Pf MICROWAVE FERRITES AND FERRIMAGNETICS BENJAMIN LAX, Ph.D. • Head, 8<Ш State Division, Lincoln Laboratory, Massachusetts Institute of Technology; Director, National Magnet Laboratory, Massachusetts Institute of Technology KENNETH J. BUTTON • Physieut, Research Staff, Lincoln Laboratory, Massachusetts Institute of Technology McGRAW-HILL BOOK COMPANY, INC. New York San Francisco Toronto London CONTENTS CHAPTER 1, History of Ferrite Research and Applications 1 1-1. Basic Properties of Ferrimagnetic Materials 1 Early History. Neel Model of Ferrimagnetism. Magnetic Spectrum. Ferrimagnetic Oxides. Single-crystal Preparation. 1-2. Ferromagnetic Resonance 15 Phenomenological Theory and Early Observations. Spectroseopic- splitting Factor (g Factor). Resonance in Ferrimagnetic Substances. Tensor Properties of Ferrites. Magnetostatic Modes. Spin Waves. Resonance in Highly Anisotropic Materials. 1-3. Ferromagnetic Relaxation 24 Ferromagnetic-resonance Line Width. Role of Spin Waves in Damping of the Precession. Line Broadening by Rapidly Relaxing Impurities. Relaxation Parameters in Ferrites and Garnets. 1-4. Microwave Applications and Related Theory 28 The Faraday Effect. Rectangular-waveguide Components. Coaxial- transmission-line Components. Nonlinear Devices. Specialized Applications. Performance of Ferrite Devices. CHAPTER 2. Paramagnetism and Ferromagnetism 47 2-1. The Magnetic Atom and Ion 48 2-2. Paramagnetism 52 The Paramagnetic Ion in a Host Crystal. The Langevin Equation. The Brillouin Function. 2-3. Ferromagnetism 63 The Ferromagnetic Solid. Phenomenological Theory of Ferromag­ netism. The Physical Basis of Domains. APPENDIX 2-1. Derivation of the Brillouin Function 89 APPENDIX 2-2. Size of Closure Domains 90 CHAPTER 3. Ferrimagnetism and Antiferromagnetism 92 3-1. Antiferromagnetism 94 Superexchange Interaction. The Phenomenological Theory of Anti- f erromagnet ism. 3-2. Ferrimagnetism 103 Molecular-field Theory; Paramagnetic Region. Molecular-field Theory; Ferrimagnetic Region. 3-3, Ferrimagnetic Oxides 114 xi xii CONTENTS Pure Ferrites. Mixed Ferrites. Substitution Ferrites. Magnetic Garnets. Magnetoplumbites. CHAPTER 4. Ferromagnetic Resonance 145 4-1. Spin Resonance 145 Precessional Motion. The Resonance Condition. The Susceptibility Tensor. 4-2. Damping 151 Bloch-Bloembergen Form. Landau-Lifshitz Form. Susceptibility, Including Losses. 4-3. Internal D-C Magnetic Field 157 Demagnetizing Factors. Anisotropy. Polycrystalline Ferrites. 4-4. Resonance Line Width 167 4-5. Spin Waves 169 The Physical Description of Spin Waves. Spin-wave Modes. 4-6. Magnetostatic Modes 180 Experimental Excitation of Nonuniform Precession. The Frequency Limits of Walker Modes. Experimental Identification. 4-7. Resonance in Highly Anisotropic Crystals 189 Smit's Generalized Resonance p'ormula. Experimental Verification in Ferroxdure and Cobalt Ferrite. APPENDIX 4-1. Outline of Derivation of Effective-susceptibility Components 194 Including Loss and Demagnetizing Factors CHAPTER 5. Relaxation and Nonlinear Effects 197 5-1. A Summary of the Problem 198 5-2. Experiments on Resonance Broadening by Spin Waves 200 Transfer of Energy to Degenerate Spin Waves. Line Broadening by Imperfections. Scattering to High-& States. 5-3. Instability of the Magnetization at High Power Levels 206 The Classical Instability. Spin-wave Instabilities. The State of the Magnetization beyond the Instability Threshold. 5-4. Initial Decline of the Main Resonance 219 5-5. Line Broadening by Rapidly Relaxing Impurities 227 5-6. Measurement of Relaxation Parameters 231 Fletcher-LeCraw-Spencer Form of the Relaxation. Relation to the Bloch-Bloembergen and Landau-Lifshitz Forms of the Damping. Measured Values. Spin-Lattice Relaxation Ti. APPENDIX 5-1. Derivation of Fletcher-Le Craw-Spencer Form of the Relaxation 241 CHAPTER 6. Ferrimagnetic and Antiferromagnetic Resonance 244 6-1. Vector Treatment of Ferrimagnetic Resonance 245 The Ferromagnetic Limit. The Ferrimagnetic Limit (Exchange Resonance). 6-2. Ferrimagnetic Equations of Motion 251 Resonant Frequencies. Magnetic Susceptibility. 6-3. Antiferromagnetic Resonance 253 Modes of Resonance. Magnetic Susceptibility. Resonance at T > 0. Demagnetizing Factors. H0 at Arbitrary Angle. 6-4. Ferrimagnetic Resonance . 259 CONTENTS xiii 6-5. Damped Resonance 262 6-6. Experimental Results 266 Antiferromagnet ic Resonance. Ferrimagnetic Resonance. APPENDIX 6-1. Approximate Ferrimagnetie-resonance Frequencies 291 APPENDIX 6-2. Resonant Frequencies of a Two-sublattice System 292 APPENDIX 6-3. Resonance in Spiral Spin Configurations 294 CHAPTER 7. Plane-wave Propagation 297 7-1. Infinite Ferrite Medium 298 Longitudinal Field. Transverse Field. 7-2. Semi-infinite Medium 306 Longitudinal Propagation. Transverse Propagation. 7-3. Metal-backed Slab 311 7-4. Unbacked Slab 313 Transmission. Faraday Rotation. 7-5. Spin Waves and Magnetostatic Modes , 317 APPENDIX 7-1. Derivation of the Absorption in a Metal-backed Ferrite Slab 321 CHAPTER 8. Perturbation Theory 323 8-1. Ferrites in Cavities 324 Fundamental Perturbation Theory. Scalar Susceptibility. Tensor Susceptibility. Depolarizing and Demagnetizing Factors. 8-2. Ferrites in Waveguides 335 The Propagation Constant. Faraday Rotation. Rectangular Waveguide. Birefringence. The Reciprocal Phase Shifter. 8-3. The Applicability of Perturbation-theory Results 352 APPENDIX 8-1. Methods of Internal-field Approximation 353 CHAPTER 9. Modes in Waveguides and Cavities 355 9-1. TE-mode Solutions for Rectangular Waveguide 356 General Solution. Ferrite Slab against the Waveguide Wall. Dielec­ tric-loaded Ferrite Phase Shifter. Twin-slab Phase Shifter. The Com­ pletely Filled Rectangular Waveguide. 9-2. Anomalous-mode Propagation in Rectangular Waveguide 383 The Ferrite-Dielectric Mode. Higher Anomalous Gyromagnetic Modes. 9-3. Cylindrical Waveguide Containing Longitudinally Magnetized b'errite 399 General Solution. The Parallel-plane Waveguide. Solution of Com­ pletely Filled Round Guide. Partially Filled Round Guide. 9-4. Ferrit e-Ioaded Cavity Resonators 414 The Parallel-plate Cavity. The Cylindrical Cavity Containing an Axially Magnetized Ferrite Rod. 9-5. Nonuniform Modes in Finite Samples 421 Electromagnetic Modes in Post Resonator. Magnetostatic Modes in Bounded Media. Volume and Surface Magnetostatic Modes. APPENDIX 9-1. The Method for Evaluating the Microwave Field Patterns 433 APPENDIX 9-2. The Transcendental Equation for the Twin-slab Nonreciprocal 434 Phase Shifter CHAPTER 10. Fundamental Measurement Techniques and Results 435 10-1. Permeability Measurements 435 Static Magnetism. High-frequency Measurements and Results. xiv CONTENTS 10-2. Microwave-cavity Measurement Techniques 450 Scalar-susceptibility Measurements. Measurement of Tensor-suscepti­ bility Components. Measurements on Single Crystals. Susceptibility Measurements by Faraday Rotation. Experimental Excitation of Non­ uniform Modes. 10-3. Physical Interpretation of Resonance Measurements 473 Fundamental Parameters. Geometrical Effects. 10-4. Measurement of Relaxation Times 488 Paramagnetic-relaxation Parameters. Ferromagnetic-relaxation Parameters. 10-5. Waveguide Measurements 496 Faraday Rotation. Rectangular-waveguide Phase Shift and Attenua­ tion. Mode Patterns. APPENDIX 10-1. Detailed Microwave-cavity Measurement Techniques 502 Crystal Matching. Balancing of Directional Coupler System. Deter­ mination of (VSWR)„ of Cavity. Cavity-coupling Hole. Effects of Sweep Speed. Vibrating Switch. Relations between VSWR and Attenuator Setting. Half-power Points; Ferrite Line-width Measure­ ments. APPENDIX 10-2. Intrinsic Susceptibilities 505 СНАРТЕЕ 11. Scattering-matrix Analysis of Ferrite Devices 506 11-1. Formulation of the Scattering Matrix 508 11-2. Properties of the Scattering Matrix 510 Conservation of Energy. Reciprocity. 11-3. Reciprocal Junctions 511 Lossless Two-port Junction. Symmetrical Directional Coupler. The Magic T. 11-4. Nonreciprocal Junctions 515 Lossless Two-port Junction. Circulators. 11-5. Analysis of Multicomponent Circulators 522 Scattering-matrix Analysis of the Four-port Circulator. Transfer-matrix Analysis of the Four-port Circulator. 11-6. Deviations from Ideal Circulator Elements 534 Deviation in Ferrite Phase Shift. Deviation in Coupling. Deviations in Both Phase Shift and Coupling. APPENDIX 11-1. Asymmetry of Scattering Matrix for Nonreciprocal Junction 537 APPENDIX 11-2. Directivity of a Symmetrical Directional Coupler 539 CHAPTER 12. Microwave Devices 54Ü 12-1. Faraday-rotation Devices: Physical Description 544 Circulator. The Faraday-rotation Isolator. Amplitude Modulator or Switch. Frequency Modulators and Carrier-suppressed Single-sideband Generators. 12-2. Faraday Rotator: Performance Characteristics 549 Temperature Limitations. Broadbanding. The Figure of Merit. Low- frequency Limit. Ellipticity. Energy-concentration Effects. High R-F Field Intensity. 12-3. Resonance Isolator: Physical Description 563 Circular-waveguide Resonance Isolator. Rectangular-waveguide Reso­ nance Isolators. Coaxial Resonance Isolator. Resonance Isolator in CONTENTS XV Helical Line. The Miniature UHP Isolator. Other Resonance-isola­ tor Structures. 12-4. Resonance Isolator: Performance Characteristics 571 Optimum Geometrical Parameters. Frequency, Power, and Loss Char­ acteristics. 12-5. Nonreciprocal Phase Shifter 589 Rectangular-waveguide Circulator. Frequency and Loss Characteristics. 12-6. Millimeter-wavelength Components 598 Antiferromagnetic-resonance Isolator. Antiferromagnetic Phase Shifter. 12-7. Reciprocal Ferrite