Nonequilibrium Magnons: Theory, Experiment, and Applications

191 Pages·2012·2.15 MB·English

by  Dr. Vladimir L. Safonov(auth.)

VladimirL.Safonov
NonequilibriumMagnons
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Vladimir L. Safonov
Nonequilibrium Magnons
Theory, Experiment, and Applications
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Thisbookisdevotedtothemoderntheoryofmagnons,quasiparticlesthatdescribequanta
ofspinwavesinmagneto-orderedmedia.
VII
Contents
Preface XI
1 HarmonicOscillatorsandtheUniversalLanguageofScience 1
1.1 HarmonicOscillator 1
1.1.1 ComplexCanonicalVariables 3
1.2 ClassicalRotation 4
1.2.1 ClassicalSpinandMagneticResonance 5
1.3 CollectiveVariablesandHarmonicOscillatorsink-space 8
1.3.1 ChainofMassesandSprings 8
1.3.2 ChainofMagneticParticles 9
1.4 Discussion 11
2 MagnonsinFerromagnetsandAntiferromagnets 13
2.1 PhenomenologicalDescription 14
2.1.1 MagnonsinaFerromagnet 14
2.1.1.1 Holstein–PrimakoffTransformation 16
2.1.1.2 TheSpectrumofMagnons 19
2.2 MicroscopicModeling 21
2.2.1 MagnonsinaTwo-SublatticeAntiferromagnet 21
2.2.1.1 Hamiltonian 21
2.2.1.2 SpectrumofMagnons 25
2.2.2 Magnon–MagnonInteractions 26
2.3 NuclearMagnons 28
2.4 MagnetoelasticWaves,QuasiPhonons 30
2.5 Discussion 33
3 RelaxationofMagnons 35
3.1 MasterEquation 35
3.2 RelaxationofBoseQuasiParticles 37
3.2.1 RelaxationProcessofHarmonicOscillators 37
3.2.2 Magnon–ElectronScattering 39
3.3 RelaxationviaanIntermediateDampedDynamicSystem 43
3.4 FerromagneticResonanceLinewidth 46
3.5 MagnonsandMacroscopicDynamicEquation 49
VIII Contents
3.5.1 LinearizedLandau–LifshitzEquation 50
3.6 RelaxationofCoupledOscillations 51
3.6.1 Example1:NuclearMagnons 53
3.6.2 Example2:MagnetoelasticOscillations 54
3.7 Discussion 57
4 MicrowavePumpingofMagnons 59
4.1 LinearTheory 60
4.1.1 FerromagneticResonance 61
4.1.2 ThresholdofParametricResonance 61
4.2 ParametricResonanceinaResonatorCavity 63
4.3 NonlinearSRTheory 67
4.4 ExperimentalTechniques 71
4.5 ExperimentalResults 73
4.5.1 EquivalentCircuit 74
4.5.2 SRTheoryandExperiment 76
4.5.2.1 ModulationResponse 79
4.6 Discussion 83
5 ThermodynamicDescriptionofStronglyExcitedMagnonSystem 85
5.1 PrincipalEquations 86
5.1.1 Hamiltonian 86
5.1.2 UnitaryTransformation 87
5.1.3 BogoliubovTransformation 88
5.1.4 EffectiveTemperatureT D0 90
eff
5.1.5 EffectiveTemperatureT ¤0 91
eff
5.1.5.1 MaximumofEntropy 93
5.2 ExactSolutions 94
5.2.1 TheEffectiveTemperature 96
5.2.1.1 InstantaneousSwitching 96
5.2.1.2 AdiabaticSwitching 97
5.2.1.3 ThermodynamicStability 97
5.2.2 CollectiveOscillations 98
5.3 MagnonPumpinginaResonator 100
5.4 Discussion 101
6 Bose–EinsteinCondensationofQuasiEquilibriumMagnons 103
6.1 BoseGasofMagnons 103
6.1.1 IdealBoseGas 103
6.1.2 MathematicalAnalogywithBEC 105
6.2 QuasiEquilibriumMagnons 105
6.2.1 IdealGasofQuasiEquilibriumMagnons 107
6.2.2 Example:IsotropicSpectrum 107
6.2.3 KineticEquations 109
6.2.3.1 TheCaseofT D T 111
eff
6.2.4 MagnonSystemwithBoseCondensate 113
Contents IX
6.2.5 MagnetodipoleEmissionofCondensate 114
6.3 FröhlichCoherence 115
6.4 Discussion 118
7 MagnonsinanUltrathinFilm 119
7.1 Model 120
7.1.1 MagneticEnergy 121
7.2 Magnons 122
7.2.1 MagnonInteractions 124
7.2.2 EffectiveFour-MagnonInteractions 125
7.3 Example 126
7.4 Discussion 129
8 CollectiveMagneticDynamicsinNanoparticles 131
8.1 Long-LivedStatesinaClusterofCoupledNuclearSpins 134
8.2 ElectronicSpins 136
8.3 Spin-EchoLogicOperations 138
AppendixA HarmonicOscillatorinQuantumMechanics 143
A.1 OperatorsofCreationandAnnihilation 143
A.1.1 UncertaintyPrinciple 144
A.1.2 CoherentStatesandUncertainties 145
AppendixB DipolarSums 147
AppendixC UnitaryTransformationsinWeaklyNonidealBoseGases 151
C.1 One-ComponentBoseGas 152
C.1.1 Three-BosonAnnihilation 154
C.1.2 TheConfluenceandDecayProcesses 155
C.2 Two-ComponentBoseGas 156
C.3 ConcludingRemarks 158
AppendixD MagnetizationDynamicEquation 159
AppendixE AParametricPairSingle-ModeRealization 163
E.1 ASingle-ModeRepresentation 164
E.2 Example 166
AppendixF SmallSignalAmplificationandPreventiveAlarmNeartheOnsetofa
DynamicInstability 167
AppendixG NoisyPumpingofCoherentParametricPairs 173
G.1 ExperimentalProcedure 174
G.2 ResultsandDiscussion 175
G.2.1 Discussion 177
References 179
Index 189
XI
Preface
In this book we shall consider magnons, which are quasi particles that describe
quantaofspinwavesinmagneto-orderedmedia.Theconceptofmagnonsasthe
elementaryexcitationsinferromagneticcrystalswasfirstintroducedbyFelixBloch
in1930.Usingthisconcept,onecanrepresentperturbationsofthemagneticma-
terialasanevolutionofmagnongasexcitedtosomedefinitelevelor,classically,as
anevolutionofspinwaves.Thistheoreticalapproachhelpstoexplainmanyther-
modynamic,dynamicandkineticphenomenainmagneticmaterials.
Magnonsareharmonicoscillatorsink-space.Inotherwords,magnonsaregood
bosons. Recent experimental discovery of Bose–Einstein condensation of quasi
equilibriummagnonsinYIGfilmatroomtemperaturedirectlydemonstratedthe
quantum natureof the elementarymagneticexcitationsin magneto-ordered sys-
tems.Theexperimentsshowedthatthereareconditionsatwhichthemacroscopic
quantum phenomenon occurs in the situation when just a classical behavior of
spinwaveswasexpected.
Inthisbookweusethegenerallanguageofharmonicmotioninapplicationto
magnetic dynamicphenomena in magneto-ordered systems. We shall be able to
analyzethemagneticdynamicsfromamoregeneralpointofviewandfindsome
useful mathematical analogies with other dynamic systems. On the other hand,
forthosewhodonotworkdirectlywithmagneticsystems,thisbookcanprovide
severalideasandanalogiesthatcanbeusefulintheirworkandresearch.
Thebookcontainsalargeamountoforiginaltheoreticalresultsandtheircom-
parisonwiththeexperimentaldata.Itcompriseseightchaptersandsevenappen-
dices.
Chapter 1 is introductory. It presents the general and universal mathematical
languageofharmonicoscillators,universallanguagethatunitesallBose-excitation
invariousphysicalsystemsandhelpsfindmathematicalanalogiesbetweenquite
differentphenomena.Ourgoalistobrieflydiscussbasicharmonicoscillatorprop-
ertiesanddemonstratesimplicityandefficiencyofthisapproach.Thehopeisthat
thisinformationwillmakeitpossibletoreadthesucceedingchapterswithoutcon-
sultingspecializedliterature.
Basiccalculationsofphenomenologicalandmicroscopicapproachestodescribe
the spin system of a ferromagnet and an antiferromagnet in terms of magnons
aregiveninChapter2.Wefindcanonicaltransformationsbetweenthespindevia-