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Competing Interactions and Patterns in Nanoworld; Contents; Preface; 1 Introduction; 1.1 How the Story Began; 1.1.1 Structure Periodicity and Modulated Phases; 1.1.2 Ferromagnetic and Ferroelectric Domains; 1.2 First Theoretical Approaches for Competing Interactions; 1.2.1 Frenkel-Kontorova Model; 1.2.2 Theoretical Models of the Magnetic/Ferroelectric Domains; 1.2.2.1 Phenomenology of the Dipolar Interaction; 1.2.2.2 Phenomenology of the Exchange and Exchange-Like Interactions; 1.2.2.3 Mechanism of the Domain Formation; 1.3 Summary; 1.4 Exercises; References |
2 Self-Competition: or How to Choose the Best from the Worst2.1 Frustration: The World is not Perfect; 2.2 Why is an Understanding of Frustration Phenomena Important for Nanosystems?; 2.3 Ising, XY, and Heisenberg Statistical Models; 2.4 Order-Disorder Phenomena; 2.4.1 Phase Transitions and their Characterization; 2.4.2 Order Below T(c); 2.4.3 Measure of Frustration: Local Energy Parameter; 2.5 Self-Competition of the Short-Range Interactions; 2.5.1 Ising Antiferromagnet on a Lattice; 2.5.1.1 Triangular Lattice; 2.5.1.2 Kagome Lattice; 2.5.1.3 Ising Antiferromagnet on Aperiodic Tilings |
2.5.2 Heisenberg Antiferromagnet on a Lattice2.5.2.1 Triangular and Kagome Lattices; 2.5.2.2 Aperiodic Tilings; 2.5.3 Three-Dimensional Spin Structure on a Periodic Two-Dimensional Lattice: Itinerant Systems; 2.5.4 Frustration Squeezed Out; 2.6 Self-Competition of the Long-Range Interactions; 2.6.1 Dipolar Interactions; 2.6.1.1 Localized Ising Moments on a Periodic Lattice; 2.6.1.2 Localized Vector Moments on a Periodic Lattice; 2.6.1.3 Localized Vector Moments on Aperiodic Tilings; 2.6.1.4 Delocalized Moments with Given Orientation: Two-Dimensional Electron Wigner Crystal |
2.6.2 Multipolar Interactions: Why Might that be Interesting?2.6.2.1 Multipolar Moments of Molecular Systems and Bose-Einstein Condensates; 2.6.2.2 Multipolar Moments of Nanomagnetic Particles; 2.6.2.3 Multipole-Multipole Interactions; 2.6.2.4 Ground States for Multipoles of Even Symmetry: Quadrupolar and Hexadecapolar Patterns; 2.6.2.5 Ground States for Multipoles of Odd Symmetry: Octopolar and Dotriacontapolar Patterns; 2.7 Summary; 2.8 Exercises; References; 3 Competition Between a Short- and a Long-Range Interaction; 3.1 Localized Particles |
3.1.1 Competition Between the Ferromagnetic Exchange and the Dipolar Interaction: Ising Spins3.1.1.1 Stripes or Checkerboard?; 3.1.1.2 Scaling Theory; 3.1.1.3 Stripes in an External Magnetic Field: Bubbles; 3.1.2 Competition Between the Ferromagnetic Exchange and the Dipolar Interaction: Vector Spins; 3.1.2.1 Films: Dominating Exchange Interaction; 3.1.2.2 Films: Dominating Dipolar Interaction; 3.1.2.3 Nanoparticles with Periodic Atomic Structure; 3.1.2.4 Nanoparticles with Aperiodic Atomic Structure; 3.1.3 Competition Between the Antiferromagnetic Exchange and the Dipolar Interaction |
3.1.3.1 Periodic Lattices |