Magnetic excitations and geometric confinement: theory and simulations
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| Language: | English |
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Bristol, UK
IOP Publishing
[2015]
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| Edition: | Version: 20151201 |
| Series: | IOP expanding physics
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| Online Access: | Inhaltsverzeichnis Klappentext |
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| ISBN: | 9780750310758 |
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| adam_text | Contents
A
Preface xv
Acknowledgements xvii
About the author xviii
Symbols xix
1 Introduction: geometrically confined magnetic systems 1-1
1.1 Magnetic order, dipoles and fields 1-1
1.2 Effects due to geometry 1-3
1.2.1 Demagnetization energy 1-6
1.2.2 Demagnetization examples 1-7
1.3 Exchange interactions 1-9
1.3.1 Exchange of an electron pair 1-10
1.3.2 Exchange and energy 1-12
1.3.3 The Heisenberg exchange Hamiltonian 1-14
1.4 Anisotropic exchange couplings 1-15
1.4.1 Space anisotropy of exchange 1-15
1.4.2 XY spin symmetry 1-17
1.4.3 Effects of XY symmetry—vortices 1-17
1.4.4 More XY symmetry—the plane rotor model 1-20
1.4.5 Uniaxial to isotropic spin symmetry—skyrmions 1-21
1.4.6 Quasi-ID magnetism—spin waves and magnetic solitons 1-22
1.5 Local anisotropies 1-23
1.5.1 Applied magnetic field 1-24
1.6 Theory for linear and nonlinear magnetism 1-25
1.7 Simulations in magnetism 1-26
Bibliography 1-29
Part 1 Theory and simulation approaches for magnetism 2-1
2 Magnetism theory: spin models 2-3
2.1 Magnetic dipoles and magnetic ordering 2-3
2.2 Atomic dipoles 2-4
2.3 Local spin interaction models 2-6
2.3.1 Spin dynamics—atomic scale local spin magnetism 2-7
2.3.2 Micromagnetics—nanoscale magnetization dynamics 2-7
vii
Magnetic Excitations and Geometric Confinement
2.4 Discrete exchange interactions 2-8
2.5 Ferromagnetic spin exchange in a continuum limit 2-8
2.6 Ferromagnetic exchange in micromagnetics 2-10
2.7 FM continuum limit on any lattice 2-13
2.7.1 Triangular lattice 2-15
2.7.2 Hexagonal lattice 2-16
2.8 Angular coordinates for classical spin direction 2-18
2.8.1 Continuum limits in angular notation 2-20
2.8.2 Models in more dimensions 2-22
2.9 Classical spin mechanics 2-22
2.9.1 Hamiltonian dynamics 2-23
2.9.2 Hamiltonian spin dynamics 2-26
2.9.3 Angular spin coordinates and canonical Poisson brackets 2-27
2.9.4 Lagrangian spin dynamics 2-28
2.9.5 Classical dynamics in a system of spins 2-29
2.9.6 Spin damping 2-30
2.9.7 A single spin in a fixed field with damping 2-32
2.10 Classical spin torques 2-34
2.10.1 Continuum limit torques 2-35
2.10.2 Continuum limit dynamics for angular coordinates 2-39
2.11 Quantum spin mechanics 2-41
2.11.1 Dynamic equations of an individual quantum spin operator 2-43
2.11.2 Solution of a quantum spin in a constant field 2-44
2.11.3 How do discrete quantum spin states exhibit themselves? 2-45
Bibliography 2-47
3 Demagnetization effects in thin magnets 3-1
3.1 The magnetostatics problem in a finite magnet 3-1
3.1.1 Green’s functions for magnetostatics 3-2
3.1.2 Uniform magnetization and effective magnetic surface charges 3-5
3.2 The magnetic field inside a cylindrical magnet 3-6
3.2.1 Longitudinal magnetization M2 3-6
3.2.2 Transverse magnetization Mx 3-9
3.3 Demagnetization fields in thin-film magnets 3-12
3.3.1 The longitudinal field in a thin magnet 3-13
3.3.2 The transverse field (77m, H^) in a thin magnet 3-15
3.3.3 Using pM to obtain (Нщ, 77^) in a thin magnet 3-19
3.3.4 The total field (77m, 77m, 77m) in a thin magnet 3-19
viii
Magnete Excitations and Geometric Confinement
3.3.5 Transverse self demagnetization—calculation of (0) 3-20
3.3.6 About the transverse demagnetization for computations 3-23
3.3.7 Example: applying Gxx to a disk with uniform Mx 3-24
3.4 Use of fast Fourier transforms 3-25
3.4.1 Finite system on a ID grid 3-27
3.4.2 Simulation of ID open boundary conditions 3-29
3.4.3 Simulation of 2D and 3D open boundaries 3-32
3.5 Demagnetization in a thin permalloy magnet 3-33
3.5.1 Energy minimization by a spin alignment scheme 3-36
Bibliography 3-38
4 Classical Monte Carlo simulation methods 4-1
4.1 Thermal equilibrium and ergodicity 4-1
4.2 Boltzmann distribution for thermal equilibrium 4-2
4.2.1 Entropy and system energy 4-3
4.2.2 General derivation of Boltzmann distribution 4-6
4.2.3 Free energy and statistical averages 4-8
4.2.4 Averages of magnetic variables 4-10
4.2.5 Averages from statistical fluctuations 4-12
4.3 Importance sampling and the Metropolis algorithm 4-14
4.3.1 Averages with importance sampling 4-16
4.3.2 Why the Metropolis algorithm? 4-17
4.3.3 Trial states for Monte Carlo simulations of magnets: single-spin 4-18
updates
4.3.4 Over-relaxation updates 4-19
4.4 Monte Carlo simulation of a 2D XY model 4-20
4.4.1 Setting up the Monte Carlo simulation 4-21
4.4.2 Monte Carlo averages and errors 4-21
4.4.3 Another approach for errors of fluctuation quantities 4-23
4.4.4 Spatial correlation function and susceptibility 4-25
4.4.5 XY model Monte Carlo results 4-26
4.5 Cluster algorithms for spin updates 4-30
4.5.1 Ising spins: Swendsen-Wang cluster algorithm 4-32
4.5.2 Heisenberg spins: the Wolff cluster algorithm 4-36
4.5.3 Cluster algorithms for Heisenberg spins with XXZ symmetry 4-37
4.5.4 2D XY model simulation data with cluster updates 4-38
4.6 Microcanonical Monte Carlo 4-39
4.6.1 Demons 4-40
ix
Magnetic Excitations and Geometric Confinement
4.6.2 Effective temperature and equipartition 4-41
4.6.3 Equipartition in different systems 4-42
4.6.4 Effective temperature in Ising models 4-43
4.6.5 ID spin chain studied with microcanonical Monte Carlo 4-44
Bibliography 4-45
5 Classical spin dynamics simulations 5-1
5.1 Landau—Lifshitz-Gilbert spin dynamics 5-1
5.1.1 Dimensionless units for spin dynamics 5-3
5.1.2 Dynamic equations for Heisenberg spins—angular variables 5-4
5.1.3 XY model 5-6
5.1.4 Planar rotor model 5-7
5.1.5 Planar rotor in Cartesian components 5-9
5.2 Numerical time evolution for spin dynamics 5-10
5.2.1 Fourth-order Runge-Kutta time evolution 5-10
5.2.2 Adams—Bashforth—Moulton fourth-order predictor-corrector 5-11
method
5.3 Hybrid Monte Carlo-spin dynamics at T 0 5-12
5.4 Stochastic dynamics—thermal fluctuations in the planar rotor model 5-13
5.4.1 Langevin equation for a free rotor in a heat bath 5-14
5.4.2 Planar rotator velocity autocorrelation function 5-15
5.4.3 Planar rotor diffusion 5-17
5.5 Numerical solutions of Langevin equations 5-19
5.5.1 Euler method for Langevin equation 5-20
5.5.2 Developing Langevin second-order methods 5-21
5.5.3 A Langevin-velocity-Verlet method 5-24
5.5.4 An out-of-phase second-order Langevin method 5-25
5.6 Langevin spin dynamics 5-29
5.6.1 Second-order Heun integration scheme 5-31
5.7 Dynamic correlation responses 5-33
5.7.1 Space Fourier transforms 5-33
5.7.2 Space correlations 5-37
5.7.3 Time correlations 5-39
5.7.4 Time Fourier transforms and correlations 5-40
5.7.5 Dynamic structure functions 5-41
5.7.6 Comparing static and dynamic correlations 5-45
Bibliography 5-47
x
Magnetic Excitations and Geometric Confinement
Part II Excitations in magnetic systems 6-1
6 Spin waves: extended but low-dimensional systems 6-3
6.1 Spin waves in ferromagnetic models 6-3
6.1.1 Isotropic Heisenberg models 6-4
6.1.2 An isotropic ferromagnet with a weak magnetic field 6-8
6.1.3 Ferromagnetic models with XY-like exchange 6-10
6.2 Spin waves in antiferromagnetic models 6-15
6.2.1 AID anisotropic antiferromagnetic chain 6-17
6.2.2 2D anisotropic antiferromagnetic model—square lattice 6-21
6.2.3 2D anisotropic antiferromagnetic model—hexagonal lattice 6-22
6.3 Dynamic correlations of spin wave fluctuations 6-24
6.3.1 Ferromagnets 6-24
6.3.2 Antiferromagnets 6-27
6.4 Nonlinear spin waves-—ferromagnets 6-29
Bibliography 6-31
7 Solitons in magnetic chains 7-1
7.1 Nonlinear excitations: solitons in FM magnetic chains 7-1
7.2 Ferromagnetic sine-Gordon kink instability 7-8
7.2.1 The Liebmann et al ansatz 7-10
7.3 tt-kinks in ferromagnetic chains 7-15
7.3.1 Easy-plane ferromagnetic chain 7-15
7.3.2 Easy-plane ferromagnetic chain with in-plane Ising symmetry 7-17
7.4 Magnetic kinks in antiferromagnetic chains 7-20
7.4.1 Antiferromagnetic chain ground state 7-21
7.4.2 Spin waves 7-23
7.4.3 Continuum limit dynamics in an antiferromagnetic chain 7-24
7.4.4 XY antiferromagnetic kinks 7-25
7.4.5 YZ antiferromagnetic kinks 7-27
7.4.6 Antiferromagnetic YZ kink stability analysis 7-29
7.4.7 Antiferromagnetic kink ansatz analysis 7-32
7.4.8 XY kink limit: 0A = 0B 7-38
7.4.9 YZ kink limit: 0A + 6B — n 7-39
7.4.10 Energy—velocity relations from the simplified antiferromagnetic 7-41
ansatz results
Bibliography 7-42
xi
Magnetic Excitations and Geometric Confinement
8 Vortices in layered or 2D ferromagnets 8-1
8.1 A 2D ferromagnet with easy-plane exchange anisotropy 8-1
8.2 In-plane and out-of-plane vortices 8-3
8.2.1 In-plane vortices 8-5
8.2.2 Out-of-plane vortices 8-5
8.2.3 Discrete lattice vortex solutions 8-6
8.3 Vortex instability 8-7
8.3.1 Numerical simulations for stability 8-8
8.3.2 Discrete energetics of vortex core stability 8-9
8.4 Moving in-plane and out-of-plane vortices 8-14
8.5 The vortex unbinding transition 8-15
8.6 Monte Carlo simulations of the Berezinskii—Kosterlitz—Thouless 8-17
transition
8.6.1 Estimations of the critical temperature—Binder’s cumulant 8-19
8.6.2 Estimations of the critical temperature—scaling of susceptibility 8-22
8.6.3 A measure of spin twist resistance—the helicity modulus 8-23
8.6.4 Dependence of critical temperature on anisotropy and vacancies 8-29
8.7 Dynamic correlations in XY models 8-32
8.7.1 Hybrid Monte Carlo-spin dynamics simulations 8-33
8.7.2 Low temperature dynamic structure function 8-34
8.7.3 Higher temperature dynamic structure function and central peak 8-36
8.7.4 Ideal gas model for vortex thermodynamics 8-39
8.7.5 Comparison of vortex ideal gas model to simulations 8-47
8.7.6 Dynamic correlations with vacancies 8-50
Bibliography 8-54
9 Magnetic vortex core motion and internal dynamics 9-1
9.1 Thiele equations and vortex motion 9-1
9.1.1 Derivation of Thiele’s equation of motion 9-2
9.1.2 Including vortex mass effects 9-5
9.1.3 Calculation of vortex gyro vector and other properties 9-6
9.2 Relation of vortex momentum to the Thiele equations 9-9
9.2.1 Poisson bracket of kinetic momentum components 9-10
9.2.2 The momentum conjugate to X 9-12
9.2.3 Momentum as a generator of translations 9-12
9.2.4 Time derivative of kinetic momentum p 9-13
9.2.5 Comparison of vortex dynamics to electric charge dynamics 9-14
9.2.6 Lagrangian mechanics for vortex core motion 9-16
xii
Magnetic Excitations and Geometric Confinement
9.3 Vortex forces and motions 9-17
9.3.1 Force due to applied field 9-18
9.3.2 Vortex pair forces 9-22
9.3.3 Vortex image forces 9-23
9.4 Some simple examples of vortex dynamics 9-31
9.4.1 An individual vortex near a straight boundary 9-31
9.4.2 A pair of vortices in a large system 9-38
9.4.3 An individual vortex in a circular system 9-43
9.5 Vortex-spin wave interactions and normal modes 9-45
9.5.1 The initial discrete lattice vortex solution 9-46
9.5.2 Perturbations of a static vortex 9-46
9.5.3 Examples of the spin wave mode spectrum on a vortex 9-51
9.6 Vortex mass obtained from vortex normal modes 9-61
Bibliography 9-65
10 Vortices in thin ferromagnetic nanodisks 10-1
10.1 Vortex states in magnetic nanodisks 10-1
10.2 Vortex-in-disk effective potentials 10-4
10.2.1 Spin-length constraint, as an example 10-5
10.2.2 Constraint on vortex position 10-7
10.2.3 Some calculated vortex potentials 10-10
10.3 T = 0 Gyrotropic vortex motion in thin nanodisks 10-18
10.3.1 Thiele equation dynamics for a thin-film system 10-19
10.3.2 Numerical simulation of vortex dynamics at zero temperature 10-20
10.3.3 Typical gyrotropic motions from simulations 10-23
10.4 Thermalized vortex motion 10-29
10.4.1 The simulation method 10-29
10.4.2 Spontaneous gyrotropic vortex motion in stochastic dynamics 10-30
10.4.3 Hamiltonian dynamics and statistics of vortex core position 10-35
Bibliography 10-44
11 Spin ices and geometric frustration 11-1
11.1 Spin ice and frustrated states 11-1
11.1.1 About frustration 11-3
11.2 Magnetic monopoles and string excitations in spin ice 11-5
11.3 Square lattice spin ice energetics and order parameters 11-8
11.3.1 Reduction to an Ising limit 11-10
xiii
Magnetic Excitations and Geometric Confinement
11.3.2 Measurement of order in square spin ice 11-13
11.3.3 Monopole charge densities in square spin ice 11-16
11.4 Dynamics in square lattice artificial spin ice 11-18
11.4.1 Island geometry and coupling parameters 11-20
11.4.2 Simulations of equilibrium for Wang islands 11-22
11.4.3 Simulations of square ice with artificial model parameters 11-26
11.4.4 Magnetic hysteresis in model C with artificial parameters 11-32
11.5 Triangular lattice artificial spin ice 11-36
11.5.1 Counting and energetics of vertex configurations 11-36
11.5.2 Comparing frustration in square and triangular spin ice 11-40
11.5.3 Thermal equilibrium in triangular spin ice 11-41
11.6 Kagomé lattice artificial spin ice 11-42
11.6.1 Kagomé vertex configurations and energetics 11-44
11.6.2 Order parameters based on ground state configurations 11-46
11.6.3 Monopole charge densities in Kagomé spin ice 11-53
11.7 Langevin dynamics for Kagomé ice in thermal equilibrium 11-54
Bibliography 11-64
xiv
■OP Expanding Physics
Magnetic Excitations and
Geometric Confinement
Theory and simulations
Gary Matthew Wysin
In this book, author Gary Wysin provides an overview of some model systems and their
behaviour and effects. Intended for advanced students and researchers in physics, chemistry
and engineering, which are interested in confined magnetics eitherto form a purely theoretical
point of view or to consider properties for potential novel applications. It is also suitable as an
auxiliary text in a class on magnetism or solid state physics. Some previous physics knowledge
is expected, including the understanding of classical Hamiltonian mechanics, basic ideas about
quantum spin, and statistical physics. Some of the later chapters consider electromagnetic
effects in magnets in metals, where some basic knowledge of classical electromagnetism and
electromagnetic waves is needed.
About the author
Gary Wysin in a professor of theoretical condensed matter physics at Kansas State University,
USA. Research interests span theoretical and simulation studies in magnetism and optics.
About Expanding Physics
Expanding Physics™ publishes high-quality texts from leading voices across the research
landscape on key areas in physics and related subject areas.
iopscience.org/books
O top I ebooks
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| author | Wysin, Gary Matthew |
| author_GND | (DE-588)1090807171 |
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| ctrlnum | (OCoLC)1024142121 (DE-599)BVBBV044397625 |
| dewey-full | 538 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 538 - Magnetism |
| dewey-raw | 538 |
| dewey-search | 538 |
| dewey-sort | 3538 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | Version: 20151201 |
| format | Book |
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| id | DE-604.BV044397625 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:51:52Z |
| institution | BVB |
| isbn | 9780750310758 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029799754 |
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| owner_facet | DE-703 |
| physical | 1 Band (verschiedene Seitenzählungen) |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | IOP Publishing |
| record_format | marc |
| series2 | IOP expanding physics |
| spelling | Wysin, Gary Matthew Verfasser (DE-588)1090807171 aut Magnetic excitations and geometric confinement theory and simulations Gary Matthew Wysin, Kansas State University, Kansas, USA Version: 20151201 Bristol, UK IOP Publishing [2015] 1 Band (verschiedene Seitenzählungen) txt rdacontent n rdamedia nc rdacarrier IOP expanding physics Geometrische Frustration (DE-588)7733821-2 gnd rswk-swf Magnetismus (DE-588)4037021-5 gnd rswk-swf Magnetische Anregung (DE-588)4202316-6 gnd rswk-swf Magnetismus (DE-588)4037021-5 s Magnetische Anregung (DE-588)4202316-6 s Geometrische Frustration (DE-588)7733821-2 s DE-604 Erscheint auch als Online-Ausgabe 978-0-7503-1074-1 Erscheint auch als Online-Ausgabe, MOBI 978-0-7503-1129-8 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799754&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799754&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Wysin, Gary Matthew Magnetic excitations and geometric confinement theory and simulations Geometrische Frustration (DE-588)7733821-2 gnd Magnetismus (DE-588)4037021-5 gnd Magnetische Anregung (DE-588)4202316-6 gnd |
| subject_GND | (DE-588)7733821-2 (DE-588)4037021-5 (DE-588)4202316-6 |
| title | Magnetic excitations and geometric confinement theory and simulations |
| title_auth | Magnetic excitations and geometric confinement theory and simulations |
| title_exact_search | Magnetic excitations and geometric confinement theory and simulations |
| title_full | Magnetic excitations and geometric confinement theory and simulations Gary Matthew Wysin, Kansas State University, Kansas, USA |
| title_fullStr | Magnetic excitations and geometric confinement theory and simulations Gary Matthew Wysin, Kansas State University, Kansas, USA |
| title_full_unstemmed | Magnetic excitations and geometric confinement theory and simulations Gary Matthew Wysin, Kansas State University, Kansas, USA |
| title_short | Magnetic excitations and geometric confinement |
| title_sort | magnetic excitations and geometric confinement theory and simulations |
| title_sub | theory and simulations |
| topic | Geometrische Frustration (DE-588)7733821-2 gnd Magnetismus (DE-588)4037021-5 gnd Magnetische Anregung (DE-588)4202316-6 gnd |
| topic_facet | Geometrische Frustration Magnetismus Magnetische Anregung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799754&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029799754&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT wysingarymatthew magneticexcitationsandgeometricconfinementtheoryandsimulations |