss. The comparison for the CPW3 and CPW4 structures is interesting and shows that the simulated data overestimates the loss; the difference in these results is examined further in the section 4.8 below. 64

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-12.0-10.0-8.0-6.0-4.0-2.00.0481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 27. Loss for CPW1 with 10 ohm-cm lid. -14.0-12.0-10.0-8.0-6.0-4.0-2.00.0481216202428323640Frequency (GHz)Loss (dB/cm)5 umno lid Figure 28. Loss for CPW2 with 10 ohm-cm lid. 65

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-16.0-14.0-12.0-10.0-8.0-6.0-4.0-2.00.0481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 29. Loss for CPW3 with 10 ohm-cm lid. -18.0-16.0-14.0-12.0-10.0-8.0-6.0-4.0-2.00.0481216202428323640Frequency (GHz)Loss (dB/cm)5 umno lid Figure 30. Loss for CPW4 with 10 ohm-cm lid. 66

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The measured and simulated normalized phase constant are compared and shown for the four structures in Figures 31 through 34. As with the loss data, the data correspond to 6 lid heights, with the symbols representing the simulated data and the solid lines the measured data. The curves in the plot advance in sequence for each of the lid heights, from high phase constant (5 m lid height) to low phase constant (no lid). Figures 31 through 34 show excellent agreement between measured and simulated data. 2.02.22.42.62.83.03.23.43.63.8481216202428323640Frequency (GHz)k/kono lid5 um Figure 31. Normalized phase constant for CPW1 with 10 ohm-cm lid. 67

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2.02.22.42.62.83.03.23.43.63.8481216202428323640Frequency (GHz)k/kono lid5 um Figure 32. Normalized phase constant for CPW2 with 10 ohm-cm lid. 2.02.22.42.62.83.03.23.43.63.8481216202428323640Frequency (GHz)k/kono lid5 um Figure 33. Normalized phase constant for CPW3 with 10 ohm-cm lid. 68

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2.02.22.42.62.83.03.23.43.63.8481216202428323640Frequency (GHz)k/kono lid5 um Figure 34. Normalized phase constant for CPW4 with 10 ohm-cm lid. 4.8 Frequency Dependent Permittivity The dielectric quasi-TEM mode is a regime where the field penetrates a lossy substrate because the relaxation time is too slow. The physical rational for this is deeper penetration of the field is equivalent to increasing the substrate resistivity for the lid in the dielectric quasi-TEM mode region. This increase in substrate resistivity affects the imaginary part of the effective dielectric constant, see equation 83. In the slow-wave region, below the transition frequency the real part of the effective dielectric constant increases as discussed in section 4.1. A frequency dependent complex dielectric constant has been presented in the literature for microstrip transmission lines [32]. To find the complex permittivity, the lid substrate dielectric constant and resistivity in the GTR algorithm were varied to find a fit to the measured data. The complex permittivity was 69

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extracted using equation 83. The complex permittivity of the lid silicon as a function of frequency and lid height is given in Figures 35 through 38 for the CPW3 and CPW4 structures. Note that one curve for the real part of the complex permittivity will fit the measured data across all lid heights. The imaginary part of the complex permittivity requires a family of curves, each curve corresponding to a lid height. This lid height dependency is non-physical but necessary to accurately represent the measured data in the full-wave simulations. 579111315171921481216202428323640Frequency (GHz)Re[Dielectric Constant] Figure 35. Real part of the lid permittivity for CPW3. 70

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05101520253035404550481216202428323640Frequency (GHz)Im[Dielectric Constant] 5 um100 um50 um25 um10 um Figure 36. Imaginary part of the lid permittivity for CPW3. 5101520253035481216202428323640Frequency (GHz)Re[Dielectric Constant] Figure 37. Real part of the lid permittivity for CPW4. 71

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051015202530481216202428323640Frequency (GHz)Im[Dielectric Constant] 5 um100 um50 um25 um10 um Figure 38. Imaginary part of the lid permittivity for CPW4. The imaginary part of the complex dielectric constant for CPW1 and CPW2 are frequency dependent but the material parameters are constant. The real part of the dielectric constant is 11.7 and the substrate resistivity is 10 ohm-cm across the frequency band. For comparison the imaginary part of the dielectric constant for CPW1 and CPW2 are shown in Figure 39. 72

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05101520253035404550481216202428323640Frequency (GHz)Im[Dielectric Constan t Figure 39. Imaginary part of the lid permittivity for CPW1 and CPW2. The propagation constant for CPW3 and CPW4 was calculated using equation 83 and the frequency dependent dielectric constants discussed above. The loss data is shown in Figures 40 and 41 for the CPW3 and CPW4 transmission lines, respectively. This data shows excellent agreement between the measured and simulated loss. 73

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-14.0-12.0-10.0-8.0-6.0-4.0-2.00.0481216202428323640Frequency (GHz)Loss (dB/cm)no lid5 um Figure 40. Corrected loss data for CPW3. -14.0-12.0-10.0-8.0-6.0-4.0-2.00.0481216202428323640Frequency (GHz)Loss (dB/cm)5 umno lid Figure 41. Corrected loss data for CPW4. 74

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The normalized phase data is shown in Figures 42 and 43 for the CPW3 and CPW4 transmission lines, respectively. This data shows excellent agreement between the measured and simulated normalized phase. 2.02.22.42.62.83.03.23.43.63.8481216202428323640Frequency (GHz)k/kono lid5 um Figure 42. Corrected normalized phase for CPW3. 75

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2.02.22.42.62.83.03.23.43.63.8481216202428323640Frequency (GHz)k/kono lid5 um Figure 43. Corrected normalized phase for CPW4. 4.9 Characteristic Impedance The classical approach to modeling transmission lines is to use a distributed RLGC lumped element model where the value of each term is frequency independent. However, substrate effects have a major impact on the performance of transmission lines over frequency. This impact is seen prominently on the distributed capacitance when the transmission line is in proximity of a highly conductive substrate. With a highly conductive substrate the penetration depth of the electric field is small because the substrate acts as a ground plane and shorts the field. This decrease in the electric field within the substrate effectively increases the distributed capacitance. Capacitance is defined as the change in charge divided by the change in voltage. The voltage is a line 76

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integral of the electric field and as the electric field decrease so does the voltage. From a physical point of view the electric field is concentrated in the CPW slot regions. The simulated characteristic impedance as a function of lid height at 4 and 40 GHz is shown in Figures 44 and 45, respectively. The solid line represents 2000 ohm-cm data while the dashed line represents 10 ohm-cm data. At small lid heights the distributed capacitance is high which would imply a low value for characteristic impedance. As the lid height increases, decreasing the distributed capacitance, the characteristic impedance increases. This trend is clearly seen in Figures 44 and 45. 354045505560050100150200250300350400Lid Height (um)Characteristic Impedance CPW1CPW3CPW2CPW4 Figure 44. Characteristic impedance as a function of lid height at 4 GHz (Solid lines 2000 ohm-cm lid; dashed lines 10 ohm-cm lid). 77

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35404550556065050100150200250300350400Lid Height (um)Characteristic Impedance CPW1CPW4CPW2CPW3 Figure 45. Characteristic impedance as a function of lid height at 40 GHz (Solid lines 2000 ohm-cm lid; dashed lines 10 ohm-cm lid). There are two interesting points to make regarding the characteristic impedance data. The first point is that the characteristic impedance converges to the same values for both frequencies as the lid height increases. This is expected because the effect of the low resistivity lid should decrease as it moves further from the CPW transmission line. Second, the characteristic impedance is the same for both the 2000 and 10 ohm-cm cases at higher frequencies. The reason for this can be seen by looking at the phase constant data at 40 GHz. Figure 46 contains plots that show the percent difference between the 2000 and 10 ohm-cm normalized phase constant for the four CPW structures as a function of lid height. The data in this figure shows that if the frequency of the propagating signal is well above the transition frequency then the substrate resistivity has little effect on the phase. This convergence is due to the fact that the data is above the 78

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relaxation frequency and the substrate is acting like a dielectric regardless of charge effects. -1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0050100150200250Lid Height (um)Difference (%)CPW4CPW3CPW1 CPW2 Figure 46. Difference in measured normalized phase between the 2000 and 10 ohm-cm lid for the four CPW structures at 40 GHz. 4.10 Summary This chapter discussed the propagation characteristics of the CPW transmission line in proximity with high and low loss silicon. The GTR algorithm discussed in chapters 1 through 3 was used to simulate these transmission line structures. The accuracy of the GTR code was demonstrated by comparing simulated data to measured data. This comparison included data showing the loss and propagation characteristics for four CPW structures using two separate silicon lids at six heights above the transmission line. Two propagation modes in the substrate were identified and discussed in sections 79

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4.6 and 4.7. It was found that CPW transmission lines in proximity to a high conductivity silicon lid show a frequency dependent complex permittivity. This frequency dependence was discussed in section 4.8. This behavior occurs for transmission line structures with deep penetration of the fields into the substrate which is a function of line geometry. Possible causes for the frequency dependent material properties could be in the electromagnetic interaction with the material. It is possible that relaxations of the charges in the substrate are dependent on signal power, interaction with other charges, and/or tensor properties. 80

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Chapter Five Characteristics of MTIS Transmission Lines The reality of CMOS integrated circuits operating into the high microwave frequency range led to strong interest in the properties of metal-insulator-semiconductor (MIS) transmission lines on silicon substrates. Such structures generally consist of planar transmission lines printed on silicon with resistivity in the 0.1 20 ohm-cm range, with a 0.1 1 m thick insulating oxide sandwiched in between. For microwave applications Co-Planar Waveguide (CPW) transmission line geometries are often used, with conductor and gap dimensions that are typically several times larger than the oxide thickness. A common goal of various theoretical and experimental studies performed starting around 1990 was to understand the geometryand frequency-dependent nature of such MIS lines, and to generate lumped equivalent circuit models to aid in their design and simulation [33][34][35]. While there remains a strong interest in the CMOS-type MIS lines, a geometrical variation involving insulating layers that are much thicker than the ~1 m oxide has received considerable attention in recent years [36][37]. Co-Planar Waveguide transmission lines fabricated on metal-thick-insulating-semiconductor (MTIS) substrates are the focus of the study in this chapter. Polyimide and bisbenzocyclobutane (BCB) are 81

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two of the more common thick insulators that are used, with thicknesses ranging from 1 to ~ 40 m. The primary advantage of such transmission lines over the CMOS IC geometry is improved attenuation properties, resulting from increased separation between the lossy silicon substrate and the conductors as well as the generally larger conductor dimensions. The MTIS structures used in this study are governed by a 3-mode formulation which is discussed in section 5.1. The fabrication of two MTIS structures on three separate substrates (BCB layered on silicon) is presented in section 5.2. Three silicon substrates were used with resistivities of 0.4-, 10and 25-ohm-cm. The dielectric quasi-TEM and slow-wave behaviors are exhibited in the 25 and 10 ohm-cm substrates and have frequency dependent complex dielectric constants, section 5.3. The skin-effect mode is discussed in section 5.4. The skin depth mode appears when the substrate conductivity is large and the frequency of the signal is low. The behavior is similar to the slow-wave mode in that the electric field is shorted at the silicon/BCB boundary. However, the magnetic field as it spreads into the silicon is bounded by the skin effect, effectively reducing the thickness of the silicon layer. This results in confining the field mostly to the low loss BCB layer. Presented in this section is measured loss data compared to simulated data. The insertion loss data for the CPW transmission lines show that at frequencies below 5 GHz there was less attenuation of the signal as the silicon resistivity decreased from 10 ohm-cm to 0.4 ohm-cm. This is due to the two different mode structures for each of the resistivities. For the 10 ohm-cm substrate a slow wave mode exists while the 0.4 ohm-cm substrate exhibits the skin depth mode. Decreasing the 82

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substrate resistivity to the point where the silicon exhibits the skin depth mode has application in creating low loss RF and high speed digital interconnects on CMOS grade silicon. The GTR algorithm discussed in chapters 1 through 3 was used to simulate these transmission line structures. 5.1 Theory Detailed research on layered insulator/low resistivity silicon transmission lines was first done by Hasegawa [25]. In this work, a microstrip transmission line was modeled using a parallel plate waveguide. Transverse resonance was used to analyze this simplified model to extract the propagation constant and characteristic impedance. From this analysis three dominate modes were identified and described using a resistivity-frequency domain chart. Three dominate mechanisms have been reported; slow-wave, skin depth, and dielectric quasi-TEM. The slow-wave mode occurs at low frequency and is dominant when the insulator layer thickness is much less than the silicon thickness. A strong interfacial polarization occurs on the silicon surface near the insulator. This polarization causes the static dielectric constant to become very large, i.e. the Maxwell-Wagner mechanism [38]. This increase in dielectric constant is caused by the electric field shorting due to the high conductivity of the substrate while the magnetic field spreads into the substrate. The skin depth mode occurs at medium frequencies where the silicon acts as a lossy conductor. In this mode, the penetration of the magnetic field into the substrate is limited by its skin depth. A detailed discussion of this mode is in section 5.4. 83

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The dielectric quasi-TEM mode occurs at high frequencies where the silicon acts as a lossy dielectric and the velocity of the dominate mode is the ratio of the speed of light to the square root of the effective dielectric constant. The boundaries between these modes are continuous, but transition frequencies can be defined as a reference for one mode to the next. The transition from the slow-wave mode to the dielectric quasi-TEM mode is given by equation 82 from chapter 4. The transition from the skin depth mode to the slow-wave mode is given by equation 87. 20SiHf (87) Where: is the resistivity of the silicon substrate in ohm-cm. 0 is the permeability of free space. SiH is the height of the silicon substrate. 5.2 MTIS CPW Fabrication The MTIS structures used in this study were fabricated on three separate substrates with BCB layered on silicon. The silicon substrate was 400 m thick with resistivities of 0.4-, 10and 25-ohm-cm. The BCB was spun-on, to a thickness of 20 m and cured in an oven with nitrogen ambient. The transmission line metal was evaporated 84

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chrome/silver/chrome/gold, Cr/Ag/Cr/Au, and formed using a lift-off process. Two CPW transmission lines were measured with the line geometries given in table 2. Table 2. MTIS CPW line dimensions. Center Conductor Width (m) Slot Width (m) Nominal Impedance MTIS1 80 45 25 MTIS2 150 30 25 To understand the mode regimes, equations 82 and 87 were used to plot the transition frequency as a function of the silicon substrate resistivity. Shown in Figure 47 are the three mode regions separated by the solid lines; the dashed lines show where each of the silicon resistivities falls on this chart. The 25 and 10 ohm-cm substrates are in transition from slow-wave to dielectric quasi-TEM mode. The 0.4 ohm-cm substrate is in the transition from slow-wave to skin depth mode. 85

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0510152025303540051015202530Silicon Resistivity (ohm-cm)Frequency (GHz)Slow-wave Region Skin Depth Region Quasi-TEM Region Figure 47. Mode regions with transition frequencies for 0.4, 10, and 25 ohm-cm silicon. 5.3 Measured Data Using the 25 ohm-cm and 10 ohm-cm Substrates The data for the MTIS structures used in this study suggest a frequency dependent complex effective dielectric constant as described in section 4.8. The dielectric quasi-TEM mode allows both the electric and magnetic fields to penetrate the substrate and the silicon acts as a lossy dielectric. This penetration of the field has the same effect as raising the substrate resistivity which affects the imaginary part of the effective dielectric constant. In the slow-wave region, below the transition frequency the real part of the effective dielectric constant increases. To find the complex permittivity, the silicon dielectric constant and resistivity in the GTR algorithm were varied to find a fit to the measured data. The complex permittivity for the silicon was extracted using equation 83 86

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and is presented as a function of frequency in Figures 48 through 51 for the 25 and 10 ohm-cm substrates. 01020304050600102030405Frequency (GHz)Re[Dielectric Constant] 0 25 and 10 ohm-cm substrates Figure 48. Real part of the substrate permittivity for MTIS1. 87

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051015200102030405Frequency (GHz)Im[Dielectric Constant] 10 ohm-cm25 ohm-cm 0 Figure 49. Imaginary part of the substrate permittivity for MTIS1. 051015202530354045500102030405Frequency (GHz)Re[Dielectric Constant] 0 25 and 10 ohm-cm substrates Figure 50. Real part of the substrate permittivity for MTIS2. 88

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051015200102030405Frequency (GHz)Im[Dielectric Constant] 10 ohm-cm25 ohm-cm 0 Figure 51. Imaginary part of the substrate permittivity for MTIS2. Figures 52 through 55 show the loss data for the 25 and 10 ohm-cm substrates in summary. Referring to Figure 44, this data represents the transition from the slow-wave mode to the dielectric quasi-TEM mode. The solid line represents measured data with the symbols representing simulated data. Two sets of simulated data appear in each graph; uncorrected and corrected. Uncorrected refers to using a constant complex dielectric constant in the GTR code. If the frequency dependence of the complex dielectric constant is ignored then the GTR code will predict too much loss as shown by the uncorrected simulated data. The corrected data uses the frequency dependent complex dielectric constant presented above. Figures 52 through 55 show excellent agreement between the corrected simulated data and measured data. 89

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-4-3-2-100102030405Frequency (GHz)Loss (dB/cm) 0 Measured Corrected Simulated Uncorrected Simulated Figure 52. Loss data for MTIS1 on 25 ohm-cm silicon. -7-6-5-4-3-2-100102030405Frequency (GHz)Loss (db/cm) 0 Measured Corrected Simulated Uncorrected Simulated Figure 53. Loss data for MTIS2 on 25 ohm-cm silicon. 90

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-10-8-6-4-200102030405Frequency (GHz)Loss (dB/cm) 0 Measured Corrected Simulated Uncorrected Simulated Figure 54. Loss data for MTIS1 on 10 ohm-cm silicon. -10-8-6-4-200102030405Frequency (GHz)Loss (db/cm) 0 Measured Corrected Simulated Uncorrected Simulated Figure 55. Loss data for MTIS2 on 10 ohm-cm silicon. 91

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5.4 Measured Data Using the 0.4 ohm-cm Substrate The skin depth mode appears when the substrate conductivity is large and the frequency of the signal is low. The behavior is similar to the slow-wave mode in that the electric field is shorted at the silicon/BCB boundary. However, the magnetic field as it spreads into the silicon is bounded by the skin effect, effectively reducing the thickness of the silicon layer. The highly conductive silicon below the skin depth region acts as a ground plane which in the GTR algorithm moves the PEC boundary closer to the BCB layer. This results in confining the field mostly to the low loss BCB layer. Figure 56 graphically shows this mode, where Hsi is the silicon thickness, Hsi is the magnetic field skin depth, and Hbcb is the BCB thickness. High conductivity silicon acts like a ground plane Hsi'= Magnetic field skin depth BCB Layer Hbcb Hsi High conductivity silicon acts like a ground plane Hsi'= Magnetic field skin depth BCB Layer Hbcb Hsi Figure 56. Pictorial of the skin depth mode. This mode was modeled in the GTR algorithm by modifying the silicon thickness, Hsi, to be equal to one-half of the magnetic field skin depth [39], equation 88. The one-half comes from the incremental inductance model. The bottom edge of this new 92

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substrate was bounded with a ground plane (PEC). Note that the substrate thickness is now frequency dependent. Figure 57 shows a plot of equation 88 for the 0.4 ohm-cm substrate. 02212SiH (88) 020406080100120140160180200481216202428323640Frequency (GHz)Skin Depth (um) Figure 57. One-half the magnetic field skin depth for a 0.4 ohm-cm substrate. The data in Figures 58 and 59 correspond to the MTIS1 and MTIS2 structures fabricated on 0.4 ohm-cm silicon and show the characteristics of transitioning from the slow-wave mode to the skin depth mode. The solid line represents measured data with the symbols representing simulated data. Two sets of simulated data appear in each graph; not accounting for and accounting for the magnetic field skin depth. When the skin depth 93

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is not accounted for the simulated data shows a significant amount of loss compared to the measured data. However, when the skin depth is accounted for the GTR algorithm closely predicts the loss. -15-12-9-6-30010203040Frequency (GHz)Loss (dB/cm) 50 Measured with Skin Effect without Skin Effect Figure 58. Loss data for MTIS1 on 0.4 ohm-cm silicon. 94

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-25-20-15-10-50010203040Frequency (GHz)Loss (db/cm) 50 Measured with Skin Effect without Skin Effect Figure 59. Loss data for MTIS2 on 0.4 ohm-cm silicon. The complex permittivity is not a function of frequency because the field is confined to the BCB layer with the low resistivity silicon acting like a ground plane. Figures 58 and 59 show excellent agreement between simulated and measured data up to about 30 GHz. Above that frequency the simulation deviates from the measured data. One possible explanation is the existence of high order current modes flowing in the surface region of the silicon. The GTR code does not account for this secondary effect. 5.5 Summary This chapter discussed two MTIS structures which were governed by a 3-mode formulation. The GTR algorithm discussed in chapters 1 through 3 was used to simulate these transmission line structures. The accuracy of the GTR code was demonstrated by comparing simulated data to measured data. The theory of these modes was discussed in 95

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section 5.1. The MTIS structures were fabricated on three separate substrates with BCB layered on silicon. Three silicon substrates were used with resistivities of 0.4-, 10and 25-ohm-cm. Two of the substrate resistivities exhibit the dielectric quasi-TEM and slow-wave behaviors and the data suggests a frequency dependent complex dielectric constants. The skin-effect mode was discussed in section 5.4 and does not have a frequency dependent complex dielectric constant. The reason for the frequency invariant complex dielectric constant is the signal propagates mostly in the low loss BCB layer with the low resistivity silicon acting as a ground conductor. The insertion loss data for the CPW transmission lines show that at frequencies below 5 GHz there was less attenuation of the signal as the silicon resistivity decreased from 10 ohm-cm to 0.4 ohm-cm. This is due to two different mode structures for each of the resistivities. For the 10 ohm-cm substrate a slow wave mode exists while the 0.4 ohm-cm substrate exhibits the skin depth mode. Decreasing the substrate resistivity to the point where the silicon exhibits the skin depth mode has application in creating low loss RF and high speed digital interconnects on CMOS grade silicon. 96

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Chapter Six Summary and Recommendations This dissertation discussed the formulation of a technique known as Generalized Transverse Resonance (GTR), which is a subset of Method of Moments. Generalized Transverse Resonance is a hybrid method combining the Transverse Resonance Method (TRM) with the Mode Matching Technique (MMT). The understanding of the generalized transverse resonance method started with a discussion of Longitudinal Section Waves and from this derives the transverse resonance method for layered media para1lel to the direction of wave propagation. It was shown that Maxwells equations can be represented as a mode function and voltage or current. This representation was used to reduce the problem of merging the TRM and MMT into the GTR method using network theory. The code was written in a module way to take advantage of this and allow very complex problems to be formulated using a CAD approach. This problem formulation philosophy is unique to this formulation of the generalized transverse resonance method. The propagation constant was found by solving the wave equation, as an eigenvalue problem, subject to the boundary conditions. Also discussed was the relative convergence phenomenon followed by the optimization strategy. 97

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6.1 Data Summary Theoretical data was compared to measured data to show the accuracy of the GTR method. Chapter 4 presented the propagation characteristics of four CPW transmission line structures in proximity with high and low loss silicon covers. Two propagation modes in the substrate were identified and discussed in sections 4.6 and 4.7. The data presented in section 4.8 suggests the substrate resistivity and dielectric constant are dependent on frequency. This behavior occurs for transmission line structures with deep penetration of the fields into the substrate which is a function of line geometry. Possible causes for the frequency dependent material properties could be in the electromagnetic interaction with the material. It is possible that relaxations of the charges in the substrate are dependent on signal power, interaction with other charges, and/or tensor properties. This study extracted the material properties from the measured data and presents a frequency dependent complex dielectric constant whose imaginary part is a function of lid height; this lid height dependency is non-physical but necessary to accurately represent the measured data in the full-wave simulations, pointing to a need for further examination of the formulation governing the dielectric property behavior in these material systems. Co-Planar Waveguide transmission lines fabricated on metal-thick-insulating-semiconductor (MTIS) substrates were the focus of chapter 5. The MTIS structures used in this study are governed by a 3-mode formulation which was discussed in section 5.1. The fabrication of two MTIS structures on three separate substrates (BCB layered on silicon) was presented in section 5.2. Three silicon substrates were used with resistivities 98

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of 0.4-, 10and 25-ohm-cm. The dielectric quasi-TEM and slow-wave behaviors are exhibited in the 25 and 10 ohm-cm substrates and have frequency dependent complex dielectric constants, section 5.3. The skin-effect mode was discussed in section 5.4. The skin depth mode appears when the substrate conductivity is large and the frequency of the signal is low. The behavior is similar to the slow-wave mode in that the electric field is shorted at the silicon/BCB boundary. However, the magnetic field as it spreads into the silicon is bounded by the skin effect, effectively reducing the thickness of the silicon layer. This results in confining the field mostly to the low loss BCB layer. Presented in this section was measured loss data compared to simulated data. The insertion loss data for the CPW transmission lines show that at frequencies below 5 GHz there was less attenuation of the signal as the silicon resistivity decreased from 10 ohm-cm to 0.4 ohm-cm. This is due to the two different mode structures for each of the resistivities. For the 10 ohm-cm substrate a slow wave mode exists while the 0.4 ohm-cm substrate exhibits the skin depth mode. Decreasing the substrate resistivity to the point where the silicon exhibits the skin depth mode has application in creating low loss RF and high speed digital interconnects on CMOS grade silicon. 6.2 Recommendations For Further Work The GTR algorithm discussed in chapters 1 through 3 was used to simulate two types of CPW transmission line structures; CPWs with superstrate layers and CPWs on metal-thick-insulating-semiconductors. The accuracy of the GTR code was demonstrated by comparing simulated data to measured data. The results of this work indicate three main areas for further work. The first is the optimization technique, followed by a 99

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rigorous metal loss model. The third area is to develop an understanding of the material properties and modify equation 83 to account for a frequency dependent complex permittivity. The simplex optimization technique used was very robust and accurate, however it was slow a typical run of the code takes 5 minutes. The slowness is due to the fact that it is a direct search algorithm which does not use a gradient to guide it. A technique has been discussed in the literature for finding the mode propagation constant using Cauchys integral formula [40]. This technique approximates the analytic characteristic equation (equation 43 from chapter 3) with a polynomial. The zeros of the polynomial are the propagation constants corresponding to the modes. This technique should speed up the process of finding the dominate mode propagation constant because the optimization algorithm is replaced by a numerical integration. Another area of research is accurate prediction of the metal loss. This model should incorporate the fabrication process and account for variations, such as surface roughness, multi-layered metal, and density. The accuracy of the GTR code was demonstrated by comparing simulated data to measured data. However, further work needs to be done to understand and model the frequency dependent material parameters, dielectric constant and substrate resistivity. This behavior occurs for transmission line structures with deep penetration of the fields into the substrate. The field penetration is a function of line geometry. A technique for 100

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measuring the complex permittivity has been developed for microstrip [32]. This technique would need to be modified for CPW geometries before permittivity data can be taken. From this data the material parameters can be extracted and a model can be developed and incorporated into equation 83. 101

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References [1] S. B. Cohn,Properties of ridged waveguide, Proc. IRE., vol 35, Aug 1947, pp 783-788. [2] S. Ramo, J. Whinnery, and T. Van Duzer, Fields and waves in Communication Electronics, John Wiley & Sons, 1965, p 23. [3] S. Peng and A. Oliner, Guidance and Leakage Properties of a Class of Open Dielectric Waveguides: Part I Mathematical Formulations, IEEE Transactions on Microwave Theory and Techniques, Vol. MTT-29, No. 9, September 1981. pp 843-855. [4] N. Marcuvitz, Waveguide Handbook, McGraw-Hill Inc, 1951, reprinted by Peter Peregrinus Ltd, 1993, p 399. [5] R. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill Inc, 1961, p 381. [6] P. Masterman and P. Clarricoats, Computer field-matching solution of waveguide transverse discontinuities, Proc. IEE, vol. 118, no. 1, Jan 1971, pp 51-63. [7] G. Schiavon, R. Sorrentino, and P. Tognolatti, Characterization of coupled finlines by generalized transverse resonance method, International Journal of Numerical Modelling:Electronic Networks, Devices, and Fields, Vol. 1, 1988, pp 45-59. [8] R. Harrington, Time-Harmonic Electromagnetic Fields, McGraw-Hill Inc, 1961, p 129. [9] R. Collin, Field Theory of Guided Waves, McGraw-Hill Inc, 1960, chapter 6. [10] M. Sadiku, Numerical Techniques in Electromagnetics, CRC Press, 1992, p 30. [11] N. Marcuvitz, Waveguide Handbook, McGraw-Hill Inc, 1951, reprinted by Peter Peregrinus Ltd, 1993, p 3. [12] M. Sadiku, Numerical Techniques in Electromagnetics, CRC Press, 1992, p 37. 102

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[13] V. Labay, J. Bornemann, Matrix Singular Value Decomposition of the Pole-Free Solutions of Homogeneous Matrix Equations as Applied to Numerical Modeling Methods, IEEE Microwave and Guided Wave Letters, Vol 2, No 2, Feb 1992. [14] R. Sorrentino, Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, edited by Tatsuo Itoh, John Wiley & Sons, 1989, Chapter 11. [15] R. Mittra, Relative Convergence of the Solution of a Doubly Infinite Set of Equations, Journal of Research of the Nat. Bureau of Standards, Vol. 67D, No 2, March-April 1963. [16] G. Eleftheriades, A. Omar, L. Katehi, G. Rebeiz, Some Important Properties of Waveguide Junction Generalized Scattering Matrices in the Context of the Mode Matching Technique, IEEE Trans on Microwave Theory and Techniques, Vol 42, No 10, Oct 1994. [17] M. Leroy, On the Convergence of Numerical Results in Modal Analysis, IEEE Trans on Antennas and Propagation, Vol AP-31, No 4, Jul 1983. [18] R. Mansour, R. MacPhie, A Unified Hybrid-Mode Analysis for the Planar Transmission Lines with Multilayer Isotropic/Anisotropic Substrates, IEEE Trans on Microwave Theory and Techniques, Vol 35, No 12, Dec 1987. [19] V. Labay, J. Bornemann, Matrix Singular Value Decomposition of the Pole-Free Solutions of Homogeneous Matrix Equations as Applied to Numerical Modeling Methods, IEEE Microwave and Guided Wave Letters, Vol 2, No 2, Feb 1992. [20] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C, Cambridge University Press, 1988, p 305. [21] D. Williams, B. Alpert, U. Arz, D. Walker, H. Grabinski, Causal Characteristic Impedance of Planar Transmission Lines, IEEE Trans on Advanced Packaging, Vol 26 No 2 May 2003. [22] L. Schmidt, T. Itoh, H. Hofmann, Characteristics of Unilateral Fin-Line Structures with Arbitrarily Located Slots , IEEE Trans on Microwave Theory and Techniques, Vol 29 No 4 Apr 1981. [23] L. Katehi, J. Harvey, and K. Herrick, -D Integration of RF Circuits Using Si Micromachining, IEEE Microwave Magazine, Vol. 2, March 2001. [24] K. Herrick, J. Yook, and L. Katehi, Microtechnology in the Development of 3-D Circuits, IEEE Trans on Microwave Theory and Techniques, Vol. 46, Nov 1998. 103

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[25] H. Hasegawa, M. Furukawa, H. Yanai, Properties of Microstrip Line on Si-SiO 2 System, IEEE Trans on Microwave Theory and Techniques, Vol. 19, No 11, Nov 1971. [26] R. Marks, A Multiline Method of Network Analyzer Calibration, IEEE Trans on Microwave Theory and Techniques, Vol. 39, No 7, Jul 1991. [27] D. Williams, J. Belguin, G. Dambrine, R. Fenton, On-Wafer Measurement at Millimeter Wave Frequencies, IEEE MTT-S International Microwave Symposium Digest, 1996, Vol. 3, Jun 1996. [28] S. Zivanovic, S. Ristic, Z. Prijic, The Effect of Static Dielectric Constant on the Ban Structure in Heavily Doped Silicon, Proc. 20 th International Conference on Microelectronics, Vol 1, Sep 1995. [29] C. Schollhorn, W. Zhao, M. Morschbach, E. Kasper, Attenuation Mechanisms of Aluminum Millimeter-Wave Coplanar Waveguides on Silicon, IEEE Transactions on Electron Devices, Vol 50, No 3, Mar 2003. [31] W. Heinrich, Quasi-TEM Description of MMIC Coplanar Lines Including Conductor-Loss Effects, IEEE Trans on Microwave Theory and Techniques, Vol 41, No 1, Jan 1993. [32] M. Janezic, D. Williams, V. Blaschke, A. Karamcheti, C. Chang, Permittivity Characterization of Low-k Thin Films From Transmission-Line Measurements, IEEE Trans on Microwave Theory and Techniques, Vol. 51, No 1, Jan 2003. [33] Y. Kwon, V. Hietala, K. Champlin, Quasi-TEM Analysis of Slow-Wave Mode Propagation on Coplanar Microstructure MIS Transmission Lines, IEEE Trans on Microwave Theory and Techniques, Vol. 35, No 6, Jun 1987. [34] E. Tuncer, D. Neikirk, Highly Accurate Quasi-Static Modeling of Microstrip Lines over Lossy Substrates, IEEE Microwave and Guided Wave Lett., Vol. 2, 1992. [35] V. Milanovic, M. Ozgur, D. DeGroot, J. Jargon, M. Gaitan, M. Zaghloul, Characterization of Broad-Band Transmission for Coplanar Waveguides on CMOS Silicon Substrates, IEEE Trans on Microwave Theory and Techniques, Vol. 46, No 5, May 1998. [36] G. Ponchak, L. Katehi, Measured attenuation of coplanar waveguide on CMOS grade silicon substrates with polyimide interface layer, Electronics Letters, Vol 34, No 13, Jun 1998. 104

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[37] G. Ponchak, A. Margomenos, L. Katehi, Low-Loss CPW on Low-Resistivity Si Substrates with a Micromachined Polyimide Interface Layer for RFIC Interconnects, IEEE Trans on Microwave Theory and Techniques, Vol. 49, No 5, May 2001. [38] A. von Hipple, Dielectrics and Waves, John Wiley & Sons, 1954, p.228. [39] H. Wheeler, Formulas for the skin effect, Proc. IRE, Vol. 30, Sep 1942. [40] R. Sorrentino, The ZEPLS Program for Solving Characteristic Equations of Electromagnetic Structures, IEEE Trans on Microwave Theory and Techniques, Vol. 23, No 5, May 1975. 105

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106 Appendices

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Appendix A: Metal Loss Closed form equations exist in the literature for predicting the metal loss [29] [30] [31]. The conductor loss for the CPW and MTIS lines were calculated using equations from [29] which are derived using conformal mapping. The conductor loss in dB/m is given by equations 89 and 90. )]()([)1)(()(240686.82dwkkKkKRreffSC (89) kktxxx114ln)( (90) Where: is the center conductor width. w is the ground-to-ground spacing. d dwk and is the geometric ratio. is the surface resistance. SR is the complete elliptic integral. )(kK 21kk is the complementary module. The surface resistance term is due to the ohmic loss from the currents flowing in the metal. This term is frequency dependent and very sensitive to the process conditions 107

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Appendix A (Continued) of the metal. The definition used in this study is based on a unit square with a thickness equal to the skin depth. The surface resistance is given by equation 91, OSfR (91) where is the resistivity of the metal and is the skin depth. The metal resistivity is significantly affected by metal density, metal adhesion layers, and surface roughness. All this parameters are functions of the fabrication process. 108

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109 Appendix B: GTR Compared to HFSS Tables 4 through 7 show the complete comparison between simulated data using GTR and HFSS. Shown in each table are the r eal and imaginary parts of the propagation constant (k) for each method and the percent difference. Table 3. 2000 ohm-cm lid with no metal loss. HFSS GTR Percent difference Re[k] Im[k] Re[k] Im[k] Re[k] Im[k] 5 um 2461.8 -2.02 2464.8 -2.03 0.12 0.25 10 um 2350.6 -1.93 2348.8 -1.93 -0.08 -0.13 25 um 2224.6 -1.89 2221.9 -1.89 -0.12 -0.10 50 um 2162.4 -1.89 2159.5 -1.89 -0.13 -0.07 100 um 2126.7 -1.90 2125.6 -1.90 -0.05 -0.01 CPW1 no lid 2104.8 -1.89 2108.2 -1.90 0.16 0.26 5 um 2504.3 -2.08 2500.6 -2.08 -0.14 -0.17 10 um 2390.6 -1.97 2383.8 -1.96 -0.28 -0.36 25 um 2255.2 -1.91 2247.0 -1.90 -0.36 -0.37 50 um 2181.7 -1.91 2175.2 -1.90 -0.30 -0.27 100 um 2138.1 -1.91 2133.6 -1.91 -0.21 -0.23 CPW2 no lid 2107.6 -1.91 2131.1 -1.95 1.11 2.27 5 um 2569.3 -2.23 2570.4 -2.24 0.04 0.12 10 um 2475.3 -2.11 2468.7 -2.10 -0.27 -0.28 25 um 2337.2 -2.01 2326.6 -2.00 -0.45 -0.32 50 um 2245.3 -1.99 2235.3 -1.98 -0.45 -0.23 100 um 2175.2 -1.99 2168.5 -1.98 -0.31 -0.16 CPW3 no lid 2098.2 -1.96 2090.6 -1.97 -0.36 0.43 5 um 2537.0 -2.31 2557.9 -2.33 0.83 0.80 10 um 2464.1 -2.20 2474.2 -2.20 0.41 0.36 25 um 2344.9 -2.09 2346.0 -2.09 0.05 0.25 50 um 2253.3 -2.06 2253.4 -2.06 0.01 0.45 100 um 2173.4 -2.04 2175.7 -2.06 0.11 0.56 CPW4 no lid 2050.0 -1.98 2038.5 -2.02 -0.56 1.88

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110 Appendix B (Continued) Table 4. 10 ohm-cm lid with no metal loss. HFSS GTR Percent difference Re[k] Im[k] Re[k] Im[k] Re[k] Im[k] 5 um 2474.4 -85.10 2477.9 -85.43 0.14 0.39 10 um 2356.0 -50.41 2357.0 -49.71 0.05 -1.39 25 um 2228.2 -22.64 2224.8 -21.82 -0.15 -3.61 50 um 2161.6 -11.99 2160.4 -11.81 -0.06 -1.47 100 um 2127.8 -6.45 2125.6 -6.79 -0.10 5.38 CPW1 no lid 2104.8 -1.89 2108.2 -1.90 0.16 0.26 5 um 2514.9 -103.42 2514.4 -101.31 -0.02 -2.03 10 um 2399.4 -63.92 2393.1 -61.44 -0.26 -3.87 25 um 2259.6 -29.44 2250.6 -27.92 -0.40 -5.17 50 um 2183.1 -15.88 2176.3 -15.30 -0.31 -3.71 100 um 2137.9 -8.49 2133.5 -8.73 -0.20 2.88 CPW2 no lid 2107.6 -1.91 2131.1 -1.95 1.11 2.27 5 um 2579.8 -148.27 2583.9 -144.88 0.16 -2.29 10 um 2484.3 -107.00 2479.5 -101.94 -0.19 -4.73 25 um 2342.6 -61.50 2331.5 -57.05 -0.47 -7.23 50 um 2247.3 -38.65 2236.4 -36.17 -0.48 -6.43 100 um 2174.9 -23.20 2167.6 -22.78 -0.34 -1.82 CPW3 no lid 2098.2 -1.96 2090.6 -1.97 -0.36 0.43 5 um 2546.8 -175.01 2568.5 -171.32 0.85 -2.10 10 um 2471.6 -138.28 2483.5 -132.15 0.48 -4.43 25 um 2348.1 -92.33 2350.2 -85.68 0.09 -7.20 50 um 2254.0 -65.79 2253.3 -60.89 -0.03 -7.45 100 um 2171.5 -44.75 2172.8 -42.66 0.06 -4.66 CPW4 no lid 2050.0 -1.98 2038.5 -2.02 -0.56 1.88

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111 Appendix B (Continued) Table 5. 2000 ohm-cm lid with metal loss. HFSS GTR Percent difference Re[k] Im[k] Re[k] Im[k] Re[k] Im[k] 5 um 2469.0 -9.14 2464.8 -13.84 -0.17 51.42 10 um 2356.0 -8.56 2348.8 -13.18 -0.30 54.08 25 um 2232.5 -8.61 2221.9 -12.53 -0.47 45.47 50 um 2169.3 -8.85 2159.5 -12.23 -0.45 38.13 100 um 2134.5 -8.18 2125.6 -12.07 -0.41 47.66 CPW1 no lid 2109.8 -8.60 2108.2 -11.97 -0.07 39.21 5 um 2509.0 -7.91 2500.6 -15.60 -0.33 97.16 10 um 2394.5 -7.63 2383.8 -14.83 -0.44 94.24 25 um 2259.3 -6.98 2247.0 -14.01 -0.54 100.73 50 um 2186.4 -6.78 2175.2 -13.61 -0.51 100.82 100 um 2142.3 -7.00 2133.6 -13.39 -0.41 91.15 CPW2 no lid 2111.7 -6.77 2131.1 -13.26 0.92 95.95 5 um 2572.6 -5.34 2570.4 -7.35 -0.08 37.63 10 um 2477.6 -5.07 2468.7 -7.06 -0.36 39.24 25 um 2341.5 -4.65 2326.6 -6.65 -0.63 43.09 50 um 2247.5 -4.69 2235.3 -6.39 -0.54 36.19 100 um 2178.9 -4.48 2168.5 -6.20 -0.48 38.40 CPW3 no lid 2102.5 -4.46 2090.6 -5.98 -0.57 34.09 5 um 2539.4 -4.69 2557.9 -5.13 0.73 9.39 10 um 2467.1 -4.03 2474.2 -4.96 0.29 22.85 25 um 2345.0 -3.73 2346.0 -4.70 0.05 26.03 50 um 2254.7 -3.76 2253.4 -4.51 -0.06 20.02 100 um 2174.1 -3.60 2175.7 -4.36 0.08 21.06 CPW4 no lid 2050.5 -3.73 2038.5 -4.19 -0.58 12.29

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112 Appendix B (Continued) Table 6. 10 ohm-cm lid with metal loss. HFSS GTR Percent difference Re[k] Im[k] Re[k] Im[k] Re[k] Im[k] 5 um 2478.7 -93.14 2477.9 -97.31 -0.03 4.48 10 um 2362.8 -57.44 2357.0 -61.01 -0.24 6.22 25 um 2235.4 -29.47 2224.8 -32.47 -0.47 10.18 50 um 2170.8 -18.64 2160.4 -22.15 -0.48 18.81 100 um 2133.3 -13.28 2125.6 -16.97 -0.36 27.83 CPW1 no lid 2109.8 -8.60 2108.2 -11.97 -0.07 39.21 5 um 2520.7 -109.39 2514.4 -114.91 -0.25 5.05 10 um 2405.7 -69.33 2393.1 -74.37 -0.52 7.26 25 um 2263.4 -34.91 2250.6 -40.05 -0.56 14.71 50 um 2188.7 -20.92 2176.3 -27.01 -0.56 29.14 100 um 2140.9 -13.79 2133.5 -20.21 -0.34 46.56 CPW2 no lid 2111.7 -6.77 2131.1 -13.26 0.92 95.95 5 um 2585.8 -151.77 2583.9 -152.27 -0.07 0.33 10 um 2489.6 -110.03 2479.5 -109.03 -0.40 -0.91 25 um 2345.3 -64.30 2331.5 -63.72 -0.59 -0.91 50 um 2250.4 -41.45 2236.4 -42.56 -0.62 2.68 100 um 2176.6 -25.85 2167.6 -28.98 -0.41 12.09 CPW3 no lid 2102.5 -4.46 2090.6 -7.95 -0.57 78.32 5 um 2546.8 -177.03 2568.5 -176.47 0.85 -0.32 10 um 2474.0 -140.33 2483.5 -137.12 0.38 -2.28 25 um 2350.1 -94.10 2350.2 -90.39 0.01 -3.95 50 um 2255.3 -67.48 2253.3 -65.40 -0.09 -3.08 100 um 2171.4 -44.30 2172.8 -47.01 0.07 6.11 CPW4 no lid 2050.5 -3.73 2038.5 -6.21 -0.58 66.42

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About the Author James Culver has over 18 years of experience in the microwave industry. Mr. Culver spent 7 years at Texas Instruments wh ere he was responsible for the development of narrow and wide band GaAs MMIC circui ts for use in radar and communication systems. Nine additional years were spent at Raytheon developing RF/Microwave space, airborne, and land based communication systems. Currently, he is an Engineering Fellow at Northrop Grumman responsible for th e technical management of miniature communication systems design. Mr. Culver holds three degrees; BA in chemistry, BS in electrical engineering and an MS in electrical engineering. Mr. Culver holds three patents and has authored several papers on MMIC/m icrowave circuit design and antennas.