Physics of Magnetic Thin Films: Theory and Simulation - Physics of Magnetic Thin Films: Theory and Simulation -

I: Basic Theory of Magnetism Spin: Origin of Magnetism Introduction Paramagnetism of a Free-Electron Gas Paramagnetism of a System of Free Atoms Diamagnetism of Many-Electron Atoms Magnetic Interactions in Solids Exchange Interaction: Origin of Magnetism Spin Models: Magnetic Materials Heisenberg Model Ising, XY, Potts Models Conclusion Problems Mean-Field Theory of Magnetic Materials Mean-Field Theory of Ferromagnets Mean-Field Equation Mean-Field Critical Temperature Graphical Solution Specific Heat Susceptibility Validity of Mean-Field Theory Antiferromagnetism in Mean-Field Theory Mean-Field Theory Spin Orientation in a Strong Applied Magnetic Field Phase Transition in an Applied Magnetic Field Ferrimagnetism in Mean-Field Theory Conclusion Problems Theory of Spin Waves Spin Waves in Ferromagnets Classical Treatment Quantum Spin Wave Theory: Holstein–Primakoff Approximation Properties at Low Temperatures Magnetization Energy and heat capacity Spin Waves in Antiferromagnets Dispersion Relation Properties at Low Temperatures Energy Magnetization at low temperatures Spin Waves in Ferrimagnets Spin Waves in Helimagnets Conclusion Problems Green’s Function Theory in Magnetism Green’s Function Method Definition Formulation Ferromagnetism by the Green’s Function Method Equation of Motion Dispersion Relation Magnetization and Critical Temperature Antiferromagnetism by the Green’s Function Method Green’s Function Method for Non-Collinear Magnets Conclusion Problems Theory of Phase Transitions and Critical Phenomena Introduction Symmetry Breaking: Order Parameter Order of a Phase Transition Correlation Function: Correlation Length Critical Exponents Universality Class Improved Mean-Field Theory: Bethe’s Approximation Landau–Ginzburg Theory Mean-Field Critical Exponents Correlation Function Corrections to Mean-Field Theory Renormalization Group Transformation of the Renormalization Group: Fixed Point Renormalization Group Applied to an Ising-Spin Chain Migdal–Kadanoff Decimation Method: Migdal–Kadanoff Bond-Moving Approximation Transfer-Matrix Method Phase Transition in Particular Systems Exactly Solved Spin Systems Kosterlitz–Thouless Transition Frustrated Spin Systems Conclusion Problems Monte Carlo Simulation: Principle and Implementation Principle of Monte Carlo Simulation Simple Sampling Importance Sampling Implementation: Construction of a Computer Program Phase Transition as Seen in Monte Carlo Simulations Finite-Size Scaling Laws Second-Order Phase Transition First-Order Phase Transition Error Estimations Autocorrelation Size Effects on Errors Advanced Techniques in Monte Carlo Simulations Cluster-Flip Algorithm Histogram Method Multiple-Histogram Technique Wang–Landau Flat-Histogram Method Conclusion II: Magnetism of Thin Films Exactly Solved Frustrated Models in Two Dimensions Introduction Frustration Definition Non-Collinear Ground-State Spin Configurations Exactly Solved Frustrated Models Example of the Decimation Method Disorder Line, Reentrance Phase Diagram Kagomé Lattice Model with NN and NNN interactions Generalized Kagomé lattice Centered Honeycomb Lattice Centered Square Lattices Other Exactly Solved Models Evidence of Partial Disorder and Reentrance in Non-Solvable Frustrated Systems Re-Orientation Transition in Molecular Thin Films: Potts Model with Dipolar Interaction Two-Dimensional Case Thin Films Effect of Surface Exchange Interaction Conclusion Spin Wave Theory for Thin Films Surface Effects Surface Effects in Magnetism Surface Magnons Reconstruction of Surface Magnetic Ordering Surface Phase Transition Semi-Infinite Solids Spin Wave Theory in Ferromagnetic Films Method Film of stacked triangular lattices Film of simple cubic lattice Film of body-centered cubic lattice Results Spin wave spectrum Layer magnetizations Antiferromagnetic Films Films of Simple Cubic Lattice Films of Body-Centered Cubic Lattice Conclusion Problems Frustrated Thin Films of Antiferromagnetic FCC Lattice Introduction Model and Classical Ground-State Analysis Monte Carlo Results Green’s Function Results Concluding Remarks Heisenberg Thin Films with Frustrated Surfaces Introduction Model Hamiltonian Ground State Green’s Function Method Formalism Phase Transition and Phase Diagram of the Quantum Case Monte Carlo Results Concluding Remarks Phase Transition in Helimagnetic Thin Films Introduction Model and Classical Ground State Green’s Function Method General Formulation for Non-Collinear Magnets BCC Helimagnetic Films Spin Waves: Results from the Green’s Function Method Spectrum Spin Contraction at T = 0 and Transition Temperature Layer Magnetizations Effect of Anisotropy and Surface Parameters Effect of the Film Thickness Classical Helimagnetic Films: Monte Carlo Simulation Simple Cubic Helimagnetic Films Conclusion Partial Phase Transition in Helimagnetic Thin Films in a Field Introduction Model: Determination of the Classical Ground State Phase Transition Results of 12-Layer Film Effects of the Film Thickness Quantum Fluctuations, Layer Magnetizations and Spin Wave Spectrum Conclusion Spin Waves in Systems with Dzyaloshinskii-Moriya Interaction Introduction Model and Ground State Self-Consistent Green’s Function Method: Formulation Two and Three Dimensions: Spin Wave Spectrum and Magnetization The Case of a Thin Film: Spin Wave Spectrum, Layer Magnetizations Discussion and Experimental Suggestion Concluding Remarks Skyrmions in Thin Films Introduction: Magnetic Field Effect, Excitations of Skyrmions Model and Ground State Skyrmion Crystal: Phase Transition Stability of Skyrmion Crystal at Finite Temperatures Skyrmion Crystal: Effect of Lattice Elasticity Triangulated Lattices and Skyrmion Model Results Conclusion Skyrmions in Superlattices Introduction Model and Ground State Model Ground State Ground state in zero magnetic field Ground state in applied magnetic field Skyrmion Phase Transition: Monte Carlo Results Spin Waves in Zero Field Monolayer Bilayer Frustration Effect: J1 − J2 model Model Ground State Skyrmion Phase Transition Conclusion Thin Films and Criticality Introduction Model and Method Model Multiple-Histogram Technique The Case of Films with Finite Thickness Results: Critical Exponents Finite-Size Scaling Larger Sizes and Correction to Scaling Role of Boundary Conditions Crossover from First- to Second-Order Transition in a Frustrated Thin Film Model and Ground-State Analysis Monte Carlo Results Crossover of the phase transition Film with 4 atomic layers(Nz = 2)Concluding Remarks Spin Resistivity in Thin Films Introduction Model Interactions Choice of Parameters and Units Simulation Method Spin Resistivity in Ferromagnets and Antiferromagnets Spin Resistivity in Frustrated Systems Simple Cubic J1 − J2 Model Fully Frustrated Face-Centered Cubic Antiferromagnet Results for the Ising case Surface Effects Results for the Heisenberg Case Remarks Surface Effects in a Multilayer The Case of MnTe Conclusion III: Solutions to Exercises and Problems Solutions to Exercises and Problems Solutions to Problems of Chapter 1 Solutions to Problems of Chapter 2 Solutions to Problems of Chapter 3 Solutions to Problems of Chapter 4 Solutions to Problems of Chapter 5 Solutions to Problems of Chapter 8 A.1 Introduction A.2 Isolated Systems: Microcanonical Description A.2.1 Fundamental Postulate A.2.2 Applications A.2.2.1 Two-level systems A.2.2.2 Classical ideal gas A.3 Systems at Constant Temperature: Canonical Description A.3.1.1 Two-level systems A.3.1.2 Classical ideal gas A.4 Open Systems at Constant Temperature: Grand-Canonical Description A.4.1 Applications A.5 Fermi–Dirac and Bose–Einstein Statistics A.6 Phase Space: Density of States A.6.1 Definition A.6.2 Density of States of a Free Particle in Three Dimensions A.7 Properties of a Free Fermi Gas at T = 0 A.7.1 Fermi Energy A.7.2 Total Average Kinetic Energy A.8 Properties of a Free Fermi Gas at Low Temperatures A.8.1 Sommerfeld’s Expansion A.8.2 Chemical Potential, Average Energy and Calorific Capacity A.9 Free Fermi Gas at the High-Temperature Limit Appendix B Second Quantization B.1 First Quantization: Symmetric and Antisymmetric Wave Functions B.2 Second Quantization: Representation of Microstates by Occupation Numbers B.2.1 The Case of Bosons B.2.2 The Case of Fermions