Preface ........................................................ IX
1 Introduction ............................................... 1
1.1 How the Story Began ........................................ 1
1.1.1 Structure Periodicity and Modulated Phases .......... 2
1.1.2 Ferromagnetic and Ferroelectric Domains ............. 5
1.2 First Theoretical Approaches for Competing Interactions .... 7
1.2.1 Frenkel-Kontorova Model ............................. 7
1.2.2 Theoretical Models of the Magnetic/Ferroelectric
Domains ............................................ 11
1.2.2.1 Phenomenology of the Dipolar Interaction .. 12
1.2.2.2 Phenomenology of the Exchange and
Exchange-Like Interactions ................ 13
1.2.2.3 Mechanism of the Domain Formation ......... 14
1.3 Summary ................................................... 15
1.4 Exercises 16 References ................................... 17
2 Self-Competition: or How to Choose the Best from the
Worst ..................................................... 21
2.1 Frustration: The World is not Perfect ..................... 21
2.2 Why is an Understanding of Frustration Phenomena
Important for Nanosystems? ................................ 22
2.3 Ising, XY, and Heisenberg Statistical Models .............. 23
2.4 Order-Disorder Phenomena .................................. 25
2.4.1 Phase Transitions and their Characterization ....... 26
2.4.2 Order Below a Critical Temperature ................. 28
2.4.3 Measure of Frustration: Local Energy Parameter ..... 28
2.5 Self-Competition of the Short-Range Interactions .......... 29
2.5.1 Ising Antiferromagnet on a Lattice ................. 30
2.5.1.1 Triangular Lattice ........................ 30
2.5.1.2 Kagome Lattice ............................ 31
2.5.1.3 Ising Antiferromagnet on Aperiodic
Tilings ................................... 32
2.5.2 Heisenberg Antiferromagnet on a Lattice ............ 36
2.5.2.1 Triangular and Kagome Lattices ............ 36
2.5.2.2 Aperiodic Tilings ......................... 38
2.5.3 Three-Dimensional Spin Structure on a Periodic
Two-Dimensional Lattice: Itinerant Systems ......... 42
2.5.4 Frustration Squeezed Out ........................... 44
2.6 Self-Competition of the Long-Range Interactions ........... 45
2.6.1 Dipolar Interactions ............................... 46
2.6.1.1 Localized Ising Moments on a Periodic
Lattice ................................... 46
2.6.1.2 Localized Vector Moments on a Periodic
Lattice ................................... 48
2.6.1.3 Localized Vector Moments on Aperiodic
Tilings ................................... 51
2.6.1.4 Delocalized Moments with Given
Orientation: Two-Dimensional Electron
Wigner Crystal ............................ 53
2.6.2 Multipolar Interactions: Why Might that be
Interesting? ....................................... 56
2.6.2.1 Multipolar Moments of Molecular Systems
and Bose-Einstein Condensates ............. 58
2.6.2.2 Multipolar Moments of Nanomagnetic
Particles ................................. 60
2.6.2.3 Multipole-Multipole Interactions .......... 64
2.6.2.4 Ground States for Multipoles of Even
Symmetry: Quadrupolar and Hexadecapolar
Patterns .................................. 64
2.6.2.5 Ground States for Multipoles of Odd
Symmetry: Octopolar and Dotriacontapolar
Patterns .................................. 67
2.7 Summary .............................................. 68
2.8 Exercises ............................................ 68
References ................................................ 70
3 Competition Between a Short- and a Long-Range
Interaction ............................................... 74
3.1 Localized Particles
3.1.1 Competition Between the Ferromagnetic Exchange
and the Dipolar Interaction: Ising Spins ........... 74
3.1.1.1 Stripes or Checkerboard? .................. 74
3.1.1.2 Scaling Theory ............................ 76
3.1.1.3 Stripes in an External Magnetic Field:
Bubbles ................................... 77
3.1.2 Competition Between the Ferromagnetic Exchange
and the Dipolar Interaction: Vector Spins .......... 78
3.1.2.1 Films: Dominating Exchange Interaction .... 78
3.1.2.2 Films: Dominating Dipolar Interaction ..... 80
3.1.2.3 Nanoparticles with Periodic Atomic
Structure ................................. 82
3.1.2.4 Nanoparticles with Aperiodic Atomic
Structure ................................. 86
3.1.3 Competition Between the Antiferromagnetic
Exchange and the Dipolar Interaction ............... 88
3.1.3.1 Periodic Lattices ......................... 88
3.1.3.2 Aperiodic Lattices 91
3.1.4 Neural Networks .................................... 92
3.2 Delocalized Particles ..................................... 94
3.2.1 Self-Assembled Domain Structures on a Solid
Surface: Dipolar Lattice Gas Model ................. 94
3.2.2 Self-Organization in Langmuir Monolayers ........... 98
3.2.3 Self-Organization in Block Copolymer Systems ...... 101
3.2.4 Self-Organization in Colloidal Systems ............ 103
3.2.4.1 Planar Colloidal Crystals ................ 103
3.2.4.2 Patterns in Ferrofluids .................. 104
3.2.4.3 Systems of Magnetic Holes ................ 107
3.2.5 Two-Dimensional Electron Systems .................. 108
3.2.6 Patterns in Animal Colors ......................... 108
3.3 Exercises ................................................ 111
References ............................................... 113
4 Competition Between Interactions on a Similar Length
Scale .................................................... 115
4.1 Two Short- or Mid-Range Interactions
4.1.1 Super-Exchange and Indirect Exchange
Interactions ...................................... 115
4.1.2 Spin Glass ........................................ 117
4.1.3 Non-Collinear Magnetism at Surfaces ............... 119
4.1.3.1 Competing Heisenberg Exchange
Interactions (Hexagonal Lattice) ......... 119
4.1.3.2 Competing Heisenberg Exchange Couplings
(Square Lattice) ......................... 124
4.1.3.3 Antiferromagnetic Domain Wall as a Spin
Spiral ................................... 125
4.1.3.4 Spin Spiral State in the Presence of
Dipolar Interactions ..................... 131
4.1.4 Two Short-Range Repulsive Interactions ............ 133
4.2 Two Long-Range Interactions .............................. 135
4.2.1 Systems with Dipolar and Quadrapolar
Interactions ...................................... 135
4.2.2 Systems with Dipolar and Octopolar Interactions ... 136
4.2.2.1 Combined Multipoles in Nanomagnetic
Arrays ................................... 136
4.2.2.2 Magnetization Reversal in Nanomagnetic
Arrays ................................... 139
4.3 Summary .................................................. 141
4.4 Exercises ................................................ 141
References ............................................... 144
5 Interplay Between Anisotropics and Interparticle
Interactions ............................................. 145
5.1 Interplay Between the Structural Anisotropy and the
Short-Range Repulsion/Attraction: Liquid Crystals ........ 145
5.1.1 Liquid Crystal Phases ............................. 147
5.1.2 Liquid Crystal Patterns: Textures and
Disclinations ..................................... 148
5.1.3 The Lattice Model of Liquid Crystals .............. 153
5.2 Competition Between the Spin-Orbit Coupling and the
Long-Range Dipolar Energy: Ultrathin Magnetic Films ...... 154
5.2.1 Shape Anisotropy from Dipolar Interactions ........ 355
5.2.2 Perpendicular Magnetic Anisotropy ................. 157
5.2.3 Anisotropy Phase Diagram .......................... 157
5.2.4 Magnetic Structure of the Spin Reorientation
Transition (SRT) .................................. 159
5.2.4.1 Regimes of Vertical and Planar
Magnetization ............................ 159
5.2.4.2 SRT via the Twisted Phase ................ 360
5.2.4.3 SRT via the State of Canted
Magnetization ............................ 161
5.2.4.4 SRT via the State of Coexisting Phases ... 164
5.3 Magnetic Nanoplatelets ................................... 267
5.3.1 Size-Dependence of Shape Anisotropy in Discrete
Atomic Approximation .............................. 167
5.3.2 Multiplicative Separation of Discrete and
Continuum Contributions ........................... 169
5.3.3 Size-Dependent Spin Reorientation Transition ...... 169
5.3.4 Size-Dependence of Crystallographic Anisotropy .... 171
5.4 Summary .................................................. 172
5.5 Exercises ................................................ 172
References .................................................... 175
6 Dynamic Self-Organization ................................ 177
6.1 Diffusion-Limited Aggregation ............................ 177
6.1.1 Computer Model .................................... 179
6.1.2 Diffusion-Limited Aggregation Altered by
Interactions ...................................... 182
6.2 Dynamic Wave Patterns .................................... 184
6.2.1 Pattern Dynamics of Spin Waves .................... 186
6.2.2 Liquid Crystals in a Rotating Magnetic Field ...... 189
6.2.3 Standing Waves in Two-Dimensional Electron Gas:
Quantum Mirages ................................... 192
6.3 Summary .................................................. 196
References ............................................... 196
Subject Index ................................................. 199
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