Covering the fruitful combination of nonlinear optics and ferroic materials!
The use of nonlinear optics for the study of ferroics, that is, magnetically, electrically or otherwise spontaneously ordered and switchable materials has witnessed a remarkable development since its inception with the invention of the laser in the 1960s.
This book on Nonlinear Optics on Ferroic Materials reviews and advances an overarching concept of ferroic order and its exploration by nonlinear-optical methods. In doing so, it brings together three fields of physics: symmetry, ferroic order, and nonlinear laser spectroscopy. It begins by introducing the fundamentals for each of these fields. The book then discusses how nonlinear optical studies help to reveal properties of ferroic materials that are often inaccessible with other methods. In this, consequent use is made of the unique degrees of freedom inherent to optical experiments. An excursion into the theoretical foundations of nonlinear optical processes in ferroics rounds off the discussion.
The final part of the book explores classes of ferroic materials of primary interest. In particular, this covers multiferroics with magnetoelectric correlations and oxide-electronic heterostructures. An outlook towards materials exhibiting novel forms of ferroic states or correlated arrangements beyond ferroic order and the study these systems by nonlinear optics concludes the work.
The book is aimed equally at experienced scientists and young researchers at the interface between condensed-matter physics and optics and with a taste for bold, innovative ideas.
Table of Contents
Preface xiii
Acknowledgements xv
1 A Preview of the Subject of the Book 1
1.1 Symmetry Considerations 1
1.2 Ferroic Materials 3
1.3 Laser Optics 6
1.4 Creating the Trinity 8
1.5 Structure of this Book 10
Part I The Ingredients and Their Combination 11
2 Symmetry 13
2.1 Describing Interactions in Condensed-Matter Systems 13
2.2 Introduction to Practical Group Theory 15
2.3 Crystals 16
2.3.1 Types of Symmetry Operations 17
2.3.2 Combinations of Operations 20
2.3.3 Nomenclature 20
2.4 Point Groups and Space Groups 21
2.4.1 Point Groups 21
2.4.2 Space Groups 24
2.5 From Symmetries to Properties 25
2.5.1 Deriving the Components of the Property Tensors 25
2.5.2 Parity of the Property Tensors 25
2.5.3 Introducing Inhomogeneity 26
2.5.4 Beyond Group Theory: Particularisation 28
3 Ferroic Materials 31
3.1 Ferroic Phase Transitions 32
3.1.1 Landau-Theoretical Description and Order Parameter 33
3.1.2 First- and Second-Order Phase Transitions 34
3.1.3 Critical Exponents 36
3.1.4 Domain States and Domains 37
3.1.5 Softness 39
3.2 Ferroic States 41
3.2.1 Conjugate Field and Switchability 41
3.2.2 Hysteresis 42
3.2.3 Curie Temperature 42
3.3 Antiferroic States 43
3.4 Classification of Ferroics 44
3.4.1 Ferromagnetism 46
3.4.2 Ferroelectricity 56
3.4.3 Ferroelasticity 64
3.4.4 Ferrotoroidicity 68
3.4.5 Other Forms of Primary Ferroic Order 76
3.4.6 Higher-Order Ferroics 78
3.4.7 Multiferroics 81
4 Nonlinear Optics 91
4.1 Interaction of Materials with the Electromagnetic Radiation Field 93
4.1.1 Hamilton Operator 93
4.1.2 Multipole Expansion 95
4.2 Wave Equation in Nonlinear Optics 97
4.2.1 Derivation of the Wave Equation with an Extended Source Term 98
4.2.2 General Solution of the Wave Equation 99
4.2.3 Four Solutions of Particular Interest 101
4.3 Microscopic Sources of Nonlinear Optical Effects 103
4.4 Important Nonlinear Optical Processes 107
4.4.1 Two-Photon Sum Frequency Generation 108
4.4.2 Second Harmonic Generation 108
4.4.3 Two-Photon Difference Frequency Generation 109
4.4.4 Optical Parametric Generation 109
4.4.5 Third Harmonic Generation 109
4.5 Nonlinear Spectroscopy of Electronic States 110
4.5.1 Transition Matrix Elements 110
4.5.2 Resonance Behaviour at the Contributing Frequencies 110
4.5.3 Local-Field Corrections 110
4.5.4 Linear Optical Properties at the Contributing Frequencies 111
4.5.5 Phase Matching 111
5 Experimental Aspects 113
5.1 Laser Sources 113
5.1.1 Nanosecond Laser Systems with Optical Parametric Oscillator 114
5.1.2 Femtosecond Laser Systems with Optical Parametric Amplifier 115
5.2 Experimental Set-Ups 116
5.2.1 Spectral Resolution 117
5.2.2 Imaging by Projection 127
5.2.3 Imaging by Scanning 133
5.3 Temporal Resolution 134
6 Nonlinear Optics on Ferroics - An Instructive Example 137
6.1 SHG Contributions from Antiferromagnetic Cr 2 O 3 140
6.2 SHG Spectroscopy 146
6.3 Topography on Antiferromagnetic Domains 149
6.4 Magnetic Structure in the Spin-Flop Phase 152
Part II Novel Functionalities 155
7 The Unique Degrees of Freedom of Optical Experiments 157
7.1 Polarisation-Dependent Spectroscopy 158
7.1.1 Basic Methodical Aspects 158
7.1.2 Resonance Enhancement of Signals 159
7.1.3 Sublattice Selectivity 162
7.1.4 Separation of Coexisting Types of Order 164
7.1.5 Spectral Identification of Symmetries 166
7.2 Spatial Resolution - Domains 167
7.2.1 Access to Hidden Domain States 168
7.2.2 Domain Microscopy at Different Resolution 171
7.2.3 Domain Topography Below the Optical Resolution Limit 173
7.2.4 Domain Topography in Three Dimensions 178
7.3 Temporal Resolution - Correlation Dynamics 181
7.3.1 Overview 181
7.3.2 Dynamical Properties of Ferromagnetic Systems 186
7.3.3 Dynamical Processes in Antiferromagnetic Systems 190
7.3.4 Nonlinear Effects in the Few-Terahertz Range 196
8 Theoretical Aspects 201
8.1 Microscopic Sources of SHG in Ferromagnetic Metals 202
8.2 Microscopic Sources of SHG in Antiferromagnetic Insulators 203
8.2.1 Chromium Sesquioxide 203
8.2.2 Hexagonal Manganites 207
8.2.3 Nickel Oxide 210
Part III Materials and Applications 211
9 SHG and Multiferroics with Magnetoelectric Correlations 213
9.1 Type-I Multiferroics - The Hexagonal Manganites 214
9.1.1 Synthesis and Crystal Structure 214
9.1.2 Lattice Trimerisation 215
9.1.3 Antiferromagnetic Order of the Mn 3+ Lattice 231
9.1.4 Magnetic Order of the Rare-Earth System 243
9.1.5 Magnetic Sublattice Interactions 247
9.1.6 Magnetoelectric Sublattice Interactions 250
9.1.7 Dynamic Correlations 259
9.2 Type-I Multiferroics - BiFeO 3 262
9.2.1 Synthesis and Crystal Structure 262
9.2.2 Ferroelectric Order 264
9.2.3 Antiferromagnetic Order 264
9.2.4 Magnetoelectric Coupling Effects 266
9.3 Type-I Multiferroics with Strain-Induced Ferroelectricity 275
9.4 Type-II Multiferroics - MnWO 4 278
9.4.1 Synthesis and Crystal Structure 278
9.4.2 Multiferroic Order 279
9.4.3 SHG Contributions - Incommensurate SHG 280
9.4.4 Types of Domains 284
9.4.5 Poling Dynamics 287
9.4.6 Multiferroic Domain Walls 289
9.5 Type-II Multiferroics - TbMn 2 O 5 291
9.5.1 Synthesis, Crystal Structure, and Magnetic Order 291
9.5.2 Decomposition of Contributions to the Spontaneous Polarisation 292
9.6 Type-II Multiferroics - TbMnO 3 295
9.6.1 Synthesis, Crystal Structure, and Magnetic Order 295
9.6.2 Domains and Poling 295
9.6.3 Optical Domain Switching 297
9.6.4 Robustness of the Multiferroic State 302
9.7 Type-II Multiferroics with Higher-Order Domain Functionalities 304
9.7.1 Magnetoelectric Inversion of a Domain Pattern 305
9.7.2 Magnetoelectric ‘Teleportation’ of a Domain Pattern 309
10 SHG and Materials with Novel Types of Primary Ferroic Orders 313
10.1 Ferrotoroidics 314
10.1.1 Ferrotoroidic LiCoPO 4 314
10.1.2 Ferrotoroidics Other than LiCoPO 4 320
10.1.3 Status of Ferrotoroidicity as Primary Ferroic Order 324
10.2 Ferro-Axial Order - RbFe(MoO 4) 2 325
10.2.1 Structure and Phase Transitions 325
10.2.2 Ferroic Nature of the Rotational Transition 326
11 SHG and Oxide Electronics - Thin Films and Heterostructures 329
11.1 Growth Techniques 330
11.1.1 Pulsed-Laser Deposition 331
11.1.2 Molecular Beam Epitaxy 332
11.1.3 Sputter Deposition 332
11.1.4 Metal-Organic Chemical Vapour Deposition 333
11.2 Thin Epitaxial Oxide Films with Magnetic Order 334
11.2.1 Ferrimagnetic Garnets 334
11.2.2 Ferromagnetic Metals 334
11.2.3 EuO - A Ferromagnetic Insulator 336
11.3 Thin Epitaxial Oxide Films with Ferroelectric Order 341
11.3.1 Crystal Structure and Domain Configurations: BiFeO 3 342
11.3.2 From Domains to Domain Walls: SrMnO 3 345
11.3.3 Internal Structure of Domain Walls: Pbzr X Ti 1-x O 3 347
11.3.4 From Domain Walls to Interfaces: LaAlO 3 on SrTiO 3 350
11.4 Poling Dynamics in Ferroelectric Thin Films 357
11.5 Growth Dynamics in Oxide Electronics by In Situ SHG Probing 361
11.5.1 Early ISHG Experiments 362
11.5.2 Experimental Set-Up for ISHG 363
11.5.3 Emergence of Ferroelectric Order in a Single Film 365
11.5.4 From Single Films to Multi-Layer Heterostructure 367
11.5.5 From Multi-Layer Heterostructures to Symmetry Engineering 368
11.5.6 Growth Dynamics - Interaction Between Materials 370
11.5.7 Growth Dynamics - Interaction Between Interfaces 372
12 Nonlinear Optics on Ordered States Beyond Ferroics 375
12.1 Superconductors 375
12.2 Metamaterials - Photonic Crystals 379
12.2.1 Optical Properties 380
12.2.2 Ferroic Properties 380
12.2.3 Quasicrystalline Metamaterials 382
12.3 Topological Insulators 384
Part IV Epilogue 387
13 A Retrospect of the Subject of the Book 389
References 393
Index 443
