# Physics of Magnetic Thin Films: Theory and Simulation

I: Basic Theory of MagnetismSpin: Origin of MagnetismIntroductionParamagnetism of a Free-Electron GasParamagnetism of a System of Free AtomsDiamagnetism of Many-Electron AtomsMagnetic Interactions in SolidsExchange Interaction: Origin of MagnetismSpin Models: Magnetic MaterialsHeisenberg ModelIsing, XY, Potts ModelsConclusionProblemsMean-Field Theory of Magnetic MaterialsMean-Field Theory of FerromagnetsMean-Field EquationMean-Field Critical TemperatureGraphical SolutionSpecific HeatSusceptibilityValidity of Mean-Field TheoryAntiferromagnetism in Mean-Field TheoryMean-Field TheorySpin Orientation in a Strong Applied Magnetic FieldPhase Transition in an Applied Magnetic FieldFerrimagnetism in Mean-Field TheoryConclusionProblemsTheory of Spin WavesSpin Waves in FerromagnetsClassical TreatmentQuantum Spin Wave Theory: Holstein–Primakoff ApproximationProperties at Low TemperaturesMagnetizationEnergy and heat capacitySpin Waves in AntiferromagnetsDispersion RelationProperties at Low TemperaturesEnergyMagnetization at low temperaturesSpin Waves in FerrimagnetsSpin Waves in HelimagnetsConclusionProblemsGreen’s Function Theory in MagnetismGreen’s Function MethodDefinitionFormulationFerromagnetism by the Green’s Function MethodEquation of MotionDispersion RelationMagnetization and Critical TemperatureAntiferromagnetism by the Green’s Function MethodGreen’s Function Method for Non-Collinear MagnetsConclusionProblemsTheory of Phase Transitions and Critical PhenomenaIntroductionSymmetry Breaking: Order ParameterOrder of a Phase TransitionCorrelation Function: Correlation LengthCritical ExponentsUniversality ClassImproved Mean-Field Theory: Bethe’s ApproximationLandau–Ginzburg TheoryMean-Field Critical ExponentsCorrelation FunctionCorrections to Mean-Field TheoryRenormalization GroupTransformation of the Renormalization Group: Fixed PointRenormalization Group Applied to an Ising-Spin ChainMigdal–Kadanoff Decimation Method: Migdal–Kadanoff Bond-Moving ApproximationTransfer-Matrix MethodPhase Transition in Particular SystemsExactly Solved Spin SystemsKosterlitz–Thouless TransitionFrustrated Spin SystemsConclusionProblemsMonte Carlo Simulation: Principle and ImplementationPrinciple of Monte Carlo SimulationSimple SamplingImportance SamplingImplementation: Construction of a Computer ProgramPhase Transition as Seen in Monte Carlo SimulationsFinite-Size Scaling LawsSecond-Order Phase TransitionFirst-Order Phase TransitionError EstimationsAutocorrelationSize Effects on ErrorsAdvanced Techniques in Monte Carlo SimulationsCluster-Flip AlgorithmHistogram MethodMultiple-Histogram TechniqueWang–Landau Flat-Histogram MethodConclusionII: Magnetism of Thin FilmsExactly Solved Frustrated Models in Two DimensionsIntroductionFrustrationDefinitionNon-Collinear Ground-State Spin ConfigurationsExactly Solved Frustrated ModelsExample of the Decimation MethodDisorder Line, ReentrancePhase DiagramKagomé LatticeModel with NN and NNN interactionsGeneralized Kagomé latticeCentered Honeycomb LatticeCentered Square LatticesOther Exactly Solved ModelsEvidence of Partial Disorder and Reentrance in Non-Solvable Frustrated SystemsRe-Orientation Transition in Molecular Thin Films: Potts Model with Dipolar InteractionTwo-Dimensional CaseThin FilmsEffect of Surface Exchange InteractionConclusionSpin Wave Theory for Thin FilmsSurface EffectsSurface Effects in MagnetismSurface MagnonsReconstruction of Surface Magnetic OrderingSurface Phase TransitionSemi-Infinite SolidsSpin Wave Theory in Ferromagnetic FilmsMethodFilm of stacked triangular latticesFilm of simple cubic latticeFilm of body-centered cubic latticeResultsSpin wave spectrumLayer magnetizationsAntiferromagnetic FilmsFilms of Simple Cubic LatticeFilms of Body-Centered Cubic LatticeConclusionProblemsFrustrated Thin Films of Antiferromagnetic FCC LatticeIntroductionModel and Classical Ground-State AnalysisMonte Carlo ResultsGreen’s Function ResultsConcluding RemarksHeisenberg Thin Films with Frustrated SurfacesIntroductionModelHamiltonianGround StateGreen’s Function MethodFormalismPhase Transition and Phase Diagram of the Quantum CaseMonte Carlo ResultsConcluding RemarksPhase Transition in Helimagnetic Thin FilmsIntroductionModel and Classical Ground StateGreen’s Function MethodGeneral Formulation for Non-Collinear MagnetsBCC Helimagnetic FilmsSpin Waves: Results from the Green’s Function MethodSpectrumSpin Contraction at T = 0 and Transition TemperatureLayer MagnetizationsEffect of Anisotropy and Surface ParametersEffect of the Film ThicknessClassical Helimagnetic Films: Monte Carlo SimulationSimple Cubic Helimagnetic FilmsConclusionPartial Phase Transition in Helimagnetic Thin Films in a FieldIntroductionModel: Determination of the Classical Ground StatePhase TransitionResults of 12-Layer FilmEffects of the Film ThicknessQuantum Fluctuations, Layer Magnetizations and Spin Wave SpectrumConclusionSpin Waves in Systems with Dzyaloshinskii-Moriya InteractionIntroductionModel and Ground StateSelf-Consistent Green’s Function Method: FormulationTwo and Three Dimensions: Spin Wave Spectrum and MagnetizationThe Case of a Thin Film: Spin Wave Spectrum, Layer MagnetizationsDiscussion and Experimental SuggestionConcluding RemarksSkyrmions in Thin FilmsIntroduction: Magnetic Field Effect, Excitations of SkyrmionsModel and Ground StateSkyrmion Crystal: Phase TransitionStability of Skyrmion Crystal at Finite TemperaturesSkyrmion Crystal: Effect of Lattice ElasticityTriangulated Lattices and Skyrmion ModelResultsConclusionSkyrmions in SuperlatticesIntroductionModel and Ground StateModelGround StateGround state in zero magnetic fieldGround state in applied magnetic fieldSkyrmion Phase Transition: Monte Carlo ResultsSpin Waves in Zero FieldMonolayerBilayerFrustration Effect: J1 − J2 modelModelGround StateSkyrmion Phase TransitionConclusionThin Films and CriticalityIntroductionModel and MethodModelMultiple-Histogram TechniqueThe Case of Films with Finite ThicknessResults: Critical ExponentsFinite-Size ScalingLarger Sizes and Correction to ScalingRole of Boundary ConditionsCrossover from First- to Second-Order Transition in a Frustrated Thin FilmModel and Ground-State AnalysisMonte Carlo ResultsCrossover of the phase transitionFilm with 4 atomic layers(Nz = 2)Concluding RemarksSpin Resistivity in Thin FilmsIntroductionModelInteractionsChoice of Parameters and UnitsSimulation MethodSpin Resistivity in Ferromagnets and AntiferromagnetsSpin Resistivity in Frustrated SystemsSimple Cubic J1 − J2 ModelFully Frustrated Face-Centered Cubic AntiferromagnetResults for the Ising caseSurface EffectsResults for the Heisenberg CaseRemarksSurface Effects in a MultilayerThe Case of MnTeConclusionIII: Solutions to Exercises and ProblemsSolutions to Exercises and ProblemsSolutions to Problems of Chapter 1Solutions to Problems of Chapter 2Solutions to Problems of Chapter 3Solutions to Problems of Chapter 4Solutions to Problems of Chapter 5Solutions to Problems of Chapter 8A.1 IntroductionA.2 Isolated Systems: Microcanonical DescriptionA.2.1 Fundamental PostulateA.2.2 ApplicationsA.2.2.1 Two-level systemsA.2.2.2 Classical ideal gasA.3 Systems at Constant Temperature: Canonical DescriptionA.3.1.1 Two-level systemsA.3.1.2 Classical ideal gasA.4 Open Systems at Constant Temperature: Grand-Canonical DescriptionA.4.1 ApplicationsA.5 Fermi–Dirac and Bose–Einstein StatisticsA.6 Phase Space: Density of StatesA.6.1 DefinitionA.6.2 Density of States of a Free Particle in Three DimensionsA.7 Properties of a Free Fermi Gas at T = 0A.7.1 Fermi EnergyA.7.2 Total Average Kinetic EnergyA.8 Properties of a Free Fermi Gas at Low TemperaturesA.8.1 Sommerfeld’s ExpansionA.8.2 Chemical Potential, Average Energy and Calorific CapacityA.9 Free Fermi Gas at the High-Temperature LimitAppendix B Second QuantizationB.1 First Quantization: Symmetric and Antisymmetric Wave FunctionsB.2 Second Quantization: Representation of Microstates by Occupation NumbersB.2.1 The Case of BosonsB.2.2 The Case of Fermions