SOLID-STATE PLASMA ELECTROKINETIC POWER AND ENERGY RELATIONS J. J. Soltis The University of Michigan Electron Physics Laboratory Approved for public release; distribution unlimited.

FOREWORD This report was prepared by the Electron Physics Laboratory, Department of Electrical and Computer Engineering, The University of Michigan, Ann Arbor, Michigan under Air Force Contract F30602-71-C-0099, Job Order No. 55730000, Task No. 557303. The secondary report number is Technical Report No. 123 under Project 037250. The work was administered under the direction of Mr. John V. McNamara (OCTE) Rome Air Development Center project engineer. This report has also been submitted as a dissertation in partial fulfillment of the requirements of the degree of Doctor of Philosophy in The University of Michigan, 1972. The author wishes to express gratitude to his doctoral committee and especially to Professor Joseph E. Rowe, who suggested the topic and guided the research development. Special thanks are also due to Professor Ronald J. Lomax, Mr. Mark K. Krage, Mr. Madhu S. Gupta, and Dr. William J. Fleming for the many fruitful discussions and suggestions. The outstanding work of the laboratory staff is greatly appreciated, particularly that of Mrs. June Corkin, Miss Betty Cummings, Mrs. Wanita Rasey and Mr. Leslie Shive. This report has been reviewed by the Office of Information (OI) and is releasable to the National Technical Information Service (NTIS). This technical report has been reviewed and is approved. jjJi XLi" l64 Approved: OH N V. MCNAMARA Project Engineer Electron Devices Section Ap —proved Approved: Khief, Techniques Branch Surveillance & Control Division -ii

ABSTRACT A general theory is developed for the electrokinetic power and energy properties associated with the basic carrier modes present in plasma media. Both hydrodynamic and kinetic theoretical models are obtained for the media in the presence of applied static electric and magnetic fields. In the hydrodynamic theory the effects of carrier collisions and thermal diffusion are properly accounted for andexplained by developing a second-order quasi-linear analysis. In this manner it is shown that the negative kinetic power property is directly related to dc slowing of the active carrier. The distinction between absolute and convective instabilities leads to the formulation of a space-averaged temporal-energy basis for determining the existence of absolute instabilities as compared to a time-averaged spatial-power basis for convective instabilities. The analysis shows that it is possible to relate the causality criteria for instabilities developed by Briggs to the conservation of power and energy in the medium. Thus useful general information is obtained on the behavior of the root trajectories in complex-k space as the imaginary part of the frequency is varied. The quasi-linear theory, as a by-product, allows the analysis of the second-order Hall effect and related phenomena in solids. In addition, a study of the physical meaning of the quasi-linear theory shows that this is a useful analytical tool for studying potential energy effects caused by the reaction of the growing RF fields on the carrier charges. This also enables the accuracy of the linear dispersion equation to be assessed. The power and energy theorems applied to the kinetic theory determine the effects of nonlocality, anisotropic carrier temperatures, and carrier heating. Whenever possible the results obtained are rigorously compared with those of the hydrodynamics theory. By obtaining the respective dispersion equations, computer results for the hybrid-hybrid electron-hole interaction are related to published experimental work on the phenomenon of microwave emission from indium antimonide. -iii

I

TABLE OF CONTENTS Page CHAPTER I. INTRODUCTION 1 1.1 Streaming Instabilities in Solid-State Plasmas 1 1.1.1 Introduction 1 1.1.2 Transverse Two-Stream Instabilities 3 1.1.3 Longitudinal Two-Stream Instabilities 7 1.1.4 Hybrid-Mode Instabilities 10 1.2 Outline of the Present Study 10 CHAPTER II. EFFECTS OF A MAGNETIC FIELD ON THE KINETIC POW'ER PROPERTIES OF CARRIER WAVES: CONVECTIVE TNS'IABILLT'Y 15 2.1 Introduction 13 2.2 Kinetic Power Characteristics of Space-Charge Waves 14 2.3 Nature of the Transverse-Field Contributions to the Kinetic Power Flow 23 2.4 Power Characteristics of Purely Transverse Modes in a Static Magnetic Field 28 2.4.1 Derivation of Electromagnetic Power for Purely Transverse Waves 28 2.4.2 Kinetic Power of the Purely Transverse Waves 33 2.5 Nature of Magnetic Field Effects on the Kinetic Power Flow 36 2.6 The Two-Stream Transverse Instability in a Longitudinal Magnetic Field 45 2.7 Effects of Collisions on the Mode Kinetic Power Properties in a Longitudinal Magnetic Field 50 2.8 Kinetic Power Properties of the Hybrid Mode 55 2.9 Effects of Collisions and Thermal Diffusion on the Kinetic Power Properties of the Hybrid Mode 59 2.10 Utility of the Kinetic-Ele:ctromagnetic Power Theorem 6~ 2.11 Summary 60 CHAPTER III. KINETIC ENERGY PROPERTIES OF CARRIER WAVES: ABSOLUTE INSTABILITY 67

Page 3.1 Introduction 67 3.2 Kinetic Energy Characteristics of Space-Charge Waves 68 3.3 Electrokinetic Energy Density of Purely Transverse Waves in a Static Magnetic Field 75 3.4 Electrokinetic Energy Density of the Hybrid Mode 83 3.5 Summary and Discussion 84 CHAPTER IV. THE EVOLUTION OF PLASMA INSTABILITIES BASED ON QUASI-LINEAR THEORY 86 4.1 Introduction 86 4.2 Application of Quasi-Linear Theory to Convectively Unstable Systems 87 4.2.1 Electron Stream-Plasma Electrostatic Interaction 87 4.2.2 Two-Stream Longitudinal Amplification 99 4.2.3 Two-Stream Transverse Amplification 101 4.3 Effects of Collisions and Carrier Diffusion 103 4.4 Application of Quasi-Linear Theory to Absolutely Unstable Systems 106 4.5 Two-Stream Electrostatic Oscillation 108 4.6 Summary and Discussion 114 CHAPTER V. THE POWER THEOREM ACCORDING TO KINETIC THEORY WITH APPLICATIONS 115 5.1 Introduction 115 5.2 Power Theorem for Longitudinal Space-Charge Waves 116 5.2.1 The Hydrodynamic Distribution Function 116 5.2.2 The Maxwellian Distribution Function 123 5.2.3 The Degenerate Distribution Function 126 5.3 Power Theorem for Purely Transverse Waves in a Static Magnetic Field 127 5.3.1 The Hydrodynamic Distribution Function 132 5.3.2 The Maxwellian Distribution Function 141 5.4 Kinetic Power Theorem for Hybrid Waves 153 5.5 Summary and Conclusions 159 CHAPTER VI. KINETIC THEORY OF SOLID-STATE PLASMAS FOR PROPAGATION - NORMAL TO THE STATIC MAGNETIC FIELD 160 6.1 Introduction 160 6.2 Distribution Functions in Applied Static Electric and Magnetic Fields 162 -vi

Page 6.3 Effects of High Electric Fields on the Carrier Distribution Functions 169 6.4 The Quasi-Static Hybrid Mode: General Solution 173 6.4.1 The "Cylindrical" Degenerate Distribution Function 176 6.4.2 Correlation of Kinetic and Hydrodynamic Theory for the "Cylindrical" Degenerate Distribution Function 187 6.4.3 The Drifted Degenerate Distribution Function 189 6.4.4 The Hybrid Dispersion Relation for Maxwellian Carriers 195 6.4.5 Effect of Collision Frequency Variation with Carrier Speed 203 6.5 Electrokinetic Energy and Power Properties of the Hybrid Mode 205 6.5.1 Carrier Distribution Function f Independent of 0 205 6.5.2 Carrier Distribution Function f Dependent on 0 210 6.6 The Ordinary Mode in Solid-State Plasmas 226 6.7 Summary and Discussion 232 CHAPTER VII. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY 234 7.1 Summary and Conclusions 234 7.2 Recommendations for Further Study 239 APPENDIX A. KINETIC POWER OF SPACE-CHARGE WAVES TO SECOND ORDER 243 APPENDIX B. VERIFICATION OF THE CYCLOTRON-MODE KINETIC POWER 246 APPENDIX C. THE MEASUREMENT OF POWER UTILIZING THE SECOND-ORDER HALL EFFECT 250 APPENDIX D. A STUDY OF DECAYING MODES BY KINETIC POWER CONCEPTS 257 APPENDIX E. ELECTROKINETIC ENERGY DENSITY OF SPACE-CHARGE WAVES TO SECOND ORDER 259 APPENDIX F. ELECTROKINETIC ENERGY DENSITY OF CYCLOTRON MODES 262 -vii

Page APPENDIX G. EFFECT OF CARRIER HEATING TRANSVERSE TO k ON THE HYBRID MODE 263 APPENDIX H. EFFECTS OF CARRIER HEATING ON THE CYCLOTRON MODES 269 LIST OF REFERENCES 274 -viii

LIST OF ILLUSTRATIONS Figure Page 2.1 Nature of the Left-Hand Polarized Mode Dispersion. 37 2.2 Comparison of Longitudinal and Transverse Modes Under Convective Growth. 41 2.3 Unstable Convective Interaction Associated with Helicon Region. 51 2.4 Convective Interaction with Backward Helicon. 52 2.5 Stable Interaction of Slow- and Fast-Cyclotron Costreaming Modes. 53 2.6 Two-Stream Transverse Interaction with the Electromagnetic Branch of the Helicon Spectrum. 54 3.1 Trajectory Study of Briggs' Mappings. 73 3.2 Comparison of Longitudinal and Transverse Modes Under Absolute Instability. 80 6.1 Growth Rate as a Function of Drift Velocity and Applied Magnetic Field. 183 6.2 Growth Rate as a Function of Magnetic Field. 184 6*3 Microwave Power as a Function of the Applied Current and Magnetic Field. (Morisaki and Inuishi78) 185 6.4 Low-Frequency Hybrid-Cyclotron Harmonic Dispersion Diagram. 202 C.1 Utilization of the Second-Order Hall Effect to Detect Electromagnetic Power of Circularly Polarized Waves. 251 - X

LIST OF SYMBOLS a a s B -o B c 1 -B D(w, k) E -o E -1 1+ E e e F (v ) o x f f = f + f o oL iL Constant used in Eq. 5.103. Plasma variable given in Eq. 6.54. Static magnetic flux density vector. RF magnetic flux density vector. Circularly polarized components of RF magnetic flux density. Second-order change in B due to the growing RF fields. -0 Velocity of light in media, c = ( ) 1 /2, m/s. Matrix introduced in Eq. 2.109. Static applied electric field vector, V/m. RF electric field vector, V/m. Circularly polarized RF electric field components introduced in Eq. 2.48. Second-order change in E due to the presence of growing RF fields. Subscript denoting electrons. Variable introduced in Eq. 5.48. Variable introduced in Eq. 2.96. Carrier distribution function in the presence of the applied fields (E,B ). Equilibrium distribution function in the absence of any applied fields. Components of the reduced distribution function given in Eq. 5.43. Perturbation from f L caused by the applied fields (E dtt. oL Reduced distribution functions given in Eq. 5.42. f,f y z

G(z) G 1 x) 1 1 H + h I(b) i I J -o J 1 L,L 1 2 J Jm(b) k 1 L2 M(a,b,c) * m N 0 N 1 N' o P(Wb,k) = P + jPi P -e ~+ The plasma dispersion function introduced in Eq. 5.29. Variables introduced in Eq. 5.49 and related by, Eq. 5.51. Circularly polarized components of the RF magnetic field intensity, A/m. Subscript denoting holes. Modified Bessel function of order I. Subscript denoting ions. Static current density vector, A/m2. RF current density vector, A/m2. Circularly polarized RF current density components introduced in Eq. 2.48. Second-order change in J due to the presence of growing RF fields. Second-order change in J due to the presence of growing RF fields for the left-hand circularly polarized mode. Bessel function of the first kind of order m. Wave vector, k = k + jki, m 1 Variables introduced in Eqs. 6.155 and 6.156. The confluent hypergeometric function. Carrier effective mass. Static carrier number density. RF carrier number density. Effective carrier number density given in Eq. 5.112. Polarization factor defined in Eq. 2.110. Circularly polarized components of the electromagnetic power vector introduced in Eq. 2.54, W/m2. Electrokinetic power, W/m2. -xi

Pk Electrokinetic power of the left-hand circularly polarized mode, W/m2. q Carrier charge, including sign, C. s Subscript denoting the sth carrier species. T Carrier temperature, ~K. u Velocity variable. v General velocity variable, m/s. v Carrier drift velocity, m/s. v RF carrier velocity, m/s. v + Circularly polarized components of RF carrier 1 velocity introduced in Eq. 2.48. v Second-order change in v due to the presence of growing RF fields. v Second-order change in v due to RF field growth 2- — 0 for the left-hand circularly polarized mode. vF Fermi speed, m/s. vF Fermi speed parallel to B, m/s. v Constant velocity introduced in Eq. 6.33. H vrs Velocity variable introduced in Eq. 6.87. rs vT Carrier thermal speed, vT = T/ m/s. v Component of thermal velocity parallel to k for the II hydrodynamic distribution function. v Component of thermal velocity perpendicular to k 1 for the hydrodynamic distribution function. VNl Component of thermal velocity parallel to k for the Maxwellian distribution function. v. Component of thermal velocity perpendicular to k for the Maxwellian distribution function. W Sum of electrostatic and electromagnetic energy densities, Jm2 densities, J/m2. -xii

W Electromagnetic energy density of the left-hand circularly polarized mode, J/m2 Wk Electrokinetic energy density, J/m2. Wk- Electrokinetic energy density of the left-hand circularly polarized mode. w Velocity variable introduced in Eq. 6.19. Z(z) Tabulated plasma dispersion function introduced in Eq. 5.31. z Normalized plasma variable used in Eq. 5.16 and 5.33. z Normalized plasma variable introduced in Eqs. 5.64 and ~~~c 55.94. a Constant used in Table 2.1. Us Variable introduced in Eq. 2.99. r(n) Gamma function, r(n) = (n - 1)'. 7, Variable introduced in Eq. 2.98. 6(x) Dirac delta function, introduced in Eq. 5.3 and used thereafter. a f(x)6(x - x )dx = f(x ) if > > 0 b = x = a, otherwise the integral equals zero. 0 CE:~ ~Lattice permittivity, e = E E, F/m. r o e Effective dielectric constant introduced in Eq. 6.49. ~o Permittivity of free space, F/m. r Lattice relative dielectric constant. Carrier charge-to-mass ratio.,8',0 eAngle variables. K Boltzmann's constant. X Wavelength, m. Xc Variable introduced in Eq. 6.170. s Variable used in Eq. 6.105 and thereafter. -xiii

~o V v PO Normalized variable introduced in Eq. 5.17. Permeability of free space. Carrier collision frequency. Effective carrier collision frequency. Static charge density, pO = qNo. p 1 p 2 p 2 -cr 0 3 4 O,. 1 2 Cpj cp1 1 D = W + jCw. r 1 C0 c LI) p Up RF charge density. Second-order change in p0 due to the presence of growing RF fields. Second-order change in p0 for the left-hand circularly polarized mode. A constant used in Eq. 5.34 and again in a different context in Eq. 5.65. RF conductivity tensor introduced in Eq. 6.49. RF conductivity of the left-hand circularly polarized mode introduced in Eq. 5.61. Variablesintroduced in Eqs. 5.12 through 5.15. Variables introduced in Eqs. 6.85 and 6.86. Angle variables. Variable introduced in Eq. 5.15. Angular frequency, rad/s. Carrier cyclotron frequency, wc = Bo, rad/s. Carrier plasma frequency, = ip /e, rad/s. Effective plasma frequency introduced in Eq. 6.207. -xiv

CHAPTER I. INTRODUCTION 1.1 Streaming Instabilities in Solid-State Plasmas 1.1.1 Introduction. There has been considerable research in recent years directed toward the generation and amplification of microwave radiation utilizing solid-state plasmas. The encouraging properties offered by these medial-lo are (1) densities of charge carriers far in excess of any that can be reasonably achieved in vacuum or in gas discharges and (2) the existence of two types of mobile charge carriers, electrons and holes, which have different effective masses; these masses are much less than the free electron mass. The high carrier densities afford large coupling strengths in any carrier interaction mechanism; the low effective masses provide both large drift velocities and very high cyclotron frequencies at moderate applied fields. On the other hand, the large Fermi (thermal) velocities of the carriers and their large collision frequencies (lattice interaction) limit the degree of spatial and temporal coherence available in any wave-carrier interaction. While the collisions in gaseous plasmas are often insignificant, in solid-state plasmas they are in many cases of vast importance. Under some conditions the presence of collisions actually induces new instabilities by permitting more general sets of carrier motions by which the system can reach lower energy states. In addition, as opposed to gaseous plasmas wherein instabilities naturally arise from the methods of plasma production, solid-state plasmas are in thermal equilibrium, and in most cases very high current -1 -

-2 - densities are required to generate instabilities. Thus the solid-state plasma experimentalist is often operating at the limit of sample power dissipation and the theoretician will, as a rule, have selfmagnetic field effects and carrier heating effects to consider. Since the present study is concerned primarily with velocity-driven instabilities, the relevant characteristics of those waves associated with the drifting carriers will be examined. A moving medium of charged carriers can support two classes of modes.1l One class carries positive kinetic energy so that energy must be supplied to the system to excite these modes. The other class of modes carries negative kinetic energy and, correspondingly, energy must be removed from the system to excite these modes. Useful instabilities arise from the interaction of a positive-kinetic-energy-carrying mode with a negative-kinetic-energy-carrying mode. If the group velocities of the interacting modes are in the same direction the instability is convective in nature; if the mode group velocities are oppositely directed, the instability is absolute (nonconvective). Regardless of the interaction under study, several criteria are available to investigate the instability characteristics of the system.12-15 Of the possible instabilities, those which have received the most attention in the literature are of the two-stream type. These can be divided into two major subsets via their distinctive wave properties. In one subset, termed the transverse instability, the related modes are essentially electromagnetic waves modified by the presence of both a static magnetic field in.the direction of wave propagation and the charge carriers. In the second subset related to longitudinal instabilities, the

-3 - natural oscillations of the charge carriers give rise to electrostatic-type modes (plasma waves), which in general are not affected by any applied magnetic field. In addition, these basic modes can be coupled together by applying the magnetic field at an angle to the direction of wave propagation (hybrid instability). These classifications will be reviewed in turn, followed by a description of the objective of the present study. 1.1.2 Transverse Two-Stream Instabilities. The possibility of transverse electromagnetic wave propagation in conducting solids employing a static magnetic field parallel to the direction of wave propagation was first discussed by Konstantinov and Perel.16 Aigrain17 termed the waves in uncompensated plasmas helicons on the basis of their circular polarization and proposed their possible amplification by drifting the charge carriers. Helicons propagate with small damping in undrifted plasmas if the wave frequency is less than the majority carrier cyclotron frequency and if the latter is large compared to the phenomenological majority carrier collision frequency. In compensated media, the helicon modes go over to Alfven waves propagating with small damping for large applied magnetic fields at wave frequencies above the mean carrier collision frequency.18 Helicons may still propagate in compensated media if the mobility of one species is much larger than that of the compensating species. Bok and Nozieres,19 on the basis of an instability analysis of the dispersion equation describing the drifted two-component system, determined that amplification could occur if the drift velocity of the more mobile carriers is greater than the phase velocity of the wave.

-4 - Also, since the contribution to the effective dielectric constant of the more mobile carriers is small near synchronism, the presence of a second carrier species is necessary to maintain the high dielectric constant. The gain mechanism cannot be described as inverse Landau damping since there is no axial RF electric field or carrier bunching. However, if the applied electric or magnetic field is inclined slightly from being parallel to the propagation vector, Misawa20 has found that a convective instability can be attributed to this mechanism. Rodriguez and Antoniewisz21 have also studied the helicon-longitudinal plasma interaction. Indeed, Baynham and Braddock22 have demonstrated gain for radiofrequency helicon waves propagating off-axis. Hasegawa23 has investigated the system of Bok and Nozieres19 under conditions of mass anisotropy wherein the electron mobility is much greater than that of the holes. In this case, the two-stream instability degenerates to a one-stream resistive instability; i.e., the hole drift can be neglected and the holes may be regarded as a resistive medium which absorbs the electromagnetic energy which the electron stream carries. As a consequence, the amplitude of the negative-kinetic-energy mode on the electron stream increases. Similarly, Akai24 has interpreted the two-stream instability of Bok and Nozieres in terms of the positive-kinetic-energy cyclotron wave supported by the drifting holes behaving as the passive wave which dissipates the energy. It was further shown via computer analysis by Bers and McWhorter25 for intrinsic InSb at liquid nitrogen temperature that above a threshold of applied magnetic field the convective instability is overridden by the occurrence of an absolute instability. Thus a necessary condition for a convective instability is that no absolute

-5 -instability occurs simultaneously. In a recent analysis, Bartelink26 has analyzed the modes of propagation and associated instabilities of transverse disturbances in solid-state plasmas of varying degrees of compensation and mobility anisotropy with drift exactly parallel to the applied magnetic field. The salient feature introduced by partial compensation was that the negative-kinetic-energy carrier helicon mode could exist with the mean carrier drift velocity less than the wave phase velocity. Thus the proper criterion for gain is that the negative-kinetic-energy mode must see a net positive resistance. In this regard, negative resistance can occur in p-type media in which the electrons cause greater collisional interaction with the wave than the holes either by having a larger drift mobility or smaller cyclotron mobility than the holes, or both. The negative resistance can lead to a second mode of amplification corresponding to a positive energy wave, as has been shown in bismuth.27 In a onecomponent and infinite plasma it is still necessary that the carrier drift velocity exceed the wave phase velocity so as to obtain a negative kinetic energy wave.28 The most promising scheme to attain amplification in onecomponent plasmas is the utilization of a multi-layered structure in which some of the layers would support the wave while the carriers in adjacent layers would be given a drift velocity. In one form of this Baraff and Buchsbaum,29,30 Wallace and Baraff,31 and Saunders and Baraff32 employ the surface waves which must exist at the layer interfaces33 to match the boundary conditions. The interaction between the surface wave and the bulk helicon wave can lead to gain even for carrier velocities much less than the wave phase velocity. McWhorter34 has explained the instability

-6 -as due to electron collisions at the interface between the drifting plasma layers. No experimental evidence of this effect has yet been obtained. In addition, Baraff35 has studied the layered device structure for bulk interaction wherein the surface wave plays a secondary role. A threshold for instability is found when the carrier drift velocity in one layer is approximately twice the phase velocity in an adjacent layer. The growth rate of the instability can be understood in terms of the balance of energy between power generated in the drifted layers, collisional losses in the undrifted layers and the losses or gains at the interfaces. The instability arises from the reversal of the collisional losses in the bulk of the current-carrying layer. Nanney et al.36 in an experimental study of this bulk phenomenon in n-PbTe have observed, from the propagation of MHz signals, a transmission increase of the order of 15 dB for the drifted vs. the undrifted case. They were not able to obtain drift velocities greater than the calculated wave phase velocity due to a limitation of the pulse current supply, so net spatial gain did not occur. For larger values of magnetic field the aforementioned surface effects will predominate whereas the bulk effects become more important as the magnetic field is decreased. One of the major problems associated with the helicon mode instabilities, in the frequency range which has been studied to date, is that the corresponding wavelengths are typically of the order of a millimeter so that the specimen will introduce a strong boundary effect on the wave propagation. The boundary effect has been observed in the helicon. wave propagation in a semiconductor,36 and Grow37 was unable to find evidence of the instability of Bok and Nozieres from an

-7 - analysis of the finite-dimension sample case. In addition the size effect can result in a significantly increased wave phase velocity over the infinite medium case.38 There is also the nonlocal effect of doppler-shifted cyclotron resonance which limits the threshold to which the applied magnetic field can be reduced before severe damping occurs.39 This latter problem will be especially significant with regard to the bulk instabilities because the upper frequency limit of the instability is effectively limited. Since large currents are generally required the self-magnetic field generated can cause a nonnegligible radially inward force to be applied to the plasma resulting in pinching.4 The effect of an external magnetic field is such that, when it is comparable to or larger in magnitude than..the self-magnetic field, the pinch does not form.43 Glicksman1 states that this is not because of effects caused directly by the applied field, but rather by the rapid rise of helical instabilities which tend to prevent formation of the pinch. At higher frequencies the helicon mode characteristic changes to that of a slow cyclotron wave. Absolute instabilities associated with the coupling between this slow wave and the fast hole cyclotron mode have been studied by H*6fflinger, 44 Gro37 and Vural and Steele.1l Experimental verification of this instability has not yet-been achieved. 1.1.3 Longitudinal Two-Stream Instabilities. In addition to the transverse polarization case, in which the waves are electromagnetic in character (their phase velocity being dependent upon the velocity of light in the medium), the plasma can also support waves arising from the collective modes of oscillation of the mobile carriers. They were first

-8 - studied by Tonks and Langmuir45 in a gaseous plasma and by Pines46 in a solid. Their electrostatic nature implies that the waves (termed plasma waves) are of purely longitudinal polarization (in the absence of an applied magnetic field). The number of distinct plasma waves is equal to the number of distinct carrier species in the plasma. Thus, corresponding to the electron-hole plasma in a semiconductor or semimetal, two collective modes of oscillation exist. One mode consists of a highfrequency oscillation in which the electrons and holes oscillate out of phase at a frequency which in the long wavelength limit is the mean plasma frequency. The other mode, corresponding to in-phase electron-hole oscillation, is typified by phase velocities of the order of the Fermi velocities (or, in a classical plasma, the thermal velocities). By analogy to the vibration spectrum of polar crystals the former highfrequency mode is termed the optical mode of plasma oscillation, the latter mode at lower frequencies, the acoustic mode of plasma oscillation. Very little experimental work has been done on the optical branch because of the high frequencies involved (optical or ultraviolet), concurrent with their longitudinal character; however, there is considerable literature related to the characteristic energy losses due to excitation of the optical branch of the plasma wave by passage of fast particles through thin metal foils.47 The acoustic branch is heavily Landau damped unless the Fermi or thermal velocities of the two carriers are widely disparate. The possibility of observing a two-stream instability in a highmobility semiconductor such as InSb was first investigated by Pines and Schrieffer48 who demonstrated that an absolute instability should occur

-9 - for the acoustic branch if the electron drift velocity relative to the hole velocity is comparable to the electron-thermal velocity and if the growth exceeds the collisional and thermal damping. Similar criteria were found by Harrison49 in his study of degenerate plasmas and Vural and Bloom50 in their study of guided plasmas. The general conclusion was that it appeared marginal whether or not the instability could be observed in practice with currently available materials. The major obstacle is the thermal condition since Glicksman and Hicinbothem5l have determined that the application of large electric fields to InSb at liquid nitrogen temperature results in hole temperatures which are of the same order as the electron temperature. In addition, since the oscillations are longitudinal in field polarization, they would have to be coupled out through some type of gradient mechanism. Efforts to experimentally observe the instability in bulk materials such as InSb, pyrolytic graphite and bismuth have not been conclusive.52 Recently, Robinson and Swartz53 and Robinson and Vural54 have analyzed layered structures of p- and n-InSb, with which the temperature condition can be more easily satisfied, with the result that the surface plasma waves can grow at rats only slightly less than those of the bulk plasma waves of the corresponding penetrating stream system. Interpreting the longitudinal plasma waves in a coupled-mode manner shows theoretically50 that a co-streaming two-component plasma can give rise to a convective instability provided that the collisional loss is not too severe. No suitable material has been suggested in the literature to observe this instability.

-10 - 1.1.4 Hybrid-Mode Instabilities. When the magnetic field is inclined from being parallel to the wave vector, a mode propagates withI characteristics of both the longitudinal plasma-wave and helicon or cyclotron modes. In drifted plasmas, a transverse component of magnetic field also gives rise to a Hall electric field which for high mobility materials can be comparable to the applied electric field. The hybrid system has been studied theoretically55-57 in an attempt to explain one type of microwave emission from InSb.58-59 An analysis by Vural60 of finite solid-state plasmas in the presence of an axial magnetic field has shown that coupling between plasma waves and cyclotron excitations occurs for this case which indicates an absolute instability associated with the cyclotron space-charge wave interactions. The hybrid-wave interaction is unusual in that Landau damping is absent.61 Hasegawa62 has shown that the hybrid-hybrid interaction leads to an instability in the limit that the holes are collision dominated. Swartz and Robinson63 consider this interaction to be responsible for coherent oscillations observed in InSb. 1.2 Outline of the Present Study The Chu kinetic power theorem64'65 for longitudinal space-charge waves is a well known general method for studying those power properties of the wave associated with a group of streaming charge carriers which enable wave growth, either spatial or temporal, to occur in electron-beam devices as well as in gaseous and solid-state plasmas. In Chapter II a kinetic-electromagnetic power theorem is derived for the basic carrier waves present in a static magnetic field. Past studies in this area65'66

-11 - have been inadequate in that the RF fields (which determine the kinetic power properties) are neglected in the analysis so that the results thus obtained only apply to the carrier wave in a region of no interest. In addition, because of their importance in the solid-state area, the effects of collisions and thermal diffusion on the carrier mode kinetic power are examined. In all past developments of power-energy theorems there has been no differentiation made between convective and absolute instabilities. Thus it is tacitly assumed that a carrier mode which is active for a convective instability (e.g., slow space-charge wave) can, under proper circuit configuration, be active for an absolute instability with the relevant carrier parameters (e.g., plasma frequency) playing the same role in both cases. In Chapter III, a kinetic-electromagnetic energy theorem is derived for both space-charge waves and the basic carrier modes present in a static magnetic field which shows that, in general, separate criteria are involved. The theorem also demonstrates the physical mechanisms whereby an absolute instability arises. Whereas in Chapter II the study of convective instabilities was formulated in a spatial-power framework, this study is undertaken in a temporal-energy framework. In this way the complete dual of the convectively unstable system is found and related to the time rate of change of the carrier kinetic energy at any point in the interaction region. In Chapter IV the power and energy theorems are examined with relation to the potential energy effects which arise due to the reaction of the growing RF fields on the charge carriers. The quasi-linear

-12 - theory employed is found to be a useful analytical technique for studying the evolvement of an instability from the point and time of its initiation. The general formulation adopted for the carrier mode electrokinetic power flow and energy density enables these concepts to be extended to the kinetic theory in Chapter V. Thus the power and energy effects of Landau damping, cyclotron resonance, and temperature anisotropy can be examined and the results compared with those of hydrodynamic theory. Chapter VI is concerned with a detailed analysis of the hybrid wave system and its application to microwave emission phenomena in solid-state materials. After the examination of the zeroth-order distribution function for a system in a transverse magnetic field the dispersion relations describing the wave propagation according to kinetic theory are obtained. Unstable cyclotron-harmonic behavior theoretically derived is compared with similar experimental results. Furthermore, from a rigorous study of carrier heating, it is found that an entirely new mode can appear. This mode is significant in that it exhibits synchronous behavior and has a small damping decrement (similar to the helicon mode). Finally, a discussion of results and conclusions is given in Chapter VII together with suggestions for further study.

CHAPTER II. EFFECTS OF A MAGNETIC FIELD ON THE KINETIC POWER PROPERTIES: OF CARRIER WAVES: CONVECTIVE INSTABILITY 2.1 Introduction There are three basic modes associated with a stream of drifting charge carriers in a solid which will be of interest; namely, the longitudinal space-charge or plasma wave, the left- or right-hand circularly polarized modes which propagate parallel to a static magnetic field (e.g., helicon or cyclotron modes), and the hybrid mode propagating perpendicular to a static magnetic field. For any of these modes to be useful as a source of power in a convective instability (i.e., be active) it must be such that the carrier mode transports less total energy than the dc stream alone in.the direction of wave propagation. The total energy flow or total power of a carrier mode is the sum of its associated electromagnetic power and its electrokinetic power. When this sum is less than the kinetic power of the dc stream alone it is designated a negative power mode. In the specific case of the slow space-charge wave of an electron beam the electromagnetic power is of the order of (cu/))2 times the kinetic power in magnitude65 with the practical result that it can be neglected and in this case the designation of negative kinetic power mode suffices. When a negative power mode interacts properly with a passive circuit, exponentially growing RF fields occur without violation of conservation of power since the carrier mode power growth is negative. In general, the circuit can correspond to a helix in electron-beam devices, to a second -13 -

-14 - carrier species in a plasma or to a mode of lattice vibration of a piezoelectric or polar solid, etc. Of particular interest in the present work is the analysis of the manner in which the RF fields may extract dc power from the relevant carrier mode. This is especially important for the cyclotron modes since no bunching processes are present and it is not immediately clear how these modes can supply power. By use of a quasi-linear analysis it is shown that the negative kinetic power property can be directly related to dc slowing of the carrier motion for all cases. In addition this method indicates the conditions under which the plane-wave method of analysis is valid. Because of its importance in solid-state plasmas special attention is directed to the effects of collisions, carrier diffusion, and a static magnetic field upon the kinetic power carried by the carrier mode. For small growth rates (ki < kr) it is shown that the collisions play a dominant role and can assist the instability process by contributing to the negative kinetic power property when (w - k v ) < O. In the study of convective instabilities in the present chapter the angular frequency of the fundamental field C is assumed pure real. 2.2 Kinetic Power Characteristics of Space-Charge Waves Chu's kinetic power theorem64 is a well known general method for studying those power properties of the longitudinal space-charge wave associated with a group of streaming charge carriers which enable wave growth to occur in electron-beam devices as well as in gaseous and solidstate plasmas. A brief review of the longitudinal theorem as applied to the traveling-wave amplification process is now given.

-15 - A Assume an electron beam in a drift region with a dc velocity v x. RF perturbations are assumed to have a space and time dependence of ej (t-x). Expressions for the electromagnetic and kinetic power flow associated with the medium can be derived from a base of Maxwell's equations and the linearized force equation, V x E = -j'B, I- 0 — 0 — I 2 jGDV + (v * V) - )Vp = E + Tv x B J 10 1. I- -1 PO -= +vo I (2.1) (2.2) (2.3) and where l vT VT v From Eqs. = the charge-to-mass ratio for electrons, = the carrier thermal velocity and = the collision frequency. 2.1 and 2.2 the small-signal Poynting theorem can be found as 1,. -X- - V ~ (E x B ) + ja1 H ~ H + jeE E + * J 0 '1 -~-1 -1 -1 -1 1 so that Re 1 V (E x B) = -Re(E J), 40 -1 -1 ~i -m I = 0 (2.4) (2.5) and the electromagnetic power is not conserved but rather is balanced by Re(E ' J ). 1 -1

A From the y component of Eq. 2.3, / kv \ j( -kvo - v)v l r - )E (2.6) and since Jly = PVly multiplying both sides of Eq. 2.6 by Ely yields, after rearranging, ElyJly - (- - kv0 - jv) lyl (2.7) lyly C[( - kv)2 + (kivo + v)2] A Similarly, the z component of Eq. 2.3 leads to * P2EV( - k v) )E12 Re(E J =p) r - (2.9) a [(w) - kyV)2 + (kv + v)2] Equations 2.8 and 2.9 show the important result that the transverse motion of the carriers provides a source mode for electromagnetic power growth because of the presence o f collisions if (a - krv) < 0. Since there is lio transverse bunching it is not immediately obvious how this can occur. This phenomenon will be explained later, however, in terms of the related second-order carrier dynamics wherein the equivalence is made between second-order dc beam slowing and the source properties of Eqs. 2.8 and 2.9.

-17 - A From the x component of Eq. 2.3, j() - kv - 0 kv2 kT VlX jp, p = rnE 1 ix (2.10) Since Ji = pv + p v, the equation of continuity V J+ = 1 ~t - 0 (2.11) gives kp v 0 ix p = 1 cw - kv 0 (2.12) Equation 2.12 permits writing Eq. 2.10 as j[(co - kv )(0 - kv - jv) - k2vT] J = p wE o 1X (2.13) from which the following may be found: C2) [( - k v )(v + 2k v ) Ip [ r i o i r T ] ixEl Re (E J ) = ix ix -{ [( k - kv) kiv (k v + v) + (k2 - ]k2 T r j + (2kivo 1 o + v) - kv ) + 2kik 2 + v)(r - k ) +2kkvJ r o r (2.14)

-18 - Since k. > 0 is the case of concern, corresponding to exponentially 1 growing fields, Eq. 2.14 shows that for Re(E J) to be a source for ix ix the field growth it is necessary that (o - krvo) < 0. Note that when this is the case the collisions assist the source power. Indeed, by inspection it can be seen that if v - -> it is then necessary that v > v in order for the longitudinal carrier mode to be a source of power. This explains why in solid-state plasmas, where v > vT is not in general possible, negative power modes are possible because of the collisional assistance provided. This will be further elaborated upon in Chapter III in the study of electrokinetic energy densities in solids. The real part of the electromagnetic power flow associated with the carrier stream is now examined. Since it is in general difficult to take into account explicitly the effects of the external circuit on the wave dispersion the results obtained for this flow are approximate and only apply to weakly coupled systems. From the assumed space and time dependence, 2k. Re V (E x B ) = - Re(E x B. (2.15) To determine this quantity from Maxwell's equation it can be found that * 1(E xE y E 1 ) E xB = +E (2.16) -1 — 1 j \ iy ox iz x / so that from Eqs. 2.15 and 2.16

-19 - ~r, ~ -| 2k.k 12 Re V (E x Bj) = r (E lyl + IE lZ), (2.17) o 0 which is of course positive for the case of interest (ki > 0). The conservation theorem is now obtained by integration of Eq. 2.4 over the volume containing both circuit and beam, which gives when the real part is taken e oV ~ (E x B) + Re | -V ~ (E x B dV e 0 o (l - circuit L - -I beam = - e(E J + Re(E Jbeam dV ~ (2.18) — 1 circuit - - beam The function Re(E * J ) corresponds to the wall losses at the -1 -1 circuit circuit and is usually negligible. In addition, since at the beam the electromagnetic power flow corresponding to Eq. 2.17 is small compared to the beam kinetic power flow (i.e., IE I >> E |, IE IE1), Eq. 2.18 reduces to Re (E x Bj +circuit (E J )beam dV - 0. (2.19) 0 o - ~ -circuit Make the definition 1 * A CP..u - - Re(E x B ) x (2.20) -c ircuit - - 0o correspond to the circuit power and

-20 - A 1 A -k 2k. Re(E J )beam x (ki 0) (2.21) 2-k. R-e -2 beam i correspond to the beam kinetic power. If Gauss' theorem is used in Eq. 2.19, the Chu kinetic power theorem is then obtained as Re S (Pircut + *dS 0 (2.22) which is a surface integral enclosing the volume of interest. From Eqs. 2.8, 2.9, 2.14 and 2.21 the most general form of the beam kinetic power should be given by wv(0w - k v )(IE 12 + |E 2) = P r o ly I.1Z k 2k.c[(C - k v )2 + (kiv + v)2] 1 r 0 ro 2 2EV [( - k ( + 2 )+ k ( + I.0. (2.25) [[( - k v)2 - kv o(kiv + V) - V (k2 - k) + [(2ki + v)(o - kv o) + 2k.k v2]2 In particular, when v = 0 = vT, the well known form is obtained: ooEwv (w-kv)E - 2 r) v v k [(a k v )2 + kv2]2 L\ 1X r 10 The latter expression has commonly been used as the definition of beam kinetic power in the presence of collisions and thermal diffusion by

-21 -Vural and Bloom.67 This is inappropriate however since this definition has no direct significance in the balance of power given in Eq. 2.19. Thus in the previous work67 the real kinetic power as defined by the right-hand side of Eq. 2.24 is determined to be negative whenever C - k v < O. Inspection of Eq. 2.23 shows that this is not in general sufficient. The fact that the definition of kinetic power of Eq. 2.24 is misleading has recently been pointed out.68 To utilize the concept of kinetic power properly in the presence of collisions and thermal diffusion the definition given in Eq. 2.21 is required. To proceed further with the study of the space-charge-wave case, from Eq. 2.22 the total real circuit power (e.g., in watts), P is circuit PT = Re S P.. dS =- Re P - dS. (2.25) circuit -circuit - -k - From the assumed spatial dependence, the following definitions can be made: E = E ej 1X 10 and Ex = E e, (2.26) IX 10 *^ where the multipliers E,E are independent of x. In the case where 10 10 collisions and thermal diffusion are ignored, use of Eqs. 2.24 and 2.26 in Eq. 2.25 gives pT pCUj2V (w - kv ) IE12 f 2k.x r o 1 o ro 1o

-22 - where only the mode of interest has been retained. Equation 2.27 indicates that for amplification to occur two requirements must be met; the nature of the exponential term requires that k. > 0 and at the same time it must be that v > (w/k ) in order that the circuit power be positive. If both these criteria are met the circuit power for a device of length L is given in terms of the input amplitude of the longitudinal RF electric field by - T pC0w2Cwv (k v - ) / 2k.L TT., = o ~ o |E I e -. (2.28)o 2 circuit k )2 + ] 11 (2.28) Hence an active or amplifying carrier mode is obtained if it has the property o/k < v,. From the dispersion equation describing the uncoupled modes on the beam this then indicates that for weak coupling at least the slow space-charge wave (w - kv = - I |p) and the fast space-charge wave (a - kv = 1| |) are active and passive, respectively. 0 p It is now shown that the carrier mode power properties can be related to the beam RF conductivity. For v = 0 = vT the longitudinal force equation gives nE X = j( - kv )v; (2.29) so employing J =p v +p v = o ngt E (20) iJx oV ix P1VO longtl ix (2.50) and the continuity equation, it can be found that c2Eok.iv (w - k v ) Re(ongt) = - I (2.531) longtl 2[(c - kv )2 +kav2]2 [( r0 o k1 o

-23 - Hence under conditions of amplification the real part of the beam RF conductivity is negative and as a result the property of negative kinetic power flow is equivalent to that of negative RF beam conductivity. This equivalence will be true in general if the kinetic power flow is determined only by the first-order fields. This has been shown to be true for the space-charge waves by retaining variables to second order in the analysis.6 The commonly held interpretation of the negative kinetic power property of the slow space-charge wave is that an excess particle density, N = N + IN I, occurs spatially localized with a decreased beam velocity, O 1 v = v - |v I, and vice versa, so that less net kinetic energy is transo 1 ported across a surface enclosing the beam than under strictly dc conditions. It is shown in Appendix A however that an alternate viewpoint can be adopted and the negative kinetic power property is simply due to secondorder dc beam slowing. The two viewpoints give mathematically identical results. 2.3 Nature of the Transverse-Field Contributions to the Kinetic Power Flow It is wondered by what physical means the transverse contributions to the kinetic power given in Eqs. 2.8 and 2.9 can be accounted for since, in the hydrodynamic model for the purely transverse modes, there is no carrier bunching present; the latter was found to be directly related to the kinetic power characteristic of the longitudinal space-charge wave. To see this a quasi-linear theory is used in which second-order effects on the carrier stream are expressed in terms of the fundamental RF fields. Write the force equation to second order to give

+v vv +v V2 a + (v * V)V + (V * V)V + Vi + V VP + e + (v *V )v + V. v + ( )v +v v + 7p = ( +E +v xB +v xB), (2.32) '-'2 ' —o '-2 P0 2 -' - _1 -1 -0 - and from this take the time-average real part, giving for the longitudinal A or x direction Sv 2 ~V2 ap, / / v + vv + = + 2 Re Re(v x B), Vo 7dx 2 P 2.x X 2 1 1 (2.335) where v, p and E are then the second-order changes in the dc state of 2 2 2 the system caused by the presence of the RF fields. From Eq. 2.7 and the assumed spatial dependence, Eq. 2.33 may be written as v2 ap k. (2kiv + v)v = - p 2 2'' 2(kr(w - k v) - ki[(c- - kr v)2 + k.vo(kiv + v)] 12 + ---- Io I rj 7 22[( - k v )2 + (kiv + v)2] ly (2.34) Inspection of this result indicates that when the source functions given in Eqs. 2.8 and 2.9 are negative this leads to a dc slowing of the beam velocity since a negative contribution is given to v. Hence any carrier 2 mode with transverse field components which are such that (ac - krvo) < O acts through collisions to reduce the dc beam velocity so that the beam kinetic power is less when such modes are excited than in their absence. In the case where the fundamental fields are purely transverse (TEM wave) the above results are still applicable with p = Iv I = 0 1 IX

-25 - since E = O. It is assumed that the mobile charge carriers drift in 1X the direction of wave propagation. The general dispersion equation may be written as50 2(W k2c2 _P k2 c-cD2 + C( -k) = 0, (2.35) C - kv - jv where c = 1/\J17 is less than the free-space velocity of light. Equation 2.35 can be written in the form ( - kv )(k2c2 - O2 + a2) = jv(k2c2 2). (2.36) Thus in the absence of collisions (v = 0) two stable uncoupled types of modes are obtained. In the presence of collisions the forward traveling waves are coupled and the solution50 for this case shows that an instability of the resistive type is obtained if v > C/kr c. This is then in full agreement with the fact that the beam kinetic power flow as given by Eq. 2.23 (with E = 0) is negative. Since the electromagnetic power flow 1X is positive when the instability occurs it is clear on physical grounds that it must be that v < 0, corresponding to beam slowing, since this is 2 the only source of power available. A problem which arises in this connection is now examined. As no RF bunching is present, the secondorder dc current is given by J = v + P v (2.37) 2 02 2 which by a consideration of continuity must be zero. Hence,

-26 - v p = -p 2 2 0 V 0 (2.38) and, in the steady state, any alteration in the dc beam velocity must be accompanied by a dc bunching given by p. From Poisson's equation the 2 function p gives rise to a second-order dc electric field given by 2 eV * E = 2k.E = p -'2 1 2 2 (2.39) Use of Eqs. 2.38 and 2.39 in Eq. 2.34 then provides TI2k v(D - k v ) -. r r o +k.iv (k.v + v)]) v _ 1 2c2 [(- - k v )2 + [2k.iv - T v 0 ki[(w - krv )2 i r o (JE 2 + EZ I2j (kiy 1zV)2z (k v + v)2] 2kiV + v + 2k v0 ]o -J (2.40) First note that the dispersion relation alone, Eq. 2.36, is independent of vT. On the other hand the alteration in the dc beam velocity given by Eq. 2.40 is dependent upon VT, particularly when the growth rate k. predicted by the dispersion relation is large. Thus, if causality is based on the requirement v < 0 in this case, it must be concluded that 2 the dispersion relation alone does not in general give sufficient information to determine causality. As an example, if vT >> v in Eq. 2.40, then v > 0 and the convective instability predicted by the solution of 2

-27 - the dispersion relation (together with any causality criteria applied thereto) does not exist. These results indicate that a basic inconsistency can be found when the dispersion relation alone is used to study the system. This inconsistency is in part removed when it is seen that if the effects of p are significant upon the steady-state system (as predicted by the 2 quasi-linear plane-wave analysis) it is necessary a priori to take into account the self-consistent possibility of a dc density gradient by solving the system starting from an assumed spatial dependence of the fundamental fields,, as C (x) ej. In addition, the time-averaged or dc 1 lo quantities become functions of x which are dependent upon the fundamental fields themselves. The problem is then highly nonlinear and cannot be solved by straightforward analytical means alone. These results indicate that essential nonlinearities can exist in the determination of the stability of a system in the steady state. In general, to satisfy conservation laws, the presence of a convective instability in a system requires that the fundamental fields extract power from the carrier mode. To solve for the stability of a system it is common to assume that no significant dc gradient of density is present so that the plane-wave type of analysis can be performed for the steadystate case. When this is done, however, the plane-wave solution to second order may show that to alter the dc beam velocity to provide source power for the instability a dc density gradient must also exist, thus violating an initial assumption made in the analysis. A balance of power which can also be obtained is found as follows. From Eqs. 2.38 and 2.59,

-28 - p V E 2 (2.41) 2 2k.v ' I 0 so that if amplification occurs (ki > 0), and v < O, therefore P22 p2~ Re(E J) = 2 > (2.42) 20 2k Thus, as in dc Poynting vector calculations related to Joule heating,70 the viewpoint may be adopted that the external source (e.g., battery) is supplying the power for the amplification in addition to the usual heating effects when collisions are present, the latter being given by E J O o 0 The plane-wave analysis is fully justified if it is assumed that recombination processes are present in the system so that no second-order density gradient is established and hence p = = = O. In this case, 2 2 J = p v 0, and Re(E * J) provides the term E J to account for the 2 02 0 2 source of the amplification. 2.4 Power Characteristics of Purely Transverse Modes in a Static Magnetic Field In this section the case of the carrier modes associated with transverse electromagnetic waves propagating parallel to a dc magnetic field is considered. If the plasma frequency is much less than the angular wave frequency, as is common with electron beams, the associated carrier waves are the slow and fast cyclotron modes. In solid-state plasmas these are the left- and right-hand circularly polarized modes. 2.4.1 Derivation of Electromagnetic Power for Purely Transverse Waves. Assume a collisionless convected medium (e.g., an electron beam)

-29 - A A in a static magnetic field B x with drift velocity v x. Transverse A A (y - z) RF perturbations are assumed with space and time dependence ej (tx). An expression is derived for the electromagnetic power flow associated with the convected medium from a base of Maxwell's equations and the linearized force equation. With the assumed spatial dependence temporarily suppressed, the following equations are then obtained: V x E = -jcB, -1 -1 (2.43) and V x B = J J + j4o E -1 o-1 -1 jwvv +(v V)v'> = nE + iv x B + rv x B t 'VI - -1 -' 1 - (2.44) (2.45) where V = From Eq From Eqs. x(8/8x) and the charge-to-mass ratio of the single 2.1 and 2.3, species medium. av jcuv + V -y ly o ax and 1IV r +v0 yE + cz v + j 1Y C 1Z jca TVo 'E - X v + ~v lZ c lY jO aE 'x (2.46) av j~vlz + Vo = aE lz 3x (2.47) where xc = nB. Introduce rotating coordinates with the following definitions:

-30 - V = v + jv 1+ ly 1z E = E + jE, 1o + Jy j = pv =J +jJ, 0 1+ J- lz v v - jv 1- ly Jz E 1 - = E - jE, Jy I 1z.9 J 1+ p v 0 1 - J - jJ, (2.48) ly lz where p0 is the charge density of the medium. If Eqs. 2.48 are then applied, Eqs. 2.46 and 2.47 become v v T+ v E + j(C) ~ )v + + v 1 qE + C 1- o t 0 = 1 E + 1 j i X (2.49) For the transverse fields of interest, the wave equation is readily obtained from Eqs. 2.43 and 2.44: a2E 2 - +- E = jciO J ax2 2-1 0-1 (2.50) From this, employing the definitions of Eq. 2.48, the independent leftand right-hand circularly polarized modes in the transformed system are shown to be a2E + 2 1 -- + E -= j-o J ax2 c2 1 ~ 1+ (2.51) * The expression E x B which is related to the real part of the electro-1 1 magnetic power flow is now examined. The equation

-31 - 1 * -E x B o -1 -1 0 -1t(E i G-)~t ly E* xy E 1z xA zx lz x (2.52) transforms according to Eq. 2.48 as 1 * - E x B 40 — i -1 E * *E ~ l- OE~X~ x.E - 2j o \ + i- (2.53) The power flows are now defined by the vectors ~E* 1 A P - - E1+. 7 x5. (2.54) -el~ 2jc i~ dx To obtain expressions for these power flows in terms of the RF carrier velocities, the following procedure is followed: a. Apply the operator (/)x to Eq. 2.49 to obtain av + o2v (O + C) X- + - = b q 21 nw o nx b. Use Eq. 2.51 now to obtain aE + Tlv a E + X > +. --- + 1- 0 1-x T I I x ( 2. 5 5 ) (2.55) 1~+ 1~ $ 0 av -2v aE + V / 2 \ j(c + ) -- + v - =. - ' + j- J -- E ) * (2.56) C7 r)x 0 ^X2 * 5x jw = V 1~ c2 1~ E ax2 d'o - ' c. The first term on the right-hand side of Eq. 2.56 is replaced using Eq. 2.49, and rearranging gives

-32 - E+ 1 v2) \1 (1- ) 2 c VO v ((C ~ OC) 2V JCVvV o v + Of 0 C7 PO_ cC 0 + i J~ v c2 v~ + w~ x 0 v2 a2V +j O + + j 1~ 6xax, (2.57) where and 2 1 2=~ C = 2 0 P C (2.58) (2.59) d. Applying the operator 6/6x to the complex conjugate of Eq. 2.57 and using the resulting expressions, together with Eq. 2.57, gives for the power flows of Eq. 2.54, = x Ip co 1+ P~f A j + a) v ~ v aPe = v2 2 c 2 2 1+ - ~x V 2 C^V. - r 2 V C2 c 0. o i~. (, p o c o0 o + ~ -O _ _ ~v d+ W - +-...... u a6x2, Ca2 x+ W ax2. ax3V (2.60) The spatial dependence assumed earlier is now invoked so that for example av,/x = -jkv +, v /x = jkv +, etc. In this manner, Eq. 2.60 1+ 1+ 1+ 1+ becomes, where k = k + jk., r J A Pe = x -e~_+ k vl+ I 2 v2 kvo k 2v2..._ ~ _ _. p 0 o c o. + -. + - _ ---— D CD - ~ - CD 2 2 c 2 (1 )c 2 2 CC 2aco 1 - c C (2.61)

-33 - Equation 2.61 gives the electromagnetic power associated with the carrier stream in the absence of interaction.. It will be found that the presence of the interaction affects this result to a negligible degree. Since examination of Eq. 2.61 indicates that the electromagnetic power is positive, to obtain a negative power mode useful in carrier wave-circuit interactions the kinetic power must be negative and exceed the electromagnetic power in magnitude. 2.4.2 Kinetic Power of the Purely Transverse Waves. In the Poynting theorem given by Eq. 2.4, if the definitions of Eq. 2.48 are applied, the theorem is transformed to the following: V7 (Peg + Pe-) + J2 Jo(HLH + HH + 2 E(E+E+ + EE ) 2 (E J + E J ) = 0 (2.62) 1 1 l- 1 -where P e are given by Eq. 2.54 and )E + H A 1 1 -H / -1 1+' (2.63) Also, because the first-order fields were assumed purely transverse, * any longitudinal terms, such as E J, are zero. Since solution of the 1X 1X dispersion equation indicates that the (+) and (-) modes are uncoupled, for our purposes assume that the (+) mode amplitude is zero, i.e., v = 0, and hence only the (-) mode is present. It will be seen that 1+ from the results obtained from the (-) mode the (+) mode results can be derived by the replacement of Xc by -w. Inspection of Eq. 2.49 provides C c c

-34 - the following table within the (-) mode framework, if electrons are assumed A as the carriers and B = calBol x, where a = +1. -o o Table 2.1 Carrier-Mode Sign Convention Sign of C- Electron Beam (en ~ en) cSign of X Solid-State Plasma Electron Beam (W << ) < 0 ( = +1) Helicon-cyclotron mode Slow-cyclotron mode > 0 (a = -1) Right-hand polarized mode (RHP) Fast-cyclotron mode To proceed with the derivation of the kinetic power the source * function E J is derived in Appendix B and verified as a proper physical 1- 1 -concept. From Eq. 2.21 and B.10, the carrier-mode kinetic power is given by 2 P1 (1 AO 1- v A -P- = - ReK(-E J) x = — |- x, (2.64) -k- 2k 2 1- 1 4rl[(o) - k v )2 + k2v2] r o 1i o which shows that the modes with wc < 0 in Table 2.1 have the negative kinetic power property. Compare this result with that obtained using coupled-mode theory.71 The latter technique applied to the pure cyclotron modes by ignoring the RF fields and requiring that the power represented by the square of the cyclotron normal mode amplitudes be conserved gives for the power carried in these cyclotron normal modes,71 _J ylv 2 P + - 0 (2.65) t 8ti iucl Except for a factor of two related to the normalization used in Poynting's theorem, this is the result obtained from Eq. 2.64 when the cyclotron

-35 - dispersion (w - kv - = 0) is inserted. Thus Eq. 2.64 is the more O C general result which reduces properly to the simple normal mode result. As was done in the derivation of the longitudinal space-chargewave case, Eq. 2.22, both sides of Eq. 2.62 are integrated over the volume of interest, Gauss' theorem is employed, and the real part is taken to obtain the conservation law, ReT (p P ) d8s = 0, (2.66) J (-ei- -k- beam + circuit as a surface integral over the volume of interest containing both carrier mode and circuit. As a simple example, assume the circuit losses given by Re(P )circuit are negligible, so that Eq. 2.66 becomes k- circu it Re ( e-circuit. dS + P * dS = 0, (2.67) Re (P el-)circuit - J -T - ' dS where PT- = Re(P_ + Pk- eam (2.68) T — beam or from Eqs. 2.61 and 2.64, w2v2 kv u k2v2 2 p o o c o r c 2 W (c 2 = c T- =V2 _L 0 ( o C2 + — c. 2. 1 (2.69) 4n[(o - k 0)2 + kv 2] r o 10

Equation 2.67 indicates that for the carrier mode to supply power to the circuit enabling wave amplification to occur it is necessary that P < 0 corresponding to a negative power mode. For the particular case of a slow-cyclotron mode interacting weakly with a circuit, if the cyclotron mode dispersion (w - kv - w O 0) is used in Eq. 2.61 it is found that Re(P ) 0. Thus the slow-cyclotron normal mode has the attribute for instability processes that it carries no electromagnetic power of significance. It is important to note, however, that the entire spectrum in cu-k space of the left-hand polarized mode has the negative kinetic power property, as can be seen in the form of Eq. 2.64. By inspection of Eq. 2.69 then, any region of carrier-mode presence near k 0 is an active mode. For example, with reference to Fig. 2.1 r wherein the left-hand polarized mode in the uncoupled state is shown, the region near C cX corresponds to an active mode even though this is 2 part of the electromagnetic branch. This important aspect of the power characteristic will be verified later in this chapter when the two-stream instability is studied. 2.5 Nature of Magnetic Field Effects on the Kinetic Power Flow As in the collisionally induced source function contribution studied in Section 2.3, since in the hydrodynamic model there is no carrier bunching in the transverse plane, it is asked what physical means account for the transverse contributions given in Eq. 2.64. To see this it is necessary to study the second-order longitudinal carrier dynamics in the steady state. The presence of the transverse fundamental field components gives rise to a nonlinear real Lorentz force given by

-37 - ELECTf RI // ROMAGNETIC RANCH.l-/ / ' I' / / o/ / HELICON / / ^ BRANCH ki: - +I>1, ' o2 2 CD2V p o D = ----... 4 DI I 0C2 40 1 k 1 V 0 22 _ C2 + =(_- = 0 Do - kv + IO 01 0 k FIG. 2.1 NATURE OF THE LEFT-HAND POLARIZED MODE DISPERSION.

-38 - 1 A FL = q Re(v x B + v x B ) x, (2.70) - - - 1 -1 where q is the carrier charge. The second part of this force which varies as (v x B ) gives rise to second-order variables which have the form — 1 — 1 e ( ). Such variables give no time-average contribution to the major power transfer (which is second order) and can be ignored. Analyzing the remaining time-averaged force in terms of Eqs. 2.48 provides IZ. + V _1+ Re(v x B ) = Re - av -X + ax (2.71) 1i -1 2 \ Ju1O - 6x Jw 1+ ~x / showing that independent contributions arise from the left- and right-hand polarized modes. The second-order longitudinal force equation, since the fundamental fields are purely transverse and collisions are assumed absent, is given in the steady state by av / \ vo = E + n e() x Re(B) A (2.72) Vo a - r- 1 1/ x which from Eq. 2.71 provides the following separate equation: v a= E2 + 2 Re ( - v + (2.73) o x 2- 2jw - 1 ax where E = (E + ) (2.74) 2 2 2+ 2 -and v (v +v ). (2.75) 2 2 2+ 2 -

-39 - Using Eq. 2.47 for the last term in Eq. 2.73 gives 1E ) 12 1v l- 1 * 2 _ 1 1 --2V10v = 2lE* J + I - I 2 _ -(2.76) 2jwo V1- ax 25o 0 1' 2rv 0- 2- 1 3X ' which from the assumed spatial dependence and Eq. B.10 gives in Eq. 2.73 for the (-) mode k. k.[v |V c - 2k.vv oE 2- + (2.77) 1 v~ 2- 2-= 2 ~1- 2[( - k v )2 + kov2] (2 ir 0 ' 1 0 Inspection of this result shows that when the kinetic power given by Eq. 2.64 is negative this leads to a dc slowing of the beam velocity since a negative contribution is given to v. Hence the circularly 2 -polarized carrier mode with xc < 0 acts through the applied magnetic field to reduce the dc beam velocity so that the beam kinetic power is less when such a mode is excited than in its absence. As no RF bunching is present, the second-order dc current is given by J = p2v +p v (2.78) 2- 02- 2- O which from continuity considerations must be zero to conserve particles. Hence, v P- = o (2.79) 2- = -o0 ~ and the dc beam-velocity alteration is accompanied by second-order dc bunching. A comparison of the fundamental mechanisms for effecting

-40 - power transfer of the longitudinal vs. the transverse modes can then be made in Fig. 2.2. For the one-carrier case, the field E is given through Poisson's 2 -equation by EV * E = 2k.EE = p. (2.80) -2- 1 2- 2 -Use of Eqs. 2.79 and 2.80 in Eq. 2.77 then provides (c - [(co - k v )2 + k2v2]} v | V Vy........ (2.81) 2- / 2 4v + 2 ) [(C - k v )2 + k2v2] o0v 2 r 0o 1 0 4k0av2 1 0 The relationship of the total kinetic power flow to the result given in Eq. 2.64 is now derived and a fundamental problem discussed. From Tonk's theorem,72 which is applicable to the present case since collisions are absent, the following may be found: Re x B + X - 2 J dS = 0 (2.82) which relates the electromagnetic power flow to the kinetic power flow. A A A In the present case, since v = (v + v ) x + v y + v z, 0 2- 1y 1z * V * V V v Re f= = 2 ) = =2 IV2 +2 v 2 v2 2v v + + - o + 0 2- 2 2 2~,.. Tr +. J... (2.83) 209 o 2,. o

-41 - I I ----- INTERACTION REGION -— I I _A I! ' Box I No.IVO '| No+NI vO+vl I NO+ vo N, I (a) DENSITY AND VELOCITY CHARACTERISTIC OF SLOW SPACE-CHARGE WAVE ---- INTERACTION REGION -- | "'__ B_ —B___x I po+p2_V0+V21 IPO.___ Po, Vo Po+P2-, Vo+ v2- I PoI Vo \ - I 4 V2 - - /~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (b) DENSITY AND VELOCITY CHARACTERISTIC OF THE (-) TRANSVERSE MODE FIG. 2.2 COMPARISON OF LONGITUDINAL AND TRANSVERSE MODES UNDER CONVECTIVE GROWTH.

-42 - Equation 2.83 leads to defining the kinetic power as lV1- 12 2v v +. T 2 2- 2 P J (2 84) k- 2T. o But from Eqs. 2.64 and 2.77 it can be found that T 1 * kP ~2k Re(E J + E J ). (2.85) k- 2 (2k2- o 1- 1 -Thus the kinetic power expression obtained from Tonk's or Chu's type of formalism is not in agreement with that obtained in Eq. 2.64. The difference lies in the fact that the field E satisfies a separate 2 -conservation law which from Eq. 2.82 may be found as Ref (E x B + E ) dS = 0 (2.86) 2- -O S -2- -- where B is the self-magnetic field due to J. The kinetic power form given inEq. 2.64 must be used for self-consistency with the dispersion given in Eq. 2.64 must be used for self-consistency with the dispersion equation obtained for the system. Note that the field E is in essence the reaction presented against the beam slowing. In the action of slowing, it must be that dc bunching occurs and via the field E setup prevents the slowing. The 2 -field E can have the aspect of an essential nonlinearity for some 2 -systems. For example, when thermal diffusion is introduced into the present model as in Eq. 2.34, Eq. 2.81 then becomes

-43 - - (cc - [(o - k v)2 + k2v2]} Iv | C r o 0 1 o - v =..-. (2.87) - 4v 1 + - ) [( kv )2 + k2v2] 0 4k2v2 v2 r o 1 o i o o 1 0 0 Assume that the linear dispersion equation for the system (which is independent of vT in the hydrodynamic model) is solved for and a growing wave (ki > 0) is obtained. Clearly v < 0 is necessary for this solution 2 -to be permissible since this is the only source of power available to balance the positive electromagnetic power growth. However, if the 1/2 solution is such that vT > v [1 + (2)/(4k0v2)], Eq. 2.87 shows that 0 p 1 v > 0. Thus for those parameters used in the system the solution obtained 2 -is not self-consistent since v > 0 indicates that in the steady state 2 -there is no beam slowing. The salient point is that in such occurrences the linear dispersion relation with homogeneous dc charge densities breaks down from the viewpoint of conservation of power and a more elaborate nonlinear analysis is required. Previous investigators have stated that the power properties of the transverse modes in a longitudinal magnetic field depend upon the secondorder RF electric field50 or upon a transverse gradient of the fundamental RF electric field.66 The former of these explanations is invalid since the second-order RF electric field has zero time average and hence does not contribute to the conservation of power to second order. The latter explanation, based upon gradients which are not in general present and are not included in the dispersion relation for the system, was invoked as a necessity since it is known that the (v x B ) force cannot alter the total carrier kinetic energy but only transfer energy between longitudinal total carrier kinetic energy but only transfer energy between longitudinal

-44 - and transverse forms. The important point however is that the (v x B ) force can alter the flow rate of the total carrier kinetic energy out of the surface of interest. It is because of this alteration of the flow rate from v to (v + v ) caused by the second-order time-averaged O 0 2 -Lorentz force that enables the fundamental RF fields to extract power from the dc carrier motion without affecting its total kinetic energy. This becomes particularly clear in the following case. Assume the system is such that, in the steady state, recombination processes are present which prevent any second-order density gradient p from forming so that E = O. In this case, which is the only case 2- 2 -which can be accurately evaluated based on the plane-wave analysis, the kinetic power forms given in Eqs. 2.64 and 2.85 are identical and if Eq. 2.83 is used the kinetic power may be written, since p = p = 0, 1 2 -as P- = Re V (2 88) k- o 2m v (2.88) where m is the carrier effective mass. By definition the total carrier kinetic energy is v * v Ek No- - (2.89) so that the net kinetic power flow out of the surface of interest is + Pk_ dS_ = Re Ekv dS. (2.90) By selecting a cylinder encompassing the interaction region whose axis is longitudinal, the nature of the carrier dynamics shows that on the

cylindrical surface v * dS = 0 (i.e., no particles escape in the transverse direction), whereas at the end sections a net contribution arises since there v * dS = [v + v (x)] x * dS. Thus even if the function Ek is -- -' O 2 - -- constant, if v < 0 the integral given in Eq. 2.90 is negative, indicating 2 -a power source for the instability. In the alternate limit, where no recombination processes are present, the field E which must then be present can alter the carrier kinetic 2 -energy and thus account for the power transfer. Again, in this case, since a density gradient p is present, an assumption used in the steady-state 2 -plane-wave analysis is violated so that strictly speaking a more complete nonlinear analysis should be performed. For such systems, although p < p, the quasi-linear theory shows that the effect of p is significant in general in the power conservation process. It is reasonable to assume, this being the case, that any study of such systems in the steady state must include the dc density gradient in the analysis of instability phenomena. The nature of the second-order field E shows that this property 2 -of the wave propagation applied to solid-state materials may be used to advantage in the measurement of carrier effective mass or in the detection of electromagnetic power. This aspect is presented in Appendix C. 2.6 The Two-Stream Transverse Instability in a Longitudinal Magnetic Field It is desired to verify some of the results obtained in connection with the power characteristics of the circularly polarized waves. A useful case which will be solved explicitly under small-signal conditions is that of the two-stream interaction. In this system a second carrier, with

-46 - superscript (2), is used as the circuit and the primary, active carrier is assigned superscript (1). The quasi-linear theory is first used to study the second-order effects when no recombination processes are present. In this case, from continuity on each stream, Eq. 2.79 gives p (S)v (S) p(S) -- v 0 s = 1,2. (2.91) From the assumed spatial dependence and Eq. 2.91, Poisson's equation for the second-order dc field, eV * E = p(s), becomes -2- S 2 - 2k. eE 1 2 - 01 02 v(2) 'V2 2 -O2 (2.92) where in general subscripts are used to delineate the carrier species in the case of zeroth-order quantities. Superscripts are used for reasons of clarity in the case of second-order quantities. Employing Eq. 2.77 for each carrier gives 2k.v v(S) 1 OS 2 - k. Iv I [oxu - ( -c - kv )2 - kv2 ] = q E + 1 1 s r os 1 s 2" 4v o[( — kv )2 + kv2 ] Os r Os I os; s = 1,2. (2.93) A further relation is provided by the first-order force equation, viz., E 1 - j(o - kv - o ) a(1) 01 C1 1 -i ( - kv ) 1 01 j(c - kv - ( ))V (2) 02 c2 1 -7r (C - kv ) 2 02 (2.94) from which it may be found that

-47y (2V () 1 i iV /) 1 (2o95) 1 - 1 1 where Z T (C kv )0(C2 (w kV C ) f r2T kv ) (_ __ 0 _ CJD (2~96) 1 01 02 (C2 Substituting Eqso 2092 and 2995 into Eqo 2o95 gives the following result Iv(-) I Vo (4k v2 + V 2) 2 2 + ) 7o 2 = ~ 1 o 01 0 1 2 P21 2 21 8p [( v )2 [.2 ]V2 (4kv ) ]+ r oi 1 01 02 P2 p 02 where s..C) F.. k ) () k2 (98) 7 s cs r oS i os and (C = k v O c ). s = 1,2 (2099) s ' r Os Cs O I Os t _ _.aat (2) By a symmetryy argument, the function v. is readily obtained by the 2.. replaceme1nt (-'L 2) througho'ut Eqo 2o97~ Inspection of Eqo 2097 shows 2 (-> 1 that'2 in the absence of recombination processes2 in the steady state the quasi-linear theory predicts a strong influence on the second-order car.rier dynamics of the primary carrier by the secondary carriers and vice versa.As an example, consider the system with CD < 02 CD > 09 Ci C2 (v ov ) > 0 and n l < 0 so that based on the purely linear theory of 01 02 2 Eqo 2064 the mode of carrier (1) is active and that of carrier (2)2 passiveo Inspection of Eqo 2971 shows that if the interaction occurs with the right-hand polarized mode of carrier (2) near (c kr v ) 0 thbe situation r o2

-48 - can easily occur wherein v( ) > 0. In this case it is clear, since 2 -v() < 0 must occur to supply power for the interaction, that in this 2 -region any convective instability predicted by the linear dispersion relation alone has dubious validity since these second-order effects which affect the power conservation are ignored. The quasi-linear theory is now used to study the two-stream transverse interaction for systems which are a priori known to possess recombination and generation mechanisms such that in the steady state no dc density gradients are permitted. For the two-carrier case, since the circuit is well defined, the electromagnetic power flow for the system can be obtained. Equation 2.56 becomes, in the (-) mode formulation, for carrier (1), av(1) a2v(l) aE1 CV l - l- E'W - O1 j(W - - T + \. l- = - \ E + ~ v ci ax x2 1 1 jc2 l- 1 0o * POSv). (2.100) S=l,2 Use Eq. 2.49 to replace the first term on the right-hand side and rearrange to give ov2 o / e2 v,( ) V v(i) - 11 -0 pi oi ci 01 1 -[1 - - E = j (ac - X vl+ - v2 S2v(1) a2 v2 2 Apply the operator a/ax to the complex conjugate of Eq. 2.101 and use the result obtained in Eq. 2.54 to provide, together with the definitions of Eqs. 2.95 and 2.96,

HI E UNkiERSi QF MGICHIGAN ENGINEERING LIBRARY k* Iv(l)I2 1 C2 0 1 ( -2 Cv2 v2 kv C k2v pi 01 o0 cl 01 Cl 2 Cd O wc C2 V2 (C - kv ) (Cw - kv - ) 2.P2 01 00 C1 C2 (w - kv ) ( - k - kC ) 01 02 C2 (2.102) The dispersion relation for as W2 (D - k2C2 _ c2 + -P w - kv 01 the two-carrier system can be readily obtained kv ) 01 - C Cl w2 (Cw - kv ) _22 _____ 02 + P2 02 C - kv - c 02 C2 = 0. (2.103) Use of this relation to replace w2 in Eq. 2.102 leads to the form P2 k lv(l)2 - kv - w 2 Re(P. ) r 1- 01 ci B e(-P = 2 2 w - 0okv 1 '0 1 (2.104) 'By a symmetry argument this result is unchanged if the replacement (1 -> 2) is made in the carrier designation on the right-hand side. Equation 2.104 indicates that the cyclotron normal modes (w - kv - wc 0) in the os CS two-carrier system carry power almost entirely in electrokinetic form. Equations 2.64 and 2.104 used in the definition of total power flow given in Eq. 2.68 show that the mode associated with carrier (1) is active if V |( 0 (X>2 k <, — ci po1 C1 2 2 1 w k v01 - w 1 1 (2.105) where C < 0 is assumed. C1

-50 -To verify that these power results are in agreement with the instability characteristics of the two-stream system, a computer. solution of the dispersion relation, Eq. 2.103, is undertaken in sample regions. In Fig. 2.3 the interaction of a passive fast-cyclotron mode on carrier (2) with the forward and backward helicon branches of the primary carrier (1) is shown. A strong convective instability occurs in this case which severely distorts the uncoupled mode dispersion. A similar case in which coupling occurs with the backward helicon branch alone is shown in Fig. 2.4. The interaction of the normal cyclotron modes is shown in Fig. 2.5. In this region the coupling between the modes becomes vanishingly small73 (p. 109) for any reasonable carrier densities so that no instability results. Equation 2.105 indicates that the negative power property is not necessarily limited to the region k v > cu. In particular, from the dispersion diagram r oi of the uncoupled left-hand polarized mode for a single carrier, Fig. 2.1, the electromagnetic branch near kr 0 may be examined utilizing the two-stream interaction. As the example of Fig. 2.6 indicates convective growth indeed occurs, although by the nature of the dispersion in this region (kr < C/c) the interaction is limited to large wavelengths which in turn invokes the necessity of including boundary conditions and self-magnetic field effects for any finite system. 2.7 Effects of Collisions on the Mode Kinetic Power Properties in a Longitudinal Magnetic Field Collisions can be taken into account via the replacement XC -<C - jv in the force equation, Eq. 2.45. When this is done Eq. 2.64 for the carrier kinetic power in the (-) mode formulation is given by

Ki 0 0.01 0.02 I I Kr pr 02 cl 01 V = -0 001 ( 0 1 krv k v FIG. 01 ' = l r..oi0 = co -k.01 02 01:' k c =- jAI K 101 r l)cl r oI FIG. 2.5 UNSTABLE CONVECTIVE INTERACTION ASSOCIATED WITH HELICON REGION.

-52 - Ki 0 0.01 0.02 0.03 0.04 0.05 Kr ' 10o |lcI J 'I - 0.001 ' vsol > 0, v - -0.167 v j C, < 0, 02 01 C0 %c2 -o.oe5 co C2 C,, ( ) 0.001, W. --,K - ID I 1' Il FIG. 2.4 CONVECTIVE INTERACTION WITH BACKWARD HELICON.

-53 - 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 K = p I =OC 11 = 3 = 2 * v01 = 0.1 V V0 2 2 (-) - 0.16 x lo4, kv K = 02 lc%2 W = lC,2 c2 < ~ co 1 > ~. C2 Cl FIG. 2.5 STABLE INTERACTION OF SLOW- AND FAST-CYCLOTRON COSTREAMING MODES.

KixlO-5 i! -0.3 -0.2 =, 0.5 l, c I a)t < ' 0 sc,C 1l -0.1 0 0.1 0.2 0.3 0.4 0.5 K,r X10-4 2 P2 0 ap2 \C/ 02 01 "p = o.01Q ( C) = ~10', V = -0.01 o > ~ kv = -0.192e I K 1, W = o INTERACTION WITH.THE ELECTROMAGNETIC BRANCH OF THE HELICON SPECTRUM. FIG. 2.6 TWO-STREAM TRANSVERSE

-55 - 1*J K ku ~ 'Y (w k rv) P ( ) R _ 2 2 2 (ca) k v )2 +) rv2 'v r o o (2 o06) For -the case where generation and recombination processes prevent the formation of any dc density gradients in the steady state, p E = 0 2' 2" so that in the presence of collisions Eqo 2o73 becomes 6v 6E v vv 2 - Re v - (2nl07) o ax 2" -\ ox which leads to (v[k (- kr- ) v k +ik-c k [(C kv )2 +- k ]} V 2l V 1 c o 2(2k.v + v)()[(w -k kv2]o '10 / N r oo 10 (2.lo8) Comparison of Eqso 2o40 and 2ol08 shows that -the general effect of collisions is to provide a power source for a convective instability if (a) k Sv ) < 0 by inducing a component of dc beam slowing 2..8 Kinetic Power Propert61ies of6 tbe Hybrid Mode A hybrid mode, which has characteristics of both the longitudinal space-charge wave and the c ircularly polarized electromagnetic wave2, is obtained when propagation occurs at right angles to ithe direction of an. applied static magnetic field0 Wit'h the applied magnetic field in the A A z direction and wave vector k in the x direction., the hybrid extraordinary wave has the field structure (E 2xE )o The remaiinng uncoupled ordinary 1x iy wave with polarization E has a dispersion relation independent of the magnetic f:ield and is thus equivalent to the purely transverse mode studied in Section 2o-o

-56 - For the hybrid mode, since no simple separation into uncoupled modes occurs as in the longitudinal magnetic field case, in general it will be found that explicit knowledge of the dispersion equation for the interacting system is required. Assume it is known then that the dispersion equation describing the system is given by D(C(k) E x) =( ) (2.109) E 0/ iy where D(c,k) is a two by two matrix. From Eq. 2.109 the following may be found: ~y _ 11 _ 2 A, E -D = - =D 2 P (w, k), (2.110) E D D ix 12 22 where the Dij's are the elements of D(,k). The cold-plasma hydrodynamic force equations, with drift velocity assumed in the direction of wave propagation only, are given for the sth carrier species from Eq. 2.45 by j(w - kv )v() csvy = 1SE1X (2.111) os ix cs lY and j(w - kv )v ) + cv(i) = _ s E (2.112) os ly cs mx 3 Use of Eq. 2.110 in Eqs. 2.111 and 2.112 provides the relations IE j(CD - kv ) + w (1 - Os )P(,vk) 1X C - I vx Os Cs. ) v() = (2.113) ix (2 - (w - kv )2

-57 - and (s) lY sExCs + a ( - kvo)2 P(k) c - (w - kv )2 CS os (2.114) The kinetic power flow of the sth carrier hybrid mode is given by p(s) 1Re k 2k.Re where J () = p () and from ly os ly x (s)* + E J(S)*) 2ix. + EaX ly ly Eqs. 2.11 and 2.12 (2.115) (s)* (s) * oos 1X IX -. tc - k v OS Equations 2.110 through 2.116 enable Eq. 2.115 to be written as (2.116) (s) k / Ipl2 X 2E I E 12 VO (C - kvo) ) + 2P. + c )2 ps ix r osi s C cs [2 - (c - kv )2 + k22 ]2 + 4k2V2 ( - k v )2 cs v r os i os i os r os (2.117) where P(w,k) = P + jPi has been used. Note that Eq. 2.117 reduces properly to the result found in Eq. 2.24 when w = 0. CS As the simplest example of the use of Eq. 2.117 consider a single carrier stream interacting weakly with an external circuit. In this case, for the present purposes, the dispersion equation can be approximated by that for the isolated single stream alone. It can readily be shown using Eqs. 2.111 and 2.112 together with Maxwell's equations that the elements of this dispersion relation are D - 11 c2c C2 2.p. c2 - ( - kv )2 (2.118)

-58 - j = W2 jczxo co. D = -D = - -(2.119) 12 21 c2[W - (O - kv )2] C 0 and 2 w2 (Cw - kv )2 D = =k2 - ~ p 0 (2.120) 22 c2 c2 [u2 - (w - kv )2] c 0 Using Eqs. 2.118 and 2.119 in Eq. 2.110 provides P(O,k) = - (2kv(o - k v) + j[(. - k v)2 (2 + c2) - k2v2] cp (2.121) This result when used in Eq. 2.117 then gives [(w k-v 2 2 (\2 + 2) -kV2] 2 -:' 1o 2E Ixi2 OO(U -krvo 1 + - r 2 C p 4k2v2 1+ - (C - kv )2 4 r o ~^P p k 2 [w2 - ( - k v )2 + k2v2] + 4k(v2( - k v )2 c r o 10 1 0 r o (2.122) Inspection of this result shows clearly that ( - k vo) < 0 is required to obtain the negative kinetic power property for this case.. That this result is in general true in the absence of collisions and thermal diffusion can be seen when Eq. 2.117 is written in the form (s) _ psE IE 12 V (o - k v )[((o + P. )2 + P2o2 ] p(S) PS ixO1X osis ros iCS rcs Pk... s sk / 2, (2.123) 2 ( - k v )2 + kV2 ] + 4kv2 (- k )2 (L cs r os i os I os r os (s) < 00 which shows directly that (wo - k v ) < 0 is required for Pk < 0. r os7

-59 - As in the case of the purely transverse modes studied, the electromagnetic power flow associated with the hybrid mode cannot be neglected. In the present model, this power flow can be approximated as ~1 J~k P 2 Re(E yB) 2 kE EX2 (2.124) Since the total power flow, Pe + Pk ) must be negative for the sth carrier hybrid mode to be active, Eqs. 2.123 and 2.124 then provide the limitation 2w2 Ev (kV - W)[(W + Pi )2 + P2W2r s k <PsOs - cs (2.125) r IP|2 [(2 - (C - krv )2 + k2v i + 4k2V2 (w - krv ) Cs r os i os r os for the mode to be active, where Vos > c/k is assumed. oS r 2.9 Effects of Collisions and Thermal Diffusion on the Kinetic Power Properties of the Hybrid Mode For completeness in the study of the basic carrier modes the effects of collisions and thermal diffusion on the hybrid mode kinetic power properties are studied, although in this case the results are particularly lengthy. Collisions and thermal diffusion are introduced into Eqs. 2.111 and 2.112 in the standard manner, whereby Eqs. 2.113 and 2.114 then become /C oS ) sEl ( - kv jv) + Cs (m - kv )P(w,jk) V(S_ x s -O S s v ix k2 ---'m - 0(3kv - jv) (2.126) - (w - kv - jv )- + Ts. 05. 5 cs Os05 s o- kv Os

-60 - and fl5E + j( - kv o)(w - kv - jvs) - k2v P( ) (s ) ly CD2 ( - ( - kv - )2 + CS OS S k22 (c - kv - jv ) w - kv os (2.127) from which Eq. 2.115 corresponding to the kinetic power flow becomes (S) k a2 E |E 12 kps ixD 2k I|D 12 i' s (p()q(s) 1 1 + (s)q() ) 2 2 (2.128) where 1 = [( - k v )2 + k.v2 ][CU2 r os i os CS - (cr - k v )2 + (v + k.v )2 r os s i os + (k - k vs] + V 2 [k.vs (k2 - ) ik v )] 'r I Ts s Ts I os r I r - r os (2.129) q(S) 1 cuvs(O - kv ) s r os CD [2kikr v + (v + 2k.v ) (a - k v )] i r Ts vs 1 os7 r os7 ( - kv )2 + k22 r os i Os (2.130) p() 2 = 2[(c r - k v )2 + k2 2 ][(c) - k v )(v + k. ) k v2 * r os i os r os s I os' r Ts + v v2 [k v (k2+ k2) + (k2 - k2)] s Ts ros r r i, (2.131) q(S) 2 cwv k.v S 1 OS = 2P.CD + co + 1 CS CD [(C - krv )2 r os (a) - kv )2 + k2v2 r os i os -kios ( + kvo ) - (k2 -k2 i)V2 i os vs i r ak, ] (2.132)

-61 - and D = + j s) S 3 3 (2.133) where (S) (C - krv )[s o - () - kvos )2 + (v + kiVO)2] + 2k.v (v + k.v ) 3 r os s r s s I Os I Os s I Os * (c - k v ) + v2 [(k2 - k2) ( - k v ) + 2k.k (v + k.v )] r os Ts r i r os (2.134) and q(s) k.v [(w - k v )2 - - (V + k.v )2] + 2(v + k.v ) (w - k v )2 3 r OS C S 1 O S 1 r os + v2 [2kk ( - k - kV ) - (v + k.vo )(k2 - ki)] Ts i r(' r os s i os r I. (2.135) As this result is difficult to analyze directly, the separate effects of diffusion and collisions are examined. First consider v = 0 and the resultant effects of thermal diffusion. s In this case, if Eqs. 2.128 through 2.132 are used, it can be shown that the kinetic power flow reduces to PkS (V O) = C IE 12 [() - k s)2 + kv2 ][( + P )2 + p2cs ] ps ix ros I Os iCs r cs ID5( - = 0) 12 * [( - k vs)V + k v2 ] r os os r Ts * (2.136) Inspection of this result shows that due to the presence of thermal diffusion (s) Pk (Vs =0) < 0 only if AYk

-62 - s > ( v ) ~ (2.157) r os so that at the very least vs > VTs is required. Thus regardless of the dispersion relation for the (isotropic) system [i.e., regardless of the polarization factor P(cu,k), number of carriers, nature of the interacting circuit, etc. ], if for all carrier species s it is known a priori that V < vT, then only collisionally assisted convective instabilities are possible in the isotropic hydrodynamic model. In a similar manner, let vTs = 0 in Eqs. 2.128 through 2.132 and obtain the kinetic power flow in the presence of collisions as (2 c E 12 P (s) = Ps 'x ' kv Q(w2 k (vTs0) (2k D (T =o) 2 (c - kv Q( krki), (2.158). il Ts'0)2 where Q (ck k. ) = [ (l - k v )2 + k.v2 ]((v + 2k.v )as + v [( - k )2 r r1 XrOs I Os IOs Cs s r os + (v + k.v o)2]}+U)V [w2 + ( - k v )2 + (V + k.v )2 + 2k.v (v + 2kvO)] s i os s Cs r os s i os i os s i os + [( - k v )2 + kvi2 ][2k.v w + 4P.c (v + k.v)] (2(159) r os i os A close inspection of Eqs. 2.138 and 2.139 assuming k. > 0 shows that again (c - k vs) < 0 is requireda to obtain ) < and that when this is the r Os K case the collisions assist the source power. These separate results would tend to indicate that in the general case, with both collisions and thermal diffusion present, the factor

-63 - (c - k v ) must be negative to obtain a negative kinetic power mode, r oS although even when this factor is negative it is still possible for the thermal diffusion to prevent this. As with the previous carrier modes studied, the hybrid mode kinetic power flow can be related in part to the steady-state second-order dc carrier dynamics. As discussed previously, the only case which can be accurately studied with the plane-wave analysis used corresponds to the dc field E = O or, equivalently, the medium is homogeneous in the steady 2 state and p(s) = 0. In this case it can easily be shown that the second2 order longitudinal time-averaged force equation in the steady state is given by k. TI (2k.v + ) + ) Ivxl = 2 Re(v B ). (2.140) 1i os s s x 2 - 2 lR(y 1z The right-hand side of this equation may be written as * Re(vB ) = Re E (2.141) 2yiz 2 Re~vly ~~( 2c2pos, ly so that when kr Re(EyJ y) < 0 a negative contribution to v is obtained r my zy as corresponding to beam slowing. 2.10 Utility of the Kinetic-Electromagnetic Power Theorem Regardless of the carrier mode used, the present study has shown that the salient function of interest from the viewpoint of convective instabilities is the kinetic power function given by Pk Re(E * J) (2.142) k 2k.

-64 - The power relations, such as Eq. 2.22, show that an instability cannot occur unless Pk < 0 if k. > 0. In a sense then, the power theorem provides a form of causality on the system indicating whether or not the root with k. > 0 indeed corresponds to the amplification of RF electromagnetic power. Since the functional dependence of Pk (ckr, ki) can in general be ascertained by analytical methods, the primary utility of the power theorem is in determining, before any solutions of the dispersion equation are attempted, which regions of (w,k) space and which carrier parameters are causal in nature. Note that this immediately delimits the investigation of the first-order dispersion relation to those carrier modes and/or those regions of (a,k) space which can support a convective instability. The present study also raises important questions related to the form of the assumed steady state of the system in convective instability analyses. If it is known at the outset that by its nature the system can support a steady-state dc density gradient of charge, such as is common in solid-state media (e.g., the Hall effect), then solving for the instability in such systems assuming a homogeneous medium in the steady state (i.e., using a plane-wave analysis) is in general inaccurate. Although the self-consistent dc density gradient is only second order in magnitude the power theorem indicates that it can still play an important role in the power exchange processes and hence that it cannot be neglected. This statement is particularly important for multiple carrier species interactions as evidenced by the discussion following and related to Eq. 2.99. In the opposite case, in which it is known that the steady state of the system cannot support equilibrium dc carrier density imbalances through

-65 - generation and recombination processes, the planee-wave dispersion equation is accurate and regardless of the carrier mode studied the kinetic power flows are in agreement with the second-order dc carrier dynamicso These considerations lend importance to the surface boundaries of the system (eogo, in solid-state devices the contact area) since these play a part in determining the steady state of the systemo These considerations lead to the speculation that for those systems in which the dc gradient of charge is pernmitted in the steady state, the possibility of nonlinearly generated instabilities exists wherein part of the power represented in the function E J is fed to the RF fields through 20 the self-consistent charge density inhomogeneityo In such cases it may also be that the power represented in the plane-wave analysis of the system by Re(E J ) need not be negative0 Verification of these hypotheses requires a full nonlinear analysis which is not attempted in this t'ract0 A further point raised by the power theorem deals with the distinction between a negative kinetic power mode and a negative power modeo For example, neglecting collisions and thermal diffusion2 the left-hand circularly polarized mode (cO < 0) was found in Eqo 2064 to be a negative kinetic power mode throughout (oDk) space. although the total mode power flow for this case, given by Eqo 20692 is negative only in restricted regions of (wck) spaceo The problem in obtaining the total power flow of a carrier mode is that a localization of the EF electromagnetic power flow is requiredo From the viewpoint of coupled-mode theory this may be permissible for weakly coupled modes, but in the actual interacting system this cannot be accomplished with certaintyo It would then appear that for weakly interacting modes a negative power mode is required for

-66 - instability, whereas for strong interactions a negative kinetic power mode may be sufficient. A negative kinetic power mode, although both necessary and sufficient to satisfy conservation of power requirements, may not be sufficient to guarantee that instability will occur when this mode interacts with a passive circuit. The techniques used in the kinetic power theorem can also be applied to gain insight into decaying mode interactions (ki < 0). This aspect is presented in Appendix D. 2.11 Summary The kinetic power theorem has now been formulated for the basic carrier modes possible. Any new mode (e.g., static magnetic field applied at an arbitrary angle) should only be a superposition of these basic modes and hence provide no new physical phenomena.

CHAPTER III. KINETIC ENERGY PROPERTIES OF CARRIER WAVES: ABSOLUTE INSTABILITY 3.1 Introduction It is generally held that the consideration of energy or power can only provide information on convective instabilities or that a mode which has negative kinetic power can be active for an absolute instability if the proper interacting circuit mode is chosen. It will be shown, however, that a separate but related conservation principle for absolute instabilities can be obtained. A kinetic-electromagnetic energy theorem will be derived for the three basic carrier modes studied in Chapter II. The theorem also demonstrates the physical mechanisms through which absolute instabilities can arise. Whereas the convective instability theory of the previous chapter used a time-averaged spatial-power framework, the present study employs a space-averaged temporal-energy basis. For example, in the convective instability process the source arises from the active carrier mode having negative power so that exponentially growing waves in space are possible. In the absolute instability process, however, it will be determined that if the carrier mode has a negative energy property, exponentially growing waves evolving in time are possible. In the latter case, in general, at each point in the interaction region, the total carrier kinetic energy increases exponentially as time progresses, but in the negative direction, For the present analysis the condition for steady-state oscillations will not be dealt with since this state is ultimately determined by saturation processes which cannot in general be taken into account. This in no way -67 -

-68 - diminishes the value of the theory for predicting which carrier modes and which regions of cu-k space are potentially absolutely unstable. Because of their importance in the case of solid-state plasmas, the effects of collisions and thermal diffusion are included. It is shown through a study of the electrokinetic energy density functions that the dynamics of the k -k. plots according to Briggs' criteria r 1 can be elaborated upon and the meaning of the root behavior explained as a function of time for both absolute and convective instabilities. The theorem is also used to show that the characteristics of the electromagnetic branch of the helicon spectrum are such that unstable wave behavior is permissible without violation of the conservation laws. This region of c-k space is usually rejected as a possible source of instabilities. Whenever possible the quasi-linear theory is included to help in understanding the carrier dynamics. It is found that the second-order effects predicted by the quasi-linear theory are intimately related to the boundary conditions of the system. In particular it is shown in which cases the linear dispersion relation is accurate and also a discussion is given of how the boundary effects can enhance or quench the instability. 3.2 Kinetic Energy Characteristics of Space-Charge Waves The development in many ways parallels that used in the convective instability study. Thus assume that fundamental variables vary as exp[j(cut - kx)],where now c = cr + jcu. and k = k + jki. The Poynting theorem is written in the form

-69 - aB aE* * (E x B*) +B* 7o- + E - + E * = o, (3.1) and the real part is taken, viz., Re [V (E x B)]+ (We + W) = (3.2) where W A 1IB12 + I E1 (533) and Wk - (3.4) W 2 ^ Re (E * J*); (3i 0 ) (3.4) k 2.The function W is the sum of the electromagnetic and electrostatic energy densities and Wk is the electrokinetic energy density. It will first be verified that the slow space-charge wave is the source mode for the backward-wave oscillator type of interaction in which a single carrier stream interacts with an external circuit wave mode. Consider the simplest case in which collisions and carrier diffusion are neglected and it is assumed a priori that k is purely real. It is shown in Appendix E that the electrokinetic energy density of the longitudinal space-charge wave to second order can be expressed in terms of the fundamental fields alone in this case as -2. t kPoo(oN -kv)e 1 v (0) 12 Wk - 00 r 02 jx (3.5) 2~[(w - kv)2 +. ]

-70 - where Iv (0)| is the longitudinal RF velocity amplitude of the carrier Ix at time t = O. Since in the present case k is purely real, if wi + 0, Eq. 3.2 gives We + = O. (3.6) Inspection of Eq. 3.3 indicates that W is positive and hence for conservation of energy to be satisfied it is necessary that Wk < O, or equivalently from Eq. 3.5 that v0 > r/k. Thus the slow space-charge wave must be used for the oscillation to grow positively in time. Note that part of the function W in Eq. 3.3 is the electrostatic energy density associated with the carrier mode. Hence in general it is not sufficient that Wk < 0 for growth to occur, but rather that the function WT defined by W w + w (3.7) WT Wes +Wk be negative, where Ws represents the electrostatic energy density of the es carriers. In order to evaluate WT, the longitudinal force equation, Eq. 2.10, gives for the present case (C( - kV )2 +.. |E (O)12 IV9= ~ lvx(o)2 I (3.8) where E (0) is the longitudinal RF electric field at time t = 0, so IX that, if Eqs. 5.5 and 5.5 are used,

-71 - ( e[( - kv )2 + c2] k v (r - kvo) iv (o2 -2wc t WT =........... - +......... ~ e.T _ vn (' ( kvo)2 + i r r ~~0 2. ~(3.9) The functional dependence of WT on cc. shows directly that xo. must be limited in magnitude and moreover indicates the important limiting parameters involved. The general form of the electrokinetic energy density of the spacecharge waves employing the definition of Eq. 3.4 can be obtained directly from Eqs. 2.6 and 2.13 as W( r-k V)(w +rk v )+(2.-k.v )(w -k.v)v +v)+(k_ -k)v 12 (r )r o i o 2kiV V)(r i o) i HE I2 k 2 I (wc-kv ) (Wj-kv -j v) -k2vT2 E 1y l2+E I2 )F( Vo kv)2+(W-Bkiv) (Li-kv-v) ( rr -k v )V ] +. I(-kv -jvI2 (-2-2i) (3.10) It will now be shown that it is possible to relate the energy characteristics of the system to the causality principle based on Briggs' criteria. For clarity assume the waves of interest are purely electrostatic and neglect the thermal diffusion (vT = 0) so that Eq. 3.2 becomes when Eq. 3.10 is used

-72 - [(r -k v )(w +k v )+(.-k.v )(.-k - v) + -k (2) v +v) ([c -k v -C. E 2 1 v r r o/ J w -k[( v-^ )2_-(wi-kivo)(.i-ki v _-v) ] +(% -kr 1)2 [22(i-k o)-v]2 (35.11) For an actual system a sum over the carrier species present can be made without altering the present relationships. If Briggs' method is applied to Eq. 3.11 by permitting c. - -oo inspection shows that for the electrokinetic energy density to remain finite and satisfy Eq. 3.11 it is necessary that ki -o O -/v-> -oo. Hence for either convective or absolute instabilities it is seen that Briggs' requirement on the amplifying mode, to satisfy causality, namely k. - -0 as o. -e -c, is directly related to conservation of energy in the system. Note that in a very real sense the state of the system defined by the operation of letting Ci. - -o is an actual potential physical state of the system according to the linear analysis since it is a solution of the dispersion equation. Thus if the wave blows up very quickly in time (C. -o -) it must be that it simultaneously decays spatially very quickly (ki - oo) or otherwise there is too much energy in the system (conservation of energy is not satisfied). This nonequilibrium behavior can be related to the dynamics of the Briggs' plot for an absolute instability as shown in Fig. 3.la. At some fixed time the system can have the solutions indicated b ~, wherein Mode 1 is the forward amplifying wave and Mode 2 is the backward passive wave. Mode 1 is then essentially in the state

-73 - 2 Wit kr x Or (a) BRIGGS MAPPING FOR ABSOLUTE INSTABILITY k i I i IA -r wr r /vo (b) BRIGGS MAPPING FOR CONVECTIVE INSTABILITY SHOWING POSSIBLE ROOT TRAJECTORIES FIG. 3.1 TRAJECTORY STUDY OF BRIGGS' MAPPINGS.

discussed above. As time proceeds reflections (and possibly considerations of entropy) drive the system to the resonant double root at point S from which the eventual steady state is reached. It is also now possible to relate the dynamics of the Briggs' plot for convective instabilities as shown in Fig. 3.lb to the dictum of conservation of energy. In particular, consider the possible forms the convectively unstable root may take in crossing the k axis (k. = 0) as r I Ci is varied, typified in the figure by potential trajectories 1 through 4. Even for convective instabilities the system must satisfy the conservation of energy equation, Eq. 3.2, which for k. = 0 indicates that Wk is 1K necessarily negative and nonzero. Inspection of Wk as given in Eq. 3.10 shows that for k = 0 it Bnust be that (o - k vo) < 0 in order that Wk < 0. Thus with reference to Fig. 3.lb trajectories such as 1 or 2 cannot occur, whereas trajectories 3 or 4 are permissible. Although this has been shown only for space-charge waves, it will be determined that this is a general result for the basic carrier modes in the hydrodynamic analysis. Similar to the result found for the kinetic power function, Eq. 3.10 shows that the electrokinetic energy density can be negative through the action of the collisions for purely transverse fields alone. To understand this, consider the case k. = 0 and take the dc part of Eq. 2.32 to obtain +v2 = + Re( X ) + =v +' Re(V x B )A (3.12) 2 2 1 1 X~~ Then from the assumed time dependence, it is found that

-75 - 2E21 IE 2krV((Cor - krV )(E 2 + I E 2) v 2 + r-ro y9z, (3.13) 2 V - 2w. 2 1 2(v 2O-)(Ci)2 + c2) |o - kv jVI r I O so that when (or - k v ) < 0 the collisions act through the second-order Lorentz force to extract energy (at all points in the interaction region) from the carriers by slowing. Note that this result indicates that collective Cerenkov radiation may be strongly influenced by the presence of collisions. In addition, as inspection of Eq. 3.10 indicates, under small-signal conditions, with xi << wr and k. < kr, it is the presence of collisions which permits the electrokinetic energy density to be negative in' the case v < vT. 0 3.3 Electrokinetic Energy Density of Purely Transverse Waves in a Static Magnetic Field Assume that the fundamental variables vary as exp[j(wt - kx)] where A A both c and k are in general complex and B = B x with v = v x as before. - - - O0 The coordinate definitions of Eqs. 2.6 are used and again, since the transverse modes, are uncoupled, only the (-) mode need be analyzed. Thus Eq. 2.7 is written in the form av Tv aE "1t ' j( - kv )v = rE + — (.14) c o i- j- (x.14) To examine the second-order longitudinal carrier dynamics in detail, consider the Lorentz force term F = qRe(v)xRe( ) = * F~ = q Re( x B + v x B) (3.15) - -1 1 — l —~ 1

which shows that two types of motion exist for the second-order variables, so that v for example may be written as 2 -2w.t 2k.x -20.t 2k.x Re(v ) = V e e + V e e cos 2( t - kx), (3.16) 2 21 22 r where V and V are independent of x and t. The sinusoidally varying 21 22 part of this dependence is related to the importance of device length for the time growth of the oscillation. In the limit k. - 0, a space integration over the device length shows that this contribution is zero if the device length, L, is an integral number of wavelengths. Also, if k. -,0, this contribution should be negligible if L >>. In general k. W 0 and it is necessary to take into account the total interaction so that at oscillation the interference between the waves present cancels the sinusoidal contribution and the expression for the ideal device length becomes dependent upon ki, the wave parameters, etc. In order to determine the kinetic energy properties of a single mode then this sinusoidal contribution is neglected. This problem does not arise for convective instabilities since x. = 0 and Eq. 3.5 is time averaged. In addition, using Eq. 2.6, if it can be rigorously assumed that v = 0, the contribution of this sinusoidal 1+ part to the second-order motion of the (-) mode is identically zero. From Eq. 2.62 the small-signal Poynting theorem can be written in the general form for the (-) mode; 6B 6E 1 7 E xB) 1 * - + c- 1 (s)* v- * (E x B 1 -) 24 o + E - t +2 - 1E s (3.17)

-77 - where the sum is over the carrier species present. Make the definitions, where the symbols have the usual meanings, as follows: Wk = - Re(E_ J) ) (3.18) 1 -- B 12 + E 1E 2 (3.19) el- 40 1 4 1 -and k c - k.w. P - r r IE 12 I (3.20) eQ- 2^1 (X2 + 2) 1 -o r 1 so that, corresponding to the real part of Eq. 3.17, the following can be written: 2k.P + - + - =. (5.21) i el- dt + t O = s The function Wk is derived explicitly in Appendix F. Now from Briggs' criteria it is clear that for an absolute instability at least one root is required with the property that c0. < 0 for k = 0. For k = 0 and 1 i 1 cx. 0 0 Eq. 3.21 takes the form W.+ X Wj ) =O. (3.22) S Inspection of Eq. 3.19 shows that W is necessarily positive (the case JE I = 0 is of no interest) so that at least one carrier species must possess Wk < 0O i.e., a negative electrokinetic energy density. Inspection

of Eq. F.4 for this case thus indicates that for Wk < 0 then c < 0 corresponding to the slow-cyclotron mode, helicon mode, etc., and that i I k v > (w - kv)2 + (3523) c r o r r 0 1 Equation 3.23 is easily satisfied by the helicon branch since (or - k v ) 0 and also by the slow-cyclotron branch since kr > Ic |I/v for positive frequencies. In addition, however, this equation can still be satisfied in certain regions of the electromagnetic branch referred to in Fig. 2.1. Hence this branch which is usually neglected in instability studies can satisfy both the criteria of conservation of power and conservation of energy and hence be a source mode for convective or absolute instabilities. It must be immediately pointed out that the presence of collisions or consideration of finite size effects (since k is small in this region) can remove this branch from the negative kinetic power or energy property. As an aid in understanding the absolute instability process the quasi-linear theory is now studied. The continuity equation for each carrier species is )p V ~ J + -- (3.24) -Z- dut o From the assumed time dependence together with physical considerations it must be that p =. Equating current density contributions from outside 2 -and within the interaction region provides P v v +OS P O S S 2 -+ (3.25) 5jOOS Os OS OS OS 25 s s

-79 - so that the displacement current balances the total current density in the interaction region. In particular, for the case of a single species of carrier interacting with an external circuit, Eq. 3525 applies if the interaction is weak aE v = - -3 -. (3.26) 2- p Ot Thus in the case of absolute instabilities the slowing down of the dc beam in time is compensated by the buildup of displacement current whose magnitude increases with time until the eventual steady state is attained. It is in this manner that continuity is achieved. In an actual device this necessitates charge buildup at the extremities x = 0 and x = L where p (t) 7 O. The device analog is a leaky capacitor which "heals" itself 2 -as time progresses. The functions v and E are spatially uniform in the 2" 2 -interaction region. The pictures obtained of the absolute instability process are thus quite different for the space-charge waves vs. the present transverse modes. The space-charge wave system evolves as shown in Fig. 3.2a, while for comparison the nature of the transverse mode is depicted in Fig. 3.2b for the simplest case, wherein k. = O. It is noted that in the most general case, w.i 0 and k. + 0, all the processes occurring in Fig. 2.3a and b and Fig. 3.2a and b may occur. A similar problem to that connected with the suitability of the linear dispersion in the convective instability process occurs here also. The field E is the self-consistent reaction of the system to the dc beam 2" slowing which must be present to conserve particles properly. This field

INTERACTION REGION " P I ^ I VI I ' Io8' I. - I. I1~~~~~~~u I~I t I (a) DENSITY AND VELOCITY CHARACTERISTIC OF SLOW SPACE-CHARGE WAVE N OF INTE RACTION REGIONS UR i| --- — Bo Vo, Po I vo+ V2- 'Po FIG. 3.2 COMPARISON OF LONGITUDINAL AND TRANSVERSE MODES UNDER ABSOLUTE INS TABILITY.

can quickly extinguish the instability so that in fact it never occurs or in some cases it can assist the instability. The quasi-linear theory can be used to show how this can occur. The second-order force equation for the active carrier can be obtained from Eq. 2.25 with collisions neglected as 6v av /E 1 -t + V a = rE + r Re v( ), (3.27) at o 0 x 2- x / where from Eq. 3.25 the field E satisfies the following: 2 -L P ( )+ E = ~. (3.28) s Clearly for the instability to proceed it must be that v of the active 2 -carrier must be negative since this is the only form of energy loss available for purely transverse waves. Equation 3.28 indicates, however, that the field E is in part determined by the properties of the passive 2 -carrier. Indeed by solving Eqs. 3.27 and 3.28 simultaneously it can be found that for the active carrier, for a two-carrier system, V1-Os O (s)2 wv Re(E (s)*) iv = - 14 2 wJ Re(E I-+ ' 2 Re(E i + ) 0 10 l~l+POS 1.OS ) v = - 4v — - Re(E l- 1 -16. + pi P2 1 (3.29) where all unscripted variables refer to the active carrier, and the last term on the right-hand side is due to the field E. The linear dispersion relation would indicate that the sign of v is completely determined by relation would indicate that the sign of v is completely determined by 2 -

-82 - the Re(E J ); however, the quasi-linear theory of Eq. 3.29 indicates that 1- 1 this is not in general true. The secondary or passive carriers can play a crucial role in determining the sign of v. The physical reason for this is that as the active carriers are slowed an imbalance occurs in the continuity of the system which must be balanced by the secondary carriers. Unless a series of generation and recombination processes occur near the ends of the device the field of the displacement current E builds up and 2 -affects the further evolvement of the active carrier dc motion. Thus the quasi-linear theory offers a method of taking into account the boundary effects of the system —something that is completely ignored in setting up the linear dispersion relationship. In particular, for solid-state devices, one set of contacts for a particular device may be such that its injection characteristics never permit any charge discontinuities such as charge buildup near the contacts and the absolute instability can proceed. For a second set of contacts with opposite injection characteristics it may be that the charge buildup at the extremities near the contacts cannot be halted and the instability is quickly quenched by the effects of the field E on the system. It is only in the former case that the results predicted by the linear dispersion equation will be valid. It would at first appear that an active mode could be obtained even if Wk < 0 by appropriate variation of the system parameters to alter E and hence obtain v < O. The difficulty with 2- 2 -this approach is that in absolute instability studies the dynamic phenomena are nonequilibrium in nature and not a steady state of the system. Thus questions of which direction the system proceeds in time leads to the conclusion that the field E (which is in a sense a measure of the effects 2'

-83 - of the system boundaries) can extinguish or enhance an instability already initiated, but cannot by itself play a role in the initiation itself. 34.Electrokinetic Energy Density of the Hybrid Mode The field directions and conventions adopted in Section 2.8 for the hybrid mode are retained. The general form for the electrokinetic energy density of the hybrid mode is Wk = - Re(E J* + E J ) (330) k 2w. i(x x jy ly(. with both C and k complex. Equations 2.113, 2.114, and 2.116 can be substituted into Eq. 3.30 for the present case to obtain the following neglecting collisions and diffusion: C2Ex IE 2 42(w - k v)(w - kv)( 2P + + W c2 p ix L r 0 ror o c r c 1w12 E(i [(r k Vo) ) (wi - kv) )2] ( p,2 + 1 Wk =2 2 ~i IW2 - (C - kv )212 (5.531) where P(0w,k) is the polarization factor defined in Eq. 2.110. Inspection shows that even for the case k. - 0 the electrokinetic energy density is strongly influenced in magnitude and sign by the polarization factor P(c,k). This is unlike the corresponding kinetic power flow of the hybrid mode given by Eq. 2.117 which indicated that (w - k vo) < 0 is always required to obtain Pk <. Thus, in general, it is necessary to study the total system, thereby obtaining explicit information on the function

-84 - P(w,k) before any specific statements can be made regarding the electrokinetic energy density. An exception to this occurs if W > |I c so that IE | ~ p c ix IE I and I P 0 in which case inspection of Eq. 3.31 shows that the function Wk approaches in value that given for the slow space-charge wave. 3.5 Summary and Discussion The electrokinetic energy density functions have been obtained for the basic carrier modes present in a static magnetic field. Since these basic carrier modes encompass all physical phenomena for isotropic media any general mode such as that obtained by applying the magnetic field at an arbitrary angle to the direction of wave propagation should simply be a superposition of the effects studied. The potentially most valuable use of the present energy studies is that it offers a method for taking into account the fact that the finite length of the system introduces end boundaries to the interaction region which can play a significant role in determining the stability of the system in the steady state. Thus if the (linear) dispersion relation indicates that an absolute instability should develop in the system, application of the quasi-linear theory to second order, together with information about the end boundaries and the carrier generation-recombination characteristics, will determine whether or not these end effects enhance or quench the further evolvement of the instability. In the latter case no instability will exist in the steady state. The energy theorems also provide useful correlations with the causality conditions of Briggs' criteria. Thus the behavior of the roots in k -k. space as w. is varied is related to the fact that the r 1

-85 - electrokinetic energy density must remain finite for arbitrary perturbations. As in Appendix D, where it was hypothesized that even decaying convective roots must satisfy a causality condition similar to that for growing waves, this can also be extended to include causality criteria for waves which are predicted to decay in time. Note that these considerations raise the question of the definition of an instability. It is for practical reasons that an instability is defined to occur when the RF fields grow in time or space. It is possible however to define an instability more generally as any interaction in which a continuous flow of power and/or energy occurs between waves in the system. It is for this reason that causality criteria should be invoked even for decaying RF fields. A central problem in'the analysis of systems is that it is never clear in which state the system resides as the time t becomes larger. This ambiguity is related to the fact that one can arbitrarily assume W complex, k complex, or both C and k complex, and obtain a variety of solutions to the dispersion relation which all satisfy causality criteria. Thus, depending on the initial perturbation, these are all potential solutions at some time in the state of the evolvement of the system. These considerations immediately invoke the question of which direction the system proceeds after the time of the initial perturbation and for what reasons it does so. Although no attempt is made herein, it is suggested that consideration of the entropy of the system will provide knowledge of the system evolvement as a function of time for arbitrary perturbations.

CHAPTER IV. THE EVOLUTION OF PLASMA INSTABILITIES BASED ON QUASI-LINEAR THEORY 4.1 Introduction In general the assessment of instability characteristics for a particular system involves the development of a dispersion relation based upon the linear wave equation. Solutions of this dispersion relation determine whether or not a small perturbation at some frequency of interest will at the perturbation onset grow spatially away from this point (convective instability) and/or build up in time from the perturbation onset (absolute instability). The dispersion relation based on the linear theory cannot be used, however, to determine how the instability evolves in space and time away from the onset (e.g., x = O, t = 0) point. The fact that the dispersion equation possesses a causal unstable root is then a necessary but not sufficient condition for the instability to persist as either the time or the spatial direction becomes finite. This is because the reaction of the carrier motion (e.g., carrier slowing) to the growing RF fields is not present in the linear equations. The quasi-linear theory is first applied to aid in developing an understanding of how convective instabilities evolve from their points of initiation. This theory is a direct extension of the linear equations in that they are retained and used as a basis for formulating the equations of carrier motion to second order. This is permissible only if the small-signal approximation still holds and, more importantly, if the wave number predicted by the linear equations can be assumed to be a constant of the system -86 -

for the root and frequency of interest. More generally it would be necessary to include the possibility that the wave number can be spatially dependent [e.g., k - k(x)]. This latter approach is analytically nontractable, however, and so the quasi-linear theory is used to obtain at least some indication of the system behavior due to the reactive forces of the growing RF fields on the carrier motion. Even in the quasi-linear theory only the very simplest of convectively unstable systems is readily analyzed and effects such as the thermal diffusion, collisions, and the spatial dependence of the wave number are only qualitatively discussed. Some useful results relating the second-order effects to the potential energy of the charge carrier system arQ obtained and discussed for both gaseous and solid-state plasmas. These methods are then applied to absolutely unstable systems. In this case the terminal boundaries of the system in the k direction can play an essential role in the development of the instability as is discussed. 4.2 Application of Quasi-Linear Theory to Convectively Unstable Systems Various examples of convectively unstable systems will be examined to determine the reaction of the growing RF fields on the particle motions in the quasi-linear theory. Except when assessing causality according to Briggs' criteria the frequency is assumed purely real. In all cases the A wave vector k is assumed to be parallel to x. 4.2.1 Electron Stream-Plasma Electrostatic Interaction. The dispersion relaion for the purely longitudinal interactionry ll i of a cold electron beam with a cold stationary plasma is well known as

-88 - 1- pb 0 (4.1) Cu2 (co - kvO)2 ob where the subscript p denotes plasma variables and b denotes the beam variables. This equation is readily solved as M2 -1/2 kvb = c Ppb 1(4.2) so that k. > 0 can exist if cu = pp and moreover this root is causal since k. — o as.i -. -oo. In particular it is noted that if co - cp, then k. -> and the growth rate is infinite. To proceed with the quasi-linear analysis it will be of interest to first study the case where Cu corresponds to a single charge carrier pp species alone denoted by subscript q so that pp = pq. The longitudinal force equation for a cold, collisionless carrier to second order is av av av av av 2 ~ ~~ ~~o ooxx 1 2 + 2 + v 0 + vi + v 0 = ( 1+ E) (4 3) The time-averaged real part of this gives 1 Sv (/v - Re v - + v Re Re (E), (4.4) O2 7x/ o \ax 2 2k.x where v, E ~ e as discussed in Chapter II since the second harmonic 2 2 contributions give a zero time average. If Eq. 4.4 is applied to the q plasma carriers, the following is obtained since v is assumed to ~~~~~~~~~~be zero: be zero:

-89-.1 ( *v * 1 Re v ( ) = j Re (E ). 2 K iq 5x / q 2 (4.5) From Eq. 4.3, by extracting the first-order equation, which is consistent with the dispersion equation of Eq. 4.1, the following is true: TI Viq =L (4.6) therefore, from Eqs. 4.5 and 4.6 it can be found that Re (E ) = 2 I k. 2 1IE2 1 2X2 1 (4.7) In a similar fashion Eq. 4.4 can be applied to the beam to obtain j(wo - kv b)vlb b l (4.8) so that Eq. 4.7 can be written in the form E k Re (E ) [ - ) \ + k2v2 ] v 12 R (E)2 r ob i ob lb (4.9) Use of Eq. 4.9 in Eq. 4.4 leads to V 2 /1 \r k v -2 k2v2i - lb q r o bi { T q -r b i o l 2b 4=. 4 o b b L w +2 I ob~~~ ~ Vob.) (4.10) Some remarks regarding the nature of the evolvement of the instability in space and time are now needed. The dispersion equation, Eq. 4.1, and

-90 - its solution, Eq. 4.2, indicates that a perturbation at any frequency W = C at any point in the system will initially be unstable at that pq point. Since the present system is purely electrostatic the Poynting vector is zero so that no power is being fed into the RF field E However, consideration of the RF electrostatic field energy density, W (x) = (c/2)jE (x)|2, shows clearly that by the definition of convective instability this quantity must be increasing with distance so that, corresponding to k > 0 and ki > O, W (x) > W (x)) if x > x Assume r i e~ a e~i 2 1 then that the instability is triggered by a continuous source at x = 0, t = O at a frequency Wot < pq and examine the system at a later (elapsed) time t = 5. Assume further that 6 is much shorter than the characteristic times of,and the operating frequency ot is much greater than the characteristic frequencies ofany possible recombination or generation mechanisms present. Since questions of recombination and generation are then inconsequential, continuity of the time-averaged charge then necessitates that ReV J + = 0. (4.11) Since X is real, Re(ap /at) = 0, and hence Eq. 4.11 leads to 2 v + 1 Re(p v) + p = 0; k. 0 (4.12) 0 2 2 11 2~ 1 For any carrier the time-averaged carrier-mode kinetic energy is defined by <K.E.> = 2 <Re(p v v)>, (4.13) 2TI.

-91 - where < > denotes a time average, e.g., <Re(p v )> = (1/2)Re(p v ). When 'I I 1 1 I J expanded to second order Eq. 4.13 gives <K.E.> = v v p + <Re P (v v )> + 2v <Re (p v )> + p v2 + p v2 2~k o 2 o 2 o i o (4.14) If Eq. 4.12 is used to replace p in Eq. 4.14, the following results: 2 <K. E.> = - v2 2d o + 2 v v + + P Re(p v ) 20 o 2 o 4i 4 1 1 (4.15) Apply this now to the electron beam-q plasma interaction. Continuity of the RF beam charge gives 7 ~J +~Pb V - lb + at = (4.16) from which it can be determined that Re (Plblb) Pob lb2 [kr (w - krv ) - k o = ob -b r rob - iob [(w - k v )2 + k2v2 I rob 1 ob (4.17) The solution with k. > 0 from Eq. 4.2 when it is inserted into Eq. 4.17 gives 1 Re (P bvb) lb lb Pob vlbI2 vob (4.18) Finally, Eq. 4.18 tsed in Eq. 4.15 provides the following for the beam: <K. E> Pob v2 + v v (4.19) beam 2 \b ob oba

-92 - Recall now that the RF electrostatic field energy W (x) is increasing in the positive x-direction away from x = 0. A simple consideration of conservation of energy shows that it must then be that the kinetic energy of the beam is decreasing in this direction, or from Eq. 4.19 since ob > 0 if Xt > 0, v < 0. ob t 2b Examine now the function v given in Eq. 4.10. Equation 4.2 shows 2b that Eq. 4.10 can be written in the form Iv 1|2 r k2v2 v = [ l (b L%.1] (4.20) 2b b Until now no assumptions about the q-plasma have been made except that it corresponds to a single species of carrier charge. Assume now that the q-plasma is an electron-ion (gaseous) plasma in which the ions have infinite mass. The ion plasma frequency, pi, is then zero and the plasma is effectively one component with pp = q = pe. Moreover ( q/b) = 1 and hence v < 0 in Eq. 4.20 only if k. < w/V = k. From Eq. 4.2, since ab ' i ob r t pp' (o2 -1/2 k.v W= aC-o~-i, (4.21) i ob pb 221 1/2 and hence v < 0 only if ct > (Pb + 2q) 2b p q The physical reasoning behind this result can be seen if the reaction of the beam electrons to the growing RF field is examined after the time of initiation. For all practical purposes no dc bunching has yet occurred at or very near to t = 0 so that E (x) = p (x)= 0 and from 2 2

-9 - Eq. 4.4, V4 b = -(1/4vOb) IVb 2. This result is in essence independent of the magnitude of k. and is really as far in time as the linear dispersion relation, Eq. 4.1, can go with full accuracy. (In general, the linear dispersion relation is still of vast importance since its solution must necessarily be unstable for the instability to be initiated in the first place.) At a slightly later time, however, the reactive effects of the growing RF field E on the beam charges lead to the spatial effect that in the region of interaction (i.e., in the region of the presence of E ) the field E is generated. In particular, the quasi-linear theory shows 2 that for the infinite-mass ion case if O (=2 + )2 the field E t (p b pq 2 is sufficiently strong to quench the further evolvement of the instability. Clearly the linear theory is in some ways extended too far by the quasi-linear theory. If the field E is large, p is then large, and 2 2b hence P ob = Pob + Pb = Pob(X), and the time-averaged carrier density is dependent on x. Self-consistently then the dispersion equation cannot be formulated assuming a dependence as exp[j(wt - kx)] for the RF variables but rather as exp(j[at - f x k(x')dx']), so that - -jk(x). (4.22) In the quasi-linear approach k was held fixed so that the linear equations could be extended. This was useful in determining the direction in which the system moves from the point of initiation by studying the reaction of the charges to the growing RF fields. It is this charge reaction which itself causes the time-averaged charge density to become a function of x

-94 - necessitating k ->k(x). The quasi-linear theory then suggests the following method for incorporating nonlinear effects directly into the dispersion relation. The general carrier drift velocity is written as 2f x ki(x') dx' (x) = VO(O) + v (0) e 0 o ~~~2 (4.23) where now k = ki(x). The force equation becomes av av av (x) " + v (x) - - + v ~1 - - nE oa O a x 1 ax 1 (4.24) so that when the dependence of Eq. 4.22 is used, wherein the ej't dependence is retained since cw is assumed real, it is found that 2f X k.(x') dx' jj[w - kv (x)]v + 2k.v (0) v e 0 1 1 2 1 = IE 1 (4.25) On the basis of Eq. 4.23 the following may also be written: 2/ po(x) = po(O) + P2(0) e Equation 4.26 coupled with the form of continuity equation takes the form x k.(x') dx' 1 (4.26) Eq. 4.22 shows that the Op J + 1t = -jk(Pov + p v ) + 2k [p(0)v + v (O)p ] -1 1~01 10 i2 1 2 1 2f x ki(x') dx' e + jcup = 0. (4.27) 1

-95 - From Poisson's equation, eV * E = Z p ),and if the dependence of Eq. 4.22 — 1 1 is used as well as Eqs. 4.25 and 4.27, a form of the dispersion relation is obtained as follows: 1 2S ) s t 2/x k (x') dx' Ek j(( - kv) + 2k.v (o) e x If k. - 0 this reduces to the usual dispersion equation, Eq. 4.1. 1 Equation 4.28 can then be regarded as a dispersion relation which incorporates the nonlinear reactive effects. Note that v and E 2S 2 still satisfy Eq. 4.4 where now these functions are assumed to vary as exp 2 f ki(x') dx'. To solve this dispersion relation it is necessary to obtain the sets [kr(x),ki(x)] which at some real frequency o of interest satisfy Eq. 4.28, bearing in mind that the functions v (0) 2S and p (0) are themselves in general dependent upon ki(x = 0), (e.g., Eq. 4.20), in such a fashion as to maintain continuity of the time-averaged charge density. This problem is then too complex to solve by any analytical means. A case in which at least the form of the solution for k can be found is if k. < k so that the terms containing v (0) and p (0) in 1 r 2S 2S Eq. 4.28 can be neglected and, in addition, it is assumed that the q-plasma is such that |q/Tb << 1 so that v (x) - 0 and p (x) - O. In this case qEq. 428 is apprioqd byq Eq. 4.28 is approximated by

-96 - 02 (~) W2 P.b., (4.29) ( - o 2 W2 kv ob()) which is readily solved as X bpb((x) k(x) =v (x) (4-30) ob In particular when v b(X) and Opb(x) are taken as constants the result of Eq. 4.2 is retrieved. Even in this simple case Eq. 4.30 is difficult to solve since Cpb (x) and vob(X) are actually dependent upon ki(x). To proceed with the physical reasoning behind the quasi-linear theory it will be instructive to consider the case where the q-plasma corresponds to ions alone. (e.g., positive ion sheath) so that in its interaction with the electron stream q/fb < 0 and from Eq. 4.10 it is seen that there is now an additional component of beam slowing due to E Recall now that the linear theory provides exponentially growing fields, k.x 1 E ~ e, so that a beam electron downstream has lost more energy than 1 one at a smaller value of x. This energy loss cannot come from RF bunching alone since there would then be a net time-averaged, spatially dependent, second-order dc current J = (1/2)Re(p v*) and this would violate 2 1 conservation of charge. Hence beam slowing and dc bunching also occur and moreover these quantities are spatially dependent (increasing in 2k.x magnitude with x) varying as e 1 to match and annul, the time-averaged RF bunching current. This dc bunching in general then gives rise to the field E. 2

-97 - In extending the linear theory it was assumed that the k value (independent of x) could be retained. Hence the quasi-linear theory imposes the constraint upon the system that it provide the same field growth as the linear theory even when second-order effects are included. In the ion q-plasma case this gave the result that the beam was losing even more kinetic energy (e.g., the additional component of beam slowing provides in Eq. 4.19 that I<K.E.>I has increased) than that required of the linear theory. The reason for this is that the electron beam now must not only slow by an amount to feed a growth rate k.to the field E but it must slow further (lose further kinetic energy) in order to account for the fact that the potential energy of the electron beam-q plasma system has increased. Stated alternatively, for the electron beam to slow in the presence of the ion-q plasma the quasi-linear theory states that the beam electrons must go uphill (do additional work and lose more energy),whereas the linear theory has the beam electrons on a flat plane. Intuitively it is clear that in this case the quasi-linear theory indicates that it is now more difficult for the RF field E to extract energy 1 from the beam since it competes for this energy with the potential energy of the charge carrier system. Since the interaction strength is determined for all practical purposes by the parameters v ob, pb, and cw (and these are effectively constant for small signals), in actuality it is expected that the growth rate of the system will be less than that predicted by the linear thery ndeed because tory.he interaction strength is independent of E -field effects the quasi-linear theory strongly suggests that when the magnitude of the potential energy part of v exceeds that of the RF field part, viz., 2b

-98 - | |k2v2 > a2, (4.31) ib i ob where Eq. 4.20 has been used, the instability does not occur in fact. In the opposite case q/rb > 0, corresponding to the electron dominated or infinite ion mass q-plasma case, Eq. 4.20 indicates a component of beam speeding results from E. In this case, to retain the growth rate 2 k. predicted by the linear theory, the beam electrons must slow less than the amount needed to feed the RF field growth in order to account for the decreased potential energy of the system. Instability occurs more readily in this system since the charge separation effects tend to keep the beam electrons and the wave field in synchronism. A limiting action (on ki) due to E is also suggested for this case, since if Eq. 4.31 is satisfied 2 v > 0, and the unacceptable situation results that the beam energy and/or 2b the field energy is increasing at the expense of the carrier charge potential energy. Rather, the instability does not ensue in this case and the primary effect of E is to alter the saturation characteristics and growth rate 2 predicted by the linear theory. The quasi-linear theory thus provides some important differences in the description of the evolvement of convective instabilities in this system as compared with the linear result. On the basis of the linear dispersion relation alone it is concluded that the sign of the charges of the interacting carriers is of no importance (in the dispersion relation the charge always enters as a squared quantity so that the sign is lost). The quasi-linear theory on the other hand stresses the importance of the charge sign.

-99 - 4.2.2 Two-Stream Longitudinal Amplification. For the two-stream convective instability the dispersion relation neglecting collisions and carrier diffusion is well known to be cO2 CO2 1 = p-E — + P2 (c - kvo )2 ( -kv02 )2 (4.32) To facilitate the analysis consider only the case where 2 2 pi P2 A P P2 = k2 and v2 V2 O V- V2 01 02 Introduce the following definitions: v,v > 0. 01 02 (4.33) A do k = - s V Os s = 1,2, (4.34) and k+ A 1 (k + k ) T + - 1 2 (4.535) The dispersion equation, Eq. 4.32, is solved for w assumed purely real as (k - k)2 = k2 + k2 + (k2 + 4k2)1/2 0 - 0 - (4.36) It can be shown by letting Cio - - co that k -> (/V, /v so that all four roots are causal for k. > 0. Inspection of Eq. 4.36 shows that for complex k to exist for real C k k2 + k k(k + 4k2)1/2, (457)

-100 - wherein the (-) sign must be chosen in Eq. 4.36 indicating that two of the roots are dropped from consideration. Equation 4.37 can be simplified to ) v v --- < /k; 01 02 V # V 01 02 (4.38) and the only root of interest is given by k = 2 v v 01 02 (4.39) (4.40) where 1/2 Y |=(k = \k2 + 4k2 - (k2 +k2 0 0 0 As discussed in the previous example assume that questions of recombination and generation do not arise in the time and frequency scales considered. For the quasi-linear theory Eqs. 4.4 and 4.12 are applicable to the present case for each of the charge carriers (s = 1,2). These relations, together with Poisson's equation in the form eV * E = Z p -2 S 2S yield k.k2 Re(E ) = 1 2 4(k2 + 2k2) 0 1 S=1, 2 Iv(s) 2 1 ~a (w - k vo )(o - 3kr ) v+ 3k0v2 r O S I Os (w - k v )2 + k2v2 r os 1 os (4.41) From the first-order equation, j( - kvo )v1 = nSE s = 1,2 ('4.42) it can be determined that

-101- i ' i'6 N c t l l 2 / T \2 (a - k v )2 + k2V2 1 () 2 Iv(22 i oi -IV( ) (4.43) v1 v12) (2a - k v )2 + k2v2 1 r 02 1 02 Use of Eqs. 4.41 and 4.43 in Eq. 4.4 for s = 1 leads to 2k2 (w - k v )(w - 3k v ) + 3kv2 ( / -V ) 0= __ o_ r 01 oi 2 2 4voi k2 + 2k2 (w - k v )2 + k2v2 nl O 1 r ol oi [( - k v )(c) 3k v ) + 3k2V2 ][(w - kro )2 + kv2 (4.44).... r- r -02 o r i02 02 r o01i i 1o) [(w - k v )2 + kV2 ]2 r 02 i 02 By symmetry considerations v(2) is obtained by the replacement (2 1) 2 throughout Eq. 4.44. The present case cannot be reduced to the system studied in Section 4.2.1 because of the assumption made in Eq. 4.33. By comparison with Eq. 4.10,however, it is seen that Eq. 4.44 introduces an additional term in v(l) due to the fact that the carrier with which it 2 interacts is now drifting. The solution given in Eqs. 4.39 and 4.40 is too lengthy to analytically (s) deal with the general behavior of v(). It is clear however that the 2 introduction of drift on the second carrier has permitted an extra degree of control over the second-order charge dynamics to be established and as a result this leads to an additional control of the maximum growth rate 2k.L k. and the saturation length (e.g., the length L such that v (0) e 1 v ) 1 2 o of the system. 4.2.3 Two-Stream Transverse Amplification. It will be assumed that one of the charge carrier species is stationary, e.g., hole or ion, and its interaction with a drifted electron stream will be studied. The dispersion relation for the purely transverse interaction is given by

-102 - y)2 2 (w - kv ) w2 - k2C2 _ Pi pe k = 0 (4.45) - W ci - kVo - ce where the (-) mode framework is used and wCs = ~SBo; s = e,i. For the undrifted species Eq. 2.26 is used to give E = -Re ( -2jv( ) - ) (4.46) 2- \c ^<1 -where v(i) E- (447) 1- j(C) - Ci) In addition it can readily be found from the first-order equations of Chapter II that 2 i2 [(W k - k v - C) + kv2] |^()|2 (uI - — r o ce io Iv(e). (4.48) ~1 — e1 [(a - kv )2 + kOV2](C - Wi)2 1 -Equations 4.46 and 4.48 then give k. iio2[(co - k v - c )2 + k2V2] E I~= 2r - %[(r o ce i o v(e) (4.49) 2(- % 2( _C)[( ) - k v )2 + kiv2] 1 -e ci r o i O From the time-averaged second-order force equation for the electron stream, viz., 2k.v ve = R E + Re el, ( 450) i 2- e2- e 2j x ' {1 ()E it can then be found when Eq. 4.49 is used that

-103 - I (e) I v(e) -= V [ (-kv( ) - [(a)- k v _ o c)2 + kv2] 2- 4v [(co - k v )2)2 + k2V2v2 ] e ci e. o r o o - [(c- v 2 +k -v 4ce], (.51) where, from Eq. 4.15, Iv(e)12 (e)v2 v(e) < and ve) | >. for <K.E.> < - 2 0 2- 2- Vo 2e Now inspection of the k-cubic dispersion equation shows that an unstable convective root (k. > 0) is expected if cu < 0 and j- ce v\ 2 > (aW+ 1c I)2( Icil (0 > (oce);- > (4.52) c 3[on e(W - OWiJ) +,2.] Pe ci p3i 1 c and this is easily satisfied near the resonance cu Ici + |161 Since (qi/e) < 0, Eq. 4.51 then shows that the reactive effects attempt to drive the charge carrier system to a higher level of potential energy. As discussed for the electron beam-q plasma interaction in Section 4.2.1, in this case instability is more difficult to achieve and the system in actuality has a growth rate diminished from the value predicted by the linear theory. 4.3 Effects of Collisions and Carrier Diffusion Up to this point to obtain useful analytical models for the quasi-linear theory the presence of collisions and carrier diffusion has been totally ignored. It was then determined that in multiple carrier interactions the presence of the growing RF fields alters the carrier dc

characteristics to satisfy the condition of charge continuity. This in turn led to an alteration in the potential energy of the charge carrier system. To obtain some understanding of the effects of collisions and thermal diffusion examine the time-averaged second-order equation given by k. v2 (v + 2k v )v = - Iv +2 + I Re (E) - 2 (453) i 0 2 2 1 2 0 0 24. which corresponds to the purely longitudinal interaction when collisions and diffusion are retained. In addition Poisson's equation, EV * E = Z p, can be used to replace E in Eq. 4.53 giving -2 S 2S 2 (v + 2k.v)V 12 + [ kvl- p 2 (4-54) where p' is appropriate to the carrier being interacted with (e.g., the 2 undrifted carrier). Also Eqs. 4.12 and 4.17 are still applicable to the present case and from these it can be found that V P Iv 12 k ( - k v ) - k2V P = - - r v)- k (4.55) o 2 Vov0 (0 - kv)2 + kv r o 1 o Finally, if Eq. 4.55 is used in Eq. 4.54 the following results: r2 4k0v2 k. rp l- 10 ' (1) 1 v 2kv + P T v IV 2 I 0 2kYv 2a2 2 1 2k.E~ c~ 1 CU2k ( - k kv ) kv2 P-m v2 --- (i -- ). (4.56) i O (CD- kv )2 +k2v2 ro z o p

-105 - Intuitively it would be expected that the carrier thermal diffusion would tend to prevent the formation of p (and hence E ) and as a result diminish 2 2 the second-order potential energy effects. Indeed,from Eqs. 4.15 and 4.17, in the present case the general result P P P |2 CI((I w - k v ) <K.E.> = 2 v- + 2 -- v2+0 o (4 57) beam 2 - 2 o 2r O 2 (-kv)2 + k2v2 r 0 1o0 shows that the beam need not be slowed to obtain ( o v2 + <K.E.>b) < 0 2il o beam if ( k - k v) < 0. r o The complexity of the problem of second-order effects is now described. It would appear justifiable to assume that if the charges have large thermal velocities concurrent with a low collision frequency (so that the diffusion coefficients are large) then v, p, E -0O. In 2 2 2 this case Eq. 4.57 would suggest that the RF bunching is solely responsible for the energy lost by the beam. However in this same case the timeaveraged charge density p is constant in the x-direction so that k is a constant corresponding to the medium being homogeneous. This then implies that Eq. 4.12 must be satisfied bringing in the contradiction that v, p # 0 2 2 to satisfy charge continuity. On the other hand, if the collision frequency is sufficiently large, the thermal diffusion effects should be greatly reduced and the importance of potential energy considerations becomes reestablished.

-106 - The quasi-linear theory then at least helps to provide some insight into the difficult problem of the nature of the evolvement of a convective instability from its point of initiation. The field E would appear to 2 be necessarily important in the purely transverse interaction since there is no RF carrier bunching in the hydrodynamic theory so that carrier dc slowing must occur to provide the energy to drive the instability. On the other hand for space-charge-wave interactions the RF bunching process can be the dominant energy extraction process. In this latter case, if the beam slowing is small, the saturation length L (where L is such that 2k.L v 0 v (0) e 1) is much larger than for the purely transverse waves. ~0o~~ 2 ~2k.x k.x 1 1 Note that since v, p,hereas v E e whereas E e,it would appear 22 2 11 that the second-order dc effects, at least in some cases, will dominate 2k.x the nonlinear behavior since the condition v (0) e 1 v can arise in k.x 2 0 k.x space prior to Iv(0O)I e 1 v 4.4 Application of Quasi-Linear Theory to Absolutely Unstable Systems In general the absolute instability process is much more difficult to analyze than the convective instability case since both C and k can be complex. For this reason only the case where k is assumed purely real is considered. This still retains the essential physics of absolute instabilities since the effect of complex k is primarily to permit timeaveraged power to be extracted from one of the device terminals. Since Briggs' criteria shows that for any absolute instability the system must possess at least one root with w. < 0 for real k, the present analysis is then meaningful in all cases. In general to study the instability characteristics of a system a linear wave equation is used to determine whether a small perturbation of

-107~ the RF field will grow from the onset point (in space or time) of the initial perturbationo Although the system boundaries transverse to the wave vector k can be taken into account in this theory (e.g., this usually introduces some additional mode structure into the solutions), this is not true of the terminal boundaries parallel to k (the end boundaries)o For convective instabilities this is not a significant problem in general because the interaction proceeds spatially so that the condition of charge continuity is satisfied locally (questions concerning reflections are outside the scope of the present work)o In the absolute instability process the end boundaries can play a fundamentally important role howevero To understand this assume that a system of.purely transverse waves is absolutely unstable (pi < 0) with k realo Since the waves are purely transverse, no RF bunching is present so that it must be that the active charge carrier is slowing as a function of time (at all points in the interaction region) as this is the only method available to feed energy to the RF field growth. This immediately raises questions about the end boundarieso For example, assume that this system corresponds to an electron beam as the active source entering and exiting a finite plasma at x = 0 and x = Lo Assume further that the RF fields are zero for x < 0 and x > L (eogo, in a gaseous plasma, coupling devices are at x = 0 and x = L and in a solid-state plasma these are the contact points). In any case there is no interaction for x < 0 and x > L so that in these regions the beam is unaffectedo On the other hand at all points x in the plasma the beam is slowing exponentially with time. Clearly this raises questions of charge continuity at x = 0 and x = Lo

If the growth rate ox. is extremely slow it would appear feasible that generation and recombination mechanisms at x = 0 and x = L could maintain the continuity. If this cannot happen the charge must build up at these terminal points and create a time-dependent field E whose 2 displacement current provides the continuity. If the former is the case the linear dispersion relation gives an accurate description of the system until such time as the small-signal approximation breaks down. To study the effects of the field E it will be assumed that the time scales of the 2 system are such that charge recombination and generation can be ignored. 4.5 Two-Stream Electrostatic Oscillation With collisions and thermal diffusion neglected,the two-stream longitudinal interaction can be described by 2 2 1 =..pi + p2 (4.58) ( - kvo )2 (o + kv )2 Ol 02 Consider the simplest case, c = C C and v = v v,which then pi p2 o01 02 o yields the solution 1/2 (kv)2 = C2 + D2 + (4D2 + 2) (4.59) 0 p p p For an absolute instability a double root of k is required with coi < 0. Inspection of Eq. 4.59 shows that this only occurs at the purely imaginary frequency o = oD = -j(cp/2), with the corresponding real wave number k = ( \//2)(op/vo). As discussed in Chapter III assume the interaction length L >>~ or L = n\ with n an integer. In that case the second-order variables of interest vary as exp(-2coit) alone.

-109 - From Eq. 4.4 therefore the following may be obtained for the terms which vary as exp(-2o).t): 1 a( (S) Re ( ) = l Re (E); \ dt Rs 2 s = 1,2 (4.60) where all other possible contributions are zero since k is purely real. The first-order equations provide the following: j() - kvo)v() = T E 1 1 1 and (2) j(cD + kv )v( ) = q E 0 1 21 (4.61) so that I() 12 1 2 2 Iv I (w + kV)2 + (lv(2) 12 (w - kv )2 + 2 1 r o i from physical considerations, in the (4.62) As discussed in Chapter III, interaction region, p(s)() = 0; 2 s = 1,2, (4.63) so that J(s) (t) =+ 2 Re (Pl ) 2 OS 2 2 ( 1 s =1,2 (4.64) and the proper form of continuity in the absence of any generation or recombination processes is

-110 - J (t) += 0. (4.65) 2 S=1 2 Use of Eqs. 4.60, 4.64, and 4.65 provides kI( 2 W Re (E) ) =( (%r - kv ) 1V)12 1 (wr + kv) Ivl 2) Re(E ) = + + p + 2 2(cU2 + CUO) r [( C - kv )2 + C2] [(D + kvo)2 + -P J 1 r o 1 2 r 0 1 (4.66) so at the unstable root o = w0, from Eq. 4.63, 2 Iv = q2 IV() O~ 2 1 ~1 1 and as a result Cw.wk2vo IV112 / Re(E ) -= 1p 0 1 (1-l ). (4.67) 2 2rl (2 + 2w2)(k2v2 + ) ) 2 p 1 0 1 Inspection of this result shows that if 2T = T, (i.e., two equivalent beams in contra-flow) then Re(E ) = 0 and,consequently,from Eq. 4.60, v (t) = 0 for each stream. Consider the kinetic energy balance which results when = r. Apply Eq. 4.13, where now p = v = 0, to obtain 1 22 2 (where carrier 1 is understood) <K.E.>> = pv2 + 2vp <<v >>+ v2 + p <p + 2v <<v >> 2j\\0 1 1 1 0 0 1 1 (4.68) -cD. t wherein since v v (0) e 1the first-order terms must be retained so that << > now represents a space average and the kinetic energy (as expected) is always a function of the time and contains variation both as exp(-W)it) and exp(-2wo.t). It is clear, however, that if the device length L = Tr,then <<v >> = p >> = 0. If from the continuity equation, Eq. 4.16, -the result -the result

-111 - kp v P = - (4.69) 1 a - kv 0 is used in Eq. 4.68, assuming L = nX, the following is true: P P0 (W - kv)( ( + k ) +2 K.E. = 22 + 2- rr <<v2>~ (4.70) <<K, *'E. >> - o 2 (% -+ o~ (4.70) 2 0 (w -kv)+ It can be seen by inspection that the <<K.E.> functions of streams 1 and 2 are identical at CD = Co and moreover (a/$t) <K.E.>> <. Indeed, on substituting the solution at w = o, for each stream, p / <<v2>\ P ED2 <<K.E. >> v2); = -. (4.71) 2ri \ O 2 5 rl 2 When Tl r,however, the field E arises and can affect the 1 2 2 evolution in time of the absolute instability. Note that now potential energy changes in the overall system will occur spatially at or near the end boundaries as opposed to the convectively unstable system in which these occurred at all points in the interaction region. The more general form of the stream kinetic energy when E W 0 is given from Eqs. 4.15 and 2 4.68 as P p K.E.> = - 2v v + - v2 + <<K.E. (Eq. 4.68)>>, (4.72) 2rI 0 2 2r o where the last term is the <<K.E.> function of Eq. 4.68 which at the unstable root is given by Eq. 4.71 for either carrier. Recall that in the quasi-linear theory the system is constrained to provide the same solution (e.g., growth rate) as the linear theory even when

-112 - second-order effects are taken into account. This alters the magnitude of the rate of loss of kinetic energy of the streams from the linear theory (Eq. 4.71) to account for potential energy changes caused by the instability. From Eq. 4.72 these effects become important at least when p P P v2 - 2v v + v2 <<K.E. (Eq. 4.68)>> — 2 2ro 02 2r] o2q (4.73) or, since in the interaction region p (t) = 0, 2 P 2vP 2ro 02 Tq 1 (4.74) To solve for v(1) in the quasi-linear theory, Eq. 2 using Eq. 4.64 gives 4.65 when expanded + kpo Iv )i (-kv ) p v + - 01 2 2(k2v2 + W2.) 0 1 kp v(2 (kvo) + o1 0 2(k2v2 +co2) 0 1 + p v(2) = 2c.eE 02 2 1 2 Equation 4.60 can be used to replace v( ) which together with 2 and 4.67, yields the following at cw = co: (4.75) Eqs. 4.62 2. ).,(1)= ~~ 1 1' 2 I ( ) ( - ) (v) _ kIv0 Ivl)<1 - (2 +.. ( 12.. + ) 2 2(k2v2 + c0) (c2 + 2C ) 0 1 P 1 (4.76) and since w. = -u /2 the last term can be simplified to the following eventual resut eventual result:

-113 - 3 IV I 2 v() = v[ 1 +_ 1 T 1 (4.77) 2 8v 3 L) 1 \ 2 2 When used in Eq. 4.74, since <<v>> = Iv( if L = n., the condition under which the effects of the field E become important (corresponding to the 2 alteration of the system potential energy becoming comparable to the alteration of stream 1 kinetic energy in the absence of E ) is 2 f( ) t | a 1 - ) 1. (4.78) 1 2 2) T 21 1 When f(rT,q ) > 1 it is expected that the instability is quickly quenched 1 2 or else driven to a much smaller growth rate. Inspection of the form f(I,T2 ) shows that if T P T it is easily possible for f 1. The method used herein for the two-stream electrostatic oscillation can readily be applied to any system described by the linear dispersion equation to be absolutely unstable. Note that since in general p (t) = 0 in the interaction region the medium remains homogeneous so that it is not expected that k - k(x) due to the RF field growth as in convectively unstable processes. Rather, the linear dispersion equation remains valid (e.g., T = T in the preceding example) until the small-signal 1 2 approximation breaks down or else the quasi-linear theory is a good approximation to take into account the end effects on the wave growth. Indeed for systems which are absolutely unstable with either k real or wi > kiv it is expected that the quasi-linear theory is the best practical method to account for the end boundaries. practical method to account for the end boundaries.

-114- - When an absolute instability occurs at a double root with k. W./v all the second-order effects studied in the convective 1 1 o instability process in Section 4.2 become important, e.g., k -4k(x). The effects of collisions and thermal diffusion are expected to play a similar role to that discussed in Section 4.3. A much more significant role may be played by the nature of the end boundaries with regard to the buildup or depletion of the carrier charge density near these regions. 4.6 Summary and Discussion The utility of the quasi-linear theory in aiding in the understanding of the evolvement of plasma instabilities has been shown. It permits studying the instability in the intermediate regime after the initiation of the instability but prior to any large-signal effects. In addition, the quasi-linear theory offers a method for taking into account the effects of the end boundaries of the system on the instability evolvement —something which can be quite important in absolutely unstable systems. In general, the quasi-linear theory shows that potential energy effects cannot be ignored in the analysis of plasma instabilities even when the thermal diffusion coefficients are large. Since some control can be effected over the parameters determining the second-order effects the quasi-linear theory can thus be used to suggest methods to help quench or diminish undesirable instabilities or enhance desirable ones by altering the growth rate and saturation characteristics.

CHAPTER V. THE POWER THEOREM ACCORDING TO KINETIC THEORY WITH APPLICATIONS 5.1 Introduction The general definition of the kinetic power flow given by 1 * Re (E * J (5-1) k 2k is readily adopted within the framework of kinetic theory for any carrier velocity distribution function. In the past the power theorem has been restricted to hydrodynamic theory by virtue of the fact that a sufficiently general formulation was not constructed. The major distribution functions examined are the hydrodynamic or square distribution, the Maxwellian, and the degenerate distribution functions. These are applied in turn to obtain the kinetic power expressions for the basic carrier modes —the space-charge, cyclotron, and hybrid modes. Whenever applicable comparison with the hydrodynamic results of Chapter II is made. Since a fundamental part of the present work is concerned with solidstate plasmas, special attention is devoted to the role of collisions in the kinetic power functions and the consequent wave behavior. It is shown that the magnetic field and the collisions play a crucially important part in active wave phenomena near the cyclotron resonance point of the carrier cyclotron modes. Applied to the phenomenon of microwave emission from indium antimonide (InSb) it is shown that even a small density of holes can lead to large wave slowing and consequent amplification by the electrons. -115 -

Comparison with the hydrodynamic theory of Chapter II shows that in many cases the hydrodynamic theory has been improperly used and the limitations on its applicability are shown. In those cases where the hydrodynamic theory is not valid, the results from the kinetic theory show that resonance and/or anisotropic temperatures can readily lead to instability. A discussion of causality within the kinetic theory is discussed and suggestions made for its implementation. 5.2 Power Theorem for Longitudinal Space-Charge Waves The Boltzmann equation is written for this case in the form j(w - k)f + E ( - (5.2) - ~ x -coll where it has been assumed that the fundamental variables ~ exp[j(ot - kx)] and where various forms for the collision term on the right-hand side will be studied. 5.2.1 The Hydrodynamic Distribution Function. Consider first the drifted-square distribution defined by N~ o > 2v y z' T~o x o T = 2vT 5(Vy)(vz) for T + v v v v f = O otherwise, (5.3) so that of N = 5(vy)5(vz) -( v + v) - (v - v - v). (5.4) dv ~ ~ 2 x y 0 x oT x

-117 - The RF current is in general given by 00 J = q J v f dv d -00 for each charge species present. In the collisionless case, (6f/6t)coi, can be readily solved to obtain v dvz (55) = Eqs 51 through 5.5 = O, Eqs. 5.1 through 5.5 cu2 wv ((o-k v ) + k p o r o r 2 V ) E 1X vo P (v = O) = 2 ((- k v )2 - k2v2 - v2(k2-k2) + ro 0 o - T i 2 4k v (w-k v ) k v2 1 o r o r T (5.6) Note that this result is identical to that obtained in the hydrodynamic analysis, Eq. 2.23, as is expected for the square distribution. If one proceeds in a similar fashion for the commonly used collision term defined by ' jcoll = -vf 1 it can be shown that v2 r o O P O |E I [(a) - kvO ) (1 + kv ) + k T p 0o E ix1 (o - k v ) k.v ~ r v + 2kv ((v + k.v )2 -( - kv )2 + (k2 - k2)V L 2k.v i o ro r iT p k 2 (( - k v )2 - (v + k.v )2 - (k2 - k2)v) ~ ro o ) 1 -r 4 ((v + kv )(- kv) + k. k v2 o1 ro i T (5.7) (5.8)

-118 - Inspection of this result shows that the collisionally assisted form of the negative kinetic power flow obtained in the hydrodynamic analysis, Eq. 2.23, when (o - k v ) < 0 is not in general present. To some extent this result is to be expected since it is known that the collision term given by Eq. 5.7 does not conserve particles properly. Although it has been asserted that this collision term is valid if Cw > v,74 Eq. 5.8 shows that even in this case Eq. 5.7 may not be an accurate representation if C1w) - k v0 < k2v2 r T The collision term which conserves particles properly is ( T - v.f -N f, (5.9) coll O where N is the perturbed number density given by 1 co N f dv dv dv, (5.10) 1 1 x y z -00 If the resulting Eq. 5.2 is integrated over velocity space and Eq. 5.10 is used, the perturbed distribution function can be written as jEf 1 w - kv- jv ) f 11 where af o r r r ov = v.x. dv dv dv (5.12) o,k o - kv - jv x y z -o00x -Co and

-119 - 00 = jN -v j. dv dv dv. (5.15) o cD- kv - jv x y z x -00 For the distribution function of Eq. 5.3, vkN o. (5.14) ((o - kv -jv)2 2 k2 To compute the kinetic power flow it is found from a transformation of variables that the following integrals in the complex 4-plane are required: = 1 d u 1,id p. - -1 (:> = -- z and 0 4 4 (5=15) where C - kv - jv z = 0. (5.16) kvT and v -v x o (5.17) vT If it is assumed that k. < kr, it is seen from Eq. 5.16 that Im (z) < 0 1 r if k. > 0 and v > k.v. Since the time-dependent Fourier transforms 1 10 exp[j(wt - kx)] it is necessary to integrate just above the singularity by choosing the contour in the complex n-plane with corners [-1, 1 - j l7, 1 - jly, 1] and lyl = -Im (z), leading to = loge ( - j1) (5.18) and = -2 + zO 4 3 (5.19)

-120 - where = 1, if IRe (z)l < 1; Im (z) > 0, = 0 otherwise. (5.20) Use of these results in Eqs. 5,.5 and 5.11 leads to the kinetic power flow of Eq. 5.1 being given as wCl|E |2 P = ____ p ix k k12 vTI( - kvO - jv)2 - k2VT2 * Re (jk [(W - - j)2 - k2vT [ - jvF(O)) (5.21) where 2k v (w - jv)> + 2jvkv T - jvoA 0 F( ) - T -- -- -T —.T v (5.22) ~~3 4|k12 v2 + v2^ D T 33 The function F(O ) is the term taking conservation of particles into 3 account. The resultant form, Eq. 5.21, can be studied in various limits. Case a: |IZ >> 1, Re (z)l > 1. For this case a = 0, log [(z + l)/(z - 1)] - j tan-(kvT/v) - 0, and hence F(O ) - 0 so that under the assumption k. << k WC2CVo IE 12k(1 r + k.v ) (w -kv)] p o 1X r- - - -- (5.23) k |k|2 v | ( kvo - jv)2 - k2v12 2 T I~ 0 T which by comparison with Eq. 2.23 shows that the present conditions are similar in power flow characteristics to the hydrodynamic case.

-121 - Case b: Iz >> 1, |Re (z) < 1 From Eq. 5.16 it can be found that in general k. - k v Im (z) = -, (5.24) k12 VT so that from Eq. 5.20 the residue only contributes if k. < 0 corresponding to damped waves with Iki > k v/o. Thus the residue cannot play any role in instability (growing wave) processes. For this case a = 1 and j - jt leading to 3 IE IEl12 4k2[( - k v )2 + v2] P p iX- r ro k 2k. Ik 2 V2 I(k - kv - jv)2 k2 12 T T o so that the self-consistent result obtained is that the interaction can only proceed if k. < 0 in which case Pk < 0 to balance the positive electromagnetic power flow. When it is assumed that Ikil < k v/c the result of Eq. 5.23 is again applicable. For any particular system it is necessary to either solve the difficult dispersion equation exactly or else assume initial constraints such as Ikil < krV/w to analytically solve the dispersion equation, in which case the results obtained must be in agreement with the initial assumptions for self-consistency. This is a general problem associated with the kinetic theory. Case c: Iz| << 1, IRe (z) < 1

-122 - Again for the case Iki > krv/w with k. < 0 the residue contributes so that a = 1. In addition log [(z + l)/(z - 1)] - jrt leading to F(O ) - 0 e and C2 E IE 2 vo k (1 k V + k2 T' "I,-p ' 0 o r V 2rv |k|2 vT K - kv| -j)2 - k2 V12 This result is of value in indicating which regions of (w-k) space will potentially provide a self-consistent solution. For example in the present case if Iki > k v/w it is required that |w - k v 12 > (3/2)krT which is not self-consistent with zl << 1 and hence no damped solutions are possible. If Ikil >> k v/w, however, a potential solution is possible (Pk < 0) only if (C - kv) < 0 and v > (3/2)kr2/(k v - ). r Q o rT r o For the unstable case (ki > O) or |kil < krv/c, k. < 0, the residue does not contribute and it can be found that, if k. << kr, i r 4W2e E |2 vk1 (2k rV (w - k v) + vT [v2 + k - (C k p -ix r rT ro k T rk P k P: I |k2 (c - kv - jv)2 - k2v2T2 (4 |k|2 vT + 2v2) (5.27) Inspection shows that since Izl << 1 the k2v2 term above dominates and r T Pk > 0 so that an instability is not possible. These results demonstrate the difference between the hydrodynamic theory of Chapter II and the kinetic theory with the square hydrodynamic distribution function. For unstable waves with Izl > 1 the two theories are in agreement provided k. > 0 or |kil < krv/O. However with I z < 1 there exist opposing results. Thus, for example, if v < vT it is only if 0

-123 - v > IkvTI that unstable solutions (such as acoustic-phonon amplification) predicted by the hydrodynamic theory are in accord with the kinetic theory. 5.2.2 The Maxwellian Distribution Function. Equations 5.11 through 5.13 can be applied to the isotropic drifted Maxwellian distribution function given by f = e exp (xp ( j ) - (5.28) 0 (2Tv 2)3/2 2v2 ex 2v2 T T Define the plasma dispersion function as poo -^2 G ) 1 -oo e d2, G(z) = He d (5.29) from which it can be shown by an integration by parts that 0oo e-2 e e d1 = J7 [zG(z) - 1] -00C Z and i m 2 e" de = \z[zG(z) - 1] (5.0) -C00 z The function G(z) is related to the tabulated function75 Z by G(z) = -Z(z*), (5.51) where the complex conjugation is due to the tabulated functions being derived assuming variations as exp[j(kx - wt)]. With these definitions and the use of Eqs. 5.5 and 5.11 in Eq. 5.1, wherein only the collision term which conserves particles properly (Eq. 5.9) is studied, the power is found as

-124 - pEI = p* Y. Re (-jk(G - 1)( H k v jvG)) k 12 V2 12 2 VGG} k 2 ki |k|2 vT(2 |k|2 vT + v2GG*) T (5.32) where G = G(z) and u - kv - jv A 0 o-7 z = (5 33) WS kvT A central problem exists in the nature of the tabulated function Z (or the function G) as has been pointed out by Montgomery and Tidman;76 namely, the function so obtained is only valid in the limit t - oo and hence cannot be applied to unstable systems. When the buildup occurs in time (ci < O) this may be the case; however, for a convective instability, the linear steady state is well defined and this problem should not occur. In addition this function is commonly used for application to unstable systems so that the kinetic power results using the G function will be obtained ir various limits. Case a: IZI >> 1 For this case, G(z) - -j '7 a exp(-z2) +: - (1. + - + + (5.34) z 2 z where 0 Im (z) < 0 C = 1 Im (z) = 0 2 Im (z) > 0

-125 - and from Eq. 5.32 kr(v + k.v ) + k.(c - k vo) Im (z) = - (55) 2 vT(k2 + k) For growing waves (ki > O) inspection of Eq. 5.35 indicates that necessarily Im (z) < 0 so that the residue term in Eq. 5.34 does not contribute (C = 0) and Eq. 5.32 becomes for this case, retaining the first two terms in the expansion, 2EwE 1| 2 (C - kv)(v + 2k )[|k14 v2 - k2[3(w - kv)2 - v2]] 111Pr io r r ro P, k 2k. |k2 vT(2 I|k2 vt + v2GG*) |( - kv - jv)212 ~!ki.(kl2 VT o (5.36) This result shows that Pk < 0 provided (c - k v ) < 0 unless the drift velocity is sufficiently nonsynchronous with the wave phase velocity that 3( - krv)2 > k2v2 + Ik4 V2. Case b: IZI << 1 For this case, G(z) - 2z[l - (2z2/3) + (4z4/15) -.. ], so that retaining only the first term IE IEI 2[v(kr[kr(w-krvo)-k (2v+kivo)] - k2u}) + 2kikr(k2+k2)vj p ix r r r o. i r P k 2k. |k|2 (2 |k|2 v2 + v2GG*)(k2 + k (5.37) Inspection of this result shows that for the hot plasma case it is again necessary that (o - k v ) > 0 and in addition that the growth rate be limited to

-126 - V Iw - k v 2k v2 T (58) rT 5.2.3 The Degenerate Distribution Function. The degenerate distribution function is defined by \v2 +2 = vT N y z T f = ~ for T1 v + vT v v - v T1 Tll l ~ T]| - x - o. T] = 0 otherwise, (539) so that If N x ^ = -— 2 — ^(^ -^ +v v v v (5.4o) x 2v [v L Vx o TI| - Vx ~ T||) ] TI Tii Note that this distribution is an improvement over the hydrodynamic distribution function of Eq. 5.3 since the latter function ignores the transverse thermal motion. The collision term which conserves particles properly is chosen so that Eq. 5.11 is applicable. Inspection of the pertinent integrals involved in forming the kinetic power function shows that in all cases the transverse thermal distribution has no effect and hence the results obtained for the hydrodynamic distribution function, Eqs. 5.14 through 5.26, are strictly applicable. This is to be expected since the space-charge wave system is one dimensional so that the random transverse thermal motion cannot alter the longitudinal RF current. This is, of course, alleviated by the presence of a transverse static magnetic field.

-127 - 5.3 Power Theorem for Purely Transverse Waves in a Static Magnetic Field The kinetic power functions for purely transverse waves with B II k II x are now developed. The Boltzmann equation for this case, assuming that the fundamental frequency variables vary as exp[j(wt - kx)], is in linearized form // af af f f j(w- k x )f + Ef -v + ZE + ri(v x B x cy z v y )v h y z v -- 0o ( f -- coll where ce = TB. Define the reduced distribution functions 0 Y S fd v00 co f = / / v f dv dv Y J J y 1 y z -00, (5.41) and 00 f = vf dv dv z J z 1 y z -00 (5.42) together with a transformation to rotating coordinates, viz., E + = Ey jE 1+ ly lz fl+ = f ~ jf 1~ ^ -y z and B+ = By jBZ 1- Jy l (5.43) Equation 5.41 is multiplied successively by v and v and integrated over the transverse velocity z the transverse velocity space (v,yVz) to obtain

-128 - j(c - kv )f - Cx f x y c z 00 - EyE f dv -00 o y -_OO dv + r z dv x 00 oo. -00o - v v B )f dv dv + Tlv B y z ly o y z X jz 00 f dv dv o y z -00 00 -0 -00 col T 'coll dv dv y z (5.44) and j(aD - kv )fZ + cf - E1 -N x. c y iEz 00 f dv dv + z o y Z d dv x 00.3 -0 (v v B - v2B )f dv dv - rv vyzJ-z zJ ly o y z jx zy oo f dv dv J 0 o y z 00 -rs v -00 coll dv dv y z (5.45) and where, if i,k subscripts represent y or z it has been assumed that f, f - 0 as v. - ~oo so that in obtaining Eqs. 5.44 and 5.45 use has On 1 been made of poov. 00 1 00 0 o; e.g., o -00 af v d. dv, k v. di k 1 0 /, dv. dv = -00 and

=129= C00 00 o 1 r r d P r v 09 dv dv = f dv. dv (5.4 J J v 1 i 6v k JJ o i k which can easily be shown via integration by partso Perform the operation [(Eqo 5044) - j(Eqo 5o45)] and use the definition of Eqo 5o43 to obtain j( kv x - C )f XA 0 1 =' - =F (v)E + J-Vv F (v )B + T d G (v ) 0 x 1i- 9Vxxodv d 1 -x x CZ) -00 v t coil (vy jv ) dv dv vy Jzy z (5.47) where F (v ) - o' and 00 f dvy dv (5o48) -00 v2) + B (v2 v2) + 2jv v B f dv dv z 1+ z y7 "y z a~ y z G (v ) I x 00 / 00 L (V2 + =00 (5o49) Now, in the presence of the constant external field B 0 f is necessarily isotropic about the direction of this field so that f = f (v 9 v2 + v2) o o x y z Thus if the B part of the integral in Eqo 5 49 is considered and written as 1+ v = u sin 0 y and v u cos 0S z the result is

-130 - 2 B j // (vz - v2 + 2jv vz) f dv dv 2 1+ z y yz o y z -00 -co = +B u3f (u2)ej dG du = (5.50) 2 1+ 0 o Thus, in general, G (v) = jB () = BB o x (v+ v )f, dv dv (5.51) iX l x 2 y z 0 y z -00 For the collision integral defined by Eq. 5.7 the right-hand side of Eq. 5.47 simplifies using Eq. 5.43 and Eq. 5.47 becomes _ dG j(w - kv )f - Fo (V)E +j- j lB L () + d (vx) = -vf x c l- Fox - l- x 1 -(5.52) From the equation V x E = -joB and Eqs. 5.43 it can be found that -1 1 B += +. +, (5.53) so that from Eq. 5.52 rkv dG I jriE [F(Vx) (1 - ) dv 1- c) -- kv - co - Jv x c Equation 5.54 shows that the circularly polarized modes separate. It can be shown by proceeding in a similar fashion that

-131-.F /kkv vxdG 1+ +x ~ dv f+ = - kv +w - jv (5'55) X C so that the results for the (+) mode can be obtained from those of the (-) mode by the simple replacement cD - -CD and only the (-) mode need be examined further. Recall that in the (-) mode formulation Cw < 0 corresponds to the helicon or slow-cyclotron mode, etc. Because reduced distribution functions have been used the RF current density is 00 J =_ q J f dv, (5.56) 1 - -eo 1- X where q is the species charge. As a simple example of the utilization of this result consider the cold plasma distribution function, f No (vy)5(v )5(v - v) (5X57) o o y z x o and use Eqs. 5.54 and 5.56 together with the definition of the kinetic power flow, Eq. 2.64, to obtain V a)?(: IE 12 L +~ V (cU - k v * * l -c k 'V r o) P. =-Re(E J ) = 2.[(.kv...... (5.58) r o c k. By using Eq. 2.94 to replace IE 12 by the appropriate Iv 12 term it can be shown that Eq. 5.58 is completely equivalent to the hydrodynamic result, Eq. 2.106. In addition, integration of Eq. 5.54 can be performed to verify that N = 0 so that only the collision term of Eq. 5.7 need be examined. 1

-152 - Prior to analyzing the various distribution functions it will be of value to examine the general requirements for instability of the purely transverse waves. For a multiple stream interaction it is readily found from Maxwell's equations, where J = J + jJi, that for real w -* r- - ~Au~ ca 2 f \ (k2 - k2)c2 - c2 J =O 0 (5.59) r 1 E 1 -and 2k.k c + J 0. (5.60) i r E r1 -The RF conductivity of a carrier is given by a = r + jci, where r- = (E_) and (. = ) (5.61) E ' Thus under conditions of amplification (kik > 0) Eq. 5.60 shows that it is necessary that the net real RF conductivity, Z ( ), must be negative. ssss Also, if k. << kr, Eq. 5.59 indicates that for slow waves to be present (which is a necessary requirement for the interaction to occur) the net reactive RF conductivity, Z c(s) must be large and positive. The latter s:L condition indicates that relatively large RF reactive currents are necessary which in turn suggest that resonance processes may be important. Note that the kinetic theory can account for resonance processes whereas the hydrodynamic theory cannot. 5.3.1 The Hydrodynamic Distribution Function. For the hydrodynamic distribution function defined in Eq. 5.3, Eqs. 5.54 and 5.56 can be used to obtain

-133 -?2E IE 201 ~P = z- i RT k- = - iki TC-v 1 T/Vo+%T j ( - kvx) o - kv - jv - co x c dv x (5.62) The pole present in the integrand can lead to a component of wave damping (cyclotron damping) or growth similar to the Landau damping of the purely longitudinal waves. By the transformation of variables = ( - v )/vT it is seen that the integrals ~ and 0c defined in Eq.1 are introduced and can be used to obtain the result Eq. 5.15 are introduced and can be used to obtain the result 2E JE 12 r =f- 1^ — Re (-jk - 4k~k.vcb 1k|2 ki)(c + jv) [log( 1 - 1} c ~(5.6) (5.63) where C - kv - jv - c O C z c kvT T (5.64) and a = 1 = 0 if IRe (zc) < 1 and otherwise Im (zc) > 0, (5.65) Case a: IlCZ Assume that either |Re (zc) > 1 or Im (zc) < 0 so that the residue does not contribute. In this case log [(zI + l)/(z - 1)] 2/z and hence from Eq. 5.63 U2e 1E1-12 O c + V (C - kvo) p2 [ c - ) + (= -. ) ] k - 2PI o k- 2 [( w -kv -b )2 + (kiv + v)2 2w[ - kro 0 V -' (5.66)

which is a recovery of the hydrodynamic results of Chapter II with the carrier thermal velocity playing no role. Indeed it will now be shown that for this case the hydrodynamic results can be retrieved exactly (e.g., helicon mode). To see the effects of the transverse thermal motion this will actually be done for the degenerate distribution function defined in Eq. 5.39 and by letting vT -0 the actual hydrodynamic distribution function result is obtained. Define the multiplier -r-L to mean that, x when this symbol appears as part of the numerator of a function, the function is multiplied by unity if (vo + vT = v = v v ) and by zero otherwise. Thus, when the integration is performed in Eq. 5.51 for the degenerate distribution function, N v o T o(Vx) B -r) (5x67) ~0 X 8vTlI so that dG (v) _ F 0~ox~x 1 dv = o 5(v - v + vT - 5( - v - vT) (5.68) Similarly from Eq. 5.48, N Fo(x) = V ( (5.69) so that in Eq. 5.54, k 2v2 jrnE N k2vT vf k(vII V - 2wvL (v - kv - j.v - v ) } 70) x o II x o cV (f;~70)~~~~~~~~~~~~~I

-1355 - and from Eqo 5o 56 k2Vv2 1 J 'p c E _p__ 1'~ 2vT L2[( K kv- jv. _)2 k22 v ] kv 0(1 ) '-', Wv -)dkT r v (Ww -k )d v + 0 T _- _ x l7 VVTIx xx c0 - k'v,j:v ,O- cw. o Tii The integral in this equation is simply related to -the functions ~ 3 and 0 of Eqs0 5o18 and 5d19 (with vT -> v ) so that in particular under 4 1 T11 the conditions of Case a, wherein z i >> 1 and either IRe (zc) > 1 or Im (z ) < 0, it is found that 1W. ~~~~~~~2CE ~k2v2 j E 1E _r __ __ _ kv T_ l~ " e CT ^ - L e-T kv jv c 4[ v ) k 0 c 4[ L( j V _ j c)2 ka2V2 ] 0 C Tii (5 72) Inspection shows that if v 0 O this result is in exact agreement with the Ti hydrodynamic -theory of Chapter II and hence gives the same dispersion relationo The second term in brackets in Eqo 5~72 is the correction term due to the finite transverse temperatureo Note that zI Zc >.1 gives the following when the substitutions are made~ o ~- kv - jv J co ~, (5 73) VT I which is independent of vT o Thus if v >> vT the correction term can hi o f Tll be large, whereas if v v it can be ignoredo In general the dispersion T T11 equation is derived from the Maxwell equations as

-136 - k22 -_ 2 + J I() =, (5.74) 1' s so that using Eq. 5.72 in Eq. 5.74 the following result is found, for the case of a single carrier: w k2v2 e (wo - kv ) p T k2c2 _ a2 + p + p 0 o - 4 kv - jv v C - kv )2 - k22 ] ~~~~~o) c T11 (5.75) where any solution obtained therefrom must satisfy Eq. 5.73 and either have IRe (zc) > 1 or Im (zc) < 0. Inspection shows that this equation c c is now fifth order in k so that two additional waves have been introduced due to the finite transverse temperature. In particular, although not attempted herein, this equation can be used to study the effects of finite vT on the drifted helicon dispersion relation with significant departures from the hydrodynamic theory occurring if vT >> vT Now from Eq. 5.64 it can easily be found that kr(o - k v - C) - ki(v + k.v) Re (z ) = r —.. c (5.76) Ck2 vT and k v + k (W - c ) Im (z) = --. (5.77) vT The latter equation shows that for ki > 0 it is always true that Im (zc) < 0 if wc < 0 corresponding to the helicon or slow-cyclotron mode, etc. Thus under growing-wave conditions the RF current density expressions never contain a residue contribution for carriers with wc < 0. c

-157 - Case b: lZJc < 1 This is often referred to as the case where nonlocal conditions apply. It is first demonstrated how a major nonlocal effect, namely cyclotron resonance absorption, occurs. To make this clear assume that v = 0 and the collisions are absent, i.e., v = O. Assume that cu -> c, i.e., o C Dc > O, so that, by inspection of Eq. 5.77, Im (z ) - 0 and hence the c c residue is absent, a = O. In this case the kinetic power flow from Eq. 5.63 is C2E |E _12 rk Cw = - r c (5.78) k- 4kivTw |Ikl2 where loge[(zc + l)/(c - 1)] jr has been used since I|zc << 1. Thus P < 0 when k. < 0 and the resonant damping occurs. The carrier mode k- with cc > 0 is as a result often termed the cyclotron resonance active mode. The RF current density is now considered for the case IZcI << 1 to compare results with the hydrodynamic theory of Chapter II. This will again be done retaining the transverse thermal motion so that Eq. 5.70 is applicable. The general result can be written, using Eqs. 5.18 and 5.19 in Eq. 5.71, as k2v2 j"c2E F cu + jv T 7 J = - I — 2 + -c,, +I I ) - 2a T1 3 2[(u - kv - jv - c))2 k2 ] I (0 C T l (5.79)

-138 - where is given in Eq. 5.18. Since z I << 1 the asymptotic expansion gives loge[(z + l)/(zc - 1)] jt + 2z in 03 leading to 1j X [1E + C+ jv / ( - kv - jv - c) +-1-. (5.80) 4[(w - kv - jv - X )2 - k2v2 ] o c T The important point now is that irrespective of whether or not the residue is present the RF current density given in Eq. 5.80 is drastically altered from the conventional result of hydrodynamic theory, viz., jceE1 (o - kv) l- = ( kv- v v - X+) (.81) +. so that if |cI << 1, i.e., [c - kv - jv - | << kvT, the hydrodynamic theory should be considered invalid. Note, however, that in many investigations of solid-state plasmas (such as the helicon-phonon interaction or the electron-hole interaction73) the results obtained from the hydrodynamic dispersion equation satisfy IZcI << 1 for any reasonable carrier densities and hence these results are invalid since a nonlocal theory should be used. The hydrodynamic theory thus breaks down at the higher frequencies, although at the lower frequencies (typically less than 1 GHz) the equality |zi >> 1 is satisfied and the hydrodynamic theory is acceptable (unless v ~> v ). From Eq. 5.80, since IZcl << 1, it can be seen very roughly that the RF current density can be approximated by

.~jcu2DE rk2v 2 3 - - _ h +- 1* (5.82) 1- c L 4[(w - kv - j- C )2 - k2v2 ] o c Tii A particularly simple result is found if it is assumed that only one species of mobile charge is present with the anisotropy vT < vT In -'-I HlI this case the second term in the brackets is negligible and use of Eq. 5.82 in Eq. 5.74 provides the dispersion relation, k2C2 _ c 2 + C2 = 0 (5.83) and undamped waves are possible if a > op. For example, if w ~ >> so that k w c/c, the condition that Eq. 5.83 be valid is, since IzI << 1 is required, |kc - kv - jv - w << |kvT |, (5.84) which can be achieved if wc > O, c kc, and the collision frequency C c satisfies v < cuv /c. These conditions can be approximated by choosing a doped-degenerate material with significant mass anisotropy. For the isotropic case the first and the third terms in Eq. 5.80 cancel and the cyclotron resonance condition is reestablished in this case. Opposite to the limit v - O0 consider vT v. Equation 5.82 Ti 10 oni I II under the assumption k. << k and of course |zcI << 1 provides the following: illvT / so( that) a- n (5d85) so that a x 0 and r

-140 - doe T V l2 i - -- ( I ). (5.86) T11 Recall now from Eqs. 5.59 through 5.61 that Eq. 5.86 is just the requirement for slow waves with k. << k provided a.i is large. The part of the dispersion equation given by Eq. 5.59 becomes in the present case V 2 (k2 -_ k2)c2 _ e2 _ ( 1 + (-ac c2) C = 0 (5.87) r m p 2v iTII r where Z represents a sum over any other carrier species present. If the r latter are absent, since in Eq. 5.86 cr 0O the dispersion equation is rv 2 k2c2 - 2 = o( 2v ), (5.88) T11 which shows that the wave is significantly slowed if the plasma frequency is large and there is a large anisotropy. Since the distribution function used actually corresponds to a degenerate' material this in turn is interpreted as being a large anisotropy of the Fermi surface between the transverse and the longitudinal directions. It is then readily conceivable that with a mobile secondary carrier present the slow wave predicted by Eq. 5.88 can be amplified. For self-consistency the solution must satisfy z c << 1 which from Eq. 5.88 becomes in the present case PVTT 1o k - kvo - jv| << (5.89) 0 2c from which it is seen that the collisions wil be the basic limiting factor since large ratios of (wo /v) are difficult to achieve in solid-state plasmas.

5.3.2 The Maxwellian Distribution Function. The Maxwellian distribution function is altered to take into account the possibility of temperature anisotropy so that Eq. 5.28 becomes N (v - v2 + v f = ~3/2 exp ) exp ( ) (5.90) (2)3/2 v. vl 2v / 2v where vil and vl are the components of the thermal velocity parallel and perpendicular to x,respectively. The integration in Eq. 5.48 is performed directly, viz., N (v )2 Fo() v= t exp (_ - ), (5.91) s1j27v v 2v2 / and similarly from Eqs. 5.51 it can be determined that dG NvJ(V-v (vv - v )2 -dv exp - --— j2v / Use of Eqs. 5.91 and 5.92 in Eq. 5.54 provides the following: f =- exp - - -1 \/2P wv I (cL - kV jv- w) 2vl / (5.93) Define for the Maxwellian case cw - kv - jv - a 2 kv c, = --- - ----- c- (5.94) and compute the current density according to Eq. 5.56 using Eqs. 5.29 through 5.31 and the definiton of Eq. 5.94 to obtain

-142 - jw2eE 00 J = P 1 -1"- \ Ukv, ' -oo v2 L( - kv ) - 4- kvl (1 - -) 0 ^~~~~~~ -I z - C e-2 d~ (5.95) or jw2eE r -= H -V - (LC - kv )G(z ) 1- 0 cukv c where again G(z ) is simply related Case a: - J kvi (l -2 cG(zC) ) - V2 - VII (5.96) to the tabulated Z function by Eq. 5.31. 1z I >> 1. For this case Eq. 5.34 is applicable with z - z c If only the first term therein of the asymptotic expansion is retained, the following results: G(z ) ~ -j a exp(-z2) +. C c z c (5.97) Apply this to Eq. 5.96 assuming Im(zc) < 0 so that C = 0 to obtain J 1 - jpcEl (o( - kv ) w (w - kv - jv - ) o c (5.98) which is exactly equivalent to the result of hydrodynamic theory. This is an unusual and important result which shows that if IzI >> 1 and c Im(zc) < 0, regardless of the distribution of the thermal velocities, even if anisotropic, to a first approximation, the RF current density, Pk', and hence the dispersion relation for Maxwellian plasmas is identical to

that given by hydrodynamic theory. Including further terms from the asymptotic expansion of Eq. 5.34 will provide only a small correction. From Eq. 5.94 it can be found that k (c[ - k v - ) - k. (v + k.v ) Re(zc) -- - -- - (5.99) \2 I kl V11 and k v + k ( - ) Im (z) = r. (5.100) c \ lk2 vll In Eq. 5.54 it is seen that for the residue to contribute it is necessary that Im (z ) - 0. Inspection of Eq. 5.100 shows that this can only occur under growth conditions (ki > 0) if w > 0 and i c kiwc > (kv + kic); (5.101) or stated alternatively, the fast-cyclotron-type mode is required and if k. < k it is necessary that (Wc - c) > v. Recall that for the purely 1. r C longitudinal waves Eq. 5.55 showed that for growing waves the residue term cannot contribute. The static magnetic field provides a method for potentially retrieving this residue. Assume then that Eq. 5.101 is satisfied with k. > 0 so that from Eq. 5.34 C = 2. Also neglect anisotropic effects or heating effects by setting vl = vii in Eq. 5.96. Use of Eq. 5.97 in Eq. 5.96 then gives the following:. v - rvo) + jk.c e + Zc j 1 [k.2v- k |k|- kV v ] + j2 e +J-z ' P1- 21 0 r (IIr + W Ik2 v1 1 c (5.102)

-144 - Let a = Re (z) and b = Im (z ) so that the exponential term can be written as exp(-z2) = exp(b2 - a2) * [cos(2ab) - j sin(2ab)]. (5.105) Recall now from Eqa. 5.59 through 5.61 and the discussion thereafter that if k. < k it is necessary that the net resistive RF current be negative and 1 r the net RF reactive conductance be large and positive (to slow the wave). Inspection of Eqs. 5.102 and 5.103 shows that large RF conductivities are obtainable if b > a because of the exponential function. Under the condition Zc I >> 1, however, inspection of Eqs. 5.99 and 5.100 indicates that if k. << k then necessarily a > b and the exponential term will be negligibly small, unless (w - k v - X ) " 0 and Im (Zc) > 1. In the latter case the exponential term is large and dominates the RF current density in Eq. 5.102, so that \ 2If eE C / [k v + k. ( - )]2 j P 1 (-k o + jk.w) exp -r, c ), (5.104) ) 1kf2 V1 r c 1 2 k14 vki (514 where it has been assumed that a 0, and from previous assumptions wc > 0 and ki(wC - w) > k v. Inspection of Eq. 5.104 shows that this current density is appropriate for unstable wave growth since J < 0 and rJi > 0 and their corresponding conductivities are large due to the exponential factor. In addition the kinetic power flow in the present case may be found from Eq. 5.104 as

-145 - 2Ci EI E =2 k rc [k v + ki(CD - )]2 p - = Re(E* J ) = - exp ( k- 2k. 1- 1- EkwD 12k2 v| I k|4 v (5.105) which shows that the kinetic power flow is negative for k. > 0 and hence assists the instability. Although v is not explicitly present it has been taken into account by the assumption a 0 O, i.e., (c - k v - c ) 0. ro c Note that without the carrier drift this latter assumption cannot be made since it would violate the assumption ki (w - w) > k v. On the other 1 C r hand it is seen that v can be replaced by -v without altering any of 0 0 the conclusions reached. The assumptions again are Iz 1, D > 0, ki(C - C) > k v, k. <<k, C C 1 c r 1 r VII = v and (w- kv -o D) O 0 or a 0. (5.106) The first of the assumptions above can be relaxed to IZc| > 1 since the asymptotic expansion is no longer important when the residue is present with a large value (b - a > 1). The condition (w - krv - c ) O, together r o c with ki(wc - C) > k v, implies that the condition b > 1 is equivalent to 1 o0 k.k k r where v < 0 is required. Hence for k. < k it is necessary that o 1 r |v| > vji. If the restriction k. < k is lifted, however, the condition hat a and b >1 from E.99 and.100 yields the following that a - 0 and b > 1 from Eqs. 5.99 and 5.100 yields the following:

IVo1 > ( k i+( ); v < O, (5.108) so that it is potentially possible for unstable waves to exist with k. > k and IVo < vil. From Eq. 5.60 such large growth rates are feasible since C can be large. rAlso under the present conditions the RF current density is altered if vii # vl. In this case with G(zc) large corresponding to the presence of the residue, Eq. 5.96 gives \/TCeE r v2 Pj_ 1 - [k Ov - k (w - k v )] + jk.i - v( [k c + k.v + j(kr v - k. c)] exp(-z2). (5.109) r c i r i c c As an example of the utilization of these results it will be shown that even a carrier species with a small number density can provide large slowing factors in solids under growth conditions. To see this assume k. < k so that Eq. 5.59 is approximated by 1 r k2c2 ( ) (5.110) r E iwhere c << k c and the presence of any other carrier species in the equation is assumed to be negligible. Equation 5.107 is then applicable so that IVo| > vii corresponding to a large drift velocity either due to a large applied static electric field or to the high field regions near the contacts in any solid-state device. If the simpler case vil = vl is assumed it is found from Eq. 5.104 that

-147 - Ca = -- p --- iexp ( -r +i(- - ) (5111) - k1 V2 2 jk|4 i where Re (z ) 0 is assumed. Equation 5.111 shows that the effect of the residue in the present case can be interpreted by defining an effective number density, / [k v + ki( w C )]2 N' = N exp 2 -k14 vii (5.112) 0 0 2 |k|4 V2 Now some typical values for a semiconductor could be chosen as v 35 x 1011 s-l 1013 S -1, 2 V 3 x 107 cm/s, so that with k.i 0.1 k Eq. 5.112 becomes if C << C 2 ( l0c N' = exp 113) Thus, for an example, if k = 5 x 103 cm-1 corresponding to a wave phase velocity of 2 x 107 cm/s at c = 1011 s-1 the effective number density is approximately a factor of 1019 times the actual number density. Clearly then, even if the actual number density is small, the resonant nature of the interaction provides that such carriers are still important to the interaction, whereas from the hydrodynamic theory such carriers may be a negligible factor in the dispersion relation. Self-consistently Eq. 5.113 can be solved to find that for the present example N x 105 cm 3. This serves to indicate that under wave growth conditions the presence of even a small density of a carrier species can be important. It is also to be noted that these results are even somewhat pessimistic for the Maxwellian distribution. This is because a constant collision frequency v has been assumed to simplify the analysis. In general,

however, the collision frequency is dependent upon the carrier energy so that at any time some of the carriers in the distribution function will be colder than the mean so that their collision frequency will be less than the mean value v. For such a subset of carriers at any particular time, then, the condition given in Eq. 5.101 is relaxed. In this case, for example, even if Cc v the residue can still be large. It is suggested that this may be the case for the holes in n-InSb, where Wchl | vh is typical and Noh is small. Case b: ZI ~ 1 For this case even if the residue is present it is limited in magnitude since exp(-z ) 1 so that 2Z2 G(z ) - -jt + 2z( - 3 - (5.114) For a comparison with hydrodynamic theory consider that the system satisfies a = 0 and vji = vl. For this case Eq. 5.96 gives, retaining only the first term in the expansion, jw2E (w - kv )(- kv - jv - ) 1J ~ j o cL — 2 ----- (5.115) 1- ck2v2 which introduces the nonlocal effect of the longitudinal carrier thermal velocity. Thus for finite temperatures the slow-cyclotron mode predicted by hydrodynamic theory does not exist since these solutions have w - kv - jv - ~cI| O. This consideration does not apply for the 0 o c

-149 - forward helicon branch since Iw - kvo| j 0 and Ic >> ~ kvill so that Z I >> 1 applies. In addition the backward helicon branch which has negative group velocity is predicted by the hydrodynamic theory to be restricted to kr < cc 1/2v for positive frequencies so that if cl |w I 2 fkvlll, i.e., v > vii, this entire branch is present. If v < vii, however, only part of the backward helicon is present and the single-carrier dispersion relation breaks away from this branch at some positive frequency near the helicon maximum frequency, uax= wpv2/(4 JI | c2) and joins a root of max P o c the kinetic dispersion relation. Note in addition that, since Eq. 5.115 will lead to a fourth-order k and second-order w dispersion equation as compared with the third order c and k hydrodynamic dispersion equation, it is clear that the root structure in (cuk) space is significantly modified. From Eq. 5.115 and Maxwell's equations the single carrier kinetic dispersion relation is found as k22(W - kv )(w - kv - C - jv) k2c2 _ -- + p 0. (5.116) k2v2 where any solution must satisfy ZcI << 1 and Im (zc) < 0, and for the helicon mode wc < 0. Although not explicitly analyzed, since Iz | < 1 and J varies proportionally to z c it is clear that the effective 1- c dielectric constant remains small and the solutions to Eq. 5.116 are characterized by heavily damped modes which have k. > k i r The case of potential interest when vil = vI corresponds to the residue being present (a + 0). The conditions: for this are as discussed for the previous Case a. For this case the RF current density is given from Eqs. 5.34 and 5.96 as

-150 - 2=2cE J o- ( - kVo); TIm (z c) > 0. (5-117) J- ac^V r O. In addition the kinetic power flow associated with this carrier mode is 1 C2Eek IE 2 2P k PRe(E J - r ( k( v ) (5.118) k- 2k..1- - 12 VI, I and the unusual result is noted that it is necessary that v < c/k for this mode to be active for a convective instability. Since Iz c << 1 this necessitates wc > 0 which is in agreement with the conditions required for the presence of the residue when k. > O. This result is independent of the sign of v in that it plays no role in the sign of P. As discussed in Case a, however, it is advantageous if v < 0 corresponding 0 to a backward wave. When Pk < 0 this indicates that the real part of the RF conductivity is negative and since the net real RF conductivity must be negative this implies that the instability is assisted. The effect of significant temperature anistropy is now considered for the case Iz I << 1. Assume that the conditions are such that the C I residue is absent, i.e., C = 0, and the longitudinal and transverse thermal velocities are widely disparate so that Eqs. 5.96 and 5.101 indicate that Py. -. —.1 () (5.119) 1- X \ 3 ~~ It can readily be seen from this result that Re(E J ) = 0 and no power transfer is expected. Indeed the dispersion equation is transfer is expected. Indeed the dispersion equation is

-151 - v 2 k2C2 _ (32 + CU2 ( - ) = 0 (5.120) so that if vl > Vli and to[(v/v 2 ) - 1] > W2, purely real solutions for k are obtained. Since an approximation has been used in deriving Eq. 5.119, namely G(z ) = 0 since z I << 1, this indicates that the system is C c actually at a point of marginal stability. This has been pointed out previously77 where the present system was analyzed by a different approach. Note that if k. = 0 then the residue is indeed necessarily absent since Im (Zc) < 0. If the correction term is included as the first term in the asymptotic expansion given in Eq. 5.114, Eq. 5.96 becomes j w2E r co - kv - jv - w J p1= -1 + 0 W1- L 2V2 k2v2 v + + - ( c- kvo - jv - (5.121) \ 2 0 C and the kinetic power flow is found from this to be p = - [.E L )1- + (v - 1) (W - kv )], (5.122) 1 r is desirable to have bc 0 and (o - k v) K 0 to drive the instability. is desirable to have c < and ( - krvo < to drive the instability. At large magnetic fields the residue can appear when xc > 0 under growth conditions. The RF current density is then given from Eqs. 5.96 and 5.114 as

-152 - jW2eE v2 j \7 (w - kv - jv - H() kw) 1- ^- p V1(i ) — l-, (5.125) ) U ~ VffH \ kv11 so that the solutions are again near k. = 0 and it is necessary that >> vwC when a + 0. Inspection shows directly that the sign of Pk is directly dependent upon the sign of [1 - (v2 /vl) ](W - kv - ) and since (w - k v - cX ) 0 it should be possible to obtain growing waves r o c with either vr > vil or vl < vi1. These results show that the Maxwellian distribution function for the purely transverse waves in a static magnetic field yields a wide variety of solutions according to the kinetic theory depending upon the assumptions made regarding their location in (w,k) space and the carrier temperatures. For the kinetic theory, since the general solutions obtained involve complex functions whose values are themselves dependent upon the nature of the solutions, it is usually necessary for the purpose of analysis to assume a priori some properties of the solution in (,k) space. This leads to the formulation of a dispersion equation with a limited range of applicability so that solutions obtained therefrom must satisfy the initial assumptions made. If a convective instability is predicted it should be verified that this convective root follows a trajectory in k -k. space such that for r 1 some ci < 0 the wave number k is purely real. The difficulty is clear, however, since the act of letting c become complex starts to alter the nature of the variables which were initially delimited by a priori assumptions. The form of the dispersion equation itself can then change

-153 - as o. is varied. This makes it difficult if not impossible to follow roots analytically as w. is varied. A necessary condition can be established by solving for the dispersion relation with k real and c complex and verifying that solutions with xi. < 0 exist. Note that an assumption such as IZcl 1 for the complex n dispersion relation does not necessarily correlate with the complex k dispersion relation with Icz >> 1 and indeed it may join onto the complex k dispersion relation with zi| << 1. The degenerate distribution function need not be analyzed further since it was incorporated into the hydrodynamic distribution function of Section 5.3.1. 5.4 Kinetic Power Theorem for Hybrid Waves As was found in Chapter III the hybrid modes are difficult to analyze since both the longitudinal, and transverse motions are coupled. Thus the hybrid modes in the present case will only be studied assuming the quasi-static assumption. In addition, only the isotropic Maxwellian distribution is examined since the hybrid mode for hydrodynamic and degenerate distribution functionsis studied rigorously in Chapter VI. For the Maxwellian distribution function, N (v v)2 ( v2 + v2 \ Sf = -~-0 exp (-; -.) exp (5.124) (2VT2)3/2 2v2 2v where only the drift velocity in the direction of wave propagation is A assumed significant. Assume that B = B z and the fundamental field varies -0o 0 as exp[j(Ct - kx)] so that if E is the applied field and Eoy is the Hall field the Boltzmann equation for the present case is

-154 - af af af af,f\ o 1 [~ + (zx v x + v E+ + (V x B )] * = t ) dt x ax ' 10 v- 6o7 dv ' -i v- -I = T - - -1coll (5.125) which becomes under the quasi-static assumption af af af j() - kv )f + qr(E + v B ) - + r (E - v B ) - - + TIE 0 1 o x y o x v 1ix v x y x - )coll. (5.126) Make the following definitions: v = u cos 0 + v X 0 v = u sin 0 y (5.127) Assume that the carrier studied has high mobility and that the magnetic field is moderate so that v o E /B and JE I << Iv B. Thus any o oy0 o yox carrier heating effects are neglected. In this case when the transformation of Eq. 5.127 is made, Eq. 5.126 takes a particularly simple form given by 6f w - kv - ku cos 0 0) c o col IEix ov - x c Coll (5.128) The differential equation is solved using an integrating factor to obtain

-155 - f 1 e (c - kvo ) - ku sin 0 p0 = exp J- - ~~~~Jc exp Kj -. C.[,E x - f coll ' x ' coll-, (5.129) where C is a constant such that f (e + 2nt) = f (0). 1 1 Equation 5.127 applied to Eq. 5.124 shows that N f =~ (2vT2)3/2 ~2 ( 2 (/Vz exp 2v exp (- ) 2v2T 2V (5.150) so that af 0 x afo ( au ~~T af = cos 0 (5.131) In addition the collision integral can be neglected to first study the collisionless case. Define w - kv o a =and c c) c ku b = u c co c (5.132) so that if Eqs. 5.131 and 5.152 are used in Eq. 5.129, TE1 =f = 2 exp[j(a c - b sin o)] 2_c. * u- ' (ej + e- ' ) ~" *' -jac ' jb sin ' e e d!', (5.133)

-156 - where the replacement ejO'.-je',+ e^'.e i cos 0 =- e + has been made. The Bessel function identities given by has been made. The Bessel function identities given by (5.1534) ~jb sin ' e 00 n=-oo Jn(b ) e~jn n' c7 (5.1535) are used in Eq. 5.133 to obtain fE 1 2 0f 00m"B ja e J( e e Jp(b )J (b ) i c m c 0 ejmG' -ja 0' e (ej ' + e - ) d ' (5.136) which gives f lx f 2 1 2 af J (b )J (b ) I c m c (exp[j(m - I + 1)] exp(j(m - - 1)0] j(m - ac + 1) j(m - a - 1) C C (5.137) Now the longitudinal RF current density is given by

-157 - 00 J = q -00 v f dv dv dv x 1 x y z 00o 27 00 = q -0 0 0 -oo (u cos 0 + v )f u du de dv, v o70 i z (5.1538) where Eq. 5.127 has been used. Now from Eq. 5.124, af o au N o (2-v2)3/2 (2tvT u v2/ T u2 exp - 2vT T exp ( V2 - Z I 2v2 T (51539) which is independent of 0, so that in Eq. 5.132 N J = -(~0 1X (2Trv2)3/2 lE1x ~Emx 00,m= Jm(bc)(b) u -;- exp 2 T ( u2 (...Lu du T ( r~ exp[-v2/2v2] dvz) z T z [ 2T L 2o ( exp[j(m - ~ +1)0] m - a + 1 C m - a - 1 c * (u cos 0 + vo)dO] (5.140) The 0 integral is given by L I 2nv = [5(m - I + 1) + 5(m I - a c - I - 1)] u u 2t ( exp[j(m - ~ + 2)] + exp[j(m - I)e] 2 rn m - a +1 C + exp[j(m - I - 2)0] + exp[j(m - 2)0] ) d (5.141) ml m - a - i c

-158 - If the Bessel function identities Jm- (bc) - Jm+l(bc) J (b )+ J+ (b ) m-1 c m+i c dJ (b ) = 2 m c db c 2m 2= J (bc) b c c and (5.142) are used in Eqs. 5.141 and 5.140, then jv op eE op ix ix 4 V. T 2 R -oo PO 2Q0C c o k(l - a) Uj2 ( ku ) u I-( - ^\^ exp ( — du + 2v2 T j cEx 4 VT 00 Q=-CO 1 ( - a)2 - 1 C 00 u 0 IW2 c [~(~ - a ) - 1] k2 2 (ku C u2 \ exp 2 -- du 2v2 T c+ f cc d + 2k du foe~a [J( ku U3 exp I w j u2 2v2 T du } (5.1435) Since 00 CO 0 J2~(t) exp( XI )2 d 2% >t dui = e I () (5.144) the following is true: J 1X - v2 k(Q - a ) I=-oo T I2 c +k2 Q(i - ac) - 1 (Q - a )2 - 1 C (5.145)

-159 - From this result the kinetic power properties can be readily derived. This is reserved for application to instabilities in Chapter VI where a more complete case can be examined thoroughly. 5.5 Summary and Conclusions The concept of the kinetic power function has been applied to the basic carrier modes within the framework of kinetic theory and the results obtained compared with those of the hydrodynamic theory of Chapter II. Analysis shows that Maxwellian plasmas can exhibit resonance behavior leading to large effective dielectric constants even for small carrier densities. This effect leads to large wave slowing and consequent wave amplification if a second carrier species is present. The possibility of temperature anisotropy is taken into account leading to several possible convective instabilities associated with the carrier modes. Whenever possible corrections or improvements have been pointed out with regard to the utilization of the hydrodynamic theory, thus defining its realm of applicability and accuracy.

CHAPTER VI. KINETIC THEORY OF SOLID-STATE PLASMAS FOR PROPAGATION NORMAL TO THE STATIC MAGNETIC FIELD 6.1 Introduction When the wave-vector k is taken perpendicular to the applied static magnetic field B, in an isotropic medium with a single species of mobile charge, a mode termed the hybrid mode results with characteristics of both the longitudinal space-charge oscillations and the transverse (electromagnetic) helicon-cyclotron mode. This mode is of special interest in solid-state plasma interactions since Landau damping is absent and the Poynting vector is nonzero so that the coupling of electromagnetic radiation to this mode is possible. A useful approximation which is commonly made when the slow waves are of interest is to assume that the RF magnetic field is negligible corresponding to the transverse component of the RF electric field being much less than the longitudinal component. In this case the resulting slow wave, termed the quasi-static hybrid mode, has a purely longitudinal RF current density which is dependent solely upon the longitudinal component of RF electric field. This quasi-static assumption will always be made and understood in this work. With two species of mobile charge present the interacting quasi-static hybrid modes of each carrier lead to the hybrid-hybrid instability. For a rigorous analysis of the waves possible with k 1 B it is first necessary to accurately ascertain the form of the carrier distribution functioniin the presence of applied static magnetic and electric fields.

-161 - This is done for equilibrium Maxwellian and degenerate distribution. functions and it is shown under which conditions the often used approximation of a drifted form of the distribution function will be a reasonable solution. In addition, for the Maxwellian case, the dc electric field is more fully taken into account in the solution and the carrier distribution function which incorporates carrier heating effects has been obtained. These results are then used to study the quasi-static hybrid mode. By introducing a modification to the degenerate distribution function, as an approximation, a particularly simple solution results which is readily analyzed. Computer results of the resulting hybrid-hybrid electron-hole instability obtained from this model are applied to the phenomenon of microwave emission from InSb and related to the harmonic structure observed in experimental studies. A discussion is also given of the transition from this plasma type of instability to acoustoelectric amplification at large magnetic fields. The dispersion relation is also obtained for the degenerate and Maxwellian distribution functions. In the latter case the full dispersion relation is solved by computer analysis and it is verified that the approximation of isolated resonances in the wave structure is quite acceptable. In all cases a comparison is made of the results obtained from the present kinetic theory with those of the hydrodynamic theory. By incorporating a collision term which conserves particles properly it is found that, in essence, the hydrodynamic theory can be retrieved from the kinetic theory at large magnetic fields.

-162 - Since in these analyses the variation of the carrier collision frequency with carrier speed has been neglected by using a constant collision frequency,a study is made of the effect of including this variation in the analysis of the waves. An examination is then conducted of the electrokinetic energy and power properties of the hybrid carrier mode, and the results applied to the RF bunching characteristics of the charge carriers. A comparison with the results of hydrodynamic theory is also made here to aid in understanding the mode behavior. The effects of carrier heating on the hybrid mode for the Maxwellian distribution function are then studied. The concept of a complex temperature is critically examined by deriving the exact dispersion relation for this case. Significant departures are introduced by the dc electric field heating from the previous kinetic analyses including collisionless modes with a negative electrokinetic energy density at k < 3/v o Finally, a second mode present in the configuration with k I B f which is electromagnetic in character and termed the ordinary mode, is examined. Unlike the hydrodynamic theory for this wave, the kinetic theory exhibits resonance behavior due to the static magnetic field. Potential instabilities of this mode are then discussed including the effects of carrier heating on this mode. 6.2 Distribution Functions in Applied Static Electric and Magnetic Fields It is desired to determine the distribution function for a drifting A stream of charge carriers in a static magnetic field B = B z and general electric field -o This distribution function is given as electric field E This distribution function is given as — o

-163 - f -fO + f - (6.1) o foL (L ' (6.1) where fL is the known equilibrium distribution function in the absence of oL any external fields and f is the perturbation due to the applied fields 1L (EoB ). From Boltzmann's equation,written in the well known form.f + v f 1 + -F f ( ) (6.2) -- -- coll where F is the external force, the function fo satisfies the following in the steady state: rTl E * I+ (v x B L * 3^) = -vf, (6.3) _ v - -o' __v 1L I where oL (v x B ) = 0 (6.4) has been used corresponding to isotropic media and the term nE * (f /v) has been ignored. The latter term brings in carrier-heating effects which are negligible unless the carrier drift velocity is comparable to the carrier A thermal velocity and this case is presently neglected. Since B = B z, -o o Eq. 6.3 then becomes ( iL ~ 1L>oL B K f = vv - (.5) B (v -v 1 + f = -E oL (6.5) ~0 y v x av yn IL -o v x y which, employing the definitions

-164 - V = w Cos x v = w sin p, (6.6) y becomes 1L _f = 1 oL -f = E oL(6.7) B B 1L Bo av (6.7) By utilizing an integrating factor, Eq. 6.7 is solved as exp -cp o c D. = -— C- exp (-^c X )E *~-~ dcp (6.8) where ac = TB and the constant c - + oo. The latter arises from the condition that f be bounded and be periodic in p with period 2ir, with 1L the choice of + determined by the sign of cc* For the case where the equilibrium distribution function is Maxwellian, viz., N ( (v2 + v2 + v2) ( -.. o y z f. = --- --- exp( —^ —2 ---z- (6.9) oL (2V)32 2v2 and A A E =Eo x + E0 y, (6.10) -o ox oy Eq. 6.8 becomes ^f ~ = -vex expc) - (p') (E cos p' + E sin p')dip' fL oxv2 oxoy oBy c 6 1 o (6.11)

-165 - so that when the integration is carried out, Eq. 6.1 gives the following f = f 1 - o oL l 1 [ (-v (- vx v B__ + + v) + E ( v v )] 2 c c(6. O l( C (6.12) Note that for foL Maxwellian, the expression 00 -00 f dv dv dv = 0 1L x y z is true, and this indicates that the solution conserves particles properly. As an example of the application of this result consider n-InSb with the applied electric field Eox For moderate magnetic fields (typically 3 kG) and rectangular bar geometries, since the Hall electric field E > E and 1 C ~> v, the distribution function becomes, from Eq. 6.12, oy ox ce e f ~ foe f + oe oL vE \ x oy B v2 o Te (6.13) A The carrier drift velocity becqmes, in the x direction, o 00 -_0 E oy dv dv dv o ~Y x y z B ' o (6.14) so that Eq. 6.13 can be written as / v v \ of f(1 +Voe x, f ri f - L + ^^ ). oe oL \ v2 Te (6.15)

Consider now the displaced Maxwellian fL given by the Taylor expansion of foL(vx) as af v2 a2f f A oL oe oL oL L x oLx oL x oe oe v 2 2' x (6.16) which for ve < vTe gives / v v \ f' I fL- + oe x. oL oL \ V Te Comparison of Eq. 6.15 with this result shows that for ve < vTe.15 with this resultoe VTe (6.17) oe ~ Noe exp (v - )2 ex (2v Te 2vTe V2 + V2 2v2 Te (6.18) thus illustrating that under the conditions assumed the drifted Maxwellian distribution function of Eq. 6.18 is an excellent approximation to Eq. 6.12. In order to study the degenerate distribution function, the following transformation is needed: v x v y = w cos cp = w sin cp = v cos p sin, = v sin cp sin 0 P and v = v cos 6 z p (6.19) where 1/2 v = (V2 + V2 + V2) P X. y Z 1/2 =.(2 + 2) z (6.20) and

-167 - 00 e r! dv dv dv x y z = F v2 Jo 0Q 0. p sin 0 dv dcp de P (6.21) The equilibrium degenerate distribution function by definition is 3N -= ~if F p - vF = 0 otherwise, (6.22) where vF is the isotropic Fermi velocity. It can readily be verified that 00 f dv dv dv dv oL x y z -00 = N o A A For the case E = E x + E y, use of Eq. 6.22 in Eq. 6.8 -o ox oy provides that f iL 1 ( oL 1 P P p p B _ + 1 cu Eox [ ox v + v y - E x y oy (v + -vY) I d c (6.23) so that Eq. 6.1 gives, directly, f = f ( 0 OL vp p 1 B ( + 1 c rEo ox (_ (ui, x yi c -E ( oy x + v c (6.24) It can be shown that the drift velocity in the x direction given by 1 V o N 0 00 -00 v f dv dv dv x o x y z (6.25) becomes, when Eq. 6.24 is used,

-168 - 1 v = 0 B v2 B ( + 1 o, W c (E +C E ) oy to ox C (6.26) which is identical to that found for the Maxwellian case. From Eq. 6.22 it can be found that af oL v foL p (vp) - 5(vp - VF)) (6.27) and from Eq. 6.19, af-O foL.x afoL oL v= P x pv p (6.28) The drifted degenerate distribution function is defined by 1/2 f A f = f (v - v - v ) oL oL x x o0 3N F40v if ((Vx - V )2 + V2 + V2 o y z K VF = 0 otherwise. (6.29) A Taylor expansion similar to Eq. 6.16 shows that, if v << vF o VF f' f oL oL - vo 7 x. (6.30) so that from Eqs. 6.26 and 6.28 f I oL - 1 f + (' oL v- p P ' P ) (-E v - - E v B / V2 + ( oy x LUc oxx B — +1 0 \2 c (6.51)

-169 - A comparison of Eq. 6.31 with Eq. 6.24 shows that if IExI < IEy and ox oy 1c l > v the drifted degenerate distribution function is an excellent approximation to the general result, Eq. 6.24. 6 Effects of High Electric Fields on the Carrier Distribution Functions The case wherein the carrier drift velocity may be comparable to its thermal or Fermi velocity is now considered by including the term rEo (f /lL/v) in the analysis. When this is done Eq. 6.5 becomes r / E Of / E Of - E of B \ ox + v --- L ~y _ 1 - oz 1L 4- v P - o x 0o y o z1 -f oL = - 7E -. (6.32) Consider first the case where E and E are negligible compared to E and define the new transformation as oy x = H + u cos 8; VH Eoy/B v = u sin, (6.33) y so that Eq. 6.32 transforms as 1L v oL ~e c L fL H=; (6.34) c y therefore, fm L V = exp wucfOL iL c expH c The equilibrium Maxwellian di' stribution, (6.35)9 provides that The equilibrium Maxwellian distribution, Eq. 6.9, provides that

-170 - foL y v - -Y f 2 oL T2 T N 0 (2irv2 (vT u sin 0 )3/2 v2 VT * exp -- / V2 x 22V T uvH exp - - 2 VT exp uQ ) 2VT cos e); (6 J.36) thus, since UVH exp - 2H V2 T 00 cos ) = n=-oo UVH) Jn n 2 vT (6.37) the following is true: foL av y = g (u) sin 0 00 n=-oo I -UVH n a 2 T (6,38) where go(u) N o (2 -v2)3/2 2 u H ) - exp 2-v V 2v2 / T T exp -u ) T / v2 z_ exp - 2v2/ T (6.39) Equation 6.35 is then readily solved as f (e) 1L 00 n=-oo / uvH I - - n v V2 / T [(jn - v/cu)sin 0 - cos e] jn e 1 + (jn- v/co)2 (6.40) where the constant c has been taken into account via the requirement fiL(0) = fl(0 + 2t).

-171 -The distribution function in the presence of the applied fields is then given from Eqs. 6.1, 6.9 and 6.40 as 2 2 +V2 ee No u + vH + v f f = exp - -- z N 0 2, e23/2 x 2V2 (2:tvT T n=-oo UVH ) jnG I (- - ) ejn n v2 T *1 uv V2 T [(jn - v/c))sin 0 - cos e] 1 + (jn- v/c)2, (6.41) which, if desired, can be transformed back to the original (Vx, y, Vz) reference frame from Eq. 6.33. Inspection of Eq. 6.41 shows that to conserve particles properly it is necessary that 00 00 — N exp Yv 2t c 0 = ~ I f dv dv dv = - H exp(_ -). 1 -00 1L x y z 2vT T 2 n=o0 l+(jn-v/c)2 — cO n =-o u2 exp (- In - H ) [(jn - v//c)sin 0 - cos O]ene du de, (6.42) ' 2v v and the integral on the right-hand side is found as co 1 2t co n- -+ (jn-v/)2 Jo n=-1 +(jn- V/Wc)2 - n=-m~~~~~ ( U2 ( n vH u2 ex^p 2)IAvT T T (exp[j(n + 1)0] [-j(jn - v/oc) - 1] + exp[j(n - 1)0] [j(jn - v/.c) - 1 ]) du d( O:0 C 2 (U2 H \ c u ex.p - I: VI (-j) - u -1 ~v ~ 2v v T T UVV vT \ 2 T = 0 (6.43)

-172 - wherein use was made of the following: I n+l(x) - I (x) nf~i n-i' 2n = - -- I (x) x n (6.44) The Thus the solution given by Eq. 6.41 conserves particles properly. drit velocity in the xdirection is found as drift velocity in the x direction is found as 00 N JJ -00o v f dv dv dv x o x y z = H + v0 (6.45) where ve vH 2rvT T 2 exp( ) 2v2 / T 00 n=-oo 1 jn- rv 00 1 + (jn - v/CJC)2 0 0 u3 exp (-_ ) 2v2 T UVH v2 v [(jn - v/co )sin 0 cos 0 - cos2 0 ejn du dO,: H 2v4 T 2 exp ( v- 2 ) T 00 0 u3 1 + v2/W2 c exp u22 T r ( U'H)U T UVV -I (I du ) 2 2 T 2vH 1 + v2/W2 c v2 exp ( H' 22T M 2;l v2 H - T (6.46) where M is the confluent hypergeometric function and vH > 0 has been assumed. For example, if E oy 1 VH = B VT C - M(2; 1; 0.1) = 2.71828 and

-173 - vo H( ) (6.47) 0 H 1 + V /2 cI If VH << vT,then it is clear that the n = 0 term will dominate in Eq. 6.41 since for small arguments II0 | > I | II,I II,.... Furthermore if |Ioc >> v, Eq. 6.41 is approximated by f f 1+ H, (6.48) o oL 2 vT which is identical to the result found previously, Eq. 6.13, which led to the drifted Maxwellian form. 6.4 The Quasi-Static Hybrid Mode: General Solution Assume that the quasi-static case is valid so that in the effective dielectric constant E = I+ jc X s ' (6.49) s where I is the unit matrix and a is the conductivity tensor of the sth carrier species, the element E is much larger than any remaining.. xx. j where a variation as exp[j(wt - kx)] is taken for the RF fields. The linearized Boltzmann equation, with B = B z, is then given for each -o o carrier species s by j(c - kv )f + Ti B (v y s - v i ) + s E + E ss Xs is s o \ ys VXs X Vys S \ OX Xs oY aVys / xs is s N os Xs O( N - r xs V Fns,o v (6.50)

-174 - where E = 0 has been assumed and the collision term which conserves oz particles properly has been taken on the right-hand side, where N is is the RF number density, 00 N = f dv dv dv. (6.51) is is xs ys zs -00 To obtain a tractable result it is now assumed that |cs | >> v, the Hall field Eo is larger in magnitude than the applied field E (or Eo is the applied field and is larger in magnitude than E ), and the carrier ox drift velocity is sufficiently small compared to the carrier thermal velocity or Fermi velocity. Then, as discussed in Section 6.2 for either the Maxwellian or degenerate equilibrium distribution function, v E E /Bo os oy The following transformation is then convenient: v = v + u cos xs OS S v = u sin, (6.52) ys s since Eq. 6.50 can now be written as af rl E / f v N is O ) S j5 Os s is -ae -j(as - b cos =)f f' (6.55) _ s ( s o 9ls Wcs - xs cs N os where o - kv -jv ku a = - 05. and b =. (6.54) S do S dO CS Cs As discussed in Section 6.2, in the present case, the drifted distribution functions given by Eq. 6.18 for the Maxwellian case and Eq. 6.29 for the

.175 -degenerate case can be used for f ~ In both instances, inspection of Eqo 6~52 indicates that f is independent of 0 so that os aSf a au OS OS S A os _ Os s xavu s v Jx Xss xs O s = Cos u 5 (6-55) Equation 6053 is solved employing an integrating factor together with Eqo 6035 to obtain f (0) exp[-j(a 0 - b sin 0)] 0 exp[-j(a 0' b5 sin 0 ')]dO' s s s os v N 0 CD N os s, CS Os ' os cos 0e exp [- j(a0' - b sin l0)]dO s s, (6o56) where c is determined from the condition f: (E + 2 ) =- f (9)o The Bessel function relations of Eqo 5o135 enable the solution of Eqo 6o56 to be written as follows fs (0) is 00 00 m=-oo n=-co n E /of [<j (a n)cos 9 + sin 0] m ( (b) n ( s s dlx f m s n s - i.cs u S 1 - (a - n)2 s S IS OS v f:f + j 1 -5s - exp j(n - m)o0 cs sn s7 I 0 (6.57) The relations dv dv dv = u dv dG dv xs ys zs s s zs (6 58) and

-176 - 2m Jm (b ) + J (b ) = Jm(b ) (6.59) m-1 s m+j s b r s s are now used in Eqs. 6.51 and 6.57 to obtain 10 OS 2om(bos mn s -b 7j Os (m - as du dv k Lu J m - a s zs -oo s s m=-oo N C5 p00,s 1 _ _v 7 _ 1 - J - - / J(bs)fo u du dv ^~~m-a, os ss s - zs m=- (6.60) At this point, prior to a study of the degenerate and Maxwellian distribution functions, a simple approximation to the degenerate distribution function is made. This will simplify the integrations involved while retaining the essential physics of the interaction. 6.4.1 The "Cylindrical" Degenerate Distribution Function. The "cylindrical" degenerate distribution function is now defined by N Os < < f -- if u = vF and | v | v o s n s Fs Zs F 2~vF aI Fi Fs = 0 otherwise, (6.61) where vF is the Fermi speed parallel to B. This is then similar to the degenerate distribution function studied previously which is given by Eqs. 6.29 and 6.52 as 35N 1/2 < f = if (u2 + v2) v OS 3 s S ZSFs 4~vFs = 0 otherwise. (6.62)

-177 - From the viewpoint of achieving instability it is expected that the function of Eq. 6061 is actually more pessimistic than that of Eq. 6.62. The distribution function of Eq. 6.61 used in Eq. 6.60 yields N is ( kvS F j2 N E cm Csm Cs voow- kv jv - mw kv2 c Cs j s s FS ----- ^-"~^ --- — o (6.65s) F s S6,, k )2 FS FS l s w jo- -kv - jv - mWo m= -00o where kvFs m w cs ku dJ s m (i cs / ku u =v d s =Fs Cs (6.64) and in obtaining Eq. 6.63 use has been made of mJ2(0) = 0 m for m integer (6.65) and J2(kx) x m dx = (x2 2 k2 m 2 m (6.66) Note that in the present case the Fermi velocity parallel to no role in the quasi-static case. From Poisson's equation, Bo vF plays 0 F1 -9 P

-178 - V ~ E = -jkE = -- N (6.67) -1 lx E lS S and the dispersion relation is found from Eq. 6.63 as (2~ 2w2~ \ mm (kvFs) ps V cs O2V2 w - kv - raV -wncs 1 = ~ Fs m=- (v, (6.68) v+ vk -iv Fs S L kv, -os j -vs m)Cs mn=-oo where Qm(kvFs/%cs) is defined directly from Eq. 6.63. The term in Q is due directly to the conservation of particles aspect of the collision term defined in Eq. 6.50. Thus if the collision term (f/)t)coll = -vfs had been used this would correspond to Q = 0. The physical effect of imposing particle conservation is seen if it is assumed that a single resonance at m = n gives the major contribution to the sums in Eq. 6,68 for some carrier "s" in which case the dispersion relation is 2ps csn (n s ) terms due to 1 = -------— + ( other rk~'vs { r kv -n w5\c ~]~ [1 ^t CSFS k carrier species kz {a - kv - nw^ - jvs - < ( ) ] } (6.69) Thus the modification introduced by the conservation of particles can be taken into account by replacing the collision frequency vs by an effective collision frequency vs, where s

-179 - n2(12 kv k I2 v ]Cs j2 Fs + ]vFs. (6.70) s s -2V n W n W c SFs Moreover, from Eq. 6.60, the more general statement for the distribution function f is, when a single resonance is assumed as shown above, OS 00 00 ku VI= _v i _ ( ) fu du dv], (6.71) Vs N J n Os s Zs C wherein it is assumed that fOs is independent of e since Eq. 6.55 has been used to obtain Eq. 6.60. This latter condition is true for the drifted Maxwellian distribution, Eq. 6.18, and the drifted degenerate distribution, Eq. 6.62, and of course the cylindrical distribution, Eq. 6.61. Hence for v < vT for the Maxwellian case and v < vF for the degenerate Os Ts os Fs case the respective distribution function is independent of 0. When k is real and fos is independent of 0 it can be shown that v', the effective os s~ collision frequency, is necessarily positive or zero. For k real, it follows that [Jo(kvFs/cs )1 1n and Jn(kvFs/Ocs )I 1/ l/f, so it must be that I't A 00 oo 0 kv \ 00 00 r kFA( s /- f u du dv f u du dv J J n Os S S Zs J Q Os s s Zs Therefore, if f is independent of 0, since Os 2jt 00 00 N = f u du de dv, Os Os s zs — oo

-180 - then the following is true: N ' 0 (6.72) Use of this result in Eq. 6.71 then provides that v' 0. Note that only s in the case n = 0 and (k/oc ) ->0 does VT ->. Cs s To investigate the properties of the theoretical model, consideration is given to high-field instabilities in n-InSb since experimental data is available for this material. The carrier resonances are well defined if Ic s| ~>> which is readily satisfied by the electrons but not by the holes. Nevertheless for the purposes of analysis the resonances will be separated out even for the holes. In addition the effective collision frequency defined in Eq. 6.70 can now be used. The dispersion equation then is written from Eq. 6.69 as 2 kve) (kvFh) j2 F2 j2 Fh mI 1 1=. pe ce + ph o ch 1 =-_ ( + ( ) m k2 - kv - j + k2v2 C - j' - m|ch| Fe oe e ce Fh h i (6.73) which describes the interaction of the fundamental electron slow-cyclotron mode with the mth harmonic of the hole cyclotron wave. The hole drift velocity is assumed negligible and the effect of the Hall electric field on the hole dispersion properties is neglected. The solution for real k is 2o0 = j(v' + v') + moh 6 (1 + ) + kve ce (+ ) (6.) e Vh ch (1+ e)+ chi, (6 74) e ~~~~~~~oe where

-181 - R = (j(v - vI) + kv - m c Ih (1 + h) - (1 + 0 )) e h oe ch h ce e - 4ml 0h | cel e0 h I chl 1w ce e h (6.75) and 2cU2 kv \ -=. pe 2 ( Fe e k22F 1 ce Fe 2w2h ^kv F > 0 2ph m2 Fh h k2vFh ch Fh (6.76) For the possibly growing solutions (WC. < 0) of interest, where o = 0r + jwi, r to the following: Eq. 6.74 simplifies, if VI v1, to the following: = koe +m|ch (1 + Oh) - Ice (1 + 0e) r oe 'c 1v h ce' (6.77) and Vt + V' r / 2 1/2 _ e v +Im ~l -2 _ h [- ( - mlchI (1 + 0h) + mI chl 1c eh] (6.78) Inspection of this result shows that growth should occur near the hole cyclotron harmonics in frequency provided that 2 kvFe >(kvFh 4m|<ohl IOel ju ~ 2 J2~ chm%' ce pe ph 1 ( ce m ch k4v2 V2 Fe Fh ( Ve + vh) > 4 (6.79) However, because of the strong dependence of the growth factor on the wave number, a solution for k from Eq. 6.77 indicates that, for moderate magnetic field strengths (Bo 0 3kG), larger growth rates may be attained

-182 - for 2co - m|.ochl, provided that oce |o ch|. Exact computer solutions r ch ce ch of Eq. 6.73 have shown this to be the case in that the emissive peaks are shifted to magnetic field valueslarger than those corresponding to the hole cyclotron harmonics. The amount of the shift increases with a decrease in carrier density. Thus in Fig. 6.1 with equilibrium electron and hole densities N = P = 4 x 1015 cm 3 the shift from the fundamental resonance for a chosen frequency crJ = 2t x 9.4 GHz (corresponding to the experimental value78) is approximately 600 G, whereas in Fig. 6.2 with N = P = 1016 cmT3 the shift is only 100 G. Other parameters used in these computations are as follows: m = 0.03 m, e 0o mh = 0.61m, mh o0 Fe = 8 x 107 cm/s, vFh = 4 x 107 cm/s This carrier density range is selected in accordance with the current density values attained in the experimental work78 under the stipulation that the drift velocity never exceeds the thermal velocity and the fact that the harmonic radiation occurs after the onset of impact ionization. 78 The theoretical results obtained in Fig. 6.1 are in general agreement with the experimental data reproduced in Fig. 6.3 with the following characteristics: 1. The quantitative position of the emissive peaks and their relation to the hole cyclotron harmonics. 2. The shift at low-current densities to higher values of magnetic field corresponding to a decrease in carrier density at low-field values.

z / / / - m=3/ m=2.m=l/ Ir 0 0.5 1.0.5 2.0 2.53.0 3.5 0 0.5 1. 1.5MAGNETIC FIELD, kG,FIG. 6.1 GROWTH RATE AS A FUNCTION OF DRIFT VELOCITY AND APPLIED MAGNETIC FIELD.

I) Iz Ir cr FQ: 'El a: 0 n: LI) Voe= 4 x07 cm/s 16 -3 N=P=10 cm m=3 m=2 I I I I I I I I I I / m=2 m=l I oD I / I \ "I I \ -..- I I I 0.2 0.8 1.4 2.0 2.6 3.2 3.8 4.4 MAGNETIC FIELD, kG FIG. 6.2 GRCWTH RATE AS A FUNCTION OF MAGNETIC FIELD.

MAGNETIC FIELD -(KG) -2 t "' D m <1 I \-' I Q. CL 20 I O 0 I 2 3 4 MAGNETIC FIELD (KG) FIG. 6. MICROWAVE POWER AS A FUNCTION OF THE APPLIED CURRENT AND MAGNETIC FIELD. (MORISAKI AND INUISHI78)

-186 - 3. The independence of the position of the emission peaks on the drift velocity. The fact that the linear theory presented here predicts higher growth rates at the m = 3 resonance than at the m = 2 or m = 1 resonances in contradiction to the experimental results may be due to any of several factors. The theory presented here does not take into account explicitly any carrier generation or recombination phenomena, whereas the experimental current densities correspond to the post-impact-ionization regime. In addition the m = 3 resonance corresponds to a larger wavelength than the m = 1 resonance and hence the experimental coupling of radiation will differ in these cases. Also in the theoretical work the effects of carrier heating have been neglected in developing the carrier distribution functions in Section 6.2. Of course there are also the possibilities of boundary effects and nonlinearities in any actual system. Also from the experimental data of Fig. 6.3 it appears that a different and stronger form of instability sets in when the transverse magnetic field becomes sufficiently large (B ~ 3 kG). There is ample evidence73'79 to conclude that this region corresponds to (electron-phonon) acoustoelectric amplification enhanced by the transverse magnetic field. Theoretically, 73 a large transverse magnetic field is necessary in high-mobility semiconductors such as InSb in order to lower the electron drift velocity to a value which maximizes the gain. Experimentally,79 transverse magnetic fields have been found to enhance greatly the acoustoelectric effect in the III-V semiconductors InSb, GaSb and GaAs. These experiments showed that the formation of high-resistance domains

can be induced by applying a sufficiently large transverse Bo, and in addition, from a qualitative comparison of the results of different materials, that for a given Bo the enhancement is more pronounced as the mobility increases. Also whereas the resonance-type emission only 78 occurred above the breakdown field for impact avalanche ionization in agreement with the present kinetic theory (i.e., below breakdown in n-InSb the hole plasma frequency in Eq. 6.79 is negligible thus preventing instability), the emission with B > 3 kG can occur below the impact 0 ionization field in agreement with the theory of the acoustoelectric effect in high-mobility materials.73 6.4.2 Correlation of Kinetic and Hydrodynamic Theory for the "Cylindrical" Degenerate Distribution Function. It is of interest to determine if the present results can be correlated in any way with the corresponding hydrodynamic theory for the quasi-static hybrid mode. Inspection of the denominator of Eq. 6.63 shows that if Ocs >> kvFs the primary contribution is from the m = 0 term. In addition the summation in the numerator may be written as c2( j k2( )Fs ) 2m2c2 J2 ( s s sm s ) -s _ 0 5m.C (6.80) - oS - - c ( - kv - v2 - a os s Cs M= 0 c M= (w-k s i Cs For |I Csl kvFs then Eq. 6.63 may be written as follows:

kv F - 00~ _ o2m2a? j2 ( Fs kv -jv j2nsO N SEix cs m xCDs os s Ns os ix cs 15 kv2 r r kv Fs 'm= [((-KVo -jV )2-m2 <s -kv s-js[l ( cs )j C S (6.81) From the asymptotic expansion for small argument, viz., 4F Z m ) Jm(Z) m ) = ( ( ), (6.82) it follows that J2 1 and J2(kVF /cs) k2v2 /4wcs2 Use of these latter o z Fs cs Fs Cs approximations in Eq. 6.81 together with Eq. 6.67 permits the dispersion equation to be found for the fundamental (m = 1) as w2 (C - kv - j ) 1 =.ps s + R, (6.83) S (- kVos)[( - kvoS -jV)2_ -2S] where R represents the contribution from any carrier species which do not s satisfy |cs| >> kvFs and hence have a different form. Now the hybrid quasi-static dispersion equation is given by73 02 (O - kv- jv ) ^1 C y _________^ " ops os' 1=7: 1ky (co- ky - j _ (o - kv )[(o - kvO jv )2 - ] - k2v T - kv - jvs) (6.84) where vTs is the thermal velocity. Comparison of these results shows good agreement and in fact becomes exact when vTs 0. Thus if I csI > kvFs Ts -CsFs

-189 - it is quite clear that the hydrodynamic analysis (for that carrier) is accurate although the kinetic theory suggests the dropping of the thermal term in the denominator of Eq. 6.84 entirely. As a result the utilization of the hydrodynamic analysis in the study of the acoustoelectric interaction in high mobility media is justified for the case of a large transverse magnetic field such that |oc >> kvT. For smaller magnetic fields, however, the kinetic theory raises serious questions about the legitimacy of the hydrodynamic analysis. In addition it is to be noted that in multiple carrier interactions, such as the electron-hole hybrid interaction, it can easily occur that the condition I|c| > kvTs for the one carrier species (electrons) is well satisfied but not for the other (holes). In this case a hydrodynamic analysis with direct substitutions in Eq. 6.84 is of dubious validity, whereas the kinetic theory incorporates departures from the |cs >> kvTs format in the term R in Eq. 6.83. Also, since the hydrodynamic and kinetic theories depart at smaller magnetic fields, it is reasonably certain that the resonance emission for B < 3 kG obtained in n-InSb from 0 the kinetic theory will not be predicted by the hydrodynamic theory. 6.4.3 The Drifted Degenerate Distribution Function. Reconsider now the drifted degenerate distribution function of Eq. 6.62 which was found to be a good approximation to the general form, Eq. 6.24, when v < vFs and 10cs >> v. Identify the integrals of interest in Eq. 6.42 by

-190 - rooroo 00 0oo 1 -'0 _-00 and / f ( 0 j2(b ) du dv (uS m s s zs J2(b )f u du dv m s os s s zs (6.85) 00 00 2 0 -_o (6.86) where it suffices to have m take any positive integer or zero value since the summation can be reduced as was done for example in Eq. 6.80. Define a new transformation, u cos 0 = vr cos e sin 0cp, u sin 6 = v sin ( sin cp s rs v = v cos cp zs rs (6.87) so that u = v sin cp s rs (r c o 0) and cc cc2 i 0 0 ~~-00 P 00 i2T I it u du de dv / v2 sin cp dv de dcp s s s J rs rs ~~~~~~ s (6.88) It can be shown that the angles e and 0 are equivalent. The distribution function, Eq. 6.62, transforms as 3N f = if os 4 3Ps - 0 othe < v = vF rs Fs rwise, (6.89)

-191 - so that af af / av _ / 35N os = os ( rs = sin cp (v ) - 5(v - vF)). (6.90) 6u = v \hs dua n in *s \8 vn r7 pv r s dSu 3 -s rs r e ions 47vFs Use of these relations in Eqs. 6.85 and 6.86 provides the following: 3Nos /2 - - _ o Fs (kvsin sin cp j2 s incp dcp 'cs (6.91) and 3N P t/2 V kv 2 -v / 0 s v2 in )( sin cp dcp dv Fs2 2 0 rs m Ccs V~S 1, (6.92) Corresponding directly to these integrals the cylindrical degenerate distribution function, Eq. 6.61, gave the values (e.g., from Eqs. 6.60, 6.63, 6.85 and 6.86) I sN kvFs \ cT = - e I2 ( 1 1 2 m \ O vFs cs (6.93) and N r / D'F = 0 2 2rtv2 L S Fs mm e kv > F kvF k_ 2cKk +vs J} ' ^2 Y ~m w cu^ ^ Fs Lm wc (6.94) (6.94) where the cylindrical 0 functions will be primed. Although Eqs. 6.91 and 6.92 can be directly integrated using the relations

-192 - r /2 -2J(o) + J m~~~ J2(z sin 0) sin 0 de = m 00 n=o J + (z), 2m+2n+l J [Re(m) > -1] (6.95) and Fs x2J (ocx) dx = 0 V2 ar r( m + 3) 2 r(m l) (m + 21 + l)r (m - 1+ ''2 r (m + 5+ J+2 I +1( ) " (6.96) where r is the gamma function, r(n + 1) = n., the results are too lengthy to be detailed here. By inspection of Eq. 6.91, however, it can be seen in general that l(-m) = (l(m) and necessarily 1 0. Similarly, from Eq. 6.92, 0 (-m) = ~ (m) and 0 = 0. For simplicity the 0 integrals will 2 2 2 only be investigated for the fundamental interactions Iml = 1 and m = 0. From Eqs. 6.91 and 6.95, and the relation sin z = 2Jl(z) - 2J (z) + 25 (z) - 2J (z) +... 3 5 7 (6.97) it can be found that 2nv [ 2 - ( cs ~~Fs ~ cs 7 %as J skv 2J cs 3 kvFs Cs 2J ( Fs) 0 11 cs * (6.98)

-193 - Similarly from Eqs. 6.92, 6.95, 6.96 and 6.97, s (m = 0) 2 Fn s vs 3C 3 k3 kvFs s Ct CULsin ( Fs "Is k2 v2 Fs Cs o kv' s Cs CS -1 j- 2 j csFs r(2n + 2)r(2n +,) n - L k r(2n)r(2n + + ) 4n+2=+2 n, =o (" CFs (6.99) and (Iml = 1) 2 3N - OS v s {2 00 n, Q=o cs Fs r(2n + 2)r(2n k r(2n)r(2n + 3 + ~I (4n + 21 + 2) + ~ ( cs (Fc 3 CS k3 r kv \ L Fs Cs sin (..) Cs k2v2 Fs 2 CS 1) ( k7Fs co s CS > 1> csFs 2( + 1) ( Jc =ok r(5 + =) ) 2 I =0 (6.100) Corresponding to these, the cylindrical degenerate distribution function gives, from Eq. 6.93, '( Iml = 1) 1 OS o2 1a ~ ~'%Fs St_ K C and from Eq. 6.75, ('(m = 0) 2 = 2S 2 k( S ) j2 2t L ~ cs 1 (Fs ) Cs

-194 - N r / kv r/ kv 2 '(Iml =l) = 0...v2 -k>(s. K v J ( ) 2 2v2 os Cs j2 + Fs + 2 \ Fs (6.101) From the point of view of the dispersion relation, the only difference between the cylindrical and the drifted degenerate functions is contained in the respective c and D integrals. 1 2 By going to the limit (kv /cs ) - 0 it can be found that 0 (m = 0) = 0'(m = 0) >~ c,O'. Thus the hydrodynamic form found for 2 2 1 1 the cylindrical degenerate distribution function, Eq. 6.83, also exists for the drifted degenerate case. For the resonant interaction, if the replacements 2 2 2 e0 = ^ (V_ ) (ImI =1) s 22 No 1 k2v2 Ns Fs and V = V (1 - 2 (Iml =1)) - (6.102) S S N 2 Os corresponding to Eqs. 6.98 and 6.100 are made in Eqs. 6.74 through 6.78, the solution for the drifted degenerate interaction is obtained directly. The results for the drifted and cylindrical degenerate systems are then quite similar and differ only in the magnitude of their respective z functions. Since these functions involve infinite sums for the drifted degenerate case these are not dealt with directly. This fact points out the utility of the cylindrical function which is much more readily analyzed yet is similar in instability characteristics to the drifted degenerate function.

-195 - 6.4.4 The Hybrid Dispersion Relation for Maxwellian Carriers. The quasi-static RF current density, Jlx for the hybrid mode with a drifted Maxwellian carrier velocity distribution function was developed in Chapter V, Section 5-5. This will be used later in the discussion of the electrokinetic energy density and kinetic power flow of the carrier modes. To develop the dispersion equation, Eq. 6.52 used in Eq. 6.18 shows that the drifted Maxwellian distribution function if IVos << vTs takes the form N u2 / v2 ef <= -0 exp ( _exp) (6.103) Os (2qv2 )3 /2 2J J22 Ts Ts Ts so that Eqs. 6.85 and 6.86 provide the following: 1. N ~ s-\ = (- = os e I () (6.104) a>1 v2 2 2rk2v4 Sm s Ts T where s = — kvT- (6.105) cs I (s ) is the modified Bessel function of order m, and the standard integral m s exp (_ ) J(L) d = x e Im() (6.0o6) 0 xp -2 m m has been used. From Eqs. 6.60, 6.85 and 6.86 the following general result is found:

-196 -00 jSEx x, 2rrm S Nk m - N J= _ ---- = ---Oc, (6.107) so that using Eq. 6.104 in Eq. 6.107 provides that jri N E C -\ S OS1Xn ) m e sI ( ) kv2 L m s m s N Ts m=-O. (6.108) 1is 00~ms,s 1 - es 1 - CD m a m( S) cs s m=-oo Use of this result in Eq. 6.67 gives the dispersion relation as 2 00 K PS _ m e S I(x) p\ V R c2 m - a m s m=-oo -\ In the limit that (kvTs/cs) -*0 the behavior of the function e I(\s) Ts/SCS m- S shows that m = 0 provides the dominant contribution in the denominator, so that when only the fundamental (|m| = 1) is retained in the numerator, Eq.. 6.109 becomes -\ 2 S) - T c _ j - ( - s )2] s ( Tt 2s (c-Vs - s -..)(6.110 ) k2V2 [C~ s (W - kv js] - kvo) S Ts Cs Os S where

-197 - m s 2m s e I()s ) =.- e I ( ) (6.111) -x m e s ) - sa( (6.112) 1behavior of e I = 0 or (6.114)small 5 s O M S~ S. m e sI )() [ - (6.113) 1 2 leading to the following result in Eq. 6.104: hydrodynamic theory is also appli cableO. This hydrodynamic limit corresponds 1 -+ ------ E^ ---- ^ ----- = 0. (6.114) (w - kv )[o)2 - (a) - kv v )2] OS CS OS S This result is identical to that found for the cylindrical degenerate distribution function under the same limit, Eq. 6.83, and is quite similar to the result of hydrodynamic theory, Eq. 6.84. Thus for the drifted Maxwellian the discussion following Eq. 6.84 comparing kinetic and hydrodynamic theory is also applicable. This hydrodynamic limit corresponds to retaining only the m = 0 term in the denominator of Eq. 6.109. If the |ml = 1 term is also retained, and having used the method of Eq. 6.111, the denominator becomes as follows (for small X so that Eqs. 6.112 and 6.113 can be used): v v k2v2 S S Ts D = + j _. (6.115) u - kv os % s cs W 2 - (w - kv - jv)2 CS OSs

-198 - When this improved form is used Eq. 6.126 becomes C2 (Lw - kv - js) 1+.......PS Os SV VS s ( - kvO )[c2 - ( - kvo - jv )2] - j k2v2T ( kv - jVS) CS = o. (6.116) This result should be compared with that of the hydrodynamic theory, Eq. 6.84. This also suggests returning to the cylindrical distribution function of Chapter V, Section 5.3.1 and retaining the Iml = 1 term in the denominator under IocsI >> kvFs. When this is done the following result is found, which is an improvement of Eq. 6.83: 2 (w-kv- -j ) vI C~~~ pS OS, S v k2v2 k2 2v2 0O OS~ 5 c s + s S CsFs s s (^^-kvo)[((f-,)k]os-+ (j 2), 4Coz cs (6.117) which also indicates divergence from the hydrodynamic result. The case is now considered where the resonance approximation can be made so that only the resonance terms contribute in the Z. The dispersion m relation then takes the following form for the nth resonance: -\ X cw2 n e I ( ) 1 + )Cs,ps ns = o, (6.118) k2v2 [nCo - ((a- kv - jv')] s Ts cs os s where v' is the effective collision frequency given by S

-199 - V = V (1 - s e (sX) Vs s n (6.119) For example, for n = 1 the maximum effect occurs when \. 1.5 in which case v' t 0.8 v. Thus the general statement can be made that v' is never altered appreciably from v for n = 1 (20 percent) and for larger n values this is even more true; for n = 2 the maximum is - 12 percent, for n = 3 the maximum is 8 percent, etc. From Eq. 6.118the dispersion equation f the electron fundamental interacting with the hole harmonic modes is %1w Ir 1 + ce e - kv - jv + w I oe e ce m chi u - jv' - h rh mi ch ch = 0, (6.120) where C)2 -_ e k2v2 1 e Te (6.121) and,p2 -' ( = h e Im (X) k2vT Th (6.122) so that re, h > O. The solution is identical to that given in Eqs. 6.74 through 6.76 with the replacement es - P s s = e,h. For self-consistency it should be verified that the resonance approximation is valid. First it is demonstrated that the term corresponding to m = 0 in the denominator of Eq. 6.109 has a negligible effect on the dispersion equation. The solution given by Eq. 6.74 for cO. < 0 indicates that kv |c for reasonable carrier densities so that \ > 1 since oe ce e voe vTe For X > 1 I (e) I e) so that defining the following as Te -e o e i e

-200 - 1 s_ D = - e Im( ); s = e,h, (6.123) s,m m a m s S it follows that ID l >> ID I, since e,1 e,o - kv jve + IceI > Ic - kv - jv I (6.124) o e ce oe e corresponding to IceI >> v. Thus the m = 0 term should have a negligible ce e effect in the electron term. For the holes e-kh W~chIo(%h) \chIm(%h) h, o h, m () - jv - ) - jV +v h - h h and since e > 1 implies that h > 1, it is true that Io(\h) s I (h). Thus for m = 1 and C << vh the right-hand side of Eq. 6.125 is approximately zero and hence, by inspection of Eq. 6.109, the resonance approximation is valid although vt should be replaced by vh. The approximation I (\h) Im(kh) becomes successively weaker for higher m values. However near Wo = IwchD m inspection of Eq. 6.125 shows that the effects should be negligible and this is the region of interest. It should also be verified that the harmonic terms can be isolated for analysis. Since v' = v and v' = vh the effects due to the sum in the e e h h denominator of Eq. 6.121 can be neglected and the denominator set to unity. From Eq. 6.123 the dispersion equation, Eq. 6.121, becomes -k s 2C/ 2 P s In(Xs)n2w2 1 = ( 2 ) e ncs -, (6.126) s n=l k s 05 s cs which can be written in the integrated form, viz.,

-201 - r r" nkV - iv Cos x o s s,2 S sin x sin x —. - j e Z / (jo \ -\ prt L \ -1 /C ps s cs 1 = _- X (-.- ) e A Cs dx. /w - kv "o s Cs s in. r - vos jvs CS (6.127) In particular since this result is amenable to complex k, Eq. 6.127 will be solved for real c and complex k. Thus, since c.i < 0 is not necessarily sufficient for instability, if amplifying roots are obtained it can be stated with good certainty that a causal instability exists. In addition, if the harmonic modes are clearly separated in the solution, the use of the resonance approximation as an analytically acceptable technique will be verified. Equation 6.127 is solved via computer for the following parameter values: N = N = 105 cm-3 oe oh e = Vh = 4 x 101l s1 m = 0.03 m =0.6m, e o Mo m B = 2 kG Te = 6 x 107 cm/s, vTh = 4 x 107 cm/s, v = 107 cm/s, voh = 104 cm/s, (6.128) which are again in the range of values corresponding to the reported experimental work in n-InSb of interest,8 namely that the instability occurs in the post-impact ionization range. The solution shown in Fig. 6.4 indicates clearly separated harmonic modes except for small wave numbers which are not in the interaction region. A convective instability at the hole

kilvoel/Ilchl 0 0.1 0.2 0.3 3 3 I Io ro lu 0 4 8 12 16 20 24 28 krlvoel/l~chl FIG. 6.4 LOW-FREQUENCY HYBRID-CYCLOTRON HARMONIC DISPERSION DIAGRAM.

-2035 cyclotron harmonic frequencies is also present, thus verifying that the CD. < 0 solution of the resonance approximation has a corresponding amplifying wave for real c)0 6~4 5 Effect of Collision Frequency Variation with Carrier Speed0 In general, the collision frequency v is a function of the carrier speed s 1/2 v = (v-2 4 v2 + v2) 9 whereas up to this point it has been assumed x y z constanto If v < vT or vF it is apparent that v is independent of O as defined in Eqo 6052~ It then follows that Eqo 6~60 is still valid if the replacement v -> v (v) is made so -that the denominator of Eqo 6O60 s S should now be written, where a (v) = [ca) kv O jv (v)]/c C as OS C OS S cS D.. g N..~ t~cs /J J oo J^ o ---- -m Z5 j, (629) m=-oo.L/2 where v = (u + v2 ) when v or v, 5 Zs 0 Is Fs Inspection shows that near resonance at small growth rates, mm c (co kv) O v (v)/[m - a (v)] I constant so that the collisional Cs v Os7 / svs s s dependence on carrier speed will have negligible effect on the value obtained for D if it is assumed that v is a constanto For hole cyclotron s s resonant interactions the resonance assumption can be made independent of the dispersion relation since it can be assumed that cr mlc| hl, |v 0> and k - IC| el/v o c |<< ILce so that in each case m cs. (0c - kv ) 0 o0 ce ~e 'ce cs o os Even away from resonance it is not expected that the replacement v -> vS(v) will seriously alter D because of the nature of the integrand in Eqo 60129o Thus it will be assumed that for any carrier velocity distribution function,

-204 - f, independent of 0 the use of a constant collision frequency v in the denominator D is justified for the collision term which conserves particles properly. With this in mind the numerator can now be examined. For the drifted degenerate distribution function of Eq. 6.62, Eq. 6.90 applied to Eq. 6.60 shows directly that because of the delta function dependence only the value of the collision frequency at v = vFs is involved. Thus the assumption of a constant collision frequency for this distribution function is well justified for resonant processes with vs = vs(v = vF). For the s Fs drifted Maxwellian distribution function for resonant processes the function Ds in Eq. 6.129 can be set equal to unity since the denominator has little effect so that Eq. 6.60 becomes o00 jSElx Is. 2/n, ocs N = kv2 Cs Ts m=-oo 2 v2/ S zS u exp -- 2 ) exp (- - s Ts Ts (ls 2Ts _ m Xs Mat Lw[ - kv - jv (v)J s Zs 0 ( __cs --- —-__ _ --— _- i__v v d v. (6.150) -o00 0 oCS C [) s os s Since the case of interest corresponds to C. < 0 for k real no singularities can arise. The effects of the variation of collision frequency with carrier speed will only be examined qualitatively because of the difficulty associated with the integrals involved. For scattering due to thermal vibrations in nonpiezoelectric materials the mean free path is a constant, I, and vs = v. Inspection of Eq. 6.130 oI0 S 0

-205 - 1/2 shows that since v - (U2 + v2 ) the carriers with low carrier speed in S ZS the distribution will interact more strongly with the wave (i.e., the resonance is more clearly defined) than in the constant collision frequency case. The net effect of this is that the values of Ps in Eqs. 6.121 and 6.122 are increased thus easing the requirements for instability without altering the essential physics of the interaction. Alternatively, if the scattering is predominantly due to lattice thermal vibrations, the analysis employing a constant collision frequency should use an appropriate constant v less than that appearing in the low field mobility. S In a similar fashion it can be determined that if impurity scattering predominates, since v (v) v, the resonances are more poorly defined s at low carrier speeds (where the exponentials in Eq. 6.130 dominate) so that the analysis employing a constant collision frequency should use a larger value of collision frequency than that associated with the low field mobility. In general, the inclusion of the variation of collision frequency with carrier speed is not expected to lead to any significant departures from the constant collision frequency theory. 6.5 Electrokinetic Energy and Power Properties of the Hybrid Mode 6.5.1 Carrier Distribution Function f Independent of 0. Since the quasi-static assumption has been made'V x B 0 0 so that it follows that J(s + j)E = 0, (6.151) IX SX S from which it can readily be found that

-206 - I ) + I I I = o, (6.132) s W(S)S where W) is the electrokinetic energy density of the sth carrier species, and is defined by W(S) 1 SRe)E* (s) - i ix ix Equation 6.132 indicates that Z W() must be negative for unstable (co. < 0) s Wk interaction to occur. From Eq. 6.131 and Poisson's equation, Eq. 6.67, it can be found that XJ - = qsNls ' (6.134) s s so that if the carrier species are assumed to be separately conserved in number, then J(s) = (c/k)q sNs. Applying this result to the case of the IX S cylindrical degenerate distribution function, Eq. 6.63, gives the following for the mth resonance: j2ps ElxO m cs ( W ) J(s) -.., (6.135) 1Xk2v ( - OS - jV - mcs) where v' is given by Eq. 6.70. If k is assumed purely real Eq. 6.133 used in Eq. 6135 povides that in Eq. 6.155 provides that

-207 - w(S) k psk2 I I- m - QTcs ) +c - m' Fs2 s ). k2v-s 1' - kVOS - jv^ - mcs L m (6.136) For example if this result is applied to the electron-hole interaction studied in Section 6.5.1, then the following results: 2 ~e |Elx12 I c J2 (kVFe r pe ix. ce W i. _(e lce' 7 I., f % D. — I Wk V'2 [ oe - wceI C / i k2v2 - kv - v + w Fe oe e ce f -1 -77\ and w(h) k ph ixx mchl m \\ Ct ih - k~22 - chih 2 h ce k2V2 D - v - I i hm I Fh uh ' ch (6.138) where m = 1,2,3,.... For the unstable solution of Eqs. 6.74 through 6.78, it is true that ci < 0 and kv > Ie I so that W(e) < 0 and W(h > 0. i oe ce k k It can also be found from Eqs. 6.134 and 6.135 applied to the electron-hole interaction that 2co2 e Ic | J2 ( Fe ) p e ce 1 l |lce I p(e) - 1 [(Ve -w.) - (w - voe + l el)]Ex e I oe ce x 2 kv2 - kv - iv + Iw I Fe oe I e ce (6.139) and

-208 - kvFh 2wcm I J2 ( ph ch ch m. (h) h- j(Vh [ - ( mlchl) IE 2 h 1hr c Ix kVFh () - jh - mlch (6.140) A comparison of this result with Eqs. 6.137 and 6.138 shows that the sign of W ) is equal to the sign of Re(p(S)/E ). This is similar k 1 Ix to the result of hydrodynamic theory for space-charge waves in cold plasma, viz., c2Ek((v - 2c )(c( - kv) - j[(U - kv )2 - C. + iv]} ~1 |~I (C - kvo)(0 - kv - jv)|2 in which if (r - kv ) < 0 then Re(pl/Elx) < 0. Thus carrier modes with negative electrokinetic energy density are characterized by the property of bunching charge carriers in regions where the passive modes (e.g., the mode with v = 0 in Eq. 6.141) have carrier depletion and vice versa. The 0 same energy exchange process occurs for the hybrid mode as for the spacecharge mode then. A basic difference, however, exists between the hydrodynamic space-charge result, Eq. 6.141, and the present kinetic case in that, although the former requires v > X r/k, in the latter case the sign of the electrokinetic energy density is dominated by the function mncs in Eq. 6.136. Results of this form also follow when the kinetic power flow is examined. Only the case k. = O+ (i.e., k. > 0 and k << k ) is considered because of the difficulty associated with handling arbitrary complex arguments of the Bessel function. Since the quasi-static assumption has

-209 - been made the Poynting vector is zero and the conservation of power takes the following form, with cX assumed real: k ' P() Re(EJ= J( ) * (6.142) s For the cylindrical distribution function then, Eq. 6.135 used in Eq. 6.142 gives, at the mth resonance, kv C2 C E 1 2 W m C J2 Fs V ps 1x cs m o s (s) cs p^ --; k. (6.143) k k.v2 Ik(o - kv - jvt - m ) -2 r 1 Fs 1 v os s cs Comparison with Eq. 6.136 shows that W(S) < 0 (o. < 0) is in correspondence k 1 with Pk() < 0 (ki > 0). Hence similar statements with regard to the energy exchange can be made regarding the sink and source of the electrokinetic power. Similar behavior is found when the drifted Maxwellian function is examined. From Eq. 5.120 in Chapter V, for the mth resonance, with the definition of v' in Eq. 6.119 having been utilized, the following results: s -2 sc32 cE m W e Im(%s) J(S),ps xj CS e (6.144) 1X. k2v2 (w - kv - jv' - mc ) Ts Os s cs so that when this result is compared with Eq. 6.135 it can be seen that the (s) (s) W and P ) functions can be obtained from the results of the cylindrical k k distribution function by the replacement kv kv 2 kvss 2 ^ Fs 1^ex p - Fs (6.145) m 2e J m e ' cs cs CS

-210 - In summation, for those distribution functions f which are OS independent of e so that Eq. 6.60 may be used, the electrokinetic energy density and electrokinetic power flow are negative if mrcs < 0 corresponding to the slow-cyclotron type of mode and hence are similar in general to hydrodynamic theory. 6.5.2 Carrier Distribution Function f Dependent on 0. As discussed in Sections 6.2 and 6.3, when the dc electric field is taken fully into account, the carrier distribution function f can become dependent on the angle 0 in the x-y plane defined by Eq. 6.33. When this occurs Eq. 6.55 (and hence Eq. 6.60) is no longer valid and as a result interesting deviations from traditional requirements for instability (e.g., v > C /k) are potentially possible." Various cases in which the carrier velocity distribution function f is dependent on e are now examined. o Up to this point the effect of the field E has been ignored for the holes so that the general formulation could be used. This neglect is, strictly speaking, only permissible if E oy/B << vTh. The effect of the inclusion of this term is examined for a carrier (e.g., hole) whose drift velocity is negligible in any direction (|vo << vT). In this case employ the transformation V = ( )Y + u cos e x B v = u sin, (6.146) y where the field E is assumed negligible compared to E. The solution ox oy given by Eqs. 6.53 and 6.54 is still valid where now, with Eqs. 6.9 and 6.146 having been used,

-211 - N f = 0- o 0 (27tV2)3 /2 [ ( E epBv T )] ( 2 F ((/uE exp - - exp - cos 2v2/ L B v2y T aT 0] ~ exp 2v2) T (6.147) so that f = f (0). From Eqs. 6.9 and 6.146, the following results: 0 0 )f.0 x vx T T _1 ( + u cos e fo(0) T (6.148) From the Bessel function identity, exp(z cos e) c0 Z II(z) exp(j2e), L=-oo (6.149) Eq. 6.147 becomes f (e) N 0 (2itV2)3/2 T 2 exp [- ( 2 oT I exp ( u ) 2v v2 exp (- 2 ) 2v2 T -C a =-Oo I -UEo ) e o T, (6.150) from which g0(u) is defined by f (e) 0 0o = g0(u) jj 2 =-co I, (- B o) e B V2 o T (6.151)

-212 - By proceeding in a manner similar to that used in obtaining Eq. 6.57, Eqs. 6.35, 6.148, 6.'150 and 6.151 provide the following: 00. m, n=-00 Jm(b)Jn(b)Il m / uE \ u_ ~y )g (u) BoVT2go a [p 1where where exp[j(m+ -n)O] _ u ecp[j(m+-n+l)0] + exp[j(m+-n-1)0] ) m+~-a 2 a m+~-a+l m+~-a-1 E E \ I -- _ __ h = --. a = —, (6.152) = 1 1 jc V2B c To jc N Co CL) c - ac C and oE p = s.1X 2c T (6.153) From Eq. 6.51 the RF number density is found 00 m, n=-o jriE N exp 1X 0 2 [- ( foy L \f7 B VT as I rI /o0 [( >-L +Lj 2 - N 1 wcvT(a - n) vexp - ( E o2] v exp - oy oT, (6.154) + j 00 L 1. a-n m, a - n m, n=-oo where

-213 - L = o 1 Jm ku) n ( m w ku uEoy( u2 u )In- (-mQ ) exp (- u du (6.155) wc n-m BV2 p ) o T - and ku ku I oy L = J( J( ' (- Y-)exp(- _)u2 du, (6.156) 2 ~ WCJ n Dc n-m BOvT 2v2T wherein use has been made of wherein use has been made of I (x) + I m (x) = 2 - (Ix = 2' (x) M-1 Mml d m (6.157) It can be verified that if E - O, Eq. 6.154 reduces properly to the result ~oy of Eq. 6.108 since only the n = m term survives in Eq. 6.155 [In(0) = 0 except for Io(0) = 1] and L (Eoy - ) - 2 -2 oy 2 2 / 2 2 a exp - u2 du T2v ( k ) n ( 0s ) () + I( ) (0)) m Cs Jn ( C ) (n-m) -1 (n-m) +1 n pcs 0ku ex 2 --- J - 2 exp -- >u du, (6.158) =~k O n ^ cs 2T T where Jn-l(x) + Jn (x) = (2n/x)J (x) has been used. Inspection of Eq. 6.154 indicates that the resonant form is still present in the factor (a - n) representing the nth resonance. In particular the n = 0 resonance can now appear, whereas this gave zero contribution for fs independent of 6 as can be seen in Eq. 6.60. However, for the n = 0 resonance in the hydrodynamic limit, kvTs cs 0, Eqs. 6.154 through 6.156 give the following:

-214 - POO (UE ( u2 EN _ L J2(0)I- exp.-. u2du vexp p 1 0 o 0 B v 2 VT 2B2v2 oT T oT O uE \ L - 0J2 (0) I - )( 2 0 ~1 B v2 0 T / 2 y /, E2 u2 ep ) du B oy exp - e 2v2 / 2B2v2 T oT (6.159) and thus N 0, wherein use was made of 1 I (z) = V exp (- vIjj JV(jZ) and 00 0 exp(-a2t2)tV+lj bV ( b 2 exp(-a2t2) t J (bt) = V+1 exp (2a2) 4a [Re(v) > -1, Re(a2) > 0] In general the integrals in Eqs. 6.155 and 6.156 are not by analytical means. Some information can be achieved by using and 6.154 in Eq. 6.133 to obtain the carrier-mode electrokinetic density for the nth resonance in the resonance approximation C IEI e - E 2j E ) 2 2ElE I exp - L-1 BoVT. (6.160) solvable Eqs. 6.134 energy kk mmMl oo [E - 2kv4 w - k - ) - jv' - ncn 0 2 - Vr+ / E, (6 * L r + k o)+ nw (6.161) W3. B c 1 i O where

-215 - v' = v - - exp - T )2 ] 00I ( oT ) =0 Tvr MJ L } 1 (6.162) so that if C. ~ W < 0 if [(E o/B )L + L ] <. By proper selection so thatif w< ky Q < if oY 0 1 2 of E, B, etc.,it is expected that this latter condition can be achieved. - -o In this case for example the hole mode can act as the energy source to drive the instability. An alternative method whereby the distribution function becomes 0 dependent occurs when the field E is taken more rigorously into account. OX The following transformation is now used to replace Eq. 6.146: V = (B )+ ucos, 0 Vy = E + u sin 0. O (6.163) Then, if the carrier drift velocity is negligible (v >~ c), Eqs. b.9 and 6.163 give the distribution function as f = 2 exp(- ) exp - exp ( 3/2 2Bo2 2v (2V^) 0oT T 2 (.r uE> 2v2 B v T oT 0c Q=-00 I Eoy ) e o0T. (6.164)

-216 - Inspection of Eqs. 6.53, 6.148 and 6.164 indicates that the term in 0 introduced by the field E can be taken into account by replacing the OX argument of Jm(b) in Eq. 6.152 via uE - ku ku ox, - b = k -- - j (6.165) cs cs B vT o0T The RF density given by Eq. 6.154 is still valid where now, however, E \2 poo uE \ k u ox ku ox ku L = exp - ( ox J 21 L\(14 B v ] m ( cU BW v2 n c o2 BVT BoT **I o - 2 exp -- u du n-m B VuE 2vu and E ~2-! poo uE L = exp[( E )] n ( ) oT 0 C B vT2 W uE o / 2 2 ( ) exp ( - )u2 du (6.166) BovT T To obtain a more tractable analysis assume that E field effects are oy negligible by letting E - 0 (e.g., the applied field can be E and oy oy the Hall field E x, with E > Eoy> without loss of generality). From an analysis such as used in Eq. 6.1 it can then be found that an analysis such as used in Eq. 6.158 it can then be found that

-217 - L = exp 1 2 ox 2 oVT I r n n ku C uE - 3 B vT 0 T J (- u co * exp (- u - du 2v2 T and n L 2 L 1 (6.167) ( k j ox c BVo ' o T The general results8 00 0 exp(-p2t2)j (oat)J (t) t dt 1 2p2 exp (- 2 + 42 \p(- 4p2 I (2p v )2p2 Re(v) > -1, larg pl <, (6.168) applied to Eqs. 6.167 and thence to Eq. 6.154 provides the following: jrlE Nix 1X O w-j v-nw c ncw -\ c e In(X) c c N 1 Re(n) > -1 (6.169) 00 1 + jv n= —oo -k e CIn (c) o- j v-nco where k2v2 T x = C 2 C kE _- i ox B C o c (6.170)

-218 - Since vT= KT/mi, where T is the carrier temperature, K is Boltzmann's constant and m is the carrier effective mass, the view can be adopted that the effect of the field E is to replace the temperature T by the ox complex temperature E mcd T = T - j (6171) c B k( o The resonance approximation is now made, corresponding to the omission of the n = 0 term in the denominator of Eq. 6.170, which is valid provided that I|\l 1 and is strictly invalid if | cl << 1. Equation 6.169 then c c becomes the following for the nth resonance: -\ jrjE kN nco e I ( c) N lxO c- | 1 (6.172) ok (W- jr ' - nw) c ( C c where -- V = v - e cI (\)). (6.175) "^c A study of the function e In(Xc) indicates that Re(v') > 0 must hold. Apply these results now to holes with IDch| << vh by comparing Eqs. 6.108 and 6.169 so that the solution given in Eq. 6.120 is still valid but wherein, -k h2 e c I(x) r rh = ph m.c., (6.174) c ch c 'Ch where from Eq. 6.170,

-219 - k2v2 kE c=. j B (6.175) h o ch ch Recalling that the field E is negligible or zero (e.g., the applied field in Eo and the Hall field E is shorted) it should still be shown ox oy that the undrifted form of the carrier distribution function is permissible in the presence of Eox when v > IclD|. By following the system of Eqs. 6.32 through 6.41, but where now E is retained and E = E = O, so that ox oy oz Eq. 6.163 replaces Eq. 6.33, it can be found that N ( 2 V2 E 2 ~ (2cv2)3/2 2vT 2v2 ) L )T o (2~vTvT 2T\ B vT ~ J~n _-j e E ( e Q+ ejn [1 _ox ( (i -.v/)os e + sin. n=-o. o T o T 1 + n - (6.176) Inspection of this result shows that for v >> 1ocl the last term should give a negligible contribution so that N 0 u2 ^ ^ ^ ^ r z / ox ~ (2cv xp ev- - exp - 2 exp - K Bv ) ] (n)32 2v 2v2 L _ - v ( o (2rr~~) /9 T T oT ^01 / uE \ /v2 + v2 + v2) \ S~ (jUEox jn 0 J * __ e exp ox y 7) L n v2 (272)3/2 2177 n=-oo oVT 2Tv

-220 - thus validating the assumption of a negligible drift velocity. The RF number density can also be found directly, without approximation, for the exact carrier distribution function f of Eq. 6.176 (although the result is not in closed form) by the following procedure. From the transformation of Eq. 6.163 with Eoy = 0 it can be determined that Eq. 6.176 can be written as N f = 0o o (2Bjv )3/2 v2 + v2+ V2 -E v2 exp x — ) z ox (- I 2v2 'By 2v1 T o0T T ep(- (vy + Eox/Bo)2 F( E](v ~ 2~p ox )-exp - BY 2xexpexp 2v2 L 2 B v %Bv> 2v2 T oT T 00 n=-oo n E E J -J OX V B v2 x oT y E 2 [ n v v + ( + ox ) )1 + (jn - - 3 r.1( v + E /B - * exp jn tan ox 0 x. (6.178) From Eq. 6.50, with E = 0, oy it follows that 00 f (e) = 1m=oo m=- o E E r exp(jaO) exp(-jb sin 0) e ix Jm(b) exp[j(m - a)o?] L c c af (0 ) vN 0.... ~ dx - cN Jm(b) exp[j(m - a)e']f (e') d' x co c where a = (w - jv)/c)c and b = ku/%0. (6.179)

-221 - Use of Eq. 6.178 in Eq. 6.179 obtains f (0) which applied to Eq. 6.51 1 provides the following: j~E N lx 0 _p exp 2w 4 c"T (Eox B 2 K ]o~0 0, Q,n=-oo 3 i =1 ei(, n) a - I N 1 (6.180) 1 +j - VT I ox exp L- B T-Qn BoVT T9n=-oo D(,, n) a - I where uE ku. ox c B v2 o T = 212 c 0 J ( kuu "i \w u2 exp(I —) 2V2exp T u du, (6.181) E k. ox Cc B v o T 2 00 0 J( ku ) n (-j Eox o T exp u ) vT uH(Q,n) du, 1 + (jn - v/Oc)2 (6.182) where H(Q,n) ru2V v2 T v "c )- jn + (n V c 2 2(I -n) b J ( ku ) - 2j (jn V (%c ) - (b) I -n + [-n (jn u2 u2 j - -- n v2 v2 T T -) C / + jnj -n-2 ( )ku ' -n-2 (cc c + n (jn c J 2v2 V2 T T (n - + n] J-n+2 ( ku) c (6.183)

-222 - B v2 1 + (n _-v ) ku, ox J( C j(u ) I 1c n B V 0 T ~ exp( 2 U3 T K(Q,n) du, (6.184) where K((,n) = 2 n -,) J (c )%c ku C__ 3[(n-^z) + J ~-n+2 ), (6.185) and D(A,n) = /PO 0 J ( ku (. Eox ^M B 0 (- -) L(,n)u du, (6.186) u2v2 where L(,n) = J-n ( ku I9)-n \ c -1 E -ox [ - - ) J_ (b) - 2jJ' (b) uBox [1a b I ( j -n /- ) -n] 2B ov [l + (jn - v/o )2] (6.187) This is a general result valid for any charge carrier species. The integrals are not readily solved except for ~ which can be obtained using 1 Eq. 6.168 and correspondsto the earlier result of Eq. 6.169. Inspection of Eqs. 6.180 through 6.187 under the condition v >> Ic | shows that even though the drift velocity is negligible. e.g., Eq. 6.177, the full

-223 - dependence of the distribution function f on 0 can introduce additional o nonnegligible terms in the RF number density beyond those of Eq. 6.169. Thus, in general, the dispersion relation obtained by first approximating the carrier distribution function f through v > IDcl can differ o significantly from the result for the dispersion relation (for the exact function f ) in the limit v >> Icl. As a consequence, in the presence of a static magnetic field the use of a complex temperature, such as Eq. 6.171, to account for carrier heating effects is not in general justified. In addition there is the problem that when v > I%|cl the resonances are not well isolated so that examining a particular resonance, i.e., selecting only one I value,is a poor approximation. This explains some of the anomalous behavior which can arise for undrifted carriers when the resonance approximation is made. For example, if Eq. 6.172 is used in Eqs. 6.133 and 6.134, it can be found that the electrokinetic energy density of such carriers can be negative due to the field E OX Conclusive statements regarding the effects of the dependence of the carrier distribution function f on e require a rigorous solution of o Eq. 6.180 which is not attempted in the present tract. This already cumbersome expression for the RF number density is further compounded by questions regarding the accuracy of a constant collision frequency assumed in the presence of carrier heating. Some qualitative statements regarding the effects of the field E are still possible for the charge carrier ox with Iocl >> v. For example, inspection of Eqs. 6.180 through 6.187 shows that the I = 0 resonance now gives a contribution to the RF number density when E # 0. In addition, the effective collision frequency ox concept is still valid where now at the Ith resonance

-224 -r / E 2 Vr o Fexp - ( ) D(,n) v' = v II Zo1 2 B V v2 n=-oo T Inspection of Eqs. 6.186 and 6.187 shows that for 1c | >> v, the function D(~,n) itself exhibits a resonance form at n = + 1; however, cancellations occur when the summation is made so that these terms have no special significance. An interesting case which can readily be analyzed is the small wave number limit (k -*0) when I = 0. If Jcl >>~ w,v Eq. 6.180 is approximated by ox jlE N exp [- ( Eo, ) ] Z [ (0,n) + (0,n)] 1X O ' 2 3 OV n=-oo NHJ~ --- o T n=-o (6.189) E \2 2 4 v -a exp ( ox D(0,n) a c v 2 - D(,n) c T oT n=-coo and by permitting k - 0 in Eqs. 6.180 through 6.187 Eq. 6.189 becomes rElxN 0 ( d Bvox) ]T E2o \/T N {EE2{e [-x(P --- )* N (k - 0) - - (-.-...... ( o ), 4 \ v2 WV3B2 2, B2v (6.190) where again M is the confluent hypergeometric function. Note that implicit in Eq. 6.190 is the result, derived from Eq. 6.188, that the effective collision frequency v' — 0 as k -0. This result is expected to be a good approximation, provided that c| >> ~kvT,, v, for finite k, and hence the field E introduces an entirely new mode, corresponding to I = 0, in which the effect of the carrier collisions on the wae dispersion can be which the effect of the carrier collisions on the wave dispersion can b e

-225 - negligible. In addition for E x 0 tables81 provide that M > 0 and ox as a result if Eqs. 6.133 and 6.134 are used in Eq. 6.190 it can be shown that this mode has a negative electrokinetic energy density. Thus, since the carrier drift velocity plays no explicit role, the field E introduces ox a negative kinetic energy wave without the traditional requirement of hydrodynamic theory that v > C /k where v II k. In the present case by using Eq. 6.176 in Eq. 6.25 it is found that F-/ E \27/ E2 2E v ox 3 ox ) ox ~ e ~xp ~ v ( + v ) c ox T 2 2x O O O oT oT B _, ~L o c phenomena also occur for the case E = El = with E + ~ (E 1 k) as 2 2C dispersion relations resulting from the general number density, Eq. 6.8, due oET 2B3v2LK( V 2 2 for the component of drift velocity II k. to its can bomplexity, but Eq. 6.190 indic stated then that the d electric fiunstable wave behavior is to betaken expected when a carrier with |Icl >> (CO,v,kvT) interacts with a secondary carrier with |cl < v. In addition, since v'(k -O) -0 in Eq. 6.190,

-226 - it may be possible that for finite wave numbers the effective collision frequency actually becomes negative. The important point here is that when carrier heating occurs in the presence of a static magnetic field the effects of collisions on the wave propagation can be significantly reduced. 6.6 The Ordinary Mode in Solid-State Plasmas A A When the geometry is retained with B II z, k II x, with the applied static electric field E and the Hall field E y the ordinary mode is A A defined as the electromagnetic mode with E I z and B = B y. For isotropic media with negligible Hall drift velocities this mode is well defined by the component zz of the effective dielectric constant, (e = I + a/jca ). If the RF magnetic field B is retained and N = 0 is ly is set, since there is no RF bunching, Eq. 6.32 becomes f is sf i f is ai j(o- kv )fi + rs B0 vys a - v Xs + Bo v - + E xs ys xs ys af af af + E os + v B Os v B = -v f. (6.192) s iz dv ys Xs v ly v S y v zs zs xs The following coordinate systems, previously used, will be needed: v - v = u cos 0 = v cos 0 sin cp xs os s rs v = u sin 0 = v sin 0 sin cp ys s rs v = v cos p, u = sin cp, (6.193) Zs rs s rs from which the following can be found:

-227 - 00 dv dv dv xs ys zs -00 oo r -tOO o 02 00 "'0~~ ~~~ co" u du dG dv S s zs 00 n2TC Ir = ~ ~ Jo/ / vs sin cp dv de dcp 0 0 0 rs rs (6.194) Only the drifted Maxwellian, Eq. 6.18, and the drifted degenerate function, Eq. 6.70, are studied in the following with their respective distribution functionssuperscripted by M and D for clarity. The term E is assumed to ox be negligible; therefore, the method of Eqs. 6.34 through 6.37 provides the following where now a = (c - kv - jv )/cw and b = ku s/co: S SS CS S SC j(a b ~e~ - j(a - b cos e) M S S iS = E- / '- f C ^cs \ " v2OJ T s Ts (6.195) and cfD ie - j(a - b cos G)fD 00 S S iS n::: E ( kvz v fD 's iz os zs os cs rs rs (6.196) wherein the following were used: fM v - v os xs os fM dV MS ~2 OS xs vT Ts afM OS ys v = ysfM v2 0 Ts afM os av zs v zs f 2 os Ts afD os (v xs xs Ms fD - v ) 0 —__s os v v rs rs ofD v /afD os ys( s t av - v \ v Y ys rs rs afD v /fD os Z= zs) ( 6 7) 0v v v (6.197) ZS rs rs

-228 - and B = -(k/w)E corresponding to variation as exp[j(ot - kx)]. Let J~y Jlz the right-hand side of Eqs. 6.195 and 6.196 be designated R and RD respectively. It can readily be demonstrated that RMD is independent of 8. Thus the solutionsfor fM, D are found in the same manner as that is for the Ns term in Eqs. 6.38 and 6.39, leading to 1is j / kv \ 0o Co jn E 1 Os s iz \ v M f () J (b )J (b ) nE (1 z -s fM exp[j(n - m)0] s m (es n s n-a =2 Os m=-oo n=-oo Ts (6.198) and 00 00 fLs(0) = m=-oo n=-oo / kv \ (-j)nE (1 - s ) w (n - a) CS S J (b )J (b ) m s. n s v zs v rs /fD ' ( r) exp[j (n Since the ordinary mode is dependent solely on e, only AY =iA density J =J (E = E z) is required where i Jz Jz -1 iz - m)0]. (6.199) the RF current 00 J = q v f (O) dv dv dv lz s ZS iS XS ys ZS S -o0o s =1 5 1z, 5 (6.200) Use of Eqs. 6.198 and 6.199 in Eq. 6.200 then gives the following: Jiz iZ, s 00 m=-co jCOps e Im(s) ow - kv - jv - cs o S s cs 0 - kv os E1z C; kv )2 c0 (6.201)

-229 - and - 0 j2itqs E (1 - kv ( s Ds D J" = ) - --- 7 — -- i 2 ( rs sin cp z)S _,Cs (m Sa )0 Om c Cs m=-oo / afD * cos cp sin cp v ) v3 dv dcp (6.202) rs and since, from Eq. 6.70, 6fD 3N [(Vs) - 5(Vr - VFS)] ' (6.203) vi rs 43vFs then jD Y Je5<^iEZr> (( - kV ) t kvos JI = + sin cos2- a sin c J si p 1Z, s Cs s 0 m( vs ( ) m=-oo (6.204) Employing Eq. 6.201 in the usual manner in Maxwell's equations leads to the dispersion relation for the Maxwellian distribution function, -x v^C2 2 ( cu - kv )e s I( k^2 - a? + ) ) P5, os. ---^ — I (6.205) k2c2 _ 2 + P O ) - kv (S (6.205) (j, kV Os~ - kv )eMC OS 5 CS s m=-oo If now (kvT /Ocs) ->0, it is true that I (X\) ->0 except for Io(\s) ->1 and Eq. 6.204 becomes W2 (wCs - kv ) k2c2 _ 02 + - - = 0 (6.206) - kvo js s s

-230 - which is identically the result found from hydrodynamic theory. Significant deviations from the hydrodynamic theory can then occur if kv >~ | cI, Ts cs the most interesting of which is the possible resonant behavior D - kv - mw C 0 leading to a slow-cyclotron-type mode. Similar OS CS statements can be made for the degenerate distribution function since if kvFs /cs - 0 in Eq. 6.204 only the m = 0 term survives in the summation and consequently the hydrodynamic result, Eq. 6.206, is obtained exactly when the integration is then performed. In the general case this integral cannot be readily evaluated; however, this integral by inspection is always positive or zero and hence qualitatively behaves similar to the -k function e Im(\). It is important to note that unlike the purely transverse mode (k II B ) studied in Chapter V the ordinary mode does not present any problems associated with nonlocal effects which alter the resonance structure. To study the instability characteristics of the ordinary mode define the effective plasma frequency, '., by the equation p 2 2 _ (p)) = p e Im(ks) (6.207) for the Maxwellian distribution function, and by 2 3 2 2 (kvF (up) = 2 sin i c sin cos sin dcp (6.208) p m cs for the degenerate case. For fundamental resonant interactions k (a + Wcs)/v s, so that if Xs 1, the dispersion relation for either cs/ 05 s

-231 - the Maxwellian or degenerate cases can be approximated for a two-carrier system from Eqso 6.205 through 6.208, with m = 1 by (W )2( - kv ) (a) )2(- kv ) k2 c _ a2 + p + p- = 0 (6.209) w - kv - jv o w - kV - iv - C) 01 '1 C1 02 2 C2 Simplify further by assuming that v - 0 and Iv << c so that the 02 01 coupled-mode form is obtained in which the coupling term appears on the right-hand side, 2 O1 1 Cl 2 02 O 2C2 01 2 C2 (w - kvo - jv c )( ( jv - Cu) = -- (I ( kv )(w - jv - Wc) k2C20 2 (cO ) - 2 ( - kv -jv - a C1), (6.210) C2 01 1 ci 9 where c << kc has been used. Inspection of this result shows that to achieve the coupling strength required for unstable wave behavior it is necessary that at least one of the carriers satisfy (c')2/k2c2 1. p Practically speaking, to avoid excessive Joule heating, this in turn requires a carrier species with large number density and negligible drift velocity in the direction of the applied electric field. For real k, Eqo 6.210 is solved as )2(1 + 0 + 0 ) = kv 0 + (kv + o )(l + + W ( + 0 ) + jv (1 + e ) 1 2 01 1 01 ci 2 C2 1 2 1 + jv (1 +0 ) +, (6,211) where RO1 + =ckvi + 62) - ((kc2 + j )(1 + C )]2 + 40e (c2 + jv )(wc + jv ) 1 2 2 c 1, (6.212)

-232 - and 2 (WO )2 0 =; s = 1,2. (6.213) s k2C2 Upon inspection of this result is is found that one of the carriers can have a very large collision frequency without diminishing the possibility for instability. For example, if v ~ v, cu < 0, Wc > 0, and 2 1 C1 < 0 2 k ~ [Icl|(1 + e ) + 0 C (1 + e )]/(1 +0 + e ) instability occurs at Cl 2 C2 2 1 2 r = Ic2 |(1 + 0 )/(l + 0 + 0 ) provided that r 2 i1 1 2 400 2w |w | I) > | v(1l + 0 )2 (6.214) 12 Ci C2 1 2 In general instabilities of the ordinary mode are more difficult to achieve than the quasi-static hybrid mode since roughly speaking in the latter case the coupling constant is proportional to (wp/kvT) or (op/kvF), whereas in the ordinary mode it is proportional to ()p/kc)2. As in the hybrid mode study it is also of interest to study the effects of carrier heating on the cyclotron modes by taking the dc electric field fully into account in the carrier distribution function f. This o aspect is presented and discussed in Appendix H. 6.7 Summary and Discussion A rigorous analysis of the wavespossible in the configuration with k I B for various distribution functions has shown several important -- '-0 results brought about by the kinetic theory which have no correspondence in a hydrodynamic analysis. This has made it possible to relate experimental data to the cyclotron harmonic structure of the hybrid-hybrid kinetic

-233 - dispersion relation. Also, the kinetic approach can take carrier heating effects into account explicitly. This has indicated that entirely new modes can appear in which the collisions play no role and unstable wave interaction is now possible in the fast wave regime (v < cDr/k). In this case the coupling of the longitudinal and transverse fields of the hybrid mode is increased, so that strictly speaking, the quasi-static assumption should be lifted and the full hybrid mode form examined. A study of the variation of the carrier collision frequency with carrier speed has demonstrated that the use of a constant collision frequency is actually an excellent approximation since it is strictly accurate for the degenerate distribution function and it only introduces small quantitative changes for the Maxwellian distribution function. An investigation of the ordinary mode in kinetic theory has revealed that cyclotron-resonant behavior is present although unless the charge carrier densities are larger and at least one of the carrier species has a small collision frequency (v << c) unstable interactions are difficult to achieve.

CHAPTER VII. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY 7.1 Summary and Conclusions The primary purpose of this theoretical investigation has been to obtain an improved understanding of instability phenomena in plasma media. This is based on a study of the electrokinetic power and energy properties of the basic carrier modes present when the plasma medium is subject to applied static electric and magnetic fields. Particular emphasis has been placed on the investigation of solid-state plasmas in which case carrier collisions and thermal diffusion must be taken into account to obtain a meaningful analysis. A discussion will be given of how the present work alleviates the problems associated with previous studies in this area. These previous studies have been incomplete and/or incorrect in the following respects: 1. The hydrodynamic expression for the electrokinetic power flow developed by Chu64 for space-charge waves has been explicitly used as a meaningful expression when carrier collisions and thermal diffusion are included. 67 82 This is incorrect since to balance the electromagnetic power flow it is not necessarily sufficient that the Chu electrokinetic power flow be negative. 2. The hydrodynamic development of the electrokinetic power for the cyclotron modes has been obtained on a normal mode basis.71 This is incorrect since the coupled-mode technique ignores both the presence of the RF fields and the fact that the normal mode described by X - kv - c c 0 is in a nonlocal regime for finite carrier temperatures so that cyclotron resonance phenomena should be included. In addition so that cyclotron resonance phenomena should be included. In addition -254 -

past interpretations of the cyclotron mode electrokinetic power properties, since carrier bunching is absent, have been based upon a second-order RF electric field50 or upon a transverse gradient of the fundamental RF electric field.66 The former is incorrect because the second-order RF electric field has zero time average and hence cannot contribute to the second-order conservation of power. Even if this field is extended to include a time-average component this would be due to a charge separation effect related to the evolvement of the instability as opposed to any direct relationship with the linear dispersion relation. In the latter case66 the explanation of the power properties based upon the RF electric field gradient is false simply because this gradient is not in general present. 3. No attempt has been made to ascertain the electrokinetic power properties of the hybrid mode which results when k 1 B and since this is a basic carrier mode the previous analyses are incomplete. 4. In the application of the electrokinetic power concept no distinction has been made between convective and absolute instabilities, although it is usually implied that a mode which is active (i.e., a source mode) in a convective instability can also be active in an absolute instability. 5. There has been no published work relating the dispersion equation root structure in complex-k space based upon the mathematical causality criteria of Briggs12 to the root behavior based upon the more physical approach of analysis of the carrier-mode power and energy properties. 6. In developing the expressionsrelated to the conservation of power and energy, no satisfactory explanation has yet been provided for the

-236 -meaning and effects of the second-order time-averaged currents and fields which arise in a general second-order analysis. 7. Because of the manner in which the Chu definition of the electrokinetic power is obtained, no attempt has been made to extend the concept of electrokinetic power flow and energy to the kinetic theory. As a result, in addition to nonlocal effects, phenomena associated with the carrier distribution function, such as anisotropic carrier temperature and carrier heating, have not been explored on a power or energy basis. The problems associated with items 1, 2, and 3 relating to convective instabilities were overcome in Chapter II by adopting a more general expression for the carrier-mode electrokinetic power flow based upon an examination of Poynting's theorem. In this manner the thermal kinetic power flow and the power flow associated with carrier collisions are incorporated directly into this carrier-mode electrokinetic power flow (Pk). Since the electromagnetic power flow must be positive for kr > 0 this expression Pk must then be negative to conserve power properly, whereas the Chu electrokinetic power can be negative without necessarily satisfying conservation. For space-charge waves the carrier-mode electrokinetic power flow, Pk, reduces to that of the Chu theorem when collisions and thermal diffusion are absent. When this more general approach is applied to the helicon-cyclotron modes discussed in item 2 it is found that the carrier waves must first be separated into their right- and left-hand circularly polarized modes. It is then self-consistently determined, by employing quasi-linear theory, that by proceeding to a full second-order analysis the source of power for the active mode is directly related to dc carrier slowing as opposed

-237 - to the presence of any second-order RF electric field or gradient of the RF electric field. The general formulation for completeness is also readily applied to obtain the electrokinetic power flow for the hybrid carrier mode. As a by-product, the quasi-linear theory applied to decaying electromagnetic waves is used in Appendix C to explain the experimental observation83 of the induction of a dc voltage in the direction of the Poynting vector when helicons propagate through a solid-state plasma. This general phenomenon is termed the second-order Hall effect. Associated with items 3 and 4, a kinetic-electromagneticelectrostatic energy theorem is derived in Chapter III which shows that a general expression (Wk) for the carrier-mode electrokinetic energy density can be obtained from the requirement of conservation of energy. This plays a similar role for absolute instabilities, as does the carrier mode electrokinetic power flow (Pk) of Chapter II for convective instabilities. The energy theorem also explains the behavior of an unstable root in complex-k space as c. is varied to test for Briggs' causality criteria. The theorem demonstrates the fact that in this dynamic state as the wave blows up more quickly in time (hi - -oo) it must correspondingly decrease its spatial growth rate (ki - -0o); otherwise the energy is not conserved. Analysis of a particular dispersion relation can now be limited to those frequencies and wave numbers which provide negative electrokinetic power and energy modes. In Chapter IV the quasi-linear theory is employed to determine the significance of the second-order products described in item 6 and shows that such products are directly related to the evolvement of the

-238 - instability from the point and time of initiation. They are also generated by the distortion of the potential energy of the system due to the reactive effects of the growing RF fields upon the interacting carrier(s). The meaning of the quasi-linear theory is explained and it is shown that this is a valuable analytical technique for handling the difficult problem of nonlinear wave propagation in inhomogeneous media. In connection with item 7 the electrokinetic power concept is extended to kinetic theory in Chapter V for the basic carrier modes and the major carrier distribution functions of interest. Comparison with the hydrodynamic theory of Chapter II shows that in many cases this hydrodynamic approach has been improperly used and the limitations on its applicability are pointed out. In particular, near the cyclotron resonance point, it is found that Maxwellian plasmas may exhibit a resonance-wave slowing in the presence of self-consistent spatially growing waves, even for small carrier number densities. This phenomenon is discussed in relation to the microwave emission observed from n-InSb in the configuration k II B - -o From a detailed analysis of wave propagation normal to the static magnetic field undertaken in Chapter VI, it is found that the harmonic nature of the microwave emission from n-InSb in the configuration where k I B is in excellent agreement with computer results based on the electron-hole hybrid-hybrid interaction. In this regard, an examination of the effects of the variation of the carrier collision frequency with carrier speed indicates that the use of the constant collision frequency approximation is well justified. A rigorous study of carrier heating shows that an important quasi-static hybrid mode can arise for Maxwellian

-239 - plasmas when carrier heating is included in the equilibrium dc state of the distribution function. This mode has a negligibly small effective collision frequency, exhibits synchronous behavior (e.g., X c kv when 0 E I k), and possesses a negative electrokinetic energy density independent -o of the sign of (a - k v). 7.2 Recommendations for Further Study From a practical viewpoint the present theoretical work is of most benefit in suggesting experimental studies related to the interaction of external radiation with matter as well as the generation of useful instabilities in solid-state plasmas. In Appendix C it is illustrated that external electromagnetic radiation will produce a second-order dc electric field in the steady state in semiconductors due to the action of the Lorentz force. Thus it is expected that a pulse of radiation in passing through the material with exponential decay will generate a second-order unidirectional transient current. In particular, assume now a material with some hysteresis in its dc I-V characteristic such that if the current exceeds some critical value, Icrit' the material switches to a second state which will persist when the current is removed. The point is now made that instead of supplying the current directly by attaching leads from the material to a current source, by the second-order Hall effect, this unidirectional current can be generated by external radiation. Indeed if the radiation has a sufficiently large power density the critical current, Icrit' can be attained. In addition, if the radiation is focused, the current

-240 - (which is in the direction of wave propagation) will be localized if the pulse length is much less than the time characteristic of diffusion processes. Such localized switching action has been observed in thin-film amorphous semiconductors by employing focused pulsed laser radiation.84 In the work cited84 the theory behind the observed effect is not clear and it is herein suggested that further study be given to the second-order Hall effect to determine if it is responsible. Such switching action is being studied for possible application to high-density memory storage systems. The second-order Hall effect occurs in any medium with mobile charge carriers under the influence of radiation, with the strength pf this effect proportional to the damping decrement k. and the radiation power density. Thus it is suggested that with the present availability of high-power radiation sources (e.g., focused laser beam) important technologies can be derived wherein either the induced unidirectional current or unidirectional voltage (induced electrostatic effect) are used to advantage. The study of Chapter IV utilizing the quasi-linear theory to study potential energy effects raises some important questions regarding the saturation length, maximum growth constant, etc., of an unstable system. As an example of this it can be asked by what means (e.g., doping) can these potential energy effects be used to enhance or quench the evolvement of the instability. The correlation that was effected between the causality criteria of Briggs and the conservation of energy is worthy of further study. For example, to test a root with k,k. > 0 (at c. = 0) for convective r i 1

-241 - instability Briggs' criterial2 requires that k. - -o as.i - -oo, whereas that of Sturrock13 requires only that C. < 0 when k. =O. The conservation of energy approach, however, would appear to state that it is sufficient proof of causality if an increase in the total energy density is balanced by a decrease in the total power (e.g., if the stored energy in a closed volume increases, it must be balanced by less energy leaving that volume). This latter approach may lead to a simpler approach for assessing causality. This is particularly important with regard to the kinetic theory since the causality criteria of Briggs is often difficult to apply. There are also fundamental problems raised between the concepts of causality and energy conservation. Thus, is there a one-to-one correspondence between all noncausal solutions and the violation of conservation of energy? Can it be shown that the advanced potential solutions of Maxwell's equations violate energy conservation? Further study is also suggested to correctly analyze the residue associated with the cyclotron resonance when k is complex and c is assumed real. The general problem of a residue is handled (even in collisionless plasmas) by introducing a small amount of damping so that the Landau pole, for example, is written as w - kv -* c - kv - jv. In x x this case for real k it is only with wc. > 0 (damped wave) that the pole can exist, as is well known. However, when the magnetic field is present and k is complex, the correct interpretation is not clear. In Chapter VI it is felt that additional theoretical work is warranted in the study of carrier heating effects on the wave behavior. Although it has been commonly asserted that such effects are negligible if the drift velocity is small compared to the carrier thermal velocity,

-242 - the present work indicates that this is not the case since an entirely new mode can appear corresponding to the extra degree of freedom in the system. Finally, in the experimental study of instabilities in solid-state plasmas, the theoretical studies of Chapters V and VI suggest strongly that attempts be made to couple directly to the bulk of the material (as opposed to an antenna approach based on surface charge oscillations) by the use of high-dielectric constant end plates to match the waveguide field to the field in the bulk.

APPENDIX A. KINETIC POWER OF SPACE-CHARGE WAVES TO SECOND ORDER It has been shown69 that the kinetic power of the slow space-charge wave is dependent only on first-order variables. Since the presentation given therein is vague the theorem is rederived. As Tonks72 has shown, the kinetic power may be written in the present case as v2 P k J (A.1) k 2-J since all variables are one dimensional. To second order, the velocity may be written as v = v + V cos(ct - k x) + v (x) + V cos2(t - k x), (A.2) where V, v (x), and V are time-independent. V varies as exp(k.x) and 1 2 2 - ----- 1 the functions v (x) and V vary as exp(2k.x). Similarly, the current density 22 1 to second order may be written as J = J + J1 cos(t - krx) + J (x) + J' cos 2(wt - k x), (A.5) 2r 2 r where J, J (x), and J' are also time-independent, J varies as exp(k.x) 1 2 2 1 1 and the functionsJ (x) and J' vary as exp(2k.x). From the time-averaged 2 2 1 continuity equation, when the current density is equated from outside the interaction region (J ) to that within [J + J (x)], it is seen that 0 0 2 J (x) = 0. (A.4) 2 -243 -

If all contributions are retained to second order, Jv2 = J 2 + V2 cos2 (ut - k x) + 2J v V cos(ct - k x) + 2J v (x) 00 0 1 r 001 r 002 + 2J v v cos 2(ct - k x) + J v2 cos(wt - k x) + 2J v V cos2(cct - k x) 002 r 10 r 101 r + v2J' cos 2(ct - krx) (A.5) 0 2 When the time average of both sides of Eq. A.5 is considered, the following is obtained: Time average (Jv2) = J v2 + 1 JV2 + 2J v (x) + V J 2o0 21 002 0 oil (A.6) MacColl85 has performed a mean-value analysis applicable to the present case which shows that V2 1 1 ^v(x) = - 7 ^0 (A.7) so that the second-order variable contribution in Eq. A.6 is annulled and the time average of Eq. A.I becomes Time average (Pk) =v2 vV ) (A.8) verifying that it is sufficient to know the fundamental fields to determine the carrier mode kinetic power. In addition, the second-order longitudinal force equation is

-245 - [cs2-k]+v+va Xv (x) 7[V co s 2(ot - k x)] + v -s — + vv [V cos 2(wt - k x)I + V cos(at - krx) H- [v cos(wt - k x)] = nE 1 r 1 r- 2 * (A.9) The differentiations having been performed and the time-average real part having been taken, Eq. A.7 then shows, as expected, that Time average [Re(E )] = 0. 2 (A.10) Equation A.2 shows that Time average (v) = v + v (x) 0 2 (A. 11) On the basis of Eqs. A.7 and 2.54, the kinetic power flow may be expressed directly in terms of the beam slowing with distance, Re(Pk) = - 4v 3J v (x) )() - k v ) 002 _. r o, n[(w - k v )2 + k2v2] r o 10o (A.12) Similarly, from Eq. 2.58, the circuit power may be written as pT 4Ecu2v2 V v (x) x dS, T o 0 + 2 circuit a - a2(w - k v ) 2 r o (A.13) which gives the power (e.g., in watts) directly in terms of the beam slowing.

APPENDIX B. VERIFICATION OF THE CYCLOTRON-MODE KINETIC POWER Since a complex transformation has been used (Eq. 2.6) it should be verified that the power expressions obtained are physically valid. From Eqs. 2.46 and 2.47, invoking the exponential dependence exp[j(cwt - kx)], j(3 - kvo)vy C - cv 0 ly c1Zz ~/ kv \ - (_ ___~ ) Eo \ c J ly / kv \ = ] ( - - lZ ) E \ 0) / 1Z and (B.1) (B.2) j(a) - kv )v +- v o 1z c iy These equations having been used, the contribution to the source function is * * E, J + E J ly ly 1Z 1Z (= (w- kv )v v -C v )(wo - kv ) o ly ly c iz ly n 0 +j(( - kv)vlzvL + cv1 (B.) Since the circularly polarized modes are uncoupled, one of these modes is selected. e.g., (-) mode, so that the following definitions can be made (in isotropic media): v = V exp[j(ct - kx)] ly 1 and v IZ = V exp j ( - kx + ) 1 2 (B.4) where V is independent of x. The use of these definitions in Eq. B.3 provides the ollowing provides the following: -246 -

-247 - Re(E J + E J ) ly ly 1z 1Z 2k.x 2cQo IV J2 e TI Re j( -kv - Lv - )c) ) ( e -6()- kv (B) 0 Use of Eq. B.4 in Eq. 2.48 shows that v = 2V exp[j(ot - kx)] From Eq. 2.7 the following result is readily derived: +, j(lJ(Cl - kVo - * Re(EJ) = Re ( j. ( - kv P v- 1~ -: - ~(~- kv )1 -P 0. (B.6) (B.7) which from Eq. B.6 may be written as 2k. x 1, IV. I 1 Re(E J*) l- 1 - 4+ IV I' p e =, ] j( ( - kvo - c) Re (, ---—, --- \ ( - kv o (B.8) Comparison of Eqs. B.5 and B.8 shows that Re(E + EJ) Re(E J ) iy iy 1Z Z 2 - ( - (B.9) which verifies the physical significance of power system and explains the factor of one half in the Eq. 2.22. Taking the real part in Eq. B.8 gives terms in the transformed Poynting expression of Re (EJ ) = l- 1 - 4) okiVwOc IV 12 Lr(0 - kv0, 1-+ k r[C(6 - k-v)2 + k.v2] r o 10 (B.10) which is to be used in the kinetic power discussion.

-248 - An alternative proof which brings into play the dispersion relation can be shown. For isotropic systems, regardless of the number of carrier species present or the form of the interacting circuit, it is in general true that the dispersion relation for the purely transverse interaction may be written as ra jb E r 0 -b ab][Ely [ L (B-11) from which the following can be found: E E-. - (B.12) 1Z Equations B.l and B.2 are solved for the RF velocities of the sth carrier species in terms of the fields as (s) r s(@ l vos)Ely [c ) + j ] (B.13) y -[ ( - (D- kV )2] Os cs Cs os and ) ~rs (jo - kv )E V(s) s kvos )ly [- (U - kv )], (B.14) 1Z L... O - ( k 2- kv )Os cs v os where Eq. B.12 has been used. Equations B.13 and B.14 are then sufficient to obtain the following: (.E JC- 2+a) E z = [ 2w2 e +E i2 k.v Re(EJ +E J ) = (~ ) ( y15) R (yly ly +z z (B.cl) vlyjy 1z o)) w[(w - kv.- W )2 + k2v2 ] r OS Cs i Os this provides the same information as that of Eqs. B.9 and B.10.

If one returns to the (-) mode formulation these results indicate clearly the dynamical reasons why the modes with xc < 0 (slow-cyclotron mode, helicon mode, etc.) have the negative kinetic power property under growth conditions (ki > O). Since there are no collisions in the present analysis, taking the real part of Eq. B.3 shows directly that the only power exchange between the fields and the carriers arises due to the current J from the field E and the current J from the field E lz ly ly 1Z When ac < 0 and k. > 0, the viewpoint can be adopted that the field O 1 E (E ) acting through the static magnetic field produces a contribution iy iz to J (Jly) which is 180 degrees out of phase with Elz(Ely). Selfconsistently then, when k. > 0 and wc < 0, the carriers supply net power to the fields. It will be nshown in Chapter II that this power is ultimately derived from the dc carrier motion. Note the important result that this phenomenon is not explicitly dependent upon the sign of (a - kv ).

APPENDIX C. THE MEASUREMENT OF POWER UTILIZING THE SECOND-ORDER HALL EFFECT The nature of the second-order time-averaged dc electric field E 2 -is studied. With reference to Fig. C.1, a circularly polarized electromagnetic waveguide mode propagates through an undrifted n-type semiconductor sample (e.g., n-InSb). From Eq. 2.26, since v = 0 and for the closed system with one carrier species J = O, it must be that for the cold plasma model 1- 5E A E = - 2 - v x (C.1) within the sample. Equation 2.7 becomes, when collisions are introduced into the analysis, rE _ = - j( v - - ) )v. (C.2) From Eq. C.2, the result may be found for Eq. C.l: (a - O)ki - vk E - 2 c - IVr v 2 (C.) The second-order field E, although varying with distance in the direction 2 -of wave propagation, can be considered a second-order Hall electric field since it is set up in the same manner.as the well known zeroth-order phenomenon. Indeed, this field has been studied86 using linear polarization, no static magnetic field, and germanium samples to construct Hall-effect wattmeters for the measurement of RF power. -250 -

-251 - SOURCE WAVEGUIDE POLARIZED FOR CIRCULARLY MODE V L A Bo Bol IRIS FIG. C.1 UTILIZATION OF THE SECOND-ORDER HALL EFFECT TO DETECT ELECTROMAGNETIC POWER OF CIRCUIARLY POLARIZED WAVES.

-252 - Equation 2.19 in the presence of collisions with v = 0 gives k |i k 2 [(w -( +)2 + 2] Re(Pe ) = r.-......-. (C.4) Using this result in Eq. C.3 gives -n% [( - a)k - )k - vk ] E (x) = - Re[P (x)] (C.5) 2- - a.)2 + v2] e^ Since 2k.x P (x) = P (O) e, (C.6) el- elwhere Pe (0) is the electromagnetic power at the top surface of the sample and is the quantity to be measured (questions of reflected power from the front or back surfaces of the sample are ignored in this analysis), and k. < 0 corresponding to a decaying wave. The voltage developed across the sample of length L (switch S in Fig. C.1 is set to the V branch) is =L [(I - cu)k - krv] 2k Re[ V 2 - 2kr[(OC - _ )2 + v] v 2 e (c.7) Alternatively, by moving the switch S to the I branch, the shortcircuit current may be measured. Since E = 0, when collisions are 2 -introduced into Eq. 2.26, the following results: v - 1= ' (1 vE (-.8) v iRe [j - -.R (C.8) 2- v ' 2jw - dx

-253 - The approximate current drawn is I = v (o)A, (c.9) 2- 0 2 -where A is the device cross-sectional area. Use of Eqs. C.5 and C.8 in C.9 shows the following: O2A[(wo - cO)k. - vk ] TI P= i r Re[P (O)]. (C.10) 2- vc2k [(C - C)2 + v2] r c As an example, exactly at resonance (a = Yc) of the right-hand polarized A mode (B is then -x directed for n-type material), the current i's o m@2A I = - PRe[Pe (0)] (C.ll) 2- v2c2 el For example in indium antimonide at liquid nitrogen temperature when typical values of a moderately pure sample87 are used, a responsivity may be found as = ARe[?eI2,-, 2- 20 (C.12) R Re Pel ~- W/CM2 (.2 The absolute power density of the incident radiation may then be measured using this technique. In addition since the current should be a maximum at Cc = w, as the magnetic field is tuned, the frequency of the radiation is also measured. Because of the tuning properties of the magnetic field the device is inherently wideband in nature. Although only a local theory

-254 - is presented here, it may be possible to determine carrier effective masses of materials of poor transmissivity (e.g., semimetals or metals) by utilization of this technique, wherein the unknown sample becomes the detector itself and the operating frequency O is known. It is to be noted that many processes are known to possibly occur which should be taken into account in a more detailed analysis. Thus the second-order Hall current or voltage may be obscured by thermoelectric effects at the metal-semiconductor junctions, generation and consequent diffusion of electron-hole pairs if the incident photon energy exceeds the bandgap energy, hot-electron effects, rectification at the contacts, etc. For the warm plasma model a useful resonance is found which can enhance the voltage across the sample. The force equation in the one carrier case is given by v2 ap ( / E" T - 1 -P =x rlE + l Re 2j v1- av (C13) P0 -where aE = P- (C.14) 2k.x Since the second-order variables vary as e, Eqs. C.13 and C.14 can be solved similar to Eq. C.3 to obtain (c - o)k - V)rk.vk E = c --- -- v 12 (C.15) 2- Z V2 1 -2w(1 - 4kivT If Eqs. C.4 and C.6 are used, the open-circuit voltage is found as

-255 - Trt-o = - 2=2k.k i r 2k. L [(wc - w)k. - vk ][1 - e 1 ] - +_ k )v22 [(%) - eC)2 + v2] (1 - Re[P (0)] e~ (c.16) Implicit in solving for E in Eq. C.14 is the assumption that the surface 2 -recombination velocity of the top and bottom surfaces is zero. Inspection of Eq. C.16 shows that by operating with ki - Cp/2vT a resonance is possible leading to large voltages, with the peak possible voltage limited by the assumption p (0) << p inherent in the analysis. Within a restricted 2- 0 frequency range it should always be possible by variation of CDc, cp, or v c p to obtain k. — p/2vT. As an example of this consider the detection of the 10-p. line of the CO laser, so that ow 2 x 1014 rad/s. At such high 2 frequencies it can be assumed that wc << o, and consequently the dispersion relation is k2(cl + jv) kC2 - C2 + v2 ---- =0 (C.17) o2 + V2 CD V2 or (C.18) ( W ci2( + jv) \1/2 k = - 1 --- c \ ((2 + V2) From Eq. C.18, to satisfy the resonance condition k. = ow /2vT, it is ~~~Irequired that required that 2 2 v2[4 1/4 C (-___ + I- P c L K?+ iv2 a2(CW2 + v 2)2 i sin r tan21 2 sin VCD j 2vT2 L t D(A 2 + V2) - CDC ) 0

-256 -where w2 + v2 > C2 has been assumed. This equation can be readily satisfied p with materials with v c w, corresponding to a low mobility of the free carriers. In addition, since Cw k c, the condition of locality, i.e., |I - jv - C >> |IkvT, is well satisfied so that the use of the dispersion equation in the form of Eq. C.18 is justified.

APPENDIX D. A STUDY OF DECAYING MODES BY KINETIC POWER CONCEPTS For k. # 0, it is in general possible to cast many carrier interactions in the form Re -circuit+ k) ds =, (D.1) where Pcircuit is always positive (kr > 0) or zero and Pk is the kinetic power flow. In general it can be assumed that the power flows A are purely longitudinal for the purposes of analysis (e.g., x-directed) so that Eq. D.1 may be written for real c as 2ki(P irc+it + Pk) = 0 (D.2) Hence even for decaying interactions (ki < 0) the function Pk 0. As an example, utilizing this, consider the kinetic power flow of the longitudinal space-charge wave, Eq. 2.23, with drift velocity = 0 I 2k. rT p -- (D.3) k [W2 _ v2(k2 - k2)]2 + [cv + 2k.k v ]2 Inspection shows that the collisions provide Pk < 0 with k. < 0 and K 1 that the thermal diffusion limits the possible damping to Iki < wv/2k v2. For purely electrostatic interactions Pir t is zero a cr o. rT circuit zero and hence from Eq. D.2, P= 0 so that K -257 -

-258 -k. = _ V (D.4) 2krV2 r T from which it is seen that the general effects of diffusion are to reduce the damping. Hence, in the definition of kinetic power flow, if k. 0, the kinetic power flow is always negative for the interaction to proceed and in the presence of collisions knowledge of the sign of ki is required to assess the meaning of the function Pk.

APPENDIX E. ELECTROKINETIC ENERGY DENSITY OF SPACE-CHARGE WAVES TO SECOND ORDER The kinetic energy properties of longitudinal space-charge waves to second order can be derived in a similar manner to the kinetic power derivation in Appendix A. Consider first the case in which it can be assumed that k is purely real. Following Tonks,72 the carrier mode kinetic energy density may be defined as Wk pv* v (E.1) Since all variables are one dimensional the velocity to second order can be written as v = v + V' exp j(cnt - kx) + v (t) + V' exp 2j(cot - kx), (E.2) 0 1 2 2 where v, V', v (t), and V' are independent of x. Similarly for the o 1 2 2 charge density, p = pO + p' exp j(cwt - kx) + p (t) + p' exp 2j(ot - kx), (E.3) 1 2 where po, p', p (t) and p' are independent of x. Physical reasoning 0 1 2 2 shows that for the charge to be conserved properly, in the interaction region p (t) =. (E.4) 2 -259 -

-260 - If all products are retained to second order, Eqs. E.2, E.3, and E.4 provide in the interaction region, -o. tRe(pv v) = p v + 2p v v (t) + 2p v Vt cos(c t - kx) e -2w. t -2w. t + 2povV e 1 cos(2c rt - 2kx) + p (VT)2 e 1 cos2( t - kx) -W. t -2w. t + p'v2 e cos(o t - kx) + 2v V'p' e cos2(Crt - kx) 10 r 0 r -2Ow.t + p'v2 e cos(2w t - 2kx). (E.5) 20 r From Eq. E.5 the function' ~1 1 L <Wk 2> <pv v> 2rL pv v dx (E.6) can be obtained, which is a space average of the total carrier-mode kinetic energy density over the device length L. If L = N(X/2), where N is any positive integer greater than zero, 2 -2w..t -2w. t 2 <Wk> = v2 + e + 1p v (t) + vp p' e. (E.7) k 0 0 2 0 1 0 0 2 0 o1 1 Note that for L # N(X/2), Eq. E.7 is still an excellent approximation if L >>. As in Appendix A, MacColl's results69'85 when interpreted as a space average provide for the mean-square velocity in the interaction region, <v2> = V2 0 (E.8)

-261 - Use of this result, together with the squared and space-averaged value from Eq. E.2,. gives 2 -2w. t v (t) = - (V) e (E9) 0 2 1 Thus Eq. E.9 can be used in Eq. E.7 to obtain -2e. t =k 2rl p + 2 VoV e (E.10) k 2~ 0 0 11 1 which shows that for the purely longitudinal space-charge modes it is sufficient to know the first-order fields alone to determine the kinetic energy density properties. Comparison of Eq. E.10 with Eq. A.8 shows that bunching plays the same role in absolute as in convective instabilities. In the more general case with k. / 0, the same space-averaging techniques may be used. Whereas the case k. = 0 provided a minimum device length L = K/2, when k. # 0, this minimum length becomes a function of k.. The spatially growing nature of the carrier mode does not appear to play any fundamental role in the oscillation, however.65 As in the developments of Eqs. A.11 and A.12 it is possible using Eq. E.9 to express the kinetic energy density function <W > directly in terms of the beam slowing.

APPENDIX F. ELECTROKINETIC ENERGY DENSITY OF CYCLOTRON MODES From Eq. 3.14 the fundamental field source function may be written as j jp (wD-kv -cW )V V - - e 0 ^' J (F 1) Re(E J ) = Re ((F.1) so that, in general, p P Iv 12 -. [( r-k v )2+ k v +(wi -k.v )2]+k.v w Re(E J ) = - * -r 1- (cL -k v )2+(cLi-k.v )2 r r o i o (F.2) From Eq. 3.18 and with the dependence assumed, k.v [( -k )v 2+ k v +(c.-k.v)2] - L 0 IV 12 r r o c r o i i o a r c = POIV LJ __________________________ r c W = -, —. (F.3) k 4r1 (L -k v )2+(W.i-k.v )2 r r o 1 i o Thus in the case k. = 0, p IV I2 (L -k v )2+LU k v +u?2 W(ki = O) = 4.rc (F.4) k 1 4i (or-k v )2+. v r r o 1 i so that in this case, once again, only the polarization Lc < 0 corresponding to the slow-cyclotron mode, helicon mode, etc.,gives a negative contribution to the carrier mode kinetic energy and synchronism is preferred to optimize the magnitude of the electrokinetic energy density. -262 -

APPENDIX G. EFFECT OF CARRIER HEATING TRANSVERSE TO k ON THE HYBRID MODE For completeness in the study of the quasi-static hybrid mode the RF number density is found for a carrier which is part of an equilibrium A Ao At The carrier distribution function which includes carrier heati=g in the presence of the dc fields was found in Eq. 6.41 as N = _( 2 + V2 + V2 0 Of = -- - exp 0 /(2V2)3 2v2 2 (L VT ) 2vT n=-oo uvH (jn - v/c. 1 - T 1 + where vH Eoy/B v = v + u cos, and v th H oy ox y H then follows exactly that used for the case I UVH ) Jn n 2 T oc) sin 0 - cos 0 ],, (G.1) (jn - v/O c)2 = u sin 0. The method used A = E x resulting in -O OX N N = 1 D (G.2) where -v2^2 D = 1+ e H/ v2 T 00 00 ~, n=-co -u/2 /UV - [ kU \ - i H VH R C~c.v T - w - kvH - jv - 2w H ~~~c,Q,_n n Clku, L~~ F, ('ku 2 - - kujuvH [2(jn - v/ )Jf ku - ) c n ku H -L- j- 'lnku.. -n +...- - d; 2v2[1 + (jn.- v/())2] (G.3) -263 -

aouts 'pue HA mr asneoaq qsaasul. jo ST apotu sFTq,.-eadde Mou ueo apomu 0 = 7 UTI;Deq UT 9qq.F q qq; ST 1uTqaq alaxeo GLq jo qadds e quqLJodmiw q.som aqTLq (. 5) r> co ny )U- { [( ) Z =( n (.-^ 9 LInu n _ IP } [Z(%O/4 - UH) + -]A J ---— A = (Ul) 2 u-) L[rc yC -~-a 0 Kro) pue f] * { L Ap (l u ),~.ii~;. _ UD UI + — A - (T - U) )C IP [I(ml/ - UP) + T]At J HA ' ( — ["I:IM -c-u- x u-I~ i" n n u A. - =- LI ] u + _ ) np ~1. n J uIP J I (L'a> T '(A/HAn-) se pooq.sapun s.R6Mese I Jo usuLnS-le aq; qq1T^ ^ou aJaqm (+ H.) -[==T oo-=iiay np n - - - H (U')v ( I) 2 A a / 0~ OX. - = N Li H a HIAZ3/z

,-265 - VH X v in high mobility materials, interactions can occur at much smaller 0 wave numbers than the resonant interactions previously studied (e.g., k Iwc |/v ). This carrier-heated mode has the additional attribute ce oe for instability that the effect of the collisions is negligible if Icl >> v. To understand this, assume that |1cl >> v so that the resonance approximation is reasonable and in particular for I = 0 the effective collision frequency is given from Eq. G.3 by -V=2 co 2 H Vsg/S /_)2vp / Y( UV i V' = v - Xe ( ku I ( — H u du 2 VT 0 "IC 0 n 2 T n=-c' T. J ( f ) + -n c CD.. / T ku JuvH 2(jn - v/c)J'n ku )+2ju k) 2v~[l + (jn - v/LOC)2] I (G.6) Inspection of this result shows that the n = 0 and n = +1 terms give the largest contributions in the sum resulting in -vV/2V2 V e T 00 -u2/2 2 uv Re(v') v - 'H ee 2 ( ) ( vH ) 2v4 0o % c v2 T T -V2/2V2 H T e 2 T oO et2/2V2 e J 0 ku TH u du It 0T w v2. c, (G.7) J For a limiting case vH ->O (which corresponds to the disappearance of the I = 0 term in N) it can be found from Eq. G.7 that Re(v') 0 O. Hence for vH VT it is clear from the behavior of the Bessel functions H T

-266 - involved that for this more general case Re(v') < v. The possibility that Re(v') < 0 for some range of (vH/VT) is also noted. In addition inspection of the.i(l,n) function in Eq. G.5 shows that if I1c >> v the collision frequency v should play an unimportant role in N. Thus for the I = 0 mode with IC | > v the deleterious collisional effects are negligible and instability is to be expected. In the constant collision frequency approximation it is expected that these results are unaltered when an increased value of v is selected, corresponding to the carrier heating, provided I|c >> v is maintained. The physical reasoning for the appearance of the (interacting) I = 0 mode due to carrier heating is now discussed in relation,to the electron-hole interaction. From Eq. 6.118, when the effects of carrier heating are not included in the analysis, the dispersion equation in the resonance approximation for the electron-hole interaction is co 2 e Ie () e Im(h) + ~~ce pe (e C + chCph e h) 1+ + p k24e [ec - ( - kvo - jve)] k2 h [ch - (co -kvoh - jv' )] Te~lwce oe e VTh ch (G.8) If this equation is solved for I = O, it is readily seen that one root is given by o - kv - jvt = 0. This is a damped noninteracting electron oe e carrier wave given by Eq. 6.119 as XC = kv r oe and -I i = ve = v[1 - e e I()], (G.9) so that as he -,0 (i.e., B -* ), i 0 0 i

-267 - On the other hand the dispersion equation in the resonance approximation, for I = O, with the effects of carrier heating included, can be obtained from Eq. G.2, together with Poisson's equation (Eq. 6.67), as pe v2 // ku 1 + T j X exp (-,n u du VT~e VH e n=-co i=1 Te + [h] = 0, (G.10) wherein [h] represents the hole term which is not of direct significance. The effect of the carrier heating then is to perturb the I = 0 electron carrier wave away from the solution given by Eq. G.9 since it is now an interacting mode. Since the solution of Eq. G.9 shows a mode approaching marginal instability as the magnetic field increases, this perturbation need not be large to drive this root into instability. Note also that the adopted viewpoint48 in which the growth rate must exceed the hole collision frequency (when the dispersion equation is solved with vh = 0) is erroneous when a static magnetic field is present. The important effect of the carrier heating of the electrons then is that an extra degree of freedom of the system is permissible which enables the electron carrier wave to interact. From a quantum viewpoint, it is expected that this effect corresponds to the introduction of additional available energy states for the system enabling more general sets of motion by which the electrons can interact with the holes. It is again pointed out that because of the strong dependence of the coupling constant on the wave number (e.g., Eq. 6.79) for this

-268 -interaction in general, the carrier heated I = 0 mode interaction (with k - wr/v ) can readily dominate over the fundamental I = 1 mode r oe interaction [with k - () + I e )/ve when o << I |. ce oe ce It is hypothesized that the carrier-heated electron carrier wave is potentially the most important carrier mode in solid-state plasmas. This is because not only does it possess a small damping decrement in the noninteracting state, but also, unlike the helicon mode, it is not severely frequency limited.

APPENDIX H. EFFECTS OF CARRIER HEATING ON THE CYCLOTRON MODES The cyclotron modes (k II B II x) were studied in Chapter V, Section -o 5.3 neglecting carrier heating. If E I k it can be found by including E in Eqs. 5.41 through 5.44 that the right- and left-hand circularly -o polarized modes are now coupled. This case is difficult to analyze since there are then at least four RF current components, e.g., J (E ), J y(E ), iy iy ly z J (E ), and J (E z). Attention is therefore directed at the case E II k. By including this field in Eqs. 5.41 through 5.52 the circularly polarized components still separate and now polarized components still separate and now f j( c - kv - c - jv) l- X C av' ~nE x ox E kv fi- = - ( ~ oE o - - ox dG 1 dv _ x J (H.1) where again F (vx) 00 - f dv dv - o y z (v2 + v2)f dv dv J J y z o y z -00 (H.2) and 1 Go (vx) = (H.3) For self-consistency Eqs. H.2 and H.3 should be solved for the carrier distribution function fo which includes any carrier effects due to Eo This is done as follows; Since E II B II, the dc equation, -o -o -269 -

-270 - - 'f. +f - E * ~oL + 1L 0 v v ] af 61L + ri(v x B ). -- + vf -o a_ 1L =0 (H.4) where f = fL + f and fL is the known equilibrium carrier distribution o oL 1L oL function in the absence of any applied fields, becomes af, 1L av x + ( B OX / ~f1L v - -f ~E 1L ' ox (H.5) where the definitions were made V y = u sin 6 v z = u cos 0 (H.6) Then the functionsFo(vx) and G (vx) can be split up as 0 X 0o F (Vx) 0'x = F L(v) + F (v ) oL x 1L x 00 = f dv dv + oL y z -00 00 k f- dv J J L y -00 dv (H.7) z and Go(vx) = GL(vx) + GL(vx) =1 00 p2rt 12 J U3foL du dO ' OJ oL 2 u ufl du de By integrating Eq. H.5 over (y, vz) space it can be found that. (H.8) aF1L x v +E FL ~Eox 1L oL = - ~v x (H.9)

-271 - wherein use was made of f0 (f 1L/S)dG = 0. In a similar fashion premultiplying by u2 and integrating over (v y,v) space Eq. H.5 generates yz G L G oL + rlE- L (H.10) x ox x In principle then since foL is known (and hence FoL and GoL) Eqs. H.9 and H.10 can be solved to obtain Fo(vx) and G (vx). For example, if the oL function foL is selected as a Maxwellian, N / v2 + v2 + v2 f = - ~ exp( — Y ), (H.ll) OL (2 v2)3/2 K 2v2 T T it can be found by proceeding in the aforementioned manner that vN vv ( v v F(vx) = 2iOexp ( EOX) exp - ) 0 X 2 2 2E ~ x ox ox 2 2E2 ox [ erf ( \j -. T )+ C, (H.12) Y T ox where erf(x) is the error function of argument x and where C is a constant 1 which can be determined by the conservation of particles requirement r F (v )dv = N The point is clear, however, that the use of a complex -oo0 X X function such as Eq. H.12 in Eq. H.1 will lead to nontractable integrals in attempting to solve for f. For the purpose of analysis assume then that the system is such that the drifted Maxwellian carrier distribution function,

-272 - N f = exp o (2rv2 )3/2 T (V - o)2 2v2 T ru2 exp - -, L 2vT T (H.13) is appropriate even if E > B VT. Equations 5.91 and 5.92 can then be used in Eq. H.1 to obtain for this case af 1 -- + jaf Xv 1 -x E1-o ) (V - 2.-.-.... — exp.... E t p L 2v2 ox T T (H.14) where (D - kv - ) - jv x, a (H.15) When the integrating factor exp[f a(vx) dvx] is noted, Eq. H.14 is solved as /Ekv E N (l — ) 1- 0 \ -C f exp (-j(a v + i X L-" ox 2 vT v x C 2 ja v' ja (v')2 e X 2 X e e - (V' - V )2 > exp (- -v) dvt T2v, (H. 16) where C1 - CO - JV a = 1 Eox OX and a 2 k = -_2Eo *ox (H.17) and C is a constant to be determined. Equation H.16 can be integrated 2 directly to obtain

-273 - k- V E N l - /- + j i-p o \ cu y / \ v 1 f =..T exp (-j(a v + a v Y erf v - + C - 1 x 2 1 x - Y s o x VT 2 (H.18).V 2 V2 where ) 2 -F1a 7 -- exp.1 Y1 = 272exp p 1/2 2 2 / N11/2 7 = (k - ja 2 2v2 2 T and C is a new constant replacing C. Since it is necessary that f -> 3 2 1 -as v - ~+00 the function f exhibits a Stokes phenomenon characteristic of 1- the asymptotic expansion of analytic functions, with the result that f f- (v > 0) f (v 0), (H.19) in which the constant C has one value for v > 0 and another value for 3 x V < 0 such that in either half-space f -> 0 as I|v ->o. It is in this x _1- x manner that the constant C is determined in each half-space. The RF current 3 density is then obtained directly from J = q J f dv. Even for the 1- -oo00 - x result for f of the drifted Maxwellian distribution function, Eq. H.18, 1 -(which is the simplest possible distribution function which retains thermal effects) this is a formidable task necessitating numerical computation and has not been undertaken.

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UNCLASSIFIED Security Classification DOCUMENT CONTROL DATA R & D (Security classification of title, body of abstract and indexing annotation must be entered when the overall report Is classified) '. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION. The University of Michigan if Electron Physics Laboratory 2b. GROUP Ann Arbor MI 48104 I/A I. REPORT TITLE SOLID-STATE PLASMA ELECTROKINETIC POWER AND ENERGY RELATIONS 4. DESCRIPTIV., NOTES (Type.of report and.inctlsive, dates) Technica. Report (.nterim) 5. AUTHOR(S) (First name, middle initial, last name) J. 3. Soltis: I. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO.. OF REFS March 1972 285 87 Ba. CONTRACT OR GRANT NO. 9. ORIGINATOR'S REPORT NUMBER(S) F30602-71-C-0099 Job Order No. Technical Report 123 55730000 Task No, s9b. OTHER REPORT NO(S) (Any other numbers that may be assigned T1~~~~~~~~~~~~~~a, sk No. ~this report) 55T7303 d. ________________________________RADC-TR-72-54 10. DISTRIBUTION STATEMENT Approved for public release; distribution unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY RADC Project Engineer: Rome Air Development Center (OCTE) John V. McNamara (OCTE) Griffiss Air Force Base, New York 13440 AC 315 330-4251 I 13. ABSTRACT A general theory is developed for the electrokinetic power and energy properties associated with the basic carrier modes present in plasma media. Both hydrodynamic and kinetic theoretical models are obtained for the media in the presence of applied static electric and magnetic fields. In the hydrodynamic theory the effects of carrier collisions and thermal diffusion are properly accounted for and explained by developing a second-order quasi-linear analysis In this manner it is shown that the negative kinetic power property is directly related to dc slowing of the active carrier. The distinction between absolute and convective instabilities leads to the formulation of a space-averaged temporal-energy basis for determining the existence of absolute instabilities as compared to; timeaveraged spatial-power basis for convective instabilities. The analysis shows that it is possible to relate the causality criteria for instabilities developed by Briggs to the conservation of power and energy in the medium. Thus useful general information is obtained on the behavior of the root trajectories in the complex-k space as the imaginary part of the frequency is varied. The quasi-linear theory, as a by-product, allows the analysis of the second-order Hall effect and related phenomena in solids. In addition, a study of the physical meaning of the quasi-linear theory shows that this is a useful analytical tool for studying potential energy effects caused by the reaction of the growing RF fields on the carrier charges. This also enables the accuracy of the linear dispersion equation to be assessed. (Over) I vmO -.. - -...,..... I. I DD,FOM 1473 I NOV 65 UNCLASSIFIED Security Classification

UNIVERSITY OF MICHIGAN 3 9015 02829 9553 3 9015 02829 9553 TTNcLASSIFIED Security Classification. ~~~~14.~~ ~KEY WORDS LINK A LINK B LINK C KEY WORDS W ROLE I WT ROLE T ROLE WT I Solid-state plasma Transverse effects Solid-state interactions Convective and absolute instabilities Generalized power theorem ABSTRACT CONTINUED The power and energy theorems applied to the kinel of nonlocality, anisotropic carrier temperatures, possible the results obtained are rigorously compr hydrodynamics theory. By obtaining the respectiv results for the hybrid-hybrid electron-hole inter experimental work on the phenomenon of microwave ic tl and c Lred u. disi Lctior nmissi I I 1 3 I:ory d rriei th ti;rsior are r in frc 1 ab term: heat: )se o: equat Ilate< a indi i f i i I ie the ig. I the.ons, to pi im ant it!ompu' lishi.moni effev ienevf c e t e d Ir ts r UNCLASS IF I UNCLASSIFIED Security Classification