# Analysis and modeling of linear and nonlinear microwave superconducting devices

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ANALYSIS AND MODELING OF LINEAR AND NONLINEAR M ICROW AVE SUPERCONDUCTING DEVICES by Mohamed Abdel Fattah Megahed A Dissertation Presented in Partial Fulfillment o f the Requirements for the Degree Doctor of Philosophy ARIZONA STATE UNIVERSITY August 1995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. DMI Number: 9538169 UMI Microform 9538169 Copyright 1995, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ANALYSIS AND MODELING OF LINEAR AND NONLINEAR MICROWAVE SUPERCONDUCTING DEVICES by Mohamed Abdel Fattah Megahed has been approved July 1995 APPROVED: . Chairperson 3 d . (lJ u J o A iMJAa 9 -----------------Supervisory Committee ACCEPTED: Dentfmhent Chairperson Dean, Graduate College Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT High Temperature Superconducting (HTS) materials have potential applications for microwave and millimeter-wave devices. The performance of superconductors is substantially superior to normal conductors and semiconductors concerning low loss, high sensitivity and low dispersion. However, new methodologies are needed for the design and analysis o f such devices. Field penetration effects must be taken into consideration, especially for high power applications. As w ith other fabrication technologies, it is desirable to simulate these devices before they are built to save money and time. Similarly, to exploit the exciting characteristics o f these new materials, accurate and flexible models have to be developed. An accurate analysis for microwave and millimeter-wave devices, which include high temperature superconductor materials, is presented in this dissertation. This study covers both low linear and high nonlinear power applications. This analysis is based on blending a fu ll electromagnetic wave model with phenomenological linear and nonlinear superconductor model, and the two-fluid model. The linear model is based on the low power London's model. On the other hand, the nonlinear model is developed using the Ginzburg-Landau theory. These models are capable o f fu lly characterizing HTS microwave devices, including obtaining the current distributions inside the superconducting material, the electromagnetic fields, and the power handling capability. These solutions are obtained using the finite-differcnce scheme. The superconductor thickness is rigorously modeled. No approximations are made to the superconductor thickness. The anisotropy associated with the superconductor is also considered. The linear problem can be solved in either the frequency or the time domain. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. However, the nonlinear solution must be performed in the time domain. This approach is employed to investigate HTS transmission lines and filter. Results have shown that the number o f superfluid electrons decreases near the edges o f transmission strips as the applied power increases, indicating the breaking o f superfluid electrons pairs. The linear model underestimates the magnetic field penetration inside the superconductor. The change in the losses with the applied field is much larger than the change in the velocity o f the wave propagating along the device. A variation in the frequency spectrum o f the applied signal resulting from the nonlinearity is seen. Also, simulation o f HTS filte r has revealed that dimension and layout o f HTS filters must be optimized in the design cycle to avoid nonlinearity effects. A novel nonlinear phenomenological tw o-fluid model for superconducting materials has also been proposed, where the thermodynamics and electromagnetics properties o f HTS are considered simultaneously. This model is very useful for computer-aided design applications. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This work is dedicated to the memory of my parents who have brought me up and given me their love, to my wife Sawsan who always comforts and consoles, never complains or interferes, and asks nothing and endures all, and to my children Ahmed, Jylan, and Samer. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS I would like to express my deepest indebtedness to Dr. Samir M. El-Ghazaly for his consent to be my thesis advisor, for suggesting the area o f research, and his constant support and patience. His valuable guidance, as well as the time and effort he spent, make this dissertation finally come true. I thank him for sending me to the IEEE MTT-S, IEEE AP-S, URSI, and PIERS Conferences, and giving me a place among other researchers. I am indebted to him for his able assistance and interest in this academic research and many other areas of my life. I would also like to thank my committee members Dr. I. Kaufman, Dr. R. Grondin, Dr. E. El-Sharawy, Dr. J. Aberle and Dr. K. Schmitt for their critical reading of this dissertation and valuable comments. Deep appreciation is due to my friends and colleagues at Arizona State University for their constant help and useful discussions. In particular, special thanks to M. A lSunaidi for many stimulating conversations and T. El-Shafiey for his eternal friendship. Finally, I acknowledge God who not only loaned me the talent and abilities necessary to complete this degree, but put me in the right circumstances so that I am where I am today. He is, at the root, responsible for all the acknowledgments above. vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page LIST OF TA BLE S .............................................................................................................. x LIST OF FIGURES............................................................................................................ xi CHAPTER 1 INTRODUCTION...............................................................................................1 1.1 Problem Definition..........................................................................................2 1.2 Motivation for the Selected Techniques.......................................................... 4 1.3 Thesis O utline................................................................................................ 7 2 LINEAR AND NONLINEAR SUPERCONDUCTOR M ODELS...................13 2.1 Material Models of Superconductors.............................................................15 2.2 The Two-Fluid M odel................................................................................... 16 2.3 Temperature Dependence of Superconductors..............................................17 2.4 London Phenomenological M odel.................................................................18 2.5 Ginzburg-Landau Phenomenological M odel.............................................. 20 2.5.1 Theory................................................................................................ 20 2.5.2 Limitations on GL Theory.................................................................. 26 2.5.3 GL for Microwave HTS Applications................................................ 27 2.6 Solution of GL Equations...............................................................................29 2.6.1 Numerical Scheme............................................................................. 29 2.6.2 Bulk Superconductor (Superconducting Half-Space)........................ 32 2.6.3 Thin Superconductor F ilm ................................................................. 35 2.7 3 Summary........................................................................................................ 40 FINITE-DIFFERENCE APPROACH.................................................... 42 3.1 Finite-Difference Frequency-Domain (FDFD)............................................. 45 3.1.1 Wave Equation for Superconducting Microwave Structure............. 45 vii Reproduced with permission of the copyright owner. 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CHAPTER Page 3.1.2 Eigenvalue Problem............................................................................ 48 3.1.3 FDFD Application to Microstrip L in e ................................................ 52 3.2 Finite-Difference Time-Domain (FD TD )...................... 63 3.2.1 Nonuniform Finite-Difference Mesh Generator................................. 65 3.2.2 Absorbing Boundary Conditions: Perfectly Matched Layers............ 69 3.2.3 Parallel Implementation of the FDTD on MASPAR Machine.......... 77 3.2.4 FDTD Application to Microstrip L in e ................................................ 86 3.3 Comparison between FDFD and FDTD Solutions........................................ 88 3.4 Summary 4 :................................................................................................92 ANISOTROPIC SUPERCONDUCOR ON ANISOTROPIC SUBSTRATES 93 4.1 Anisotropic High Temperature Superconductor Model................................. 95 4.2 Anisotropic Finite-Difference Time-Domain Approach................................ 97 4.3 Anisotropic Superconductor Microstripline on Anisotropic Sapphire Substrate....................................................................................................... 100 4.4 Anisotropic Superconductor Coplanar Waveguide on Anisotropic Sapphire Substrate....................................................................................................... 108 4.5 5 Summary........................................................................................................112 FULL-W AVE NONLINEAR ANALYSIS OF MICROWAVE SUPERCONDUCTING DEVICES..................................................................113 5.1 Time-Domain versus Frequency-Domain Numerical Techniques................116 5.2 Nonlinear Full-Wave Superconductor M odel...............................................117 5.2 Nonlinear Full-Wave Superconductor Simulator..........................................121 5.4 Nonlinear Analysis o f Superconducting Microstrip Lines........................... 124 5.5 Nonlinear Analysis o f Superconducting Filters............................................138 5.5.1 Simulation of Microstrip Resonator Array HTS F ilte r.................... 139 viii Reproduced with permission of the copyright owner. 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CHAPTER Page 5.5.2 High Power Design Consideration for HTSF ilters.......................... 147 5.6 6 Summary...................................................................................................... 148 NOVEL NONLINEAR PHENOMENOLOGICAL TW O FLU ID M O D E L. 150 6.1 Nonlinear Phenomenological Two fluid model........................................... 152 6.2 Macroscopic Model o f Nonlinear Constitutive relationsin H T S ................. 157 6.3 HTS Nonlinear Surface Impedance...............................................................159 6.4 Nonlinear Model Validation and Verification..............................................163 6.5 Summary....................................................................................................... 167 7 CONCLUSIONS.................................................................................:............ 168 7.1 Summary of Findinds and Conclusions........................................................ 169 7.2 Recommendations for Future Research........................................................ 172 REFERENCES..................................................................................................................174 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table 3.1 Page Number o f Processors and Machine Size for available options........................82 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Page Figure 2.1 Superconducting Half Space (bulk superconductor)..........................................33 2.2 Normalized superfluid current density in one-dimensional bulk Y B A 2Cu3 0 7 .x XA T-ITO i l l J ......................................................................................................................................................,....................................................................................................................._ > * + 2.3 Y B A 2CU3O7-X HTS strip (thin film superconductor).........................................35 2.4 Tangential magnetic field intensity to a YBa2Cu307-x HTS strip in microstrip line configuration............................................................................................... 36 2.5 Normalized superfluid current density distribution in YBa2Cu307-x HTS strip in microstrip line configuration at 0.2 Pcrf applied power................................ 37 2.6 Normalized superfluid current density distribution in YBa2Cu307-x HTS strip in microstrip line configuration at 0.45 Pcrf applied power.............................. 38 2.7 Normalized superfluid current density distribution in YBa2Qi307-x HTS strip in microstrip line configuration at 0.9 Pcrf applied power................................ 39 2.8 Normalized superfluid electron density distribution in YBa2Cu307-x HTS strip in microstrip line configuration at 0.45 Pcrf applied power.............................. 40 3.1 Mesh indices ij and the corresponding xy axis..................................................49 3.2 Superconductor microstrip line geometry..........................................................52 3.3 Phase constant of superconducting microstrip line filled with air at different temperatures...................................................................................................... 54 3.4 Attenuation constant of superconducting microstrip line filled with air at different temperatures....................................................................................... 55 3.5 Propagation constant of superconducting microstrip line on lossless dielectric............................................................................................................ 56 3.6 Phase constant of superconducting microstrip line on lossless substrate at different temperatures........................................................................................57 3.7 Attenuation constant of superconducting microstrip line filled with air at different temperatures....................................................................................... 58 3.8 Attenuation constant of superconducting microstrip line on a lossless substrate at different penetration depth.............................................................................60 3.9 Relative phase constant of superconducting microstrip line on a lossless substrate at different penetration depth..............................................................61 xi Reproduced with permission of the copyright owner. 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Page Figure 3.10 Attenuation constant o f superconducting microstrip line on a lossless and lossy substrates at different temperatures....................................................................62 3.11 Nonuniform mesh for half microstrip line structure................................. 3.12 Electric field calculated at the same physical positions using the uniform and nonuniform discretizations with the same number of mesh points....................68 3.13 Perfectly matched layers absorbing boundary condition in three-dimension cartesian coordinates..........................................................................................73 3.14 Perfectly matched layers absorbing boundary condition for waveguiding structures that include different dielectric materials and metallic conductors.. 76 3.15 Virtual Layers in the processing elements memory........................................... 83 3.16 CPU time for MASAPR and DBM RS/6000 machines for problems that have different sizes and equal number ot time steps.................................................. 85 3.17 Effective dielectric constant (er = 13). Comparison of the results obtained fromthe FDTD with the empirical formula and the results presented in [52]... 87 3.18 Attenuation constant for copper and superconducting microstrip lines with different strip thickness using the FDTD and FDFD approaches......................90 3.19 Effective dielectric constant for lossless, copper and superconducting microstrip lines with different strip thickness using the FDTD and FDFD approaches.......................................................................................................... 91 4.1 Anisotropic microstrip line on anisotropic sapphire substrate......................... 101 4.2 The propagation characteristics of anisotropic HTS on isotropic substrate using the anisotropic FDTD....................................................................................... 102 4.3 Normalized normal-fluid, super-fluid, and total current densities at the bottom surface o f the strip, for both the isotropic and anisotropic HTS cases, on isotropic substrate............................................................................................. 103 4.4 Propagation characteristics for anisotropic HTS on different r-cut sapphire substrates and on isotropic substrate with er = 10.03..................................... 106 4.5 Normal-fluid, super-fluid, and total current densities for anisotropic HTS on different r-cut sapphire substrates and on isotropic substrate er = 10.03.......107 4.6 Anisotropic HTS coplanar waveguide on anoistropic substrate...................... 108 4.7 Propagation characteristics for anisotropic HTS coplanar waveguide on different r-cut sapphire substrates and on isotropic substrate with er - 10.03110 Xll Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Figure Page 4.8 Normal-fluid, super-fluid, and total current densities for anisotropic HTS coplanar waveguide on different r-cut sapphire substrates and on isotropic substrate sr = 10.03......................................................................................... I l l 5.1 Flow chart o f the nonlinear analysis algorithm................................................. 123 5.2 HTS microstrip line geometry........................................................................... 124 5.3 Normalized tangential magnetic field intensiy under the strip probed at 60 /im and 150 } im .......................................................................................................126 5.4 Normalized tangential magnetic field intensiy under the strip probed at 150 fim at different levels of applied power...................................................................127 5.5 Effective dielectric constant for the HTS microstrip line at different levels of applied power................................................................................................... 128 5.6 Fractional change in the effective dielectric constant for the HTS microstrip line with applied power w.r.t. the linear model................................................ 129 5.7 Attenuation constant for the HTS microstrip line at different levels of applied power.................................................................................................................130 5.8 Fractional change in the attenuation constant for the HTS microstrip line with applied power w.r.t. the linear model..............................................................131 5.9 Normalized longuitudenal super fluid current density at the bottom surface of the HTS strip at different applied power levels................................................ 133 5.10 Normalized longuitudenal super fluid current density at the top surface o f the HTS strip at different applied power levels..................................................... 134 5.11 Normalized longuitudenal super fluid current density at the side surface o f the HTS strip at different applied power levels..................................................... 135 5.12 Normalized tangential magnetic field intensity at the top and bottom surface of the HTS strip at different applied power levels................................................ 136 5.13 Fractional change in the amplitude of the frequency spectrum o f the output pulse w.r.t the dc component at different applied power levels....................... 137 5.14 HTS microstrip resonator array filter structure................................................ 139 5.15 The calculated S21 parameter for the HTS microstrip staggered resonator array filter................................................................................................................... 143 xiii Reproduced with permission of the copyright owner. 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Figure Page 5.16 Comparison between the calculated and the measured S21 parameters for the HTS microstrip staggered resonator array filter............................................... 144 5.17 Output input power relation o f the HTS microstrip resonator array bandpass filter...................................................................................................................145 5.18 Electric field distribution in the substrate along the longitudinal direction of the bandpass filter...................................................................................................146 6.1 Variation of the number of superfluid electrons and the magnetic field H near a flux lin e ............................................................................................................ 153 6.2 The function ( l — )as function of h for different values of the variable a ........................................................................................................ 162 6.3 Comparison between the calculated and the measured [106] critical magnetic field for YBCO HTS as a function of temperature.......................................... 164 6.4 Comparison between the calculated and the measured [106] surface resistance for YBCO HTS as a function of temperature at zero magnetic fie ld .............. 165 6.5 Comparison between the calculated and the measured [106] surface resistance for YBCO HTS as a function of temperature and magnetic fie ld ..................166 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 1 INTRODUCTION In late 1986, J.G. Bednorz and K.A. Muller o f IBM's Zurich Lab reported possible superconductivity in La and Ba copper oxides at temperature o f 30°K. These materials are known as High critical-Temperature Superconductors (HTS). Following their discovery, the HTS phenomena have been found in various ceramic oxides having critical temperatures Tc as high as 123°K. The importance of these discoveries is based on the fact that these materials can be cooled by using inexpensive liquid nitrogen, with a 77.4°K boiling point, rather than liquid helium so that superconductivity applications become economically viable. This research opened a broad range o f applications in electronic systems. The performance of superconductors is substantially superior to normal conductors and semiconductors concerning low loss, dispersion, low noise, high sensitivity, and highest frequency of operation. Some HTS materials can be easily deposited on a substrate in a thin film form. These films have critical fields and currents that are well within the operating region of most microwave and microelectronics circuits. In addition, they provide lower resistance than either copper or bulk superconductors and are able to carry much higher currents per cross section than other metals. The low resistance o f superconducting materials is attractive for applications in antennas, filters, delay lines, interconnects, microwave matching networks and other sub systems [1]. The workhorse HTS materials are YBa 2Cu 3 0 * and TIBaCaCuO. Good quality HTS epitaxial films can be deposited on low loss dielectric substrates, such as Silica, Sapphire, Lithium Niobate, MgO, or LaA 1 03 , to form planar microwave structures Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 [2]. For these reasons, thin HTS films make a good choice for microwave applications, and pave the way for extremely small conductor patterns and circuits. Research and development performed on superconducting devices has accelerated since the discovery of HTS materials [3]-[ 15]. One goal o f this research is to develop devices that w ill have lower losses and better operating characteristics than normal metal devices. The technology used to fabricate such devices is not mature yet. It is difficult to build devices with consistently good results. However, many devices have been designed, built, and tested. As with other fabrication technologies, it is desirable to simulate these devices before they are built to save time and money. To exploit the exciting characteristics of HTS materials, accurate and flexible numerical models have to be developed. Moreover, many fundamental electrical parameters of the superconductor material, such as the surface resistance, conductivity, critical temperature, magnetic field strength, and fie ld penetration are determined by measuring the propagation characteristics and the quality factors of microwave devices [ 16]-[ 17]. Therefore, it is also important to develop accurate numerical models to determine these electrical parameters precisely from the measurements. 1.1 Problem Definition The use of HTS in microwave and millimeter-wave devices presents new challenges which are not relevant in the present design of normal metal devices. The first major difference is that the currents in HTS strips are not limited to the surface o f the conductors. The superconducting current flows along the cross section of the entire strip. Hence, the field penetration effects on the device performance must be considered. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Another important aspect of the physics of the HTS is the layered structure and the associated large anisotropy [18]-[20]. It is generally believed that the two- dimensional C u0 2 network is the most essential block of the HTS materials. The main macroscopic parameters o f the material, the magnetic field penetration depth and the normal conductivity, have been measured and found to be anisotropic according to the layered crystal structure of the HTS. Experiments also reported that the critical current density and the upper critical field are anisotropic [21 ]-[23]. Moreover, anisotropic substrates seem to be appropriate substrate materials for HTS applications, such as sapphire and boron nitride [24], Crystal lattice matching of the sapphire with the c-axis oriented YBCO, small dielectric loss, and high thermal conductivity can be achieved simultaneously [25]-[28]. The microwave device designer is now faced w ith several choices, as the type o f material and film direction, to obtain the optimum configuration that enhances the characteristics of the HTS material on anisotropic substrate. Hence, the isotropy assumption for HTS materials is inappropriate for the accurate analysis o f HTS microwave and millimeter-wave devices. Passive HTS devices, such as filters, multiplexers, and delay lines, can provide better performance than conventional thin-film technologies [29]-[38]. However, with increasing input power, typical HTS devices become nonlinear and their losses increase. Although, the high current value may not exceed the HTS critical current densities of high quality YBCO films, they are high enough to drive the HTS into nonlinear behavior [39]-[44], The nonlinear characteristics o f the HTS results in the generation of harmonics and spurious products created by the mixing of multiple input signals [45]. To date, HTS technology has only been able to address "receive" but not "transmit" applications due to nonlinearity effects. In order to efficiently use HTS in microwave and millimeter-wave applications, it is crucial to understand the dependence o f the field penetration depth, as Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 well as the superconductor electron density, on the electromagnetic field inside the HTS. Therefore, studying and simulating the nonlinearity associated with the new HTS materials w ill open a new frontier for HTS applications. Nonlinearity is a complicated issue in all areas o f engineering applications. Typically, modeling and simulation of the nonlinearity is very involved and requires extensive numerical processing. However, it is very important to include the nonlinearity in the design process o f high power HTS applications. A simple nonlinear model, yet accurate over a wide range o f material parameters, applied power, temperature, and frequency, is needed for Computer-Aided-Design (CAD) o f microwave and millimeterwave devices. In addition to accuracy, speed is required so that the design process can proceed at a reasonable rate. Until now, no nonlinear HTS models exist that are both fast and accurate. 1.2 Motivation for the Selected Techniques Most recent microwave and millimeter-wave applications problems are not tractable to closed-form analytical expressions. Moreover, solutions involving the more approximate variational and perturbadonal techniques are not suitable for these problems. Thus, they can only be solved using numerical techniques. In some of the numerical techniques such as the integral equation and spectral domain approaches, the methods are known to be efficient but are restricted, in general, to structures that may involve infinitely thin conductor patches [46], On the other hand, the Finite Difference (FD) method is considered as one of the most flexible numerical methods used in describing awide range of structures, especially those including finite thickness conductor patches. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Recently, this has become a very attractive method with the advance in speed and memory size o f computer systems [47]-[48]. To account for the field penetration effects in HTS, the finite thickness o f the strips is rigorously modeled using the finite difference approach. No approximations are made to the strip thickness. The mesh size is adjusted according to the physical characteristics o f the HTS materials. If the mesh is uniform, it would lead to a very dense mesh that requires a unrealistic memory storage. A graded nonuniform mesh is adopted t in all the work presented in the thesis [49]. An arbitrary three-dimensional structure can be embedded in a FD lattice simply by assigning desired values o f electrical permittivity and conductivity to each lattice electric field intensity component, and magnetic permeability and equivalent loss to each magnetic field intensity component. The material parameters are interpreted by the FD program as local coefficients [50]-[51]. Specification o f the media properties in this component-by-component manner provides a convenient algorithm to. represent the anisotropy in a media, and assures continuity of tangential fields at the interface of dissimilar media with no need for special field matching. Analysis and modeling the nonlinearity imposes some restrictions on the selected numerical techniques. The frequency domain approach is based on analysis in the Fourier transform domain. It provides an elegant tool for the reduction of the partial differential equations of mathematical physics into ordinary ones, which in many cases are amenable to further analytical processing. The time-dependent partial differential equation is decoupled into a series of frequency-dependent ones. Hence, the solution is separately carried on each frequency component. The time-domain solution can be Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 obtained by the superposition of the results calculated at each frequency component. This approach is widely used in problems containing linear materials. However, when a nonlinear material is used, the partial differential equation can not be transformed to the frequency domain. The equations for the various harmonics are no longer separable, and the superposition technique is not allowed. Hence, the equations must be solved in time domain. One should notice that this is a fundamental issue. It is not a matter of approximation or simplification. Therefore, the Finite-Difference Time-Domain method (FDTD) [52]-[58] is adopted in simulating the nonlinearity associated w ith the HTS materials. Recent advances in FDTD modeling concepts and software implementation, combined with advances in parallel computers, have expanded the scope, accuracy, and speed of the method. The field components at each point in the simulation domain of the FDTD are calculated frorn their nearest points. The FDTD technique has an explicit or semi-implicit scheme, which makes parallel computers an excellent environment to execute such schemes. Thus, execution of a FDTD computer code on parallel machines w ill decrease the required simulation time [59]-[61]. The Ginzburg-Landau (GL) theory is used to model the nonlinear mechanism in the HTS materials [62]-[65], It is the only available approach where the macroscopic parameters are field and temperature dependent simultaneously. Thus, GL theory provides an appropriate framework to describe nonlinear effects in the HTS materials. However, the GL type scheme needs to be modified due to the large anisotropy associated with the HTS materials [66]. In applying GL approach to the HTS, the structure discreteness becomes important and one expects a crossover from anisotropic three-dimensional behavior to quasi-two-dimensional behavior. The GL solution is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 conducted in two-dimensions, and the nonlinear macroscopic parameters are calculated for the currents flowing in the longitudinal direction o f the strip. It is well known that the currents in the transverse plane are small for most of the planar microwave and millimeter-wave applications. Hence, the simple linear model is used to calculate the currents in the transverse plane [67]-[69]. One must note that a microscopic theory describing the physics of HTS is unavailable at the present time. Numerous issues are still in part controversial, especially those dealing with phase transitions o f the vortex lattice [66]. For applications such as CAD, a model is needed which incorporates the accuracy o f the microscopic theory with the speed and intuitive nature o f the phenomenological simple model. The expressions obtained from the linear two-fluid and London models are elegant and simple in general. They are still in use today as a qualitative model. However, they do not consider the bidirectional coupling between the thermodynamics and electrodynamics in a superconducting system. Hence, developing a nonlinear twofluid model which incorporates the physics inherent in the nonlinear GL theory and the speed of the linear models would be very useful for CAD o f microwave and millimeterwave HTS devices. 1.3 Thesis Outline The main goal of the research is to conduct a nonlinear analysis o f microwave HTS devices using a full-wave electromagnetic simulator, which incorporates the anisotropic behavior of the superconducting strip and the substrate simultaneously. This approach is not only useful to predict the nonlinearity effects on high power microwave devices performance but also can be utilized in the characterization o f HTS materials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 This work presents the first rigorous effort in modeling the nonlinearity o f HTS in the time domain. A novel nonlinear tw o-fluid model is developed based on a phenomenological observation. This model w ill be very useful for CAD. It is the first simple model that blends the electrodynamics and the thermodynamics o f the superconducting material. Chapter 2 introduces the linear and nonlinear models o f superconducting materials. Material models, traditional two fluid model, and London's linear model are briefly described, while the nonlinear GL theory is explained in details. It is used to model the nonlinearity in HTS bulk and thin film forms. The GL equations are derived by minimizing the total free energy with respect to the complex order parameter and the magnetic vector potential near the critical temperature o f the superconductor. The resulting two coupled complex vector nonlinear differential equations govern the spatial distribution o f the order parameter and the magnetic vector potential in equilibrium. The anisotropic three-dimensional behavior is reduced to quasi-two-dimensional behavior for the superconducting strip used in microwave devices operating in the low gigahertz range. The current is assumed to flow mainly in the longitudinal direction. In this case, it can be shown that the GL equations can be simplified to two coupled real scalar nonlinear equations. First, the one-dimensional GL equations are solved numerically for the superconductor half-space. The finite difference method is applied to approximate the differential equations. An iterative scheme is adopted for the solution o f GL equations where the first nonlinear GL equation, which resolves the order parameter, is solved using a Newton-SSOR iteration scheme. The second equation, which corresponds to the superconducting current, is manipulated using a linearized scheme. Numerical results for the super fluid current density and the order parameter at different applied power levels are presented. It is observed that as the applied power increases, the super Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 fluid electrons near the edges decreases, the magnetic field penetrates more into the HTS, and the edge enhancement of the current density is suppressed. The HTS material can even loose its superconductivity near the edges at moderate values of applied power. This algorithm is used to estimate the rf power handling capability for HTS microstrip lines. Chapter 3 provides the implementation of the finite difference approach in the thesis. The finite-difference frequency-domain for structure containing nonuniform dielectric material is presented in detail. The general wave equation governing the longitudinal fields is derived. The boundary conditions are inherently satisfied inside the wave equation, and the field discontinuity across the interface between different dielectric materials is smoothly treated. The finite difference approximation is applied to the general wave equation.. This leads to a finite difference scheme which is valid everywhere in the simulated sU'ucture. The full wave analysis is carried on by solving the eigen value problem for the propagation characteristics o f the transmission line. The algorithm is applied to a linear HTS microstrip line. The attenuation and the phase constants are calculated at different temperatures, magnetic field penetration depths, and lossy substrates. The slow wave effects of HTS are obsen/ed along the microstrip line. The losses increases with increasing the temperature or the penetration depth. The attenuation associated with a lossy substrate dominates the losses in HTS structures. Therefore, dielectrics with very low loss tangents are required for HTS applications. The FDTD is briefly described since it was developed through the last 30 years. Only interesting features peculiar to its implementation are presented. These features are necessary to successfully model the HTS microwave devices, where the field penetration effects need to be taken into consideration. They are the nonuniform graded mesh generator, the Perfectly Matched Layer Absorbing Boundary conditions (PM L-ABC), Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 and the execution of the computer code on a Massively Parallel Processors machine (MPP). Chapter 4 presents a technique based on the three dimensional finite-difference time-domain method, to model transmission lines incorporating an anisotropic superconducting material deposited on sapphire substrate. The anisotropy of both the HTS material and the sapphire substrate are taken into account simultaneously. The equations are derived directly from Maxwell's equations. The approach fits the needs for accurate computation of the dispersion characteristics o f an anisotropic superconducting transmission line. Also, effects of anisotropy on the field distribution inside the structure and on the current distribution inside the HTS are investigated. Interesting comparisons between isotropic and anisotropic structures, as well as a comparison between the characteristics of the microstrip versus coplanar waveguide is presented In chapter 5, a nonlinear full-wave solution, based on the GL theory is developed using the Finite-Difference Time-Domain (FDTD) technique. The nonlinearity in the HTS is modeled by the GL equations. The anisotropic three-dimensional behavior of HTS superconductor is reduced to a quasi-two-dimensional one. The physical characteristics o f the HTS are blended with the electromagnetic model using the phenomenological two flu id model. Maxwell's and G L equations are solved simultaneously in three-dimensions. This time-domain nonlinear model is successfully used to predict the effects o f the nonlinearity on the performance of HTS transmission lines and filters. This approach takes into account the field penetration effects. The spatial distribution of the total electrons and the number of the super electrons compared to the normal electrons vary with the applied power. A study o f the nonlinearity effects on the propagation characteristics, current distributions, electromagnetic field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 distribution, and frequency spectrum is conducted. Numerical results show a change in the phase velocity of the wave propagating with the applied power. The corresponding increase in the attenuation is dramatic. The presented results show that the attenuation constant is more nonlinear than the phase velocity. The effect on the electromagnetic field distribution is also studied. It is more pronounced near the edge o f the HTS strip. The superfluid current density distributions change dramatically with the applied field. The change in the frequency spectrum is successfully depicted. The scattering parameter o f microstrip resonator array bandpass HTS filte r is calculated and the results are compared with experimental data. The output input power relation is also depicted. The maximum operating power for the filter without nonlinearity effects is estimated. The field distribution of the wave propagating along the array filter is studied. The field at the input o f the filter near the connection of the feeding line with the microstrip resonator is high. The dimension and the layout of resonator array filter must be optimized to reduce the nonlinearity effects on HTS filters performance. Chapter 6 introduces a novel nonlinear phenomenological two-fluid model for superconducting materials. The model is based on experimental observation for superconductors. Both the temperature and field dependence is taken into consideration simultaneously. The nonlinear main macroscopic parameters for superconductors are derived. An empirical formula for the surface impedance of HTS that agrees very closely with experimental measurements for YBCO superconductors is developed. These compact models are validated and verified by comparing the calculated results with data obtained from experimental measurements. This model combines the physics associated with the G L phenomenological model and the required sim plicity obtained from the linear London's model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 Chapter 7 summarizes the results and conclusions o f this dissertation. Possible extensions o f the research are given. An extensive reference list is attached at the end. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 2 LINEAR AND NONLINEAR SUPERCONDUCTOR MODELS An effective use o f superconductor material in engineering applications necessitates a better understanding o f its underlying physics. In addition, practical system design requires the development of tractable models o f the phenomenon. The classical model was the first attempt at describing superconductivity. This model incorporates the fundamental superconducting properties of zero resistance and perfect diamagnetism into electromagnetic constitutive relations known as the London equations [68]. The mathematical expressions, proposed by Fritz and Heinz London in 1935, are based on empirical observation rather than theoretical analysis. They are not deduced from any microscopic mechanisms within the material. Nevertheless, these relations are extremely useful. Just as it is possible to design a system using Ohm’s law without a detailed knowledge of conduction processes, a superconducting system's relevant parameters can be calculated and estimated using the London's equations. Despite the power of the classical model, it is limited in several ways. First, it does not provide a comprehensive understanding o f the superconducting phenomenon. Second, many properties of superconductors can not be explained by the classical model. In fact, superconductivity is a manifest quantum mechanical phenomenon. Fritz London developed a macroscopic quantum model for superconducting materials in 1948. He showed that the two London equations were a result of the quantum mechanical nature of superconductivity. This description not only encompasses the results of the classical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 model but also self-consistently describes other properties of superconductors that are important in many applications. The macroscopic quantum model is related to the superconducting model proposed by Vitaly Ginzburg and Lev Landau in 1950 [63]. The Ginzburg-Landau (GL) phenomenological theory tied the thermodynamics and the electrodynamics o f the superconductor intimately together. This theory contributed greatly to the understanding o f superconductivity in general. Despite their usefulness, all the models discussed thus far are phenomenological in nature. In other words, these models do not give any explanation as to how superconductivity occurs. A microscopic theory as the one provided by John Bardeen, Leon Cooper, and Robert Schreiffer in 1957 is required in this context [62]. However, it may not be necessary for describing most applications. Indeed, it should be noted that in 1959, L. P. Gorkov showed that Bardeen-Cooper-Schreiffer (BCS) theory reduced to the more tractable Ginzburg-Landau theory near the critical temperature of superconductors [65], As a result, it is known that the important conclusions of the phenomenological models are consistent with the microscopic theories. In late 1986, J. G. Bednorz and K. A. M uller of IBM 's Zurich lab reported possible superconductivity in La and Ba copper oxides at temperature o f 30° K. These materials are known as High critical-Temperature Superconductors (HTS). However, it appears that the BCS theory may not be adequate in explaining the phenomena of hightemperature superconductivity [66]. Despite the present lack o f a fu lly successful microscopic explanation o f the HTS, the phenomenological theories seem to work reasonably well. Moreover, there is experimental evidence that the superelectrons in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 HTS materials also carry a charge that is twice that of an electron. Consequently, the general ideas presented in the phenomenological models are just as applicable to these newly discovered superconductors as they are to the conventional ones. In this chapter, we shall adopt the two-fluid model as the basis for our analysis, focusing on the macroscopic features of superconductors. The two-fluid model assumes that the conduction current in a superconductor comprises two separate fluids, the normal and the super electrons fluid. The temperature dependence of the superconductors is presented. London phenomenological linear model is briefly described. A detailed presentation o f the nonlinear GL theory is given. The limitations of the GL model for microwave applications as applied to the new HTS materials, and the appropriate approach overcoming them are explained. The solution algorithm for GL equations is discussed. Application of GL two-dirpensional solution to bulk and thin film HTS is performed. The spatial distribution o f the superfluid current density and the superfluid electrons density is presented for different applied magnetic field intensity. 2.1 Material Models of Superconductors The material parameters of superconductors can be derived from the MattisBardeen formula based on the microscopic BCS theory or a classical two-fluid model. The Mattis-Bardeen formula predicts the sudden increase o f loss at or above the gap frequency, but the two-fluid model doesn’t [62], Despite its failure at the gap frequency, fc, and at temperatures close to the critical temperature, Tc, the two-fluid model provides reasonable material parameters at frequencies significantly lower than the gap frequency. In fact, it is believed also to be a good approximate model for HTS. On the other hand, the applicability o f the Mattis-Bardeen formula to HTS is debatable, since it only Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 describes the extreme anomalous lim it where the coherence length, £, is large compared with the penetration depth, A . This lim it is not realized, for example, in the YBaCuO ceramic superconductor. Measurements show a typical value o f A(0) = 1500 A 0 fo r the penetration depth for current flow w ithin the copper-oxygen planes but reveal considerably larger values for current flow perpendicular to the planes. Estimates o f the coherence length are£ = 5 - 2 0 A 0, depending on the crystalorientation [66]. Therefore, the two-fluid model may be considered as one of the most appropriatechoices, currently available for modeling HTS materials. 2.2 The Two-Fluid Model The two-fluid model postulates that the conduction current in a superconductor consists o f two separate fluids, the normal and the super electrons fluids, J = Jn + J s (2.1) with 7, = n„q{vm) (2.2) J, = n,qv, (2.3) where Jn and Js are the normal and super currents, respectively; nn and ns are the densities of the super and normal electrons, respectively; v, is the velocity o f the super fluid electrons, and (v„) is the average velocity of the normal electrons. The total number Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 of electrons n in the superconducting material is constant, according to the conservation of particles law, and equals to (2.4) n = n„+n, The superconducting fraction o f the conduction electrons is in the lowest energy state, while the normal fraction is in the excited state. Under the influence o f external electric fields, the motion o f normal electrons includes the effects o f both resistance and inertia. : The movement o f superconducting electrons is inertial only. This phenomenological model was originally used by London to explain the first microwave experiment with superconductors and it is still the framework in which many physicists and engineers visualize many superconductive phenomena [68]. This model is satisfactory for microwave engineers who wish to develop a simple, yet intuitive, formulation for the superconductor phenomena without having to delve too deeply into the underlying theory of superconductivity, which does not exist for HTS. 2.3 Temperature Dependence of Superconductors The most successful temperature dependence of superconductors was developed by Gorter and Casimir in 1934 [67]. They assumed that the fraction o f the conduction electrons in the superfluid state ns varies from unity at T = 0 to zero at the temperature of transition to the completely normal state Tc. They found that the best agreement with the thermal properties o f superconductors was obtained when the fraction o f the super electrons and normal-electrons was chosen to have the form, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. IE 4 n = n (2.5) 1- It is known that application of a magnetic field stimulates the flow o f a current in the superconductor, and energy is added to the superconductor. This can be represented by an equivalent magnetization. A t some value o f magnetic field, the energy o f the magnetization is larger than the condensation energy, which is the difference o f energies between the superconducting and normal states. So, it is more favorable for the superconductor to be in the normal state. Since the energy o f condensation into the superconducting state depends on the temperature, the critical value of the magnetic field H c does also. The relation between H c and T is known experimentally to follow to within a few percent o f the relation (2.6) which is consistent w ith the tw o-fluid model o f Gorter-Casimir developed for conventional superconductor. 2.4 London Phenomenological Model London equations are derived by combining the hydrodynamic o f a superconductor with one o f Maxwell's equations. The hydrodynamic equations of the superfluid, which is assumed to be collision free, is expressed Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 dt (2.7) and the normalfluid, which is governed by the momentum conservation equation, is written as (2 .8) where vs is the velocity of the super fluid electrons, (vn) is the average velocity o f the normal electrons, which have an average momentum relaxation time t„ , and m and e are the mass and charge of single electron, respectively. The first London equation is derived by combining Eqs. (2.3) and (2.7), yielding where A is the field penetration depth in the superconducting material, and equals to A = /n/nJe2. The second London equation is obtained by substituting Eq. (2.7) into Faraday's Law, which yields (2.10) These equations when coupled with the two-fluid model represent the fu ll constitutive model that describes the total current in terms of the electric and magnetic fields. It is the role o f Maxwell's equations to predict the electric and magnetic field in terms of these Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 currents. The complete solution is achieved when the outputs o f London's equations, the two-fluid model and Maxwell's equations are consistent. 2.5 Ginzburg-Landau Phenomenological Model Ginzburg and Landau proposed a phenomenological extension o f the London theory to take account of spatial variation of the macroscopic parameters o f the superconductor. The construction of the GL theory is independent o f the microscopic mechanism and is purely based on the ideas o f the second order phase transition. In the G L theory, the macroscopic parameters o f the superconductor are field-dependent, which provides an appropriate way to describe non-linear-effects. 2.5.1 Theory The macroscopic electromagnetic London's equations are able to satisfactorily account for the current persistence and the magnetic flux exclusion (Meissner) effect [67], However, they do not give a completely satisfactory macroscopic picture o f all superconducting phenomena in a magnetic field, because they regard the superconducting material as being entirely superconducting or entirely normal. These deficiencies were overcome in 1950 by Ginzburg and Landau, who proposed a phenomenological set o f equations, allowing for spatial variations in the superconducting order due to the presence of a magnetic field. The accuracy of GL equations have been validated by Gorkov based on the microscopic theory for conventional superconductor materials [65], and extended beyond their region of validity by other researchers [70]. However, an exact theory describing the new HTS has not yet been developed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 The construction of GL theory is independent of the microscopic mechanism and is purely based on the ideas o f the second order phase transition only. G L theory begins by introducing a quantity to characterize the degree of superconductivity at various points in the material. This quantity is called the "order parameter" and denoted by y/(r). The local order parameter is defined to be zero for a normal region and unity for a fu lly superconductive region at zero temperature with zero magnetic field. Clearly, y / { f ) must be closely related to the superfluid fraction in a two-fluid model, but the two quantities are not identical. Rather, to allow for supercurrent flow, ijf( r) is taken as a complex function and interpreted as analogous to a "wave function" for superconductivity, so that its magnitude square can be identified with the superfluid density Ns(r), (2.11) It should be noted that y/(r) is not the system wave function for the electrons in the material, since it is defined to be zero in the normal state. However, pursuing the interpretation o f the order parameter as a wave function, it is reasonable to write the expression for the supercurrent J3, in the absence of a magnetic field, as J. = 2 in (2. 12) where e* and m’ are the charge and mass of the entities whose wave function is y/(r), h is the reduced Plank's constant, and i is V - l . It is equally natural to include the magnetic field via the vector potential A : J. = 1 2m < /( p)| y V - e ‘A ( r ) j i/ ( r ) + ^ ( r ) f y V - e ’ A ( r ) \ v ' ( r ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.13) From the point o f view o f the two-fluid hydrodynamic model, the superfluid component has been regarded as a "quantum fluid" and a quantum wave function is associated with it. Reinforcing this interpretation, a more suggestive form o f Eq. (2.3) is J,(r) = e N s( r ) V s( r ) (2.14) where the superfluid velocity v3(r) can be related to the phase o f the order parameter by (2.15) Furthermore, i f there is no position dependence to the superconductivity and the order parameter yr(r) is independent of r , then Eq. (2.14) reduces to the London's relation Js(r) = ^ N sA(r) (2.16) m Next, a relation determining y/ must be constructed. Ginzburg and Landau focused on the free energy of the material Fs, which they assume as a functional of y/ and y/*. The equation determining yr is obtained by requiring that Fs, be a minimum with respect to variations o f yr *. The condition that Fs be a minimum can be shown to be equivalent to requiring that the superfluid and normal components o f the two-fluid model be in stable equilibrium with respect to each other. Thus, the functional Fs plays a role analogous to a Lagrangian o f Schrodinger wave mechanics, while its minimum value with respect to y/ and y/* is just the free energy o f the superconducting phase in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. thermodynamic equilibrium with the magnetic field. Ginzburg and Landau chose the following form for Fs, 2 (r )|! ) + |H (r [ y V - e'A (r )1 ^ ( r ) (2.17) The term F„ is the free energy, at the same temperature, of the underlying normal phase, which could be obtained by increasing the magnetic field above its critical value. The function F > 0 represents the free energy lowering of the system due to having formed the superconducting correlation. For temperatures just below the zero field transition temperature, Tc - T « T C, it should be sufficient to expand F in a power series and keep just the first two non vanishing terms: (2.18) The next term in Eq. (2.16), the magnetic field energy, represents the increase in superconducting free energy due to the expulsion of magnetic flux. The final term in Eq. (2.16) represents the increase in superconducting free energy coming from the spatial variations in the order parameter and from the current flow. It could also be written, using Eq. (2.15), as (2.19) which illustrates more clearly that the term is in the form of a supercurrent kinetic energy plus a stiffness against rapid changes in the superfluid density. It is the latter contribution Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 which tends to prevent the formation o f superconducting-conducting-normal domain boundaries. A further argument for the plausibility of the form of these terms is that Eq. (2.13) for the supercurrent follows from a variation o f the free energy functional with respect to A . Thus minimizing Fs, with respect to A is equivalent to solving Maxwell's equation, V x / / = y, (2.20) GL phenomenological theory results in a set o f two equations relating the order parameter yr and the magnetic vector potential A . These equations can be reduced to a dimensionless form by taking all the lengths in units o f the weak-field penetration depth X, measuring the magnetic field in terms of the thermodynamic critical magnetic field H c, and introducing a reduced order parameter normalized by its zero-field positionindependent value. Then, the dimensionless GL equations can be expressed as follows, (2.21) (2.22) The subscript N denotes normalized quantities. The boundary conditions for AN and y/N at the superconductor-insulator interface are given by (2.23) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 n x ( V i r x A N) = n x n 0H ft (2.24) Eq. (2.23) forces the normal component o f the conduction current to vanish at superconducting-insulator boundary, resulting in the boundary condition for the order parameter yr. Eq. (2.24) indicates that the boundary conditions imposed on the magnetic vector potential A correspond to the magnetic field tangential to the superconductor surface. The dimensionless parameter k , known as GL parameter is defined as : K = 4 l [ e l t i ) n 0H cX \ (2.25) where XL = ^ m l n 0e \ l \v~\2=fh Hn = H /H c an = a / x lh ,h c V „ = A ,V V'v = V / Y ~ k l and nL are the low field London penetration depth and the low field superconducting electron density. The subscript N w ill be omitted in the rest of the thesis for simplicity. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 Even though the Ginzburg-Landau theory relies on the two experimental parameters H c and X , in fact it is just a one parameter theory. Only the differences o f k from one material to another prevent the equations from scaling perfectly into a single law of corresponding states valid for all superconductors. It can be seen that whereas the magnetic field varies spatially over characteristic length o f order X , the reduced order parameter y/ has spatial variations with a quite different characteristic length, called the coherence length which equals X / k . The GL coherence length is expressed as * (2.26) which is the characteristic decay length for a disturbance o f y/ from its value in the absence of currents and magnetic fields, yr„. The ratio of the penetration depth and coherence length, and thus k itself, determines the relative balance between rapidity of the variation o f the magnetic field and the order parameter in the final solution. 2.5.2 Limitations on GL Theory The ordinary superconductors prior to the new revolution which began 1986 had Tc < 25K and coherence lengths at zero temperature much larger than the interatomic or interelectronic distance (of the order o f 500 A 0 or more). It is quite accurate to use near Te mean field theory such as the GL macroscopic theory for this reason. In the new high Tc systems, in contrast, the coherence length is small (of the order o f 15 A 0 in x-y parallel plane and 2-3 A° along z-perpendicular axis) and the GL type schemes need to be modified. This means that we must take account o f the strong anisotropy o f the new Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 systems in developing the G L type scheme. In the original scheme one obtains a local differential equations for the order parameter whereas in the new systems, besides introducing the strong anisotropy, one may want to develop a differential equations in x-y plane different than the z-plane. 2.5.3 G L fo r Microwave HTS Applications The GL theory discussed so far has concentrated on a time-independent thermal equilibrium situation, where the superconductor is in a static magnetic field. However, in a non equilibrium situation where an electric field is also present in the superconductor, the time-dependent G L has to be considered. The order parameter relaxes to its equilibrium value in the temperature dependent relaxation time xo which is given by [62] T = ----- ^ — 7 0 8ArB|r e- T | (2.27) I f the characteristic time scale of the electromagnetic field ( 1 // where / is the frequency) is much larger than x0, we can neglect the time dependence of yr and use the time-independent GL equations. In this case, the order parameter ys responses to A so rapidly that we can assume that it adjusts itself instantaneously to A . For example, for Tc - T = 13K, x0 ~ 10"13s and for operating frequency in the order of tens gigahertz 1 // ~10-10.y which means that x0 is about three orders o f magnitude less than 1 //. A common feature of the family of HTS, including YBaCuO and TIBaCaCuO, is that they all have layered crystal structures. It is generally believed that the two- dimensional C u02 network is the most essential building block of the HTS materials [21]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 The metallic Cu-0 layers are separated by other layers of atoms. This structure causes a pronounced anisotropy o f the electronics properties of HTS. The anisotropy in HTS materials is a complicated issue, which is analyzed fully in a chapter 4. A model taking this into account is the Lawrence-Doiach model. It describes the system as a stack of superconducting layers with spacing s, each treated within a 2D Ginzburg-Landau theory, coupled by interlayer Josephson tunneling [71]. However, th in -film microwave transmission lines w ill favor films in which conducting sheets lie in the plane of the film. To overcome the lim itation in applying the GL theory to the new HTS materials, the solution for GL equations is only performed for the longitudinal superfluid z-current component in the HTS strip. The transverse x- and y-current components are calculated using London low field model since the current in the transverse plane is relatively small for most microwave and millimeter-wave applications working in the low gigahertz band. Using the London gauge V.A = 0 and assuming y/ = |y/|exp(id), the normalized GL equations fo r the superfluid z-component current density can be sim plified to the following expressions, V p L = |v /|2A; (2.28) (2.29) where r stands for the transverse x-y direction and V,0 = 0. The required boundary conditions become n x ( V t x z A . ) = MoH, (2.30) n- V , \y \ = 0 . (2.31) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 2.6 Solution of GL Equations The G L coupled nonlinear differential equations are solved simultaneously to obtain the superconducting current and the order parameter. The first nonlinear G L equation, which resolves the order parameter, is solved using a Newton-SSOR iteration scheme. The second GL equation, which corresponds to the superconducting current, is manipulated using a linearized scheme. These two equations are solved iteratively until convergence, starting with initial conditions = 1 and ,4. = 0 . The described procedure is rapid and robust, and is successfully applied to the solution of G L equations both in one- and two-dimensions. The solution converges in a few number o f iterations, which depends on the applied magnetic field intensity. 2.6.1 Numerical Scheme The Newton-SSOR iteration scheme for solving a system of nonlinear equation is based on the following Newton algorithm Xw given f o r k = 1, 2 r (i_1) = - F ( ^ (i_)) x^ = x ^ + d^ -» end (/') Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 which is modulated by the successive over relaxation method for faster convergence and robust solution. The Newton-SSOR algorithm can be summarized as follows x w given f o r k = 0,1 r ik) = - F ( x ik)) dw = 0 f o r j = 1,...... m (*) ii \ l<> (>< y. = Q){k'j)y. + ( l - CQ^-’ ^ d j( / - I ) end (/) d: = - /(*) ri \ (<i l> i d\j) =CO^i)di + ( l - ( D a 'i ) )yi end (/) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 end O') x iM) = x (k) + d imi>)) end ( k ) where the Jacobi J is defined as follows r?L dxl dx2 dxm (2.32) J - F'(x) = d fn d fn Kdxy dx2 d fn dx.n J The k iteration is the outer, or primary iteration, and the j iteration is the inner, or secondary iteration. The values for m(t) depends in general on k. It could be chosen independent of k and equals m, since the system y (t).va+1) = b w does not need to be solved exactly for lower values o f k. G L coupled nonlinear equations are discretized by using a centered difference scheme for the second order derivatives : d 2f fi+i ~ 2 f i + f i - dx A.v 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2-33) 32 where / stands for y/ and A . The boundary conditions are applied to the system of equadons. The solution procedure starts by the iniual conditions y/ = 1 and A = 0. Eq. (2.28) is manipulated using a linear solver to calculate the magnetic vector potential. The updated A is substituted into Eq. (2.29) which is solved using the Newton-SSOR numerical scheme with F(x(i)) < lO ^F ^x10*) as a stopping criteria. The calculated value for y/ is then substituted back into Eq. (2.18), and the whole process is repeated until A and y/ converge. 2.6.2 Bulk Superconductor (Superconducting Half-Space) In this section, GL equations are solved for the bulk superconductor, shown in Fig. 2.1 in one dimension. The applied field is parallel to the plane surface of the superconducting half space, which is of infinite extent in this dimension. The 1-D GL equations are (2.34) (2.35) with boundary conditions (2.36) dx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.37) 33 L y ► Hy Fig. 2.1 Superconducting Half Space (bulk superconductor) By imposing the boundary condition on the tangential field Hy at the boundary o f the half space, G L equations are solved numerically fo r y/(x) and A.(;t) inside the superconductor. The superconducting current is calculated using the following relation (2.38) Js{ x ) = r ( x ) * A :{x) In our analysis, the HTS macroscopic parameters are those measured for YBa 2Cu3C>7-x atT = U K with Tc equals to 90K. The GL parameter k equals to 44.8. The corresponding penetration depth A (T) and critical magnetic flux density n 0H c(T) equals to .323 fim and 0.1 T, respectively. Fig. 2.2 presents the variations o f the superfluid current density with distance for one-dimensional superconducting slab at different magnetic fields. These results are in excellent agreement with those in Lam et al. [72], The peak values observed in the plots of the normalized current density are about the same and approximately equal to 0.544. This value is very close to the GL depairing critical current density [67]. As a matter of fact, the critical current density for type II superconductor is about one order of magnitude less than the GL depairing critical current density. This is explained by the vortex pinning which is not present in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 conventional superconductor [18]. The nonlinearity in the superconducting material is clear, especially near the edge of the material. The superconducting current near the edge is suppressed in favor of the normalfluid current, which explains the increase in the losses as the applied field increases. Also, the predicted penetration depth using G L theory is higher than the low-field value calculated from the linear London model, which can explain the more pronounced slow wave effects associated with the superconducting material as the magnetic field intensity increases. Thus, the field penetration effects on the superconducting material is represented more rigorously in the GL theory compared to the London model. These results confirm the success of the phenomenological GL model to evaluate a field and position dependent macroscopic parameters for the superconducting material. • Lam etal. - - - H = 1.0H, H = 0.9Hc a - - H = 0.7h ' H = 0.5H u 0.6 3 2 *3 *3 o §•0.4 t/5 -o io.2 o z 0 0.5 1 2 1.5 Normalized distance (x^J 2.5 3 Fig. 2.2 Normalized superfluid current density in one-dimensional bulk YBa2Cu30 7_x HTS. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 2.6.3 Thin Superconductor Film The solution for the thin HTS film is performed in the transverse two-dimensional plane. It is reasonable to assume that the local order parameter y/ depends on the total magnetic field at coordinates (x,y), and denoted by y/(x,y). To demonstrate the versatility of our numerical scheme, two-dimensional solution for a typical HTS strip, shown in Fig. 2.3, used in microwave and millimeter-wave devices is presented. The strip width, W, and thickness, t, are 7.5 fim and 1.0 \±m, respectively. The strip is divided into numerical grid cells. The generated mesh is uniform in the vertical direction and nonuniform in the horizontal direction. The horizontal mesh size decreases near the edges o f the strip where mere rapid change in the macroscopic parameters of the superconductor is expected. The mesh size is chosen smaller than the low -field penetration depth. The HTS parameters are the same as previously described. Fig. 2.3 YBa 2Cu3 0 7 -x HTS strip (thin film superconductor). The typical tangential magnetic field shown in Fig. 2.4 is applied to the strip. This Field distribution is obtained for the HTS microstrip line using a full-wave electromagnetic simulator, which w ill be explained later. The corresponding applied power is calculated. GL equations are solved at different levels o f applied power. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 36 maximum r f power P crf , where the HTS m icrostrip com pletely looses its superconductivity, is predicted. Its value equals to 920 W / cm2. strip side strip thickness strip bottom 0.5 - half-strip width - -0.5 strip top 0 0.75 1.5 2.25 3.0 3.75 Distance (pm) Fig. 2.4 Tangential magnetic field intensity to a YBa 2Cu3 0 7 _x HTS strip in microstrip line configuration. The normalized superconducting current distributions are shown in Figs. 2.5-2.7 for different applied power levels : 834 W / cm2, 410 W / cm1, and 181.8 W I cm2 denoted by 0.9 Pcrf, 0.45 Pcr/, and 0.2 P crf. respectively. Fig. 2.5 shows that the HTS may be considered linear when the applied magnetic field is low, i.e. approximately less than 0.2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 Pcrf- As the applied power increases, the dependence of the macroscopic parameter of the superconducting material on the magnetic field becomes nonlinear. Applied power = 0.2 Pcrf Jsz 0.6 0.4 0.2 —■ — 0.6 0.4 0.75 half-strip width (um) Fig. 2.5 2.25 0.2 strip thickness (um) Normalized superfluid current density distribution in YBa2Cu307-x HTS strip in microstrip line configuration at 0.2 Pcrf applied power. A typical distribution for the normalized superconducting current density in this nonlinear region at 0.45 Pcrf is presented in Fig. 2.6. London's equation fails to predict the superconductor behavior in this nonlinear region, even the type II superconducting material is still in the mixed state, and possesses a relatively good superconductor nature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 Applied power = 0.45 Pcrf Jsz y- 0.8 0.6 0.4 s -/ 0.2 0.6 0.4 0.75 half-strip width (um) Fig. 2.6 2.25 0.2 strip thickness (um) Normalized superfluid current density distribution in YBa2Cu307-x HTS strip in microstrip line configuration at 0.45 Pcrf applied power. Fig. 2.7 demonstrates that the superconducting material partially looses its superconductivity at high rf power 0.9 Pcrf. It is obvious that the material lost its superconductivity near the edge of the strip where the singularity in the field is expected. On the other hand, the material behaves as a good superconductor at the center of the strip. This behavior w ill not only introduce nonlinearity effects but it w ill also increase the noise. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 Applied power = 0.9 Pcrf strip thickness (um) half-strip width (um) Fig. 2.7 Normalized superfluid current density distribution in YBa2Cu307-x HTS strip in microstrip line configuration at 0.9 PCTf applied power. Fig. 2.8 presents the normalized superfluid electron density at 0.45 Pcrf. It is clear that the bottom part of the strip looses its superconductivity much faster than the top section. This can be explained by the effect of the dielectric substrate underneath the strip, which increases the field intensity at the bottom side. Thus, superconducting applications w ill favor low dielectric substrate to decrease the nonlinearity effects in microwave and millimeter-wave applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Applied power = 0.45 Pcrf y< Ns(x,y) 0.8 0.6 0.4 0.2 0.6 0.4 0.75 half-strip width (um) Fig. 2.8 2.25 0.2 strip thickness (um) Normalized superfluid electron density distribution in Y B a 2Cu307-x HTS strip in microstrip line configuration at 0.45 Pcrf applied power. 2.7 Summary In this chapter, the linear and nonlinear models of superconductor are presented. The London model is briefly reviewed. Detailed analysis of Ginzburg-Landau theory is illustrated. The application of GL theory to the nonlinear modeling o f HTS superconducting bulk and thin films is discussed. The anisotropic three-dimensional behavior of HTS superconductor is reduced to a quasi two-dimensional one. The solution of GL equations is performed under the assumption of negligible transverse current. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 complex vector GL nonlinear differential equations are simplified to a coupled set of real scalar differential equations governing the spatial variations o f the order parameter and the magnetic vector potential. The simplified equations are normalized fo r the convenience o f the numerical calculations. The GL coupled nonlinear differential equations are solved simultaneously to obtain the superconducting current and the order parameter. The first nonlinear GL equation, which resolves the order parameter, is solved using a Newton-SSOR iteration scheme. The second GL equation, which corresponds to the superconducting current, is manipulated using a linearized scheme. These two equations are solved iteratively until convergence. The described procedure is rapid and robust, and is successfully applied to the solution o f GL equations both in one- and two-dimensions. The required boundary conditions for the thin film solution is obtained from the full-wave simulator described in chapter 5. Numerical results show that as the magnetic field at the boundary increases, the order parameter near the edges decreases, indicating the breaking of superfiuid electron pairs. The HTS material can even loose its superconductivity near the edges at moderate value of applied power. The rf critical power density for HTS strip used in microwave applications depends not only on the physical characteristics o f the HTS but also on the structure configuration. Special attention has to be placed in predicting the efficient operating region of the HTS materials used in microwave and m illimeter-wave applications. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 3 FINITE-DIFFERENCE APPROACH Most partial differential equations describing recent physical phenomena of science and engineering are not amenable to closed-form solution. Solutions involving the more approximate variational and perturbational techniques are not suitable for these problems. Thus, they can only be solved using numerical techniques. In some numerical techniques such as the integral equation and spectral domain approaches, the methods are known to be efficient but are restricted, in general, to structures that may involve infinitely thin conductors. On the other hand, the finite difference (FD) method is considered as one o f the most flexible numerical method used in describing a wide range of structures, especially those including finite thickness conductors. Moreover, the FD solution of Maxwell's equations is one of the most suitable numerical modeling approaches for the electromagnetic analysis of volumes containing arbitrary shaped dielectric and metal objects. FD is relatively simple in its concept and execution. However, it is remarkably robust, and provides highly accurate modeling predictions for a wide variety of electromagnetic wave interaction problems [50]-[58]. The FD technique is based on approximations which permit replacing partial differential equations by finite difference equations. The formulation is algebraic in form, relating the value o f the dependent variable at a point in the solution region to the values at the neighboring points. In this analysis, both the finite difference frequency domain (FDFD) [49], [80] and the finite difference time domain (FDTD) [74]-[84] are employed to study the linear and nonlinear electromagnetic characteristics of microstrip lines, coplanar waveguides, and microstrip multiarray resonators filters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 In previous research work, the transmission line characteristics are described in terms of their integrated quantities, and strip thickness is mostly treated by some approximation. Also, some studies of transmission line properties of strip and microstrip lines have been made with the assumption that TEM modes are supported by both structures. This assumption may be valid for the strip line structures w ith uniform dielectric filling. However, it is not tenable when an air-dielectric interface is present, as in microstrip lines, for example. Moreover, these calculations may be valid at low frequencies, but not at higher frequencies. Modeling superconducting microwave and millimeter-wave devices imposes new restrictions on the selected numerical approach. The strip thickness can not be approximated by infmitesimally thin perfect conductor. The field penetration inside the strip must be considered. The problem o f calculating the propagation characteristics o f superconducting microstrip lines are tackled by different approaches. The microstrip line is modeled using a modified spectral-domain immitance approach, based on the transverse resonance method, which takes into account the complex resistive boundary conditions [4]-[5], The spectral domain formulation is also used in conjunction with the method o f moments [ 6 ]. An equivalent single strip, which has the same internal impedance as the original superconducting line, is utilized in the phenomenological loss equivalence method [7]. The superconducting strip is replaced by a frequency-dependent surface impedance boundaries in the characterization of a thin film line [ 8 ], Monte-Carlo method is used to calculate the propagation characteristics of superconducting interconnects [33]. Despite the usefulness of the above mentioned approaches, the field penetration effects are not taken into account. This effect must be considered, especially when evaluating losses for the propagating wave inside the superconducting microstrip line [ 10 ]-[ 11 ]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 In this chapter, technique based on the FDFD method is presented to model a microstrip line incorporating a superconducting material. The equations are derived directly from Maxwell's equations. The approach fits the needs for accurate computation o f the dispersion characteristics of a superconducting transmission line. The model is flexible, and can be used for any planar transmission line structures containing HTS materials. It incorporates all the physical aspects o f the HTS through London's equations. The electromagnetic characteristics and the required boundary conditions in the structure are represented using Maxwell's equations. The physical characteristics o f t the HTS are blended with the electromagnetic model by using the phenomenological two fluid model. The complex propagation constant is calculated. Hence, the losses inside the superconductor material are considered. The technique is also used to analyze microstrip line structures at different temperatures and different frequencies. The effect o f the losses associated with a lossy dielectric substrate on the performance of the superconducting microwave transmission line is also investigated. The FDTD is briefly described in this chapter. presented. Only interesting features peculiar to our approach are They are the nonuniform graded mesh generator, the Perfectly Matched Layer Absorbing Boundary Conditions (PML-ABC), and the execution o f the computer code on Massively Parallel Processors machine (MPP) [83]. These features are necessary to successfully model the HTS microwave devices, when the field penetration effects must be considered. Finally, a comparison between the FDFD and FDTD solutions is conducted. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 3.1 Finite-Difference Frequency-Domain (FDFD) 3.1.1 Wave Equation for Superconducting Microwave Structure The relation between the HTS current density and the fields inside the superconducting material is described by London's equation [ 6 8 ]. The dependence of the effective penetration depth (XjJ, as well as the ratio of the densities o f superconducting and normal electrons, on the temperature are expressed using Gorter-Casimir expressions t [67]. According to this model, the superconductor has almost constant penetration depth at a temperature well below the critical temperature Tc and the field penetration remains almost unchanged with frequency. In general, the effective penetration depth is greater than the one depicted in the two fluid model [18]. Combining Ampere's Law with the two fluid model, and using London's equation, we get the follow ing relation between the electric and magnetic fields inside HTS materials [19] V x H = j( 0 £0£sE (3.1) where the complex relative dielectric constant ej is £. = j ~ CO£„ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.2) 46 Here, the electrical properties of the superconductor are assumed isotropic. The negative dielectric constant for the superconducting strip may be explained by the stored electric kinetic energy associated with the superconductor electrons pair motion. The negative real part of the dielectric constant means that the electromagnetic wave is expelled from the HTS material without attenuation. The negative imaginary part o f the dielectric constant shows that the structure expels the wave from its inside with attenuation due to the presence of normal electrons. Consider an electromagnetic wave propagating in the +z direction with a propagation constant y = a + //}, that is yet to be determined. The 3-D problem is transformed into a 2-D one by assuming that the transmission line is infinite in the +zdirection. A ll the fields have an e~r' dependence on z, so that d/dz can be replaced by —yz. The problem may be formulated for either T M Z or TEZ. In this section, the equations are derived for the T M Z mode, for simplicity only. The TE Z derivation is analogous to the T M Z case. The wave equation governing the longitudinal electric field E . in the case of uniform dielectric constant at the points inside air, dielectric, or superconductor regions may be written as, i r + 1r +,^ = 0 where hp is the transverse wave number, and is expressed as follows; Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (33) 47 h; = y 2 + co2p o£p (3.4) and the subscript "p" designates air, dielectric, or superconductor regions. The dielectric constant, ep, equals e0er in case of lossless dielectric, equals £0er( l- y 'ta n 5 ) in case of lossy dielectric, and equals £o£r[ ( l - l / t u 2/z0£o£rA ^ ) - 7'(crn/G)£(,£r )j in sid e the superconductor. At the interface points of the FD simulation domain, the dielectric constant, £p, is equal to the average of the dielectric constants of the materials across the interface. The boundary conditions are inherently satisfied inside the wave equation, and the field discontinuity across the interface is smoothly treated. When the interface between two different materials lies parallel to the x-axis, the wave equation has the following expressions, d 'E . d 2E. Bx" CO~jJ.0£p ■) d ' 1 b e By2 ~ r ~fy h i dy ' + h2 pE: = 0 (3.5) and when the interface is parallel to the y-axis, the corresponding wave equation can be written as, t d d 2E. 03 I10£ op dx2 ~ T 1Tx 1 BE, yh2 x dx +^J± 2 + h2 'P„ E. = 0 By (3.6) The general wave equation for a nonuniform dielectric constant can be obtained by combining (3.5) and (3.6). It is expressed as follows, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 d 2E. co^o 0£p dx2 d 2E, *— r dy' 1 d E .] d / 1 dE ^ dx h: dx + h lE = 0 (3.7) \ hyi dy / / where subscripts "x" and "y" designate x and y directions, respectively. These equations are derived in a form convenient for formulating an eigenvalue problem using the finite difference approach. Also, it successfully describes the points lying on the interface between the different material constituting the microwave structures. The fu ll wave analysis can be carried out by applying the same procedure to the TEz mode, then combining both modes together. We may conclude that this approach, presented above, results in a consistent formulation for the wave equation throughout the structure, including the interface points. The technique is general, and can be applied to any superconducting microwave structure. 3.1.2 Eigenvalue Problem In general the wave equations governing linear transmission line structure result in an eigenvalue problem when discretized using the finite difference approximations. The eigenvalues are the propagation constants of the structure while the corresponding eigenvectors are the possible modes in the structure. This section explains how to calculate the complex propagation constant at a given frequency. The finite difference method is used to approximate the wave equations for Ez presented in the previous section. Employing the notation shown in Fig. 3.1, Ez becomes Ezij= E : {iAx + jA y) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.8) 49 Substituting in the general wave equation shown in (3.7), and after some manipulation, we get the following general finite difference equation, 1 Ax 2 1 ,2 ( £ ‘+ l.j hi.j X p 1 ^3+ 1,j (e Ay 2 1 ftw j u Ax 2 1 f t ^P £. . *' ft 1 f £u -l f t 1 £ M ■; h j j 1 Ax" £ i.j h i 1,;. ^£fc i . / + l fti2. j Ay2 V P + ft.J ‘ Ay 2 I £.v f t - J f t +I, l2 f£ h2 i,;+ l ^ £; E - j . j -1 + /l; j £ , ;. /I* ,.,. £ h* ^ fci . ; - l n i.j P i.j h2 i,j-l y i.j -1 i+1, j i- l.j i, j +1 Fig. 3.1 Mesh indices ij and the corresponding xy axis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.9) 50 The finite difference equation is consistent with the partial differential equation representing the wave equation. The coefficients inserted are only used at the interface between the different materials constituting the structure. The discontinuities at the interface are carefully handled to satisfy the matching conditions without generating undesired numerical singularities. The discontinuity in the field in the direction perpendicular to the interface, as a result of different dielectric values at both sides, is treated using the electric field-dielectric constant product. Similarly, the finite difference equations for the homogeneous media, the interface parallel to the x-axis, and the interface parallel to the y-axis can be easily derived. This results in a finite difference scheme which is valid everywhere in the transmission line structure. There is no need to impose unnecessary boundary conditions anywhere inside the structure. , It is well known for the case considered in this analysis for the microstrip line that the modes may be subdivided into the TEz and TM Z modes. It is apparent that the lowest T M Z mode is of great interest. Therefore, we w ill focus our analysis to this mode. Using the even symmetry o f this fundamental mode, the numerical model is simulated over half of the structure. A perfect magnetic wall is inserted at the plane o f symmetry. The ground plane is represented by a perfect electric conductor, for simplicity. The open boundaries can be closed with perfect conducting walls, either electric or magnetic. The effect of these walls on the final solution is minimized by placing them relatively far away from the strip. A nonuniform mesh is generated over the transmission line cross-section. The numbers of patches increase in the area where rapid field variations are expected. The electric parameters are averaged over the patches lying at the interface between different materials, as explained before. The mesh is constructed such that all the interfaces and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 the boundaries lie exactly on one side o f a patch. The nonuniform mesh generator is explained later. The finite difference approach presented above results in a finite set o f algebraic equations, which have the form o f a matrix eigenvalue problem, A E .= X E . (3.10) where A is the eigenvalue of the matrix. In our case, the eigenvalue value A equals to the negative of the square of the propagation constant Some elements of the matrix A are functions o f y2. These elements represent the points on the interface between the different materials in the structure. Physically, they couple the solution of the wave propagating in the different regions of the structure. The eigenvalue problem is solved using a direct method. First, a reasonable value for the propagation constant y is estimated and supplied to the numerical solver, which produces a more accurate estimate for y. The updated propagation constant y value is supplied again to the numerical solver. This iterative process continues until the solution converges. Thus, an accurate propagation constant is obtained iteratively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 3.1.3 FDFD Application to Microstrip Line Numerical results fo r the finite difference method are generated fo r two configurations. One configuration is a superconductor microstrip line w ith a lossless dielectric substrate o f £d = 23, as shown in Fig. 3.2. The other structure is completely fille d w ith air. The latter structure is relatively simple. It is used to validate the generated results, since it is a pure TEM structure. Therefore, it provides an accurate tool for comparing the results. Both configurations have the same dimensions; strip width W = 500 pm, substrate thickness d = 450 pm, and superconducting strip t = 1.0 pm. The characteristics of the superconducting material are as follows, penetration depth at T = 0 K is X(0) = 0.18 pm, the conductivity for the normal electron gas at T = Tc equals to <Jn = 104 S/cm, and Tc = 100 K. The results are calculated at two different temperatures, the liquid nitrogen boiling point T = 77 K, and T = 89 K. Air W/2 Dielectric x Fig. 3.2 Superconductor microstrip line geometry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 The phase constant and relative phase constant shift for the air-filled structure are shown in Fig. 3.3. The propagating mode is a quasi-TEM mode, since E. must exist due to the losses. The percentage change in the phase constant is expressed as follows, M P = W O - / ? ( I r . o ) . x ioo (3.U ) P(T=0) The slow-wave effect o f the superconducting material can also be observed. Fig. 3.4 shows the corresponding attenuation constant. The attenuation increases with frequency and temperature as expected. As the critical temperature for the superconductor material is approached, the attenuation increases dramatically. The phase constant o f the superconducting microstrip line as function o f frequency, at T = 0 K, is shown in Fig. 3.5. The results are verified by comparing them with the phase constant of a microstrip line with perfect conductors [85]. As expected, the two curves are very similar. The phase constant of the superconducting structure is slightly greater than that of the perfectly conducting structure due the internal inductance of the superconducting material. The attenuation characteristics for the superconducting microstrip line on a lossless dielectric substrate o f Ed = 23 at different temperatures, are illustrated in Fig. 3.6. The attenuation increases with frequency and temperature as shown before. The corresponding phase constant and relative phase constant shift are shown in Fig. 3.7. The propagating mode is T M Z. The low dispersion of the superconducting transmission line is clear. The percentage changes in the phase constant at T = 77K and T = 89K are also depicted in Fig. 3.7. It should be noted that the slope of the phase constant /3-curves is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 always positive in Fig. 3.3 and Fig. 3.7. The fact that A (3J[3 curves have negative slope in Fig. 3.3 does not imply that the phase constant /}-curve itself has a negative slope. As the temperature increases, the phase constant slightly increases. The increase in the attenuation constant with the temperature is due to the increase of the normal electron current penetration in the material. The increase in the phase constant with temperature may be explained by the slow wave effect resulting from the increase in the internal inductance associated with the superconducting material. Although, the phase constant shift is small, it is predicted by our calculations. 0.01 1400 73 Phase constant 1200 0.008 2sT cr. < o 5 1000 0.006 3 o o o t/5 o U o T =0K 800 eu T= 89K T=77K 600 0.004 0.002 > "GO 400 10 20 30 40 50 60 70 Frequency (GHz) Fig. 3.3 Phase constant of superconducting microstrip line filled with air at different temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.0004 0.0003 s To § T=89 K 0.0002 3 C 3 3 C O < 5 T=77K 0.0001 10 20 30 40 50 60 70 Frequency (GHz) Fig. 3.4 Attenuation constant of superconducting microstrip line Filled with air at different temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6000 5000 4000 3000 perfect conductor 2000 1000 10 20 30 40 50 60 70 Frequency (GHz) Fig. 3.5 Propagation constant of superconducting microstrip line lossless dielectric. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.004 /H 0.003 T=89 K aC*5 o u 0.002 s o •3 3 5 < T=77 K 0.001 10 20 30 40 50 60 70 Frequency (GHz) Fig. 3.6 Attenuation constant of superconducting microstrip line filled with dielectric at different temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 6000 0.01 Phase constant (rad/m) 0.008 Phase constant 4000 0.006 T=0K 3000 0.004 T= 89K T=77K 2000 0.002 Relative Phase constant shift (Ap/p %) 5000 1000 10 20 30 40 50 60 70 Frequency (Ghz) Fig. 3.7 Phase constant o f superconducting microstrip line on lossless substrate at different temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 To study the effect o f different HTS materials on the line performance, structures with different London's penetration depths are investigated. A ll other parameters are unchanged. The attenuation and phase constants at T = 77K are shown in Fig. 3.8 and Fig. 3.9, respectively. The increase in the attenuation constant with the penetration depth is due to the increase of the normal electron current penetration in the material, and the corresponding decrease in the super electron current in the HTS material. The increase in the phase constant can be explained by the slow wave effect resulting from the increase in the internal inductance associated with the superconducting material, as previously stated. The former results are for HTS microstrip lines on a lossless dielectric substrate, to emphasize the effect of a superconducting material on a microwave structure (e.g., the slow wave effect and the low losses of the superconducting material). The characteristics o f a microstrip line, deposited on LaGa0 3 /LaAlC >3 substrate, are shown in Fig. 10. The substrate loss tangent equals 5.x 10‘5. The microstrip line has the same dimensions and parameters as previously described. The attenuation constants o f the superconducting microstrip line, with both lossless and lossy substrates, at two different temperatures, T = 77K and 89K, are shown in Fig. 3.10. The attenuation is dominated by the dielectric substrate losses, since the attenuation constant for the lossy substrate case at the two different temperatures of interest is almost the same. Also, one may note that the attenuation constant is drawn on a logarithmic scale. The effect o f the losses, due to the lossy dielectric substrate, on the phase constant is extremely small. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.02 T = 77K s 0.015 A. ( (Jm ) = ,558g - i 1 CJ 0.01 c 0 ars 3 1 < Am 0.005 .3553 .2235 10 20 30 40 50 60 7 0 Frequency (GHz) Fig. 3.8 Attenuation constant of superconducting microstrip line on lossless substrate at different penetration depth. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.01 T = 77K 0.008 k (pm) = .5588 c/5 | 0.006 c/5 a 0 u 1 .4471 0.004 s > I "o .3353 0 .0 0 2 10 20 30 40 50 60 _ 70 Frequency (GHz) Fig. 3.9 Relative phase constant o f superconducting microstrip line on lossless substrate at different penetration depth. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ILossy dielectric] T =89kl 0.01 I Lossless Dielectric! 1 < 0.001 0.0001 10 20 30 40 50 Frequency (GHz) 60 70 Fig. 3.10 Attenuation constant of superconducting microstrip line on lossless and lossy substrates at different temperatures. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 3.2 Finite-Difference Time-Domain (FDTD) The FDTD applies second-order accurate central-difference approximations for the space and time derivatives of the electric (E) and magnetic (H) field intensities directly to the differential operators of the curl equations. It is a marching-in-time procedure which simulates the continuous actual waves by sampled-data numerical analogs propagation in a computer data space. An arbitrary 3-D structures can be embedded in an FDTD lattice simply by assigning desired values o f electrical permittivity and conductivity to each lattice electric field intensity (E) component, and magnetic permeability and equivalent loss to each magnetic field intensity (H) component. These are interpreted by the FDTD program as local coefficients for the time-stepping algorithm [51]. Specification of the media properties in this componentby-component manner assures continuity of tangential fields at the interface of dissimilar media with no need for special field matching. Moreover, recent advances in FDTD modeling concepts and software implementation, combined with advances in computers, have expanded the scope, accuracy, and speed of FDTD modeling to the point where it may be one of the best choice for large electromagnetic wave interaction problems. Also, FDTD technique has an explicit or semi-implicit scheme, where parallel computers provide a good environment to run such schemes [61]. The FDTD is well known [50]-[58], Hence, it is only briefly discussed. Specifically, only the features peculiar to its implementation in the thesis are described. These features are necessary to successfully model the HTS microwave devices, where the field penetration effects need to be taken into consideration. They are the nonuniform graded mesh generator, the perfectly matched layer absorbing boundary conditions, and the execution of the computer code on massively parallel processors machine. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 64 The follow ing finite difference equation for Ez is obtained by combining Ampere's law with the two fluid model, A/ e : +1(/, j , k + 1/ 2 ) = — e : (/, j , k + 1/ 2 ) + £,z O 'M -+ i/ 2 ) ; 1 , <fz{U.k + l/2) A ’ J’ ' } 1( az{i,j,k + \l2) 2e~(i,j,k + 1/ 2 ) 2ez(i,j,k + l/2) (H n y+'/2{i +1/2 J , k + 1/2) - / / ; +V2(/ - 1/ 2 , 7 , k + 1/2)) z i r ( / - l / 2 ,;,A: + l/ 2 ) +1/2,/: +1/2) - H n x+V2{ i , j -1 /2 , k + 1/2)) (3.12) A y (/,7 -l/2 ,* + l/2) J”+i/2( i\ M + 1/2 )] where the superconducting current density Js; is obtained from the discretized form of London equation [67], which can be written as, • C V2( i . M + 1/2) = + 1/2) + At VoK{iJ,k + 1/ 2 ) En z { i , j , k + 1/2) (3.13) It is obvious that the physical parameters of the different materials are defined at each point in the three dimensional simulation domain. Similarly, the finite difference equations required for Ex and Ey can be obtained. The finite difference equations for the H field components obey the modified Yee's algorithm [51]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 65 3.2.1 Nonuniform Finite-Difference Mesh Generator The developed three-dimensional finite-difference scheme is capable o f modeling the finite thickness of the HTS strip. No approximations are made to the strip thickness. This leads to a very dense uniform mesh that requires unrealistic memory storage. To alleviate this problem, a graded nonuniform mesh is implemented along the cross section o f the waveguiding structure. This nonuniform discretization imposes some restrictions which do not exist with the uniform discretization. The computational domain is discretized according to the following expression (3.14) where Ax is the mesh size, W distance to be discretized, n number of points, and p is the mesh resolution factor. The smallest mesh size is chosen inside and around the HTS strip. It is equal to a fraction of the magnetic field penetration depth for the HTS material, or the skin depth for normal lossy conductors. The mesh resolution factor p must be optimized to minimize the dispersion introduced by the nonuniform discretization. In general, the ratio between two consecutive mesh step distances must not exceed 2. The Courant stability condition, which determines the time increment, At, for the FDTD algorithm, equals to where is the maximum wave phase velocity within the structure under investigation, 8 ^ the smallest mesh size, and n is the number of space dimensions. For practical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 reasons, it is best to choose the ratio of the time increment to spatial increment as large as possible yet satisfying Eq. (3.14). I f the physical mesh points are aligned with the electric field components, the mesh spacing for the magnetic field components could be calculated according to the following expressions + (3.16, To ensure the accuracy o f the-computed results, the spatial increment must be small compared to the wavelength. As a rule of thumb and to reduce the truncation and grid dispersion errors, the smallest wavelength A ^ , i.e. at the highest frequency, existing in the computational domain must be at least 2 0 times greater than the maximum step size (5max in the discretized mesh, ^•min - 20<5max (3.17) One must note that the mesh is uniform ly discretized along the direction o f wave propagation for most of the guided wave problems considered in this thesis. Fig. 3.11 shows the nonuniform finite-difference mesh for half microstrip line structure along its cross-section. It is seen that the conducting strip is approximated by an adequate number of points, which corresponds to the skin depth for the normal conductor and the magnetic field penetration depth for the superconductor. In our case for strip of dimension of 15 pm x 1 pm, the number o f nodes are 24 x 10. The mesh is dense around the strip edges where the variation in the fields is expected to be large. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Fig. 3 .11 Nonuniform mesh for half microstrip line structure Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 The calculated electric field using the FDTD approach for a microstrip line is presented in Fig. 3.12. The substrate height is 10 pm, and the conducting strip o f width 15 jim is assumed to be perfectly conducting w ith zero thickness. The results are obtained for the uniform and nonuniform discretization meshes. The mesh size for the uniform case is 0.625 pm, while the smallest mesh size in the nonuniform discretization case equals to 0.125 pm. The number of points used are 127 x 64 x 127 for both cases. Both cases have the same simulation time with equal time step. 0.8 Nonuniform mesh — — Uniform mesh S 0.575 tS c_> •s b 3 0.35 ■N 3 Z 0.125 - 0.1 0 0.2 0.4 0.6 0.8 1 Time (ps) Fig. 3.12 Electric field calculated at the same physical positions using the uniform and nonuniform discretizations with the same number of mesh points. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 The electric fields are probed underneath the center o f the conducting strip at different positions along the direction of propagation. It can be observed that the nonuniform mesh discretization gives better convergence than the uniform case. There is no need to decrease the mesh size in an uniform scheme, which in turn increases the number nodes required to represent the same structure. This w ill lead to unrealistic memory allocation. 3.2.2 Absorbing Boundary Conditions: Perfectly Matched Layers The FDTD has been applied to various electromagnetic problems such as scattering, radiation, and integrated-circuit component modeling [50]-[58], Many applications involve modeling electromagnetic fields in an unbounded open space. Due to the limited storage space o f computers, numerical computation domains must be finite. A certain type of boundary condition, the so-called the Absorbing Boundary Condition (ABC), must be applied on the outer boundaries of the computation domain to simulate the unbounded physical space. Practical absorbing boundary conditions usually cannot absorb outgoing waves completely, and generate some error in numerical solutions, especially the frequency domain parameters. With better absorbing boundary conditions, not only is the numerical solutions more accurate, but the outer boundaries may be brought closer to the modeled targets, resulting in considerable savings in computer memory space and computation time. Special attention must be given to unshielded microwave devices. The side and top walls must be placed relatively far from the device to include all the physical waves propagating in the device. It is well known that the Fourier transform of the time domain results is very sensitive to numerical errors, notably those resulting from the imperfect treatment of the absorbing boundary conditions used to truncate the numerical computations of an open domain [52], The available absorbing Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 boundary conditions for the discretized wave equations are either not accurate enough or require impractical large computer memory to model microwave devices. Also, they are originally developed for lossless uniform dielectric configuration, where no metal strips exist. The effects of these ABC on the evanescent waves which may be present in some structures are not studied, and in general do not work for these cases. A dispersive ABC is implemented in [ 86 ], but it effects are not perfect. Most o f the implemented ABC are based on the one-way wave equations suggested by Mur [87]. Actually, the stability o f the absorbing boundary conditions can not be achieved exactly; there w ill always be some non-physical reflected wave returning to the computation domain due to the boundary treatment. Most of the accurate numerical computation performed for microwave devices are obtained by making the computation domain sufficiently large and stopping the computation after the useful information has been obtained [56], Recently, a new type o f absorbing boundary condition algorithm has been developed, which greatly improves the accuracy of the local absorbing boundary conditions. The new technique, called Perfectly Matched Layer (PML), was developed by Berenger in 1994 [ 8 8 ], It was validated and extended by Taflove et al. [89]-[90]. This approach is based on the use of an absorbing layer especially designed to absorb without reflection the electromagnetic waves. The theoretical reflection factor of a plane wave striking a vacuum-layer interface is null at any frequency and at any incidence angle. So, the layer surrounding the computational domain can theoretically absorb without reflection any kind of wave traveling towards boundaries, and it can be regarded as a perfectly matched layer. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 3.2.2.1 Theory of the PML Medium A reflectionless transmission of a plane wave propagating normally across the interface between free space and an outer boundary layer is achieved by satisfying the following condition a o (3.18) where cr and cr* denotes the electric and magnetic conductivities, respectively, o f the outer boundary medium. In other words, impedance o f the outer boundary medium is matched to the impedance o f the air. Layers o f this type have been previously used to terminate FDTD grids. However, the absorption is thought at best to be in the order of the analytical ABC's because o f increasing reflection at oblique incident angles [ 8 8 ], The PML technique introduces a new degree of freedom in specifying loss and impedance matching by splitting the fields components into sub-components. The magnetic field component H . can be splitted into H .x and H 0 , as example. In three-dimensions, all six Cartesian field vector components are split, and the resulting PM L modification of Maxwell's equations yields 12 equations, as follows: (3.19) (3.20) (3.21) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 dE„ £ iL + ( jE = £° dt ° y‘b» £ — h .+ a E = 0 dt z yz dt +* a E' - r w r -“ d iH ^ + H ) v ^ dy ’ dE d (H yx + H r ) £ ^ + £ 7 . ^ = - - L2L__2i2 dt dz dz ° dt dx f (3.22) (3.23) ( 3 .2 4 ) dx The wave in the PML layers propagates with exactly the same velocity as the interior medium, but decays exponentially with distance. Also, the wave impedance of the PML medium exactly matches that of the interior region regardless of the angle of propagation or frequency. This makes the PML approach an excellent boundary truncation for microwave and millimeter-wave devices. Berenger proposes a lossless free-space FDTD computational zone surrounded by a PM L backed by perfectly conducting (PEC) walls, as shown in Fig. 3.13. A t both the upper and lower sides of the grid, each PML has a t and ax matched according to Eq. (3.18) along with a y = cr* = cr. = o ’. = 0 to permit reflectionless transmission across the interior region-PML interface. A t both the left and right sides o f the grid, each PML has a y and cr* matched according to Eq. (3.18) along with a x = cr'x = cr. = cr[ = 0. A t both the front and back sides of the grid, each PML has cr. and cr* matched according to Eq. (3.18) along with csx - cr* = crv = <j* = 0. A t the four normal edges o f the grid (along xaxis), where there is overlap of two PML's, each PML has o \, cr*, cr. and cr*, which are set equal to those of the adjacent PML's, along with a x = a x = 0. A t the four transverse edges of the grid (along y-axis), each PML has crx, cr*, cr. and cr‘ , which are set equal Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 to those o f the adjacent PML's, along with a y = cr* = 0. A t the four longitudinal edges of the grid (along z-axis), each PML has a x, cr*, a y and cr*, which are set equal to those of the adjacent PML's, along with a. = cr* = 0 . A t the eight comers o f the grid , where there is overlap of three PM L’s, all eight losses, a x, cr‘ , ay, cr*, a , and cr*, are present and set equal to those o f the adjacent PML's. Fig. 3.13 Perfectly matched layers absorbing boundary condition in threedimension cartesian coordinates. Beringer suggests that the loss should increase gracefully with depth, p , within each PML. The electric loss rises from zero at p = 0 , which is the interface between the interior region and the PML load, to a maximum value of crmax at p = 8, the location of the conducting wall backing the PML, as following Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 n (3.25) *(P ) = < * » * ( ! ) where S is the PM L thickness and a is either o x, a y or a. ., and n determines the rate o f increase of the losses in the PML. Beringer assumed that the loss increases quadratically, i.e. n - 2. The corresponding magnetic loss is expressed as *v (3.26) where p* is shifted half spatial step size with respect to p to account for the physical location of the fields in the FDTD lattice. Then, crmajt can be chosen to bound the apparent reflection coefficient 2 0 ^ 5 cosfl R(9) = e n+1 cc (3.27) where 9 is the incidence angle defined with respect to the interface between the interior region and the PML's, and equals zero for normal incidence. Numerical experiments confirm that the reflection does not depend on the incidence angle [ 8 8 ]. The reflection coefficient R reduces to a key user-defined parameter. In fact, sharp variations o f conductivity create numerical reflection. Numerical studies have shown that the optimum value of R depends on the number of PML's, the permittivity of the interior medium, the factor n , and the types of waves incident on the PML. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 The attenuation to outgoing waves afforded by a PML medium is so rapid that the standard Yee time-stepping algorithm cannot be used. The field components may be advanced by using the explicit exponentially difference equations [91]. W ith the usual FDTD notation, the following equations are presented as examples: 1/2{ i J + 1 / 2 , k + 1 / 2 ) = ; + 1 / 2 ,Jfc+ 1 / 2 ) + az{ i , j + 1/2,k + l/2 ) A z (3.28) [£ ;( /,; +1 / 2,* +1 / 2) - En y ( i J +1 / 2,k - 1 / 2)] - a z(i+ \ll,j,k )& i/c p n E ”« ( i + l / 2 J , k ) = e e ; ( ; + i / 2,m )+ cr. (/' + ! / 2 ,j,k)A z (3.29) 3.2.2.2 PML ABC for Waveguiding Structures The PM L ABC is applied to terminate all the microwave devices simulated in the thesis. A ll o f these devices consist of different dielectric medium, and include metallic strips. The FDTD lattice is terminated by extending the dielectric layers into its matching PML [89]. The conductors are extended without PML region. This configuration, shown in Fig. 3.14, simulates extremely lossy waveguiding structure. The propagation wave is highly attenuated in the PML without reflection at the interface between the PM L and the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 actual structure. This approach fits the need for an appropriate ABC for dispersive multimodal propagation, which was not available. It is essential for calculating the dispersion characteristics o f a transmission line, especially the losses. The enhancement applied to the PML-ABC eliminates any possible discontinuity effects at the end of the line. Air PML-Air Conductor Dielectric ; P M L-D iel; Fig. 3.14 Perfectly matched layers absorbing boundary condition for waveguiding structures that include different dielectric materials and metallic conductors. It is obvious that the reflection coefficient calculated using the PM L is very small compared to the one-way wave equation [90]. Although the PML is successfully applied at the back of the FDTD lattice, special precautions have to be taken into considerations when applying the PML to the top and sides walls. It is essential to place the PML's far enough from the device to prevent absorbing the physical power propagating along the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 device. The simulation domain can also be closed at the front surface using PML. The source plane is inserted next to the PML layers. 3.2.3 Parallel Implementation of the FDTD on MASPAR Machine The past decade has witnessed a tremendous explosion o f research on various aspects o f parallel processing covering parallel architecture, parallel models and complexity classes, parallel algorithms, programming languages with parallel construct, compilers, and operating systems for parallel computers. Also, with the recent advances in VLSI technology, it has become feasible to build computing machines with hundreds or even thousands o f processors cooperating in solving a given problem. Computing machines with various types and degrees of parallelism built into their architecture are already available in the market, and many more are in various stages o f development. Examples include CRAY research machines, and Massively Parallel Processors (MPP) . Parallel computation for electromagnetic problem analysis has been successfully implemented for many applications [60]-[61]. FDTD technique has an explicit or sem i-im plicit scheme, where parallel computers provide a good environment to run such schemes. The electric and magnetic fields components are calculated using their nearest magnetic and electric field components, respectively. This algorithm avoids the time consuming communication within the MASPAR machine. The implementation of the parallel program code is summarized in the following steps. First, a FORTRAN77 code for the problem is developed, executed on a serial machine, and compared with previously published work. Second, the serial code is parallelised using the VAST translation software. The parallel Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 code is executed on the MASPAR, and the efficiency of the code is investigated. Then, the parallel code is optimized.using MASPAR FORTRAN, which is sim ilar to FORTRAN90, in order to increase the efficiency o f the code on the MASPAR . Finally, the computation time on the serial IB M RS6000 cluster machine and that o f the MASPAR are compared. 3.2.3.1 FDTD Parallel Algorithm The implementation o f the FDTD on parallel machine requires a different algorithm for defining the media parameters from the serial code. The media parameter in FORTRAN77 code are defined for each domain constituting the structure as follows, c Compute the media parameters do 1 m = 1, media eaf 1 c - r * sig(m) / epr(m) ca(m) = (l.OeO - eaf)/(1.0e0 + eaf) cb(m) = ra / epr(m) / (I.eO + eaf) continue define the dielectric substrate do 2 k - 0, ksub-1 ; do 2 i = 0, im a x; do 2 j = 0, jm ax ixmed ( i, j, k ) = 2 2 continue But, media parameters can not be defined as domain in MASPAR FORTRAN, they need to be represented as point parameters. They are written as, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 c Compute the media parameters e a fl: media ) c = r * sig( :media ) / epr( :media ) ca( :media ) = (l.OeO - eafl :media )) / (LOeO + eafl .-media )) c b (: media ) - r a / e p r ( :media ) / ( I.eO + eafl :media )) define the dielectric substrate cax( :imax, :jmax, :ksub-l ) = ca(2) 3.23.2 MASPAR FORTRAN versus FORTRAN77 MASPAR FORTRAN is based on the well-known FORTRAN77 standard, with extensions from DEC FORTRAN and the FORTRAN90 ISO standard. These enhancements are designed to take advantage of the massively parallel, SIMD power of the MASPAR family of computers. The most significant enhancements provided with MASPAR FORTRAN are in the areas of processing arrays and new' intrinsic functions. Next, some of these features w ill be discussed. Their implementation in the computational program w ill be shown. The differences with the serial code w ill be addressed. These characteristics are: (a) Arrays as a First Class Objects (b) Attribute specification (c) Array Assignments (d) Intrinsic functions In FORTRAN77, operations on arrays are programmed explicitly, using iterative DO loops. In MASPAR FORTRAN, arrays are first class objects and arrays operations can be written as simple expressions rather than necessarily as iterative loops. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 M ASPA R FORTRAN compiler generates parallel SIM D code fo r these array calculations. The following serial code do 3 k = 1, kmax-1 ; do 3 j = 0, jm a x t; do 3 i = 0, imaxt m = iymed ( i, j, k ) ey(i,j, k) = ca(m) * ey(i,j, k) + cb(m) * (h x (i , j, k) - hx( i, j, k -1 )hz(i+1, j, k) + hz(i, j, k) ) 3 continue is changed to the corresponding parallel code as, ey(: imaxt,:jmaxt, 1: kmax-1) = cay(: imaxt,:jmaxt, 1:kmax-1) * ey( :im axt cby(:imaxt,:jmaxt, 1 :km ax-l)*( ( hx( :im axt jmaxt, 1:kmax-1)+ ,:jmaxt, 1:kmax-1 )- hx(: imaxt,:jmaxt, 0: kmax-2)) - (hz(l: im axt+1,:jmaxt, 1:kmax-1 )+ hz(:im axt,:jm axt,l:km ax-l))) MASPAR FORTRAN allows to specify attributes of arrays in a compact form as, real, array ( O'.imax, O.'jmax, 0:km ax) : : ex, ey, ez, hx, hy, hz instead o f FORTRAN77, which has the following form real ex ( O'.imax, 0:jm a x, 0:kinax) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 M ASPAR includes a FO RALL statements to specify an array assignment statement in terms of array elements. The FORALL statement is the parallel equivalent o f a DO loop; it specifies parallel calculation for the included assignment statement. There are some restrictions on its use. The FORALL statement didn't efficiently work in our FDTD implementation for replacing the DO loop. Regular assignment expressions are used instead. MASPAR FORTRAN offers significant additions to the FORTRAN77 intrinsic function libraries. Most of these new intrinsic functions are derived from the proposed FORTRAN 90 standard. The array reduction functions, for example the statement SUM, are very helpful in writing a compact computer code for many applications. The array manipulation functions, as CSHIFT, are very useful for finite difference solution applications, however it didn't work efficiently when used for updating the field expressions. To perform array operations on certain array elements, WHERE statement can make array element assignments conditionally. 3.2.3.3 Data Allocation and Array Mapping in MASPAR FORTRAN The MASPAR system is composed of a front end and a Data Parallel Unit (DPU), each of which has its own processors and memory organization. On the DPU, memory is distributed among each Processing Element (PE) in the PE grid. Serial operations are performed on the front end. Parallel operations are performed on the PE array in the DPU. The key to good performance on a data-parallel machine is to use as many PEs as possible and to use them efficiently. This can achieved by using (a) Mapping Directives (b) Array Mapping Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 Mapping directives give control over how arrays are allocated on the PE grid. Arrays for the fields and the media parameters are allocated as follows, cmpfmap ex(memory, xbits, ybits) These compiler directives allocate the first dimension to memory of the corresponding PE, the second dimension in the X dimension, and the third dimension in the Y dimension. These allocations are changed with the size of the problem in each Cartesian coordinates or dimensions. Arrays are mapped onto the PE grid in columns and rows on the MASPAR machine. The target PE array size is chosen according to the size o f the problem. The -pesize compilation option specifies the number of parallel processors in the DPU that can be dedicated for the executed program code. Table 1 shows the number o f processors, and the machine size for the available options. Table 3.1 Number of Processors and Machine Size for available options Options Number of processors Machine size 1 1,024 processors 32 columns x 32 rows 2 2,048 processors 64 columns x 32 rows 4 4,096 processors 64 columns x 64 rows 8 8,192 processors 128 columns x 64 rows Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 The PE grid is physically two-dimensional. The PE local memory on each processor provides a virtual third dimension, as it is shown in Fig. 3.15. Layer 0 corresponds to the PE array, and layer 1 to n represents the local memory. It is crucial to adapt the number o f processors to the problem size. layer n layer 1 ybits layerO xbits Fig. 3.15 Virtual Layers in the processing elements memory. 3.2.3.4 Communication in MASPAR FORTRAN Communication on the MASPAR machine is performed either through X-Net or a global route. Programming style and array access have a direct effect on the type of communication used. Therefore, when PE-PE communication is required the type o f communication used is determined by how the arrays are aligned. Regular column-to- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 column or row-to-row operations are translated into X-Net instructions by M ASPAR FORTRAN. This is the primary benefit o f the MASPAR FORTRAN array mapping. The fields arrays and the corresponding media parameters arrays are mapped the same way in the computational program. cm pfm ap ex(memory,xbits,ybits) cm pf map cax(mernory,xbits,ybits) x The communication-to-computation time needs to be minimized on the MASPAR to increase the execution efficiency of the code on the parallel machine. The FDTD algorithm exhibits nearest-neighbor communication pattern, which makes the approach well suited for parallelization. 3.2.3.5 Comparison between Serial and Parallel FDTD Codes A comparison between the required CPU time to execute the serial code on the IB M RS6000 and the parallel code on the MASPAR machines is presented. The dimension o f the problem is fixed in the x and y directions to (64x32), while the third dimension, which corresponds to the z direction, varies from 30 to 120. The number of time steps in the FDTD calculation is constant for all simulations. The results are shown in Fig. 3.16. The improvement in the CPU time acquired by running the code on the parallel machine is clear. The speedup achieved in the FDTD calculations increases as the problem size increases. It increases from 3.13 for the (64x32x30) problem to 10.39 for the (64x32x120) problem. The results demonstrate the potential o f parallel computation for FDTD calculations in electromagnetics. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. □ MASPAR M IBM RS6000 30 Fig. 3.16 60 90 3rd dimension size (kmax) 120 CPU time for MASPAR and IB M RS/6000 machines for problems that have different sizes and equal number of time steps. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 3.2.4 FDTD Application to Microstrip Line A microstrip line with lossless material on isotropic substrate is analyzed using the FDTD technique. The thickness of the perfectly conducting strip is rigorously taken into consideration. The strip is presented with adequate number o f mesh points as previously explained. The calculated effective dielectric constant for the finite lossless strip is compared with the result obtained using the following analytical expression [92]: - p r*ff = er +1 e - 1 T ^ O —L______ |------1_____ 1+ + 0.04 1 - ('W/h) - errff reff = £. + <-) 1 £ -1 — 1+ '■) 1/2 12 12 W C, W /h < 1 (3.30) 1 /2 -C , W /h ^ 1 (3.31) (W/A). and, C= e , ~ l t/h 4.6 V WJh (3.32) where £r is the relative dielectric constant for the substrate, h is the substrate height, and W and t are the strip width and thickness, respectively. Numerical results are obtained for microstrip line with W = 7.5 fim , h = 10 fim , and £r = 13. The simulation is performed for zero, 0.5 fdm, and 1 (im strip thicknesses. These dimensions are selected to allow comparing our results with those obtained in [52], The calculated effective dielectric constant using the FDTD is plotted in Fig. 3.17. The results depicted for the zero thickness perfectly conducting strip case shows good agreement with the previously published ones using the FDTD. The results obtained for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 the finite thickness lossless strip are compared w ith the ones calculated using the analytical expression presented in Eq. (3.31). Excellent agreement can be observed from the graph. The decrease in the effective dielectric constant with increasing the strip thickness can be explained by considering the increase in the total power propagating in the air by increasing the strip thickness. This leads to an increase in the phase velocity of the wave propagating along the microstrip line, which in turn decreases the effective dielectric constant. One may note that the slow wave effect of the field penetration inside the conducting strip is not considered due to the perfectly conducting strip assumption. 9 _ § 88 ------------ ,i------------ i,------------, ------------- ------------" r-----------1 - - ’ thickness = 0 pm to __ > , - i ~ —------- • ---------------- 8.4 thickness = 0,5 0.5 pm um z_______ *________ 0_________ 82 ----------- .----------- •----------- .----------- •--------- — -------- = •a o w t -- I O ZJ 'M O ----------- FDTD ......Analytical Analytical ......Published ............. Published .. .________ .-------------= thickness = 1 pm 8 0 I1 I1 10 20 I1 I1 30 40 Frequency (GHz) I1 -----------50 60 Fig. 3.17 Effective dielectric constant (er = 13). Comparison o f the results calculated using the FDTD with the empirical formula and the results presented in [52], Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 88 3.3 Comparison between FDFD and FDTD Solutions The finite-difference technique offers many advantages as a modeling, simulation, and analysis tool, in general. Both the FDFD and FDTD could deal w ith arbitrary structure geometries. The interaction with an object o f any conductivity either real metal or perfect conductor can be handled. The main advantage o f the FDTD is its broadband response predictions capability. However, the time domain method does not have the capability to distinguish between modes. The technique typically assumes that the only mode which can propagate is the dominant mode. Actually, the solution does not correspond to the exact dominant mode in general, although how great an influence the higher order modes have on the dominant mode is still to be determined. The FDFD differentiates between the different modes propagating in the structure. The problem is formulated in an eigenvalue form, where each mode corresponds to a specific eigen value. Then, the deterministic problem can be solved for each eigen value or mode. However, the Fields and currents distributions obtained from the FDTD solution are more realistic than those of the FDFD. Also, FDTD is most suited to computing transient responses. Moreover, it may be the method of choice for some problems. Interestingly enough, interior coupling into metallic enclosures is a situation where FDFD w ill most likely fail to capture the highly resonance behavior of a metallic enclosure, even when made at many frequency points. Especially, the nonlinear analysis can only be conducted using the FDTD and never the FDFD. Finally, the FDTD is computationally more efficient than the FDFD. It has an explicit or semi-implicit scheme where no matrix operation is required. To conclude our discussion, the choice between the FDTD and FDFD depends on the problem on hand. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 Numerical results obtained using the FDTD and FDFD approaches are presented in Figs. 3.18 and 3.19. The simulation is performed for a microstrip line with 50 £2 impedance. The microstrip line dimensions are: substrate height = 10 /dm, strip width = 7.5 /im , strip thickness = 0.5 /im and 1 /dm. The relative dielectric constant o f the substrate equals to 13. The substrate is assumed lossless, since the effect o f the conducting strip needs to be analyzed. The conducting strip is studied for the lossless, lossy copper with conductivity o f 5.87 x 107 (S/m), and YBCO superconducting material with magnetic field penetration depth of 0.223 /im and normal conductivity o f 3.51 x 105 (S/m) at T = 77 K cases. The attenuation constant for the copper and superconductor strips is presented in Fig. 3.18. As expected, the losses introduced by the copper are much higher than the losses associated with the superconductor. The losses increase with the decrease of the strip thickness. This can be explained by the decrease in the number o f the conducting electrons as the strip dimension decreases. Also, the fringing field at the edges of the strip increases as the strip thickness decreases. The results depicted for the attenuation constant are simulated first using the FDTD technique, then duplicated at 10 and 30 GHz using the FDFD approach. The same results are obtained using both methods. Fig. 3.19 shows the effective dielectric constant o f the microstrip line for three different cases: lossless , lossy, and superconductor strips. The lossy and superconductor strips are copper and YBCO, respectively. The structure is investigated for the 0.5 um and 1 um strip thicknesses. As explained before, the effective dielectric constant decreases as the strip thickness increases. The effective dielectric constant for the copper strip is greater than the lossless case. This can be explained by the increase in the inductance o f the transmission line as the field penetration inside the conducting strip is included. The effective dielectric constant for the superconductor is larger than the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 copper conductor. This is due to the addition of the internal kinetic inductance associated with the motion of the paired superconducting electrons. This kinetic inductance w ill be added to the total inductance o f the line, and results in an increase in the effective dielectric constant o f the transmission line. These results are obtained first using the FDTD technique, then duplicated at 10 and 30 GHz using the FDFD method. Typical results are obtained using both approaches. Finally, one can conclude that both methods could be used to calculate the dispersion characteristics of waveguiding structures with the same accuracy. 0.5 B CQ "3 0.4 £3 S 0.3 .o copper .5 um copper 1 um super .5 um super 1 um FDFD solution o V c o 0.2 a o 0.1 - * - 10 20 30 40 Frequency (GHz) 50 60 Fig. 3.18 Attenuation constant for copper and superconducting microstrip lines with different strip thickness using the FDTD and FDFD approaches. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9.5 super lossy lossless FDFD solution £ O 8.5 'S Z J > o W thk = luml 7.5 0 Fig. 3.19 10 20 30 40 Frequency (GHz) 50 Effective dielectric constant for lossless, copper and superconducting microstrip lines with different strip thickness using the FDTD and FDFD approaches. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 3.4 Summary The FDFD and FDTD approaches are presented to execute HTS transmission lines with nonuniform dielectric filling. The conducting strip is rigorously modeled to account for the field penetration effects. This approach is essential to accurately model and simulate transmission lines which incorporate superconducting material. Slow wave propagation is observed along the superconducting microstrip line. The increase in the attenuation with temperature and frequency is clearly demonstrated. The geometric effects on the propagation characteristics of microstrip line are shown. As the conducting strip dimensions increase, the field penetration effects decrease. The nonuniform mesh generator is described. Results show better convergence and less memory space requirement by using the nonuniform discretization compared to the uniform one. Implementation of the algorithm on serial and parallel supercomputers are presented. The improvement in the execution time is examined. A substantial speedup is obtained by running the FDTD code on the MASPAR machine. A novel implementation for the PML ABC is suggested. The approach is suitable to terminate the simulation domain for structures with metallization. The enhancement in the reflection coefficient obtained by using the PML ABC compared to the one-way wave equation is shown. A comparison between the FDFD and FDTD methods is conducted. The main of advantage of the FDTD is its broadband response capability. The FDFD is more suitable for single frequency analysis, where mode differentiation is required. The accuracy obtained from methods is o f the same order, since both are based on the finite difference approximations. The technique presented in this chapter are used in the rest of the thesis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 4 ANISOTROPIC SUPERCONDUCTORS ON ANISOTROPIC SUBSTRATES A common feature o f the family of high temperature superconductors (HTS), including LaBaCuO, YBaCuO, BiSrCaCuO, TIBaCACuO, is that all have layered structures. It is generally believed that the two-dimensional Cu 0 2 network is the most essential building block o f the HTS materials [18]. One expects the quasi-two- dimensionallity associated with the layered crystal structures to be manifested not only in the superconducting properties but also in the electronic properties in the normal state. Anisotropy, associated with HTS, analysis and modeling are also essential for studying the nonlinearity associated with these materials. Moreover, there has been a growing interest in the use of low loss anisotropic substrates such as sapphire and boron nitride in microwave and millimeter-wave integrated circuits [24], Although the sapphire substrate is anisotropic, the r-cut single crystal sapphire seems to be an appropriate substrate material for HTS applications. Crystal lattice matching o f the sapphire with c-axis oriented YBCO, small dielectric loss and high thermal conductivity can be achieved simultaneously [27], Rigorous analysis of anisotropic HTS on anisotropic substrate is essential, regarding both utilization the anisotropy characteristics and elimination of its undesirable effects. In general, the analysis o f isotropic microstrip lines on anisotropic substrates has been reported in a few full-wave studies [93]-[94], Results have been presented for the tilted optic axis lying on the transverse plane [95]-[96] and the horizontal plane [97]-[99]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 For the HTS microwave transmission lines, a full wave analysis which takes into account either the anisotropy in the HTS material or in the substrate itself based on the spectral domain/volume integral equation approach (SDVIE) is represented [13]-[14], A ll the papers employed spectral-domain techniques except [95], who formulated Bergeron's method in the time domain to deal w ith tilted optic axis. However, no dispersive characteristics of microstrip lines were presented. Full-wave analysis of such structures, which takes account o f both the anisotropy in the metal strip and the dielectric substrate, is not performed or presented in any of the literature. In this chapter, we present a technique based on the three dimensional finitedifference time-domain method, to model transmission lines incorporating an anisotropic superconducting material deposited on sapphire substrate. The anisotropy of both the HTS material and the sapphire substrate are taken into account simultaneously. The equations are derived directly from Maxwell's equations. The approach fits the needs for accurate computation of the dispersion characteristics of an anisotropic superconducting transmission line. Also, effects o f anisotropy on the field distribution inside the structure and on the currents distribution inside the HTS are investigated. The model is flexible, and can be used for any o f the planar transmission line structures containing HTS material. It incorporates all the physical aspects of the HTS through London's equations. The electromagnetic characteristics and the required boundary conditions in the structure are represented using Maxwell's equations. The physical characteristics of the HTS are blended with the electromagnetic model using the phenomenological two fluid model. To validate the accuracy of our algorithm, the complex propagation constant is calculated and the results are compared with the published spectral domain technique data. The effect o f the anisotropy orientation on the characteristics of the microstrip lines and the coplanar waveguides are studied. Interesting comparisons between isotropic and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. anisotropic structures as well as comparison between the characteristics o f the microstrip lines versus coplanar waveguides w ill be presented. 4.1 Anisotropic High Temperature Superconductor Model The two fluid model assumes that the electron gas in a superconductor material consists of two gases, the superconducting electron gas and the normal electron gas [67]. The main parameters o f the superconducting material are the London penetration depth X l and the normal conductivity a n. The physical nature o f the superconducting phenomena is included in the dependence of the charge carrier densities and the effective masses o f the superconducting and normal states, as well as the normal electrons relaxation time, on the temperature. The temperature dependence is approximated by the well known Gorter-Casimir model. In fact, this model is in good agreement with the measured penetration depth for conventional superconductors, but not for HTS [21]. The total current density in superconducting material is expressed as follows (4.1) where Jn and Js are the normal state and super state current densities, respectively. The normal fluid current density obeys Ohm's law (4.2) The superconducting fluid current density is obtained using London equation [ 6 8 ], — = E dt \ l 0XL Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.3) 96 These expressions are valid assuming that the HTS material is isotropic. However, experiments show that the properties o f the HTS materials are anisotropic. Such anisotropy has significant effects on the device performance. The designer is faced with several choices, as the type o f material and film direction, to obtain the optimum configuration that enhances the characteristics of the HTS material. A common feature of the family o f HTS, including YBaCuO and TIBaCaCuO, is that they all have layered crystal structures. It is generally believed that the two-dimensional Q 1O2 network is the most essential building block o f the HTS materials. Thin-film transmission lines w ill favor films in which conducting sheets lie in the plane o f the film . The anisotropy for the HTS can be represented by an anisotropic conductivity for the normal state, and an anisotropic London penetration depth for the superconducting state. The diagonal conductivity tensor cr is given by 0 o' 0 °b 0 0 0 tfc. (4.4) Also, the diagonal London penetration depth tensor A is diagonal, and is written as 0 0 0 0 ' 0 0 (4.5) K. where a , b and c are the principal axes of the anisotropic superconducting material. The normal conductivity and the London penetration depth in Eqs. (4.1)-(4.3) w ill be replaced by their corresponding tensors to formulate the anisotropic superconductor model. Note Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 that the subscripts a, b, and c do not refer to the direction of the magnetic field, but to the direction, in which the screening currents flow. Experiments show that, for YBCO, the HTS parameters along the a- and b -axes are approximately equal. In our discussion, these parameters are the normal fluid conductivity onab and the superconducting fluid London penetration depth The normal conductivity o nab equals 10-80 times anc, and the penetration depth Xc equals 3-5 times Xab [21]. Experiments also report that the critical current density and the upper critical field are anisotropic [22], One must note that this superconductor model assumes that the material has almost constant penetration depth at a temperature w ell below the critical temperature Tc and the field penetration remains almost unchanged with frequency. In general, the effective penetration depth is greater than the one depicted in the two fluid model [ 2 1 ], 4.2 Anisotropic Finite-DifTerence Time-Domain Approach The finite-difference time-domain (FDTD) solution of Maxwell’s curl equations is one of the most suitable numerical modeling approach for the electromagnetic analysis of volumes containing arbitrary shaped dielectric and metal objects. An arbitrary 3-D structure can be embedded in a FDTD lattice simply by assigning desired values of the material parameters (e.g. electrical permittivity, conductivity, and London penetration depth) to each lattice point. These parameters are utilized in the calculation of the respective electromagnetic field components. These are interpreted by the FDTD program as local coefficients for the time-stepping algorithm [51]. Specification o f the media properties in this component-by-component manner provides a convenient algorithm to represent the anisotropy in a media, and assures continuity o f tangential Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 fields at the interface o f dissimilar media with no need for special field matching. The developed three-dimensional finite-difference time-domain scheme is capable of modeling the finite thickness of the HTS strip. The finite thickness is represented by adequate number o f mesh points. A graded non uniform mesh generator is used to discretize the simulation domain [49]. The ground plane is chosen as a perfect electric w all, for sim plicity. The computational domain is closed by the PM L absorbing boundary conditions [82]. The program code for our analysis is written in FORTRAN 90 and is executed in massively parallel machine (MASPAR) environment. The anisotropy is included in the FDTD code without significantly affecting its efficiency or the required memory size. The FDTD applies second-order accurate central-difference approximations for the space and time derivatives of the electric (E ) and magnetic (H ) field intensities directly to the differential operators of the curl equations, when the anisotropy of the material is along its principal axis. The same algorithm can be implemented when the principal axes o f the anisotropic material are Lilted with respect to the coordinates. In this case, the solution w ill be carried on the electric ( D ) and magnetic ( B ) flux densities. The algorithm w ill be extended one step further, to obtain the field intensity using the appropriate constitutive relations. In our discussion, the YBCO HTS strip is assumed to be anisotropic along its principal axis (i.e. c, b, and a along x, y, and z respectively). The r-cut sapphire substrate is considered for two cases at 0 and 90 degrees rotation about its principal axis (i.e. rotation about x-axis). The configuration simulated in this study corresponds to the practical structure used in [27]. However, the analysis presented here is flexible and can be extended to any anisotropic configuration. The follow ing finite difference equation for Ez is obtained by combining Ampere's law with the two fluid model, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 t Al-<rzz{ i J , k + l / 2 ) 2£zz(i,j,k + 1/ 2 ) Al ezz( i , j , k + 2| 1/2) ^ A t - f f a { i , j , k + 1/2) 2e-z(i,y,Jfc+ l/2) //;+1/2(/+ 1/ 2 ,7,A'+ 1/ 2) - #;+1/2(/ - 1/2 J , k + 1/ 2 ) A x ( i- l/ 2 , j, k + l/2 ) +1/2,/: +1/2) - ^(Z, j -1 /2 ,£ +1/2) (4.6) Ay(/,y -1 /2 , £ +1/2) J“ v2(;,;.* + V2)] where the superconducting current density Jsz is obtained from the discretized form of London equation, which can be written as, J : ; v2{ i j , k + 1/ 2 ) = J :r l/2( i j , k + 1/ 2 ) + ~ e: ( /j,a - + 1/ 2 ) (4.7) K It is obvious that the physical parameters, electric perm ittivity, conductivity, and penetration depth, of the different materials are defined at each point in the three dimensional simulation domain. Similarly, the finite difference equations required for Ex and Ey can be obtained. The finite difference equations for the H field components obey the modified Yee's algorithm [51]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 The developed three-dimensional finite-difference time-domain scheme is capable of modeling the finite thickness of the HTS strip. The finite thickness is represented by adequate number of mesh points. A graded non uniform mesh generator is used to discretize the simulation domain, where the smallest mesh size is chosen inside and around the HTS strip. The Courant stability condition is based on the smallest mesh size. The ground plane is chosen as a perfect electric wall, for simplicity. The computational domain is closed by the PML absorbing boundary conditions. The symmetry o f the structure is used, and the simulation is carried only on half of the structure. The excitation pulse used in the analysis has been chosen to be Gaussian in time. Unfortunately, it is not possible to excite the structure with a pulse having the field pattern o f the appropriate mode due to the anisotropy investigated in our analysis. A very simple field distribution can be specified at the excitation plane using our knowledge of the modes in the simulated structure. As long as a suitable distance is allowed for the Gaussian pulse propagation, the pulse pattern in the transverse direction evolves to its actual physical form. This can be verified graphically. The Gaussian pulse must be wide enough in time to cover adequate number of space divisions to obtain a good resolution. The turn-on amplitude of the excitation ought to be small and smooth. 4.3 Anisotropic Superconductor Microstrip line on Anisotropic Sapphire Substrate A microstrip line with anisotropic YBCO HTS material on a sapphire substrate, as shown in Fig. 4.1, is analyzed using the previously described technique. The HTS strip has penetration depth \ { 0 K ) = 0.14 pm, normal conductivity o n(Tc) = 0.5x106 S/m, and critical temperature Tc = 92.5 K. The frequency independent penetration depth and the normal conductivity equal to 0.2 pm and 1.0xl0 6 S/m, respectively at 77K. The dimensions o f the microstrip line are as follows : strip width 2W = 2 pm, substrate height Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 101 h = 1 (im, and HTS strip thickness t = 0.5 pm. These dimensions are selected to allow comparing our results with those presented by Lee et al. [13]. Although, our analysis is carried out on this stfucture, the FDTD formulation is flexible and can virtually be applied to any microwave device w ith arbitrary dimensions and/or anisotropic orientation. 2W H-----------M Fig. 4.1 Anisotropic microstrip line on anisotropic sapphire substrate. The simulation is performed for a microstrip line with isotropic substrate with dielectric constant £r = 3.9. Two cases of the HTS strip are considered, the isotropic and anisotropic HTS. The anisotropic characteristics of the HTS are presented by anisotropic penetration depth, Ac = 5 A ^ , and anisotropic normal conductivity, a nab = 50 o nc [ 2 1 ]. The principal axes of the HTS Film is aligned with the coordinate axes. The c-axis is assumed to be in the z-direction. The results calculated for the effective dielectric Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 constant and attenuation constant, for both the isotropic and anisotropic HTS strip on isotropic substrate are shown in Fig. 4.2. i W5 a 8 0.75 O 3 u § 2 Si ~a 5 o > •3 fcs w FDTD SDVE 0.5 0.25 * 8 I 12 3 C 3 cr. o 3 o 0 3 c/i E ? 3 01 CO A 16 20 Frequency (GHz) Fig. 4.2 The propagation characteristics of anisotropic HTS on isotropic substrate using the anisotropic FDTD. These results are also compared with these obtained by Lee et al. [13] using the SD/VIE approach. Our results are in good agreement with the previously published ones. A slight increase in the effective dielectric constant accompanied by a small increase in the attenuation constant is observed in Fig. 4.2. Obviously, this slight change is due to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 larger field penetration in the FDTD treatment compared to the SDVIE approach and the way the current distribution is represented in both methods. This clearly demonstrates the accuracy and the consistency of our model. The effect of the HTS anisotropy, on the propagation characteristics o f the microstrip line, is negligible due to the orientation o f the thin film HTS strip. In the microstrip line, the current in the x-direction is small compared to the other two components. istropic HTS/istropic SUB anisotropic HTS/istropic SUB N _ 0.6 Total current Super current SJ 2 0.2 Normal current 0 0.2 0.4 0.6 Half strip width (pttrv) 0.8 1 ► x Fig. 4.3 Normalized normal-fluid, super-fluid, and total current densities at the bottom surface of the strip, for both the isotropic and anisotropic HTS cases, on isotropic substrate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 The distribution o f the normalized longitudinal current density component ( / . ) at the bottom surface o f the HTS strip for the normal and the super gases are shown in Fig. 4.3. The values and the distributions are approximately equal for both cases, isotropic and anisotropic HTS strip. This can explain the sim ilarity in the effective dielectric constant and the attenuation constant for both cases, isotropic and anisotropic HTS films. Also, one can observe that the current is doubled at the strip edge with respect to the center o f the strip. The superfluid represents almost 80 % o f the total current at 77 K, according to the two fluid model used in our analysis. The study o f the anisotropy is carried on an anisotropic HTS strip on sapphire anisotropic substrate. The anisotropic YBCO HTS strip has the same characteristics as before. Anisotropic microstrip line with the same dimensions as presented above is used in this study. The anisotropy of the sapphire substrate is represented by the dielectric permittivity tensor. For the 0° r-cut sapphire, the relative permittivity tensor is e „ = 10.03, £>y - 10.97, and e „ = 9.4 along x, y, and z directions respectively. The tensor is £ „ = 10.03, £)y = 9.4 and e „ = 10.97 for the 90° r-cut [27]. The angle is calculated with respect to the rotation about the x-axis. To assess the effect of the substrate anisotropy, a comparison w ill be made between the performance of the microstrip line on a sapphire substrate with a similar structure on an isotropic substrate with a relative permittivity £r = 10.03. It should be noted that the chosen relative dielectric constant of the isotropic substrate equals to the element of the anisotropic dielectric tensor £ „. In the microstrip line configuration, the electric field in the x-direction is the strongest, thus the propagating wave velocity strongly depends on the value of the relative permittivity in that direction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 Fig. 4.4 shows the attenuation constant for the anisotropic HTS on r-cut sapphire substrate as a function of the rotation angle about the x-axis at two different angles, 0 ° and 90° and for anisotropic HTS on isotropic substrate with er = 10.03. The microstrip line on anisotropic sapphire substrate with 0° r-cut has the highest attenuation. The 90° rcut produces the lowest attenuation of the three structures. The effective dielectric constants ereff of the three structures are also depicted in Fig. 4.4. The 0° r-cut sapphire substrate results in the highest e reff, while the 90° r-cut substrate produces the lowest. The change in the propagation characteristics can be explained by the value o f the permittivity tensor element £)y in the y-direction. The y-field component o f the fringing field increases with the increase of eyy from 9.4 (90° rotation) to 10.03 (isotropic) to 10.97 (0° rotation) which in turn increases the energy stored in the substrate, and results in the decrease of the propagating wave velocity on the line or increase in the effective dielectric constant. The increase in the attenuation constant can be explained by considering the various current distributions presented in Fig. 4.5. It is shown that the normal current density is the largest for the 0° r-cut and it is the lowest for the 90° r-cut substrate. This explains the slightly higher attenuation in the 0 ° r-cut case. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 10 9.6 a o U u •5 o _CJ "o o > •a CJ S3 W 9.2 0.6 88 0.4 -< 8.4 0.2 8 0 4 8 12 Frequency (GHz) 16 20 Fig. 4.4 Propagation characteristics for anisotropic HTS on different r-cut sapphire substrates and on isotropic substrate with er = 10.03. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Attenuation Constant (dB/m) anisotropic HTS/sapphire SUB (Cf) anisotropic HTS/isotropic SUB anisotropic HTS/sapphire SUB (90°) 107 Total current Super current T5 anisotropic HTS/sapphire SUB (0 ) anisotropic HTS/isotropic SUB anisotropic HTS/sapphire SUB (90°) Normal current 0.1 0 Fig. 4.5 0.2 0.4 0.6 Half strip width (pm) 0.8 1 N orm al-fluid, super-fluid, and total current densities for anisotropic HTS on different r-cut sapphire substrates and on isotropic substrate £r = 10.03. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 4.4. Anisotropic Superconductor Coplanar Waveguide on Anisotropic Sapphire Substrate A coplanar waveguide with YBCO HTS material on sapphire substrates are analyzed using the previously described technique. The frequency independent penetration depth and the normal conductivity of the HTS equal to 0.2 (im and l.OxlO^ S/m, respectively at 77K for both structures. The dimensions of the coplanar waveguide are as follows : strip width 2W = 2 pm, substrate height h = 1 pm, HTS strip thickness t = 0.5 pm, and slot width o f s = 0.5 pm. These dimensions are selected to allow comparing our results with those presented for the microstrip line. The coplanar waveguide dimensions are demonstrated in Fig. 4.6. s 2W s Fig. 4.6 Anisotropic HTS coplanar waveguide on anisotropic substrate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 The anisotropic characteristics of the HTS strip and the sapphire substrates are chosen the same as the microstrip line case. The effective dielectric constant and attenuation constant, and the current densities distribution are calculated as the microstrip line case and shown in Figs. 4.7 and 4.8, respectively. The coplanar waveguide on anisotropic sapphire substrate with 0 ° cut has the highest attenuation and highest effective dielectric constant, while the 90° r-cut substrate results in the lowest ones. The change in the propagation characteristics can be explained by the value of the perm ittivity tensor element e}y in the y-direction. It is known that the electric field in the y-direction is the strongest field component in the coplanar waveguide. This results in an increase in the energy stored in the substrate as the permittivity tensor element £yy increases, which in turn decreases the phase velocity of the propagating wave along the transmission line. The increase in the attenuation constant can be justified by considering the various current distributions presented in Fig. 4.8. It is observed that the normal current density is the largest for the 0° r-cut and it is the lowest for the 90° r-cut sapphire substrate. This explains the slightly higher attenuation in the 0° r-cut case. Comparison of the anisotropy effects on both the microstrip line and the coplanar waveguide can be depicted by analyzing Figs. 4.4 and 4.7. One should observe that the change in the propagation characteristics for the presented coplanar waveguide with the anisotropy of the materials is more pronounced than the microstrip line. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 0 0.0 0.76 ■ -0 .7 2 a o o o CJ o > 'Z o3 & w 3 7.8 - “ anisotropic HTS/sapphire SUB ((f) ------------- anisotropic HTS/isotropic SUB — anisotropic HTS/sapphire SUB (90°) 7 .4 - 0.68 -0 .6 4 7- - 0.6 - 0.56 0 Fig. 4.7 4 8 12 Frequency (GHz) 16 20 Propagation characteristics for anisotropic HTS coplanar waveguide on different r-cut sapphire substrates and on isotropic substrate with £r = 10.03. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Attenuation constant (dB/m) 8.2 - Ill N y—** Total current Super current. a P anisotropic HTS/sapphire SUB (d ) anisotropic HTS/isotropic SUB anisotropic HTS/sapphire SUB (9(J) u ■N3 1 1 Normal current 0. 1 - 2 0 Fig. 4.8 0.2 0.4 0.6 Half strip width ( pn) 0.8 ► 1 y N orm al-fluid, super-fluid, and total current densities for anisotropic HTS coplanar waveguide on different r-cut sapphire substrates and on isotropic substrate er = 10.03. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 4.5 Summary A full-wave analysis for anisotropic HTS planar microwave structures deposited on anisotropic substrates, is presented. The FDTD technique, which takes the finite thickness o f the anisotropic HTS film into consideration, is developed using a graded non uniform mesh generator. The propagation characteristics of the anisotropic HTS microstrip line and coplanar wave guide, on r-cut sapphire substrate, are calculated as functions o f different r-cut angles. It is shown that the 90° r-cut sapphire substrate structure has lower loss and lower effective dielectric constant than the 0 ° r-cut one for both structures. These observations are explained by the current distributions on the HTS strip. The approach presented can be used not only to obtain the characteristics of HTS microwave structures but also to determine the optimum design that exploits the anisotropic conductor characteristics on anisotropic substrates. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 5 FULL-WAVE NONLINEAR ANALYSIS OF MICROWAVE SUPERCONDUCTING DEVICES The recent discovery of HTS has fundamentally changed the prospects for applications o f superconductive electronics and has generated considerable effort to apply these materials in a number of new areas [29]-[31]. The first successful applications are in the area o f passive microwave and millimeter-wave transmission structures such as resonators [34] and [43], filters [32] and [35], and delay lines [28] and [35]. Computer simulation is needed to analyze and design superconductor components, devices, and circuits. These simulations must be valid for both low and high power applications. Most of the theoretical research, developed for superconducting devices, focused on the linear theory. London equations were simultaneously solved with Maxwell's equations [4]-[12], Quasi-TEM methods were employed to calculate the propagation characteristics o f HTS transmission lines [3], [7] and [37], and the resistance, the inductance, and the current distributions of striplines [38], The nature o f the quasi-TEM approximation in these calculation restricts its application to the lower frequency range (0 to 10 GHz for 150 um striplines). The impedance boundary condition approach, in which the strip is assumed to be either much thinner or much thicker than the magnetic penetration depth, was also used [8]-[14], The application of these methods are restricted by the dimensions o f the structures. To accurately model the characteristics o f superconducting microwave devices over a wider range of frequencies, a rigorous full- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 wave approach has to be used. The spectral-domain volume-integral equation method [14] and the elaborate mode matching technique [11] have been used. The methods presented before are for the linear response o f superconducting materials, and are thus valid for low power levels only. However, significant nonlinear response can be observed in measurements of the resonance curve and the quality factor in stripline resonators [39]-[40] and [43], microstrip resonators [41], and cavities [44]. Moreover, the high current value existing in some applications may not exceed the HTS critical current densities of high-quality YBCO films, but they are high enough to drive the HTS into nonlinear behavior. As an example, HTS transmission line resonators in narrow band filters have high peak current densities, which result from the high standingwave ratios on the resonator lines [45], The nonlinear characteristics of the HTS result in generation of harmonics and also spurious products created by the mixing o f multiple input signals. Therefore, the magnitude and the detailed nature o f the nonlinear effects must be understood in order to facilitate widespread application of HTS in microwave and millimeter-wave applications. It is generally believed that a major source o f the nonlinearity in superconductors is due to the breaking of superelectron pairs in a high field environment. Better understanding for the dependence o f the penetration depth, as well as the superconductor electron density, on the electromagnetic field requires a rigorous full-wave nonlinear model. In addition, accurate modeling of the nonlinearity in superconducting resonators can serve as a useful tool to characterize the critical parameters o f superconductor thin films. The nonlinear model must only be developed in time-domain and never in the frequency domain or phasor form o f fields [83]. The problem of modeling the nonlinearity in HTS microwave and millimeter has been tackled before using different approaches. An iterative method combining the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 spectral domain approach and the impedance boundary condition model is applied in [100] and [101]. The Ginzburg-Landau (GL) theory was used to predict the nonlinear behavior in a superconducting stripline resonator as a function of the input current [ 102 ]. These approaches were based on frequency domain calculation. A macroscopic model of the nonlinear constitutive relations in superconductors is derived from a velocity distribution assumption [103]. To our knowledge, modeling and full-wave analysis of the nonlinearity associated with microwave HTS devices were never performed in the time domain. In this chapter, a nonlinear full-wave solution, based on the GL theory is developed using the Finite-Difference Time-Domain (FDTD) technique. The GL theory is independent of the microscopic mechanism in superconductor and is purely based on the ideas of the second order phase transition only. The physical characteristics o f the HTS are blended with the electromagnetic model using the phenomenological two fluid model. Maxwell's and GL equations are solved simultaneously in three-dimensions. This time-domain nonlinear model is successfully used to predict the effects o f the nonlinearity on the performance of HTS transmission lines and filters. This approach takes into account the field penetration effects. The spatial distribution of the total electrons and the number of the super electrons compared to the normal electrons vary with the applied power. A study o f the nonlinearity effects on the propagation characteristics, current distributions, electromagnetic field distribution, and frequency spectrum o f microstrip lines is conducted. The s-parameter o f HTS filte r is predicted. The nonlinear output input power relation is estimated. The results are compared with experimental data. The field distribution along the filter resonators is studied. Our model is flexible and can be used for any of the planar microwave and millimeter-wave devices that include HTS material. This approach is not only useful to predict the nonlinearity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 effects on microwave devices performance but also can be utilized in the characterization of the HTS materials. 5.1 Time-Domain versus Frequency-Domain Numerical Techniques Numerical Characterizations and modeling o f guided-wave components has been an important research topic in the past three decades. When a specific structure is analyzed, one has to make a choice which method is best suited for the structure. Obviously, the choice may not be unique. One must make a critical assessment for every possible method. Analysis and modeling of the nonlinearity imposes some restrictions on the selected numerical techniques. It is known that the application of a signal to a waveguiding structure including nonlinear material causes frequency mixing to occur. This results in generation o f harmonics and spurious products. The frequency domain approach is based on analysis in the Fourier transform domain. It provides an elegant tool for the reduction of the partial differential equations of mathematical physics into ordinary ones, which in many cases are amenable to further analytical processing. The time-dependent partial differential equation is decoupled into a series o f frequencydependent ones. Hence, the solution is separately carried on each frequency component. The time-domain solution can be obtained by the superposition of the results calculated at each frequency components. This approach is widely used in problem containing linear materials. However, when a nonlinear material is used, the partial differential equation can not be transformed to the frequency domain. The equations for the various harmonics are no longer separable, and the superposition technique is not allowed. Hence, the equations must be solved in time domain. One should notice that this is a fundamental issue. It is not a matter of approximation or simplification. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 5.2 Nonlinear Full-Wave Superconductor Model The nonlinear superconducting model, described in Chap. 2, is incorporated into the three-dimensional full-wave electromagnetic simulator, presented in Chap. 3, using the two fluid model postulate. The two fluid model assumes that the electron gas in a superconductor material consists of two gases, the superconducting electron gas and the normal electron gas [67]. The main macroscopic parameters o f the superconducting material are the magnetic filed penetration depth Xs and the normal conductivity o n. The superfluid characteristics depends on the penetration depth, while the normal conductivity determines the normalfluid characteristics. The physical nature o f the superconducting phenomena is included in the dependence of the charge carrier densities and the effective masses o f the superconducting and normal states, as well as the normal electrons relaxation time, on the temperature. Also, it is generally believed that the origin of the nonlinear behavior of superconducting materials is the breaking of the superelectron pairs inside the superconductor w ith the applied field. This results in inhomogeneous superfluid and normalfluid currents distributions with the nonuniform magnetic field associated with most of the superconducting strip used in microwave and millimeterwave applications. Since the current-field relation is assumed to be local for HTS, the nonlinear phenomena can be modeled by spatial, field, and temperature dependents magnetic field penetration depth X s{ t ,\h \,T^ and normal conductivity where r, |//(r)|, and T are the position, magnetic field intensity magnitude and temperature inside the superconductor, respectively. The total current density in superconducting material is expressed as follows J = 7„ + 7S Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.1) 118 where Jn is the normal state current density, which obeys Ohm's law, (5.2) and Js is the superconducting fluid current density, which follows the modified form of GL equation, dJ, __ dt 1 E (5.3) H 0X ] { r \ H ( r ) \ T ) The temperature dependence is approximated by the well known Gorter-Casimir model. In fact, this model is in good agreement with the measured penetration depth for conventional superconductors, but not for HTS [18]. The field dependence is obtained from the solution o f the phenomenological GL equations. The normalized order parameter, which corresponds to the fraction of the superfluid electrons density is calculated. It is field, position and temperature dependent. The normalfluid and the superfluid electrons densities, nn and ns, are calculated from the conservation o f the total number of electrons n, in the superconductor. The relations between nn, ns, and n, is derived based on the superconductor state at T = 0, Tc. The two fluid theory assumes that the superconducting material is in the superconducting state at T = 0, and in the normal conductor state at T = Tc, i.e. n, = n J(0,0) = nn(0,Tc) (5.4) (5.5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 x4 n , ( H , T ) _ nt (H ,T ) n, (5.6) ns(0,T) e/ which can be written in form o f the order parameter 4' ¥ (H ,T ) V (H ,T ) V. ■KO,T) 1 - f r l {T c j (5.7) and using the low field temperature dependence of the London penetration depth [67] A^o.o) A f (o .r) = - r (5.8) r r N4\ iT V \ C/ y The spatial, field and temperature dependent magnetic field penetration depth Xs for the superconductor can be calculated from the following expressions : A^o.o) y(»(Q.r) y /(o .r) Y ( (T 1\ (5.9) Tc / where /l/o.o) is the low field penetration depth measured at T = Oand H = 0 and Tc is the critical temperature for the superconductor calculated at H = 0. The parameters a and P may be obtained from experimental studies. In our analysis, we chose a = 2 and P = 4 fo llo w ing G L model fo r the field dependence and Gorter-Casimir for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 temperature approximations. The normalfluid electron density nn is derived from the following relations n J H J l= i-iW I) n, ( 5 . 10) n, The corresponding normal conductivity is expressed as follows : crn(w (F).r) = crn(nc/Tc) 1 - y(H(F),r) \ f / ( o ,r) 1- (5.11) T where o„{nciTe) is the maximum normal conductivity measured either at T = Te or H = H C and H c is the critical magnetic field for the superconductor calculated at T - 0. Eqs. 5.9 and 5.11 are one o f the main results of this chapter. They represent the dependence of the main macroscopic parameters of the superconducting material on the normalized order parameter, which in turn depends on the spatial distribution o f the applied field. The superfluid and normal fluid currents densities can be updated by using Eqs. 5.2 and 5.4. Their dependence on the applied field fu lly represent our nonlinear problem. A general iterative scheme for solving the nonlinear problem is suggested. The fields are firs t initialized using the low field macroscopic parameter o f the superconductor. The appropriate boundary conditions for solving G L equations are extracted from the electromagnetic simulator. The nonlinear G L equations are solved, and the normalized order parameter is calculated. The nonlinear magnetic field penetration depth and normal conductivity are updated. Then, the electromagnetic fields Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 are recalculated. This procedure is repeated until the total current in the superconductor converges. 5.3 Nonlinear Full-Wave Superconductor Simulator Experiments show that the HTS macroscopic parameters, the field penetration depth and the normal conductivity, are anisotropic, as explained in Chap. 4. Moreover, thin-film transmission lines w ill favor films in which conducting sheets lie in the plane of the film . So, the superconductor current w ill be considerably high in the direction of the wave propagation. Therefore, the nonlinearity effect is expected to arise from this longitudinal current component much earlier than the other two current components. Hence, two-dimensional solution for the nonlinear macroscopic parameters in the plane perpendicular to the direction of propagation is'satisfactory to model the nonlinearity in superconducting films used in microwave and millimeter-wave planar structures. The two-dimensional spatial distributions of the macroscopic parameters of the HTS, Ai (x,y;|H (x,y)|;T) and (*, y)\; r ) , can be obtained by solving the Ginzburg-Landau nonlinear differential equations inside the HTS strip. The time- independent GL equations w ill be used since the characteristic time scale of the propagating electromagnetic fields is much larger than the relaxation time o f the order parameter. In other words, the order parameter responds almost instantaneously to the time-varying electromagnetic field. It is straight forward to show that the normalized GL equations for the z-component current density can be sim plified to the following expressions, (5.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 122 -rV,2H=Hfw2-i+ K (5.13) V where t stands for the transverse x-y direction. The required boundary conditions become n x { V , x z A z) = n 0H, (5.14) » -v ,H = o • (5.15) where H t is the normalized magnetic field component tangential to the superconductor surface. This tangential magnetic field at the boundary of the strip is deduced from the full-wave electromagnetic simulator. So, the superfluid and normalfluid currents densities in the z-direction can be calculated. The currents densities in the x- and ydirections are calculated using the low field London model. The solution is performed by dividing the signal strip conductor into numerical grid segments. A nonuniform gridding scheme, explained in Chap. 3, is adopted to maintain a good resolution near the edges of the strip. Smaller segments are used near the edges where the changes in the order parameter and the field are rapid. The grid size is chosen at least one-fifth o f the penetration depth at the edge and increases gradually towards the center. Our nonlinear analysis algorithm is summarized in Fig. 5.1. First, the excitation pulse is launched into the HTS microwave device. Then, the magnetic field components are updated using the FDTD approach. Next, the tangential magnetic field to the HTS strip surface are used as boundary conditions for the nonlinear HTS simulator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( START ) ,, t = 0, n = 0 Launch Excitation t = n.dt Update Magnetic Fields Update Boundary Conditions forGL Magnetic vector Potential Solve Nonlinear HTS model Update Xs & ^ Order Parameter Update Electric Fields No n- No id t> T n =n +1 Fig. 5.1 Flow chart of the nonlinear analysis algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 The normalized superfluid electron density is calculated as previously described. The spatial, field, and temperature dependent macroscopic HTS parameters cr„ and Xs are updated. Finally, the electric field components are updated using the electromagnetic simulator. Then, a convergence test fo r the current flowing in the superconductor is conducted. This procedure is repeated for a time period suitable to analyze wave propagation characteristics inside the structure. The temporal fields are probed during the simulation process. The applied power is measured at the first probe. 5.4 Nonlinear Analysis of Superconducting Microstrip Lines The nonlinear characteristics o f a YBa 2Cu3 0 7 _x superconducting microstrip line, shown in Fig. 5.2, is analyzed in this section. The microstrip line is an inhomogeneous transmission line since the field lines between the strip and the ground plane are not contained entirely in the substrate. Therefore, the microstrip line can not support TEM mode. A full-wave analysis is required to rigorously analyze these devices. 2\v Fig. 5.2 HTS microstrip line geometry. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 The YBa 2Cu 3 0 7 _x superconducting material has a critical temperature o f 90 K, critical magnetic flux density n oH c( T ) o f 0.1T, and GL parameter o f 44.8 at 77 K [102], The superconducting microstrip line has a 50 Q impedance with a strip width o f 7.5 jim , and a thickness o f 1 fim . The substrate thickness is 10 /rm, with er= 13. The transmission line characteristics are simply chosen to demonstrate the nonlinear wave propagation along the line. The maximum rf power, Pcrf, where the HTS microstrip looses completely its superconductivity, is predicted using the G L solution described in Chap. 2. Its value equals to 920 W / cm 2. Numerical results are obtained at different levels of the applied power : 834 W / cm2, 410 W / cm2, and 181.8 W / cm2 denoted by 0.9 PCrf, 0.45 Pcrf, and 0 .2 Pcrf respectively. The temporal magnetic field propagating along the line are probed at 60 jdm and 150 iim , and shown in Figs. 5.3 and 5.4. The amplitude of the wave is attenuated as the applied field increases. Also, the slow wave effect can be observed from the figures. Fig. 5.4 also shows that London's model incorrectly predicts higher pulse amplitudes and lower attenuation even at high power. This qualitative discussion gives a good insight into the behavior o f the wave propagating along the transmission line. Fig. 5.5 compares the effective dielectric constant o f the transmission line at different applied power levels. The effective dielectric constant predicted using London's equations is smaller than the one obtained from G L solution at low applied power. This can be explained knowing that the penetration depth predicted by London theory is smaller than the one obtained from GL model, which agrees well with the experimental observations [21], The effective dielectric constant increases as the applied power increases. This slow wave effect is due to the increase in the internal inductance Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 126 o f the line introduced by the increase in the field penetration as the applied power increases. 1 input London - - - G L 0 .2 P - — GL 0.45 I*1 . - - - GL 0.9 p cr; erf 0.8 S o 2 0.6 “o E o 0.4 e o 03 2 •g s 0.2 o 0 z - 0.2 0 1 2 3 time (ps) 4 5 6 Fig. 5.3 Normalized tangential magnetic field intensity under the strip probed at 60 jim and 150 jim. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 127 Normalized Magnetic Field Intensity 0.94 0.93 ■London GL 0.2 P ' erf — GL 0.45 P . G L 0.9 P Crf erf [- A ' 0.92 // // 0.91 h !> / h .• j / i 0.9 3.2 3.25 / \ u ... \ i 3.3 time (ps) \ 3.35 3.4 Fig. 5.4 Normalized tangential magnetic field intensity under the strip probed at 150 Jim at different levels of applied power. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 128 8.5 1 ...... 1 ------ — — ------ “ ■ V 8.45 co U u '£ "3 5 o > •a u is w ““ ------------ London ------ — GL 0.2 P , erf ------------GL 0.45 P . ---------- GL 0.9 P crJ erf 8.4 8.35 ------ ------ ------ ------ ------ ------ — 8.3 8.25 Fig. 5.5 I 1 1 8 12 Frequency (GHz) 16 1 20 Effective dielectric constant for the HTS microstrip line at different levels o f applied power. The fractional change in the effective dielectric constant at different power levels obtained from GL model with respect to the one calculated using London's theory is drawn in Fig. 5.6. The change in the effective dielectric constant increases with the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 129 increase in the power level up to 0.7 Pcrf. It is approximately constant for higher power levels, where almost complete field penetration occurs. Freq = lb GHz /—s o 9 CO. a. < 0.6 0.3 0 0.2 0.4 0.8 0.6 Normalized Applied power (P.pP p 1 Fig. 5.6 Fractional change in the effective dielectric constant for the HTS microstrip line with applied power w.r.t. the linear model. The attenuation constant for the HTS at different levels o f applied power is presented in Fig. 5.7. As the applied power increases, the attenuation constant increases as well. This can be explained by the increase in the normal electron density in favor of the super electron density as the field penetrates the HTS. Also, it is observed that the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 130 HTS looses its superconducting characteristics earlier than its static critical power. This is due to the field singularity associated with the planar microwave and millimeter-wave devices. It is noticed that the change in the propagation characteristics o f the HTS microstrip line is not linear with the applied field. The change is more pronounced as the applied power increases. This is understood from the nature o f the superconducting material, which deteriorates very quickly as the applied power approaches its critical value. 100 London GL 0.2 P , GL 0.45 P , GL 0.9 P f erf 10 1 1 0.01 0.001 0.0001 0 4 8 12 16 20 Frequency Fig. 5.7 Attenuation constant for the HTS microstrip line at different levels of applied power. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 131 The fractional change in the attenuation constant obtained from GL model with respect to the value predicted by London model at 10 GHz is depicted in Fig. 5.8. It is clear that the HTS loses its superconductivity very quickly as the applied power approaches the electromagnetic critical power. Also, the nonlinearity associated with the HTS appears very early, even with the material in fairly good superconducting stage. 200 Freq = 10 GHz 160 120 80 40 0 0 Fig. 5.8 0.2 0.4 0.6 0.8 Normalized Applied Power (P y P crP 1 Fractional change in the attenuation constant for the HTS microstrip line with applied power w.r.t. the linear model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 132 The change in the losses is faster and more nonlinear than the change in the phase velocity. This can be explained knowing that the losses are ohmic, which are induced by the moving particles. On the other hand, the effect on the phase velocity is a variation in the stored kinetic energy inside the superconductor, which is mostly a wave effect. Hence, particle related effects can be stronger and faster than the wave related effects. The effect of the applied field on the current distribution is presented in Figs. 5.95.11. Fig. 5.9 shows the superfluid current distribution at the bottom o f the HTS strip. It is observed that the London model under estimates the current carrying capacity o f the HTS material. The HTS strip looses part of its superconductivity at the edges as the applied power increases from 0.2 Pcrf to 0.45 Pcrf. For 0.9 PCr f case, the partial loss of the superconductivity is induced across the entire cross section o f the HTS strip. It is more pronounced at the edge o f the HTS strip, where the superfluid current for the 0.9 Pcrf case is almost equal to the 0.45 Pcrf one. The superfluid current distribution at the top o f the HTS strip is presented in Fig. 5.10. The current values are less than the one obtained for the bottom o f the strip as explained before. For the 0.9 Pcrf case, the superfluid current increases at the top surface because the applied field is less than the critical magnetic field of the HTS material. Thus, the superconductor redistributes the superfluid as the applied field increases. The normalfluid current density behavior w ill be opposite to the superfluid one based on to the conservation in the total number of electrons in the HTS. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 a 0 London — GL 0.2 P GL 0.45 F*" - - - GL 0.9 PCTl erf - 3.5 Q w a S 3 <3 <nT 2.5 2 § 1o 'O s a. 3 2 o T— CO X o •o 3 co co 1 0.5 0 0 0.75 1.5 2.25 'y-axis' strip bottom (pm) 3.75 Fig. 5.9 Normalized longitudinal super fluid current density at the bottom surface o f the HTS strip at different applied power levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 134 London - GL 0.2 Pcrf GL 0.45 Pcrf - - - GL 0.9 Pcrf 3.5 u °C ■S 6 '3 SO vo^ Q* O 53 S ^ .X. - 2.5 2 1.5 •o 0.5 0 0.75 1.5 2.25 'y-axis' strip top (pm) 3 3.75 Fig. 5.10 Normalized longitudinal super fluid current density at the top surface of the HTS strip at different applied power levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 135 Fig. 5.11 shows the superfluid current distribution at the side of the HTS strip. The superconductivity o f the HTS material decreases ais the applied power increases. The distribution obtained from GL model at low power levels equals to the one predicted by the London model. Then, the effect of the nonlinearity on the electromagnetic field distribution in the microstrip line configuration is studied. >> — London — GL 0.2 P , - GL 0.45 Dcrtf Pcrf - GL 0.9 P . 3.5 erf •o « S 'S < O VO c. o 3 i Vi 2.5 ^ X top bottom 0.5 0 0.2 0.4 0.6 ’x-axis’ strip side (pm) 0.8 1 Fig. 5.11 Normalized longitudinal super fluid current density at the side surface o f the HTS strip at different applied power levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 136 The normalized tangential magnetic field intensity at the top and the bottom o f the strip are presented in Fig. 5.12. The effect of the applied power on the electromagnetic field distribution is small. It is only observed near the edge o f the strip. This explains the small change in the phase velocity of the wave propagating along the line. *3 5O L5 a &0 -t1 S ■a •a co 0.5 o Bottom C3 W H *3 O N - n London — GL 0.2 P . GL 0.45 p“ GL 0.9 P " f e rf -0.5 g Top 'z. 0 0.5 1 1.5 2.5 2 3 'y-axis' halt-strip width (pm) 3.5 4 Fig. 5.12 Normalized tangential magnetic field intensity at the top and bottom surface of the HTS strip at different applied power levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 137 Finally, the effect of the nonlinearity on the frequency spectrum o f the wave propagating along the line is analyzed in Fig. 5.13. The fractional change in the amplitude o f the output pulse frequency components increases with the applied power. It is observed that as the power level increases, the amplitudes o f the different harmonics change, which is one o f the primary characteristics of nonlinear devices. This confirms our point that the nonlinearity has to be modeled in the time domain. The results calculated form GL model for low applied power are approximately the same as the ones obtained from the linear London model. This validates our treatment for the HTS as linear material below the low applied power value. 0.05 London GL 0.20 P , GL 0.45 P " GL 0.90 P f 0.04 erf < 0.02 0.01 0 4 8 12 16 20 Frequency (GHz) Fig. 5.13 Fractional change in the amplitude of the frequency spectrum of the output pulse w.r.t the dc component at different applied power levels. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 138 5.5 Nonlinear Analysis of Superconducting Filters High temperature superconductor materials have potential applications in the design o f small size, light weight, and high quality factor microwave filters [32] and [45]. However, transmission line resonators in narrow band filters have peak current densities, which result from the high standing-wave ratios on the resonator lines. This high current value may not exceed the HTS critical current densities o f high-quality YBCO films, but they are high enough to drive the HTS into nonlinear behavior. The maximum r f power level that a filte r can handle without changing its characteristics depends on the HTS properties, the cross-sectional dimensions of the transmission line resonators, and the bandwidth o f the filte r. Moreover, most microwave CAD software requires an approximation in order to analyze filters circuits. They may fa il to predict the performance o f resonator array filters, especially for low coupling resonator filters. Thus, filte r design and analysis need a more rigorous simulator tool. For high power applications, nonlinearity associated with HTS materials w ill effect the filte r characteristics. In this case, the simulator has to be implemented in the time domain. If such a model is implemented in frequency domain, unreliable results may be obtained. In this section, the non linear full-wave model, based on the GL theory, is used to analyze and study the nonlinearity effects on HTS microwave filters. The s-parameter of microstrip line array resonator filte r is estimated. The nonlinear output input power relation is predicted. The field distribution along the filter resonators is studied. The calculated results are compared with the experimental data. This approach is not only useful to predict the nonlinearity effects on microwave filters performance but also can be utilized in the characterization of the HTS materials. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 139 5.5.1 Simulation of Microstrip Resonator Array HTS Filter Simulation for the microstrip resonator array filte r using the full-w ave 3D nonlinear HTS FDTD electromagnetic simulator, which rigorously includes the finite thickness o f the HTS film , is performed. The filter configuration is shown in Fig. 5.14. Fig. 5.14 HTS microstrip resonator array filter structure The filte r consists of 6 staggered resonators and two feeding lines. The substrate is LaAlO w ith dielectric constant of 23.5 and loss tangent of 5xlO '3. The dielectric height is 10 mils. The feeding lines have 50 £2 impedance. The width o f the feeding lines equal to 3.8 mils. The microstrip resonators have lengths of 163,165 and 166 mils. Their widths are 9.7, 10.6 and 8.3 mils, respectively. The spacing between the resonators is 17.7 mils at both ends of the staggered filters. It equals 76 mils between the 4 center Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 140 resonators. One must note that the filter is symmetric in the z-direction. The ground plane of the filter is assumed perfect electric conductor in our simulation. The HTS strip thickness is considered to be 0.3 pm. The macroscopic parameters o f the YBaCuO films are as follows: low field penetration depth AJ(0) = 1600A° and normal conductivity o n(Te) = 1.6 x 10sS / m. The critical temperature Tc for YBCO equals to 89.6 K. The GL parameter is assumed to equal 44.6. This filter is designed by David Samoff Research Center. A ll the measurements are performed in their labs. In general, the frequency dependent scattering parameters, Sy, can be calculated as follows [56]: (5.16) where V; and Vj are the voltages at ports i and j, respectively, and Z 0j and Z0, are the characteristic impedances of the line connected to these ports. The voltages, V(co), are obtained by taking the Fourier transform o f the time record of the voltage underneath the center of the strip. The voltage equals the average line integral o f the vertical electric field under the whole strip. The characteristic impedances are calculated using the following relation, as described in [52] where I(a>) is defined as the Fourier transform of the time record o f the loop integral of the magnetic field around the metal strip. Eq. (5.16), for the scattering parameters, takes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 141 into account the variation o f the characteristic impedance of the feeding lines with the frequency and the applied power, which is the case for superconducting structures. The S21 parameter for the array filter is calculated from 0 to 24 GHz. The results are shown in Fig. 5.15. The FDTD extracts the frequency response over a wide frequency band in one simulation. The simulation is not only capable to predict the fundamental resonant response o f the bandpass filter but also to calculate the higher order resonance. The second resonance peak appears at double the operating frequency of the filte r due to the periodic design of the array filter, as expected. The higher order intermodulation distortion can be easily predicted by the developed electromagnetic simulator. As a matter o f fact, the 3rd-order two-tone intermodulation performance of a microwave device is often used as a figure of merit for the linearity o f a microwave device. The calculated S21 depicted by our simulation is compared with the measured S21 . The comparison is shown in Fig. 5.16. The agreement between the measured and calculated scattering parameters is clear. The resonance frequency at 8.67 GHz as well as the small bandwidth of the bandpass filter are successfully simulated. The discrepancy between the measured and the calculated S21 at the upper frequency tail o f the filter response characteristics does not affect the accuracy of the calculation. The output input power relation for the bandpass filter is calculated and presented in Fig. 5.17. The predicted power relation is compared with the measured one. The nonlinearity associated with the HTS filte r appears at approximately 10 dBm. The developed nonlinear HTS electromagnetic simulator is able to estimate the nonlinearity in the HTS microwave filter. It is clear that the filter response deteriorates very quickly, as the applied power approaches the critical r f power, which is previously defined. The Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 142 critical r f power varies for the same HTS material according to the dimension o f the structure and its configuration. The electric field distribution in the dielectric along the longitudinal direction of the filter is depicted in Fig. 5.18. The input signal propagates along the input feeding line and is reflected at the end o f the feeding line. The wave resonates along the first microstrip resonator. Part o f the wave transfers to the second resonator, while the reflected wave returns to the input end. The resonant component transfers to the third microstrip resonator. The coupling between the resonators continues, and the wave reaches the output feeding line. Although, the field distribution gives a qualitative picture for the filte r performance, it is very important for optimizing the filter design. It is clear that the field levels are very high at the connection between the input feeding line and the first resonator. The 90° junction at this connection point may not be the optimum design configuration, especially for HTS materials. It is known that the field distribution is highly nonuniform for microstrip resonators, usually with very high peaks at the edges of the line. This behavior is clearly shown in Fig. 5.18. Moreover, one can observe that the fields at the microstrip resonators close to the input side are higher than the resonators near the output end. This behavior needs to be taken into account during the preliminary design o f the filter, especially for estimating the power handling capability o f the filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.8 0.6 0.4 0.2 0 4 8 12 16 20 24 Frequency (GHz) Fig. 5.15 The calculated S21 parameter for the HTS microstrip staggered resonator array filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a 21 ma g S]_ i Khh -2 0 .0 dB i, i. 0 . 0 dB/ v -3.G 9 0 S dB 3 C 7 3 7 T 5 1 J0 LAM35B2 . 0 + log REF 0 .0 dB log A 5 .0 dB/ MAG 77 M AR < E R 375 GHz CENTER SPAM Fig. 5.16 8 .7 35 0 0 0 0 0 0 0 .2 50 0 0 0 0 0 0 GHz GHz 1 Comparison between the calculated and the measured S parameters lor the HTS microstrip staggered resonator array filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission linear resp' — -*• —calculated measured S CQ T3 V -/ u O & o CL, -10 -20 -30 -30 -20 -10 0 10 20 30 Input Power (dBm) Fig. 5.17 Output input power relation of the HTS microstrip resonator array bandpass filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 5.18 Electric field distribution in the substrate along the longitudinal direction of the bandpass filter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 147 5.5.2 High Power Design Consideration for HTS Filters HTS microstrip line resonator array filter has very high unloaded quality factor as a result o f the very low surface impedance o f HTS materials. The HTS thin film s characteristics depend on the operating frequency, temperature, and r f power. A t high frequencies, fi)> 1012^-1, the macroscopic parameters o f the HTS strip, the normal conductivity and field penetration depth, become functions of frequency [19]. This limits the maximum operating frequency to the low millimeter-wave region. It is well known that the current distribution in microstrip lines is highly nonuniform. As a matter of fact, the current has a very high peak at the edges of the strip. For high power applications, the HTS may lose its superconductivity near the edge of the strip. This may drive the HTS material to exhibit a nonlinear behavior. The nonlinearity w ill result in the generation o f harmonics and also spurious products created by the mixing o f multiple input signals. Also, calculation of multiresonator filters power handling capability using the multiplicity approach is not adequate, especially for high power applications. The power handled by each resonator is not equal. The maximum operating r f power for HTS filters not only depends on the HTS materials characteristics but also varies with the filter dimensions. The connection between the feeding lines and the resonators must be carefully chosen. To avoid the nonlinearity effects, the filte r design must avoid any bend which creates high field region. Microstrip line resonators have significantly different even- and odd-mode effective dielectric constants. This difference w ill result in considerable forward coupling, which w ill reduce the dimension o f the filte r in the lateral dimension. However, the small bandwidth of the high quality filters requires a relatively large spacing between the strips in the longitudinal direction. Hence, microstrip line resonators Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 148 are characterized by their small size, even when made o f conventional conductors. Therefore, the main advantage o f using HTS in designing high quality filters is the better performance compared to conventional metal. However, the size and the layout o f the high power HTS filters must be optimized to avoid the nonlinearity effects associated with the HTS materials. 5.6 Summary A nonlinear full-wave three-dimension time-domain analysis for the HTS microwave devices is presented. This approach takes into account the variation of the macroscopic parameter of the superconducting material with the applied power, position, and temperature simultaneously. The wave penetration effects are rigorously included using the features of the three-dimensional finite-difference time-domain approach. The nonlinearity in the HTS is modeled by the GL equations. The anisotropic three-dimensional behavior of HTS superconductor is reduced to a quasi two-dimensional one. The GL solution gives the spatial distribution of the order parameter. The deduced order parameter is used to update the main macroscopic parameters o f the HTS, the magnetic field penetration depth and the normal conductivity in the longitudinal direction. The normal and super fluid current densities are calculated using Ohm's law and London's equation respectively. The currents in the transverse plane are very small. They are depicted using the simple linear model. The physical characteristics of the HTS are blended with the electromagnetic model using the two fluid model. Hence, the superfluid electron density is calculated at different levels o f applied power. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 149 The maximum r f power for HTS microstrip line is predicted. Numerical results show that a change in the phase velocityof the wave propagating along the line of about 1.5% occurs as the applied power reaches 0.9 o f the maximum r f power. The corresponding increase in the attenuation is dramatic. It increased 170 times compared to the low power case. The presented results show that the attenuation constant is more nonlinear than both the phase velocity. The effect on the electromagnetic field distribution is studied. It is more pronounced near the edge of the HTS strip. The superfluid current density distributions change dramatically with the applied field. The t change in the frequency spectrum is successfully depicted. The linear London model underestimates the field penetration inside the HTS material. It also overestimates the current crowding effects. Thus, the nonlinearity associated with.the HTS material is successfully modeled. A complete study concerning the effects o f the nonlinearity on the performance of HTS transmission lines is presented. The full-wave 3D nonlinear HTS electromagnetic simulator is successfully used to simulate microstrip resonator array filter. The scattering parameter, S21 , is predicted and compared with the measured data. The results show a good agreement for the resonant frequency and bandwidth. The output input power relation is also depicted. The nonlinearity associated with the HTS filter appears at approximately 10 dBm. The field distribution in the dielectric is presented. The high field values at the input o f the filter, as well as, the edges of the resonator are shown. The dimension and the layout o f high power HTS filters needs to be optimized in the design cycle to avoid the nonlinearity effects. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 6 N O VE L N O N LIN E A R P H E N O M E N O LO G IC A L TW O F L U ID M O D E L The greatest use of phenomenological models, such as the two fluid model, is to reveal experimentally verifiable interrelations among physical properties, such as the temperature dependence of the penetration depth. Then, it is the task of the microscopic theory to justify the phenomenological model. The phenomenological model can also help in a qualitative visualization of experimental phenomena. The two-fluid model in superconductivity has had only limited success in these respects. The point here is that the Gorter-Casimir and related two-fluid model were set up as to have the observed thermodynamics properties [62], But the model gave no information on, and indeed were not intended to have anything to do with, the hydrodynamic or electrodynamics aspects of the two fluids, the superconducting and normal fluids. F. London and H. London developed an electrodynamics model for superconducting materials on a phenomenological basis [ 6 8 ]. The theory gives a consistent description of essentially all the electromagnetics properties of superconductors. However, London model lent itself primarily to the electrodynamics, and not to the thermal properties of the superconducting materials. Thus, the Gorter-Casimir two-fluid and London models did not consider the bidirectional coupling between the thermodynamics and electrodynamics in a superconducting system. However, the expressions obtained from those models are elegant and simple in general, and are in use today as a qualitative model. The next level o f phenomenological description were presented by the GL theory o f superconductivity [63], The G L phenomenological model ties the thermodynamics and electrodynamics of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 151 the superconductor intimately together. It represents a plausible extension o f the London theory and two-fluid model to situations where the super electrons density is field- and position-dependent. But, the theory is encapsulated in a set of nonlinear equations. The solution o f G L equations is very involved, which makes them unsuitable for a computeraided-design environment. One important aspect of the physics of the HTS is their layer structure and the associated large anisotropy. It turns out that the structure and dynamics of flux-lines in such layered systems are markedly different from those o f isotropic conventional superconductors. Another important aspect is the unusual range o f parameters, such as short coherence lengths, large penetration depths, and high operating temperatures allowed by the HTS. In view of this, it is not surprising that a rich variety o f unusual behavior is found. A microscopic theory describing the physics o f HTS is unavailable at the present time. Numerous issues are still in part controversial, especially those dealing with phase transitions of the vortex lattice, explained later. In this chapter, we present a novel nonlinear phenomenological two-fluid model for superconducting materials. The model is based on experimental observation of superconductors. The temperature and field dependence are taken into consideration simultaneously. The main nonlinear macroscopic parameters for superconductors are derived. An empirical formula for the surface impedance of HTS that agrees very closely w ith experimental measurements for YBCO superconductors is developed. These compact models are validated and verified by comparing the calculated results with data obtained from experimental measurements. This model combines the physics associated w ith the G L phenomenological model and the required sim plicity obtained from the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 152 linear London's model. This new model fits the need for nonlinear consideration in computer-aided-design of microwave and millimeter-wave HTS devices and circuits. 6.1 Nonlinear Phenomenological Two fluid model HTS are classified as type II or mixed state superconductors. The first successful theoretical explanations of type II superconductors was derived by Abrikosov in 1957 [103]. The flux is observed to enter a type II superconductor in a discrete array of entities known as vortices. Each vortex has a quantized amount of flu x d>0 = h/2e associated with it, where h is the Plank's constant and e is the electron charge. As the vortices enter the superconductor, it has already been confirmed experimentally that they form a triangular lattice with a lattice constant a0 =(<E>0 / B)yi [104]-[105]. The radius of the vortex, called the coherence length £, is another im portant length scale in superconductivity. As the applied magnetic field increases, the density o f the vortex lattice increases. A t the upper critical field the normal cores of the vortices overlap, and the material becomes normal. The formation of the vortex lattice fundamentally changes the way that externally applied currents flow through the superconductor. Instead o f being confined to a penetration depth A near the surface, the currents flow uniformly throughout the superconductor, thereby greatly increasing the current carrying capacity of the material. HTS are characterized by small coherence lengths and large penetration depths. The typical values of the coherence length £ and of the penetration depth A in HTS are of order C = 10 A 0 and A ~ 1000 A°, respectively. This yields a G L parameter k = A/C of about 100, i.e., the HTS materials are extreme type II superconductors. The structure o f a flux line is depicted in Fig. 6.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 153 -X X Fig. 6.1 Variation of the number of superfluid electrons and the magnetic field H near a flux line. The magnetic field decays over a length scale set by the penetration depth A. The superfluid electrons density is zero at the center o f the flux line and approaches rapidly its equilibrium value over distances of order the coherence length £ When the coherence length £c in the c-direction is shorter than the layer spacing s, the structure discreteness becomes important and one expects a crossover from anisotropic 3D behavior to quasi 2D behavior. The structure of the vortex cores quasi 2D behavior occur for almost all temperatures T < Tc. This condition holds for temperatures o f about 4% below TC(B=0) for YBa 2Cu 3 0 7 .x [ 6 6 ]. When the magnetic field is applied parallel to the CuO-layers, the vortex cores fit between the superconducting layers. This situation is rather similar to that of a single wide Josephson junction in a parallel magnetic field [71]. The HTS could be represented by a stack of Josephson coupled layers. The Josephson vortices differ from the usual Abrikosov vortices mainly in that they have no normal vortex core, since the superfluid electron density is not modified by the presence of the Josephson vortex. In conclusion, the central point o f the London Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 154 theory, that the supercurrent is always determined by the local magnetic field, is an appropriate postulate for HTS materials. The main macroscopic parameters o f superconductors are the magnetic field penetration depth and the normal conductivity. GL theory results in a spatial, field and temperature dependent macroscopic parameters. The magnetic field penetration depth can be calculated form the following expression: (6.1) where /l/o.o) is the low field penetration depth measured at T = 0 and H - 0. The normalized order parameter is obtained from 4“ ¥ (H ,T ) W{H, T) V~ ¥ ( 0 ,T) ( t ) 1 - { T c) where Tc is the critical temperature for the superconductor calculated at H = 0. Hence, the magnetic field penetration depth can be written in the following form /i,(o.o) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.3) 155 The parameters a and /3 may be obtained from experimental studies. In our full-wave analysis, we chose a = 2 and ( 3 - 4 following G L model for the field dependence and Gorter-Casimir for the temperature approximations. Gorter and Casimir assumed that the fraction of the conduction electrons in the superfluid state ns varies from unity at T = 0 to zero at the temperature o f the transition to the completely normal state Tc. They found that the best agreement w ith the thermal properties o f conventional superconductors was obtained when this fraction was chosen to have the form f j. \ 4 ^ - = 1- (6.4) vT -w where nsx is the number of superfluid electrons at zero temperature and zero applied field, which is equal to the total number o f electrons in the system. The magnetic field dependence was ignored in their formulation. A low field condition is assumed. However, GL theory coupled both the thermodynamics and electrodynamics o f superconductors together. This results in simultaneous spatial, field, and temperature magnetic field penetration depth dependence, as shown in Eq. (6.3). It is known that fraction of the conduction electrons in the superfluid state ns could be assumed to vary from unity at H = 0 to zero at the field of the transition to the completely normal state H c. This postulate is valid at temperature T = 0. The critical magnetic field H c is computed at T = 0. This argument is expected to lead to a dependence similar to the Corter-Casimir thermal dependence. Then, the fraction o f the superfluid electrons can be obtained from the following expression Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The parameter a may be obtained from experimental studies. A temperature and field dependence formulation can be obtained by combining Eqs. 6.4 and 6.5, and using the result obtained from GL theory. The fraction o f the conduction electrons in the superfluid state ns can be written as H 1- (6.6) H C(T) where 2“ H c{T )~ H a 1 - f T] for T < T (6.7) {TcJ where H co is the thermodynamic critical field at zero temperature. Actually, H C( T ) w ill be assumed as a given experimental quantity. Having provided a plausibility argument based on experiment describing the presence o f the superfluid electrons as a function of temperature and magnetic field, we next consider the electrons as particles. Flux quantization inside a superconductor allows us to deduce that the carrier, called a superfluid electron, is in reality a correlation o f two actual electrons. This observation is consistent with the notion of a Cooper pair. Using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 157 the conservation o f electrons law, the fraction o f the normalfluid electrons can be deduced from the following expression (6 .8) The factor 2 doesn't appear in this expression because we are dealing w ith ratio of superfluid electrons, not absolute values. Eq. ( 6 .8 ) can be rewritten in terms of temperature and field dependence as n 1- f ~ T V5' \ H 1- H C(T) \ Tc J (6.9) 6.2 Macroscopic Model of Nonlinear Constitutive relations in HTS London's equations are used to derive the constitutive relations for electromagnetic fields. The London's equations are derived from the fundamental Newtonian dynamics and the Meissner effect. They can also be derived from quantum mechanics by introducing a canonical momentum [62], I f the Lorentz force due to magnetic field is not considered, the linear London's equations are valid from the Drude model and the Newton's second law. E . dt E_ A Vx / =- B Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (6.10) (6. 11) where E and B are the total electric and magnetic fields, respectively, Js is the current density due to the superfluid electrons, and A= (6.12) where ms and qs are the mass and charge o f the superfluid electrons respectively, ns is the number of the superfluid electrons. The nonlinearity w ill be included in A (H ) = n X ( H ) (6.13) The spatial, field and temperature dependent magnetic field penetration depth Xs for the superconductor can be calculated from the following expressions : (6.14) where A,(o,o) is the low field penetration depth measured at T = 0 and H = 0, h = H ( r ) / H c(T ), t = T /T c, and Tc is the critical temperature for the superconductor calculated at H = 0. The parameters a and /? may be obtained from experimental studies. The norm alfluid current density is deduced from Ohm's law, and the corresponding normal conductivity is expressed as fo llo w s: (6.15) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 159 where o „ { hcit c) is the maximum normal conductivity measured either at T = TC or H - H c and H c is the critical magnetic field for the superconductor calculated at T = 0. Eqs. 6.14 and 6.15 represent the dependence o f the main macroscopic parameters o f the superconducting material on the spatial distribution o f the applied fie ld and the temperature simultaneously. Their dependence on the applied field fu lly represent our nonlinear tw o-fluid model. The constitutive relations for time harmonic fields can be represented using either the complex conductivity or complex dielectric concepts. 6.3 HTS Nonlinear Surface Impedance The proposed nonlinear two-fluid model is verified using the experimental results obtained for the surface impedance. In our calculation, the surface impedance formula of a good conductor is adopted (6.16) Using the complex conductivity concept for superconductors ( T = ( T „ + (T, (6.17) ZJ =0.5co2n 2 0X3an + ja> ngA. (6.18) Eq. (6.16) becomes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The real part gives the surface losses per unit area per unit surface current-density amplitude, and the imaginary term represents the surface inductive reactance o f the superconductor. Then, the surface resistance Rs can be approximated as (6.19) Rs =0.5Q )2n ; l 3Gn The field, temperature, and frequency dependent surface resistance is expressed as follows R,(h,t,a>) = 0 . 5 ^ ] ( 0 , 0 ) a n{H J T c)co where h = H ( r ) / H C(T ), t - T/T c, Tc is the critical temperature for the superconductor calculated at H = 0, H c. is the critical thermodynamic magnetic field depicted at temperature T, and co is the operating frequency. The surface resistance may be normalized to the material characteristics factor Q .5jjlA.](0,0)an(H c/T c), that yields to a general normalized surface resistance phase parameter expression, (6.21) This expression is valid for all superconductor materials, and is independent o f their physical characteristics. The surface resistance phase parameter is only dependent on the operating conditions. The dependence o f HTS on the temperature and field can be easily deduced from this expression. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 161 An empirical formula proposed by Pippard that agrees very closely with experiment measurements for conventional superconductor [67] gives However, this expression fails to predict the temperature dependence for the surface resistance o f the YBCO HTS. The formula that fits mostly with the experimental results depicted for the temperature and field dependence o f the YBCO HTS, as w ill be shown later, can be expressed as follows (6.23) where a and ft are obtained empirically. Their values depend on the operating region of interest. The function ( l - h a) is plotted in Fig. 6.2 as a function o f h for different values o f the variable a . The rate o f change in the function value increases with the decrease in a close to h = 0. It increases with the increase in a near h = 1. The value of the function itself decreases with the decrease in the value of a . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 162 1 a =4 a = 2 0.8 a = 3/2 0.6 a =3/4 0.4 0.2 a = 1/2 0 0 0.2 0.4 0.6 0.8 1 h Fig. 6.2 The function (l - / t “ ) as function o f h for different values o f the variable a . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 163 6.4 Nonlinear Model Validation and Verification The surface resistance o f YBa 2 Cu3 0 7 -x HTS film with critical temperature o f 86.4 K , and magnetic field penetration depth Ai (0) equals 0.167 fim is estimated using the proposed nonlinear two fluid model The calculated results are compared with the measured data in [106]. Fig. 6.3 shows the critical magnetic field, calculated using Eq. (6.7), as a function o f the normalized temperature. Although, this formula is assumed for conventional superconductor, it fits well the measured data for the YBCO in this operating region. The temperature dependence for the surface resistance at zero applied magnetic field is calculated using Eq. (6.23), with (5 = 3/2. The operating frequency equals to 1.5 GHz. The results are compared with the measured data presented in [106], and shown in Fig. 6.4. It is seen that fair agreement is obtained between t = 0.6 and t = 0.95. This operating region includes the liquid nitrogen boiling temperature, 77 K, where all the new HTS material operates. The surface resistance as a function of both the temperature and magnetic field at 1.5 GHz is depicted in Fig. 6.5. The surface resistance is calculated using Eq. (6.23), with a = 3/4 and [5 = 3/2. A comparison between the calculated results and the experimental data is conducted and presented in Fig. 6.5. A good agreement between the measured and the calculated data in the practical operating region of the YBCO HTS can be observed. As the operating temperature decreases a slight difference between the measured and the calculated results appears. This is due to the imperfection associated with the HTS material, which is not taken into consideration in the analytical formula. The presented results show that the effect of the magnetic field on the HTS material is Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 164 more pronounced than the temperature effect. The material looses its superconducting characteristics much faster with the increase in the magnetic field intensity. 1400 — Calculated 1200 o O — Measured 1000 800 600 400 ■c u 200 0.5 0.6 0.7 0.8 0.9 1 Normalized Temperature ( t = T /T ,) Fig. 6.3 Comparison between the calculated and the measured [106] critical magnetic field for YBCO HTS as a function of temperature. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 165 7 Calculated Measured PeakH f = 0 6 cs 'o r—< 5 X o u 4 o o u 3u a 3 CA on 2 1 0 0.5 0.6 0.7 0.8 0.9 Normalized Temperature (t = T/T ) Fig. 6.4 Comparison between the calculated and the measured [106] surface resistance for YBCO HTS as a function of temperature at zero magnetic field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 12 -8 4 K 81 K — Calculated - - Measured 78 K 10 8 63 K 6 53 K 4 2 0 0 100 200 300 400 500 600 700 800 PeakHr f (Oe) Fig. 6.5 Comparison between the calculated and the measured [106] surface resistance for YBCO HTS as a function o f temperature and magnetic field. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 167 6.5 Summary A novel nonlinear phenomenological tw o -flu id model is proposed fo r superconducting materials. A macroscopic model for the nonlinear constitutive relations is also suggested. The nonlinear main macroscopic parameters for superconductors, the superconductor magnetic field penetration depth and the normal conductivity, are derived. An empirical formula for the surface impedance of HTS that agrees very closely with experimental measurements fo r YBCO superconductors is developed. These compact models are validated and verified by comparing the calculated results with data obtained from experimental measurements. A fairly good agreement is seen in the practical operating region of the new HTS materials. This model combines the physics associated with the G L phenomenological model and the required sim plicity obtained from the linear London's model. It is observed that the HTS looses its superconducting characteristics faster with the applied field than the temperature. This new model is very useful for computer-aided-design of microwave and millimeter-wave HTS devices and circuits. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER 7 CONCLUSIONS This dissertation presents a comprehensive analysis of linear and nonlinear planar microwave devices using a full-wave electromagnetics approach, the finite-difference technique. The technique was applied to High critical Temperature Superconducting material (HTS). The goal of the research is to conduct a nonlinear analysis of microwave HTS devices using a full-wave electromagnetics simulator, which incorporates the anisotropic o f the superconducting strip and the substrate simultaneously. This study is crucial for high power applications of HTS microwave and millimeter-wave devices. A nonlinear phenomenological model for HTS materials is also developed. This model w ill be very useful for CAD. The preliminary material of chapter 1 introduced the goals, the importance o f the research, and set guidelines for achieving our objectives. Chapter 2 described the present linear and nonlinear phenomenological models for HTS. The solution for the nonlinear Ginzburg-Landau (GL) was presented for bulk and thin film s HTS to provide better understanding of the nonlinearity and the RF power handling capability associated with HTS materials. Chapter 3 provided the finite difference approach as it is the numerical technique used in the dissertation. Solution of nonlinear problems using the Newton's SSOR method was presented. The finite difference frequency (FDFD) and time (FDTD) domains compared. algorithms as applied to electromagnetics problems were discussed and Chapter 4 introduced the analysis and sim ulation o f anisotropic superconductors on anisotropic substrates. The anisotropic FDTD was applied to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 169 microstrip line and coplanar waveguide structures. Chapter 5 showed the full-wave nonlinear analysis o f HTS microwave HTS devices. The two fluid model blended the G L nonlinear solution with the FDTD electromagnetics simulator. A complete study on the nonlinear effects o f HTS microwave devices was conducted. In chapter 6 , a novel nonlinear phenomenological two fluid model was suggested. An empirical formula for the nonlinear surface impedance o f HTS thin films was also proposed. 7.1 Summary of Findings and Conclusions This thesis has reported the first rigorous effort in modeling the nonlinear characteristics o f superconducting microwave and millimeter-wave devices in the time domain using a full-wave technique. The anisotropic three-dimensional behavior o f the HTS was reduced to a quasi-two-dimensional analysis for applications operating in the low gigahertz range. G L theory has been used to model the nonlinear mechanism o f superfluid electron pair breaking. The longitudinal superfluid current along the direction of wave propagation was calculated using the nonlinear model. The currents in the transverse plane, which are small for planar microwave structures, were considered with a linear model. The anisotropic characteristics of HTS and the effect o f anisotropic substrates on the performance o f HTS devices were also studied. This work presented the first attempt to handle the anisotropy both in the conducting strip and in the dielectric substrate simultaneously. A novel phenomenological nonlinear two fluid model and an empirical formula for the nonlinear surface resistance have been reported. This model is the first simple model that combines the electrodynamics and thermodynamics o f the superconducting material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 170 The major results in the dissertation are summarized below: 1. It has been shown that the nonlinearity behavior must be modeled in the time domain, and not in the frequency domain. A full-wave simulator was suggested for the nonlinear analysis of anisotropic HTS. 2. The nonlinear Ginzburg-Landau model could be applied to the new HTS materials, despite the anisotropy associated with them. The anisotropic threedimensional behavior o f HTS could be reduced to a quasi-two-dimensional behavior for HTS used in planar microwave and millimeter-wave applications. 3. Numerical results for HTS strip used in microwave applications showed that the number of superfluid electrons decreased near the edges w ith the increase in the applied power, indicating the breaking o f superfluid electrons pairs. The shielding ability of the HTS was weakened near the edges, resulting in enhanced penetration o f the magnetic field. The linear model underestimates the magnetic field penetration and overestimates the edge enhancement o f the current density. 4. The power handling capability o f HTS used in microwave and m illim eter applications depend not only on the material characteristics but also on the structure under investigation. In this case, the maximum power is determined by the rf critical power. 5. The complete nonlinear study of HTS materials showed that the phase velocity and the attenuation o f the wave propagating along HTS transmission lines changes nonlinearly with the applied power. The change in the losses is much larger than the change in the phase velocity. The change in the electromagnetic fields is more pronounced near the edge of the HTS strip. The superfluid current density distributions change dramatically with the applied power. The variation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 171 in the frequency spectrum of the applied signal resulting from the nonlinearity was obvious. 6. HTS microstrip resonator array filte r was successfully simulated using the developed nonlinear approach. Results showed that the dimension and the layout o f high power HTS filters need to be optimized in the design cycle to avoid the nonlinearity effects. 7. A novel nonlinear phenomenological tw o-fluid model fo r superconducting materials was proposed. This model blended the thermodynamics and the electrodynamics properties of HTS. 8. An empirical formula for the nonlinear surface resistance o f HTS was suggested. The results calculated using this formula was successfully compared with experimental data. A good agreement is obtained in the region o f interest. In order to achieve the goal of this dissertation, several numerical aspects for the finite difference algorithm are considered. The main features, that are tackled in the thesis, are: 1. A general wave equation for nonuniform dielectric structures was derived. There is no need to impose unnecessary boundary conditions anywhere inside the structure. 2. Finite thickness conductors in planar microwave and millimeter-wave devices was rigorously modeled using nonuniform mesh generator for the finite difference scheme. Results showed that field penetration effects were successfully predicted in our analysis. 3. The Perfectly Matched Layers (PML) absorbing boundary condition was modified and successfully applied to microwave devices that include finite conductivity Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 172 conductors. Results presented the improvement in the finite-difference timedomain lattice truncation by using the PM L compared to the one-way wave equation ABC. 4. A parallel finite-difference time-domain algorithm was demonstrated. Results showed that a tremendous run time improvement is obtained by using parallel machine environment compared to powerful serial machine as the mesh size increases. 5. An anisotropic finite-difference time-domain scheme was suggested. This technique was able to take the anisotropy in the conductors and the substrates simultaneously. Results showed the importance of considering the anisotropy for the optimum design o f microwave and millimeter-wave devices. 7.2 Recommendations for Future Research Extensions or possible improvements in this work are numerous. This dissertation had contributions in several areas such as numerical techniques, microwave and millimeter-wave devices, nonlinear phenomena analysis, computer-aided-design o f microwave devices, analysis and modeling of HTS materials. In the computational aspect, the perfectly matched layers (PML) absorbing boundary conditions can be extended and applied to truncate lossy materials, where no appropriate absorbing boundary conditions exist. A study for the use of the PML at the side and top walls of the finite-difference lattice that include microwave devices can be conducted. A rigorous dispersion analysis for the nonuniform mesh generator used with the finite difference approach can be performed. The nonuniform mesh finite-difference algorithm can be applied to model and study other phenomena besides the wave penetration effects, such as the diffusion or injection o f electrons in thin oxide in active semiconductor devices. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 173 The anisotropic finite-difference time-domain technique can be applied to study other materials where the anisotropy can not be neglected or approximated such as ferrite materials. Application o f the full-wave simulator to other HTS microwave and millimeter-wave devices could be performed. Effects of the nonlinearity and anisotropy associated with HTS materials on the performance o f those devices can be studied. In the HTS materials characterization, the nonlinear two fluid model needs to be verified with measurements from different samples. Empirical formulas fo r the nonlinear surface impedance o f different HTS materials can be obtained using the same algorithm r presented in the dissertation. Also, a macroscopic model o f frequency dependent constitutive relations in HTS needs to be developed. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. REFERENCES [1] K. B. Bhasin, S. S. Tonich, C. M. Chorey, R. R. Bonetti, and A. E. W illiam s, "Performance o f a Y-Ba-Cu-0 superconducting filter/GaAs low noise amplifier hybrid circuit," in IEEEM TT-S Int. Microwave Symp. Dig., 1992, pp. 481-483. [2] M. S. Schmidt, R. G. Forse, R. B. Hammond, M. M. Eddy, and W. L. Olson, "Measured performance at 77K o f superconducting microstrip resonators and filters," IE E E Trans. M icrowave Theory Tech., vol. 39, no. 9, pp. 1475-1479, 1991. [3] O. R. Baiocchi, K. S. Kong, H. Ling, and T. Itoh, "Effects o f superconducting losses in pulse propagation on microstrip lines," IEEE M icrowave Guided Wave Lett. , vol. 1, no. 2, pp. 2-4, 1991. [4] J. M. Pond, C. M. Krowne, and W. L. Carter, "On the application of complex resistive boundary conditions to model transmission lines consisting o f very thin superconductors," IEEE Trans. Microwave Theory Tech., vol. 37, no. 1, pp. 181189, 1989. [5] Z. Cai, and J. Bornemann, "Generalized spectral-domain analysis for multilayered complex media and high-Tc superconductor applications," IE E E Trans. Microwave Theory Tech., vol. 40, no. 12, pp. 2251-2257, 1992. [6] D. Nghiem, J. T. Williams, and D. R. Jackson, "A general analysis of propagation along multiple-layer superconducting stripline and microstrip transmission lines," IEEE Trans. Microwave Theory Tech., vol. 39, no. 9, pp. 1533-1564, 1991. [7] H. Lee, and T. Itoh, "Phenomenological loss equivalence method for planar quasiTEM transmission lines with a thin normal conductor or superconductor," IE E E Trans. Microwave Theory Tech., vol. 37, no. 12, pp. 1904-1909, 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 175 [ 8] T. E. van Deventer, P. B. Katehi, J. Y. Josefowicz and D. B. Rensch, "High frequency characterization of high-temperature superconducting thin film lines," in IE E E M TT-S Int. Microwave Symp. Dig., 1990, pp. 285-288. [9] A. Fathy et al., "Microwave properties and modeling of high-Tc superconductor thin film meander line," in IE E E MTT-S Int. M icrowave Symp. D ig., 1990, pp. 859-862. [10] S. M. El-Ghazaly, R. B. Hammond, and T. Itoh, "Analysis o f superconducting microwave structures: application to microstrip lines, " IEEE Trans. Microwave Theory Tech., vol. 40, no. 3, pp. 499-508,1992. [11] J. Kessler, R. D ill, and P. Russer, "Field Theory Investigation o f H igh-Tc Superconducting Coplanar Waveguide Transmission Lines and Resonators," IE E E Trans. Appl. Supercond., vol. 3, no. 9, pp. 2782-2787, 1992. [12] L. Lee, S. A li, W. Lyons, D. Oates, and J. Goettee "Analysis o f superconducting transmission line structures for passive microwave device applications," IE E E Trans. Appl. Supercond., vol. 3, no. 9, pp. 2782-2787, 1992. [13] L. H. Lee, S. M. A li, and G. Lyons, "Full-wave characterization o f high-Tc superconducting transmission lines," IEEE Trans. Appl. Supercond., vol. 2, no. 3, pp. 49-57, 1992. [14] L. Lee, W. Lyons, T. Orlando, S. A li, and R. Withers, "Full-wave analysis of superconducting microstrip lines on anisotropic substrates using equivalent surface impedance approach," IEEE Trans. Microwave Theory Tech., vol. 41, no. 12, pp. 2359-2367, 1993. [15] C. H ilbert, D. Gibson, and D. Herrell, "A comparison o f lossy and superconducting interconnect for computers," IEEE Trans. Electron Devices, vol. 36, no. 9, pp. 1830-1839, 1989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 176 [16] L. Drabek, J. Carini, G. Gruner, T. L. Hylton, A. Katulnik, and M . R. Beasley, "Surface impedance of high Tc superconductors," in IEEE MTT-S Int. Microwave Symp. Dig., 1989, pp. 551-553. [17] A.B. Pippard, "The surface impedance of superconductor and normal metals at high frequencies-III: The relationship between impedance and superconducting penetration depth," Proc. Roy. Soc. London, vol. A191, pp. 399-415, 1947. [18] M. R. Beasley, "High-Temperature Superconductive Thin Films," Proc. o f the IEEE, vol. 77, no. 8 , pp. 1155-1163,1989. [19] K. K. Mei, and G. Liang, "Electromagnetics of superconductors," IE E E Trans. Microwave Theory Tech., vol. 39, no. 9, pp. 1545-1552, 1991. [20] S. A. Long and J. T. Williams, "High temperature superconductors," IEEE Trans. Potentials, vol. 10, no. 8 , pp. 37-40, 1991. [21] A. Narlikar, ed., Studies o f H igh Temperature Superconductors. Vol. 2 of Advances in Research and Applications series. New York: Nova, 1989. [22] T. P. Orlando, K. A. Delin, S. Foner, E. J. M cN iff, J. M. Tarascon, L. H. Grene, W. R. Mckinnon, and G. W. Hull, "Upper critical fields and anisotropy limits of h ig h - T c superconductors RiBa 2Cu3 0 7.y where R -- Nd, Eu, Gd, Ho, Er, and Tm and YBa 2Cu30 7.x," Phys. Rev. B, vol. 36, no. 7, pp. 2394-2397, 1989. [23] D. Wu and S. Sridhar, "Pinning forces and lower critical fields in YBa 2Cu3 0 7.x crystals: Temperature dependence and anisotropy," Phys. Rev. Lett., vol. 65, no. 8 , pp. 2074-2077, 1990. [24] A. C. Anderson, R. S. Withers, S. A. Reible and R. W. Raltson, "Substrates for superconductive analog signal processing devices," IEEE Trans. Magn.., vol. 19, no. 8 , pp. 485-489, 1983. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 177 [25] B. F. Cole, G.-C Liang, K. Char, G. Zaharchuck, and J. S. Martens, "Large area YBaCuO thin film s on sapphire for microwave applications," Appl. Phys. Lett., vol. 61, no. 3, pp. 1727-1729, 1992. [26] K. Char, N. Newman, S. M. Garrison, R. W. Barton, R. C. Taber and B. F. Cole, "Microwave surface resistance of epitaxial YBaCuO thin film s on sapphire," A p p l Phys. Lett., vol. 57, no. 5, pp. 409-411, 1990. [27] I. B. Vendik et al., "CAD model for microstrips on r-cut sapphire substrates," submitted to Int. Joun. o f Microwave and M illim eter-W ave Computer-Aided Engineering, 1994. [28] G. C. Liang, R. S. Withers, and B. F. Cole "High-temperature superconductive devices on sapphire" IEEE Trans. Microwave Theory Tech., vol. 42, no. 1, pp. 3440, 1994. [29] T. Van Duzer, "Prospects for applications of high-temperature superconductors," presented at 2nd workshop on H igh Temperature Superconducting Electron Devices, R & D Association fo r Future Electron Devices, Japan, June 1989. [30] A. I. Braginski, "Progress and trends in high-Tc superconducting electronics," IEEE Trans. Magn., vol. 27, no. 5, pp. 2533-2536, 1991. [31] J. C. Ribber, M. Nisenoff, G. Price, and S. A. W o lf, "High temperature superconductivity space experiment (HTSSE)," IE E E Trans. Magn., vol. 27, no. 3, pp. 2533-2536, 1991. [32] S. H. Talisa, M. A. Janocko, C. Moskowitz, J. Talvacchio, J. F. Billing, R. Brown, D. C. Buck, C. K. Jones, B. R. McAvoy, G. R. Wagner, and D. H. Watt, "Lowand high-temperature superconducting microwave filte rs," IE E E Trans. Microwave Theory Tech., vol. 39, no. 4, pp. 1448-1454, 1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 178 [33] O.K. Kwon, B.W. Langley, R.F. Pease, M.R. Beasley, "Superconductors as very high-speed system-level interconnects," IEEE Electron Device Lett., vol. 8 , no.3, pp. 582-585, 1987. [34] M. M . Driscoll, J. T. Haynes, R. AS. Jelen, R. W. Weinert, J. R. Gavaler, J. Talvacchio, G. R. Wagner, K. A. Zaki, and X. P. Liang, "Cooled, ultrahigh Q, sapphire dielectric resonators for low-noise, microwave signal generation," IE E E Trans. Ultrason. Ferroelec., Freq. Contr., vol. 39, no. 3, pp. 405-411, 1992. [35] W. G. Lyons, R. S. Withers, J. M. Hamm, A. C. Anderson, P. M. Mankiewich, M . t L. O'Malley, and R. E. Howard, "High-Tc superconductive delay line structures and signal conditioning networks," IEEE Trans. Magn., vol. 27, no. 6 , pp. 25372539,1991. [36] M. J. Burns, K. Char, and B. F. Cole, "M ultichip module using m ultilayer YBa2Cu307-delta interconnects," Appl. Phys. Lett., vol. 62, no. 5, pp. 2074-2077, 1993. [37] A. C. Cangellaris, "Propagation characteristics of superconductive interconnects," Microwave and Optical Tech. Lett., vol. 1, no. 2, pp. 153-156, 1988. [38] D.M. Sheen, S. M. A li, D. E. Oates, R. S. Withers, and J. A. Kong, "Current distribution, resistance and inductance for superconducting strip transmission lines," IE E E Trans. Appl. Supercond., vol. 1, pp. 108-111, 1991. [39] C. C. Chin, D. E. Oates, G. Dresslhaus, and M. S. Dresslhaus, "Nonlinear electrodynamics of superconducting NbN and Nb thin film s at microwave frequencies," Phys. Rev. B, vol. 45, no. 3, pp. 4788-4798, 1992. [40] D. E. Oates, A. C. Anderson, and P. M. Mankiewich, "Measurement o f the surface resistance o f YBa2Cu307-x thin film s using stripline resonators," J. Superconduct. , vol. 3, no. 4, pp. 251-259, 1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 179 [41] R. B. Hammond, G. V. Negrete, M. S. Scmidt, M. J. Moskowitz, M . M. Eddy, D. D. Strother, and D. L. Skoglund, "Superconducting TI-Ca-Ba-Cu-0 thin film microstrip resonator and its power handling performance at 77 K," in IEEE MTTS Int. Microwave Symp. Dig., 1990, pp. 867-870. [42] P. H. Kes, J. Aarts, J. van den Berg, C. J. van der Beek, and J. A. Mydosh, "Thermally assisted flux flow at small driving forces," Supercond. Sci. Technol., vol. 1, no. l,p p . 242-248, 1989. [43] D. E. Oates, A. C. Anderson, D. M. Sheen, and S. M. A li, "Stripline resonator measurements o f Zs versus H rf in YBa2Cu307-x thin film s," IE E E Trans. Microwave Theory Tech., vol. 39, no. 6 , pp. 1522-1529, 1991. [44] A. M. Portis et al., "Power and magnetic field-induced microwave absorption in Tl-based high-Tc superconducting films," Appl. Phys. Lett., vol. 58, no. 8 , pp. 307-309, 1991. [45] G.L. Matthaei and G. Hey-Shipton, "Concerning the Use o f High-Temperature Superconductivity in Planar Microwave Filters ," IEEE Trans. Microwave Theory Tech., vol. 42, no. 7, pp. 1287-1294, 1994. [46] M. N. O. Sadiku, Numerical Techniques in Electromagnetics. Boca Raton: CRC, 1992. [47] C. A. Hall, and T. A. Porsching, N um erical analysis o f p a rtia l d iffe re n tia l equations. New Jersey: Prentice Hall, 1990. [48] K. S. Kunz, R. J. Luebers, The Finite Difference Time D om ain M ethod f o r Electromagnetics. Boca Raton: CRC, 1993. [49] M. A. Megahed, S. A. El-Ghazaly, "Finite difference approach for rigorous fullwave analysis o f superconducting microwave structures," in IE E E MTT-S Int. Microwave Symp. Dig. , 1993, pp. 718-721. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 180 [50] K. S. Yee, "Numerical solution of initial boundary problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagation, vol. 14, no. 6 , pp. 302-307, 1966. [51] A. Taflove and M. Browdin, "Numerical Solution of Steady-state electromagnetic scattering using time-dependent M axwell's equations," IE E E Trans, on Microwave Theory Tech., vol. 23, no. 8 , pp. 623-630, 1975. [52] X. Zhang, J. Fang, K. Mei, and Y. Liu, "Calculations o f the dispersive characteristics o f microstrips by the time-domain finite difference method," IE E E Trans. Microwave Theory Tech., vol. 36, no. 2, pp. 263-266, 1988. [53] X. Zhang, and K. Mei, "Time-domain Finite difference approach to the calculation of the frequency-dependent characteristics of microstrip discontinuities," IE E E Trans. Microwave Theory Tech., vol. 36, no. 12, pp. 1775-1787, 1988. [54] G. Liang, Y. Liu, and K. Mei, "Full-wave analysis o f coplanar waveguide and slotline using the time-domain finite-difference method," IE E E Trans. Microwave Theory Tech., vol. 37, no. 7, pp. 1949-1957, 1989. [55] J. Railton and J. McGeehan, "Analysis of microstrip discontinuities using the finite-difference time-domain method," in IEEE MTT-S Int. M icrowave Symp. Dig., 1989, pp. 1009-1012. [56] D. Sheen, S. A li, M. Abouzahra, and J. Kong, "Application o f the threedimensional finite-difference time-domain method to the analysis of planar microstrip circuits," IEEE Trans. M icrowave Theory Tech., vol. 38, no. 2, pp. 849-857, 1990. [57] L. Wu and Y. Chang, "Characterization of shielding effects on the frequencydependent effective dielectric constant of a waveguide-shielded microstrip using the finite-difference time-domain method," IEEE Trans. M icrow ave Theory Tech., vol. 39, no. 3, pp. 1688-1693, 1991. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 181 [58] I. W olff, "Finite difference time-domain simulation o f electromagnetic fields and microwave circuits," Int. J. Numer. M odeling: Electronic Networks, Devices and Fields, vol. 5, no.l, pp. 163-182, 1992. [59] D.B. Davidson, " A Parallel processing tutorial," IE E E Trans. Antennas Propagation Mag., vol. 32, no. 2, pp. 6-19, A pril 1990. [60] A. King, " Massively Parallel Solutions o f Complex Electromagnetic Problems," in Electromagnetic Code Consortium Symposium Proc., A pril 1992, pp. 96-97. [61] W. Kumpel, U. Effing, M. Rittweger, and I. W olff, "Parallel FDTD simulator for microwave structures," in Proc. European Microwave C onf, 1993, pp. 676-677. [62] R. D. Parks, Superconductivity (in two volumes). New York: Marcel Dekker, 1969. [63] M. Cyrot, "Ginzburg-landau theory for superconductors," Rep. Prog. Phys., vol. 36, no. 1, pp. 103, 1973. [64] S.A. Zhou, Electrodynamic theory o f superconductors. London: Peter Peregrinus, 1991. [65] L. P. Gor'Kov, "Microscopic derivation of the Ginzburg-Landau equations in the theory o f superconductivity," Sov. Phys., JETP, vol. 36, no. 3, pp. 1364-1367, 1959. [6 6] L. C. Gupta and M. S. Multani, ed., Selected Topics in Superconductivity. Vol. 1 o f Frontiers in Solid State Sciences, New York: Nova, 1991. [67] T. Van Duzer and C. W. Turner, Principles o f Superconductive Devices and Circuits. New York: Elsevier, 1981. [68] F. London, Superfluids. New York: Wiley, vol. 1, 1950. [69] O. G. Vendik and A. Y. Popov, "Can the bipolaron model be used for a description o f microwave and infrared properties o f a high-temperature superconductor?," Philosophical Magazine B, vol. 67, no. 6 , pp. 833-845, 1993. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 82 [70] K. Maki, "The magnetic properties of superconducting alloys I," Physics, vol. 1, no. 1, pp. 21-30, 1964. [71] W. E. Lawrence and S. Doniach, "Theory of layer structure superconductors," in Proc. Twelve International Conference on Low Temperature Physics, 1971, pp. 361-362. [72] Cheung-Wei Lam, D. M. Sheen, S. M. A li, D. E. Oates, "M odeling the nonlinearity o f superconducting strip transmission lines," IE E E Trans. App. Superconduct., vol. 2, no. 2, pp. 58-65, 1992. t [73] M . A. Megahed and S. M. El-Ghazaly, "Full-W ave Nonlinear analysis o f Microwave Superconductor Devices : Application to Filters," in IEEE MTT-S Int. Microwave Symp. Dig., 1995, pp. 245-249. [74] M. A. Megahed and S. M. El-Ghazaly, "Analysis of Anisotropic Conductors on Anisotropic Substrate," in IE E E MTT-S Int. M icrowave Symp. D ig., 1995, pp. 671-675. [75] M. A. Megahed and S. M. El-Ghazaly, "Effects of microwave power levels on high temperature superconductor systems" presented at Systems Applications o f H igh Temperature Superconductors and Cr)’ogenic Electronics workshop, IE E E MTT-S Int. Microwave Symp., Orlando, May 1995. [76] M. A. Megahed and S. M. El-Ghazaly, "High power microwave nonlinear performance of high-temperature superconductor filters" in Proc. European M icrowave C onf, 1995, pp. 303-308. [77] M. A. Megahed and S. M. El-Ghazaly, "High power effects on high temperature superconductor devices" in Progress in Electron!. Res. Symp. Dig., July 1995, pp. 206. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 183 [78] M. A. Megahed and S. M. El-Ghazaly, "Modeling nonlinear wave propagation on microwave superconducting transmission lines," in IEEE AP-S Int. Symp. Dig., 1994, pp. 577-580. [79] M . A. Megahed and S. M . El-Ghazaly, "Analysis of Waveguides w ith Nonlinear Material: Applications to Superconductor Microstrip lines" in Proc. European M icrowave Conf., 1994, pp. 1124-1128. [80] M . A. Megahed, Shapar Shapar, Jung Pyon, and S. M . El-Ghazaly, "High T c superconductor microwave lines: coplanar waveguide versus microstrip line," in, URSIRadio Science Meeting Dig., July 1994, pp. 97. [81] M . A. Megahed and S. M. El-Ghazaly, "Direct calculation of attenuation and propagation constants in superconducting microwave structures," in IEEE A P S Int. Symp. Dig., 1993, pp. 987-991. [82] M. A. Megahed and S. M. El-Ghazaly, "A rigorous technique for the analysis of the dispersion characteristics of superconducting microstrip lines," in N a tional Radio Science Meeting Dig., Jan. 1993, pp. 78. [83] M. A. Megahed and S. M . El-Ghazaly, "Analysis of anisotropic high temperature superconductor planar structures on sapphire anisotropic substrates," to appear in IE E E Trans. Microwave Theory Tech., August 1995. [84] M . A. Megahed and S. M. El-Ghazaly, "Nonlinear analysis of microwave superconductor devices using full-wave electromagnetic model" accepted for publication in IEEE Trans. Microwave Theory Tech., 1995. [85] C. A. Balanis, Advanced Engineering Electromagnetic. New York: W iley, 1989. [8 6 ] Z. Q. B i, K. L. Wu, C. Wu, and J. Litva, "A dispersive boundary condition for microstrip component anlysis using the FD-TD method" IEEE Trans. Microwave Theory Tech., vol. 40, no.4, 1992, pp. 774-777. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 184 [87] G. Mur, "Absorbing boundary conditions for the finite-difference approximation o f the time-domain electromagnetic-field equations" IEEE Trans. Electroinagn. Compat., vol. 23, no. 4, 1981, pp. 377-382. [ 88 ] J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," 7. Comp. Phys., vol. 114, no. 7, pp. 185-200, 1994. [89] D. S. Katz, E. T. Thiele, and A. Taflove, "Validation and extension to three dimensions o f the Berenger PML absorbing boudary condition for FD-TD meshes," IEEE Microwave Guided Lett., vol. 4, no. 8 , pp. 268-270. [90] C. E. Reuter, R. M. Joseph, E. T. Thiele, D. S. Katz, and A. Taflove, "Ultrawideband Absorbing Boundary condition for Termination o f Waveguiding Structures in FD-TD Simulations," IEEE M icrowave Guided Lett., vol. 4, no. 10, pp. 334-346. [91] R. Holland, "Finite-difference time-domain (FDTD) analysis o f magnetic diffusion" IEEE Trans. Electromagn. Compat., vol. 36, no. 2, 1994, pp. 32-39. [92] I. J. Bahl and R. Garg, "Simple and accurate formulas for a microstrip with finite strip thickness" Proc. IEEE, vol. 65, no. 11, 1977, pp. 1611-1612. [93] J. L. Tsalamengas, N. K. Uzunoglu, and N. G. Alexopoulos, "Propagation characteristics of a microstrip line printed on a general anisotropic substrate," IE E E Trans. Microwave Theory Tech., vol. 33, no. 6 , pp. 941-945, 1985. [94] A. A. Mostafa, C. M. Krowne, and K. A. Zaki, "Numerical spectral matrix method for propagation in general layred media: application to isotropic and anisotropic substrates," IEEE Trans. Microwave Theory Tech., vol. 35, no. 4, pp. 1399-1407, 1987. [95] T. Q. Ho and B. Beker, "Effects of misalignment on propagation characteristics of transmission lines printed on anisotropic substrates," IE E E Trans. M icrowave Theory Tech., vol. 40, no. 3, pp. 1018-1012, 1992. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 185 [96] S. Koike, N. Yoshida, and I. Fukai, "Transient analysis o f microstrip line on anisotropic substrate in three-dimensional space," IEEE Trans. Microwave Theory Tech., vol. 36, no. 1, pp. 34-43, 1988. [97] R. Marques and M. Horno, "Dyadic green's function fo r m icrostrip-like transmission lines on a large class of anisotropic substrates," IE E Proc., vol. 133H, no. 2, pp. 450-454, 1986. [98] M. Geshiro, S. Yagi, and S. Sawa, "Analysis of slotlines and microstrip lines on anisotropic substrates," IEEE Trans. Microwave Theory Tech., vol. 39, no. 1, pp. 64-69, 1991. [99] M. Homo and R. Marques, "Coupled microstrips on double anisotropic layers," IE E E Trans. Microwave Theory Tech., vol. 32, no. 2, pp. 467-470, 1984. [100] C. Kuo and Tatsuo Itoh, "An iterative method for the nonlinear characterization o f the high Tc superconducting microstrip line " in Proc. European M icrow rave Conf., pp. 655-660, Sep. 1991. [101] Y. Liu and T. Itoh, "Characterization of power-dependent high-7'c superconductor microstrip line by modified spectral domain method " Radio Science, vol. 28, no. 5, pp. 913-918, 1993. [102] J.J. Xia, J.A. Kong, and R.T. Shin, "A macroscopic model o f nonlinear constitutive relations in superconductors," IEEE Trans. Microwave Theory Tech., vol. 42, no. 10, pp. 1951-1957, 1994. [103] A. A. Abrikosov, "On the magnetic properties of superconductors o f the second group," Sov. Phys., JETP, vol. 5, no. 12, pp. 1174-1182, 1957. [104] P. L. Gammel, D. J. Bishop, G. J. Dolan, J. R. Kwok, C. A. Murray, L. F. Scheemeyer, and J. V. Waszcak, "Observation o f hexagonally correlated flux quanta in YBa 2 Cu3 0 7 ," Phys. Rev. Lett., vol. 59, no. 22, pp. 2592-2595, 1987. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 186 [105] R. N. Kleiraann, P. L. Gammel, L. F. Scheemeyer, J. V. Waszcak, and D. J. Bishop, "Evidence from mechanical measurements fo r flux-lattice melting in single crystal and B i 2 .2 $r2Cao.8Cu2 0 8 " Phys. Rev. Lett., vol. 62, no. 19, pp. 23302331, 1989. [106] D. E. Oates et a l„ "Power handling capabilities o f YBCO films in microwave devices" presented at H igh Power Superconducting M icrowave workshop, EEE MTT-S Int. Microwave Symp., San Diego, May 1994. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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