DISPERSION AND INSTABILITY CHARACTERISTICS OF SOLID - STATE CARRIER WAVES IN MICROWAVE INTEGRATED SYSTEMS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY CHANG EON KANG 1974 This is to certify that the thesis entitled DISPERSION AND INSTABILITY CHARACTERISTICS OF SOLID- STATE CARRIER WAVES IN MICROWAVE INTEGRATED SYSTEMS presented by Chang Eon Kang has been accepted towards fulfillment of the requirements for Ph.D. Electrical Engineering degree in I, / *7 Majcyrofessor Date February 5, 1974 0-7639 DISPERSIO CARRI The in is studied in characteristi Maxwell's, co in a general investigated interaction ‘1 the slow circ 0f interactio 1'0 adj acent the carrier e in $0]id5 are In the a hydrodI’IIaIIIi motion. The thickness, ar general diSp( finite Semicc non'triViéil s ABSTRACT DISPERSION AND INSTABILITY CHARACTERISTICS OF SOLID-STATE CARRIER WAVES IN MICROWAVE INTEGRATED SYSTEMS by Chang Eon Kang The interaction of solid-state carrier waves in semiconductors is studied in order to predict the wave instabilities and wave characteristics of propagating modes. The analysis is based on the Maxwell's, continuity, and Boltzmann transport equations, and is deve10ped in a general manner. Here, two models of wave interactions are mainly investigated in the frequency range of 1 m 10 GHz: the first model of interaction is obtained between a carrier stream in semiconductors and the slow circuit waves guided by meander-tape line, and the second model of interaction is obtained due to velocity-modulated carrier waves in two adjacent streams pr0pagated in the same direction. By introducing the carrier effective mass and collision frequency, the scattering effects in solids are taken into account. In the interaction of the carrier stream with rf circuit waves, a hydrodynamic model is adopted to describe the behavior of the carrier motion. The effects of diffusion, insulator thickness, semiconductor thickness, and the substrate material are included in the analysis. A general dispersion expression of the coupled waves traveling along a finite semiconductor slab is developed in a formal way so that for a non-trivial solution of the fields the determinant of the fields matrix m5: vanish. T net gain for ma continuous mean circuit. The i far a range of collision freqt Judging from t] and the practiu fines not heavi fabrication an shnus that the ratio play an The fen is investigate IIIo stream ins transmission 1 6(Nations for instability C} and doping 18\ results, Vhic} Chang Eon Kang must vanish. This analysis aims to evaluate the possibility of obtaining net gain for materials such as InSb, GaAs, Si and Ge by choosing the continuous meander-tape circuit and capacitively-coupled meander-tape circuit. The instability characteristics of the growing wave are analyzed for a range of material parameters in terms of frequency, circuit velocity, collision frequency, carrier drift velocity and actual layer thickness. Judging from the theoretical analysis varied by the material constants and the practical point of view, the best result of this type of device does not heavily depend upon materials themselves but proper design, fabrication and maximization of their drift velocities. The result shows that the collision frequency and the drift to thermal velocity ratio play an important role in the nature of the carrier wave. The feasibility of wave amplification in the two-valley model is investigated, leading to a clearer description of Gunn instability. Two stream instability is analyzed first by establishing an equivalent transmission line and then by writing the appropriate partial differential equations for the interactions between two valleys. The dispersion and instability characteristics are obtained as a function of frequencies and doping levels. The analysis is in agreement with the experimental results, which confirm the validity of this approach. DISPERS] CARP DISPERSION AND INSTABILITY CHARACTERISTICS OF SOLID-STATE CARRIER WAVES IN MICROWAVE INTEGRATED SYSTEMS BY Chang Eon Kang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1974 l, I \J WHO D TO MY PARENTS MR. AND MRS. BYUNG HYO KANG WHO DID SO MUCH TOWARD THE COMPLETION OF THIS PROJECT The aut Professor, Dr. through his tc stimlating di and research. Dr. J. Asmussc suggestions th The autl Electrical Eng; Division at Mic Dr. G, Haddad, OfMiChigan, fc faCilities in t Financial guppo and the Divisio acknowledged. Finally, Parents for encc Vife, Jung Hee f lite and good Ch been a gem,“ of ACKNOWLEDGMENTS The author would like to express his heartfelt appreciation to Professor, Dr. B. Ho for bringing the topic to the author's attention through his teaching, his interest, good humor, invaluable guidance, stimulating discussions and encouragement during this period of study and research. Thanks are also due to committee members, Dr. S. Frame, Dr. J. Asmussen, Dr. D. Nyquist and Dr. K. Chen for their guidance and suggestions through all phases of the project. The author wants to extend thanks to Dr. D. Fisher, Professor of Electrical Engineering and Mr. J. Hoffman, Director of the Engineering Division at Michigan State University, for helpful assistance, and to Dr. G. Haddad, Director of Electron Physics Laboratory at the University of Michigan, for the permission of the use of the integrated circuit facilities in the course of the preparation of this investigation. Financial support was granted by the Department of Electrical Engineering and the Division of Engineering Research. This support is gratefully acknowledged. Finally, an expression of immeasurable gratitude is given to his parents for encouraging him to undertake a life long education, to his wife, Jung Hee for her continuous encouragement, patience, understanding, live and good cheer, and to his children, Jeannie and Joseph for having been a source of joy. ABSTRAC ACKNOKI LIST OF LIST OF Cheater I. INTRODU IL PRELIMI 2.1 2.2 Hi IIL FIELD DI 3 l 3 2 Met 3.3 Wav 3-4 Bou 3 5 Fie 3 6 Fie 3.7 Ele. C; FIELD AN; 0F LINE Intr BOUn AA Lo _. 4‘3 Elec ane Two Gene: F017“ AA in :5 CC V' DISPERSIQ BETWEEN TABLE OF CONTENTS Chapter I. II. III. IV. Page ABSTRACT ACKNOWLEDGMENTS iii LIST OF FIGURES vi LIST OF TABLES ix INTRODUCTION 1 PRELIMINARY ANALYSIS AND THEORETICAL BACKGROUND 4 2.1 Introduction 4 2.2 Historical Observation of Wave Interaction Phenomena in Semiconductors S 2.3 Physical Description of Mathematical Model 8 2.4 Fundamental Equations 10 2.5 Linearized Small Signal Analysis and Wave Theorems Relevant to Slow-wave Circuit 14 2.6 Negative Resistance Effects and Two Valley Instability in Solids 17 FIELD DISTRIBUTION IN SOLID-STATE MATERIALS 22 3.1 Introduction 22 3.2 Method of Solution and Configuration of Sample 23 3.3 Wave Equation of Charged Carriers in Solids 28 3.4 Boundary Conditions in Semiconductors 31 3.5 Field Interpretation in the Solid-State 33 3.6 Field Analysis in the Influence of Collision Effect 38 3.7 Electromagnetic Field Description for Streaming Carriers in the Absence of Collisions 40 FIELD ANALYSIS OF SLOW WAVE CIRCUIT AND FORMULATION 0F LINEAR EQUATIONS FOR THE COMPLETE SYSTEM 43 4.1 Introduction 44 4.2 Boundary Conditions about a Slow-wave Circuit Structure 45 4.3 Electromagnetic Field Solutions in the Slow- wave Circuit Region when the Permittivities of Two Insulating Layers are the Same 46 4.4 General Field Analysis in the Circuit Region 48 4.5 Formulation of Linear Equations and Determination of Unknown Coefficients for the Coupled System 49 DISPERSION AND GAIN CHARACTERISTICS OF INTERACTION BETWEEN CARRIER WAVES AND SLOW CIRCUIT WAVES 55 Chapter UIU‘ NH O H 5.3 Re] 5.4 Wan 5.5 Sol 5.6 Gai 5.7 The V1. TWO STRE 000‘ (”Ne—- '71 O '1 65 CII‘I 6.6 The 6-7 The 6‘8 SOIL Chapter Page 5.1 Introduction 55 5.2 Dispersion Relations for Carrier Wave Interactions 56 5.2.1 General Dispersion Relation 56 5.2.2 Dispersion Relation in a Collision Dominated Stream 61 5.3 Relations of Dispersion Characteristics and Gain 63 5.4 Wave Interaction Analysis with Numerical Method 65 5.5 Solution of Dispersion Equations for Different Materials 69 5.5.1 Continuous Type of Tape-Circuit Model 69 5.5.2 Capacitively Coupled Tape-circuit Model 74 5.6 Gain Characteristics as a Function of Collision Frequency, Circuit Velocity and Carrier Drift Velocity 77 5.7 The Functional Dependence of Device Gain Upon the Variation of Insulating-layer Thickness 82 VI. TWO STREAM INSTABILITY IN SOLIDS 85 6.1 Introduction 85 6.2 Formation of Two-valley Model 88 6.3 Population Densities of Carriers in Two-valley Model 91 6.4 Carrier Interaction in a Two-valley Semi- conductor. 93 6.5 Circuit Equation of Carriers in the Upper Valley 96 6.6 The Electronic Equation of the Lower Valley Carriers 96 6.7 The Dispersion Characteristic Equation in Carrier Wave Interaction 98 6.8 Solution of the Dispersion Characteristic Equation for Gunn Devices 100 6.8.1 Collisionless Analysis 100 6.8.2 General Analysis 104 VII. DESIGN AND FABRICATION CONSIDERATIONS OF PRACTICAL DEVICES 116 7.1 Introduction 116 7.2 Effect of the Insulating Layer Between Circuit and Semiconductor 117 7.3 Selection of Solid-state Materials 118 7.4 Design of Slow-wave Circuit Structure 120 7.5 Fabrication Considerations of Devices 122 VIII. CONCLUSIONS AND SUGGESTIONS FOR FURTHER DEVELOP- MENT 127 BIBLIOGRAPHY 130 APPENDICES 137 5.6.1 54.2 Ener numb Velo for Two I diffl One l wave Sever Geome diele Field confi M. a Varia‘ IDSb : feu d mater; Compal insule Semicc normal solid- Gain c IIOrmal Sena, Gain Ch IDSUIat I POSsit IHStabil Figure 2.6.1 2.6.2 2.6.3 3.2.1 4.1.1 4.3.1 5.2.1 5.5.1 5.5.2 5.5.3 5.5.4 5.6.1 5.6.3 5.7.1 6.1.1 LIST OF FIGURES Page Energy, velocity and effective mass vs. wave number for electrons in conduction band. 18 Velocity (or current density) vs. energy curve for voltage controlled negative resistance. 19 Two valley model separated by an energy difference AE. 20 One possible structure for solid-state traveling- wave amplifier 26 Several typical tape slow-wave structures 44 Geometry of the device structure with the same dielectric constant around the circuit 46 Field coefficients matrix of the structure configuration. 57 . 14 3 f-B diagram for a n = 10 /cm InSb sample 70 Variation of attenuation with frequency for a InSb sample ‘ 72 f-a diagram of the growing mode for four different materials 75 Comparison of attenuation constants when the insulating layer between the circuit and the semiconductor exists and not exists 76 Gain characteristics with respect to the normalized collision frequency for several solid-state materials 79 Gain characteristics with respect to the normalized circuit velocity for several solid- state materials 80 Gain vs. normalized drift velocity for various solid-state materials 81 Gain characteristics for various values of an insulating layer thickness (d) 83 A possible configuration for two—stream instability devices 87 vi Figure 62.1 L2.2 C4.l 6.8.7 GE‘s 63.9 6-8.10 6.8.11 Simpl model GaAs TheOI with An or vallI to t] f-S ( GaAs uppeI f-SL GaAs Uppe: f‘fi ( samp Stre; Varh four V.§eq cm° Attm When Atto f~g samp cOns f‘BI samp f-a u I samp Vari; n=101 Cons; Varie n=10: Vall£ Figure Page 6.2.1 Simplified energy band structure of two valley model. The numerical values are taken from a GaAs sample 89 6.2.2 Theoretical and experimental drift velocity with electric field of GaAs sample 92 6.4.1 An over—all equivalent representation of two- valley models. A displacement current is flown to the circuit 95 6.8.1 f-B diagram of four waves for a n= 1013/cm3 GaAs sample when the collision frequency of an upper stream is neglected 101 6.8.2 f-B diagram of four waves for a n=1014/cm3 GaAs sample when the collision frequency of an upper stream is neglected 102 6.8.3 f-B diagram of four waves for a n=1015/cm3 GaAs sample when the collision frequency of an upper stream is neglected 103 6.8.4 Variation of attenuation with frequency for 2 four waves where vu=0, E: 7 KV/cm, u = 5000 cm / v.sec, u =100 cm /v.sec, e = 12.5, and n = 1013/ u r cm 105 6.8.5 Attenuation vs. frequency curves of four waves where er = 12.5, vu = 0, and n= 1014/cm3. 106 6.8.6 Attenuation vs. frequency of four gaves where cr = 12.5, Vu = 0, and n = 1015/cm . 107 6.8.7 f-B curves of four waves for a n=1013/cm3 GaAs sample when all collision frequencies are considered 108 6.8.8 f-B curves of four waves for a n=1014/cm3 GaAs sample when collision frequencies are considered.109 6.8.9 f-B curves of four waves for a n=1015/cm3 GaAs sample when collision frequencies are considered.110 6.8.10 Variation of attenuation with frequency for a n=1013/cm3 GaAs device when collisions are 111 considered. 6.8.11 Variation of attenuation with frequency for a n=1014lcm3 GaAs sample when the lower and upper valley collision are considered. 112 vii Figure h8.12 DJ Variz 8 n=1 are I A me; Slow Stat' Flow Figure Page 6.8.12 Variation of attenuation with frequency for a n=1015/cm3 GaAs device when all collisions are considered 113 7.4.1 A meander-tape line 118 7.5.1 Slow-wave circuit and test structure of a solid- state traveling-wave amplifier 121 D.1 Flow chart for computer solutions of a polynomial 149 viii Table 5.4.1 Typic mater 12.1 Diele Appendix B LIST OF TABLES Table Page 5.4.1 Typical numerical values for several solid—state materials 67 7.2.1 Dielectric permittivity 118 Appendix B Table of electromagnetic field solutions for all regions 141 ix The cle amplificatior interelectmc In order to c brothers pror electron bunc OPerc’ition the Cat'lty gap, h mdulation an The sch Wave amPlifie a great deal . toPotential ; mic“. At generatiOn or because it She althougn1 the e Poser operatic considerEltion ) CHAPTER I INTRODUCTION The classical electron vacuum device in the generation or amplification of high frequency signals has many drawbacks, such as the interelectrode capacitance and long transit time between electrodes. In order to overcome such deleterious effects the Heils and the Varian brothers proposed the scheme of velocity modulation, in which the electron bunching process produces the signal. However, in the klystron operation the electron beam interacts with the rf field over a short cavity gap, hence the electric field must be intense to provide proper modulation and the frequency band is rather narrow. The scheme, extended interaction space, is the wellknown traveling- wave amplifier or backward—wave oscillator. In the past several years, a great deal of research effort in solid-state physics has been devoted to potential application between electron beam devices and solid-state devices. At present the solid-state device has some merits in the generation or amplification of microwave and millimeter-wave signals because it shows excellent noise figure and high frequency response, although the electron beam device has no competition in considering high power operation. Furthermore, especially due to cost, weight and space consideration, some solid-state sources for applications are in the stage of taking over. Besides, many potential applications in the future are expected in various aspects. In this thesis the properties of carrier-wave interactions in several solid-state materials will be investigated —— mainly concerning two types of carrier wave Two st which explai some similar bear-.5 in a v physical ins coupling, th in electron instability Anothe to a driftin‘ line on the ; Solid-state 1 meander type Stream of a I relation is 1 are dEtermiIu some C1‘UCial and the "31’ c eprOred, “OI surface charg its finite di dispersiOn 1‘8 Wave Pr: theoretically. possibly some two types of interactions, namely, the two stream instability and carrier wave interaction with the circuit. Two stream instability in solids could cause wave amplification, which explains the Gunn instability mechanism. Possibly there exist some similarities between the streams of carriers in solids and electron beams in a vacuum. Therefore, in order to understand the important physical insights of the phenomena such as energy conversion and wave coupling, the concept of wave interaction, which has been successful in electron beam devices, will be used in describing the various instability characteristics in solid-state materials. Another interaction is when a slow electromagnetic wave is coupled to a drifting stream of carriers by depositing the slow-wave meander line on the semiconductor slab with integrated circuit technology. The solid-state traveling-wave amplifier is based on the principle that a meander type of slow-wave circuit is suitably coupled to the carrier stream of a negative kinetic power. Conventionally, the dispersion relation is first formulated and then the instability characteristics are determined by examining its propagation constant. In such approaches, some crucial aspects of the interactions such as the limiting conditions and the way of finding the dispersion relation were neither fully explored, nor rigorous enough. For example, the reflected waves and surface charges on the semiconductor were totally ignored in spite of its finite dimension. A formal and straightforward way of deriving the dispersion relation is presented here. Wave prOpagation in solid-state materials will be examined theoretically, then solved by a computer technique. From the solutions, possibly some instabilities can be obtained. In CI preliminaI') device strL in semiconc‘ II' presents computer so function of collision f instability experimental Chapte devices. Th Finall Ch3Pter VI I l In Chapter II, the literature survey for solid-state devices and preliminary analysis are presented with descriptive explanations of the device structure. Electromagnetic field solutions of wave propagation in semiconductor and circuit are derived in Chapter III and IV. Chapter V presents the characteristic equations and gain expressions, and the computer solution of gain and dispersion solution are plotted, as a fUnction of the carrier drift velocity, circuit wave velocity and collision frequency. Chapter V1 is devoted to investigating two stream instability and the theoretical results of gain are compared with the experimental results. Chapter VII presents the design criteria of solid-state microwave devices. The design data of experimental devices are illustrated. Finally, a discussion of results and conclusions is given in Chapter VIII, together with suggestions for further study. 2.1 ”tr Bef' might be ‘ backgrounC This Chapt required b“ matical "’0‘ Section 2'2 action {1190 is conCerne the problem will be W equationS an small sisnal equations Ca structures at significantly Finally two valley mo phenomena, buj intense intert CHAPTER II PRELIMINARY ANALYSIS AND THEORETICAL BACKGROUND 2.1 Introduction Before getting started with the development of the main theory it might be well not only to understand the theoretical and historical background of the study but also to establish directions for the research. This chapter contains what is believed to be the most fundamental details required by the following chapters. Some basic equations and a mathe- matical model that will be used in subsequent studies are discussed. Section 2.2 is mainly devoted to the literature review of wave inter- action theory in the solid-state and semi-metal materials. Section 2.3 is concerned with the mathematical description of our model in attacking the problem. Descriptive explanations and developments of the model will be treated within the context of this model configuration. Maxwell's equations and Boltzmann's equation are introduced in Section 2.4 and small signal analysis is employed in Section 2.5 so that the nonlinear equations can be linearized. Several theorems pertaining to periodic structures and wave propagation are also explained; these will be significantly helpful in understanding the main problem. Finally, the basic theory of negative resistance effects and the two valley model are reviewed in Section 2.6. Of several physical phenomena, bulk effects within solid-state materials have received intense interest. Devices employing these phenomena are quite different from junction devices, such as transistors, and one would expect greater power output from bulk effect devices since the interaction regions are one or twc are two t) instabilit cal work 0 analysis a bulk effec 2.2 Hi_sto_ In ti suggested t oSOusly be I~‘as not the semimetals ; COnflned to PTEdlCtion t solids with KonstantinoV EAnnette fie After , EXperimental 1961 by 30m one or two orders of magnitude larger in dimension. In solids there are two types of bulk effects, negative resistance and two valley instability which are explained in Section 2.6. However, the theoreti- cal work on these effects has not been well developed. Two stream analysis and a new type of wave interaction mechanism based on these bulk effects will be investigated utilizing the contents of this chapter. 2.2 Historical Observation of Wave Interaction Phenomena in Semi- conductors In the 1960's a great deal of interest in solid-state plasma suggested the possibility that the gaseous plasma theory could anal- ogously be applied to the solid-state materials but a clear description was not then possible. The solid-state plasma exists in semiconductors, semimetals and metals commonly —- being a gas of conduction electrons confined to the volume of the specimen. Interest began with the prediction that certain electromagnetic waves would be propagated in solids with relatively little attenuation. The idea was proposed by Konstantinov [K01] and independently by Aigrain [All] in applying dc magnetic fields to certain semiconductors. After first reports of wave propagation in semiconductors, an experimental observation was successfully carried out with sodium in 1961 by Bowers, et a1 [B01]. The problem in solid-state plasma, there- fore, is not how to generate a plasma but how to disturb the existing plasma from thermal equilibrium and induce instabilities, both convective and absolute. In 1963 Gunn [GUI], [cuz] observed that, with GaAs and InP, a periodic oscillation could be obtained at a threshold electric field. This was completely different from previous negative resistance effects which occu: measured b: capacitive moving higl Krone properties differentia also insist differentia field on wh states. Th incremental been Pr0pos. Ridle: DEgative m0] dmamic “81 Sintle laye] discontinun c°ntr011ed , another tn): llmction and had Verified responsible &”9 [8L2], [8E2] treate which occurred in p-n junction semiconductor devices. The velocity was measured by a hetrodyne detection system, removing any possibility that capacitive probe experiments [GUZ] measured something other than a moving high field region in bulk GaAs devices. Kromer [KR4], [KRS] proposed in late 1964 that the observed properties of the Gunn effect [GUI], [GUZ] could be explained by a differential negative conductivity. The negative mass amplifier was also insisted upon earlier by Kromer [KRZ], [KR3] in 1959. The differential negative mobility changes sign at the critical electric field on which sufficient electrons are transferred to low mobility states. This is caused by a decrease in carrier drift velocity with an incremental increase in electric field. Actually this mechanism had been proposed three years earlier by Ridley and Watkins [R11]. Ridley [R12]extended the earlier analysis [R11] of differential negative mobility of the voltage controlled type by general thermo- dynamic arguments. He neglected carrier diffusion and assumed that a single layer of mobile charge would accumulate or be removed from the discontinuity in the longitudinal electric field. While the voltage. controlled negative resistance occurs in bulk Gunn effect devices, another type of current controlled negative resistance may occur in p-n junction and impact ionization devices [GUl], [GUZ]. Several workers had verified experimentally that the transferred electron mechanism is responsible for the voltage controlled negative conductance observed in GaAs [8L2], [F01], [GIl], [H12], [HU4], [R01]. Additionally, Betjemann [BEZ] treated temporal effects on carrier mobility. McCumber and Chynoweth [MCl] undertook the solution of the relevant transport nonlinear equations for bulk GaAs by numerical methods. satellit Maxwelli; amplifier and expei several 2 [A03], [II [T01]. N mechanism that an e conductior conduction In t Pierce [P1 interactio requiremen‘ Ndel. Thim Stable. lin GaAs device carrier den Recent instabilitiE IWhich has be [SH] ‘ ana transmiss 10” methods. They assumed that the relative population of the central and satellite conduction band valleys was given instantaneously by a Maxwellian distribution with a single electron temperature. Microwave amplification and negative conductance have been discussed theoretically and experimentally for materials such as GaAs, InP, InSb, and CdTe by several authors - [BL3], [8L4], DHAz], [HA3], [HA5], [HEI], [H12], [x03], [KRS], [LAl], [101], [MAl], [NAl], [PEI], [P11], [R13], [8A2], [T01]. No one, however, has clearly justified the transferred electron mechanism in semiconductors. It is very obvious quantum mechanically that an electron is transferred from a high mobility, low mass, conduction sub-band of low energy valley to a low mobility, high mass, conduction sub-band of high energy valley when the excitation is given. In this work we adopt a similar approach to the method used by Pierce [P12] for the traveling-wave tube amplifier where the mutual interactions between beam and circuit are subjected to a self consistency requirement to explain the new interaction mechanism of the two valley model. Thim and Barber [THl] demonstrated that it is possible to achieve stable, linear and comparatively high microwave amplification in bulk GaAs devices. The devices tested had n2 = 2 x lOlz/cm2 where n is carrier density and 2 active length. Recently Ho [H01],[H02], [H03] attempted to describe the various instabilities in solid—state plasmas by the concept of wave interaction, which has been successfully used in electron beam devices [RAl], [SE1], [8T1] - analyzing the problem by both the coupled mode approach and transmission line analog [FUI]. The conductors [c011, [CAI power carri Sol) substitutin carrier in explanation ultrasonic I industry ha: [0521. [E01] ISAII. [8A2] transverse s 0f dc longitI 2'3 M In des microscopic o sc0p1'c treatm. Boltzmann equa ”thanically. Particle SYSte. be dealt with 1 transport equaz AlthOUgh long wavelength mathematically The amplification of acoustic waves in piezoelectric semi- conductors such as CdS has been analyzed by many research workers [BU4], [C01], [GAL], [HA1], [HUI], [HUZ], [1N1], [KRl],[LOl], [M11]. Kinetic power carried by carrier waves was given by Vural and Bloom [VUZ]. Solymer and Ash developed [501] a one dimensional analysis by substituting the electron beam in a traveling-wave tube for the drifting carrier in a semiconductor. Sumi [SUl], [SUZ] treated the theoretical explanation of solid-state traveling-wave amplifier by comparing it to ultrasonic wave amplification. Even though the growth of semiconductor industry has many workers concentrating on this type of research [DEl], [052], [501], [ENl], [ETl], [FUZ], [H11], [K01], [MUl], [NEZ], [p11], [8A1], [8A2], [VUl], a successful result has not been reported. The transverse solid-state device has also been studied with the application of do longitudinal magnetic fields [BAl], [302], [Hus], [TUl]. 2.3 Physical Description of Mathematical Model In describing any physical phenomenon on some object, either a microscopic or a macroscopic approach is in general used. The micro- sc0pic treatment, which uses Maxwell's equations with the microscopic Boltzmann equation, puts emphasis on each individual particle quantum mechanically. On the other hand, the macroscopic approach treats the particle system collectively. That is, a large number of particles can be dealt with statistically, using the same Maxwell's equation with the transport equation. Although the same result from both treatments is obtained in the long wavelength range, the first approach is usually much more difficult mathematically and requires serious physical restriction be placed on the model to make the problem tractable. In a self-create characteris determined I appropriate should be it scopic attac The b parameters, collision frI phenomenologi wave and elec the charged 5 Also, in the ': excitations w The int SOlids is take effeCt of can carrier Effect Untthe Semlc( carriers due tc "Source” or "Si FurtDErm larger than the hydrodynamical] In a semiconductor a large number of carriers interact with their self-created or externally applied electromagnetic field or both. The characteristics and behavior of such a system are experimentally determined by the average behavior of the ensemble. Therefore, the appropriate theoretical model description may be statistical and it should be in general a quantum-statistical description. The macro- scopic attack of the hydrodynamic model will be used throughout. The behavior of the hydrodynamic model is characterized by several parameters, such as mean velocity v, mean plasma frequency mp, mean collision frequency V and pressure p (or mean thermal velocity Vt). The phenomenological expressions describing the interaction of the circuit wave and electromagnetic wave can be obtained combining the equation of the charged stream of Maxwell's equations, with boundary conditions. Also, in the model the dispersion relations of the medium for different excitations will be derived. The interaction of carriers with lattice and thermal vibration in solids is taken into account in the model by including the collision effect of carriers, v = 13 and the environmental effect by considering 2 2 -l . . carrier effective mass, m* = n (315—) . 1n the model. It 15 assumed that the semiconductor is heavily doped such that the change in number of carriers due to generation of recombination, usually referred to as "source" or "sink" terms, can be neglected. Furthermore, the wave length of any disturbance is comparatively larger than the Debye length, A so that one may treat the electron stream D hydrodynamically. 2.4 532 K ., field B and time. through. 81 density p intensity it is nece the solid- equations, are needed 311 chapte dimensions MaXI Ithere u is t 311 VeCtOr q1 In add cantinuity eq Charge der 10 2.4 Fundamental Equations When the electric field E (f, t) is applied, it builds a magnetic field B (E, t) and charge density p (f, t) which are functions of position and time. Simultaneously, a current with density 3 (f, t) will flow through. Similarly, a charged carrier in solid-state posseses a charge density p (f, t). These densities will produce an electric field intensity E (f, t) and a magnetic field intensity H (E, t). Therefore, it is necessary to interrelate all quantities in the rf circuit and in the solid-state. In order to describe the relationship, Maxwell's equations, continuity equation, force equation and Boltzmann's equation are needed. Conveniently, M.K.S. rationalized units are used throughout all chapters except for describing carrier densities and small sample dimensions. Maxwell's equations are given by: 315 VXE--'a-£'B (2.4.1) vitii=3+9— (2.4.2) at + v - 0 = 0 (2.4.3) v - I = 0 (2.4.4) If the medium is isotropic, the B and B field vectors become: 3 = nil (2.4.5) B = tii (2.4.6) where u is the permeability and e is the permittivity of the medium and all vector quantities are functions of space and time. In addition, the carrier and rf waves are related by the continuity equation and the equation relating current density, velocity, and charge density: 4 . where v is Not due to den however, It (electrons The where q an Performing are velocl electron s In Thel’et‘ore , Provided tl The I0 the equa Wher 9v ”.9 attachment lI'h‘ 11 V°Zi+§%=o (2.4.7) 3:. p 3 (2.4.8) + O o o where v is velocrty of particle. Note that charge density p and current density 3 generally are due to densities of electrons and holes. Only heavily d0ped semiconductors, however, will be used for the devices here, therefore one type of carrier (electrons here) will be considered. The Lorentz force equation is: g%=—§;(I€+i7x§) (2.4.9) where q and m* are the charge and effective mass of a particle respectively. Performing the analysis in Eulerian variables, the dependent variables are velocity, charge and current density at a fixed position within the electron stream [B03]. In Eulerian variables the Operator, d/dt in Eq. (2.4.9), is given by: dV 3 + + R’ .3? ., V . v) v (2.4.10) Therefore, the Lorentz force equation can be rewritten for most microwave devices as: (-—+V'V) Y=9—(E+-\7x—B) (2.4.11) provided that collision and pressure terms are not involved. The zeroth moment of the Boltzmann equation (i.e., o = 1) leads to the equation of continuity for carriers an 51-1. vr-(nii) = 11 (vi - v - v) (2-4-12) a r where vi, Va’ vr represent the collision frequency due to ionization, attachment and recombination. Usually one assumes vi — Va - Vr = 0 which reduces Eq. (2.4.12) to a simple continuity equation. H0‘ in the ga state and results i inputitie another 1 will usua create an can act t depends u and the w scatterin Or even 1 An the atoms 3P€riodic scat:er t they nigh t0 the th 15, the S of Scatte is the do: 118erCt c- On. NhEre R i< 12 However, the collision effect in solids is more complicated than in the gas. That is one of the most important differences between solid- state and gaseous plasmas. The collision mechanism in solids which results in randomizing the carrier distribution must be associated with impurities, structural imperfections or aperiodicities of one sort or another in the crystal. The presence of an impurity atom in a crystal will usually alter the electrostatic potential in the neighborhood and create an aperiodicity in the potential field within the crystal which can act to scatter conduction electrons. This impurity scattering process depends upon the nature of the impurity atom, its ionic size, its valence, and the way it is bonded into the crystal lattice. The impurity scattering mechanism is dominant in crystals which are relatively impure, or even in very pure samples at very low temperatures. Another scattering mechanism is due to the thermal vibrations of the atoms in a very pure material. At any given time, a slight aperiodicity of the potential exists within the crystal which serves to scatter the conduction electrons, dissipating whatever drift velocity they might have acquired from externally applied fields and retaining them to the thermal equilibrium state. Obviously the higher the temperature is, the stronger the lattice vibrations and the higher the probability of scattering per unit time become. This lattice scattering mechanism is the dominant scattering process in relatively pure and structurally perfect crystals, especially in the higher temperature ranges. One more important scattering parameter due to the material permittivity is the dielectric relaxation frequency which is defined by 1 Va = ER' (2.4.13) where R is the resistivity of the material. Of : (i.e., 1&1t1 to thermali three colli the impurit respectivel equation be R cf 3t ( 1 C where v = v and f 0 reprI If mc effective cc fTEQUency v The f I Where <3) and T; ASSumi O V 0n e may I“Writ applfing Eq ( 01 13 Of several scattering mechanisms, if three scattering factors (i.e., lattice, impurity and dielectric relaxation) operate simultaneously to thermalize the carrier distribution function, then there will be , associated with the lattice, three collision frequencies, v , v , v t p d the impurity, the dielectric relaxation, scattering mechanism respectively. Under these conditions the collision term of the Boltzmann equation becomes 3f (——- = (-v - v - v ) (f - f ) = -v (f - f ) (2.4.14) at coll t p d o o w r v = + + he 3 vt vp ”d (2.4.15) and f0 represents the equilibrium state of the distribution function f. If more than three scattering mechanisms are involved, the effective collision frequency can be written by a single collision frequency v given by v = §\In (2.4.16) The first moment of the Boltzmann equation becomes + d 5* + 1 + dt = %;-(E + xv> x B)-v-nm* Vr'p (2.4.17) + where average veloclty of carriers intrinsic pressure tensor and p nmtfv ( - T) - ( - T) fdvxdvydvz Assuming that‘p*is isotropic and can be expressed as p = nykt where k is Boltzmann's constant, T is the absolute temperature of the carriers (°K) and y = l'b3, then H V - p = Vp = V(nykt) = yktVn (2.4.18) One may rewrite Eq. (2.4.17) by conveniently dropping the bracket <> and applying Eq. (2.4.10). 2 a + -+ q E + g Vt ~ (fi— +V.V+ \j) v=-m-Tk— ( +Vx )-fi-Vfl (2.4.19) where Vt =/ If an electrI y: 1. For 1 appears in tl In ser the electron where I 2.5 Lineari; Slow-war Unfort exact solutiI equéltions, a SYStem. A11 part) and an asSW3d to b. all Produqs Only One fret Problem Can 1 ACCOrI QuantitieS c I there the Sul CORpOnent. men I in SectiOn 2 14 where vt =J’%Eii = mean thermal velocity of carrier. If an electron stream is considered to be under an isothermal condition, y = 1. For higher frequencies Y = 3 since the environmental circumstance appears in the adiabetic state. In semiconductors the resistivity can be expressed in terms of the electron charge density n and the mobility u. l R = -nue (2.4.20) where - e “ ‘ vm* (2.4.21) 2.5 Linearized Small Signal Analysis and Wave Theorems Relevant to Slow-wave Circuit Unfortunately most equations are nonlinear in nature and hence, exact solutions are very difficult to obtain. In order to simplify the equations, a small signal analysis is introduced in linearizing a given system. All quantities are broken up into a dc part (or time average part) and an ac part (or time varying part). Further the ac part is assumed to be very small in magnitude compared with the dc part, so that all products of second or higher order in ac quantities can be neglected. Only one frequency needs be considered, and the general time varying problem can be solved using the Fourier transform technique. According to the small signal simplification, all various quantities can be written in the following form: K (;’ t) = Ko (¥) +‘Xl (;’ t) = K0 (5) + K1 (f) eth (2.5.1) where the subscript "0" refers to the dc component and "l" to the ac component. When the above equation is substituted into the equations given in Section 2.4, one obtains the dc equations: <1 <1 )4 >4 32+ "1+ <1 _C3+ <3 C—Q 9+ II 0 '1 V <+ o . and ac equat V x E Vxl] d ('74 C4 H H 15 v x ab = 0 (2.5.2) v x H - 3 ’ 5 3 0 - o (2 . ) v 3 - 2 5 4 O - °o ( . . ) v 5 - 0 2 5 5 - o - ( . . ) v . 30 = 0 (2.5.6) + + Jo - povo 2 (2.5.7) I v? - 3- (E’ + 3 S ) V‘ v T 2 s 8 v0 ° 0 -nfi' o o x o - IT' no - Wo ( ° ' ) and ac equations v x E, = -jmuH1 (2.5.9) + , + V x H1 = 31 + deil (2.5.10) v . B, = pl (2.5.11) v . E, = 0 (2.5.12) V . 31 = -jwpl (2.5.13) + + 31 - oov1 + plvo (2.5.14) jw-Jl + (I; o V)-\71 4’ (III . V)I/> + VIII =q‘; (Eli-T; xgl + O 2 0 III 0 31x35) - XE. VH1 (2.5.15) n All products of ac quantities have been neglected in the ac equations; thus, the ac equations are linearized. However, the dc equations are still nonlinear. To solve the linear ac equations, the dc equations must first be solved. Various artifices are used to circumvent the non- linear nature of the dc equations. These will be cleared later in specific cases. For the moment, merely assume that they have been solved so that $0 and 00 are known functions to be used in the ac equations. From this point on, in most cases, one needs only be concerned with ac equations of interest. The wave form traveling along an axially periodic structure is described by the concept of space harmonic functions commonly known as Fquuet's to linear different French ma out by BIL are often waves tha Floquet's [T P a P In other Period to proof of Th as the fu COTTeSpon attenuati 16 Floquet's theorem. This theorem actually constitutes a generalization to linear partial differential equations of a theorem in ordinary linear differential equations with periodic coefficients established by the French mathematician Floquet. Such a generalization has been carried out by Bloch. Accordingly, waves that pr0pagate along a periodic structure are often called Bloch waves by analogy to the quantum-mechanical electron waves that propagate through a periodic crystal lattice in solids. Floquet's theorem is stated as follows: [Theorem 2.5.1] Floquet's Theorem: "For a given mode of propagation and at a given steady state frequency, the fields at two points on a transmission system, separated by one period, differ by a complex constant." In other words, the waves regardless of the choice of origin differ from period to period only in phase and not in wave-form or magnitude. The proof of the theorem is shown in Appendix A. The general complex Floquet wave number k = B - ja is referred to as the fundamental pr0pagation constant where B and a represent the corresponding phase and attenuation constants. For lossless circuits the attenuation constant a is zero. The wave propagates with a phase constant znn B + -B—-where the n'th term is called the n'th space harmonic or Hartree harmonic. For a lossless system, since k = 80 we define n' o p (2.5.16) where 8n is termed the phase constant for the n'th space harmonic. To understand the wave characteristics of slow waves in given media, a diagrammatic representation of their properties, similar to the one by Brillouin, is required. This plot shows the functional relation- ship between the Operating frequency w and the phase constant B. The diagrammatic representation is called a Brillouin diagram or w-B diagram. If a Brilloui either experi to predict th except for a give similar used. Anotht as a functior be derived fI interaction. dispersion C] In re. [Theoz "The the ' and I The proof of 2.6 It is lead to OSCi with negatix, to travelinE with separat convert ed t C 17 If a Brillouin diagram for any specified medium or circuit can be prepared, either experimentally or theoretically, then there is enough information to predict the possible outcome of the interaction with carrier streams, except for a knowledge of field distribution and power. Many other forms give similar information, but the Brillouin form is the most commonly used. Another common diagram is the dispersion curve which is plotted as a function of frequency or wavelength. The dispersion equation will be derived for the given problem and utilized in the analysis of wave interaction. The Brillouin diagram is sometimes referred to as a dispersion curve. In regard to power flow there is a theorem pertaining to systems. [Theorem 2.5.2] Power Flow Theorem: "The time average power flow in the passband is equal to the group velocity times the time average stored electrical and magnetic energy per period divided by the period." The proof of this statement is shown in Appendix A. 2.6 Negative Resistance Effects and Two Valley Instability in Solids It is well known that nonconvective (or absolute) instabilities lead to oscillator devices which can be represented as one-port devices with negative internal resistance, and that convective instabilities lead to traveling wave amplifiers which are generally used as two port devices with separate input and output ports. However, any amplifier can be converted to an oscillator by applying positive feedback. First, look at voltage-controlled negative effects due to negative effective mass. The earliest proposal for such an effect involved negative effective masses for electrons. In solids, the energy E vs. wave number k curve in the conduction band is shown in Figure 2.6.1 (3). The energy (b) (C) (d Figure 2 . 6 . 18 A Energy E . s i I I h ' I --Lp-- : I . i . | ' I I (a) J— ' ' ' e‘1-»k 0 k 1 (b) ‘ | *- I. | I I I 9.3 dkz (C) : I + k l I l I l (d) k Figure 2.6.1 Energy, velocity and effective mass vs. wave number for electrons in conduction band. of the elec where p = I Sin: above equa where h = First and and effect or 01‘ The elecu and 2-6.1 Velocity C driven in a VEIOCit) one Shown dEViCe 9X} Figure 2_€ 19 of the electron is given by E ‘— v — =.£ 2 2 (2.6.1) where p = mv Since the electron momentum is related to the wave number k, the above equation can be rewritten as hzkz E = 2m (2.6.2) where 5 = 2““ Planck's constant First and second differentiations of Eq. (2.6.2) yield electron velocity and effective mass respectively as follows: BT’T'ET =fiv 01' V = l.§§. fi dk (2.6.3) or dZE _ 53 dkz’- n1 2 . m. = 52 (it?) -1 (2.6.4) The electron velocity and effective mass are plotted in Figures 2.6.1 (b) and 2.6.1 (d). For energies greater than B m* is negative, and the 1. velocity decreased with increasing energy. If enough electrons can be driven in energy levels with the application of external electric field, a velocity versus electric field intensity curve is obtained, like the one shown in Figure 2.6.2. Therefore, for energy greater than 5', the device exhibits negative resistance effects. + + v (or J) E I I l I I, 1-. E Figure 2.6.2 Velocity (or current density) vs. energy curve for voltage controlled negative resistance. vhere 1110' The averag Eq. (2.6.5 An: Iatkins [F experiment Cor separated lover val; (; = Li) , 10w mobi1 AS the el the upper tham LE. Valle)’ de current f DEgatiVe FiEUre 2. 20 In bulk semiconductor material the effective mass may be expressed as[HAS] 912,; dB 1 (2.6.5) me = mo* + where m0* is effective average carrier mass at the dc bias field E0. The average carrier mass m* varies with ac electric field E1 as shown in Eq. (2.6.5). Another type of instability in solids was predicted by Ridly and Watkins [R11] and Hilsun [H12] theoretically and was demonstrated experimentally in 1963 by Gunn [cuz], as indicated in Section 2.2. Consider a semiconductor having a conductive band with two minima separated by an energy difference AE as shown in Figure 2.6.3. The lower valley has electrons with a low effective mass and high mobility (T = ug), while the upper valley electrons have large effective mass and low mobility. Initially all electrons occupy the lower state, in L valley. As the electric field increases, some electrons gain energy and get into the upper valley if the energy supplied to the lower valley is greater than AE. When this situation happens, the electron velocity in the U valley decreases sharply due to the low mobility there. This reduces current flow in the U valley, and hence the semiconductor exhibits a negative resistance in the bulk material. fi \\\\\_‘///0 valley Lvaley ___. -__ Figure 2.6.3 Two valley model separated by an energy difference AB. The explained Frc The where r = It the charg aliplifica Ir. The max in device. by elect: For 3 gr greater FOr a ty thall 101 21 The instability induced by the negative conductance effect can be explained as follows: From the continuity equation and Poisson's equation, we have 30 - _ + = _ ‘T ._-. _ p at . J V . (0E) 02 (2.6.6) The solution of p (F, t) has the following form: 1 p('{-, t) = p0 (f, o) e f (2.6.7) where T = _1:1 and po (F, o) is the initial charge density. It can be seen from Eq. (2.6.7) that a negative conductance makes the charge density p grow exponentially. This growth can lead to amplification, and with the preper feedback, to oscillators. In solids the conductivity 0 can be expressed as o = qnu (2.6.8) The maximum growth of p is at t = T where T is the transit time of the device. The growth factor is equivalent to the sample length 1 divided by electron velocity v . O T e— i— 1’ = .2.‘ V0 = £3911- (2.6.9) 6 73' EV _ __ O O qnp For a group of charge to form and grow, the growth factor should be greater than unity, i.e., ;'> 1. In other words, the n2 product is defined for fixed constants n, u, a, q and v0 as: ev n2 > o qu (2.6.10) For a typical Gunn device material such as GaAs, the n1 product is greater than 1012/cm2. ll 131g! Thi interpreta dominated to figure is require range of 1 Th cmmentra llth a te is order the dOpir and $1, 2 (Lip) is ; material 101“ C/s 0” Play temperat “90°10 . the 5011 Of any 3 16. POI CHAPTER III FIELD DISTRIBUTION IN SOLID-STATE MATERIALS 3.1 Introduction This chapter deals with the general sample configuration, wave interpretation, wave equation and field analysis in general, collision dominated and collisionless cases. Every field solution is too complicated to figure out the wave picture at a glance. Appropriate approximation is required to understand the propagating wave —— keeping the typical range of parameters in mind. The electron density (n) in solids is determined by doping concentration and limited by materials, though the density can be varied with a technical manner. The density of typical solids, GaAs and InSb, is order of magnitude 1613 m 1016 electrons/ems. 'For other materials the doping concentration is over the range of 101“ electrons/cm3 for Ge and Si, and in the range of 1022 electrons for Cu. The plasma frequency (up) is a function of the dielectric constant and effective mass of a material with a range of 105 m 1012 c/sec for typical semiconductor and 101“ c/sec for materials. Besides microwave frequency, collision frequency (v) plays an important role in solids, ranging over 107 m 1013 c/sec as temperature increases from liquid nitrogen (77°K) to room temperature (300°K). Because the crystal lattice is polarizable the permittivity of the solid is determined by the constituents and the lattice configuration of any specific materials. The relative dielectric constant range is 4 m 16. For 810 Er = 4, for mica er: 6for InSb cr = 15, and for GaAs Er = 2 22 defined times t mast pa grows b1 instabiL and in I System I SUpport used as 3.2 ESE R to the c solid.st many Imp for “ave relation OI‘ersimp “Challis Clearly onthe 5 fl.‘ 12.5. Since the bunching and debunching effects in most microwave devices are indispensable, whenever these effects occur, the diffusion effect (D) is consequently involved. For typical materials its magnitude is D = 10.3 m 10"1 mZ/sec. Another factor for solids is debye length (AD), defined as AD = 10.8 m 10'5 m and for metals, AD = 10'10 m 10.6 m. Some- times the AD can be replaced by Bv = cup/vt (/ m) in semiconductors, since most parameters for solids are interrelated functions. If a propagating system is unstable such that the disturbance grows but is prepagated away from the origin, this system has convective instability. On the other hand, if the disturbance grows in amplitude and in extent but always embraces the original point of origin, this system has nonconvective instability. The convective instability can support amplifying waves while the nonconvective instability can only be used as an oscillator, as mentioned earlier. 3.2 Method of Solution and Configuration of Sample Recently several authors investigated the wave amplification due to the coupling between the space charge wave and the circuit wave in solid-state materials, as was mentioned in Section 2.2. In most cases many important aspects of the analysis such as the limiting conditions for wave interaction and the way of finding roots of the dispersion relations are not clearly justified. The approaches used were either oversimplified or theoretically unfounded. Furthermore, the physical mechanism of wave interaction and the energy conversion scheme were not clearly revealed. For instance, the reflected waves and surface charges on the semiconductor were totally ignored in spite of the finite dimensions of the slab. In a simple one-dimensional analysis, these approaches propagatiOI The the slow-w: physical di problem mor of the prir action a th As I. as InSb, Ga voltage. I t)Tles of se physical st arises from to generate The . 1960' S! eSPI amPlificatic [not] in let The c amplifiers h agreement Hi sional anal), justifiable. magnetic fie] 24 approaches are adequate as far as the transverse direction for wave propagation is neglected. The difficulty lies in the fact that for any practical devices the slow-wave structure and the semiconductor active region have finite physical dimensions, i.e., finite boundary conditions which make the problem more complicated. Consequently, to obtain a better understanding of the principle of operation and the physical insight of a given inter- action a two or three dimensional analysis should be carried out. As was previously mentioned, the n-type bulk semiconductor such as InSb, GaAs, InP, and CdTe, have negative conductances above threshold voltage. It was verified theoretically and experimentally that different types of semiconductors have different threshold potentials due to the physical structure of the material. This kind of convective instability arises from the negative differential conductance and has been utilized to generate or amplify microwave signals in the past few years. The solid-state traveling-wave amplification is similar to the ultrasonic wave amplification which has been developed in the early 1960's, especially in the wave coupling mechanism. The ultrasonic amplification in CdS observed for the first time by Hutson and McFee [HUl] in 1961 was an example of traveling-wave amplifications in solids. The one dimensional analysis of the ultrasonic and traveling-wave amplifiers has been developed, and the results are in satisfactory agreement with experimental work. In most cases the use of one dimen- sional analysis in traveling-wave and ultrasonic wave amplifiers are justifiable. The former device generally uses a strong focusing static magnetic field and as a result there exists a negligible transverse motion of charged particles. In the latter device, if the wave propagates longitudi remain la coupling types of dimension 0n deposited two or th analysis A analyzed is Shown to obtain Bicrowave slow wave and 50mic Th the adjac If the Ca i“Sulatin The 'DOre Us the inSul an inSUIa Prepagati} Charged C; 3r e appro; 25 longitudinally, a variation in the transverse dimensions -— where they remain larger than the acoustic wavelength - makes no change in the coupling of carrier waves with the rf circuit waves. Therefore, both types of wave interaction mechanism were essentially treated as a one dimensional problem. On the other hand, the coupling through a slow wave circuit deposited on semiconductor by the integrated circuit technology is of two or three dimensional nature, and requires two or three dimensional analysis for the right solution of wave interaction phenomena. A two dimensional problem of wave interaction in solids will be analyzed through this chapter. The two dimensional structure considered is shown in Figure 3.2.1. This is one of the possible structures used to obtain coupling between semiconductor space charge wave and external microwave. The device configuration consists of mainly four parts: slow wave circuit, two insulating layers at top and bottom of the circuit, and semiconductor. The meander line is adopted for the slow wave circuit in which the adjacent tape elements are coupled capacitively or continuously. If the capacitively coupled type of slow wave circuit is used, one insulating layer between solid-state and meander line may be excluded. The more descriptive detail will be given in design consideration. Using integrated circuit technology the circuit is deposited on the insulating layer or directly on the semiconductor. The exclusion of an insulating layer in capacitively coupled circuit, might give slowly propagating-waves a better chance to interact strongly with moving charged carriers if the rf wave propagation and carrier drift velocities are approximately synchronous. Energy is also easily transferred from rf input _ Gob-wave ’ itructurc Y 26 *Scule is not considered top insulating layer ,___., rf output t- Z bottgpcgnsulating K\\\ ohmic contaczs at both ends X ./ . II II / I I I I l I / )' ------------- I " n - II rr Input I i I -’ - / I I'll ’ 11% . , r-----—-------i slow-wave , .. . I ' d structure ./ 1 [7 I7 | l I, #cmiconductorth ” k -------- - ------ I ’I / Substrate I I I Y . (H) [II I'M—1 IL bias poten6fhl \\\\ low pass filter 0 [I IV (b) Figure 3.2.1 One possible structure for solid-state traveling-wave amplifier. (a) Sketch showing coupling between space charge waves and external microwave fields (b) Schematic planar layout of (a). the drift distance . Su lapping b For gener analysis distance have circ with the infinite with subs insulator surface. T} laI'ers be the sum TI maghetic the Carr VEIOQity a Seinico velocity is rough and the 510w “3V thOUSand A undemEa 27 the drifting carriers to the slow wave, resulting in wave growth with distance along the device. Such a circuit structure could be made from metal layers of over- lapping bars separated by a capacitive material such as SiO2 or mica. For general purposes, the insulating layer will be considered in a field analysis which includes the effect of the dielectric constant and the distance d between the rf circuit and solid-state material, i.e., a slow wave circuit is located at x = d. A semi-infinite region x . _ _ j _BZD = v E1+ 32 (1 3w ) £1 JBZVO v E1 (1 ——-)vv E 0 (3.3.15) where ci = 33:. = speed of electromagnetic wave in medium:i o i 8i = g- = wave number in medium i. (3.3.14) and (3.3.15) are called the general wave equation and the collision dominated wave equation of a majority carrier in solid-state with dielectric constant 82. As seen in the two equations, both equations include collision and diffusion constants. The more general form of Eq. (3.3.15) can be derived in terms of the ac velocity and dc velocity for the analysis of transverse wave device. A little algebraic manipulation leads to: 22 2 2 2 B w v E1+32E1+j—:Eg— {71 -j%?-vo (v-ia’l) =0 (3.3.16) where 2 2 32 t‘ 32 V E1 = X (5—2' f 5—2951x + Y (5’2' 5—29 E1y + Z (5-2' 5‘29E12 * 32 ' + “ 32 = x (3.2? - kzieix + Y (a 2— - k2)Ely + z (537 - k ) E12 (3.3.17) 351 . o .-_-_ .__X .. 4 vEl ax Jkfiiz (3.3.18) A 2 . A wo-El) = x (SEEM — jk 23—5—12.) + z (41(ng - 18512) (3.3.19) 0 3.4 Boundary Conditions in Semiconductors The finite solid substance (or waveguide) which are bound to a dielectric material are used in various microwave components. A typical configuration is an example which is shown in Figure 3.2.1. The propagating modes in dielectric substances of this type are not, in general, TE or TM modes but hybrid modes. In other words, a finite structure of the substances containing mobile charged carriers cannot support independent TM and TE modes. This inability results from the coupling between two 32 types of waves brought about by the motion of the charges. Such coupling between TM and TE waves vanishes when B0 = O or B0 = e3, and also for waves with infinite phase velocities (cutoff points in the dispersion diagram), [8T2]. The coupling can be also neglected if the carrier density is very small. For the traveling wave amplifier tube, the analysis falls into the category of pure TM and TE waves since an infinite dc magnetic field is assumed. In our structural configuration, TM and TE waves can exist independently because dc magnetic field is not applied to the device. Then, the wave solution of the carrier wave equation can be subdivided into two types, namely TE and TM modes. According to the two dimensional boundary, the TE wave has only the transverse component of the electric field and hence cannot couple with the drifting carriers. Therefore we are not interested in the TE solution where interaction is concerned. We will only consider the TM mode since a longitudinal electric field exists which couples the circuit and the carriers. However, to understand the complete motion of the charged carrier and to determine unknown coefficients of field solutions in a coupled circuit system, the TB solution is also necessary. From the two dimensional assumption we made earlier, §%-= 0. Hence the TM wave consists of E , E , and H and the TE wave E , 1x 12 1y 1y H12, and H1x in solids. Field solutions will be derived in the following three sections. Just as the diffusion constant in wave equations was included, the diffusion effect of the carrier in semiconductor should be taken into account. Further, the conductivity of semiconductor is finite. Therefore, no surface current flows and the tangential component of magnetic field I'E'CI vani C091 Sid? pref Us (.11 str 33 veCtor fl is continuous at the boundary. In addition, the rf velocity vanishes at two semiconductor boundaries. The boundary conditions used can be summarized as: 1. The tangential electric field component is continuous at the boundary x=-5. 2. The tangential magnetic field component is also continuous at the boundary x = -5. 3. The normal rf velocity vanishes at the boundaries x = o and x = -5. These three conditions will be used in formulating the field coefficients matrix and in determining the admittance functions in solid- state region. Another five conditions about slow-wave circuit will be presented in Section 4.2. 3.5 Field Interpretation in the Solid-State From Eq. (3.3.14) a general wave equation for a charged carrier stream is: 2 2 :3: 831’. + 83v. .. v 61+ 32 (1 — “’“o )31 - J m ”$1411.37” v-El =0 (3.5.1) Using algebraic operation of' 2 Eq. (3.3.17) through Eq. (3.3.19) the wave equation is split into three component form. Since the direction of a bias potential is easily varied, the direction of carrier velocity may be adjusted such that V0 = no 2 for convenience. A. The wave equation for x-component: 2 8251 mm 2 up 2 . wmv 3 1z _ .._._.. 4, __ [82 (l ’35—) " k JElx ‘ JR (1 --B'Z—v—) 3X—- - 0 (3.5.2) 6x2 Bgvt2 V 2 t B. The wave equation for the y-component: EEEI. 2 2 sz 3x ' [k - 82 (1 - $d‘9] E1x = 0 (3'5'3) V 34 C. The wave equation for the z-component: 2 2 . 2 2 U 82v2 E 325,; 82 “p 2 vt . 82 o 2 t a 1x _ 8x I m (m - w - kuO - k mv ) E12 + Jk (1 -m k - mm ) 3x - 0 (3.5.4) For simplicity, let (I) a2 = k2 - 82 (Li) (3.5.5) 2 mwv w w2 kZVZ bZ=——‘3-(-w+k ._P_____* t) (3.5.6) 2 0 w vt v and WV 8 = j 8 v. (3.5.7) 2 t Then the three wave equations are reduced to simple form as: 32 3512 (5;: + 3832) Elx = jk (1 + jg) 5x— (3.5.2) 3 2 1 (5:7" a ) Ely = 0 (3.5.3) and 2 BE 32 . b2 _ . 82110 1 1X (3x2 - J Y) E12 - Jk (1 — wk + 3—8" ax (3.5.4)‘ Before obtaining the solution one may compare this analysis with a previous simple one-dimensional analysis [VUZ]. For one dimensional a 3 2 2 . —=——= = = 3: case 3x 3y 0. Hence, a b 0 Since E1x . 0, E1), 5 O and Elz # 0 even for the one dimensional analysis. Therefore, 2 2 w 2=w_ -_E_ k '5? (1 2 v and w2 ku = w --JB- 0 mwv The last equation holds under the assumption that D = 0. If we can further assume Im-ku0|< to x give E1x l = j_k_ ax _ -ax jbk(l-j‘~'_) bx _ -bx Elx £1 (A1 6 Aze ) + —;§:3—§5 (A36 A4e ) (3.5.15) 5' For the TM wave the only nonvanishing component of the magnetic field is Hly while H1x = Hlz = 0 due to the two dimensional geometry. The evaluation of H1), is done by substituting the relations Eqs. (3.5.14) and (3.5.15) into the curl equation of E1, Eq. (3.3.1)- l‘——3 ln—J 37 2 2 2 .1 2 2 6 c 2 2 (a -k )-J-(b —k l _ 2 2 a -k ax -ax g bx -bx Hly - jw [ a (Ale -A2e ) + 32 - jbz b (A3e -A4e )] (3.5.16) 8’ Similarly the solutions of TE mode is obtained from Eqs. (3.5.10) and (3.3.1) as Ely = Ble + Bze (3.5.17) -k€2c2 ax -ax H1x = m (Ble + Bze ) (3.5.18) E2c2a ax -ax “12* i w (Ble — Bze ) (3.5.19) where 81 and B2 are all constants -— noting that the eJ(Wt-kz)is suppressed in all field equations of Eqs. (3.5.14) through (3.5.19). All fields in solid-state are obtained in terms of x- and z-coordinates. Next fields in the substrate region should be found to determine unknown coefficients. The conduction current in the insulator region is zero, so the wave equation is reduced to 02 51+ '1 251 = 0 (3.5.20) 3x7 u where Y 2 = k2 .- (1)211 CL. (3.5.21) 5 0 Keeping in mind that Y has a principal real positive value and that fields vanish at x = -w , the electromagnetic fields in the substrate region of dielectric permittivitys:4 are: TM wave Elz = F4eth (3.5.22) Elx = Jy—k thew!“x (3.5.23) H1y = If“ 4&4)‘ (3.5.24) TE wave 51), = 6464" (3.5.25) 1x ='k_€:)_ca 264.94" (3.5.26) = jump” eh." (3.5.27) .‘15. a.) 5y 38 where 3 1 c4 Vuo€u All of unknown coefficients will be interrelated in Section 4.5. Waves can be interpreted by investigating the wave equation, Eq. (3.5.1). PuttingV- E = 0 in Eq. (3.5.1) produces the solution of eaxand e-ax therefore this wave associated with a is called the solenoidal wave or + transverse wave. Similarly, since V x E = 0 yields the solution of ebx bx, the wave with b is named as the irrotational wave or longitudinal and e- wave. The longitudinal wave is associated with space charges in the semiconductors and this wave is originated from diffusion in the equation of motion of the carriers. According to the device structure both of the fundamental modes must be excited. It is also noted that without the carrier motion, two modes cannot be coupled at all in the solid-state. 3.6 Field Analysis in the Influence of Collision Effect In the previous section general electromagnetic fields in solids are derived without any simplifying assumptions. Practically, a large amount of collisions in solids exists due to the ever-present thermal vibrations of the lattice. Besides photon scattering of the carriers, scattering effects are from ionized impurities and neutral impurities which let the collision frequency incarese. The analysis in some collision effective materials may be assumed to be Iw-kuol<—9 41 where 2 2 ”p2 = k .. — z"2 2 3%; l (w-kuo] (3.7.5) 2 . m b2 = £34.}... -( w - 1010)] (3.7.6) m w- ku O 2 2 BE Expressing 110 in terms of a2 and b2, and substituting of'5;—} from Eq. (3.7.2) into Eq. (3.7.4) yield 32512 _ 2E _ 5;?“' 32 12 - 0 (3.7.7) According to arguments similar to Section 3.5, electromagnetic fields can be obtained as: TM wave E = A e32x + A 6.82X 12 1 2 (3.7.8) . a X -a X = 15. 2 2 51x 32 (Ale - Aze ) (3.7.9) 2 . LU a X -a X H = 1&2— _ .__._2__ 2 __ 2 1Y 32 (1 w(w- kuo)) (Ale A29 ) (3.7.10) TE wave E = B eazx + B -a2x a X -a X = -kC C2 2 2 H1x ——63;—-(Ble + Bze ) (3.7.12) - .8 C-a2 82X -32X le - 3—254——-(31e - Bze ) (3.7.13) It is noted that for one dimensional analysis by putting %;-= 0, this solution can be reduced to the wellknown Hahn-Ramo space charge wave solution [vu1]. In this chapter the semiconductor slab is assumed to be so thin that reflected waves should be considered. However, if the slab is 42 thick enough such that the reflected waves (e-ax and e'bx) may be neglected for electromagnetic field solutions, all equations can be reduced to the simple forms, which are tabulated in Appendix B after applying boundary conditions. CHAPTER IV FIELD ANALYSIS OF SLOW WAVE CIRCUIT AND FORMULATION OF LINEAR EQUATIONS FOR THE COMPLETE SYSTEM 4.1 Introduction In the preceding chapter the complete field solution was obtained in the region of the solid—state medium. This chapter will be devoted to the field solution of a periodic array of slow-wave tape structures by symmetrically spaced fingers. It is of course impossible to obtain an exact solution to such a complicated geometry; however, by idealizing the circuit it is possible to get a closed form of the solution for the model depicted in Figure 3.2.1. The analysis is restricted to idealized tapes of uniform width and zero thickness. An idealized model of the meander line circuit is then constructed by introducing electrical shorts between tapes at appropriate positions along the tapes although the model may not be easily fabricated practically. It is desirable at this point to review several typical slow-wave structures before analyzing the main subject. Figure 4.1.1 shows a variety of slow-wave tape structures which can be used for solid-state traveling wave amplifier. Figure 4.1.1 (a) is a helix tape line, which is a popular structure for a travelling wave amplifier tube and Figure 4.1.1 (b) an interdigital tape line. Figure 4.1.1 (c) represents meander tape line which is used for our purpose of the analysis and Figure 4.1.1 (d) tape ladder line. The two structures illustrated in Figure 4.1.1 (b) and (c) are complementary or dual structures. At millimeter wavelengths 43 44 a heix has too small a diameter to be a useful slow-wave structure and various forms of interdigital lines, meander lines, and ladder lines are preferred. mm // // . 2 2 2 //////l/////// \“ ////. (C) (d) Figure 4.1.1 Several typical tape slow-wave structures. (a) tape helix line (b) interdigital tape line (c) meander tape line (d) tape ladder line. In general, a periodic array of uniform tapes can propagate a variety of TM and TE waves corresponding to arbitrary excitation of individual tape elements. The power of various modes is delivered in the direction of group velocity of the system, which was shown in theorem 2.5.2. In a general slow-wave structure, there are Hatree spatial harmonics associated with wave propagations as indicated in Eq. (2.5.16). For simplicity, only a fundamental harmonic will be treated in the 45 analysis. One more remark on evaluating the gain and the field distribution of the output is important. The values of the gain and the field at the output terminal can be determined by applying the Floquet theorem that the output signal is different from the input signal in phase by the length of the active region of the slow-wave circuit. In this chapter, fields about the circuit structure are analyzed and then fourteen linear equations are also formulated by applying all conditions, while unknown coefficients are interrelated in the ratio of the first unknown coefficient, namely A1. 4.2 Boundary Conditions about a Slow—wave Circuit Structure For a tape structure type of slow-wave circuits, two specific boundary conditions are used to obtain unknown coefficients of the field solution. Those conditions were originated by Chu [CH1] and afterwards were used by several authors [C02], [HUS], [P12]. Boundary conditions are: 1. Along the direction of the tape-line, the tangential electric field at the edge of the tape must vanish since the tape is considered to be a perfect conductor. 2. The component of magnetic field intensity E1 tangent to the tape-line must be continuous since no current flows perpendicular to the tape. In addition to these two conditions some typical boundary conditions are also used in this case. 3. The tangential component of electric field intensity E1 at the boundary surfaces x = 0 and x = dgis continuous. 4. The tangential component of magnetic field intensity El at the boundary x = 0 is continuous under the assumption that there is no surface 46 current, which is the same as the second boundary condition in Section 3.4. One more condition should be added up according to the chosen geometry with the top insulating layer (or free space) in the positive space. 5. In transverse direction, the electric field intensity at positive infinity will vanish since no growing field exists in the passive chosen circuit geometry. These five boundary conditions about the slow-wave circuit region and another three boundary conditions in the solid-state region will be used together to evaluate unknown coefficients of field solutions and to construct the field coefficients matrix for the coupled system. 4.3 Electromagnetic Field Solutions in the Slow-wave Circuit Region when the Permittivities of Two Insulating Layers are the Same Here, the problem will be solved in the case of same permittivities of two insulating layers surrounding the slow-wave tape structure. The geometry of the structure is illustrated in Figure 4.3.1. The infinitely thin meander tape line is located in the direction with an angle 4 to the y-axis between two insulators with the same relative diolOCLric (iii) / /:/ / p——n————-———--———--———--———-—4-- Z O “2 (11) y x=-(§ E4 (IV) Figure 4.3.1 ueometry or the device structure with the same dielectric constant around the circuit permittivity 61- Then the circuit phase velocity in the medium I becomes c1 tan w where c is the medium wave velocity. 1 47 The wave equation of the electric field E1 in the insulating layer with permittivity El becomes VZEI + BIZEI = 0 (4.3.1) 2 or %;§1-- YIEI = 0 (4.3.2) where yf = k2 - sf: 1.2 _ wzu 6 (4.3.3) 0 1 Eq. (4.3.2) was obtained on the basis that the wave is traveling in the positive z-direction of the type ej(wt’kz) and that the Laplacian v2 is separated into gig-~183for a two-dimensional analysis. In the analysis of the traveling-wave tube amplifier, the helix circuit can be approximated as the helical sheath model and its result is in quite good agreement, but the meander circuit model is a more complicated structure. The complexity of the meander model may be deduced if the meander tape can be reduced by taking an infinite radius in the helical tape. The quasi-helix model will then be chosen for our structure. The TM and TE modes can be determined by the fact that they are coupled by the boundary conditions of the meander circuit tape. It means that both the TB and TM waves must be excited in the actual device due to the meander circuit structure, but these modes in the semiconductor need not be strongly coupled. This will be discussed later. The electromagnetic field in various regions can be obtained by solving the second order partial differential equation, Eq. (4.3.2) with the given boundary conditions. In rectangular corrdinates, the electro- magnetic fields in circuit regions are: A. Region III (xzd) TM wave - ‘Y x 4.3.4 Elz Fl8 1 ( ) . = .13 -‘Y x 4.3.5 EIX Y1 F10 1 ( ) H = 1331: 6‘le (4.3.6) 48 TE wave _ ‘le E1y - Gle 2 (4.3.7) H =-__J__chc G e-le 4 3 8 1x w 1 ( ° ° ) 2 =-j)r) 6)“ -le Hlz w Gle (4.3.9) 8. Region I ( Ofixid ) TM wave = ’Y x Y x Elz er 1 F30 1 (4.3.10) =’_j_k_ ”le _ le E1x Y1 (er F36 ) (4.3.11) =.E_€_L . -Y X __ YX H1y Y1 (F20 1 F3e1 ) (4.3.12) TE wave _Yix Y x Ely - Gze + G30 1 (4.3.13) =‘KF1C12 ‘Y'X Y_< H1x w (Gze 1 + Gsel ) (4.3.14) —°Y g 2 .. : 4.13.1. Y X _ X- le w (626 1 5386 ) (4.3.15) where F1, F2, F3, 61’ G2 and G3 are all unknown coefficients, and eJ(wt-kz) is suppressed in all solutions. 4.4 General Field Analysis in the Circuit Region In the preceding section, the electromagnetic fields were solved when the same insulating materials are used on both the top and bottom of the slow-wave circuit. In general, different materials with different permittivities are used on different parts of the structure. The other insulator (or free dielectric) with permittivity £3 for various generalizations is assumed in Region III as shown in Figure 4.3.1. Similar to the previous steps, the field solutions in Region III can be written as follows: 49 TM wave .1 'Y,—X Elz Fle J (4.4 l) E =13. p e'Ya" (4 4 2) 1x Y3 1 ' ° Hly = 11321 Fle 1’3" (4 4 3) 3 where 2 73 = k2 - a: = k2-w2u063 (4 4 4) TE wave = ’Y3x Ely Gle (4.4.5) - 2 - HI" = LES—3— Gle Y2." (4.4.6) . ~ 2 _ Hlz {lg—27391 Gle Ya“ (4.4.7) The solution type of electromagnetic fields in Region I is identical to those in Eq. (4.3.10) through Eq. (4.3.15). Field solutions around the circuit region are listed in Appendix B. 4.5 Formulation of Linear Equations and Determination of Unknown Coefficients for the Coupled System We have discussed field solutions for the whole system in Chapters III and IV, but we have not dealt with the determination of unknown coefficients in the field solutions. Now we are in a position to apply all boundary conditions in Section 3.4 and 4.2, to formulate linear equations for the coupled system and then to interrelate all coefficients in terms of Al. For convenience, let us consider the boundary conditions for the lower half region of the system, including the semiconductor slab. The noraml rf velocity vahishes at the semiconductor boundaries, accordingly two linear equations are obtained, SO (Al-A2) - (As‘A4) R = 0 (4.5.1) (Ale-aé- A2636) - (Ase-bé- A4eb6) R = 0 (4.5.2) _ l-jl/g Vt 2 b2(1 - j 1/g) where R - (ab)[ - 52: j bZ/g + (6;) ( 32- j bz/g 1)] (4.5.3) Further, the tangential boundary condition of electric and magnetic fields at x = -6 gives four equations. For the TM waves one has :a6 a6 -66 66 _ -Y,6 Ale + Aze + A36 + A46 - F4e (4.5.4) £2C22 a2- k2 -36 a6 32— k?. j (b - k2)/g -66 66 jw a (Ale - A2e ) + 32- j bZ/g b(A3e - A4e ) _ 123.}? e-YQG y 4 1, or -a6 a6 -66 66 _ —y 6 hl (Ale - Aze ) + h2 (A30 - A4e ) - F4e 6 (4.5.5) where 62 Eg_2 Yu 2 2 111 - (C?) (m) ("g-Mk - a) (4-5-6) ‘ 2 k2 E C k2— a2) + ' b - -2 _22 ( J “2 * (cu) (m ) (l,b) 2 g (4.5.7) aZ— jb 8' and for the TE waves -a6 a6 _ -y 6 Ble + Bze - G4e a (4.5.8) 2 2 2 ' C - 'Y . .. 2E£f334(81e 35 - 82836) = l_sEaE&—G4e Via (4.5.9) Thus, for the semiconductor region, six equations have been formulated, which determine the characteristics of a carrier stream in solid-state materials. By manipulating Eqs. (4.5.4) and (4.5.5), the set of six equations can be written in two simple forms L 51 ._ l R -R A2 1 - ad 1 Re-(a+b)6 Re(-a+b)6 A3 = e 2 A1 (4.5.10) - 6 5 - 5 (1+h1)ead (1-h2)e b (1+h2)eb A4 thl-l) e a .‘ h J A and for the TB waves e-ad ea6 B1 1 _ 6 a6 36 - Y e Y“ 64 (4.5.11) - _ — .3. e e B2 [.8 i After a few algebraic operations Eq. (4.5.10) leads to a coefficient expression in terms of A1, namely A2 n2 A3 = "3 Al (4.5.12) A4 I H1, is J n. A where a6 . e - (h2 + R(h1-1)). SlHJ b6 - cosh b6 - a6 H2 = -ad ' . ._., e e - (62 + R(hl+l))osinh 66 - cosh 66 (4.5.13) e(-2a+b)5 (1 + 62 + R(h1-1)) - eb6 (1 + 62 + R(h1+l) + 2Re'35 113 = - 2R [e a5 - (62 + R(hl+1) sinh 66 - cosh 66] (4.5.14) e(.2a+6)6 (61-1 + R(hl-l)) - e'b6 [(62-1) + R(h1+1)] + 2Re'35 H6 = 2R[ e‘ a5 - (h2+R (h1+1) sinh 66 - cosh 66] (4.5.15) Similarly, all relations in case of a collision dominated stream are reduced to little simpler forms, replacing a by a b by b and g l’ l by g1 in the above equations. The collisionless case is also obtained even if it is not realistic in most semiconductors. The expression for F may be obtained directly from Eqs. (4.5.4) 4 and (4.5.12), in the ratio of A1, E4 = champs, H2e(a+yu )6 + A1 H3e(-b+Yu)6+ H6e(b+yu)5 (4.5.16) 52 Up to this point, four unknown coefficients in the ratio of A1 have been determined. The remaining coefficients can be obtained by supplementing linear equations about the slow-wave circuit. Around the slow-wave circuit, the first condition is that the tangential component of the electric field is continuous. Hence -Yd ~Yd Yd = 1 . Fle 3 F26 + F30 1 (4.5.17) Gle'Ysd = GZe‘YId + G3eY1d (4.5.18) The next condition is that the electric field perpendicular to the direction of conduction is continuous; thus Gle-Y3dcos 4 + F1043d sin V = 0 (4.5.19) 01‘ G1 = - F1 tan W and that the tangential component of the magnetic field parallel to the direction of conduction is continuous, which yields w c1Y3 (F2 - F3 e a ) + Cl clyiy3 (G2 - 636 . ) tan W e(Y1'Y3)d + Y3 G e(Y1-Y3)d tan W (4.5.20) = 2 w 5171 F1 1 c3 CsYl Finally one applies the condition that the tangential component of fields at the semiconductor slab x = o is continuous and hence four more equations are obtained, for the TM waves A1 + A2 + A3 + A4 = F2 + F3 (4.5.21) th1 (Al-A2) + Alh2 (AS-A4) = FZ—F3 (4.5.22) where A =?.‘£.L o e y 1 H and for the TB waves B + B = G + G (4.5.23) B - =--E} (62-63) (4.5.24) 53 Fourteen independent homogeneous equations for fourteen coefficients have been obtained for the configuration sample. Algebraic manipulation of the above equations determines general forms of the rest of unknown coefficients, with the result 2(1 * H2"“34'“4) - 31+ 32 (4.5.25) 212° II where N = (13:12-6236 cosh y d + 3-(1 — 3113- e'236)sin6 y d (4.5.26) 1 ‘ l 1 a+Yh 1+y d 2 = N1 6Y1 _ QYld l m— (l + C ) (4.5.27) -288 _ l. -3. ‘3 u-Ya. .3_ K1 2 (1 Y1) + 2 a+Y l + Y ) (4.5.28) 2:15 1 - l 9. <__ a-YH 3.. . K2 —22 (1+Y ) + 2 3W“ (1 - Y1) (4.5.29) 2 . d c C t. Y Y Y Y 32 [ijr——(K2e K1) + QQEf—L_“ }] tan €1Y3 tan W -Zyld (l - e ) (4.5.30) 22. = 0.236 a -Y“ 1+n25'nl+'nk A1 ° 3 +y1+ ° '31 + 22 (4.5.31) El. = ZNl 6’Y3 d 1+H2;+H3-+Hq 1 tan W ' 21 - 22 (4.5.32) E; = 1+HL+ (13441;, L -__.__._ - Z 1‘ - A -3 + 3 - ( ;2, (, —J—:;;Y 6“) (4.5.33) 1 1 2 l-e 1 l+e 1 £3. ___ 1+H2: 1134-111, (Z ‘1 + £1 . —27W A1 -31+ 32 1 _1 C'EV]J+1 (4.5.34) 2N 1 Y3 d 91 = 1( +n2+n3+ng 6 (4.5.35) A1 '21 + 32 92_ = 2K1(1+H2f“3Hha 4 . A1 -31 + 22 ( .5 36) 91’ = 2K2(1+H2+H3flhp A1 -31+ 22 (4.5.37) and 54 9!. = 4a (1+H2+ 113+ UL, ) 6(Y4-a)5 (4 5 38) A1 ('31+ Z2) (a +Y6) ' ' Therefore, there are fourteen unknown coefficients in field analysis of the system and they are all interrelated as shown in this section. CHAPTER V DISPERSION AND GAIN CHARACTERISTICS OF INTERACTION BETWEEN CARRIER WAVES AND SLOW CIRCUIT WAVES 5.1 Introduction In order to investigate the behavior of the coupling effect of a tape circuit with carrier waves, it is necessary to have a detailed description of their dispersion characteristics. In the characteristic equation most of the important information of the waves is obtained. As was mentioned in Section 2.5, all quantities are assumed to be separated + + + J.(mt-kz) into dc and ac terms of the form A = A0 + A e . The effective 1 mass of carriers m* is considered to be a scalar constant. In addition, the carrier collision frequency is assumed independent of carrier velocity. The dispersion characteristics can be obtained by matching the wave admittance functions of two systems at the circuit-semiconductor boundary. Another alternative of deriving the dispersion relation is to set the determinant of the coefficients matrix equal to zero for a non- trivial solution. This chapter deals with two ways of finding characteristic equations, compares these equations, and afterwards gain relations are derived. Numerical solutions will be plotted using a CDC 6500 computer. Dispersion characteristic equations for some cases are derived in Section 5.2. Section 5.3 is concerned with some appropriate approximations made for some typical cases, and dispersion and gain relations are accordingly obtained. Section 5.4 introduces normalized constants. The 55 56 last two sections investigate the effects of collision frequency, slow- wave circuit velocity, carrierdrift velocityand insulating layer thickness. 5.2 Dispersion Relations for Carrier Wave Interactions In this section, we first try to formulate a general dispersion equation relating to solid-state traveling-wave amplifier devices. To achieve that, two methods are used, namely, the field coefficients matrix method and the wave admittance matching method. As has been discussed earlier, in connection with the method of solution, these two methods are based upon the fact that the slow circuit waves are propagating with a phase velocity equal to the carrier drift velocity (i.e., they are synchronized). Following that we consider the special case of carrier interaction from the point of view of a collision dominated stream. 5.2.1 General Dispersion Relation A. Field Coefficients Matrix As was derived in Section 4.5, application of the boundary conditions to the structure configuration produces fourteen homogeneous linear equations subject to fourteen coefficients. These linear equations may be written in a simple matrix form. [A] [X] = 0 (5.2.1) T where [x] =[ A1,A ,A ,A ,B ,B ,F ,F ,F ,F G 62,6 ,6 ] and 2 3 4 l 2 l 2 3 4’ 1’ 3 4 [A] is the field coefficients matrix defined in Figure 5.2.1. A necessary and sufficient condition for a nontrivial solution to Eq. (5.2.1) is that the determinant of the field coefficients matrix [A] be zero. :ofiumuswwmcou opsuosuum on» we xwuumz mucofioflmmoou vHon H.m.m ouzmfim a m>H 57 H a m H m 3 m e N 3 H u o a can > >oliv rllv n c u < 9 can A U u < 1 u < owes: me N m H>mm m H>Ho H>zo 6 <- 6- H 6N6- eHmsc>v HUHsm HH>H>V a :3 O m IO.- e 6 0 0 0| 6 eH> 6H»- an». .n:. H- H O O H H- N; a N; < Haoq Hcoq H H H- H- H H H H 0! mm mm- 0 6a 66- N N H H 0| 0 0 O 0 63>- on a on- a on n we. ; 0| 0 O O 0 as»- an an- we on. menu men-6- 666- 66-6 m m- H- H det 58 Using row operations of the matrix algebra to reduce the size of a 14 x 14 square matrix, we obtain the following 9 x 9 matrix, after long algebraic rearrangement, 1 1 -l -R R 1/f2 -1 -R/fg Rg/f (1-61)/f (1+hl)/f (1-h21/g (1+h2)/8 thl -th1 thz -th2 -1 l l l 1 1 -1 -1 '1 1 “2 (5.2 2) Kl+u2K2 -1 tan ¢ 1 A1(K1-u2K2) -A2 1 -u2 -A3 where f = e36, g = eba, and u = eyld. This equation does not seem to be appropriate in taking it as a final form for the characteristic equation and hence further reduction should be carried out. Reducing the sixth column and the ninth column in order and taking the determinant the matrix yield a comparatively simple 7 x 7 matrix of the form, - l -l -R R T 1/f2 -1 -R/fg Rg/f (1-611/f (1+hl)/f (1-h2)/g (1+h21/g = 0 1 -1 hz/hl -h2/h1 -1/th1 1/th1 1 l l l -1 -1 31/(1+ié) 1 u2 32/(1-12) l -u2 II 59 Applying the Laplace expansion theorem to Eq. (5.2.3), this determinant can be expanded as follows: Dl . D2 - 03 . D4 = 0 (5.2.4) where 1 -1 -R R l/f2 -l -R/fg Rg/f h = = r _ _2. _ Dl (1-h1)/f (1+h1)f (1-h2)/g (1+hz)g det B 51 II2+h1U13 H4)] 1 -1 62/61 -h2/h1 -1 -R R det B = -1 -R/fg Rg/f (hr-11f (1-h21/g (Ia-2):; o -1 -1/4 _ zu2/ 2 1 1 _ -z- z 3 3 02’1““) _-- 162*???)‘(5157'1‘3‘571 2 - zzu /(1-u2) 1 1 1 -1 -R R 03 = I/f2 -1 —R/fg Rg/f = det B (1+n2+n3+n() (l-h11/f (1+hl)f (l-h2)/g (1+h2)g 1 1 1 1 and 2 _ A A 0 1/ Ohl l/ ohlu _ 2 2 _ 1 _ 2 z 3 £1. _ 3 D4 ' Z1“,/(1*“ ) 1 1 ‘thl E ” (1+uz + 1-u2)+(1+07' l-u2)] 2 Zzu/(l+u2) l -l Since Eq. (5.2.4) is a result of det [A] = 0, it is clear that this equation represents the dispersion relation of the system. We now 60 substitute four determinants, D1, D2, D3’ D4 into the dispersion equation then the corresponding expression is thl(1-H2) + AOhZUI3- UL.) - K3(I‘l+tanh Yld) - K4(I‘2-tanh Yld) - _ (5.2.5) 1 +IIZ+II3+II., -K3(I‘ltanh Yld 4' 1) +K4(F2 tanh .Yld I) where _2Y1d Y _. Y _ _2a6 K3=(l+c )(__1_‘.1__._li__§.e ) Y +a Y +8 1 1- 2Y1d _ _ _ _2a6 K = (1 + e ) (1 (YHA?) (Yl-a) 4 (We) (- +a) Y - e y c p .J.+ _fl_l_.i._l__ 5: Y1( 2 82 L1 E'1Y3) 51 S3 - ...m_..: -_.L..= ° ' - . . 851 - v91 Cltanw Circuit propagation constant 1n medium I 853 = -9-= -&L—-= circuit prepagation constant in medium III v53 c3tanw and V51, v53 represent circuit phase velocities in each medium. Eq. (5.2.5) is an exact final form which has been derived without making any approximations. The dispersion equation can be simplified by arranging the left hand side of Eq. (5.2.5). Hence, one has K3(F1+tanh ydd)-K4(F2-tanh yld) Yubl -K3(F1tad1 yld)-K4(F2tanh Yld:l) Y1€u = (h1+h2/R)cosh 66 + R(h1+h2/R)(h1+h2/R+1) sinh b5 -8e'a6+(hith2/R+2)cosh b6+[R+l/R+R(h1+h2/R)]sinh 66 provided that a5<>|w-kuol, the dispersion relation is obtained by letting wv -jv. With the same type of insulating layers on the top and bottom of the slow-wave circuit, the dispersion equation becomes (1+e2kd)k2 — 8:1 e de E cosh (mp/Vt) + j S sidi (mp2vt) kd 2 2 2kd = K . w 5 . . (u 6__ (5'2'7) (l-e )k + 851 e T cosh ( p /Vt) + 3 W Sinh ( p /vt) where C K =31- (5.2 8) 2 E = EO + Elk + Ezk (S 2 9) s = so + 31k + 521-2 (5.2.10) 'r = To + le + 1*sz (5.2.11) W: W +w1< + Wk2 (5.2.12) 62 _p_2t v 2 muo 2 vt2 u o(3'11 [( 1 (11 +13?ng +wZ(C—1] u 2 v w v 2 w 2 2 v 1* v w 2 2 1% w[(C—°1 + (1-(‘3-1 (:91) - 2(;§—‘—1 ] - 1[;,1’- (1-(53—1 (El—11 +2;1;(C—‘—1 ] 2 2 2 2 2 v 2 me + 5 “C(31) (ESE) 1+ [112V (11 82 — [— v(1+ -—1 - ~13-][1(-1 (—-—1 J —2 ——2——R(1 + 2—+ 14 V: (11p Cu (:L‘v VC2 E2l+ 02 1+ [2 2(—1(;§1(J§— - ‘—"—(1+:——11- (Jinn—10421 (—--1 1] 2 (”P 2 m'vt wuo[(1+2 —1(J31(:—.E 1 2(1 :— (.- V1] +1u0[(1+2-:-—1(—R+ 3,721+ C2 1+) P t ‘* Vt P 2 vt m (31—) (32-) 2] P V11 2 E 3 (1+ 11 mp t CL. C U) V 2 E mu 8 E: (DV 2 (11 l 2 t 2 . O 2 2 t uO(C—-1[ 2339—1 - (2+ :1] + 1 35,157+ (2:1(71 1 2 2 i 2 2 412+ —1 [1+(—°1 - (—1(—-—1 ] -2(-—14(:§-v-t—1 +1(—1(—1 [:— (cc—£1 w- -2w-;,P—1 c2 C2 C2 51+ CV2 -4w] 2 E V uo(2+ 23101 + jig—($1 1 ‘1 2 1. v (.02 82 mzvt th-_6_2 :2_ _u_o_2_. (3,31 (3:1(I+;:1(1- (g1 (C—Z-1 1 — LV (C21 [2(a) + (C21 1 14 u E:2 Vt <11 2 E:2 :03 (112:1; [2 (1+ :1 (g) (3;) + E: (Vtm- (;1(C21 1 82 w v mu 2w2v £2 2 2 Zwuofl + ——)[v_P_V£ C2 5:3E2—E—2 + juo [m Czt (1+ Z—) - (53) . 2 thP 2 P 2 '1‘ 3 (——-1 (i1 + ——(—P1] C2 c2 Eh Vt 4hr»: 0 3.1 .311” .2! ‘12. w . fix .I. .4. .Pv V 63 and e u 2 w v v ” w v 3w 2 B t . 2 N2 = (1+ 5 )2 3 - v t(l-(%9 (c ) ) + 32 VZCE N p t 2 2 It is, however, a wellknown fact that the wave can best be amplified when the drift velocity is nearly equal to the velocity of the rf slow circuit wave. When this occurs, it is usually referred to as synchronous coupling. Therefore, if a device is used under this situation, the value of Im-kuol may not be a significant quantity. Actually the collision frequency in most cases could always be considered as much higher than Iw-kuol and thus a general case might be reduced to the case of collision dominated stream. 5.3 Relations of Dispersion Characteristic and Gain In practice, when the device is fabricated it is much easier to choose a single material for the insulating layers. Therefore, this case will be taken for our analysis hereafter. The bottom insulating layer of the circuit is not also required when the ideal capacitively coupled circuit is to be used. Hence, with d = 0, the dispersion relation becomes ‘§__’ K.E+jS tanh (wpdlvt) B )2 ' 1 = w 5 v 51 T+jW tanh ( p / t) In actual device fabrication, the insulating layer, d, is a few 2( (5.3.1) microns thick and then one may use the approximation of exé 1+x for small x (x< | > I 3 - 2[d(rO-kEo) -T1) + j 2[d(wo-Kso) - Wl]tanh(wp6/vt) (5.3.6) . 6 V = 2 . - ° 2 - (1) A2 351(T2+r52) 2T0 + )[esl(w2+K52) 2Wo]tanh( p / t) (5.3.7) 6 v = 2 - ' 2 “3 Al 851(T1+KEI) + J 851(W1+Ksl)tanh( p / t) (5.3.8) and A0 = 82 (T +KE ) + j 82 (W +KS )tanwa 5/vt) (5 3 9) 51 o o 51 o o p ' ' For better approximation one expresses the exponentials in their continued fraction approximation ex = l + 12x 12—6x+x2 provided that x 5 l. and then the sixth order polynomial is obtained as: 2d2k4+(6-d23§1)k2—3d5§1k-3s§1 E+jStanh(wp6/vt) (5.3.10) 3 _ -6dk +d23§1k2+3ds§1k+3s§1 T+thanh(wp6/vt) It can be easily seen that Eq. (5.3.10), of three dispersion relations, gives the best result in the frequency range of x-band, for our chosen semiconductor materials, InSb, Ge, Si, GaAs. In general, an explicit analytical solution for fifth or sixth order polynomial may not always be feasible. Even if it is possible, it is too complicated to derive the final form. One way of solving higher order equation is to approximate with the aid of a computer, using numerical methods. Once complex roots of k are obtained, in any way, the gain is also computed from the imaginary roots of k and is expressible as: Gain (db) = 20 log e(Im k).z Since loglo e = 0.434 and Im k = -a, it becomes Gain (db) = 8.68-(-a)-z (5.3.11) 65 The gain in db per mm becomes Gain (db/mm) = 8.68 x 10‘3-(-a) (5.3.12) where<1 is the metric unit. This gain expression can also be expressed in db per wavelength. 2H Gain (db/Y ) = - Gain (db/mm) x 103 (5.3.13) 0 Instead of using db unit, the gain per micrometer is also used for the microwave devices in an exponential form such as Gain (.nm) = Exp (-a x 10‘6) (5.3.14) In each expression, the gain is proportional to the imaginary part of the propagation constant k. 5.4 Wave Interaction Analysis with Numerical Method The dispersion relation of Eq. (5.3.10) will be solved by a CDC 6500 computer. It is useful to introduce normalized quantities for computer programming. The propagation constant is chosen as a dependent variable and the other constants as independent variables. The dispersion equation can be put in terms of normalized constants. The normalized Operating frequency and collision frequency are defined as (1) q = ;— (5.4.1) P — AL 5 - w (5.4.2) P Also the normalized drift velocity and slow circuit wave velocity are defined as c: O p _ V; (504.3) I. _ "s (s 4 4) - Vt o 0 With the use of these normalized constants, the independent variables for device parameters will be manipulated as required in calculation. 66 The purpose of solving the equation is to find the complex roots of the propagation constant k with arbitrary complex coefficients. Each root is located approximately by the Lehmer method and then improved upon by Newton-Raphson method. Then a reduced polynomial is obtained by removing the first root. This process is repeated until all of the roots are removed. The detailed explanation of this method is given in Appendix D. Some typical values pertinent to the description of solid-state materials that will be used for the numerical analysis are illustrated in Table 5.4.1. Redefining coefficients of k in Eq. (5.2.9) through Eq. (5.2.12) in terms of real and imaginary coefficients as E0 = Eor + JEoi E = E + jE 1 1r 11 E2 = EZr * jEZi S0 = Sor + jSoi S1 = Slr + 3.811 S2 = Szr + jSZi To = Tor + jToi T1 = T1r * jT11 T2 = T2r + jTZi wo = wor + jwoi wl = w1r + jwli and N2 = w2r + iji we arrived at the best approximate form of the dispersion characteristic equation with complex coefficients, which are rearranged by 67 Hr zczmwmn>r <>rcmm mow mmr mohacuma>am Z>meu>rm a>wrm m.a.H zmnwewmw msmemx wow wow. coma. a»\ao zocwwwnx mmmwdwm H.mu H~.m o.oo~ m.mxaow o.om oxaoo-pxwoo papa .m.mxaou N.mo doom u.meHom o.mo HoHu Hume o.Hm Hm.a o.oHa m.amxpom o.ou~ oxaoq-mxpoa Haas a.oxwom o.oHu deem 5 4 2 68 6 3 - 86k + 85k + 84k + 83k + 82k + Blk + BO - 0 (5.4.5) where _ 2 - _ 2 2 _ 2 X0 - 3851, X1 - dxo, X2 -6-(d351) , X4 = 2d , X5 = 6d, X6 - ((1851) Q = tanh(m 5/”t) p Eol = Eor - Soi.Q’ E()2 = 13oi + Sor “1 E11 = Klr ' Slj'Q’ K12 = K11 + Sir “2 E21 = E2r - S21 Q’ E22 : E2r + S2r 41 T01 = Tor - woi Q’ T02 = Toi + wor 42 T11 = Tlr ' "li'Q' T12 : T11 + wzr-'Q T21 = T2r ' ”21°Q’ T22 = T21 * ”21°Q B6 = x4(T21 * 3 T11) B5 ‘ x4(T11 * 3 T12) * Kx5(521 * J 522) B4 ‘ x2(T21 * 3 T22) * X4KTol * 3 T02) + K[xsmn + 3 512) ‘ X6KE21 I 3 522)] B3 ='x1[(T21 + 3 T22) I K‘EZI * J 522)] + x2(T11 * J T12) + K(X5(Eo1 + 3 £02) ' X6 (511 + J 512)] B2 = 'xo[(T21 K J T22) + K (E21 + 3 522)] ' X1[(T11 K 3 T12) + K (E11 + j 512)] + x2 (T01 + j T02) ' K X6KEol + j 502) Bl = “x0 [(T11 * J T12) * K (511 + 3 E12)] ’ XIKKTol + 3 T02) + K (E01 * J 502)] B0 =’xouTol + 3 T02) + K (£01 + 3 E02)] Eq. (5.4.5) is the approximate dispersion relation which satisfies the boundary conditions of the device structure chosen. 69 5.5 Solution of Dispersion Equations for Different Materials Numerical results of the dispersion equation in the previous section are presented and discussed in this section. Solid-state materials chosen for that analysis are InSb, GaAs, Ge and Si. The wave propagation and attenuation characteristics corresponding to the structure configuration should be investigated numerically for several materials. The f-B diagram is useful in interpreting the behavior of a wave propagation and the attenuation curve is helpful in determining whether growing modes exist. Both wave propagation and attenuation characteristics are determined by finding complex roots of the dispersion equation in k where k =[3- ju. The thickness of the semiconductor slab will be assumed 6 = lxlO-4m throughout the numerical analysis. 5.5.1 Continuous Type of Tape-circuit Model In connection with the sample configuration, if a continuous tape-line is used instead of a capacitively coupled tape-line, an insulating layer between the semiconductor and the circuit line should be deposited to prevent the dc field from being bypassed. This implies that the characteristic equation remains an infinite order polynomial in k since the thickness of the insulating layer d can not be neglected in the range of high frequencies. For the thickness of the insulator between the slow wave circuit and the semiconductor slab, d = 1 micron is used in the continuous tape model. Figure 5.5.1 shows the functional dependence of B vs. f, determined by Eq. (5.2.5) and Eq. (5.4.5) for a device of InSb wafer. The drift velocity was taken as u0 = 6 x lOSm/sec, the relative permittivity as €2r = 15.7, the carrier density as n = 1014/cm3, the normalized collision ~10 7O 10° p 105 ‘ I” C -—- approximate solution 10 )- ’ a ’ I --......_ exact solution V T )- E h \ Q 3 510 A A A L A 1 l A 1 .. 53 i 3 7: 5 B ‘ 5 '9 10 2-10 ‘ I o N L) g _ \D FREQUENCY (GHZ) \ °‘ \ T = 77°K ’ \\ \ s \\ u = 6x10 m/sec Figure 5.5.1 f-D diagrn for a n=10“/cm3 InSb sample 71 frequency as = v/mp = 1.0, the normalized circuit velocity as vs/uo = 1.0 and the carrier temperature as T = 77°K. Real roots of the exact equation are obtained by taking six roots of the approximate equation as starting solution of the numerical iteration method. Then both the exact and approximate solutions of the possible six modes are plotted for comparison. As can be seen from the graph, both solutions have the same general shape of variation, except that the exact solution indicates a slight faster phase velocity. The exact dispersion equation requires tremendous computing time and preparative effort to obtain the solution for each operating frequency. However, as far as two dispersion equations for an amplifying mode exhibit similar general trends in the nature of wave propagation and considering only the amplifying wave, the approximate equation may be used in order to investigate effects of material parameters for convenience. Figure 5.5.2 illustrates the variation of attenuation constant corresponding to the phase constant of Figure 5.5.1. Here, both the approximate and exact solutions of the attenuation constant are plotted. Since the attenuation constant is attained from the imaginary part of the wave propagation constant k, for forward propagating waves a negative value implies growing wave. It is noted that the exact dispersion equation gives a smaller growing factor than that of the approximate dispersion equation and only one mode indicates amplification. The results showed that the rest of the five modes are highly attenuated. In order to understand the wave propagation phenomena and to identify the growing wave, it is necessary to investigate the nature of the propagating modes from the f—B diagram. By comparing the phase velocities of the waves with the carrier drift velocity, one can identify A TZ'E/Vl/A 7704’ «.4 /~./ CONJI’A’V 7' 01): 0 U 1102 .~.-r '00.. 72 ——— --- —_———- —-——-———— .— ~— £1 -——-—---——- :---"--'"""""p___—_—-——______________________——- n = lOld/ch u a 6x10S m/sec o v ,/u = l s o I'RIL. HJLNCY ({le) —I‘ ”— .—— 10‘ 73 ——————————————- approxilate solution “ -o-_ —_.____ A \ Figure 5.5.2 Variation of attenuation with frequency 30 I _ _— — -— -—_ '—_ —— ——- _— — r a InSb sample 74 which mode is actively coupled to the carrier stream. The growing mode should be approximately synchronized (the same phase velocities) with the dc drift velocity of the carriers. As can be seen in Figure 5.5.1, the phase velocity of the forward propagating mode is approximately equal to the stream velocity of carriers. Therefore, the forward propagating mode which produces amplification is traveling along the device in the same direction and with approximately the same velocity f as the carrier stream. Using Eq. (5.4.5), also shown in Figure 5.5.3 is the variation of attenuation constant for four different semiconductor wafers, those are, InSb, Si, Ge, and GaAs. Parameters are chosen to have T = 77°K, n = E 1014/cm3, q = l, s = l and p/r = 1. Typical values for the silicon are l x 104 m/sec, c u0 = 2r = 11.8 and m* = 0°33"b and for the germanium u0 = 6 X 104 m/sec, K2r = 16 and m* = 0.04me while for the gallium arsenide, u0 = 6 x 104 m/sec, le = 12.5 and m* = 0.072me are used. 5.5.2 Capacitively Coupled Tape—circuit Model As was presented in Figure 3.2.1, if capacitively coupled tape lines are used ideally for a slow—wave circuit, the insulating-layer thickness between the circuit and the solid-state in the device can be reduced to zero because a bottom insulating layer may be removed. Under such a structure the dispersion equation is then reduced to a fourth order polynomial from Eq. (5.4.5). The fourth order equation jpredicts four possible waves to be traveled along the device. Two additional modes due to the effect of the insulating layer, and turned out a pair of forward and backward traveling modes. Figure 5.5.4 contains the results for the forward growing mode when the layer does and not exist. The upper curve indicates the gain A'I'I'ENUATION CONSTANTS (1(lm) -6x10 -5x10’ -4x10 -3x10 —2x105 -lx10 b n l 1 1 75 __ lnSl) Si h D b Figure 5.5.3 1 5 (1 7 8 9 10 FREQUENCY ((3112) f—u diagram of the growing mode for four different materials ATTENUATION CONSTANT.1( /m) 76 408 E- ( ~107 h I . v/mp = l .5 u = 6x10 m/sec . 0 material: InSb -106 - . d=1u 5 1i 1 l I l l l 1 1 44, '10 1 2 3 4 5 o 7 8 9 10 *" FREQUENCY ((illZ) Figure 5.5.4 Comparison of attenuation constants when the insulating layer between the circuit and the semiconductor exists and not exists. 77 of the InSb model without the insulating layer while the lower curve indicates the layer effect of one micron thick. For high frequencies the insulating layer makes a big difference in the growing rate but for lower frequencies it approaches a small loss. As a result of increasing the frequency of operation, the decrease of the net gain corresponds to the increase of the dielectric loss. A comparison of the continuous tape model and the capacitively coupled ideal tape model shows that although the insulating layer significantly affects the coupling of carrier waves with the rf waves, the behavior of the growing mode is nearly identical. 5.6 Gain Characterisitcs as a Function of Collision Frequency, Circuit Velocity and Carrier Drift Velocity In this section the effects on gain due to various collision frequencies, circuit velocities, and carrier drift velocities will be examined. For the purpose of this analysis a heavily doped N-type material is used with a majority carrier n = lOlS/cm3 and the lattice temperature T = 77°K. Gain will be plotted for various parameters of four semiconductor materials, InSb, Ge, GaAs and Si. The plot of gain as a function of the normalized collision frequency v/wp is presented in Figure 5.6.1, where the normalized circuit velocity for the continuous circuit model is unity. In gaseous plasmas the higher collision frequency causes gain to decrease. There are, however, two hypotheses for the effect of collision frequency in solid-state plasmas: Birdsall, Brew, Whinney, Haeff and Misawa argued that collisions would induce instability in solids and that collisions are another mechanism of a solid-state amplification [MIZ],[BI1],[BI2]. On the other hand, Vural and Bloom proposed that 78 collisions tend to decrease the amplification while collisons may lead to amplification in the presence of a magnetic field [VUl]. The first hypothesis does not physically interpret amplification phenomena. Even in the case of no collisions, the instability is due to interaction between the positive-energy-carrying circuit wave and the negative-energy—carrying slow space-charge wave supported by a stream of carriers. It is seen that the plot of gain has analoguous shapes for inaterials, Si, GaAs, Ge and InSb, and that with collision effects alone the gain tends to decrease, which supports second hypothesis. The effects of the normalized circuit velocity vS/uo is displayed in Figure 5.6.2, where the capacitively coupled model is used and f = 1x109 112. A larger value of the normalized circuit velocity indicates lower .gain. Therefore, the circuit velocity, for the optimum operation, should always be smaller than the carrier drift velocity. The relative gain for InSb, Ge, GaAs and Si as a function of the inormalized drift velocity is shown in Figure 5.6.3. The highest gain was obtained as the normalized drift velocity varied between 2.9 and 3.2. 'This statement disagrees with Sumi's simplified approximate analysis [SUI] in which the maximum gain occurs at uO/vt = 7.3. In reality, the highest possible value of the normalized drift velocity uO/vt is limited due to the hot-carrier effect [GL1]. The attainable highest normalized drift velocity in the liquid nitrogen temperature is around 2.0 for the indium antimonide and 0.5 for the gallium arsenide. Therefore, it can be concluded that for a good solid-state traveling-wave amplifier device, the lower collision frequency and higher carrier drift velocity material, and the lower temperature operation are all desirable. GAIN (db/mm) 10 l() 1() ---- InSb ...._-—. GaAs - . \ o \ ‘x '\ , x \ \ \ \‘ o \ \ -\ \\\\~ ‘.‘. \ \\ . ‘\\\\ “\ \ ‘5 ~““~ --- \\~ \\ \~ ‘5‘~ --—' Si 1— . “‘ (i6 : ‘~ GaAs : InSb . 13 3 n = 10 /cm ' = sun; ' T = 77°K . vgluo = l . . . l - . o-I I I I I J 11 [I l I I 0.5 l 2 3 4 5 10 15 20 30 NORMALIZFD COLLISION runounncr (v/mw) 1 Figure 5.6.1 Gain characteristics with respect to the normalized collision frequency for several solid-state materials. GAIN (db/mm) T = 770k 9 n = 1013/cm0 10’ n. (i 0 1" = lxl()"::3: . v/mp = 1 . InSb 104 __ Q 3 l 1 1 1 IO . l ' ‘ ° ‘ 0 0 o o O o e ‘ 4 S l S 10 15 20 NORM/H.121“) CIRCUIT V15|.()(II'1'Y (Vs/“0) Figure 5.6.2 Gain characteristits with respect to the normalized circuit velocity for several solid-state materials. GAIN (db/mm) 81 T = 770K 3 r n = lOlS/cm 10° - v /v = 0.5 . s t ° v/m = I o P 9 . f = 1x10 “2 InSb I? I . GaAsl I I I 4 10 __ I . Si I ° I ° I . l I ‘ I I ' I I I ‘ I I I 3 I 10 Q J J o o 01 J I l l 0 ‘ ' 'I I l .5 1 2 3 4 5 10 15 20 NORMALIZED DRIFT VELOCITY (no/Vt) figure 5.6.3 Gain Vs. normaIizofl friit velocity for various solid-state materials. 82 5.7 The Functional Dependence of Device Gain Upon the Variation of Insulating:layer Thickness The carrier wave is supported by the applied dc field. As was mentioned in the continuous type model, the degree of coupling is a strong function of the insulating layer thickness which separates the tape-line and the solid-state material. In the tape-line circuit structure the field configuration of the slow-wave circuit decreases transversally to zero in a finite depth. Consequently the excited field intensity in the tape-line becomes very weak, which results in weak coupling. Numerix:ally, extensive calculations were performed for four materials at several values of the thickness d. The functional dependence of gain is plotted in Figure 5.7.1 for the variation of the thickness of an insulating layer. From the result it cari be seen that a rather high gain is obtainable at a thickness of d = 167m. With today's integrated circuit technology, depositing such a thin layer on the semiconductor surface is difficult. Besides that, the other problem is in practically achieving a reasonable uniformity of a thin layer. The gain decreases drastically as the layer thickness increases as illustrated in Figure 5.7.1. The possible fabricated thickness ranges 10-6n.10’4m. Judging from the theoretical and practical point of view, the best result: of the device does not heavily depend upon materials themselves bUF hCHn to fabricate the device and to maximize their drift velocities. As £31? as the drift velocity is concerned, indium antimonide seems to be thfia favorable material for this type of device. It is also interesting to “Ote that the solid-state type of device is a potential device for high fre(Ilalency applications, which may be an attraction feature, as seen in GAIN (db/mm) InSb GaAs 103 h. Si 3 4— Ge .. .."“"' GaAs 2 n = IOU/cmJ 1” 1.. t = 30112 “‘5" I 'T = 77"K . \) 1) = I . /(p ' vs/u0 = l 101 1 1 l 1 17-, 10'7 10'6 10'5 10'4 10" 10' INSULATING LAYER THICKNESS (m) Figure 5.7.1 Gain characteristics for various values of an insulating layer thickness (d) 84 the analysis of this chapter. However, the circuit and semiconductor losses are a serious problem, and as a result a high gain as anticipated in the curve should not be expected. CHAPTER VI TWO STREAM INSTABILITY IN SOLIDS 6.1 Introduction The theory of traveling-wave amplification in a thin semiconductor layer and a slow electromagnetic tape line deposited on the layer was treated in Chapter III through Chapter V. As was mentioned in Section 2.6, instability can also be obtained between the upper and lower valley electrons without using a rf circuit. The instability mechanism in solid-state materials is explained by 'the concept of wave interaction which has been successfully used in (electron beam devices. Kino [K11] described the charged carrier motion in semiconductor by a space charge wave concept, referring to them as "carrier waves." The kinetic power carried by these carrier waves were given by Vural and Bloom [VUZ]. Assume that a Gunn sample of length I has been biased with a constant ‘voltage so that the net internal electric field reaches a negative- sloPe region which starts from a little over threshold voltage. Ridley [R11] [R12] proposed that this condition is unstable and leads to a theory «of lower and higher valley model. This type of transferred-electron device is completely different from the established devices such as transistors, tunnel diode, or varactor. The mechanism comes from the bulk property of GaAs and is not based upon the ordinary p-n junction theory. The device structure consists of an electrically uniform doping and geometrically regular semiconductor with two ohmic contacts for the application of a 'bias voltage. The instability occurs during a transit time between two 85 86 contacts; thus, the higher operating frequency is,the thinner the sample length becomes. Consequently, the power output for a given device decreases as the Operating frequency is raised. Several parameters such as material resistivity, temperature, and the low conversion efficiency induce a high power loss throughout the device which requires efficient heat sinking. Although, for continuous wave Operation, heat sinking is possible for small size devices, the limitation restricts practically the lowest operating frequency to around 4 GHz and the total active length limitation restricts the output jpower to several hundred milliwatts. However, the pulsed operation lets the power loss be reduced with the reduction of the duty cycle. Hence, the low frequency restriction is removed and pulsed output power can be increased by enlarging the sample length and thickness. Unfortunately, there is a limit to the pulse width to protect the danger of overheating (during the pulse period. The existing pulse widths and duty cycles are vdthin these limits for systems such as mobile radars, aircraft direction- and height-finding equipment and aircraft identification. This type of solid—state device can be designed to Operate in any part of the micro- 'wave equipments and it is small in size and simple. For these reasons the solid-state device coupled with the newly developed integrated circuit technology in electronics is compatible with the high voltage microwave devices. A simple configuration for transferred electron devices is illustrated in Figure 6.1.1. Since the reactance of the device is usually capacitive, an inductive reactance must be provided to obtain resonance at the Operating frequency. This mechanism is obtained by using waveguide, coaxial or microstrip cavities which are wellknown in conventional methods. 87 .meo_>oa xu_~_2=umcm Ecouumnozp com co_unc:wmmcco ofisdmmo: < ~.~.c oc:w_; zocom m:_::p moud>ep cocuoo~o noncommzscu .lll. /////UYIV (HIIH NW \\. ass—use n.1, .\\\\ m_;_ . osmuozvcm :»_3 - can”; mcmuczoe acozm o_2sum:ade coufimm mans zo~ #:mu:6uoz made up \\\\\|\ mumoa dauoa 88 The reactance may be supplemented by constructing the inductive iris in the waveguide. In addition, tuning can be accomplished either electronically or mechanically using varactors or YIGs. Typical varactor-tuned and YIG- tuned devices covers a 10 m 20% range for the frequency of 1 m 10 GHz. This chapter is devoted to describing the formation of two-valley model and to developing the two-stream instability analysis. Finally, a computer solution will be carried out, based on the theoretical analysis. 6.2 Formation of Two-Valley Model The transferred electron mechanism of some semiconductors such as GaAs, InP, CdTe and InSb was predicted theoretically by several workers. The Gunn effect is a typical example found in GaAs by Gunn in 1963. This effect has not been observed at room temperature and normal pressure in Si and Ge, mainly due to not being able to get a sufficient number of carriers to populate the negative effective mass states. When a large enough potential is applied across the sample, negative differential conductance is obtained. When this occurs, a phenological transfer mechanism of carriers from one energy band to the other is followed up in the physical sense. Once they are separated in two energy states, these carriers in two different states have completely different physical characteristics. Thus, two distinct groups of carriers moving across the sample can be considered, namely the upper and lower valley carrier. Then it is possible to explain the instability by two stream formation. The two valley structure in GaAs is illustrated in Figure 6.2.1 The energy gap separation between the conductance band minimum and valence band is 1.53 ev. Carriers are distributed between light-mass lower valley and heavy-mass upper valley whose minima are separated by an energy displacement of 0.36 ev. As shown in Figure 6.2.1, the mass of a carrier 89 Energy 1 * *1! = . 8 111K 0 O 09:50 Ocn /v sec L(lower valley) * mu=l . 2m uu=100cm2/v. sec U (Upper V3110Y) AE=0.SOCV conduction band B =1.53ev 8 __I._ /— Figure 6.2.1 Simplified energy iI-k (1001 -~“‘\\\\::::nce hand gand structure of two-valley model. The numerical values are taken from a GaAs sample. 90 in the lower valley is m; = 0.08 1118 at the minimum of conduction band and the mass of a upper-valley carrier m; = 1.2 m8 at the bottom of the upper valley. Subscripts 2 and u refer to the lower valley and upper 5000 cm2/ valley, respectively. Carrier mobility in lower valley (“1 100 cm2/volt- volt-sec) is much greater than that in the upper valley (nu sec). The numerical values used here pertain to GaAs. GaP has a similar band structure to GaAs material but its upper-valley mass is lighter than the lower-valley mass (AE uu_cp _auces_cozxo was acumuocooze N.N.c eezwmg mEu\>xv >+Hmzzkz_ :;m_; Uszuxgu 22~412< mm mg - sq m a A c m e m m H c d d u q d q q d u d q a n O 0 O G . . G . nu nu O ._ 325.898 93 no can: 000 90 . xuoezu m.Hm we pocouzm oom\ao no~x~.~uxaeaosv c h n m n h c~xm.c sfixo.~ :me.~ o~xc.m o.xm.m )OTHA IJINU I 8 (nos/m3)on 111 93 Under the assumption of Maxwell-Boltzmann distribution of carrier densities, McCumber and Chynoweth [MCl] calculated the carrier densities of lower valley and upper valley as n = n {l - (1+§§9 exp (-éE)} (6.3.1) 2 AE F T 1 + “ exp (' TFO AE “ exp (' Tr) AB n = (1 + a + ——) (6.3.2) u AB T L +Jl£ a carrier stream of lower valley J a II #0 Z Figure 6.4.1 An over—all equivalent representation of two valley model. A displacement current is flown to the circuit. The sum of the lower valley convection current J 2 and the 1 displacement current J in a volume of a carrier stream should vanish, 1d because the total current flowing out of the volume of a carrier stream is zero. In a stream 3” + 31d = 0 (6.4.3) Over the surfaces of a stream, SJ —> — 12, -> fsd 31d°dA‘ 'fsr, ["f12+(312+—5sz)] dA 3J11 K = - - 6.4.4 ’36 I 82 dz) d ( 1 Eq. (6.4.4) may be rewritten as OJ 31 1d _ 12 96 6.5. Circuit Equation of Carriers in the Upper Valley A one dimensional analysis will be assumed for simplicity through- cnrt this chapter. In Figure 6.4.2 a set of circuit equation can be written as 8V1 Z —7E?—-D- s Jlu (6'5'1) 3.} aJ lu _ V 12 az _ -Ysh 1 - 82 (6.5.2) where VI = rf induced voltage Jlu rf upper circuit current J12 rf lower-valley convection current. . j(wt_kz) . . . . . Since e type of var1ation for all quantitles is assumed, the above equations become -jkvl = Zleu (6.5.3) -Jleu = YSh V1 + JkJ the following circuit equation is obtained, 11 (6.5.4) Eliminating J lu’ 2 J12 = k + Yshzs (6 5 5) V1 szS In Eq. (6.5.5) series impedance Z5 and shunt admittance YSh were defined in Eqs. (6.4.1) and (6.4.2). Note that the convection lower valley current produces the circuit voltage in the equivalent transmission line. 6.6 The Electronic Equation of the Lower Valley Carriers From force equaqgon, rf velocity of lower valley carriers is: —» "g + Vtg + v” = 3;)— ( - El + T7 v V-El) (6.6.1) 02 pg where 97 I1) = - - ' VB w knot 302 v8 = collision frequency of lower-valley carriers 1102 = drift velocity of lower-valley carriers 02* = effective charge to mass ratio of lower-valley carriers vtg = thermal velocity of lower-valley carriers mp2 = plasma frequency of lower-valley carriers E1 = rf E field due to rf potential V 'The continuity equation is written as 1 -JkJ12 + prl E 0 (6.6.2) where pl2 refers to lower valley rf charge density. The lower valley convection current is written in terms of charge density and velocity of carriers. 3:6" u) 11 OIL V12 + 019.1102 (6°6’3) where p02 stands for lower valley dc charge density. From Eqs. (6.6.2) and (6.6.3), rf charge density and current density are given by .3 -kp V 912 = W (6.6.4) ‘*’ oz _wp _ OI e 312 - Wku v” (6.6.5) 0 Substituting Eq. (6.6.1) into Eq. (6.6.5) yields 'wpotnI + Vtiz 3” =-.———1—(——) (.131 + ._.—2- v v-El) (6.6.6) JKK' not wot mp2 For one dimensional analysis V VoEl = -k2E12 and + _ 3V A A _ , . . El - 321 z = jkvlz.where e3(wt-kz) lS recognized. Hence, Eq. (6.6.6) is reduced to a simple scalar equation as _ * V 2 2 kwpoznz (“Byl- k ) V1 p12 J = . (6.6.7) 12 (w- kuog) (m-kuog - 3V2) 98 Therefore, the electronic equation can be obtained as 2 vt J Be 19 HR” *k(1+mL iZRZ ) 12 P1 (6.6.8) v1 "otIK'Bez)(K’Bei+ijg) 6.7 The Dispersion Characteristic Equation in Carrier Wave Interaction Equating the circuit, and electronic equation, Eqs. (6.5.5) and (6.6.8) gives the following characteristic equation. k2¥Y z a k (1 + -33? k2 ) shs=e£ ””621 Mp2 (671) szs 1bRKk Bel) (R-Be£+JBv£) The characteristic equation is a fourth order polynomial in k with complex coefficients. The normalized velocity and operating frequency are defined as u p =—J51 (6.7.2) U V tu Q = ‘” (6.7.3) u w pu Collision frequency is also normalized in the following way, v S = _2. u w pu The thermal prOpagation constants are, (l) Btu = 322' (6.7.4) I111 and B = m 2 (6.7.5) t1 v t9. 99 With these definitions, the equivalent series impedance Z5 and shunt admittance YS of the upper valley carrier stream may be rewritten h as z = 1. (if): _ 3L .4(1-Qu2) + Puzcsuz-o (6 7 6) s we p2-1 we 4(P 2_1) u u UZ Y z jflfi (6.7.7) sh p 2_1 u and 2 2 2 2 —-- Y Z = 4(1.Qu )+Pu (Su 4) B 2 + .(Eusuetu 6 7 8 r sh s 2 2 tu J 2 2 ( ' ' ) 4(Pu -1) (Pu -1) The characteristic equation is obtained after combining Eq. (6.7.1) with Eqs. (6.7.6) through (6.7.8). 0 a 4 O 3 O O O 2 O O (l-EZE) k - (b +86%) k + (bBCQ + C - a) k - (b + 862) C k 0 O + Beth c = 0 (6.7.9) Where 0 n * 4(1-Q 2) + P 2 (s 2-4) Q 8 ° _ 02 2 2 ~ u u u . u u a - 7.7::— Bell 2 *3 2 (6-7-1") 4 (P -1) P -1 u u 0 b = Be2 - 1 ng (6.7.11) - 2 2 2- 2 ° 4K1 Qu ) + pu (Su 4) 2 . QuSuBtu C = 2 Btu + J -——7?———§' (6.7.12) 4 (Pu -1) (Pu -1) As mentioned in Chapter V, an analytical solution for a fourth order equation cannot easily be, in general, obtained. Computer solution of such an equation is more practical. Complex roots of k with complex coefficients will be solved by both Lehmer method and Newton-Raphson method. With these roots, the gain of a device is obtained. Gain (db/mm) = 8.68 x 10'3 (-a) (6.7.13) and 100 Gain (db/1e) = if 3 where k = B - ju and a is the metric unit. ° Gain (db/mm) x 10 (6.7.14) Note that the device length of a typical Gunn device is on the order of ten microns for x - band frequencies. The operating frequency of a Gunn diode can be extended to the range of millimeter wavelength. 6.8 Solution of the Dispersion Characteristic Equation for Gunn Devices The device material chosen for this analysis is GaAs. The applied 1_ electric field E is taken as E = 7KV/cm, the mobilities of the upper stream and the lower stream as “u = 100 cmz/v.sec and “g = 5000 cmZ/v.sec, respectively and the relative dielective constant is er = 12.5. Various doping concentrations are considered in investigating the effect of various levels. The frequency dependence of the propagation constant B and the attenuation constant a will be plotted. The drift 5 velocities of the lower and upper stream are assumed to be110£=2.2x10 m/sec and“ou = 9x104 m/sec respectively. 6.8.1 Collisionless Analysis The collision frequency of the upper valley is neglected here, considering the fact that the upper-stream density is much smaller than that of the lower stream. The functional dependence of f - B are displayed in Figure 6.8.1, Figure 6.8.2 and Figure 6.8.3 for three different carrier densities n = 1013, 1014 and lOls/cm3 as determined by Eq. (6.7.9) when Vu = 0. From these figures, it can be seen that four possible waves are propagating along the device. It is also noticed that the f - 8 curves are slightly shifted by carrier densities, although the wave forms remain almost the same. #215103 j‘i‘ I. .9; 8.1:... PHASE CONSTANT B ( /m) 101 lO6 (- - é. S 10 7' ' 7 KV/cm - “9 = 5000 cmZ/v.sec . “u = 100 cmzlv.sec e = 12.5 . r T = 77°K 104 5 1 1 1, 1 1 1 1 1 .1 1.- -10 1 2 3 4 5 6 7 8 9 10 FRIEQUIENCY (Guz) . 6 2 ~10 r ,. . , .. 1 1 . 13 3 . Figure 6.8.1 i-B uldeau or rear a;ves for a n=10 /cm GaAs samnle ”he" the collision frequency of an upper stream is neglected. PHASE CONSTANT 8 ( /m) !()2 106 .— b 1 I I I r_~ 5 I 10 __ . I E = 7 KV/cm 7 . n = 5000 cm“/v.sec II. 2 . “u: 100 cm /v.sec L = 12.5 , r T = 770K 4 10 I I J _l S l l l l p -10 1 2 3 4 5 6 7 8 9 10 b 406 L. Figure 6.8.2 f-B diagram of four waves for a n=1014/cm3 GaAs sample when the collision frequency of an upper stream is neglected. file“!!! but, In. . B ( /In) CONS'I'AN'I‘ PHASE II) II)5 10 —1() -IO ______ . E = 7 KV/cm 7 u = 5000 cm“/v.sec K 2 p”: 100 cm /v.sec ' I 3‘- 12.5 : " 'I' = 77”K _ 1 1 1 1 1 1 1 1 1 1 ,, I 2 3 4 5 6 7 8 9 10 EREQIIIINCY (CHE) .. _. ,. | . . . . IS 3 Figure 6.8.3 x-b diagram 01 tour waves for a n=10 /cm GaAs s'aple when the collision frequency of an upper stream is neglected. 104 In addition to the f - 8 curves the variations of attenuation constant are illustrated in Figures (6.8.4), (6.8.5), and (6.8.6). By comparing the f - B diagrams with the f - a curves, one forward wave can be used for amplifier devices and one backward wave for oscillator devices while the other two waves drastically attenuate. The lattice temperature of semiconductor materials restrict the amplification rate of devices due to the diffusion and the collision effects, as shown in figures. The dotted line for room temperature of 300° K is shifted down from the liquid-nitrogen temperature line of 77°K. As pointed out in connection with the solid-state traveling-wave devices, the transferred electron device also has better response of the gain at the higher frequencies. 6.8.2 General Analysis The general solution will be obtained by including the effect of collision frequencies in both the upper-and lower-valley from Eq. (6.7.9). The normalized collision frequencies at both valleys are assumed to be unity. In Figures (6.8.7), (6.8.8) and 6.8.9) the numerically computed phase constant is plotted as a function of operating frequency in the range of l m 10 GHz for three different carrier densities. Four waves correspond to the solution of fourth order equation. As discussed earlier, the wave shapes for several doping levels are nearly identical while the phase curves are slightly shifted. Three figures are similar to the curves shown in Figures (6.8.1), (6.8.2) and (6.8.3) which were calculated by neglecting the upper-valley collision frequency. Numerical solutions of the attenuation constant are plotted in Figures (6.8.10), (6.8.11) and (6.8.12). Seeing the possibility of ATTENUATION CONSTANT a( /m) 103__ Figure 6.8.4 195 77°K 300 K A o II ----- l ———---- -— ’ a Amplifier ‘ -~~ --____—- Oscillator 1 1 1 1 l 1 I j 3 4 5 6 7 8 9 10 FREQUENCY (OUZ) Variation of attenuation with frequency for four waves where v =0, E=7 KV/cm, n =5000 cm /v.sec, n =100 Emz/v.sec, c =12.5, and n=1613/cm5. " K A'I'I'ENUATION CONSTANT 11 ( /m) -10 -10 -l() Amplifier Oscillator _ --"' T = 77°K _____ I = 3000K 1 1 1 1 1 1 1 1 FREQUENCY ((1112) .~~ ‘~~ --.~~_ Figure 6.8.5 Attenuation vs. frequency curves of four waves where sr=12.5, vu=0, and n=1014/cm3. ATTENUATION CONSTANT 6 ( /m) 107 -1.106p T = 77°K _____ r = 300°K ~8X105- 5 Amplifier 1.. '6X10 P "_———"' '— ------- 5 ----‘------—--.“---"--—————T? -4x10 n 0sc111ator -2x105. 1 l 1 l I 1 1, 1 13, ’ 3 4 5 6 7 8 9 IO 101 P FREQUENCY (GHz) 102 _ 103 _ 104 b 105 _ 1061- I Figure 6.8.6 Attenuation vs. frequency curyes of four waves where cr=12.5, vn=0 and n=101“/cn3 PHASE CONSTANT B ( /m) 103 V /II) =1 .1) 106 _ II. pit 1 VII/(”llll:1 O 105 __ Z 7 KV/Cm - H1: 5000 cmZ/v.sec ° “u: 100 cmz/v.sec . I = 12.5 r , T = 77“K 104 ‘ I 1 1 I 1 I I ~105 l L" 1 2 3 4 5 6 7 8 9 10 FREQUENCY ((1112) -106 _ .. - n - . 13 3 Figure 6.8.7 I—B curves 01 (our waves Ior a n=10 /cm GaAs sample when all collision frequencies are considered. I /m) 3 CONSTANT I’I IASE 10 10 -l() -1() 109 b \ ‘ E = 7 KV/cm ‘ o . n = 5000 cm“/v.sec '7 ' “1: 100 cm“/v.sec . 1 ‘3 12.5 1 2 3 I 5 6 7 8 9 II) I 1 1 1 1 1 1 1 1 L ' FREQUENCY ((1112) o \ C k h- .. , , P . 1 l4 3 . Figure 6.6.6 1-8 curves of tour waves Tor a n=lO /cm GaAs sample when collision frequencies are considered. PHASE CONSTANT B ( /m) 110 ' E = 7 KV/cm u2= SOOO cmz/v.sec 106 '7' "u: 100 cmZ/v.sec ' t; = 12.5 . r - T = 77"K C r— Vl/mpgz 1.0 10 L_ Vu/(Opu: 1 . 0 10” | j l L l l 1 l I I -106 1 2 3 4 5 6 7 8 9 10" FREQUENCY ((2112) -107 ;_ .. . 15 3 . Figure 6.8.9 f—B curves of four waves for a n=10 /cm GaAs sample when collision frequenCTOs are censidercd. ATTENUATION CONSTANT a( /m) -leO -10 —5x10 -10 10 IO 10 10 111 V I E = 7 KV/n 7 u = 5000 cm“/v.sec " 2 ' “u: 100 cm /v.sec ( = 12.5 r I = 77 K Amplifier Oscillator . fi 22 ‘5 53 (1 77 Ii 5) l I) 1 I 1 1 1 1 I 1 .I .- FREQUENCY ((1112) p. p \ b Figure 6/8/10 Variation of attenuation with frequency for a n=IOI“/cm” GaAs device when collisions are considered. ATTENUATION CONSTANT a ( /m) ~2x10 -1.Sx10 -1x10 10 10 10 10 10 112 V Figure 6.8. E = 7 KV/cm 7 p u2= 5000 cm”/v.sec ') “u: 100 cm“/v.sec 5 c = 12.5 T = 77°K P Amplifier \ - Oscillator l 3 4 S 6 7 8 9 10 I. 1 I 1 J 1 1 1 1 1.. FREQUENCY (0112) b. b P 11 Variation of attenuation with frequency for a n=1014/cm3 GaAs sample when the lower and upper valley collision are considered. ATTENUATION CONSTANT a ( /m) 113 -4.8x10 _ Amplifier --I.6XIOS I- vi/wp2= 1 0 /u) = 1.1) 11 n1 , .1...“ "I.[X105 b 5 I Oscillator E -4.2x10S P t -4.0x10 10 l 2 3 4 5 6 7 8 9 10 102 __ FREQUENCY (C112) :03 - .104 _ 1 = 77 k a E = 7 KV/cm 2 5 u = 5000 cm /v.sec 10 __ 9. ,, “u: 100 cm“/v.sec 6 tr: 12.5 10 b \ Figure 6.8.12 Variapion of attenuation with frequency for s n=101"/cmZ GaAs device when all collisions are considered. 114 operating the device as either an amplifier or an oscillator, one has to investigate two curves shown in the upper quadrant. In order to find out which curve is fit for an oscillator or an amplifier - device, the dispersion curce is required to search the direction of wave propagation. In other words, the distinction between the amplifier and the oscillator can be determined from the B vs. f curves. Again higher carrier density sample shows higher amplification as anticipated. For oscillator devices the 1013 and ION/cm3 sample is more stable than the lOlS/cm3 sample, while for amplifier devices the lOls/cm3 sample is suggested. This statement supports the basic criterion of n2 product described in Eq. (2.6.10). The results of the analysis may be checked with the calculation of gain by using Eq. (6.7.13). For a frequency of 10 GHz and an active length of approximately 15 microns, gain is calculated directly from the curve as follows: (i) lOls/cm3 sample Gain (db/mm) = 8.68 x 10’3 (-«) Gain = Gain (db/mm) x 15 x 10‘3 For amplifier: Gain = 1.74 x 103 x 15 x 10‘3 = 26 db For oscillator: Gain = 4.68 x 102 x 15 x 10'3 = 7 db (ii) ION/cm3 sample For amplifier: Gain = 1.86 x 103 x 15 x 10'3 = 28 db For oscillator: Gain = 1.13 x 103 x 15 x 10‘3 = 17 db (iii) lOlS/cm3 sample For amplifier: Gain = 4.21 x 103 x 15 x 10"3 = 63 db For oscillator: Gain = 3.61 x 103 x 15 x 10-3 = 54 db 115 Perlman showed [PEI] the evidence of microwave amplification experimentally with GaAs device and measured gain of the device. If a typical noise figure of 15 db for the amplifier devices, which was suggested by Perlman, is taken into consideration, the above gain calculations is in good agreement with his experimental results, which confirms the validity Of this approach of analyzing transferred electron devices. CHAPTER VII DESIGN AND FABRICATION CONSIDERATIONS OF PRACTICAL DEVICES 7.1 Introduction The theoretical analysis based on the computer simulation has revealed a possibility of a high-gain solid-state amplifier in the micro- wave frequency range. Several important factors must be taken into account when the experimental device is fabricated. Of these the most important are the tape lengths, tape pitches, semiconductor materials and the slowing factor. In addition, other factors may be essential in designing and fabricating the device. The criterion stated in Eq. (2.6.10) for wave amplification will be followed in the subsequent design procedure. Here solid-state traveling-wave amplifiers will mainly be treated. Finally, some guidelines will be established after the design factors are qualitatively discussed. The design of transferred electron devices is rather simpler than the solid-state traveling—wave devices. The Gunn devices are comprised of an active gallium arsenide layer, with or without a GaAs substrate, and two ohmic contacts. The thickness of the solid-state material is the major factor which determines the optimum operating frequency. The output power is a function of the cross sectional area of the wafer and the conversion efficiency. Usually the ohmic contact on the active layer is an evaporated film of silver tin alloyed in at several hundred degrees of celsius. 116 117 In designing the traveling-wave solid-state devices, extra care must be taken in selecting the slow-wave circuit and insulating layer since they are critical to the device operation. The Operating frequency in Gunn devices is tuned by using waveguide techniques mechanically such as iris tuning or an adjustable short. The tuning can also be obtained electronically with the use of YIGs or varactors. However, the range of Operating frequencies is limited. In solid-state traveling-wave devices .31 the Operating frequency is quite broad and generally such a tuning is not required. 7.2 Effect of the Insulating Layer Between Circuit and Semiconductor The purpose of using an insulating layer between the slow-wave circuit and the semiconductor slab is to prevent a short circuit for the applied drift field. In the capacitively coupled circuit, such a layer is not required but a similar layer must be deposited between the fingers of the slow-wave structure. As was mentioned in Section 5.7, the total gain of the device is greatly influenced by the insulating layer whose thickness, limited by integrated circuit technology, should be minimized. A thickness of 1 micron can be achieved by putting the wafer on the high-speed rotating disk but it is questionable how precise a degree of uniformity can be obtained in an average laboratory setup. Besides the effect of the layer thickness, the dielectric constant of the insulating layer influences the circuit velocity which becomes a factor in determining the device size. The permittivity of several insulating materials are given in Table 7.2.1. As can be seen from the table, the permittivity of most materials for this type of device ranges 118 between 3 and 10. Fortunately, the effect of permittivities does not significantly affect the net gain. TABLE 7.2.1 DIELECTRIC PERMITTIVITY Material Dielectric permittivity Si02 3 5 Alumina silitate 4.8 THE. Muscovite mica 7 7.3 Synthetic mica 6.3 Alumina 8.1 10.2 Methylethacrylate 3.6 Kodak KMER 3.8 4.5 It can be concluded that the use of high permittivity material reduces the transverse dimension of the circuit structure and hence eliminates some difficulties in the circuit fabrication. In the photo—etching process, if the tape length is too long, compared with its width dimension, the tape width lengthens into a concave form due to lense effect and results in the disconnection of the tape line. 7.3 Selection of Solid-state Materials Many factors are involves in choosing an appropriate semiconductor material, which affects the performance of the solid-state traveling-wave devices. Selection of the solid-state material is basically concerned with the semiconductor losses, the carrier drift velocity and the carrier concentration density. In addition, the conversion efficiency losses, circuit losses, contact losses and transmission losses should not be overlooked. Most of these losses can be minimized if an appropriate material together with an excellent technique of fabrication is selected. 119 Further, cryogenic operation such as at liquid helium (4°K) or liquid nitrogen (77°K) temperature plays an important role in reducing some losses. Solymer expressed the semiconductor loss [801] as: L5. = 4'35 [ESE [db/m] (7.3.1) E:0 VCt where L5 is the loss of semiconductor materials with relative permittivity Cr’ in db per meter and 0t the transverse conductivity in the presence of the applied drift electric field intensity, i.e. a material which has the characteristic of a sharp saturation, as far as the transverse conductivity is concerned. However, when 0t is reduced the longitudinal conductivity is possibly reduced simultaneously in themetallurgical process and therefore extreme care must be taken in such a treatment. Secondly, a material with high relative permittivity would give low loss, as indicated in Eq. (7.3.1). Referring to Table 5.4.1, Ge is the best, considering this aspect. For optimum operation the carrier drift velocity to the thermal ratio is 3.1 as discussed in Section 5.6. This range does not fall into that of most solid-state materials even under cryogenic Operation. Therefore, the problem is in choosing a reasonable material which has a nearly close ratio and in deriving a maximum carrier drift velocity. The carrier density or the carrier resistivity should be considered since they are related to each other. Theoretically Eq. (5.2.3) indicates that the gain is saturated at a high doping level. Due to the debunching effects of carriers in physical sence, too high doping concentration will not increase the net gain. The appropriate resistivities of semiconductor materials range from 1 O-cm to 20 O-cm. Finally, the physical dimensions of the semiconductor wafer must be minimized to reduce the losses. 120 7.4 Design of Slow-wave Circuit Structure The main function of the periodic structure is to provide an adequate slowing factor of the propagating waves to match the carrier velocity in a reasonable length of structure. The slowing factor (s.f) in the construction of the slow-wave circuit is expressed by the ratio of the light velocity in the medium to the carrier drift velocity: C. s.f= KC: Vg (7.4.1) where c is the velocity of the light and vg the group velocity of the circuit. The slowing factor for the traveling-wave amplifier ranges from 103 to 104. In selecting a slow-wave circuit, qualitative estimates of the slowing factor can be made by checking the permittivity of the medium and the drift velocity of the material. In order to overcome the difficulty in constructing such a large slowing-factor, the permittivity of the material and the carrier drift velocity should be increased. However, the drift velocity is limited and therefore increasing the medium permittivity to a maximum is desirable. Under any circumstances the slowing factor must always be large enough to make the circuit traveling-wave synchronize with the drifting carrier stream and be fairly constant over the circuit length. Also, the complete system of the slow-wave structure should be matched to a usual transmission line over thr broadest possible frequency range. A 50 0 transmission line is commonly used for the micro-strip line structure. Here, a meander-line or a helix-tape line will be convenient to match the microstrip-line. The effect of a ground plane is negligible if the groumiplane spacing is much greater than the tape spacing. If the circuit is very close to the ground, some field lines will terminate on the ground plane, and such effects must be taken into account. The meander-tape line is shown in Figure 7.4.1. The longitudinal velocity down the tape-line is written in terms of the circuit dimensions as: _ d+s . c V8 h E (7.4.2) 121 Since the dispersion curve is not generally a straight line the dispersion factor fs also should be accounted for in an actual design. Then the slowing factor can be rewritten as: h s°f=deT' fs (7.4.3) P—I’—’I /’ p—lfi:fi \ fgk \ _f— nos-q- \\\\ \\\\\\‘ \§\ \ Xfi S\\\ §\ \\\1 \Y\\\ Figure 7.4.1 A meander-tape line In practice, fS ranges from 1 m 3 since it varies with the circuit condition. When 5 = d, the value of f5 is approximately 2. Furthermore, Butcher suggested that the field distribution down the tape-line is maximum when s = d and h = 1/41, where A is the wavelength of the operating frequency [BU3]. From this statement the optimum relation between the tape length and the operating frequency can be obtained as: 3" II Ali—I 5%: (7.4.4) r One more consideration is the effect of surrounding dielectric materials around the tape surfaces. When two different types of dielectric materials are used for the upper and lower surfaces of the tape-line for insulation, the effective dielectric constant can be defined by: (7.4.5) 122 The above relation holds provided the dielectric deposition of finite regions is treated. The connecting devices between rf connectors and the tape-line use an exponentially or linearly increasing tape until the final width of the increased sides equals that of the microstrip line. The meander-tape line is designed in accordance with the above criteria. A design example of the meander line is given as: Parameter Design value conductor thickness t = 2m10u conductor width 5 = lSu conductor spacing d = 15u pitch p = 60p conductor length h = 7500a total width of conductor g = 1200u material used InSb optimum frequency 2.5 GHZ The above example was calculated on the basis that u0 = 6 x 107 cm/sec, fg = 2, Cr = 15.7. Similar calculations will be done when different semiconductors are used. 7.5 Fabrication Considerations of Devices The first step of constructing the slow-wave circuit consists of drafting a large scale version Of the desired circuit, reducing it to a proper size by a reduction camera and etching an image of the circuit onto a piece of glass. For constructing a capacitively coupled slow-wave circuit, one side of the fingers of the meander-line is required for the first layer of deposition on the semiconductor, and then the other side of the fingers can be deposited by turning the mask glass upside down -— making a complete slow-wave structure. 123 The semiconductors are cut in a (111) crystal surface orientation and polished by several grades of sandpaper on rotating disc. The possible size of the semiconductor slab of InSb is 9x3i<0.l mms. The surface damage layer is also removed by an ordinary etching technique with the solution of CP4a ( HNO CH CO H; HF = 5:3:3). 3’ 3 2 The wafers are evaporated with gold of 24:5u thickness in a vacuum chamber. Then a uniform photoresist pattern is formed on them with Kodak Shirpley 1350. The photoresist pattern is uniformly deposited when the disc is rotating at a rate of about 3000 rpm. First meander type of circuit is manufactured by the photo etching process. To make a capacitively coupled meander circuit, some dielectric material (mica, SiO2 or Kmer) is deposited on the first type of fingers. The thickness of the dielectric material is approximately one-half to one micron, which gives C = .02aa.05 PF/sq.mil. Another gold evaporation is made to form the other side of the meander circuit. Such structure can be made from two metal layers of overlapping bars separated by the dielectric material. Actually three layers are deposited successively with the metal layers being etched after deposition. One side of the circuit pattern is shown in Figure 7.5.la and the small portion of the complete circuit is taken by the enlargement of a microscope as shown in Figure 7.5.1b. A final assembly of the wafer is connected through a commercial microstrip line with 50 O OSM fixture. The circuit is then placed onto the middle part of the wafer to make carriers flow parallel to the direction of slow-wave propagation. The substrate under the wafer is made by alumina. One of the final test structures is presented in Figure 7.5.lc. (b) (c) Fig 7.5.1 124 Slow-wave circuit and test structure of a Solid-state traveling-wave amplifier. (a) One side of the meander circuit. (b) A small portion Of the complete circuit after fabrication. (c) A final assembly of the test structure. 125 To establish the optimum drift field Ohmic contacts at both ends of the wafer are formed by evaporating a film of silver tin which is alloyed in at 600 degrees celsius. Indium ohmic contacts may be substituted at both ends of semiconductor. Unfortunately fabricated samples of the InSb, Ge and Si did not give qualitatively reasonable results as anticipated from the theoretical analysis. Under the liquid nitrogen temperature, only several dbs of electronic gain have been recorded. Physically, the result showed the evidence of coupling between rf circuit waves and carrier waves in the circuit. For better results several carefulness should be taken into account and they will be outlined.. Sandpaper polishing is not adequate for this type of device which requires a high degree of precision. The rough surfaces might be resulted in non-uniform circuit and be broken at several places. The input impedance will be high at most frequencies - making a highly mismatched circuit. However, minor circuit imperfections are unavoidable. The other factor comes from point contacts for the drift field, which make a weak coupling of a carrier stream with the rf wave, since the point contact induces the carrier waves along a thin line between two contacts. For a stronger coupling the contacts of dc potential would be better extended -— covering the contact surfaces with silver plates. Practically, regardless of design or preciseness of fabrication technique, it is almost inevitable in the laboratory to expect a flat and ideal characteristics without the accompanying reactive component. Reactive effects are largely asSociated with the gap and tape widths, connections of the system and discontinuity at the junction, with additional reactance being due to the capacitively coupling of the 126 meander-tape line in the original design. However, some reactive elements can be eliminated by adjusting the input and output double-stub tuners. To improve the mismatch, an additional modification of the device circuit is suggested. The modification might be made by utilizing a tapered region at each end of the meander line. The tapering raises the impedance of the line to a nominal value close to 50 ohms and further reduces the dispersion in the end regions; thus the impedance is nearly constant over the operating frequency. This technique can be employed empirically. CHAPTER VIII CONCLUSIONS AND SUGGESTIONS FOR FURTHER DEVELOPMENT The purpose of this study is to investigate the carrier wave interactions in solids. Two types of interactions are considered: the first type of interaction is between the carrier wave in solids and the circuit wave propagating along the surface of the solids, and the second type of interaction is due to carrier waves in two adjacent streams propagated in the same direction. In these analyses, the effect of lattice vibrations in solids are taken into account by introducing carrier effective mass and thermal diffusion. The main work embodied here can be divided into three parts: theoretical analysis, computer simulation, design and fabrication considerations. The general theory of two stream instability was developed —— leading to a clearer description of Gunn devices. The dispersion and attenuation curves are presented. This two stream analysis given here was checked closely with the experimental results published recently. The results Show that the gain of the interaction is proportional to the doping level of the semiconductor, which was expected since a higher power output is possible with more charged carriers taking part in the interaction. However, it is also expected that the gain will level off as the dOping reaches a high level, which means the collision effect will be dominated. Also, investigated was the second type of wave interaction due to the coupling of a drifting stream of carriers with the rf circuit 1.27 128 attained by evaporating gold on the solid-state materials. The slow wave circuit used was a capacitively coupled meander-tape line. Such a complicated structure is advantageous because the dc drift field will. not be disturbed by the presence of the slow-wave circuit. As a result, the drift velocity of the carriers will be reasonably uniform. A two dimensional boundary problem of slow-wave circuit has been carried out. Numerical solutions for commonly used materials such as Si, Ge, GaAs and InSb were obtained. Based upon dispersion equations, derived from the two dimensional boundary conditions, instability characteristics of carrier waves propagating along the semiconductor were obtained for continuous type of tape circuit model and capacitively coupled tape circuit model. It has been demonstrated that an optimum gain of the device is a strong function of insulating layer thickness, circuit velocity, collision frequency and carrier drift velocity. The insulating layer impairs operation drastically because the electromagnetic field of the tape line decreases rapidly in the transverse direction. The variation of the net gain as a function of circuit velocity was anticipated. It was also found that the ratio of carrier drift to thermal velocity is 3.0 for the highest attainable gain. The collision frequency dependence of the gain confirmed the Vural's hypothesis which collisions tend only to decrease the amplification. Theoretically, traveling-wave interaction in solids shows good gain-frequency characteristics, compared to most of the classical micro- wave active device. This is an attractive feature for the solid-state traveling-wave amplifier device, which may be a potential application for space communication. In reality, however, the inevitable circuit loss, fabrication loss and the reduction of carrier mobility in the surface will reduce the gain significantly. Under Optimum conditions, both InSb and 129 Ge give highest gain. Furthermore, InSb has the best velocity vs. field characteristic which is crucial in design criterion concerning device fabrication. As has been mentioned, the device has potential possibilities in broad band high frequency operation. Therefore, solid-state devices appear promising as a means of improving the operation efficiency when considering its future applicability. At the present stage of development a few outlines are suggested for further improvement based upon the analysis. Primarily the best result of this type of device will heavily depend upon prOper design and fabrication. Further work in fabricating comparatively perfect meander circuit lines may be developed with electron-beam scanning techniques. This method will reduce the tape size in length and width, and then increase the optimum Operating frequencies of the device up to nearly 40 GHZ. It is also desirable to have a smoothly lapped semi- conductor surface before evaporating gold for making fine circuit structure. Matching the device structure to the external circuit system is also a serious problem. The gold bond wire technique presently used is definitely not an ideal method. 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From the periodic translation symmetry condition, A (x,y,zl) = A (x,y,z1 + np) which implies that: -J' knP K21x.y.zl+np) = K11x.y.zl)e (M) The field vector A at two points on a periodic transmission line separated by n periods differs by the complex constant e-sz q.e.d. 137 138 A.2 Power flow theorem Consider the structure to be divided by a series of planes perpendicular to the axis spaced by the periodic distance p. These surfaces extend perpendicular to the axis to infinity for an unbounded structure or terminate on perfectly conduction boundaries if the structure is so bounded. Over these surfaces + +* -> jExH - ds = 0 (A.5) s where the superscript * denotes complex conjugate. The integral is zero on any metal boundaries where the tangential component of E is zero or at infinity the fields fall off at least as fast as %-. Substituting Eq. (2.5.9) into Eq. (A.5), and applying the divergence theorem, one obtains if Ennis”) - d‘sf =Jv ~ [Ex(Vx5*)]dv S V J{.[(ng*)'(VXE) - E ' (VxVxE*)]dv v ['ijH ' (jwufi*) - E ' (w2u€E*)]dv V (A.6) . 2. DIVIdIng the above equation by Wm u gives * * %j%uH-Pdv=%—jc-E°Edv (A.7) V V It can be seen that in the pass band the time average stored electrical energy per period is equal to the time average stored magnetic energy per period, from Eq. (A.7). Based on the statement, the wave theorem will be shown. Applying Maxwell's equations, Eq. (A.7) is rewritten as, f[(VxE*) . (VxE) - 5 - (wzueE*)]dv = 0 (A.8) v Differentiate Eq. (A. 8) with respect to w. J(Vx§*)-(ang -a—)dv +fv (mg—E ) mm, _ w Zuefgg , ydv (A.9) V V Rewriting Eq. (A.9) gives + 2 Reer (Vx%§5 ' (VxE*)dv - EE-° (VxVxE*)dv} v v l . 117:: "‘1‘ ‘ l'_-' 139 'ZNUEJI E - E* dv = O v Using vector identity B~VxA-A-Vx§ = V-(AxB), 2Re£JrV- a v - Zwue E3E* dv = 0 v and using the divergence theorem on the first term, 2Re{} g—E— x (VxE*) dS - 201116} E°E* dv = 0 (A.lO) s v Neither metal walls nor surface at infinity contribute to the surface integral term, so only consider the surface integral over the two surfaces perpendicular to the axis bounding one cell of the structure. Let subscript 1 and 2 designate the quantities at these two surfaces. From the Floquet theorem the following relations hold: ._) _' 52 = E1 e 3“? (A.ll) My 35 BE . . 2 .. ___1_ ~38p - dB 1 -JBp duo - 001 e - J p d0) 1:1 e (A.lZ) Separating the surface integral of Eq. (A.lO) and using Eqs. (A.ll) and (A. 1%). a * 2Re{ -,-—x(VxEl *1+)-d§ sz—xwxfi; ) (132-20115 E-E dv=0 S‘I d§ 3'51 :ReJlgi-flxflxiil *°) l+f:—w1— x(VxE1*-)-2dS j—p—j$2 El x 528 (VxEl'k) °dS2-2wuc-f E-E’k dv (A.13) V Since S1 and S2 are in opposite directions, the first two integrals cancel ‘ * out. From Eqs. (A.ll) and (A.lZ) E1 x (VxE1*) = E2 eJBpx (VxE2 ) e-j8p_ —E2 x (VxE2 ) Then Eq. (A.l3) is rewritten as: ,dB -> .-+* +_ .—>* 2Re{- Jp 35;]; E2 x (JmuH2 ) dS2 - Zwuc V E E dv * 211m; 543’ dv (A. 14) V .J I.,h<.fl4 $4.. 140 Multiplying Eq. (A.l4) by 4m: p gg-and applying Eq. (A.7) leads to: i * * %Ref Exfi ~2=c1's’-v%l gjv(—eE-E +%ufi°fi)dv (A.lS) s P 2 where V = §2_= group velocity (A.l6) g d8 which shows the main wave theorem. APPEND IX 3 TABLE OF ELECTFHHACHETIC FIELD SOLUTIONS FOR ALL REGIONS Region case-s TH Modes TE Myles ax tax (wt-k2) ,. ax. jhrt-kz) x - I; General F AU: (1: )r‘] 1 ' [L )0 12 Y H [- ‘ I '11 f C 21' r, . ”5“" ‘77 1 . .. 2 '2 ax (wt-k7.) :51 ,1-. an- E fir! nay. ‘ .1 bx e: (at kz) H : -k Ce 2) R-uion 1x a . 5 12 1x u ,‘ nt-J; ' I. J 1 2 2 2 2 f -k — -— -' C r a s, C ‘ r2 J: a ) )0": K ) jun-kt) 2 2 ax j(wt-kz) H 2 ;‘ i -k :11 t u bx e H - 1 Ce 1'; -—-—:‘s ._.—1, ——————T—-‘ e 12 U ‘ )w a 2 ,b a - )- 9 . , 81" r1", jlwt-kz) 5 ‘ 6I" (j (wt-k2) . _ .~ ; e . _ I" 1.. e '. ‘w kuaK . £12 A11: r e j: 1y 1 r 1 , nab (l-j“) . . -_ ' — C t a x 1x 1" 1 l 1 (1I b1x «QM k“ 2 2 1 ' jtwt—kz) F e + e H - -k——1‘ e. e lx a l. I; ‘ 11: w l 1 L a 2 j l l 1 \ wzax 'kuw2 tx iczfa ax 1'2'1 1' . 1 3 0 .- 1 “wt-la) 2 2 I l _j(wt-kz) H w—; {e - —————-—«.~ (1 H = ————c1e 1.. 1y a . , .2 12 w 1 (w-ku -3;-. o) o 2 01115101119“ T'- a2 . ‘(wt-kz) 2 )(wt-kz) Elz ‘ ”‘2" I '1 1y - '2 c 4 (v-o) L. ‘ 2 r'k ‘1’ -' k 1 C2 "2 2 '(w-kz) E 1—A e L e” (Ht 2 H = -k———-C e e1 I 1x a 2 1 w L2 J r, 2 ' jr‘ 2’ a d X 1H! H a X -_ l._ . ’ « '1 2 ‘ - ‘ ‘ J 2 2 2 (Vt-kl) H -——12 11 - ——_‘)I e a)“.t k2, H - ————r" - e. 93 1y 3 2 12 a 2 2 wtw-ku_) U “’X\d General '- ‘l x T i . -le 11x j(Vt-kz) .,_ _ . - E F o 1 + F e 1 ejhrt-kz) E E 678 + 53,3 e 1: 2 3 _ 1y - Circuit Fegicn m -V x . x ‘ -, x ’v I . '1 1 1 . j(ut-kz) . I . . vt-kz) .. —— k[ t ” I = -2521: e 1 - F e 1 e]( H — - G e u e . L EIx L» 2 3 ' 11: ;‘~. 2 2 3 1 1 ' _ V X jVL ‘y' —! X )ch '1x '1 . j(wt-kz) H . _ 11(0 e l _ G H - ——- r e - r e r e u 2 2 3 lY Y 2 3 = 1 1 - X , _.' . ‘ —k' de General B F e ‘3 ejh-rt-Kz) .1 [a e 3jejtvt a) Circuit - j _ HE k -1 x , Region 13 NIX] j(wt-kz) H .. - 3 G e. 1 ejhrt-kz) I51x - I:1” 0 1x , 2 1 1} £43 F '14:: jun I. *1 X _ I 1 13 jh-It-kz) H _ - 3 3 c e 3 ejhvt kt) H1 - Fla (2 12 2 1 Y 13 33 NOTE: Field solutions in solid-state region are obtained under the assumption of the thick semiconductor slab for convenience. However, these solutions are not used in the numerical analysis. 141 APPENDIX C DERIVATION OF DISPERSION RELATION BY ADMITTANCE MATCHING METHOD First, the admittance functions in the solid-state and the circuit regions will be obtained from solutions of coefficients inter- related in Section 4.5. The admittance functions in both spaces are connected on the boundary surface x = 0 and will determina a dispersion characteristic equation of the complete coupled system. By manipulating Eq. (4.5.11), one can find the ratio of B2 to B1, B a+y e (5.1) 4 In the solid-state region the general admittance function for the TE waves evaluated at the semiconductor surface x=o- is, as the ratio of H12 3 tanh a6 +74 YTE = quo . a +Y4 tanh a6 (C.2) By using Eq. (4.5.12) the corresponding admittance function for the TM waves is then, as the ratio of H1y to Elz 2 2 . b -k 2 2 (32’k2)'3 T a -k (l - H ) + C c2 a 2 a2_.b2 = 2 2 . 38" J0) 1+ “2+ 113+ “1+ ((3.3) and by simplifying Y5 = 31115“ E Y D (C.4) 1, where -2a6 h2 2 “2 N = (e -1)[(h1+§—) cosh b6 + (2h1h2+Rhl *R’3 51nh b6] - (1+e’235)(Rhl+h2) sinh b6 142 143 h D = e 35+(2e'235-1)[(h1+§39 cosh b6 + (R+%Q sinh 551—(2e’236+1) [2 cosh b6+(Rh1+h2) sinh 55] Eq. (C.3) may be reduced to hz hz Y5 = chu . (hl+E—)cosh b6 + R(h1+§— + l) Slnh b6 Y4 -a6 h2 1 h2 - e +(h1+§—-+2)cosh b6+[(R+§)+R(h1+§—)] sinh b6 (C.4) previded that a6>>1 which is practically true. Similarly, the admittance function of the circuit region at the boundary x = 0+ can be obtained. Accordingly, from Eqs. (4.5.33) and (4.5.34) the admittance of the circuit region is, for the TE waves, a-y1+ _ . 1_(;:__) e 2a6 Y6 = 1:1.(2_9 Y“ TE mu Y a'Yu _ o 1 1+( ) e 230 (C.S) a+Yu The circuit propagation constants are simplified by yléyaéyué k 2 2 2 since for slow waves k22>81, k >>B§ and k >>Bu~ From definitions of a2 and b2 they may be approximated by a 5 k and b écup/vt where 2 2 w 32: kz[1-(§Eza (1-1;f§p 2 2 92_ 2 w 2 b = (h ) [1 + k / p I Vt (T) t The last statement can be justified as: m + K V ‘p t >> -w+ku (1) v i9_.= .23.: 10 3«.10 4 C2 C2 and k 144 With these assumptions, notice that the TE admittance in the solid-state region and circuit region are exactly the same and are not affected by the presence of either the circuit or the semiconductor, hence there is no discontinuity of the TB fields at x = 0. The corresponding admittance function for the TM waves in the circuit region is, in straightforward manner, calculated with the . . . . c . coeff1c1ents solutions. At the boundary surface the admittance Y 15 . , _ _ Y yc = Jwel . K3(Fl+ tanh Yld) KH(F2 tanh 1d) (C.6) Y1 -1<3(I‘l tanh Yld +1) +KQ(F2tanhYld) The admittance, where the same dielectric materials surrounding the meander line are used, can be obtained from a special case of the I general solution with 81:5 and Y1=Y3. which are reduced to 3 . 2 d 2 2 2 d o 0 Y Y ye: 30151 . -K3+KuF(1+C 1 )Yl/BlsA-e 1 Y1 -Kg‘*K2[(l-e2Yld)Y12/B:l + e2Y1d] (C.7) where Y -a _ 1 -236 K3 — Y1+3 (l-e ) (C.8) 2 Y -a o _ 1 -230 K3 -1- (Y +21) e (C.9) 1 In practice, when the device sample is fabricated it is easier either to choose a single material for the insulating layers or to leave the top region as a free space. Eq. (C.13) can be more simplified, depending upon the following situations: (1) Case 1: d = O 1 .k ~[2(—-B ) -1] (0.10) 51 14$ (2) Case 2: |y1d|<< 1 For small x, exp(x) é l+x, then the admittance becomes 2 . le - 2k (l-kd) jOe k C Y = 1 2k3d-B:1 (0.11) (3) Case 3: IYldlf 1 This case may be better approximated exp(x) by 1 2 1+—x + 5—- 2 12 1 x3. 1'2" I 12 Hence the admittance is .- 2 4 2 2 2 2 YC _ Jwel . 2d k +(6—d le)k '3stlk-3851 _ k 3 2 2 2 2 2 _6dk +d 851k +3d851k+38 $1 ._"7 (4) Case 4: extremely large Yld such that e 5Y1d<<1 jmc YC =' k 1 (0.13) The case 4 implies that very thick thickness of insulating layer are involved, and hence there is no bhance for the carriers to interact with the circuit. The admittance of the slow-wave circuit space and the solid-state space must be connected at the interface of x=0, and thus Eqs. (C.3) and (C.6) yield the general dispersion relation. APPENDIX C THE COMPUTING METHOD FOR THE COMPLEX ROOTS OF A DISPERSION EQUATION WITH COMPLEX COEFFICIENTS As a matter of fact, all classical methods require a large amount of skill and judgement for isolation and separation of complex roots of a polynomial equation. Quite often one encounters an unexpected difficulty in taking a root as a starting interaction solution. With these difficulties in mind Lehmer has constructed a method which can easily capture approximate locations of roots. Therefore, Lehmer's method is used to find approximate values of the roots and then the faster successive interaction by Newton - Raphson method is used to locate the roots within the specified convergence criterion. Let a function P (k) be analytic inside and on a closed contour C except for at most a finite number of poles interior to C. Also, let P (k) have no zeros on C and at most a finite number of zeros interior to C. Then, if C is described in the positive sense, it is a well- known fact that 2H P(k) 1 1. I p——(—k)—dk=N-N (0.1) J c 0 P where No is the total number of zeros of P (k) inside C, a zero of order 1110 being times, and Np is the total number of poles inside C if a pole of order mp is counted mp times. As shown in the dispersion equation, N = O and then the integral gives the number or roots for the polynomial function. 146 147 To capture a root of the polynomial P (k) = O by Lehmer's method, first remove zero roots, c.f. flow chart, P (O) = 0. One starts with the unit circle and begin to process in doubline (or having) the radius at each step. Since the roots are bounded, one soon finds an annulus R<|k|<2R where the circle of radius 2R contains a root of P (k) while the inner circle is free of roots. This annulus can be completely covered by eight overlapping circles each of radius R1 = SHR C2Hnj/ SR/6 with centers at —3—- these circles in turn, one soon finds a circle containing a root of P (k). , (n=0(l)7). Repeating the process on This completes step 1. Calling the center of this circle C1, one finds 2, such that 2R2 contains a root of P (k) but whose inner circle is free of roots. Eight smaller circles as before, an annulus R2 <|k-C1I< 2R of radius 5R2/6 at center C2 cover this annulus and find a circle containing a root. This completes second step. After n steps one has a circle of radius not exceeding 2R (S/lZ)“, and probably smaller containing a root. This procedure gives the small roots first. In this routine one obtains roots as a first approximation, upon completion of 8 steps with Lehmer's method. Newton-Raphson's method is applied to improve our estimation of the root obtained by Lehmer's method, Taylor series expansion is then used to a polynomial. h h2 P(k+h) = P(k)+P'(k)—l-! + p"(k)F + (0.2) LGt a ToOt of the polynomial be k = kl’ which can be obtained by the Lehmer method. Suppose, however, that the root is actually k1 + h. Then, in the Taylor expansion of P (k) becomes P(kl+h) = P (k1) + hP' (k1) + (0.3) ....L-:1IQ.\ . if“! ...N . J'VU The deviation in the real root is h. If h is small, h can be written as h = P(k1) Pl(kl) (0.4) Then the iteraled approximation to the actual root gives the general formula _ p(k2) 1+1 ' k1 ' Pl(ki) (D S) ‘— Iteration steps will be continued until Relkz-ki_l| f DELTA 2 and 1m] ki—ki-IA f DELTA 2 where DELTA 2 is a permissible error which is taken 10-7in our computer program. This new approximation determines one root of the polynomial and p P(k) 1 k-k same fashion where k1 is a first root of p (k). Continuing the procedure, a reduced polynomial P1(k) = is then computed and solved in the all roots are finally obtained. The flow chart explaining the procedure is illustrated in Figure D.1. 149 C Stait D FR,FI,NORDER RR,RI, DELTAZ Set by calling program Remove zero roots, set I: P(k)= P (k) number of zero k-(RR(I-l)+jRI(I-l)) roots +1 I Scale polynom- ial so that highest order coefficient is equal to 1 Obtain first approximation for root I by completing 8 steps of Lehmer method Improve root I with Newton-Raphson method until _7 DELTAZ 10 Figure D.1 Flow chart for computer solutions of a polynomial. ERSI M'clliilimfiliiliiliiiiWI IllHilililiil'ifiilifis 3 1293 03013 1063