ELECTROMAGNETIC PHYSICAL MODELING OF MICROWAVE DEVICES AND CIRCUITS by Yasser A. Hussein A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy ARIZONA STATE UNIVERSITY August 2003 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3094964 UMI UMI Microform 3094964 Copyright 2003 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ELECTROMAGNETIC PHYSICAL MODELING OF MICROWAVE DEVICES AND CIRCUITS by Yasser A. Hussein has been approved July 2003 APPROVED: , Co-Chair , Co-Chair Supervisory Committee ACCEPTED: a M Dean, Graduate College Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT Electromagnetic physical modeling (global modeling) is the accurate approach to model today’s high frequency active devices. The technique couples the device physics and electromagnetics into a single package. There are many challenges in the implementations o f the global modeling techniques. The most prominent ones are CPU time requirements, stability, and accuracy. Accordingly, global modeling techniques should be based on efficient computer aided design (CAD) tools. In this dissertation, a complete description of the global modeling technique is provided, with an emphasis on a hydrodynamic model (HD) coupled with Maxwell’s equations. Moreover, electromagnetic physical modeling is carried out for complex microwave structures. This includes closely packed microwave transistors and multifinger transistors. The second part of this dissertation deals with introducing new numerical techniques to efficiently perform the global modeling approach. The new numerical techniques are based on wavelets and genetic algorithms (GAs). A genetic algorithm is employed to solve the equations that describe the semiconductor transport physics in conjunction with Poisson’s equation. An objective function is formulated, and most of the GA parameters are recommended to change during the simulation. Furthermore, the effect of different GA parameters is analyzed. The technique is validated by simulating a submicrometer field effect transistor (FET), and then compared to successive over relaxation (SOR); showing the same degree of accuracy along with a moderate speed of convergence. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Also in this dissertation, a new wavelet-based time-domain simulation approach for large-signal physical modeling of high frequency semiconductor devices is presented. The proposed approach solves the complete hydrodynamic model and Maxwell’s equations on nonuninform multiresolution self-adaptive grids. The nonuniform grids are obtained by applying wavelet transforms followed by hard thresholding. A general criterion is mathematically defined for grid updating within the simulation. In addition, an efficient thresholding formula is proposed and verified. Different numerical examples are presented along with illustrative comparison graphs showing more than 75% reduction in CPU time, while maintaining the same degree of accuracy achieved using a uniform grid case. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I dedicate this dissertation to my parents for their deep love and support. v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS This dissertation would not have been completed without the will and blessing of God, the most gracious, and the most merciful. Next, I would like to express my gratitude to my advisor, Prof. Samir El-Ghazaly, for his continuous support and guidance. This dissertation would not have been finished without Prof. El-Ghazaly’s invaluable ideas, devoted time, and support. Also, I will always be indebted to Prof. El-Ghazaly for accepting to be my advisor and helping me get admitted to ASU. I was privileged working under his supervision and being a member of his research group. I would like to express my appreciation and gratitude to Prof. Steven Goodnick for accepting to be my co-advisor. I would also like to thank Prof. Goodnick for his support and interest in my research along with valuable discussions and suggestions during the research presentations. My appreciation also goes to my committee members, Prof. George Pan, Prof. Hans Mittelmann, and Dr. Rodolfo Diaz for their interest and time devoted to the considerations of this dissertation. All of them taught me something and I will always be grateful. I would also like to express my appreciation to Prof. Saad Eid of Cairo University, who introduced genetic algorithms to me. I am also grateful to my teachers at ASU, especially Prof. Carl Gardner, Prof. Christian Ringhofer, and Dr. Badawy El-Sharawy. Sincere appreciation and respect are due to my past and present friends and colleagues of the Telecommunication Research Center for their help and useful discussions. In particular, I would like to thank Dr. Emmanuel Larique, Dr. Sebastien vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Goasguen, Dr. Kai Liu, Mr. Munes Tomeh, and Mr. Ekram Bhuiyan. Many thanks to my colleagues Mohammed Waliullah, Sung-Woo Lee, Yong-Hee Park, and Aly Aly. This research has been supported by a grant provided by the U.S. Army Research Office. I would like to thank them generously for their contributions. vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS Page LIST OF TABLES................................................................................................................... xi LIST OF FIGURES................................................................................................................ xii CHAPTER 1 INTRODUCTION..................................................................................................................1 1.1 Motivations and Obj ective of This Study............................................................... 1 1.2 Dissertation Overview..............................................................................................5 1.3 Original Contributions............................................................................................. 7 1.4 Publications............................................................................................................... 9 2 GLOBAL MODELING OF COMPLEX MICROWAVE STRUCTURES....................12 2.1 Introduction..............................................................................................................12 2.2 Equivalent Circuit Models..................................................................................... 13 2.3 Physics-Based Models ...........................................................................................14 2.4 The Hydrodynamic Model (H D )...........................................................................15 2.4.1 Hydrodynamic Model Limitations................................................................. 17 2.4.2 Single Gas Approximation.............................................................................. 19 2.4.3 The Relaxation Time Approximation........................................................... 20 2.4.4 Transport Parameters......................................................................................20 2.4.5 Boundary Conditions Implementation.......................................................... 22 2.4.6 Discretization of the HD Model.....................................................................25 viii Reproduced with permission of the copyright owner. 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CHAPTER 2.4.7 2.5 Page 2.4.6.1 Discretization of the Continuity Equation.......................................... 26 2.4.6.2 Discretization of the Energy Conservation Equation......................... 27 2.4.6.3 Discretization of the Momentum Conservation Equation.................27 Hydrodynamic Model DC Results.................................................................28 Full-Wave Physical Simulations............................................................................ 33 2.5.1 Error and Stability Analysis........................................................................... 36 2.5.2 Electromagnetic-Physical Coupling...............................................................37 2.5.3 Absorbing Boundary Conditions...................................................................38 2.5.3.1 MUR Absorbing Boundary Conditions.............................................. 38 2.5.3.2 PML Absorbing Boundary Conditions................................................39 2.6 Microwave Characteristics......................................................................................39 2.7 EM-Wave Propagation Effects.............................................................................. 42 2.8 Summary.................................................................................................................. 51 3 MODELING AND OPTIMIZATION OF MICROWAVE DEVICES AND CIRCUITS USING GENETIC ALGORITHMS (GAS)......................................................................... 52 3.1 Introduction..............................................................................................................52 3.2 Genetic Algorithms: Literature Overview............................................................ 53 3.3 Problem Description............................................................................................... 58 3.4 Optimization Using Genetic Algorithms.............................................................. 61 3.5 Results And Discussions.........................................................................................69 3.5.1 DC Simulation Results....................................................................................77 ix Reproduced with permission of the copyright owner. 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CHAPTER 3.5.2 3.6 Page AC Simulation Results...................................................................................79 Summary................................................................................................................... 81 4 A NEW WAVELET-BASED TIME-DOMAIN TECHNIQUE FOR MODELING AND OPTIMIZATION OF HIGH-FREQUENCY ACTIVE DEVICES.................................... 82 4.1 Introduction...............................................................................................................82 4.2 Fundamentals of the M RTD ................................................................................... 84 4.2.1 Two Dimensional MRTD Scheme................................................................. 85 4.2.2 Battle-Lemarie Expansion Basis.................................................................... 88 4.3 Problem Description................................................................................................ 93 4.4 The Proposed Algorithm..........................................................................................95 4.5 Results And Discussions........................................................................................107 4.5.1 Hydrodynamic Model DC Simulation Results........................................... 107 4.5.2 Hydrodynamic Model AC Simulation Results........................................... 113 4.5.3 FDTD Simulation Results............................................................................117 4.6 Scheme Errors and Stability Analysis................................................................... 119 4.7 Summary................................................................................................................. 121 5 CONCLUSIONS AND FUTURE WORK...................................................................... 122 5.1 Conclusions............................................................................................................. 122 5.2 Future W ork............................................................................................................124 REFERENCES......................................................................................................................125 x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table Page 2.1 Transistor parameters used in the simulations....................................................... 29 2.2 Multifmger transistor optimized-parameters used in the simulation................... 50 3.1 GA parameters used in the simulation.................................................................... 69 4.1 Coefficients a(i),b0(i),c0( i ) ...................................................................................91 4.2 Transistor parameters used in the simulation......................................................... 95 4.3 Grid adaptability o f the different variables for T0 = 1% ......................................103 4.4 Effect o f the threshold value on error and CPU-time for FDTD simulations..! 17 xi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES 1.1 Output power versus frequency of millimeter-wave devices: solid lines, tubes; dashed line, solid-state devices. After Sieger et al. [1]................................4 2.1 A typical FET cross-section view............................................................................ 23 2.2 Carrier density distribution....................................................................................... 29 2.3 Electron energy distribution (eV).............................................................................30 2.4 The distribution of the x-direction electric field (V/cm).......................................30 2.5 The distribution of the y-direction electric field (V/cm).......................................31 2.6 The distribution o f the x-direction velocity (cm/sec.).........................................31 2.7 The distribution of the y-direction velocity (cm/sec.)......................................... 32 2.8 Potential distribution (V).......................................................................................... 32 2.9 I-V Characteristics.....................................................................................................33 2.10 Genetic view o f the electromagnetic computational domain.................................34 2.11 Attenuation constant as a function of frequency at different points along the device width for the gate electrode......................................................................... 40 2.12 Effective dielectric constant as a function of frequency at different points along the device width for the gate electrode........................................................ 41 2.13 Phase velocity as a function of frequency at different points along the device width for the gate mode........................................................................................... 42 2.14 3D view of the simulated transistors (not to scale).................................................43 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 2.15 Page Drain voltage (normalized) of the simulated transistor when EM-wave propagation and electron-wave interaction are considered at different points in the z-direction.......................................................................................................44 2.16 The potential of a passive electrode at different points in the z-direction induced due to the proximity of an operating transistor excited by a Gaussian signal.........................................................................................................45 2.17 Drain voltage (normalized) at z = 62.5mm when EM-wave propagation and electron-wave interaction are considered. Solid line: transistor is simulated alone. Dashed line: Source electrode of a second operating transistor is 0.5jum apart from the drain of the simulated transistor........................................ 45 2.18 Drain voltage (normalized) at z = 62.5mm when EM-wave propagation and electron-wave interaction are considered. Solid line: transistor is simulated alone. Dotted line: Drain electrode of a second operating transistor is 0.5fim apart from the drain of the simulated transistor........................................ 46 2.19 Generic 3D view of the simulated multifmger transistors (not to scale), (a) Single-finger transistor (lx 450pm). (b) Two-finger transistor (2x 225pm). (c) Four-finger transistor (4x112.5pm).................................................................. 49 2.20 Output voltage for the simulated multifinger transistors when EM-wave propagation and electron-wave interactions are considered.................................50 3.1 Cross-section of the simulated transistor................................................................ 60 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure Page 3.2 Generic flowchart of the genetic algorithm.............................................................64 3.3 Flowchart of the randomly generated solution G .................................................. 65 3.4 A 5 by 5 grid example illustrating how the algorithm in Fig. 3.3 works...........66 3.5 Distance from the optimal solution versus number of generations for different mutation values......................................................................................... 73 3.6 Distance from the optimal solution versus number of generations for different number of crossover points..................................................................... 73 3.7 Distance from the optimal solution versus number of generations for different values of crossover widths....................................................................... 74 3.8 Distance from the optimal solution versus number of generations for different numbers of mutated elements.................................................................. 74 3.9 Distance from the optimal solution versus number of generations for different selection criteria........................................................................................ 75 3.10 Distance from the optimal solution versus number of generations for different probabilities of mutation..........................................................................75 3.11 Distance from the optimal solution versus number of generations for different population sizes........................................................................................ 76 3.12 Sample DC results obtained using the proposed algorithms, (a) Potential distribution, (b) Carrier density distribution..........................................................77 3.13 AC gate and drain voltages obtained using the proposed algorithm ................... 78 4.1 Cross-section of the simulated transistor................................................................94 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 Generic flowchart of the proposed algorithm......................................................... 97 4.3 (a) Normalized details coefficients for the electron energy at a certain transverse cross-section, (b) Grid points marked on the actual curve for the electron energy at the same transverse cross-section, (c) Normalized details coefficients for the x-momentum at a certain transverse cross-section, (d) Grid points marked on the actual curve for x-momentum at the same transverse cross-section........................................................................................ 100 4.4 (a) Normalized details coefficients for the electron energy at a certain longitudinal cross-section, (b) Grid points marked on the actual curve for the electron energy at the same longitudinal cross-section, (c) Normalized details coefficients for the x-momentum at a certain longitudinal crosssection. (d) Grid points marked on the actual curve for x-momentum at the same longitudinal cross-section............................................................................ 101 4.5 Demonstration o f the procedure employed to obtain the nonuniform grid for the y-direction electric and magnetic fields for FDTD simulations..................106 4.6 Remaining number of unknowns as a percentage versus the iteration number for different initial threshold values......................................................................108 4.7 DC drains current convergence curves for the uniform grid and the proposed wavelet-based non-uniform grids for different initial threshold values 4.8 109 DC potential distribution obtained by the proposed algorithm using a value of the initial threshold equals to 1%..................................................................... I l l xv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.9 Comparison between the uniform-grid and the proposed algorithm with a value of the initial threshold equals to 1% (a) Gate-to-source capacitance versus gate-to-source voltage, (b) Transconductance versus gate-to-source voltage. .114 4.10 Large-signal result obtained by the proposed algorithm for a value of the initial threshold equals to 1%................................................................................ 116 4.11 AC output voltage for the uniform grid and the proposed wavelet-based nonuniform grids with different values of the initial threshold................................ 116 4.12 Potential of the gate at a specific cross-section versus time for the uniform grid case and the proposed MRTD algorithm with different values of T0 xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 Chapter 1 Introduction 1.1 Motivations and Obj ective of This Study The quest for high-power, lightweight, compact-size millimeter-wave components is becoming more demanding. Fig. (1.1) shows a comparison of power levels of various tube devices and solid-state devices. It is noteworthy to say that tubes produce highpower at millimeter-wave frequencies. However, they are becoming less desirable because o f their reliability, bulky-size, and the need for high-voltage DC power supplies. As the millimeter-wave components evolve, new modeling techniques should be developed to accurately design and optimized these devices. As the frequency increases, the wave-length of the operating frequency becomes comparable to device dimensions. Moreover, the time-period of the operating frequency becomes comparable to electron relaxation times. This forces us to address topics such as the electromagnetics and device physics on more than an individual basis. This is achieved by developing coupled electromagnetic-physics-based simulators or global modeling simulators [2], Electromagnetic physical simulation is carried out by solving Maxwell’s equations in conjunction with a physical device model. There are different physical models with different range of validity and CPU-time requirements. For instance, the hydrodynamic model (HD) and the drift-diffusion model (DD) are both derived from the Boltzman transport equation (BTE) with different truncation order of moments and are considered fluid-based physical models. The DD model is an approximation for the HD model, which can be employed only if hot-electron and velocity overshoot phenomena do Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 not take place. On the other hand, Monte Carlo simulations of the BTE are considered particle-based models, which are much more accurate than fluid-based models. BTE simulations are implemented for ultra small devices, where the assumptions to model electrons as fluids break down. It is thus imperative to carefully choose each of these physical models depending on the problem under consideration. It is not optimum to employ, for instance, BTE simulations for devices where HD models are still valid and accurate. The reason is BTE simulations need more CPU-time compared to HD models. From the above, one can conclude that modeling is a very powerful tool that should be used to design and optimize microwave devices and circuits. Accordingly, new numerical modeling techniques should be developed to efficiently model today’s microwave components and chips. The issues of stability, accuracy, and CPU-time of the new numerical techniques should be carefully addressed and maintained. In this dissertation, efficient numerical techniques have been developed and successfully applied to Maxwell’s equations and the highly nonlinear HD model. The numerical techniques are based on wavelets and genetic algorithms. A genetic algorithm has been developed and successfully applied to solve Poisson’s equations in conjunction with the HD model. The proposed technique solves the equations that describe the semiconductor transport physics in conjunction with Poisson’s equation, employing an adaptive real-coded GA. An objective function is formulated, and most o f the GA parameters are recommended to change during the simulation. In addition, different methods for describing the way the GA parameters change are developed and studied. The effect of GA parameters including mutation value, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 number o f crossover points, selection criteria, size of population, and probability of mutation is analyzed. The technique is validated by simulating a submicrometer field effect transistor (FET), and then compared to successive over relaxation (SOR); showing the same degree of accuracy along with a moderate speed of convergence. The purpose o f this study was to introduce a new vision for a genetic algorithm capable of optimizing real value functions with a considerably large number of variables. This study also represents a fundamental step toward applying GAs to Maxwell’s equations in conjunction with the hydrodynamic model (HDM), aiming to develop an optimized and unconditionally stable global-modeling simulator Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 H i" Klystrons 106 10 Gyrotrunsr 5 Gridded Tubes 4 TWT’s _L 10 S' X Urn VFET io 3 £ 2 1 Free- - SiBJT -h*N N \ io 2 b---------------- 10* = i \ j ^Electron \\ ----------------- CT - EffSi&f- ' '^ ^ 'V J m E S F E T , r^ v H = I 0 1 a- \ s * \ \ 10“ ' 10 ‘ r j 0.1 \ ■ i ' ' ' " i i --------------- NIMPATT PH EM Tf -i ^ i i ................ i " ~ ^ \p ^ r-G u n n : i i \\\ \ 11111i i 10 100 I. 1000 FREQUENCY (Gita) Fig. 1.1. Output power versus frequency of millimeter-wave devices: solid lines, tubes; dashed line, solid-state devices. After Sieger et al. [1]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 Wavelets have been applied, for the first time in literature, to the HD model along with extending it to Maxwell’s equations. This has achieved the ultimate goal by developing a fast and unified modeling approach for FDTD and physical simulations suitable for the global modeling technique. The developed approach solves the partial differential equations PDEs of either the HD model or Maxwell’s equations on nonuniform multiresolution self-adaptive grids obtained using wavelets. A CPU-time reduction of 80% is achieved while maintaining the same degree of accuracy obtained using the original techniques. Finally, this dissertation also represents electromagnetic physical modeling results of complex microwave structures. This includes closely packed millimeter-wave transistors simulated simultaneously. In additions, electromagnetic-physical modeling of high-power and frequency multifinger transistors has been carried out and the preliminary results are given based on ad-hoc optimizations. 1.2 Dissertation Overview This dissertation is organized as follows: Chapter two presents global modeling results of complex microwave structures along with an overview of the details of how global modeling is performed. The chapter begins with an introduction followed by a comparison between circuit and physics-based models. Then, the HD model equations are presented in details including the implementation of single gas and relaxation-time approximations. Transport parameter estimation is also provided. Different discretization schemes employed for the HD model are conferred along with boundary condition implementation. Elements of the full-wave Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 physical simulation are also described. Implementation of the different boundary conditions for FDTD simulation is provided. Stability and accuracy are discussed. The coupling between the FDTD model and physical model is then explained. The second part of chapter two presents full-wave physical simulations of complex microwave structures. This includes two closely packed millimeter-wave transistors simulated simultaneously and multifinger transistors. This chapter also presents the results of the characteristics of high-frequency transistors including attenuation, phase-velocity, and effective-dielectric constant. The chapter is then wrapped up with a summary. Chapter three presents an approach for global modeling of microwave devices and circuits using genetic algorithms. The chapter begins with an introduction and an overview of genetic algorithms. The details of the proposed genetic-based algorithm are provided. Complete set of results along with their discussions are given. Different numerical examples are presented to study the effect of various algorithm parameters. Finally, chapter summary is provided. Chapter four presents a new time-domain simulation approach for large-signal physical modeling of high frequency semiconductor devices, using wavelets. The proposed approach solves the complete hydrodynamic model and Maxwell’s equations on nonuniform multi-resolution self-adaptive grids. The nonuniform grids are conceived by applying wavelet transforms followed by hard thresholding. Chapter four begins with an introduction followed by an overview of the multiresolution time-domain (MRTD) techniques. Then, the problem under consideration is described. Details of algorithm implementation for both the HD model and FDTD simulation are also presented. A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 comprehensive set of results is included along with illustrative comparison graphs. Chapter summary is then provided. Chapter five gives the overall conclusions of this dissertation. Future work and possible research extensions are also provided. 1.3 Original Contributions The research work presented in this dissertation has led to the following contributions: A complete genetic-based software package to optimize functions with a large number of unknowns has been developed. An entirely new concept, namely fitnessdependent GA parameters, has been introduced and implemented. A novel vision for formulating the objective function is also provided. Two versions of the software package were developed using C/C++ and Fortran programming languages. This research represents a fundamental contribution in which GAs can be employed to optimize functions with a large number of unknowns or to solve problems that have stability constraints in order to have unconditionally stable algorithm. Genetic algorithms are also superior over traditional gradient-based algorithms in that they do not get stuck in local minima. Thus, genetic algorithms are suitable of finding a global solution for problems with multiple extreme. A wavelet-based algorithm has been developed from the scratch. The algorithm solves the hydrodynamic partial differential equations on nonuniform self-adaptive grids obtained using wavelets. This is the first time in literature to introduce a wavelet-based HD model simulator. Moreover, an efficient grid-updating criterion has been introduced. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 The algorithm is general and independent of the problem and equations under consideration, which is found suitable to solve a wide range of problems. The developed software package is implemented in Fortran. An 80% reduction in CPU-time has been achieved using the proposed algorithm. The reason is the proposed wavelet-based algorithm removes the redundancies of the original formulations. The same wavelet-based algorithm developed for the HD model is extended to three-dimensional (3-D) FDTD simulation. The developed algorithm is general that it can be applied to any type o f problem including inhomogeneous and anisotropy media. This is in contrast o f the developed MRTD approaches available today. These approaches require careful formulation for each type of problem along with being very difficult to be extended to 3-D problems. This underlines the generality and versatility of the proposed technique. Fortran is used to develop the software package of the algorithm. Electromagnetic physical modeling of complex microwave structures is presented in this dissertation. For the first time in literature, two closely packed millimeter-wave transistors are simulated simultaneously employing a coupled electromagnetic-physical model. The results show that EM-wave propagation effects should be considered, not only inside the device, but around it as well. This study is a fundamental step toward electromagnetic physical modeling of several microwave components simultaneously. Electromagnetic physical modeling of high-power and frequency multifinger transistors is presented in this dissertation. The preliminary results of this dissertation show that at very high frequency, several phenomena with strong impact on the device Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 behavior start to emerge, such as phase velocity mismatches, electron-wave interaction, and attenuation. The results suggest that contemporary microwave devices should be optimized to minimize these effects or possibly take advantage of in favor of improved device characteristics. The results also recommend multifinger transistors as potential alternatives to conventional transistors. This is achieved by using multiple-finger gates of less width instead of a single-gate device. Furthermore, this dissertation underlines the enhanced microwave characteristics of multifinger transistors attributable to reducing attenuation and EM-wave propagation effects along the device width. 1.4 Publications The work associated with this dissertation resulted in the following publications [2]-[17]: • Y. A. Hussein, M. Wali, and S. M. El-Ghazaly,” Efficient Simulators and Design Techniques for Global Modeling of High-Frequency Active Devices,” in Advances in RF Design, Editor: J. Kiang, Kluwer Academic Publishers, in press (to appear 2003). • Yasser A. Hussein and Samir M. El-Ghazaly, “ Extending Multiresolution Time Domain Technique (MRTD) To The Simulation of High-Frequency Active Devices,” IEEE Transactions on Microwave Theory and Techniques, in press and to appear July 2003. • Yasser A. Hussein and Samir M. El-Ghazaly, “ Global Modeling of Microwave Devices and Circuits Using a Genetic-Based Optimization Technique,” IEEE Transactions on Microwave Theory and Techniques, to appear January 2004. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Yasser A. Hussein and Samir M. El-Ghazaly, “ Global Modeling of Microwave Devices Using a New Multiresolution-Time Domain (MRTD) Technique,” IEEE Microwave and Wireless Components Letters, submitted. • Yasser A. Hussein, Samir M. El-Ghazaly, and Stephen M. Goodnick,“ An Efficient Electromagnetic-Physics-Based Technique For Accurate Modeling of High- Frequency Multifinger Transistors,” IEEE Transactions on Microwave Theory and Techniques, to appear December 2003. • Yasser A. Hussein, Samir M. El-Ghazaly, and Stephen Goodnick, “A New WaveletBased Technique for Full-wave Physical Simulation of Millimeter-wave Transistors,” presented at the MTT-s International Microwave Symposium (long paper), Philadelphia 2003. • Yasser A. Hussein, Samir M. El-Ghazaly, Yong-Hee Park, and Stephen Goodnick, “EM-Wave Effects On Closely Packed Microwave Transistors Using a Fast TimeDomain Simulation Approach,” presented at the MTT-s International Microwave Symposium (longpaper), Philadelphia 2003. • Yong-Hee Park, Yasser A. Hussein, Samir El-Ghazaly, Vijay Nair, and Herb Goronkin, “Effect o f Bonding-Wire On Electrically Tunable Microstrip Antennas,” invited for Presentation at APS, Columbus-Ohio 2003. • Samir M. El-Ghazaly, Stephen Goodnick, Yasser A. Hussein, et-al., “Hierarchy of Global Modeling Simulations: From Circuit-Based to Physics-Based Models,” presented at the Asia-Pacific Microwave Conference Workshop, Japan, Nov. 2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • Yasser A. Hussein and Samir M. El-Ghazaly, “Global Modeling of Active Microwave Devices Incorporating a Novel Time-Domain Large-Signal FullHydrodynamic Physical Simulator Using Wavelet-Based Adaptive Grids,” presented at the MTT-s International Microwave Symposium (longpaper), Seattle, WA, U.S.A., June 2002. • Samir M. El-Ghazaly, Stephen Goodnick, Yasser A. Hussein, et-al., “Discretization and Circuit-Based Simulation of High Frequency Devices and Circuits Including Distributed Effects,” presented at the MTT-s International Microwave Symposium Workshop, Seattle, WA, U.S.A., June 2002. • Yasser A. Hussein and Samir M. El-Ghazaly, “Global Modeling of Active Microwave Devices Using Genetic Algorithms,” presented at APS/URSI, San Antonio, TX, U.S.A., June 2002. • Yasser A. Hussein and Samir M. El-Ghazaly, “ Global Modeling of Active Microwave Devices Using Wavelets,” presented at APS/URSI, San Antonio, TX, U.S.A., June 2002. • Yasser A. Hussein and Samir M. El-Ghazaly, “ Large-Signal Physical Modeling of Active Microwave Devices Using an Adaptive Real-Coded Genetic Algorithm,” presented at APS, San Antonio, TX, U.S.A., June 2002. • Yasser A. Hussein and Samir M. El-Ghazaly, “Global Modeling of Compact HighSpeed Circuits,” US Army Research Office Annual Reports (DAAD19-99-1-0194), May 2002/2003. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Global Modeling of Complex Microwave Structures 2.1 Introduction Modeling is a very powerful tool since it is very helpful to know the device characteristics before fabrication. This would save us time and money because fabrication will be carried out at the end of the design cycle. Accordingly, optimization and design of microwave devices should be based on modeling techniques that incorporate every aspect including the physics of the device as well as electromagnetic-wave propagation effects. Modem high performance electronics are based on technologies such as monolithic microwave integrated circuits (MMICs), with a large number of closely packed passive and active structures, and several levels of transmission lines and discontinuities. These devices operate at high speeds, frequencies, and often over very broad bandwidths. It is thus perceptible that the design of MMICs should be based on robust design tools that would simulate all the circuit elements simultaneously. The possibility of achieving this type of modeling is addressed by global circuit modeling that has been demonstrated in [18]-[45]. Device Models can be classified into two main categories: physical device models and equivalent circuit models. Each model has its own advantages and disadvantages. The choice of each approach depends on the problem under consideration. Generally speaking, circuit models are more suitable for large-scale circuit design and optimization. On the other hand, physics-based models are used for device development, optimization, and characterization. However, the increase of the operating frequency along with the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 small size of today’s chips have led to the importance to employ physical models for the design and optimizations of MMICs. In the following sections, an overview of both models, namely the physical and equivalent circuit models, will be provided. Accuracy and range of validity of each model will also be given. 2.2 Equivalent Circuit Models Equivalent circuit models are based on the electrical performance of the device at its terminals. The elements of the equivalent circuit models can be extracted either by measurements or by simulation using physics-based models. One of the advantages of equivalent circuit models is that they are easy to implement. Thus, they are suitable for large-scale design and optimization. Furthermore, equivalent circuit models are very efficient in terms of CPU-time, i.e., computationally efficient. However, at very high frequency, circuit models cannot obtain the correct device characteristics. The reason is circuit models elements are extracted at a specific operating condition. This includes, biasing, operating frequency, power, and temperature. Another limitation of circuit models is they do not relate circuit element values to the physical and process parameters, such as doping profile, device geometry, mobility, and effective mass. The equivalent circuit models also break down for high-frequency devices for the following reasons [46]: • Device dimensions become very small and comparable to the operating frequency wavelength. The distributed effects become very important and must be modeled. This is achieved by electromagnetic simulation, i.e., full-wave simulations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 • The electron transit and relaxation times become comparable to the operating frequency time period. Accordingly, the coupling between the electromagnetic wave and the physics of the device becomes very significant. • For high power application, in which time-varying fields are comparable to the dc bias fields, the EM-wave and electron interactions become highly nonlinear with several harmonics. This can be accounted for by employing a coupled electromagnetic-physics-based simulator. 2.3 Physics-Based Models Physical models provide insight to the device operation and able to accurately obtain the device characteristics at different operating conditions. The physical models provide the important link between the physical and process parameters (doping profile, gate length, mobility, etc.) and electrical performance parameters (dc characteristics, RF tranconductance, resistances, capacitances, etc.). From the above-mentioned reasons, physics-based models are very suitable to model and optimize today’s devices and circuits over a wide-band of frequencies. Physics-based models can be classified into two categories: particle-based models and fluid-based models. The first category is represented by the Monte-Carlo technique. On the other hand, the second category is based on a set of conservation equations obtained by estimating the different moments of the Boltzmann’s Transport Equation (BTE). These models usually involve several approximations, from the more complex to the simplest, they are: Full-Hydrodynamic models, Energy models, and Drift-Diffusion models. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 The drift-diffusion model includes a drift velocity controlled by the electric field and diffusion down carrier density gradients. In a homogeneous system, the drift- diffusion model is reduced to Ohm’s law. This assumes that the microscopic distribution of momentum and energy over the charge carriers at any time and location equals to that found in a large sample with a DC field equals to the local instantaneous field. This assumption breaks down for submicron devices, where carrier transport is predominantly non-stationary. For submicron devices, non-stationary effects such as hot electron effect and velocity overshoot start to emerge and should be accounted for in device modeling. Semi-classical device models have been developed to include energy and momentum relaxation effects at the same time provide CPU-time efficient models. These models deal with charge carriers as classical particles, which are derived from quantum models. In the next section, a complete description of the Full-hydrodynamic model (HD) will be provided. The HD is accurate for devices with gate length less than 0.5 micrometer and larger than 0.1 micrometer. The HD is a semi-classical model because the descriptions used for the band structure and scattering processes are generated by quantum mechanics. 2.4 The Hydrodynamic Model (HD) The transistor model used in this chapter is a two-dimensional (2-D) full- hydrodynamic large-signal physical model [47]. The active device model is based on the moments o f the Boltzmann’s Transport Equation obtained by integrating over the momentum space. The integration results in a strongly coupled highly nonlinear set of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 partial differential equations, called the conservation equations or the HD. These equations provide a time-dependent self-consistent solution for carrier density, carrier energy, and carrier momentum, which are given as follows [48]-[50]. • current continuity — + V.(»w) = 0. dt C2-1) energy conservation d(ne) » • + qnu.E + V.(nu(£ + K BT )) = - n(e ^ tb00 (2.3) x-momentum conservation ^ dt + ^ 4ox ^ xm(s) <2-4> In the above equations, n is the electron concentration, o is the electron velocity, E is the electric field, e is the electron energy, s0 is the equilibrium thermal energy, and p is the electron momentum. The energy and momentum relaxation times are given by rs and zm_respectively. Similar expression is obtained for the .y-direction momentum. The current density J is estimated from the FHD using (2.5). J (t) = -qnv(t). (2.5) The low field mobility is given by the empirical relation [51]: ,, _ 0 “ ~1 + 8000 ^ (A V 10-") cm2 V i’ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.6) 17 The above model accurately describes all the non-stationary transport effects by incorporating energy dependence into all the transport parameters such as effective mass and relaxation times. 2.4.1 Hydrodynamic Model Limitations It is very crucial to choose the appropriate semiconductor device model to accurately and efficiently model ultrasmall semiconductor devices. The reason is each model has its own range of validity and CPU-time requirements. In general, semiconductor device modeling can be categorized into four approaches: classical transport, semi-classical transport, quantum transport, and atomic-level transport [51]. The classical transport is valid for devices of 1 pm-scale where all transport phenomena are assumed to change more slowly than the carrier energy relaxation and the electron temperature is assumed to be equal to the lattice temperature. The drift-diffusion model is an example o f a classical transport modeling approach. On the other hand, semi-classical transport models are employed for devices of scale between 0.1 pm and 1 pm. These models include some phenomena that take place in ultrasmall devices such as hot electron and velocity overshoot. The hydrodynamic model and Monte Carlo method are considered semi-classical transport approaches and are derived from Boltzmann’s transport equation (BTE). They are semi-classical because scattering processes and descriptions of energy bands are generated using quantum mechanics. However, for devices of scale less than 0.1 pm where the device size approaches the coherent length of electrons, semi-classical transport models will no longer be valid. Accordingly, quantum transport models (QT) should be employed to accurately model such devices. These Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 models perfectly capture quantum interference effects such as tunneling and energy quantization. Furthermore, since the electron’s flight time across the device region becomes extremely short, the uncertainty relation in terms of energy and time are also included in QT models. Transmission coefficient, Wigner function, and Green’s function are examples of quantum transport modeling approaches. Further downscaling of devices requires employing Atomic-level transport modeling approaches. These are valid for devices o f scale less than 10 A0. It is noteworthy to say that quantum correction terms can be added to the HD model and BTE to have a CPU-time efficient model that includes quantum transport. These models are quantum hydrodynamic (QHD) or BTE with quantum corrections [51]-[53]. The HD model presented in this dissertation is a semi-classical model approximation to many particle quantum-mechanical problems. The HD model is an approximation for BTE, which is valid only under the following assumptions [54]: • The free carriers in the device are considered as point particles, with well-defined position and momentum. This is contrary to the quantum first exclusion principle. Quantum effects are included implicitly in the effective mass and other properties in the scattering rates. • The number of carriers in the device is large enough that a statistical description is appropriate. • The carriers can be considered uncorrelated and thus the n-particle distribution function can be written as a product of n single-particle distribution functions. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 • Collisions instantaneously change the momentum of carriers but not their positions. It is noteworthy to underline the fact that the hydrodynamic model is an approximation for BTE, which has further approximations. For instance, truncated moment expansion in terms of local variables and problems in non-stationary transport such as velocity overshoot. 2.4.2 Single Gas Approximation The conservation equations are obtained by suitable integration of the Boltzmann’s transport equation over the momentum space and averaging over the multivalley conduction band [55]. Certainly, in semiconductors with multi-valley energy structures, such as GaAs, there exists one electron gas with a different distribution function for each of its conduction band valleys. As a result, the BTE and the hydrodynamic equations are separately valid for each of the conduction band valleys. To reduce the number of partial differential equations that need to be solved, the single electron gas approximation is usually used. This consists of using average quantities over all main valleys for the carrier density momentum and energy [56]. For instance, the new quantities used in the single gas approximation are defined as follows: n = Y s ni (2.7) T =Y r i Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 The summations are taking over all conduction band valleys. 2.4.3 The Relaxation Time Approximation The time relaxation approximation used for the collision terms is given by: (O f} V / collission f~ fo (2.8) ^ Where / 0 is the distribution function under equilibrium conditions and z is the characteristic time that describes how the distribution function decays or relaxes back to its equilibrium state as the driving forces are removed. This approximation is very essential because it makes the solution of the BTE much easier. In this study, minority charge transport as well as generation and recombination are neglected since we are dealing with unipolar devices such as MESFETs. 2.4.4 Transport Parameters The solution of the balance equations requires different transport parameters such as the ensemble energy and momentum relaxation times, the electron effective mass, mobility and temperature. Two major approaches are employed to estimate these parameters. The first approach is to guess the form of the distribution and then use the balance equations to solve for the parameters in this function. The most generally used guess is the displaced Maxwellian. The other approach to evaluate these transport parameters is based on Ensemble Monte Carlo simulations. The parameters are evaluated under steady-state conditions in bulk GaAs. Several simulations are carried out using different electric field Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 and impurity concentration profiles. The resulting statistical information can then be empirically fitted in analytical expressions suitable for numerical simulations. Following this, the electron energy can be expressed in terms of the steady-state applied field E as: e = 0 .3 1 6 - 0.316 fo r l + (0.2£„)3! g = 0.308 +3.353 fe * £ ,,< 1 2 .5 kV cm (2.9) ! 2 '5^ 10J fo r E ss> 12.5 — cm While, the electron saturation velocity is given by: 1.53 + 0.9 (lO"4 Ess ) Dss (2.10) 19 1+ 10'4 £ ss and the electron drift velocity is given by: MqE ss + ( 4 5 0 0 ) ~ 6 u ^ ( £ w ) vd = (2 .11) 1 + (4 5 0 0 )-6 ( E s s ) where ju0 is the low-field mobility and is given by: 8000 Mo - (2.12) i + V ^ . i o - 17 The electron temperature Te is expressed as: T = 0.085 1 - for s< 03eV r e + 0.08x23 1+ 0.115 T. = 0.05 + 0.03^1 + ^ - 0 . 3 x16 0.075 . (2.13) elsewhere Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 The electron effective mass is approximated by: 0.0623m0 for Ess < 1.9944kV/cm [0.0623 + 0.00526 (Ess - 1.944)] m0 for 1.944 < Ess < 4.167kV/cm m* = J [0.074+ 0.0216 (Ea - 4.167)] m0 for 4.167 < E SS < 6.194kV/cm [0.1172 + 0.01053 (Ess - 6.194)] m0 for 6.194 < Ess <8.33kV/cm [0.1397 +0.005 for Ess > 8.33kV/cm -8.33)]m 0 (2.14) Finally, the electron mobility and the energy and momentum relaxation times are respectively evaluated as: (2.15) (2.16) T m { s h rnH s)M s) (2.17) q 2.4.5 Boundary Condition Implementations In addition to the transport equations, a set of consistent boundary and initial conditions is needed in order to have a unique solution to the problem. The surface and contact properties provide the type boundaries to be used. These are generally mixed type boundaries due to the presence of current free boundaries as well as Ohmic and Schottky contacts. At the surfaces were no current flow is assumed to exist, Neumann boundary conditions are used for both the potential and carrier density. In this case, the gradient of both the potential and the carrier density are set to zero in order to ensure zero drift and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 diffusion currents, respectively. Referring the 2D MESFET geometry of Fig. (2.1), those boundaries are given as follows: dx ^= o dx dV dn for boundaries parallel to the y axis, (2.18) = 0 for boundaries parallel to the x axis Source Gate Drain X Active Layer Buffer Layer Fig. 2.1. A typical FET cross-section view. The values for the potential at the metal contacts of the different device electrodes are described by Dirichlet boundary conditions defined by the applied bias voltages. The potential value at the semiconductor side of the Schottky contact is given by V = Vg - Vb. Where Vg is the voltage applied externally at the gate electrode and Vh is the potential barrier height. Voltages at the source and drain are defined by the applied bias voltages at the two electrodes. The boundary conditions for the carrier density at the source and drain contacts can be defined using either Neumann or Dirichlet boundaries. In the Dirichlet boundaries, a constant equilibrium carrier concentration, usually equal to the doping concentration, is used. The boundary condition at the gate is often specified using Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 thermionic emisson-diffusion theory [57], where the equilibrium electron density is given by (2.19). (2.19) where N c is the effective density of states. This type of boundary condition, however, results in a net gate current, which is found to slow the convergence rate to the steadystate solution [61]. Therefore, as an alternative, the boundary condition is set such that there is zero gate-current. In other words, the electron density at the gate is computed using (2.20). qn/uE + D — = 0 dy (2 .20) where /u is the electron mobility and D = K bT/u. For the electron energy, Neumann type boundaries are used everywhere. The final solution is obtained in a self-consistent evaluation of the three conservation equations in conjunction with Poisson’s equation. It was found that the order in which these equations are solved is critical to the stability of the solution. Poisson’s equation is used to update the electric field. Because the momentum relaxation time is about one order of magnitude shorter than the dielectric and energy relaxation times, the momentum equation is solved first using the new values of the electric field. Then, the carrier energy is updated and used to compute the new transport parameters. Finally, the continuity equation is solved for a new carrier density distribution. This process is repeated until a self-consistent solution is obtained. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 2.4.6 Discretization of the HD Model In this section, a finite-difference (FD) discretization scheme will be presented for the hydrodynamic model. Stability and accuracy will also be discussed. To have a numerical solution of the semiconductor model, a two-dimensional rectangular grid that covers the computational domain is generated. The variables of the transport equations are defined on a staggered grid. The scalar variables such as potential, electron density, electron energy, and transport parameters are defined at the basic nodes ( i ,j ) . On the other hand, vector quantities such as electron velocity, electric field, and current density are defined at the complementary nodes (z +1 / 2, j +1 / 2). These arrangements separate the component of the vector variable and facilitate a convenient way for decoupling the variables and expressing the approximations of the space derivatives using the FD method. Before carrying out a discretization scheme, a very important question needs to be answered. What discretization scheme should we use: Implicit or Explicit. The fastest answer would be to use implicit methods since there are no restrictions on the time-step value used. Accordingly, one can use a large time step and the CPU-time will be reduced dramatically. However, explicit methods have been found to outperform implicit methods despite the time-step size restrictions for the following reasons: • Employing implicit discretization methods result in full matrices that need more computational effort to be solved, i.e., more CPU-time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 • Error is a function of the time-step used. Accordingly, a large time-step should not be employed because this increases error. • Implicit methods introduce diffusion. For a long time of simulation, this diffusion accumulates and damps our solution. Eventually, the simulator will have a zero output. From the above, it is clear that there is no benefit achieved using implicit methods. Thus, all equations in this study are represented or discretized using explicit FD schemes. 2.4.6.1 Discretization of the Continuity Equation The continuity equation is expanded and written as: dn _ d(nvx) d(nvy) (2.21) dt dx dy Using an FD scheme along with employing Upwinding, the space derivatives are approximated as: d(nox) dx d(nuy) ^ dy 1 f H U j) v x (i, j ) - n(i -1 , j)u x (i -1 , y)] [1 + S(i,j)] +1 2Ax\[n(i + l , j ) vx(i + l j ) - n ( i , j ) u x( i , j ) ] [ l - S( i , j ) \ J i fH U j ) v y(Uj ) ~ < i , j ~ l ) o yO', j ~ 1)][1 + T (i, y')] +1 (2.23) 2 A y [ H U j + l)vy(i, j + T) -n( i , j ) uy( i , j ) ] \ l - T( i , j ) ] J S(i, j) = 1 ifox( i, j )>0. S(i, j ) = - l T(i,j) = 1 i f vx 0, y) < 0. i f uy(i,j) > 0. T(i,j) = - 1 i f vy(i,j) < 0. On the hand, the time derivative is approximated using forward Euler as: (2.24) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 2.4.62 Discretization of the Energy Conservation Equation To enhance the stability of the FD scheme of the energy conservation equation, the following procedure is followed. The energy conservation equation can be written as: - =a -b s (2.25) dt The above equation is a first order differential equation and has a closed form solution given by: * 0 = £ + m /„) b b (2.26) Rearranging the above equation and setting At = t - t 0 and t0 = k A t , along with using a first order approximation of the potential leads to: (2.27) 2.4.6.3 Discretization of the Momentum Conservation Equation The momentum conservation equation is highly nonlinear compared to the other two equations. This makes the momentum conservation equation very sensitive to the smallest errors. In this study, LAX method is employed to achieve a higher degree of accuracy. For instance, the derivatives of the electron velocity in x and y directions are carried out as follow s. f e l l I dt ) i+l j 2 At L “ (i +1 / 2, j ) - i [ y (i + 3 / 2, J ) + u ‘ (i - 1/2, j ) ] | Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a28) 28 f \ \ diPy) dt 'IJ+,. i 2 i - | u,‘" (i, j +1 / 2 ) - i [ o / ( i j + 3 / 2) + u ‘ ( I J - 1/ 2)] [ At (2.29) It is noteworthy to mention here that Upwinding is employed for the last two terms on the right hand side of (2.28) and (2.29). 2.4.7 Hydrodynamic Model DC Results DC results obtained by coupling Poisson’s with the HD model are presented in this section. The DC results represent the initial condition for ac simulations, which are carried out by coupling Maxwell’s equations with the HD model. To demonstrate the potential of this approach, it is applied to an idealized MESFET structure, which is discretized by a mesh of 64 Ar by 32 Ay with At = 0.001 p s. Forward Euler is adopted as an explicit finite-difference method. In addition, Upwinding is employed to have a stable finite-difference scheme. The space step sizes are adjusted to satisfy Debye length, while the time step value At is chosen to satisfy the Courant-Friedrichs-Levy (CFL) condition. First, DC simulations are performed, and the current density is calculated using (2.5). DC excitation is performed by forcing the potential to be equal to the applied voltages to the electrodes (i.e., Dirichlet boundary conditions). Table (2.1) summarizes the transistor parameters used in the simulation. Figures (2.2) -(2.9) show the DC distributions and results obtained using the HD model presented in the previous section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 TABLE 2.1 TRANSISTOR PARAMETERS USED IN THE SIMULATIONS Drain and source contacts Gate-source separation Gate-drain separation Device thickness Device length Gate length Device Width Active layer thickness Active layer doping Schottky barrier height DC gate-source voltage DC drain-source voltage 0.5 pm 0.5 pm 1.0 pm 0.8 pm 2.8 pm 0.3 pm 250 pm 0.2 pm 2 x l0 17 cm' 0.8 V -0.5 V 3.0 V *10 2 1 0,5 0 Y X Fig. 2.2. Carrier density distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Fig. 2.3. Electron energy distribution (eV). x 10 3 2 ■o 1 u- 0 a ■6 Fig. 2.4. The distribution of the x-direction electric field (V/cm). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 * x 10 Fig. 2.5. The distribution of the y-direction electric field (V/cm). 7 X 10 ■ : r" Fig. 2.6. The distribution of the x-direction velocity (cm/sec.). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 x 10 £ Fig. 2.7. The distribution of the y-direction velocity (cm/sec.). x Fig. 2.8. Potential distribution (V). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Vgs= -0.25 200 £ 150 £ Vgs= -0.75 c <D C 100 s Q Vgs= -2.75 0 0.5 1 1.5 2 2.5 Drain Voltage (V) 3 4 3.5 Fig. 2 .9 .1-V Characteristics. 2.5 Full-Wave Physical Simulations The full-wave or electromagnetic simulation presented in this section is based on Yee’s algorithm [62]. The algorithm begins with descritizing Maxwell’s equations along with arranging terms. Brief description of the algorithm is provided below. More details can be found in [63]. Maxwell’s equations are given by (2.30). V x jE = - j u ^ dt V x H =e $ £ - + J dt i Expanding, N r .... dz 1 I dE y = _Lr dH X dt e dz dE y dx dE = j_ dt e dHz dx - J , S’ 1 dEx 1 dz i _L ' d E X M [ dy II = I M dx 1 I = ±_ f dE z dt dEx dt I d H x = JL dE y dt M dz dHz dt (2.30) dHx - JZ dy Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 SOURC GATE DRAIN ACTIVE LAYER BUFFER LAYER Fig. 2.10. Genetic view of the electromagnetic computational domain. Yee defines the grid coordinates as: (.i , j , k ) = ( i A x , j A y , k A z ) (2-32) where Ax, Ay, and Az are the actual grid separations. Any function of space and time is then written as: F " ( i , j , k ) = (iAx, j A y , k A z , n A t ) . (2.33) Where At is the time increment and n is the time index. The spatial temporal derivatives of F are written using central difference approximations as: & = ----------- 2 -------Ax---------- 2 ------- d F ' ( i , j , k ) _ dF" * ( i , j , k ) - d F n^ ( i , j , k ) dt At Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. <2-34) (2.35) 35 The above equation are applied to the six scalar equations resulting in six coupled explicit finite difference equations: K k i j +i* + + i ) = K ' H i j + 1 ,1 + 1 ) Az At ■ (2.36) E ; ( i , j + i , k + ± - ) - E : ( i , j , k + ±-) H ; h n i + L j ik + i.) = + L j , k + 1) E : ( i + i , j , k + ± ) - E : ( i j , k + ±-) At + (2.37) E nx (i + 2 , J , k + 1) - E nx (i + ~ k , j , k ) n ;n H i + ± j + 1 *) = # ; " * ( / + 1 / + i1 *) E:(i + ± , j + i , k ) - E : a + ± , j , k ) At At + £;(/■ + i , j + \ , k ) - E ; ( i , j + L , k ) (2.38) Ax E:"(i + ^ J , k ) = E : ( i + L j , k ) + At H ] \ i + j1- , j. -+^1i *) - #;**(,■ + l j - 1-,*) at H n;H i + + + ) - H n;H i + i y , t - h Az - 2(i + ± , j , k ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.39) 36 At H "H iJ + \,k + \)~ H p Q .j+ ^ .k - \) AZ 1 1 (2.40) E : +' ( i , J , k + ± ) = E ; ( i , j , k + ±:) + At e (i,j,k + ^ ) H 7 J(i + \ , j , k + |1) - H p ( i - j - , j , k + £ ) Ax H n; k h j + ± r , k + b - a : m , j - \ i, k + \ ) (2.41) Ay 2.5.1 Error and Stability The FDTD Yee-Based numerical scheme is second-order accurate, both in space and time, because second difference discretization is employed. Accordingly, the error introduced by this scheme is dispersive. This error is minimized by choosing the spatial step-size to be much smaller than the wavelength X . As a rule of thumb, the spatial increments are chosen to be less than A /1 0 . On the other hand, the stability of the Yeebased FDTD algorithm is based on the Courant-Friedrichs-Lewy (CFL) condition given by (2.41). At = ------- 1 Ay2 Az2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 2.5.2 Electromagnetic-Physical Coupling The full-wave solution is obtained by coupling the semiconductor and the electromagnetic solutions with the appropriate initial conditions. The semiconductor solution provides the electronic state inside the active device and evaluates the current density. The electromagnetic solution, on the other hand, updates the electric and magnetic fields everywhere inside the structure in response to the electron current densities and applied excitation. The semiconductor part is considered a lossy dielectric with a nonlinear conductivity, which is time, field, and space dependent. This conductivity is obtained from the solution of the active device model. To incorporate the initial conditions of the solution, the above equations are modified to: Ay 6 (/ + J , j , k ) + Az T o ta l + (2.42) Az + T o ta l( n + \ ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.43) 38 E r ( i , j , k + ±-) = E : ( i , j , k + ± ) + At e (if j + L }k) H n;* ( i + l ; ' t + 1 ) - H ^ ( J - \ , j , k + Ax (2.44) H">V,j + j 2.5.3 + f> - K h i , j - \ , k + 1) + Ky : c ( i , j , k + ± ) ~ j ; " ' 1"* ' (,•, j , k + 1 ) Absorbing Boundary Conditions Boundary conditions should be implemented since it is imperative to have a confined computational domain. The main idea of the absorbing boundary conditions (ABCs) is to have no reflection boundary surfaces. In this chapter, two methods that have been developed to implement ABCs will be discussed, namely, MUR and perfectly matched layer (PML) absorbing boundaries. 2.5.3.1 MUR Absorbing Boundary Conditions Conceptually, the MUR ABC is based on the one-way wave equation. This allows EM-wave propagation only in one direction [58]. The authors in [58] proposed two different schemes based on accuracy, namely first order and second order MUR ABCs. The first order and second order MUR ABCs are obtained using one term and two terms of the Taylor expansion, respectively. Despite working very well for scattering problems with uniform media, MUR ABCs are found to result in significant reflection for multilayered media. On the other hand, the PML method is found to have better performance for a broad type of problems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 2.5.3.2 PML Absorbing Boundary Conditions PML absorbing boundary conditions has been proposed by Berenger [62]-[64]. Berenger adds a new degree of freedom to control loss and impedance matching by splitting the scalar field components into two sub-components. Following this, regardless of the angle incidence, all waves observe the PML region to have the same characteristic impedance as of the computational domain immediately before the PML domain. 2.6 Microwave Characteristics To study the characteristics of transistors at high frequency, a time-domain Guassian signal is applied between the source and gate electrodes. The input and output time-domain signals are observed at different points along the width of the device. The characteristics of the device are then estimated. For example, the propagation constant y can be evaluated as: F ( g?, z + 1 ) (2.45) Where F(co,z) is the Fourier Transform of the time-domain signal. The attenuation and propagation constants are evaluated as the real and imaginary parts of y, respectively. Fig. (2.11) shows the attenuation constant as a function of frequency at different points along the device width. Considering Fig. (2.11), it should be noticed that the attenuation constant increases with frequency as well as from point to point along the device width. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 60 co 40 0 30 — 100 120 z1 = 200 um z2 = 300 um z3 = 400 um 140 Frequency (GHz) Fig. 2.11. Attenuation constant as a function of frequency at different points along the device width for the gate electrode. The phase velocity oph and effective dielectric constant s r can be estimated using Equations (2.46)-(2.47), respectively. »ph = J3 - 8 = (2.46) (2.47) (2 n f ) 2 Where /? is the propagation constant, c is the free-space wave velocity, and co is the frequency in rad./sec. Figures (2.12)-(2.13) show the effective dielectric constant and phase velocity versus frequency at different points along the device width, respectively. The results shown in these figures are mainly due to the change of the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 distribution of the electric field as a function of frequency and distance. — — 3.25 z1 = 200 um z2 = 300 um z3 = 400 um 3.2 | 3.15 </) s o 0 0) a> Q a> •§ 3.1 1 3.05 £W 2.95 20 40 60 80 Frequency ( GHz) 100 120 140 Fig. 2.12. Effective dielectric constant as a function of frequency at different points along the device width for the gate electrode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 x 10s 1.75 — z1 = 200 um ■— z2 = 300 um —" z3 = 400 um 1.74 1.73 8 172 1.68 1.67 1.66 20 40 60 80 Freauencv (GHz) 100 120 140 Fig. 2.13. Phase velocity as a function of frequency at different points along the device width for the gate mode. 2.7 EM-Wave Propagation Effects In this section, a full-wave physical simulator is developed to model two closely packed millimeter-wave transistors and multifinger transistors. Fig. (2.14) gives a 3D view of the simulated transistors. The simulated devices are biased to V.as = 3.0V and Vgs = -0.1V. The gate-length for the transistors is set to 0.2jum . The DC distributions are obtained by solving the active device model only. A sinusoidal signal is employed in the AC simulations with peak value of 100 mV and frequency of 80 GHz, respectively. The two transistors shown in Fig. (2.14) are identical. First, full-wave simulations are carried out for one transistor only, and the results are depicted in Fig. (2.15). Considering this figure, one should observe the variations of the output voltage with distance along Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 the device width. This demonstrates the importance of coupling the EM-waves with the semiconductor transport physics for accurate modeling of millimeter-wave transistors. Now, we turn our attention to full-wave simulation of the two transistors shown in Fig. (2.14). First, we assume that one of the transistors is operating, while the other transistor is not. Fig. (2.16) depicts the simulation results, which emphasize the significance to include the EM-wave propagation effects, not only inside the device, but around it as well. In fact, this is the basic theory of operation of multifinger transistors. Ideally, the non-operating transistor should have a zero drain potential, however due to the proximity of an operating transistor, an induced voltage that varies along the device width is introduced. TrmsistoE Separation Transistor#! Transistor #2 2.1 pm ^ 0 3 (Jm 0.4 (fin 0 2 (fin 0 3 (fin 0 3 (fint 0 3 (fin; Two possible oases: «-^ Scnircel Gatel ------------- +1 Brain2 or Souios2 D ran l . , Doped GaAs (2xl0l?end) Undqped GaAs «* ** • OS (fin I Fig. 2.14. 3D view of the simulated transistors (not to scale). Next, the two transistors in the configuration shown in Fig. (2.14) are simulated, assuming that both transistors are now operating. There may be two cases to consider. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 The first case is to assume the drains of the two transistors are adjacent to each other (the case of multifinger transistors). While, the other case is to consider the drain of one of the transistors is adjacent to the source of the other transistor. Figures (2.17)-(2.18) show the simulation results. The first conclusion that can be drawn out of the two figures is that the proximity of an operating transistor affects the output voltage due to the EM-wave propagation. Furthermore, the EM-wave effects for the case of two adjacent drain electrodes is much larger than the other case. This is expected, since the drain electrode has the amplified output signal. It is important to mention that the results in Figures (2.15), (2.17), and (2.18) are normalized such that the effect of the increase of Transconductance with width is not included. The reason is that we are interested only in investigating the effect of EM-wave propagation. 0 .0 8 0 .0 6 0 .0 4 0.02 0 Q -0 .0 2 i\ -0 .0 4 -0 .0 6 — — — -0 .0 8 0 z z z z = = = = 6 2 .5 um 1 2 5 um 1 8 7 .5 um 2 5 0 um 0 .5 1 1.5 T im e (s) 2 2 .5 3 x 1Q-n Fig. 2.15. Drain voltage (normalized) of the simulated transistor when EM-wave propagation and electron-wave interaction are considered at different points in the zdirection. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 45 0.01 z1 —7 8 .1 2 um z 2 - 5 8 .5 9 um z 3 = 1 9 .5 3 um 0 .0 0 5 -0 .0 0 5 5- CO - 0.0 1 -0 .0 1 5 - 0.02 -0 .0 2 5 - -0 .0 3 T im e (s) Fig. 2.16. The potential of a passive electrode at different points in the z-direction induced due to the proximity of an operating transistor excited by a Gaussian signal. 0.06 0.04 0.02 <D O) CO Ia Q - 0.02 -0 .04 -0.06 T im e (s) •11 x 10' Fig. 2.17. Drain voltage (normalized) at z = 62.5mm when EM-wave propagation and electron-wave interaction are considered. Solid line: transistor is simulated alone. Dashed line: Source electrode of a second operating transistor is 0.5/um apart from the drain of the simulated transistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 0.1 0.08 0.06 0.04 0.02 '«c - 0.02 - 0.04 - 0.06 - 0.08 - 0.1 0.2 0.4 0.6 0.8 1 .2 1. 6 T im e (s) 1.8 x 10'11 Fig. 2.18. Drain voltage (normalized) at z = 6 2 . 5 when EM-wave propagation and electron-wave interaction are considered. Solid line: transistor is simulated alone. Dotted line: Drain electrode of a second operating transistor is 0.5ftm apart from the drain of the simulated transistor. Multifinger transistors have proven better performance over conventional transistors, especially at very high frequency [65]-[75]. However, till now, modeling of such devices did not account for EM-wave effects as well as electron-wave interactions using a fully numerical simulator. Accordingly, it is indispensable to present analysis of multifinger transistors based on a coupled electromagnetic-physics-based simulator. It is clear from the previous section results that EM-wave propagation and electron-wave interaction change the device characteristics at high frequency. Accordingly, different structure shapes and configurations need to be employed to minimize these effects, aiming to improve the device performance, especially at high operating power and frequency. A possible solution is to use multiple gate-fingers of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 shorter lengths. In this manner, EM-wave propagation effects along the device width are minimized. Moreover, attenuation is reduced as a result of reducing the gate metallic resistance. Thus large number of fingers is better in terms of reducing attenuation and wave-propagation effects along the device width. However, large number of fingers means that attenuation and EM-wave propagation effects are increased along the feeding line. Moreover, more fingers may cause more EM-waves interference. Thus, EM-wave synchronization for the multiple fingers is crucial for maximum power and minimum interference. It is noteworthy to say that EM-wave phase-velocity mismatches is due to the different applied voltages to the electrodes and also due to the unsymmetrical shape of the structure. Therefore, the number of fingers and distance between gate-fingers should be optimized simultaneously. Moreover, for the case of the four-finger transistors, the shape and size of the air bridge connecting the different fingers affect the high-frequency characteristics of the transistor. Considering Fig. (2.19.c), it should be noticed that new capacitances between the air-bridge and transistor electrodes Cair bridge are created. This would definitely change the EM-wave phase velocities and as a result change the device behavior. Thus an optimal air-bridge structure and size should be employed as well. Furthermore, the air bridge should not be fragile in order not to break easily, which represents an extra constraint that needs to be included in our optimization problem. The feeding line shape represents also a parameter that needs to be considered for circuit-matching issues. In this study, ad-hoc optimization is performed to obtain near-optimal transistor parameters based on the above criteria. Table (2.2) shows the new parameters for the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 optimized multifinger transistors, and Fig. (2.19) gives a generic 3D view of the simulated multifinger transistors. Output voltages for the simulated multifinger transistors are shown in Fig. (2.19). Considering this figure, one should observe that the voltage-gain increases when using four-finger transistors. In addition, the shape of the output signal for the four-finger transistor case appears to be much better, which means fewer harmonics. Design and optimization of high-frequency multifinger transistors need a tremendous research work. It requires, as a backbone, a very efficient numerical simulator that includes EM-wave propagation and electron-wave interactions. The simulator should be accurate and most importantly fast in order to be suitable for optimizing complex microwave structures. It is our belief that this paper presents this type of simulator. This paper also presents, for the first time in literature, the preliminary numerical results of electromagnetic physical simulation of multifinger transistors, based on ad-hoc optimization. The future research work will employ rigorous optimization techniques to obtain the optimal multifinger transistor structure based on the model presented in this paper. Moreover, measurements will be carried out and compared to the results achieved by our model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O u u n a l Lijmar (D o p id G*As) B»n*> U jm lU n fcfw d -jaA fi (a) C h a n n e l L ay « f (D o p e d G aA s) (b) HuftVr 1 *y**i f •"'l.d im 'J L .iy e r d ' 0 {>«d G a A s ! Fig. 2.19. Generic 3D view of the simulated multifinger transistors (not to scale). (a) Single-finger transistor (lx 450|jm). (b) Two-finger transistor (2x 225p.m). (c) Four-finger transistor (4x112.5|am). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE 2.2 MULTIFINGER TRANSISTOR OPTIMIZED-PARAMETERS Drain and source contacts Gate-source separation i 0.5 pm 0.5 pm 0.4 pm Gate-drain separation 0.4 pm D evice thickness D evice length Gate length 2.1 pm 0.2 pm 1x450, 2x225, 4x112.5 pm D evice W idth A ctive layer thickness i 0.1 pm 2x1017 cm'3 0.8 V -0.2 V 3.0 V (50 GHz A ctive layer doping S chottky b amer height DC gate-source voltage DC drain-source voltage Operating frequency □ .4 0 .3 0.2 0.1 0 - 0.1 - 0.2 S in g le Fin ger: 1 x 4 5 0 um - 0 .3 T w o F in g e rs : 2 x 2 2 5 um F o u r F in g e rs : 4 x 1 1 2 .5 um - 0 .4 0 0.2 0 .4 0.6 0.0 1 1.2 1 .4 T im e (s e c o n d s ) 1.6 ^g-11 Fig. 2.20. Output voltage for the simulated multifinger transistors when EMpropagation and electron-wave interactions are considered. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 2.8 Summary In this chapter, electromagnetic physical simulation is described in details. The different parts o f the problem are provided. The emphasize is on an HD model coupled with Maxwell’s equations. A complete set of DC and AC results is presented. The preliminary results of this chapter show that at very high frequency, several phenomena with strong impact on the device behavior start to emerge, such as phase velocity mismatches, electron-wave interaction, and attenuation. The results suggest that contemporary microwave devices should be optimized to minimize these effects or possibly take advantage of in favor of improved device characteristics. The results also recommend multifinger transistors as potential alternatives to conventional transistors. This is achieved by using multiple-finger gates of less width instead of a single-gate device. Furthermore, this chapter underlines the enhanced microwave characteristics of multifinger transistors attributable to reducing attenuation and EM-wave propagation effects along the device width. The future research work will involve employing rigorous optimization techniques to obtain the optimal multifinger transistor structure based on the electromagnetic-physical model presented in this chapter. Moreover, measurements will be carried out and compared to the results achieved by our model. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Modeling and Optimization of Microwave Devices and Circuits Using Genetic Algorithms 3.1 Introduction This chapter presents a new approach for the simulation and optimization of microwave devices, using a genetic algorithm (GA). The proposed technique solves the equations that describe the semiconductor transport physics in conjunction with Poisson’s equation, employing an adaptive real-coded GA. An objective function is formulated, and most o f the GA parameters are recommended to change during the simulation. In addition, different methods for describing the way the GA parameters change are developed and studied. The effect of GA parameters including the mutation value, number of crossover points, selection criteria, size of population, and probability of mutation is analyzed. The technique is validated by simulating a submicrometer field effect transistor (FET), and then compared to successive over relaxation (SOR); showing the same degree of accuracy along with a moderate speed of convergence. The purpose o f this chapter is to introduce a new vision for a genetic algorithm capable of optimizing real value functions with a considerably large number of variables. This study also represents a fundamental step toward applying GAs to Maxwell’s equations in conjunction with the hydrodynamic model (HDM), aiming to develop an optimized and unconditionally stable global-modeling simulator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 3.2 Genetic Algorithms: Literature Overview The early pioneers of science were as much interested in biology and psychology as in electronics, and they looked to natural structure as guiding metaphors for how to achieve their visions. It should be no surprise that from the earliest days computers were applied not only to calculating missile trajectories and deciphering military codes but also to modeling the brain, mimicking human learning, and simulating biological evolution. These biological motivated computing activities have been developed over the years, but since the early 1980’s they have undergone a rebirth in the computation research community. The first has gone into the field of neural networks, the second into machine learning, and third into what is called evolutionary computation of which genetic algorithms are the most prominent example. GAs were invented by John Holland in the 1960s, in which the author described the genetic algorithm as an abstraction of biological evolution, and gave a theoretical framework for adaptation under the GA. Holland’s GA is a method for moving from a population of chromosomes to a new population by using a kind of natural selection along with the genetics-inspired operators of crossover, mutation, and inversion. Each chromosome consists of genes, each gene being an instance of a particular allele (e.g., 0 or 1). The selection operator chooses those chromosomes in the population that will be allowed to reproduce, and on average the fitter chromosomes produce more offspring than the less fit ones. Crossover exchanges subparts of two chromosomes, roughly mimicking biological recombination between two single-chromosome ‘haploid’ organisms. While, mutation randomly changes the allele values of some locations in the chromosome, and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 an inversion reverses the order of a contiguous section of the chromosome, thus rearranging the order in which genes are arrayed [76]. In this section, an overview of GAs will be provided. The emphasize will be on the studies carried out in our field, i.e., microwave and antennas. GAs have proven a great deal o f flexibility for solving different types of problems. The genetic-based algorithms are unconditionally stable and not likely to get stuck in local minima when employed for solving any type of problems. In [77], the authors presented a new technique to obtain the material parameters for each single layer in a multiplayer structure, using GAs. The GA for parameter extraction from the reflection or transmission measurements is based on a simplified evolutionary strategy. The GA is briefly described and verified by measurements in the frequency range from 115 to 145 GHz. The parameters obtained using the GA show good agreements with reference values estimated by other researchers. In [78], the authors introduced a GA for parameter extraction for RF on-chip inductors. The details of applying the developed GA in extracting the model parameters are discussed. For a set of inductors, the meaningfulness of extracted parameters is considered in the procedure of extraction. The S-parameters, inductance, and Q value can then be set as the fitting targets simultaneously. The values of the model parameters are extracted by a fully automatic programmed GA procedure, which shows excellent agreement with measured data. In [79], the authors presented a novel procedure for synthesizing multilayered radar absorbing coating, using GAs. The algorithm was successfully applied to the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 synthesis o f wide-band absorbing coating in the frequency ranges 0.2-2 GHz and 2-8 GHz. hi [80], a computational approach to the imaging of an imperfectly conducting cylinder is presented, using GAs. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is formulated into an optimization problem. The GA is then employed to find the global solution of the cost function. Numerical results demonstrated that, even when the initial guess is far away from the exact one, good reconstruction is obtained. In such a case, the gradient-based methods often get trapped in a local minimum. Numerical results show that multiple incident directions permit good reconstruction of shape and, to a lesser extent, conductivity in the presence of noise in measured data. In [81], topology and dimensions of line-segment circuits are expressed by a set o f parameters, which describe the way the structural growth of line-segment circuits. The sets of parameters are then optimized by GAs to specify specifications. In the GA process, to reduce computational time, a circuit is decomposed into lines and discontinuous elements. The S-parameters are then synthesized to obtain the response of the circuit. Three filters and a power divider are designed and tested. In [82], a graphical analysis of the impedance matching problem for the multilayer dielectric and magnetic coating of metallic surfaces, for normal plan wave incidence, is presented using GA. Methods for visual design using Smith-chart-type graphical tools, which can complement computationally intensive optimization, are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 derived. The problem of estimating the required permittivity and permeability for a given frequency and thickness is also discussed. In [83], a possible application of global optimization techniques based on a GA to imaging biological bodies exposed to interrogating microwaves has been evaluated by means o f numerical forward simulations. For the text case considered (a human abdomen), it is shown that a reduced investigation domain can be used with limited errors on the measured scattered data. Once the domain is selected, the global optimization technique can then be applied. The authors also proposed a hybrid technique that employs both gradient methods and GAs. In [84], the authors presented a procedure for synthesizing broadband microwave absorbers incorporating frequency selective surface (FSS) screens embedded in dielectric media using a binary coded GA. The GA simultaneously and optimally chooses the material in each layer, thickness of each layer, FSS screen periodicity in the x- and ydirections, its placement within the dielectric composite, and the FSS screen material. Furthermore, the GA generates the cell structure of the FSS screen. The result is a multilayer composite that provides maximum absorbing of both TE and TM waves of a prescribed range of frequencies and incident angles. This technique automatically places an upper bound on the total thickness of the composite. In [85], the authors report on the structure of a large-signal neural network (NN) high electron mobility (HEMT) model as determined by a pruning technique and a genetic algorithm. By representing the configuration of a standard multilayered neural network as a chromosome, the optimum configuration of a model is obtained through a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 simulated evolution process. For this approach, the configuration of an NN that simultaneously represents seven intrinsic elements of an equivalent circuit was also shown for comparison to previous work. In [86], the study is aimed at exploring the possibility of using a microwave approach based on a genetic algorithm to detect a defect inside a known host object, starting from the knowledge of the scattered field, the problem solution is recast as a twostep procedure. After defining a cost function depending on the geometric parameters of the crack, a minimization procedure based on a hybrid-coded genetic algorithm is applied. The influence of the noise as well as the geometry of the defect on the crack detection and reconstruction is investigated. Moreover, the numerical effectiveness of the iteration approach is examined. More research studies employing GAs can be found in [87]-[120]. The above overview represents a very good introduction to GA application in our field. This also represents an excellent starting point for those interested to use GAs. Some optimization problems have multiple local minima. Where methods based on steepest descent would fall in one of these local minima, resulting in a different solution. GA’s are random algorithms and researchers have found their generality and that they are unconditionally stable. GAs are thus suitable to find the global solution for problems having multiple minima [76]. Furthermore, in many problems requiring solving systems of linear equations Ax = b, the matrix A has a large Condition Number. For these problems, standard methods will not be able to get the correct solution. For instance, solving Poisson’s equation on a nonuniform grid. In this case, genetic-based algorithms Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 would outperform standard methods. This is because standard methods work fine only for well-posed problems (problems with A having a small condition number). On the other hand, a genetic-based algorithm converges independent of the condition number o f A . From the above, it is motivating to make an effort to apply GAs to Maxwell’s equations or the hydrodynamic model (HDM), aiming to develop an optimized and unconditionally stable algorithm. It is noteworthy to say that the main purpose of this chapter is to lay the foundation of a genetic algorithm capable of optimizing real value problems, with a considerably large number of unknowns. In this stage of the work, we will demonstrate that genetic algorithms can be applied to the hydrodynamic model (HDM) in conjunction with Poisson’s equation to accurately model submicrometer gate devices, with less stability constraints. Ultimately, a hydrodynamic model should be implemented with equations that would have numerical stability restrictions such as Maxwell’s equations rather than Poisson’s equation in order to obtain a self-consistent simulation of electromagnetic-wave propagation effects, employing an optimized and unconditionally stable algorithm. 3.3 Problem Description The transistor model used in this work is a two dimensional (2-D) full- hydrodynamic large-signal physical model. The active device model is based on the moments of the Boltzmann’s Transport equation obtained by integrating over the momentum space. The integration results in a strongly coupled highly nonlinear set of partial differential equations, called the conservation equations. These equations provide Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 a time-dependent self-consistent solution for carrier density, carrier energy, and carrier momentum, which are given as follows. • current continuity — + V.(«u) - 0. dt • (3.1) energy conservation (3.2) • x-momentum conservation (3.3) In the above equations, n is the electron concentration, v is the electron velocity, E is the electric field, s is the electron energy, s0 is the equilibrium thermal energy and p is the electron momentum. The energy and momentum relaxation times are given by z£ and rm, respectively. Similar expression can be obtained for the y-direction momentum. The three conservation equations are solved in conjunction with Poisson’s equation: (3.4) where <j) is the electrostatic potential, q is the electron charge, e is the dielectric constant, N d is the doping concentration, and n is the carrier concentration at any given time. The total current density distribution J inside the active device at any time t is given as: J ( t) = -qnu(t). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.5) 60 The low field mobility is given by the empirical relation: (3.6) On the other hand, the mobility for large-signal simulations is calculated as ju = vd / Ess, where vd is estimated using (3.7). (3.7) In the above equation, vd is the electron drift-velocity, ju0 is low field mobility given by Eq. (3.6), uss and Ess are the steady-state electron velocity and electric field, respectively. It is significant to note that both u„ and Ess are functions of energy, and they get updated each time a new energy distribution is estimated using the hydrodynamic model. The above model accurately describes all the non-stationary transport effects by incorporating energy dependence into all the transport parameters such as the effective mass and relaxation times. Fig. (3.1) shows the cross-section of the simulated structure. 05 pm 0 3 Mm Drain Doped GaAs C2iclOl , cm'^ t 28 Mm Fig. 3.1. Cross-section of the simulated transistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 3.4 Optimization Using Genetic Algorithms In this section, we apply a genetic algorithm for the solution of the boundary value problem for the distribution of potential across an FET. A generic flowchart of the algorithm is shown in Fig. (3.2). The first step is to read the matrix A and vector b of the system of linear equations, Ax = b, which are derived from Poisson’s equation [121]. A population of random solutions (chromosomes) representing the vector solution x is then initialized. Next, the objective function (fitness) is estimated for all the chromosomes that have been randomly generated. Based on the fitness, two parents are generated either by roulette wheel selection or by tournament selection methods. Mutation and crossover are then performed on the selected parents to generate two children. Replacement is conducted by comparing the fitness o f children with their parents, and the worst two chromosomes are removed from the population. The best chromosome is identified based on fitness, and finally a check is carried out against a certain stopping criteria to either stop the simulator, or to perform another iteration. The details o f implementing the proposed algorithm are as follows. • Step One: Initialization Initialization is done by randomly generating M chromosomes representing the GA population. Real encoding is adopted for our problem. Each chromosome contains n genes, which corresponds to the variables in the vector solution x . The generated random numbers have a range associated with the applied DC voltages to the device electrodes. • Step Two: E valuate Fitness Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Each chromosome is evaluated based on an objective function. The objective function is developed in a way that it accurately determines how close the randomly generated solutions are to the optimal solution. Dealing with 2D Poisson’s equation means that the matrix A has one to five elements in each row (sparse matrix) [121]. Based on this, it was found that the vector solution x with size n - N xN y should be fully constructed by knowing only the 2N x elements close to the electrodes following the flowchart given in Fig. (3.3). Fig. (3.4) shows a five-by-five-grid to illustrate the implementation of the algorithm shown in Fig. (3.3). Each square in Fig. (3.4) represents a grid-point. In Fig. (3.4), the elements inside the gray squares are used to estimate the next element, which is inside the darker square. For instance, .% is estimated as a linear combination of % , and Xj5, following the algorithm in Fig. (3.3). This process is repeated until all elements in our domain are estimated, as shown in Fig. (3.4). It is worth mentioning that the proposed numbering sequence is crucial for the correct estimation. On the other hand, the 2N x elements needed to implement the algorithm in Fig. (3.3) can be randomly generated with a minimal error based on the following. Since boundary conditions must be satisfied, then the values of the potential are known precisely at the boundaries. Moreover, an estimate for the value of the potential near boundaries can be randomly generated bearing in mind that they would have very close values to the potential at the electrodes. The randomly generated solution g along with the second norm of A x - b are included in the objective function given by Eq. (3.8), which needs to be maximized for a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 minimum value o f the dominator. In Eq. (3.8), g is the randomly generated solution, while a is a scalar representing the weight of the norm criterion. The value of a is chosen to be 10 %, by trial and error. F itn es s = --------------- = = = = = ---------- 1+ — +D n Z D = I*/ - 8: ------------ ci = Y aiJXJ~bi 7=1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 3 .8) 64 Start In itia liz a tio n E valu ate F it n e s s R e g e n e r a tio n C r o ss o v e r M u ta tio n R e p la c e m e n t B a s e d o n F itn ess Id e n tify B e s t C h r o m o so m e F itn e s s < STCR No Ye Fig. 3.2. Generic flowchart of the genetic algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2NX Values for Randomly Guess Potential I Obtain the Remaining N - 2 N X Potential Values as Follows: J = 1:^-2 i = N - j : N y - j : s t ep = - N y 8 l ~l ~ elseif O Al ~ i if (i + N y < N and i X At.kSk ^ i,i :k = i,i + l , i - N y ,i + N y (i + N y > N ) b> ~ S t-1 - Z A>,kSk * : k = i, i + 1, i - N i,i e ls e bt Si-i en d ~ Z Ai.k8t * : k = i, i + 1, / + N Ai,i loops r Supply the Initial Guess g to the Main Program ( Stop ) Fig. 3.3. Flowchart of the randomly generated solution G. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. *-x *5 *10 *15 *4 *9 *14 *5 *4 *10 i *9 66 *13 *20 *25 *14 *19 *241 |* 1 8 1 *5 *10 *4 *9 , *15 ^ *14 *23 *20 ,.....*25 ,*19 *24 f ir * " ..... *131 *18 *23 *25 *24 *23 *5 *10 §X A Xq |* 3 | *8 L. y ' *15 *20 *25 X]A X\ n iy *24 *13 *18 *23 *5 *10 *15 *20 *25 *4 *9 *14 *19 *24 *3 *8 *13 *18 *23 *2 *7 *12 *17 *22 *1 *6 *11 *16 *21 Fig. 3.4. A 5 by 5 grid example illustrating how the algorithm in Fig. 3.3 works. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 Step Three: Regeneration • Two methods are used for parent regeneration, namely, Roulette Wheel and Tournament selections. The details of implementing each method are described below. A. Roulette Wheel Selection Parents in the Roulette Wheel method are chosen randomly according to their fitness. As the name implies, the method imitates the Roulette Wheel game, where the thrown dice would most probably end being in the slot with the largest area. Following this, one can conclude that the chromosome with the largest fitness value is most likely to be chosen because it has the largest slot size or equivalently the largest fitness. B. Tournament Selection In this method, two groups from the population are randomly selected, sub populations. It is worth mentioning that the population size is chosen randomly as well. The best chromosome from each of the randomly generated sub-populations is chosen to represent a parent. • Step Four: Crossover Now, two parents have been selected for their genes to be crossed over and mutated. Crossover is conducted by first randomly selecting a crossover point within the chromosome. Two children are then conceived by mixing the genes of the two parents at the crossover point. At this moment, two different parameters can be analyzed. The first parameter is the number of crossover points, i.e., more than one crossover point can be achieved. The other parameter is the number of genes involved in each crossover point, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 and will be denoted by the crossover width. The effects of both parameters are studied and included in the results section. • Step Five: Mutation Mutation is carried out by randomly changing one or more genes (variables) of the created offspring. We then have two mutation parameters to study their effect. The first one is the number of mutated genes or variables within the chromosome. The other parameter deals with the value of mutation. • Step Six: Replacement Replacement is performed by comparing the fitness of the parents with their offspring. The best two chromosomes out of the four are included in the population for the next iteration. • Step Seven: Ending the Algorithm The best chromosome is identified at each iteration and error is calculated as the second norm of A x - b . This error is checked against a predefined value, if satisfied, the simulator stops and prints the final results. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 3.5 Results and Discussions In this section, the effect of different GA parameters on the algorithm behavior is investigated. The GA parameters used in the simulation are summarized in Table (3.1). TABLE 3.1 _______ GA PARAMETERS USED IN THE SIMULATION_________ MU NC CW NM SC NPOP PM Mutation factor or value Number o f crossover points Crossover width Number o f mutated variables Selection criterion Size o f population Probability o f mutation_____ The default values are 0.1, 1, 1, 1, 1, 100, and 1 for MU, NC, CW, NM, SC, NPOP, and PM respectively. It is significant to note here that the developed algorithm is implemented as a subroutine to solve the system of linear equations, Ax = b. Poisson’s equation is then coupled to the HDM equations as a subroutine. The coupling is carried out as follows. First, the hydrodynamic equations are solved to get the updated value for carrier density n . The updated carrier density is then plugged into Poisson’s equation, resulting in a new system of linear equations. The new system of linear equations is passed to the geneticbased Poisson solver to solve for x , i.e., the updated value of the potential. The potential is differentiated to get the electric filed. Finally, the updated value of the electric field is plugged into the HDM to estimate the updated value of carrier density. This process is repeated until the stopping criterion is satisfied. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 Fig. (3.5) shows the distance from the optimal solution versus number of generations for different values of MU. The mutation value of any gene (variable) is proposed as follows. Childnew(i) = Childold(i) + M U ■RND (3 .9 ) Where MU is the mutation factor and RND is a random number between -1 and 1. Fig. (3.5) shows that the best result is obtained when the mutation value is dependent on the fitness. The reason is that as the value of the fitness increases, which indicates being very close to the optimal solution, the mutation factor decreases. In this manner, the mutation value is changed in the correct way for a faster convergence. Moreover, introducing a random feature along with the dependence of MU on the fitness does not enhance the convergence. A general conclusion is that smaller values of MU are observed to have better convergence curves. Figures (3.6) and (3.7) illustrate the effect of the number of crossover points and crossover width, respectively. From Fig. (3.6), it can be concluded that larger number of crossover points is the right choice for better accuracy along with higher speed of convergence. Moreover, choosing the number of crossover points to change randomly within simulation does not improve the algorithm. On the other hand, considering Fig. (3.7), it is apparent that rate of convergence of the genetic algorithm is independent of the crossover width value. Fig. (3.8) shows the effect of the number of mutated elements on the convergence and accuracy of the proposed algorithm. It can be observed that as the number of the mutated variables decreases, the convergence and accuracy of the algorithm are Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 improved. The best curve is obtained for a number of mutated variables equals to one percent. This complies with nature, since biological mutation hits only a very small number of genes. Moreover, changing the number of the mutated variables randomly throughout the simulation introduces a reasonable improvement over the one percent mutation case. Fig. (3.9) demonstrates how the choice of parent selection method affects the algorithm performance. It can be pointed out that roulette wheel selection has the best performance. Furthermore, employing a hybrid technique does not improve the algorithm. For instance, error reaches 10”3 in almost 500 generations when roulette wheel selection is employed, whereas 1200 generations are needed for tournament selection to reach the same value of error. It is worth mentioning here that roulette wheel selection inherently uses some sort of elitism. Employing elitism may or may not be useful depending on the problem under consideration. The main reason for roulette wheel selection producing better results is the choice of the objective function given by (3.8). It is important to mention that tournament selection is known to produce better results over the roulette wheel method. However, this is not general, and the numerical example provided in this chapter emphasizes that. Fig. (3.10) shows the effect of the probability of mutation. It should be noticed that increasing the probability o f mutation has a positive effect on the performance o f the algorithm, and the mutation probability is not crucial to the algorithm as long as it is relatively high. Fig. (3.11) shows the effect of the population size on the algorithm convergence. This figure emphasizes that the population size is not a critical parameter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 The reason is that the proposed objective function given by (3.8) inherently allows elitism. This makes the proposed genetic algorithm independent of population size. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 10 — ■ MU MU — MU MU 10'1 = = = = 0.9 0.1 RND*(1-FIT) 0.1*(1-FIT) 10'2 10" 10' -5 10' -7 0 500 1000 1500 2000 2500 3000 3500 number of generations Fig. 3.5. Distance from the optimal solution versus number of generations for different mutation values. — NC = NC = NC = NC = 1 3 10 5*RND o 10 500 1000 1500 2000 number of oenerations 2500 3000 3500 Fig. 3.6. Distance from the optimal solution versus number of generations for different number of crossover points. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 — — CW = 1 CW = 10 CW = 30 o 10' 10 500 1000 1500 2000 number of generations 2500 3000 3500 Fig. 3.7. Distance from the optimal solution versus number of generations for different values o f crossover widths. NM = 1 — NM = 3 — NM = 10 — NM = RND*5 o 10' 500 1000 1500 2000 number of generations 2500 3000 3500 Fig. 3.8. Distance from the optimal solution versus number of generations for different numbers of mutated elements. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 SC = 1, random group selection SC = 2, roulette wheel selection SC = INT (1.5*RND+1), hybrid o 10' 500 1000 1500 2000 number of generations 2500 3000 3500 Fig. 3.9. Distance from the optimal solution versus number of generations for different selection criteria. N POP N POP N POP — - N POP = = = = 10 30 70 100 10 o 10' 500 1000 1500 2000 number of generations 2500 3000 3500 Fig. 3.10. Distance from the optimal solution versus number of generations for different population sizes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 76 — PM = 0.1 PM = 0.5 PM = 0.8 o 10' 500 1000 1500 2000 number of generations 2500 3000 3500 Fig. 3.11. Distance from the optimal solution versus number of generations for different population sizes. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 3.5.1 DC Simulation Results To demonstrate the potential of the proposed approach, it is applied to an idealized MESFET structure, which is discretized by a mesh of 32 Ar by 32 Ay with At = 0.001 p s. Forward Euler is adopted as an explicit finite difference method. In addition, upwinding is employed to have a stable scheme. The space step sizes are adjusted to satisfy Debye length, while the time step value At is chosen to satisfy the Courant-Friedrichs-Levy (CFL) condition. While, Poisson’s equation is solved using the proposed algorithm. Fig. (3.12.a) shows the potential distribution obtained using the proposed algorithm. This graph demonstrates that boundary conditions are satisfied at the electrodes. For instance, the value of the potential at the gate equals to -1.3 volts, which is the applied DC voltage minus the Schottky barrier height. While, Fig. (3.12.b) shows the carrier density distribution. It is significant to indicate that the proposed algorithm gives precisely the same results obtained when SOR algorithm is employed. The comparison results between the algorithms are not provided because their results coincide exactly on each other. It is noteworthy to say that the purpose of this section is to show that genetic algorithms can be applied to solve real value problems having a large number of unknowns, with a very high degree of accuracy. The speed of convergence is not an objective at this stage o f the work. Ultimately, the developed genetic algorithm needs to be applied to equations that have stability constraints in order to have unconditionally stable algorithm, or to solve problems that traditional optimization techniques cannot solve (problems with multiple local minima). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 x 10 2.5 2(/>- 1.5 c b J<D a; 'E S 0.5 •0.5 0 0 X 2.5 |<o c 4) o Q. 1.5 0.5 •0.5 Fig. 3.12. Sample DC results obtained using the proposed algorithms, (a) Potential distribution, (b) Carrier density distribution. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 3.5.2 AC Simulation Results The AC excitation applied to the gate electrode is given as: v ss( 0 = Vgs0 + A v gssin (a )t) (3.10) where Vgs is the DC bias applied to the gate electrode, Augs is the peak value of the AC signal (0.2 volts), and co is the frequency of the applied signal in rad. I sec. The frequency used in the simulation is 60 GHz. First, the DC distribution is obtained by solving Poisson’s equation, using the proposed algorithm in conjunction with the three hydrodynamic conservation equations. Then, a new value of Vgs is calculated using (3.10). The new value of Vgs is used to update Poisson’s equation to get the new potential distribution. The electric field is then estimated and used to update the variables in the conservation equations. This process is repeated every A/ until t = tmax. The current density is obtained using (3.5). The current density calculated on the plan located midway between the drain and gate is integrated to obtain the total current. The output voltage is estimated by multiplying the total current by the resistance that defines the DC operating point (Q point) of the transistor. Fig. (3.13) shows the AC gate and drain voltages. A maximum gain of 9 dB is achieved. Moreover, it is observed that there is an output delay of about 1p s that represents the time required for the transistor to respond to the input signal. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 D.e - - vds vgs 0.6 D.4 0.2 s E 05B “ 0n "5 > - 0.2 -0.4 - 0.6 -0.0 0 5 10 15 20 25 30 35 Tim e (ps) Fig. 3.13. AC gate and drain voltages obtained using the proposed algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 81 3.6 Summary In this chapter, a new technique is developed for solving the hydrodynamic model (HDM) in conjunction with Poisson’s equation, using an adaptive real-coded genetic algorithm. Several GA design parameters have been studied to illustrate their effects on the algorithm convergence. The novelty of the proposed technique comes from the genetic algorithm itself. This has been achieved by developing a very efficient objective function, along with introducing completely new concepts such as fitness-dependent GA parameters. Moreover, the problem this chapter presents is a new application of genetic algorithms. In addition, the proposed genetic algorithm should outperform standard methods for several types of problems. For instance, finding the global solution of optimization problems having multiple local minima, and problems where matrices have large Condition Numbers. This study also represents a fundamental step toward applying GAs to Maxwell’s equations in conjunction with the HDM, aiming to develop an optimized and unconditionally stable global-modeling simulator. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 A New Wavelet-Based Time-Domain Technique For Modeling and Optimization Of High-Frequency Active Devices 4.1 Introduction In this chapter, a new time-domain simulation approach for large-signal physical modeling o f high frequency semiconductor devices, using wavelets, is presented. The proposed approach solves the complete hydrodynamic model and Maxwell’s equations on nonuninform multiresolution adaptive grids. The nonuniform grids are obtained by applying wavelet transforms followed by hard thresholding. This allows forming fine and coarse grids in locations where variable solutions change rapidly and slowly, respectively. A general criterion is mathematically defined for grid updating within the simulation. In addition, an efficient thresholding formula is proposed and verified. The developed technique is validated by simulating a submicrometer FET. Different numerical examples are presented along with illustrative comparison graphs, showing more than 75% reduction in CPU time, while maintaining the same degree of accuracy achieved using a uniform grid case. Tradeoffs between threshold values, CPU-time, and accuracy are discussed. Global modeling is a tremendous task that involves advanced numerical techniques and different algorithms. As a result, it is computationally expensive [21-49]. Therefore, there is an urgent need to present a new approach to reduce the simulation time, while maintaining the same degree of accuracy presented by global modeling techniques. One approach is to adaptively refine grids in locations where the unknown Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 83 variables vary rapidly. Such technique is called multiresolution time domain (MRTD), and a very attractive way to implement it is to use wavelets [122]-[123]. The MRTD approach has been successfully applied to fmite-difference timedomain (FDTD) simulations of passive structures [124]-[134]. However, for the active devices, that are characterized by a set of coupled and highly nonlinear partial differential equations, applying the same approach would become quite time consuming [135]. Several different approaches for solving partial differential equations (PDE’s) using wavelets have been considered. It has been observed by several authors that nonlinear operators such as multiplication are too computationally expensive when conducted directly on a wavelet basis. One of the approaches for solving PDE’s is the Interpolating Wavelets technique presented in [136], in which the nonlinearities are dealt with using the so-called sparse point representation (SPR). Interpolating Wavelets have been successfully applied to the simple drift diffusion active device model [137]-[139]. Being primarily developed for long-gate devices, the drift diffusion model leads to inaccurate estimations of device internal distributions and microwave characteristics for submicrometer devices [54], It is worth mentioning that in [137], the author proposed a new technique to solve simple forms of Hyperbolic PDE’s using an Interpolating Wavelet scheme. These PDEs can represent Maxwell’s equations or the simple drift-diffusion model but not the complete hydrodynamic model. Thus, a new approach to apply wavelets to the hydrodynamic model PDEs is needed, along with extending it to Maxwell’s Equations, for accurate modeling of submicrometer devices, while achieving a CPU-time reduction. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 In this chapter, a unified approach to apply wavelets to the full hydrodynamic model and Maxwell’s equations is developed. The main idea is to take snapshots of the solution during the simulation, and apply wavelet transform to the current solution to obtain the coefficients of the details. The coefficients of the details are then normalized, and a threshold is applied to obtain a nonuniform grid. Two independent grid-updating criteria are developed for the active and passive parts of the problem. Moreover, a threshold formula that is dependent on the variable solution at any given time has been developed and verified. A comprehensive set of results is included along with illustrative comparison graphs. 4.2 Fundamentals of the MRTD The construction o f biorthogonal wavelet bases relies on the notation of multiresolution analysis [140]. This notation gives a formal description of the intuitive idea that every signal can be constructed by a successive refinement, by iteratively adding details to an approximation. The coefficients of the approximations are given by: +00 a x[n ,m ]= $ x(t)<pnm(t)d t (4 ^ -oo where <pnm (t) is the family of dilates and translates of the scaling function formed as (pnm{t) = 2 m,1(p{2mt - n ) (4.2) On the other hand, (4.3) gives the coefficients of the details. +00 d x[n ,m ]= \ x ( t) y / nm(t)d t (4.3) —oo where y/nm (t) is the family of dilates and translates of the wavelet function defined as: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 ( / „ ( 0 = 2’ ' V ( 2 ’ < - « ) (4'4) While some wavelets such as Daubechies are asymmetrical [140], it is possible to create symmetric wavelets with compact support by using two sets of wavelets, one to compose the signal and the other to construct it. Such wavelets are called Biorthogonal [141]. Complete coverage of the theory and applications of wavelets in electromagnetic and device modeling can be found in the first book on the subject by George Pan [142]. 4.2.1 Two Dimensional MRTD Scheme For simplicity, the authors in [124] proposed a two-dimensional scheme for a homogeneous lossless medium with the permittivity e and permeability p. Assuming no variation along the y-direction, the Maxwell’s equations for the 2D TMZ mode were written as: dEx 1 dH s r =- s ^ r SHy l dt p dEz dEx ) - r - T 1 ox dz I dEz 1 dH - r - =— dt e dx (4-5) (4-6) (4. 7) To derive the 2D MRTD scheme, the electric and magnetic field components incorporated in these equations were expanded in a series of scaling and wavelet functions in both x- and z- directions and in pulse functions in time [125]: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 Ex(x ,z,t) = +00 £ (*¥.(*) 2 k,l,m =-oo +QO +00 2rz - 1 £,/,m =-oo r2=0 p z =0 +oo +oo 2r* —1 + Z E L £ /-w A ( o^ 2( ^ , ( z) k ,l,m = -oo ^ = 0 p x=0 +oo +oo (4.8) 2 ^ /z - l +£ £ £ k,l,m=-x> rx ,rz =0 p x ,p z =0 Ez(x ,z,t) = +oo k ,l,m = - oo +00 + +00 2rz- l I k j , m = - oo r2=0 p z =0 +oo +oo 2 ^ -1 Z Z k K ^ T 2 K ( 0 ^ X(X)<f>m-U2(z) + Z k,l,m=-<x> rx =0 p x =0 +oo + Z +oo ^ 2'*,/z —1 Z Z k ^ : ^ K { t ) w ^ PS x ) y 'tm .P,^ ') k j , m =-oo ^ , ^ = 0 p x ,p z =0 H y(x ,z,t) = +00 k+inHi'-lf2> m-l/2K+l/2(t)$l-l/2(X)< />m-\n(Z) Z k ,l,m = - oo +Q0 +oo 2 ^ -1 +£ I £ k ,l,m = - oo f2=0 p z=0 +oo + Z +oo 2j* - 1 Z Z * + l / 2^ / - ^ A +l/2(0 ^ /2W ^ ,-l/ 2(^) k ,l,m =-oo rx=0 p x=0 +co +oo 2r* ,fz -1 +£ £ £ k ,l,m =-oo r*,r2=0 p x ,p z =0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.10) 87 Where cp„ (x) = cp((x / Ax) - n) and \yrn (x) = 2 r/2 v|/0 (2r [(x / Ax) - « ] - / ? ) represent the Battle-Lamerie scaling function and r -resolution wavelet function, respectively. Since higher resolutions of wavelets are shifted and dilated version of the zero resolution, their domain will be a fraction of that of the zero-resolution wavelet. Thus there is going to be more than one higher resolution coefficient for each MRTD cell. Specially, for the arbitrary r resolution for the n cell to the x -direction, there exists 2r wavelet coefficients located at x / Ax = n + (p / 2r+l), p = 0,.... ,2r -1 . This is the reason for the summation of the p terms for each resolution r in the above expansion. and k+l with k = x ,y ,z and p,v = <j),v)/ are the coefficients for the field expansions in terms of scaling and wavelet functions. Schemes based on a displacement of H and E by (I!2 )(rx'rz)+1 instead of 54 could provide slightly improved numerical dispersion and stability characteristics. However, for simplicity reasons, the authors in [125] used Yee’s convention for the derivation of the their MRTD equations. The indices /, m, k are the discrete space and time indexes related to the space and time coordinates. For an accuracy of 0.1%, the above summations are truncated to a finite number of terms determined by the dispersion and stability requirements (typically their maximum value is between 22-26). The time domain expansion function hk (t) is defined as: = — *) At with the rectangular pulse function: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.11) The magnetic-field components are shifted by half a discretization interval in the space and time domains with respect to the electric field components (leap-frog). Upon inserting the field expansions, Maxwell’s equations are sampled using pulse functions as time-domain test functions and scaling/wavelet functions as space-domain test functions. For the sampling in the time domain, the following integrals were employed [124]: +00 (4.13) -00 and (4.14) -0 0 where 5k is the Kroenecker symbol 4.2.2 1 fo r k = k 0 fo r k ^ k (4.15) Battle-Lemarie Expansion Basis Sampling in the space domain is obtained by use of orthogonal relationships for the Battle-Lemarie scaling and wavelet functions [123]: + 00 (4.16) —00 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 89 +00 J <Vm(.x W m \P'(x ) dx = W r , p and (4.17) +00 J<P m‘,p'(x >prm ,p(x ) dx = ^r,r^m,m^p,p'A x (4.18) -00 The integrals containing derivatives can be approximated by the following expressions: +•00 jV * (x) d(Pm' ^ 1 ~X) dx K Y s t - n (4-19) -CO with oo A a ( 0 = -^ Jl <KI) l2^ sin[ W + 1 / 2 ) ] ^ (4.20) and + l\k / \ d§m'+H2,p(x ) , V ’ "''-''2 < L 0 ) ------------------ d x * y j / ■ xs d r ^ P ^ m+i,m' (4.2 1) with 00 d r ( i , p ) = - f 2 - ’-/ 2 $ „ ( 5 ) J 0 ( 5 / 2 ' - ) s i n [ « i + 0 . 5 + P / 2 ' + l / 2 ' ' * 1) ] ^ n J (4.22) and +00 Jf d $m'+ l / l ( * ax ) e r O ,P )&M ^ J~Ul=~nc,rl1 <4-23> — 00 with 00 cr ( i , p ) = - f 2 - W2 i „ ( O v 0 ( V 2 ' ' ) s i n [ « ( + 0 . 5 + p / 2 r + l / 2 " 1)]a'4 (4.24) 71 J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 90 and too 1 rl ( \ ^ + 1 /2 ^ 2 ( x ) , V m , p l ( X ) --------------- ~ -----------d x dx b,r\,r2,2 m+i , m rib , r l , r 2 , l (4.25) with br\,r2(i’P l ’P T >= 00 - f|vj)0 ( ^ / 2 ' ' 1) 11 v|;0 (^ /2 '" 2) | ^sin [^ (i + 0.5 + p 2 ! 2 rl - p H 2 rX + l / 2 r2+1 - l / 2 /’1+1)]c/^ 71 J 0 (4.26) For the remainder of this section, the authors in [124] considered only an expansion as a series of scaling and zero-resolution wavelet functions. Hints for the enhancement of additional wavelet resolutions will be presented where needed. Since for the zero resolution ( r = 0 ), there is only one wavelet coefficient per cell ( p = 0 ), the p symbols will be omitted from the definition of the b,c,d coefficients. The coefficients a{i), b0(/), c 0 (/) are given in Table (4.1) [125]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 TABLE 4.1 COEFFICIENTS a(i),b0(i),c0(i) 1: «(*) 0 L m n m u 7 m m m m 2.47253977327429 0. I -0.153878843323872 0.9502282774123074 -4.659725793402785E-O2 2 S.96O630332468729OB-02 0.1060591000788887 M s a m o s is s s m i & o a 8 -2 9291577898008906*02 9.392437777679437B-02 *3.69t9&774O974982B-02 3.14M 4447S216036& 03 2 .05744M I98775452E-02 4 5 -8.1844023252837126-03 1.3493860087091086*02 •ia isa o 3 ia o 8 0 4 9 6 7 & 4 n S 4Jf57585fi52354830& 03 .2.8S89418100947S2B4J3 SJ7O 877725270031E-03 I -2.3423053566494616-03 2.7780805141155296-03 -3.202621363952005 B-03 8 1,182877717O42020E-03 4 .m M 6 t§ 7 W S S 6 E - « 1.71408684956O89OE-03 0 -6.716635O68590737&O4 7J718O 730937770JE-04 -0.1765O84384& 4196604 10 3.583500907489797& 04 4.4M 267305845043B-O 4 4.9117M 74807M 8B 4M 11 '1 .9 3 1 3 2 M 8 4 7 U 7 N E -M 1.9527114191049006-04 -2.629253O13538502E-04 12 1.0198277670578696-04 -1.021304423384722E-04 1.407386855875626E-04 13 *5.0139431835184346*05 5.531259273804269E-05 -7.5338406895730006*05 14 2M 4 m m m m m E rm -2.94733O468OME31E-05 4.0331462310996741-05 In [124], the stability limit for the 2D MRTD scheme based only on the scaling functions’ expansion (S-MRTD) is given by: At < -----------------cH " , C l a i n l (4.27) (Ax)2 (Ay)2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 92 where Ax and Ay are the cell dimensions and c = —} = is the velocity of the light in the yjjuz modeled medium. For Ax = Ay = A , the above stability criterion gives: A‘ur,m £ (4.28) cV 2 £ M K i ' ) l It is known [60] that A' fdtd ~ j~ t------------ — (4.29) Cf ( A x F + (Av)r For Ax = Ay = A , the above FDTD stability criterion reduces to ^ fd td — ;V2 (4.30) The above equations show that, for the same discretization size, the upper bounds of the time steps of FDTD and S-MRTD are comparable and related through some factor. The stability analysis can be generalized easily to three-dimensional case [124]. It is very clear from the above analysis that the proposed MRTD technique is not simple to derive and implement. Moreover, the author applied the technique to a very simple problem. This means that more complex derivations are needed for practical problems. Accordingly, general approaches should be developed in order to remove the redundancies of the original formulations. This can be implemented by solving the original PDEs without modifications on nonuniform grids obtained using wavelets. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 4.3 Problem Description The transistor model used in this work is a 3D full-hydrodynamic large-signal electromagnetic-physical model. The active device model is based on the moments of the Boltzmann’s Transport Equation obtained by integrating over the momentum space. The integration results in a strongly coupled highly nonlinear set of partial differential equations, called the conservation equations. These equations provide a time-dependent self-consistent solution for carrier density, carrier energy, and carrier momentum, which are given as follows. current continuity f + V'<ro) = 0' (4.31) ■ energy conservation - + qnv.E + V.(ffl>(s + K bT) ) = dt x-momentum conservation Ot dx ~ Sq) x8(e) = (4.32) (4.33) In the above equations, n is the electron concentration, u is the electron velocity, E is the electric field, e is the electron energy, e0 is the equilibrium thermal energy and p is the electron momentum. The energy and momentum relaxation times are given by ze and r mj respectively. Similar expression is obtained for the y-direction Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 94 momentum. The three conservation equations are solved in conjunction with Maxwell’s equations: (4.34) W xH = $ £-+ J dt (4.35) where E is the electric field, H is the magnetic field, D is the electric flux density, and B is the magnetic flux density. The fields in Maxwell’s Equations are updated using the current density /estim ated by Eq. (10). J(t) = -qnu(t). (4.36) The above model accurately describes all the non-stationary transport effects by incorporating energy dependence into all the transport parameters such as effective mass and relaxation times, along with including EM-wave effects. Fig. (4.1) shows the crosssection of the simulated structure with parameters summarized in Table (4.2). 05 pm 1 Source □3 pm Gate Drain DopedGaAs (2k10' ' cm’') t y □S pm UndopedGaAs 21 pm Fig. 4.1. Cross-section of the simulated transistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 95 TABLE 4.2 TRANSISTOR PARAMETERS USED IN THE SIMULATION Drain and source contacts 0.5 pm Gate-source separation 0.5 pm Gate-drain separation 1.0 pm Device thickness 0.8 pm Device length 2.8 pm Gate length 0.3 pm Device Width 250 pm Active layer thickness 0.2 pm Active layer doping 2 x 10 1 cm' Schottky barrier height 0.8 V DC gate-source voltage -0.5 V DC drain-source voltage 3.0 V 4.4 The Proposed Algorithm Fig. 2 shows the flow chart of the proposed algorithm. A uniform grid is defined at the beginning of the simulation. Equations 5 through 9 are then solved in the sequence shown in the flow chart to update the grid of the different variables at the new iteration with the following criterion. 1 I C r — x I max,min max,min xm l a x , m in ■ £ Hydro The updating criterion checks if the solution of the variable x has changed by 8 since last iteration using wavelet transform. The subscripts c and / designate quantities defined in the current time and last time where wavelet transform is performed, respectively. The subscript “max, min” indicates that the maximum and minimum of the variable x are checked with Relation (4.37) at the same time. It is worth mentioning here Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 that boundary grid-points are not included for the maximum or minimum checking. The value o f S used in the simulation is 0.1. If (4.37) is satisfied, wavelet transform is performed on the current variable solution followed by thresholding to obtain a new nonuniform grid for the variable x. Biorthogonal wavelets are used with notation BI03.1 to point out three vanishing moments for the mother wavelet and only one vanishing moment for the scaling function. The nonuniform grids of the different variables are then combined into only one nonuniform grid for the next iteration. The above steps are repeated until the stopping criterion is satisfied. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 Initialization Define a Uniform Grid (Nx by Ny) Evaluate Fields Solve Momentum Eq. Solve Energy Eq. Update Transport Parameters Solve Continuity Eq. No Perform Wavelet Transform (BIO.3.1) on the Current Solution of the Variable x and Obtain the Coefficients of the Details. Normalize the Coefficients of the Details. Calculate the Threshold Value. Remove Grid Points of Values Less Than the Threshold. Combine Variable Grids into One Nonuniform Grid No Stopping Criterion Satisfied? Yes ( ^ ^ S to p ~ ~ ^ ) Fig. 4.2. Generic flowchart of the proposed algorithm. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 98 It should be noted that magnitude ranges of the variables used in the simulations vary dramatically. For instance, carrier density per cm'3, is on the order of 1017, while, energy expressed in eV, is on the order of 0.5. Accordingly, the threshold value should be dependent on the variable solution at any given iteration. The proposed threshold formula is given by (4.38). T = I jl N I t * (4.38) /=i In this equation, T0 is the initial threshold value, d? s are the coefficients of the details, and A is the number of grid points in the x- or y-direction. Hence, the value of the threshold T depends mainly on the variable solution at any given time rather than being fixed. The values of T0 used in the simulation are 0.001, 0.01, and 0.05, respectively. In this paper, a new technique to conceive the nonuniform grids using wavelets has been developed. The main idea is to apply wavelet transform to the variable solution at any given time to obtain the coefficients of the details, which are then normalized to its maximum. Only grid points where the value of the normalized coefficients of the details are larger than the threshold value given by Eq. (4.38) are included. Figures 4.3 and 4.4 illustrate different examples of the technique employed to obtain the nonuniform grids for electron energy and x-momentum solutions at a specific cross-section. Fig. (4.3) exemplifies how the proposed algorithm obtains the nonuniform grid using transverse compression only. For instance, Fig. (4.3.a), shows the normalized amplitude of the coefficients of the details for the electron energy. While, Fig. (4.3.b.) marks the grid Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 points remaining after thresholding the normalized coefficients of the details using (4.38). It is observed that the proposed technique accurately removes grid points in the locations where variable solutions change very slowly. Fig. (4.4) shows the method adopted to obtain the nonuniform grid using longitudinal compression only. Considering Figures (4.3) and (4.4), one can conclude that the compression in the longitudinal cross-sections is much more than that in the transverse cross-sections. This is consistent with the fact that the physical changes in the longitudinal cross-sections are much slower compared to those in the transverse crosssections. Fig. (4.5) shows the procedure employed to obtain the nonuniform grid of the electron energy. The process is achieved by obtaining two separate grids for the transverse and longitudinal compressions, respectively. Then the two grids are combined together using logical ‘AND’ to conceive the overall grid for the electron energy at this given time. The same process is conducted for the other variables including xmomentum, y-momentum, carrier density, and potential whenever (4.37) is satisfied. The separate grids of our variables are then combined using logical ‘OR’ to obtain the overall grid for the next iterations. The overall grid obtained needs further processing in order to define a finitedifference scheme on it. The simplest way to achieve that is to have the same number of grid points for the parallel cross-sections, while the number of grid points in the longitudinal cross-sections and the transverse cross-sections need not to be the same. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 Following the above procedure, it was found that boundary conditions implementation, including Ohmic and Schottky contacts, does not need special treatment. They can be treated similar to the standard finite-difference (FD) scheme. The only issue the algorithm needs to keep track of is identifying the new boundaries of the metallic contacts for each new grid, which is straightforward. 0.8 0.8 0.6 0.6 0.2 0.2 Normalized A m p litu d e 0.4 0.4 =5 - 0.2 z -0.4 - 0.2 - 0.6 -0.4 - 0.6 - 0.8 10 20 30 40 50 80 10 20 30 Grid P aints G rid P oints (a) (c) 40 50 40 50 80 0.16 0.14 0.12 Electron Energy (ev) V 4 0.08 0.06 0.04 0.02 10 20 30 40 Grid P ointi (b) 50 80 10 20 30 Grid Points (d) Fig. 4.3. (a) Normalized details coefficients for the electron energy at a certain transverse cross-section, (b) Grid points marked on the actual curve for the electron energy at the same transverse cross-section, (c) Normalized details coefficients for the x-momentum at a certain transverse cross-section, (d) Grid points marked on the actual curve for xmomentum at the same transverse cross-section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Normalized Amplitude 101 0) "O 3 Q. E < 73 Q N) "E!5 Z a -0.5 10 20 Grid Points 30 40 50 60 40 50 60 Grid Points (c) x 10 0.3 Electron Energy (ev) 0.25 0.2 H 10 0.15 0.05 10 20 30 40 Grid Points (b) 50 60 10 20 30 Grid Points id ) Fig. 4.4. (a) Normalized details coefficients for the electron energy at a certain longitudinal cross-section, (b) Grid points marked on the actual curve for the electron energy at the same longitudinal cross-section, (c) Normalized details coefficients for the x-momentum at a certain longitudinal cross-section, (d) Grid points marked on the actual curve for x-momentum at the same longitudinal cross-section. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 102 Table (4.3) shows the evolution of the nonuniform grids. It can be observed that the number of grid points for the overall grid increases as time advances. The reason is that at the beginning of the simulation the solution is not completely formed yet. As the time marches, more grid points are needed to incorporate the changes in the solution. Furthermore, the different variable grids should not be updated at the same rate. For instance, it is apparent that the potential needs not to be updated at the same rate as the other variables. Notice that Table (4.3) is used for illustration purposes to demonstrate the way the different variable grids change. In the actual simulation, the potential grid is updated a few times at the beginning of the simulation, and then it remains unchanged. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 103 TABLE 4.3 GRID ADAPTABILITY OF THE DIFFERENT VARIABLES FOR T0 = 1% Variable Potential Carrier Density Energy x-Momentum y -Momentum All Variables Potential Carrier Density Energy x-Momentum y -Momentum All Variables Potential Carrier Density Energy x-Momentum y -Momentum All Variables Potential Carrier Density Energy x-Momentum y -Momentum All Variables Potential Carrier Density Energy x-Momentum y -Momentum All Variables Unknowns Unknowns Remaining After Remaining After Longitudinal Transverse Compression Compression (%) (%) Time Iteration # 1 2 0 7.74 5.69 14.92 6.54 17.63 39.65 16.06 43.39 17.53 16.11 22.36 65.14 Time Iteration # 250 8.59 5.88 16.70 13.69 39.21 23.00 19.09 43.65 20.02 19.46 61.94 28.93 Time Iteration # 480 6.27 9.23 21.51 17.16 43.99 28.88 23.44 38.57 26.20 26.76 58.84 36.25 Time Iteration # 5 9 0 6.04 9.64 29.88 18.24 48.85 31.88 41.91 29.08 37.04 32.91 62.36 44.73 Time Iteration # 730 7.01 11.13 34.08 16.55 39.77 35.64 41.91 27.98 51.46 34.84 62.84 58.96 Total Unknowns Remaining (%) 0.63 2.64 8.54 9.18 3.78 14.43 0.76 5.18 12.26 10.64 7.95 20.58 0.90 7.71 15.72 12.72 13.53 25.05 0.93 10.61 18.43 16.21 20.36 31.74 1.15 8.42 18.43 13.32 25.00 36.43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 104 Now, we turn our attention to Maxwell’s Equations. The passive part of the FET represents a co-planar structure in which a 3D FDTD is developed to solve for the electric and magnetic fields. The current density estimated from the active device conversation equations is used to update the variables in Maxwell’s Equations. It is importance to state that the same approach developed to obtain the nonuniform grid for the variables of the conversation equations is applied to Maxwell’s Equations as well. However, a different updating mechanism should be developed to keep track of the wave propagation within the passive structure. The following is the algorithm developed for the grid updating of FDTD simulations. Step 1: Construct a 3D matrix M that has only 0’s and l ’s, based whether or not we have a non-zero solution of the field at this location. For example, “ 1” is assigned if a non-zero field solution exists, and “0 ” elsewhere. Step 2: Estimate the value of dFDTD (FDTD grid-updating factor) as: y (v m new ® m old''i,j,k ..).., jLmd I i * _________ (4.39) FDTD Where M newand M old are the matrices constructed using step one for the current, and old solutions of the fields, respectively. N xd, N yd, and N zd are the number of grid points in x , y , and z directions, respectively. Step 3: Check c ^ ^ ’s value against a predefined value, for example 5%. Step 4: If satisfied, move the grid to z = z + d z . Where dz is proportional to 4FDTD- Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 105 Step 5: t = t + dt Fig. (4.5) illustrates examples of how the nonuniform grids are obtained for the magnetic and electric fields at a specific cross-section for FDTD simulations. For instance, Fig. (4.5.a) shows the normalized amplitude of the coefficients of the details for the electric field. Fig. (4.5.b) marks the grid points remaining after thresholding the normalized coefficients of the details using (4.38). It should be observed that the proposed technique accurately removes the grid points in locations where variable solutions change very slowly. This would have an effect of reducing the CPU-time by removing the redundant grid-points introduced by the original formulation. The total grid of the electric field can be achieved by obtaining two separate grids for the transverse and longitudinal compressions, respectively. Then the two grids are combined together using logical ‘AND’ to conceive the overall grid for the electric field at this given time. It is worth mentioning here that the excitation wave exists at the source plan at all times, and the technique proposed here is generic that can be applied also to a short pulse propagating in the computational domain. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 106 0.5 0.5 04 03 02 % 0:3 0I 1 02 0 -0.3 too 120 d istan ce mtwms «f distance, in term s of grid points (a) (c) xtCF 1,005 f 2, 4]015 -002 E -0.03 -0.04 a 40 si ao too distancemtermsofgridpoints (b) 120 20 40 60 80 distance in ttrm s of grid points 100 120 (d) Fig. 4.5. Demonstration of the procedure employed to obtain the nonuniform grid for the y-direction electric and magnetic fields for FDTD simulations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 107 4.5 Results And Discussions 4.5.1 Hydrodynamic Model DC Simulation Results The approach presented in this chapter is general and it can be applied to any unipolar transistor. To demonstrate the potential of this approach, it is applied to an idealized MESFET structure, which is discretized by a mesh of 64 Ar by 64 Ay with At = 0.001 p s . Forward Euler is adopted as an explicit fmite-difference method. In addition, Upwinding is employed to have a stable fmite-difference scheme. The space step sizes are adjusted to satisfy Debye length, while the time step value At is chosen to satisfy the Courant-Friedrichs-Levy (CFL) condition. First, DC simulations are performed following the flow chart given by Fig. (4.2), and the current density is calculated using (4.36). DC excitation is performed by forcing the potential to be equal to the applied voltages to the electrodes (i.e., Dirichlet boundary conditions). Fig. (4.6) shows the percentage remaining number of unknowns or grid points versus the time-iteration number for different initial threshold values. Notice that as the threshold value increases, the remaining number of grid points decreases. At the end of the simulation and for an initial threshold value of 0 . 1%, the remaining number of unknowns is almost 70%, whereas for an initial threshold value of 1%, the remaining number o f unknowns is about 30%. The remaining number of unknowns is very sensitive to the initial threshold value in a way that small changes in T0 results in considerable changes in the remaining number of unknowns. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 108 100 -V- To=0.001 To=0.01 T d= 0 .0 5 to c so c c -> 01 -Q E 3 z c c CO E 0! □e Iteration N u m b er Fig. 4.6. Remaining number of unknowns as a percentage versus the iteration number for different initial threshold values. Furthermore, it is observed that the remaining number of unknowns change during the simulation and this is associated with the grid adaptability used in the simulation. In the next section, we will study the effect of the initial threshold value on the final result accuracy as well the trade off between accuracy and CPU-time. It is important to note that a suitable approach to investigate the capabilities of the proposed technique is to compare it to the uniform-grid algorithm. In this case, the new simulator will be accurately evaluated. Since both algorithms, the wavelet-based and the uniform one will run on the same computer. In addition, both algorithms will have the same discretization schemes and the exact semiconductor parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 109 Fig. (4.7) shows the drain current convergence curves versus the CPU-time in seconds for the cases of the uniform grid and the proposed wavelet-based adaptive grids with different initial threshold values T0. Fig. (4.7) demonstrates that using the proposed wavelet-based grids approach reduces the CPU-time dramatically. For instance, there is a reduction of about 75% in CPU-time over the uniform grid case for the initial threshold value of 1%, while the DC drain current error is within 1%. In addition, increasing the initial threshold beyond certain value has a negative effect on the accuracy of the final results. - 0.1 - 0.2 -0.3 2-04 c -O.i -0.7 - 0.8 -0.9 — 0 20 40 60 Uniform Grid W avelet-Based Adaptive Grids With To=0.1% W avelet-Based Adaptive Grids With To=1% . W avelet-Based Adaptive Grids With To=5% 80 100 CPU-Time (Seconds) 120 140 160 180 Fig. 4.7. DC drain current convergence curves for the uniform grid and the proposed wavelet-based non-uniform grids for different initial threshold values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 110 This is apparent for T0 equals to 5%, where there is no agreement between the results achieved using the uniform grid case and the wavelet-based nonuniform grids. The reason is that using large values of T0 implies that more grid points are removed, including important grid points that will have a negative effect on the final result. On the other hand, using a very small threshold values implies redundant grid points. In summary, there should be an optimal value of T0 such that both the CPU-time and error are minimized. In this work, T0of 1% is suggested to have a considerable reduction in CPU-time, while keeping error within an acceptable range. Fig. (4.8) shows the potential distribution obtained using the proposed algorithm with T0 equals to 1%. This graph demonstrates that boundary conditions are satisfied at the electrodes. For instance, the value of potential at the gate is equal to -1.3 volts, which is the applied DC voltage minus the Schottky barrier height. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Ill Fig. 4.8. DC potential distribution obtained by the proposed algorithm using a value of the initial threshold equals to 1%. The values of the elements of the small-signal equivalent-circuit model are computed from the variations in voltages, currents, and charges due to small changes in the DC bias voltages and/or currents. For instance, the small signal gate-to-source capacitance and transconductance are computed as: AQ C g 8s AV gs (4.40) AI (4.41) gs Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 112 where Q o is the charge on the gate electrode and V o*5 is the gate-to-source voltage. The charge Qg is calculated using the integral form of Gauss’s law. An important figure of merit that can be evaluated from these parameters is the device unit-gain cut-off frequency given by (4.42). f _ J t ~ 2nC The values of C gs and g m (4.42) gs are plotted against the applied gate-to-source voltage for both the uniform-grid case and the proposed algorithm. Fig. (4.9) shows the comparison, where the good agreement between the proposed algorithm and the uniformgrid case should be observed. Using the proposed algorithm and for V found to be 263 mS/mm, and C = -0 .5 V , g m is to be 0.47 pF/mm. In this case, the unit-gain cut-off frequency is calculated using (4.42) to be 90 GHz. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 113 4.5.2 Hydrodynamic Model AC Simulation Results The AC excitation applied to the gate electrode is given as: V (t) = V +Au sin(*yf) gs gso gs (4.43) where VgS0 is the DC bias applied to the gate electrode, A ogS is the peak value of the AC signal (0.1 volts), and a is the frequency of the applied signal in rad./sec. The frequency used in the simulation is 60 GHz. AC excitation is implemented in the same manner as in DC excitation. However, values of the gate potential are obtained at the new time t using (4.43). First, the DC solution is obtained by solving Poisson’s equation in conjunction with the three hydrodynamic conservation equations. Then, a new value gate-source voltage is calculated using (4.43). This new value is used to update Poisson’s equation for the new voltage distribution, and consequently the new electric field. The electric field is then used to update the variables in the conservation equations. This process is repeated every A t following the proposed algorithm given in Fig. (4.2), until t - t m a x . The current density is obtained using (4.36). The current density calculated on the plan located midway between the drain and gate is integrated to obtain the total current. The output voltage is estimated by multiplying the total current by the resistance that defines the DC operating point (Q point) of the transistor. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 114 0.5 Uniform Grid Wavelet-Based Grid 0.48 0.46 0.44 0.42 E E u_ & 0.4 (/) O) o 0.38 0.36 0.34 0.32 0.3 - - 0.8 1.6 fal 280 260 240 220 200 160 140 120 - Uniform Grid " Wavelet-Based Grid 100 80 •1.8 - 1.6 -1.4 •1.2 •1 - 0.8 .6 -O .i Vgs (V) fbl Fig. 4.9. Comparison between the uniform-grid and the proposed algorithm with a value of the initial threshold equals to 1% (a) Gate-to-source capacitance versus gate-to-source voltage, (b) Transconductance versus gate-to-source voltage. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 115 Fig. (4.10) shows the output voltage obtained using the proposed algorithm with T0=\% . A gain of 11 dB is achieved. Moreover, it is observed that there is an output delay of about 1 ps that represents the time required for the transistor to respond to the input signal. Fig. (4.11) shows the output voltage for the uniform-grid case and the proposed algorithm with different initial threshold values jf. The purpose of the figure is to emphasize that an optimal value of the threshold should be employed to maintain the required accuracy, while keeping CPU-time as minimum as possible. It is observed that using different values of T0 affects the accuracy of the solution. For instance, using a large value of T0 results in a completely different solution, and this means the scheme for this special case is inaccurate. This is apparent for the case of T0 equals to 5%. Because employing a large value o f T0 results in removing significant grid points, which degrades the final results. Similar to DC simulations, the existence of an optimal value for T0 is suggested. Furthermore, it is noticed that there is no significant difference in terms of accuracy between the two cases of T0= 0 . 1% and T0=\% . The mean relative error obtained for the two cases is in the order of 3 ~ to 4%. This suggests that using T0equals to 1% be the right choice in terms of both accuracy and CPU-time. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 116 0.5 — Vgs ..... Vds 0.4 / 0.3 0.2 s ±2 o > - 0.1 - 0.2 -0.3 -0.4 -0.5 0.5 1.5 Time 2.5 ■it x10' Fig. 4.10. Large-signal result obtained by the proposed algorithm for a value of the initial threshold equals to 1%. — — Uniform Grid W avelet-Based Adaptive Grids To=0.1% W avelet-Based Adaptive Grids To=1.0% . . . . . W avelet-Based Adaptive Grids To=5.0% 0.5 0.4 0.3 0.2 0.1 - 0.2 -0.3 -0.4 -0.5 0 0.5 1 1.5 Time (Seconds) 2 2.5 3 x 1 0 '11 Fig. 4.11. AC output voltage for the uniform grid and the proposed wavelet-based nonuniform grids with different values of the initial threshold. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 117 4.5.3 FDTD Simulation Results A 3D Yee-based FDTD code is developed, with the proposed algorithm employed. A Guassian excitation pulse is applied to evaluate the algorithm over a wide range o f frequencies. Table (4.4) shows that as threshold value increases, CPU-time and error introduced decreases as well. It is noteworthy to point out that using an initial threshold value equals to 10% seems to reduce error along with the CPU-time. However, considering Fig. (4.12), one should conclude that using T0 equals to 10% introduces dispersion, which is a serious type of error. Accordingly, an initial threshold value of 5% is recommended in terms of both CPU-time and error for FDTD simulations. It is important to emphasize that the passive and active parts of the problem have different optimal threshold values. This is expected since the variables in the conservations equations are highly nonlinear compared to the fields obtained when solving Maxwell’s Equations. Research leading to the work presented in this chapter can be found in [2][15]. TABLE 4.4 EFFECT OF THE THRESHOLD VALUE ON ERROR AND CPU-TIME FOR FDTD SIMULATIONS T -Lo CPU-Time (Seconds) 0.0 (Uniform Grid) 0 . 1% 1 .0 % 5.0% 10 .0 % 744.90 300.17 205.92 155.10 111.05 Error 2-norm co-norm 0.0873% 0.0871% 0.0778% 0.0473% 8.80% 8.75% 7.69% 3.66% Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 118 0.8 — — 0.7 Uniform Grid MRTD, To=1% MRTD, To=5% MRTD, To=10% 0.6 0.5 S 0.4 0.2 0.1 - 0.1 0.5 2.5 3.5 time in seconds Fig. 4.12. Potential of the gate at a specific cross-section versus time for the uniform grid case and the proposed MRTD algorithm with different values of T0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 119 4.6 Scheme Errors and Stability Analysis It is important to mention here that the simulation and physical times are completely separate entities. The simulation time required to model a specific physical process should vary depending on the technique implemented in the simulation. The purpose of this section is to demonstrate that the mechanism by which error is introduced when employing the proposed wavelet-based technique is different than of the uniform-grid case. The local truncation error for the uniform grid case is dependent, in general, on the mesh spacing ( Ar and Ay), and the time step used A t . On the other hand, the local truncation error for the wavelet-based nonuniform grids approach depends on how accurately the important grid points are reserved as well as the time step used. This suggests that the local truncation errors, due to spatial discretization, for the uniform grid case and the wavelet-based nonuniform grids are different. The local truncation error accumulates from iteration to iteration. The total truncation or discretization error is thus dependent on the number of iterations used (space and time iterations combined). Accordingly, one can conclude that the total error introduced by the wavelet-based technique due to the local discretization errors accumulating during the simulation, may or may not be larger than that of the uniform grid case, at least for the two cases of T0= 0 . 1% and 7^=1%. The reason is the number of iterations required reaching the steady state solution for the uniform grid case is much larger than that of the proposed algorithm. In summary, the total error introduced depends on the local truncation error along with the number of iterations required to reach the final solution. This explains the results in the paper comparison figures, where it would be difficult to draw a precise Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 120 conclusion of which technique is more accurate. This is because for each case or curve, the number o f iterations required to obtain the steady state solution and the local discretization errors are different. The problem of identifying the most accurate solution becomes even more difficult since we are dealing with a highly nonlinear problem. It is worth mentioning here that the proposed algorithm does not have any stability constraints if At is chosen to satisfy the CFL condition at the beginning of the simulation. The reason is, as the simulation progresses, the spatial distances employed become even larger than the ones introduced at the beginning. This represents an extra benefit o f using the proposed algorithm that it does not need any time-step At change while the simulation is in progress. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 121 4.7 Summary In this chapter, a new wavelet approach has been developed and successfully applied to a 2D full hydrodynamic large-signal simulator and 3-D FDTD simulator. The proposed algorithm solves the highly nonlinear PDEs that characterize the semiconductor device behavior and Maxwell’s equations on nonuniform grids. The nonuniform grids are conceived by applying wavelet transforms to the variable solution followed by thresholding. It is found that each variable has its own grid at any given time, and the grids of the different variables need not to be updated at the same rate. A reduction of 75% in CPU-time is achieved compared to a uniform grid case with an error of 2% on the DC drain current for a 1% initial threshold value. Furthermore, the same CPU-time reduction has been achieved for AC simulations with a mean relative error of order 3 to 4%. Moreover, an 80% CPU-time reduction is obtained for FDTD simulations with a 0.1% average error on the potential. It has been observed that tradeoffs exist among the threshold value, CPU-time, and accuracy, suggesting an optimal value of the threshold. The proposed algorithm efficiently solves both DC and large-signal AC problems. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Conclusions and Future Work 5.1 Conclusions This dissertation presents a new approach for the simulation and optimization of microwave devices, using a genetic algorithm (GA). The proposed technique solves the equations that describe the semiconductor transport physics in conjunction with Poisson’s equation, employing an adaptive real-coded GA. An objective function is formulated, and most of the GA parameters are recommended to change during the simulation. In addition, different methods for describing the way the GA parameters change are developed and studied. The effect of GA parameters including the mutation value, number of crossover points, selection criteria, size of population, and probability of mutation is analyzed. The technique is validated by simulating a submicrometer field effect transistor (FET), and then compared to successive over relaxation (SOR), showing the same degree of accuracy along with a moderate speed of convergence. The purpose o f this study is to introduce a new vision for a genetic algorithm capable of optimizing real value functions with a considerably large number of variables. This study also represents a fundamental step toward applying GAs to Maxwell’s equations in conjunction with the hydrodynamic model (HDM), aiming to develop an optimized and unconditionally stable global-modeling simulator. Moreover, this dissertation presents a new wavelet approach that has been developed and successfully applied to a 2D full hydrodynamic large-signal physical Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 123 simulator and 3D FDTD simulator. The proposed algorithm solves the highly nonlinear PDEs that characterize the semiconductor device behavior on nonuniform multiresolution self-adaptive grids. The nonuniform grids are conceived by applying wavelet transforms to the variable solution followed by thresholding. It is found that each variable has its own grid at any given time, and the grids of the different variables need not to be updated at the same rate. A reduction of 75% in CPU-time is achieved compared to a uniform grid case with an error of 2% on the DC drain current for a 1% initial threshold value. Furthermore, the same CPU-time reduction has been achieved for AC simulations with a mean relative error of order 3 to 4%. Moreover, an 80% CPU-time reduction is obtained for FDTD simulations with a 0.1% average error on the potential. It has been observed that tradeoffs exist among the threshold value, CPU-time, and accuracy, suggesting an optimal value of the threshold. The proposed algorithm efficiently solves both DC and large-signal AC problems. Furthermore, in this dissertation the potential of high power and frequency multifinger transistors is demonstrated, using a completely numerical coupled electromagnetic-physical simulator. The preliminary results of this study show that at very high frequency, several phenomena with strong impact on the device behavior start to emerge, such as phase velocity mismatches, electron-wave interaction, and attenuation. The results suggest that contemporary microwave devices should be optimized to minimize these effects or possibly take advantage of in favor of improved device characteristics. The results also recommend multifinger transistors as potential alternatives to conventional transistors. This is achieved by using multiple-finger gates of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 124 less width instead o f a single-gate device. Furthermore, this dissertation underlines the enhanced microwave characteristics of multifinger transistors attributable to reducing attenuation and EM-wave propagation effects along the device width. 5.1 Future Work There are many possible future research ideas that can be carried out. For instance: • A possible future research work is to employ rigorous optimization techniques to obtain the optimal multifinger transistor structure based on the electromagneticphysical model presented in this dissertation. Moreover, measurements can be carried out and compared to the results achieved by our model. • To accurately model nano-devices, a possible future research work may involve developing a global modeling simulator by carrying out Monte Carlo simulations in conjunction with Maxwell’s equations. Moreover, measurements can be performed to validate the algorithm. • Another possible work extension is to model complete microwave chips having several components simultaneously, using the coupled electromagnetic-physicsbased simulator introduced in this dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 125 References [1] K. Sieger, R. Abram, and R. Parker, “Trends in solid-state and millimeter-wave technology,” IEEE MTT-S Newsletter, pp. 11-15, Fall 1990. [2] Y. A. Hussein, M. Wali, and S. M. El-Ghazaly, ’’Efficient simulators and design techniques for global modeling of high-frequency active devices,” in Advances in RF Design, Editor: J. Kiang, Kluwer Academic Publishers, in press (to appear 2003). [3] Yasser A. Hussein and Samir M. El-Ghazaly, “Extending multiresolution time domain technique (MRTD) to the simulation of high-frequency active devices,” IEEE Transactions on Microwave Theory and Techniques, July 2003. [4] Yasser A. Hussein and Samir M. El-Ghazaly, “Global modeling of microwave devices and circuits using a genetic-based optimization technique,” IEEE Transactions on Microwave Theory and Techniques, to appear January 2004. [5] Yasser A. Hussein and Samir M. El-Ghazaly, “Global modeling of microwave devices using a new multiresolution-time domain (MRTD) technique,” IEEE Microwave and Wireless Components Letters, submitted March 2003. [6 ] Yasser A. Hussein, Samir M. El-Ghazaly, and Stephen M. Goodnick, “An efficient electromagnetic-physics-based numerical technique for accurate modeling of highfrequency multifinger transistors,” IEEE Transactions on Microwave Theory and Techniques, to appear December 2004. [7] Yasser A. Hussein, Samir M. El-Ghazaly, and Stephen Goodnick, “A new wavelet-based technique for full-wave physical simulation of millimeter-wave transistors,” presented at the MTT-s International Microwave Symposium, Philadelphia 2003. [8 ] Yasser A. 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