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Nonlinear Optics on Ferroic Materials
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.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
Preface
Acknowledgements
1 A Preview of the Subject of the Book
1.1 Symmetry Considerations
1.2 Ferroic Materials
1.3 Laser Optics
1.4 Creating the Trinity
1.5 Structure of this Book
Part I: The Ingredients and Their Combination
2 Symmetry
2.1 Describing Interactions in Condensed‐Matter Systems
2.2 Introduction to Practical Group Theory
2.3 Crystals
2.4 Point Groups and Space Groups
2.5 From Symmetries to Properties
3 Ferroic Materials
3.1 Ferroic Phase Transitions
3.2 Ferroic States
3.3 Antiferroic States
3.4 Classification of Ferroics
4 Nonlinear Optics
4.1 Interaction of Materials with the Electromagnetic Radiation Field
4.2 Wave Equation in Nonlinear Optics
4.3 Microscopic Sources of Nonlinear Optical Effects
4.4 Important Nonlinear Optical Processes
4.5 Nonlinear Spectroscopy of Electronic States
5 Experimental Aspects
5.1 Laser Sources
5.2 Experimental Set‐Ups
5.3 Temporal Resolution
6 Nonlinear Optics on Ferroics – An Instructive Example
6.1 SHG Contributions from Antiferromagnetic Cr
6.2 SHG Spectroscopy
6.3 Topography on Antiferromagnetic Domains
6.4 Magnetic Structure in the Spin‐Flop Phase
Part II: Novel Functionalities
7 The Unique Degrees of Freedom of Optical Experiments
7.1 Polarisation‐Dependent Spectroscopy
7.2 Spatial Resolution – Domains
7.3 Temporal Resolution – Correlation Dynamics
8 Theoretical Aspects
8.1 Microscopic Sources of SHG in Ferromagnetic Metals
8.2 Microscopic Sources of SHG in Antiferromagnetic Insulators
Part III: Materials and Applications
9 SHG and Multiferroics with Magnetoelectric Correlations
9.1 Type‐I Multiferroics – The Hexagonal Manganites
9.2 Type‐I Multiferroics – BiFe
9.3 Type‐I Multiferroics with Strain‐Induced Ferroelectricity
9.4 Type‐II Multiferroics – MnW
9.5 Type‐II Multiferroics – TbMn
9.6 Type‐II Multiferroics – TbMn
9.7 Type‐II Multiferroics with Higher‐Order Domain Functionalities
10 SHG and Materials with Novel Types of Primary Ferroic Orders
10.1 Ferrotoroidics
10.2 Ferro‐Axial Order – RbFe(Mo)
11 SHG and Oxide Electronics – Thin Films and Heterostructures
11.1 Growth Techniques
11.2 Thin Epitaxial Oxide Films with Magnetic Order
11.3 Thin Epitaxial Oxide Films with Ferroelectric Order
11.4 Poling Dynamics in Ferroelectric Thin Films
11.5 Growth Dynamics in Oxide Electronics by In Situ SHG Probing
12 Nonlinear Optics on Ordered States Beyond Ferroics
12.1 Superconductors
12.2 Metamaterials – Photonic Crystals
12.3 Topological Insulators
Part IV: Epilogue
13 A Retrospect of the Subject of the Book
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Denomination of point‐group symmetry operations.
Table 2.2 Vanishing tensors in the 122 colour point groups.
Chapter 5
Table 5.1 Optical filters for SHG spectroscopy.
Chapter 6
Table 6.1 Types of SHG contributions in antiferromagnetic Cr.
Chapter 9
Table 9.1 Decomposition of atomic displacements in YMn contributing to the...
Table 9.2 ED‐SHG contributions in the h‐Mn system.
Table 9.3 Accessible SHG susceptibilities in YMn.
Table 9.4 Non‐zero SHG contributions in MnW inside and outside the multife...
List of Illustrations
Chapter 1
Figure 1.1 The potential of symmetry analysis. (a) Deciduous tree from above...
Figure 1.2 Magnetic and electric order of materials. (a) Atoms in a crystal ...
Figure 1.3 Nonlinear optical processes with lasers. (a) Design of the ruby l...
Chapter 2
Figure 2.1 Site symmetry and the Wyckoff notation. For the same crystallogra...
Figure 2.2 Association of time reversal with magnetisation reversal. Sketche...
Figure 2.3 Point‐group symmetry and space‐group symmetry. Exemplified on the...
Figure 2.4 Field distributions and the inversion operation. (a) Homogeneous ...
Chapter 3
Figure 3.1 The concept of the prototype phase. (a) Sequence of phase transit...
Figure 3.2 First‐ and second‐order phase transitions. (a) Conceptual sketch ...
Figure 3.3 Domains and domain states. (a) Unit cell of a cubic crystal in it...
Figure 3.4 Orientation and translation domain states. (a) Arrangement of ori...
Figure 3.5 Basic types of domain walls. (a) In Ising walls, the spontaneous ...
Figure 3.6 Hysteresis in materials with ferroic and antiferroic orders. (a) ...
Figure 3.7 Concept of full, partial, and non‐ferroic order. (a) In a full fe...
Figure 3.8 Classification of the types of primary ferroic order. At least fi...
Figure 3.9 Classical meso‐ and macroscale ferromagnetism. (a) A periodic arr...
Figure 3.10 Antiferromagnetism as spin wave. (a, b) Commensurate antiferroma...
Figure 3.11 Magnetic ordering phenomena. (a) Weak ferromagnetism. A non‐cent...
Figure 3.12 Polarisation‐lattice model for ferroelectricity. (a) Because of ...
Figure 3.13 Mechanisms driving ferroelectricity. The source of the spontaneo...
Figure 3.14 Domain formation and strain release in ferroelastics. (a) At the...
Figure 3.15 Concept of the magnetic toroidal moment. (a–f) Different structu...
Figure 3.16 Types of magnetic multipoles. (a) Monopole, described by a scala...
Figure 3.17 Possible new types of primary ferroic order. (a) Spontaneous uni...
Figure 3.18 Origin of the linear magnetoelectric effect in Cr...
Figure 3.19 Conceptional phase diagrams of type‐I and type‐II multiferroics....
Chapter 4
Figure 4.1 Electronic excitation in a two‐level system. (a) In the absence o...
Figure 4.2 Polariton as mixed light–matter state and phase matching. (a) Dis...
Figure 4.3 Solution of the nonlinear wave equation and phase matching. (a) I...
Figure 4.4 Leading contributions to resonant two‐photon sum frequency genera...
Chapter 5
Figure 5.1 Set‐up of two‐photon sum frequency generation. (a) Generation of ...
Figure 5.2 Sample holders for cryostat operation. (a) Entire sample rod desi...
Figure 5.3 Types of polarisation‐dependent nonlinear‐optical measurements. A...
Figure 5.4 Normalisation procedures in nonlinear optics. (a–e)
Intensity nor
...
Figure 5.5 Effect of high temperature on the SHG yield. The amplitude of the...
Figure 5.6 Comparison of image focusing with lasers and daylight. (a) Focuse...
Figure 5.7 Concept of phase‐sensitive nonlinear optics. (a) Experimental set...
Figure 5.8 Phase‐sensitive SHG with achromatic beam imaging. (a) Experimenta...
Figure 5.9 SHG imaging by scanning techniques. (a, b) Far‐field SHG microsco...
Figure 5.10 Set‐up for pump–probe spectroscopy. (a) Two‐colour reflection se...
Chapter 6
Figure 6.1 Crystal structure of chromium sesquioxide, Cr
2
O
3
. (a) Primitive a...
Figure 6.2 Electronic states of the Cr
3+
ion in an octahedral ligand fie...
Figure 6.3 Domain states in antiferromagnetic Cr
2
O
3
. In zero magnetic field,...
Figure 6.4 Linear and nonlinear optical spectroscopy on Cr...
Figure 6.5 SHG spectra on antiferromagnetic Cr...
Figure 6.6 Magnetic and crystallographic contributions to SHG in Cr...
Figure 6.7 Simulation of SHG interference in Cr
2
O
3
. (a) The magnetic and cry...
Figure 6.8 Spatially resolved images of antiferromagnetic 180°...
Figure 6.9 Manipulation of the Cr...
Figure 6.10 Cr
2
O
3
in the spin‐flop phase. (a) Spectral and circular‐polarisa...
Chapter 7
Figure 7.1 Resonance‐enhanced 2P‐SFG on antiferromagnetic NiO. (a) Sketch of...
Figure 7.2 Site‐selective SHG on antiferromagnetic Cu...
Figure 7.3 Site‐selective magnetic order in Cu...
Figure 7.4 Ferroelectric and antiferromagnetic SHG contributions in hexagona...
Figure 7.5 Spectroscopic identification of magnetic symmetry in hexagonal ma...
Figure 7.6 Domains in a ferrimagnetic garnet by linear and nonlinear optics....
Figure 7.7 Visualisation of magnetic orientation and translation domains by ...
Figure 7.8 Visualisation of domains in novel types of ferroics by SHG. Ferro...
Figure 7.9 Spatially resolved SHG imaging with two types of lenses. (a, b) D...
Figure 7.10 Spatially resolved image of domains with an extension below the ...
Figure 7.11 Sub‐wavelength resolution with nonlinear near‐field microscopy. ...
Figure 7.12 SHG images of ferroelectric domain walls. (a) Spatially resolved...
Figure 7.13 Interference of the nonlinear signal at domain walls. (a) Sketch...
Figure 7.14 Visualisation of domain walls in ferroelectric YMn...
Figure 7.15 Attenuation of nonlinear signal by multi‐domain configurations. ...
Figure 7.16 Three‐dimensional domain‐wall imaging by Čerenkov...
Figure 7.17 Three‐dimensional domain topography on LiCoPO...
Figure 7.18 Precession of magnetisation with Gilbert damping. (a) Motion of ...
Figure 7.19 Exchange‐coupling dynamics in the ferromagnetic Eu...
Figure 7.20 Model for the exchange‐coupling dynamics in the Eu...
Figure 7.21 Spin and lattice dynamics in antiferromagnetic Cr...
Figure 7.22 Universal relation between spin–lattice relaxation and anisotrop...
Figure 7.23 Ultrafast magnetisation dynamics in antiferromagnetic NiO. (a) T...
Figure 7.24 Model for the photoinduced reorientation of the antiferromagneti...
Figure 7.25 Effect of pump‐pulse shaping on the NiO relaxation dynamics. Tim...
Figure 7.26 Harmonic generation in the few‐terahertz regime. (a) Set‐up for ...
Figure 7.27 The concept of two‐dimensional terahertz time‐domain spectroscop...
Chapter 8
Figure 8.1 Early observation of magnetically induced SHG. (a) SHG intensity ...
Figure 8.2 Antiferromagnetic contributions to SHG in the hexagonal manganite...
Chapter 9
Figure 9.1 Prototype structure of the h‐...
Figure 9.2 Trimerisation and domain states of the h‐i...
Figure 9.3 Nature of the improper‐ferroelectric phase transition in YMn...
Figure 9.4 Monte‐Carlo simulation of the formation of trimerisation–polarisa...
Figure 9.5 Vortex‐like distribution of polarisation domains in the h‐...
Figure 9.6 System size dependence of the perceived temperature for the trans...
Figure 9.7 Improper ferroelectricity in ultrathin YMn ...
Figure 9.8 Ferroelectric SHG spectra of YMn...
Figure 9.9 Graphical interpretation of domain‐vortex formation in the h‐...
Figure 9.10 Comparison of the lattice‐trimerising order of ErMn...
Figure 9.11 Types of antiferromagnetic order in the h‐...
Figure 9.12 Relation between lattice trimerisation and antiferromagnetic ord...
Figure 9.13 Magnetic phase diagram of the h‐...
Figure 9.14 Magnetic SHG spectra in the h‐...
Figure 9.15 Distribution of antiferromagnetic domains in YMn. (a, b) Spat...
Figure 9.16 Mn...
Figure 9.17 Mn...
Figure 9.18 Antiferromagnetic domain pattern across the Mn...
Figure 9.19 Magnetic‐field‐induced phase transition of the Mn sublattice. (...
Figure 9.20 Prototypical magnetic phase diagrams of the Mn order in the rM...
Figure 9.21 Temperature‐ and magnetic‐field‐induced phase transition in HoMn
Figure 9.22 Types of magnetic orders compatible with the high‐symmetry mag...
Figure 9.23 Magnetic order of the rare‐earth ions in ErMn. (a–c) Specific F...
Figure 9.24 Sample dependence of the magnetic rare‐earth order. Specific Far...
Figure 9.25 Prototypical magnetic transitions of the....
Figure 9.26 Interaction between magnetic sublattices in ErMn. Just below ,...
Figure 9.27 Order–parameter coupling of SHG contributions in the h‐Mn fami...
Figure 9.28 Coupling between ferroelectric and antiferromagnetic orders in t...
Figure 9.29 Model for the Mn spin reorientation across the boundary between...
Figure 9.30 Electric‐field‐induced transition to ferromagnetic order in HoMn
Figure 9.31 Ultrafast three‐dimensional motion of an antiferromagnetic order...
Figure 9.32 Dynamic antiferromagnetic Mn order in the basal plane. (a, b) C...
Figure 9.33 Ferroelectric and antiferromagnetic orders in room‐temperature m...
Figure 9.34 Distribution of ferroelectric and antiferromagnetic domains in B...
Figure 9.35 Magnetoelectric SHG spectra of BiFe. (a) Spectral dependence of...
Figure 9.36 Coupling of ferroelectric and antiferromagnetic orders in BiFe....
Figure 9.37 Coupled polarisation and magnetisation reversal in a BiFeOCoFe
Figure 9.38 Probing buried domains by SHG. (a) Sketch of a BiFe(001) stripe...
Figure 9.39 Operational electric‐field poling of BiFe. (a, d, g) Spatially ...
Figure 9.40 Dynamic magnetoelectric coupling between BiFe...
Figure 9.41 Strain‐induced multiferroicity in EuTi...
Figure 9.42 Strain‐induced ferroelectricity and multiferroicity in SrMn...
Figure 9.43 Crystallographic and magnetic structure of MnW...
Figure 9.44 Ferroic characterisation of MnW...
Figure 9.45 Conceptual origin of incommensurate SHG in MnW...
Figure 9.46 Ferroelectric and antiferromagnetic domains in MnW. (a) Spatial...
Figure 9.47 The concept of multiferroic hybrid domains. (a) Spatially resolv...
Figure 9.48 Magnetoelectric poling dynamics of MnW. (a) Response of the bub...
Figure 9.49 Response of MnW domains to a multiferroic phase transition. (a)...
Figure 9.50 Crystal structure and magnetic phases of TbMn...
Figure 9.51 Identification of three contributions to the ferroelectric polar...
Figure 9.52 Magnetic‐field dependence of the three polarisation contribution...
Figure 9.53 Temperature and magnetic‐field dependence of the magnetically in...
Figure 9.54 Crystal structure and ferroic order in orthorhombic TbMn. (a) C...
Figure 9.55 Multiferroic order and poling in orthorhombic TbMn. (a) Tempera...
Figure 9.56 Optically induced reversal of the multiferroic TbMn order. (a, ...
Figure 9.57 All‐optical antiferromagnetic switching. (a) Monte‐Carlo simulat...
Figure 9.58 Stability of the photoinduced switching process. Monte‐Carlo sim...
Figure 9.59 Multiferroic order in epitaxial TbMn thin films. (a) Temperatur...
Figure 9.60 Response of the TbMn domains to a multiferroic phase transition...
Figure 9.61 Inversion of magnetic and electric domain patterns. (a, b) Spati...
Figure 9.62 Model of the domain‐inversion process....
Figure 9.63 Three‐order‐parameter coupling in multiferroic...
Figure 9.64 Magnetoelectric transfer of a domain pattern. (a) Multi‐domain c...
Chapter 10
Figure 10.1 Crystal structure and magneto‐toroidal order of LiCoP. (a) Crys...
Figure 10.2 Distribution of ferrotoroidic domains in a LiCoP sample. (a) Sp...
Figure 10.3 Ferrotoroidic poling of domains. (a, b) Spatially resolved SHG i...
Figure 10.4 Toroidal and non‐toroidal contributions to the linear magnetoele...
Figure 10.5 Magnetic coordinate system in CoTe. (a, b) Spatially resolved ...
Figure 10.6 Hysteretic magnetoelectric poling of CoTe. Dependence of the S...
Figure 10.7 Ferro‐axial (ferro‐rotational) phase transition in RbFe(Mo). (...
Figure 10.8 Indications for the presence and mobility of ferro‐axial domains...
Chapter 11
Figure 11.1 Interference of crystallographic and magnetic SHG contributions ...
Figure 11.2 SHG on ferromagnetic EuO films. (a) Temperature dependence of th...
Figure 11.3 Origin of ferromagnetic SHG in EuO. (a) Temperature dependence o...
Figure 11.4 Crystal structure of thin BiFe films in dependence of substrate...
Figure 11.5 SHG precision measurement of monoclinic distortions. (a) Depende...
Figure 11.6 Distinction of the T‐ and R‐like phases by SHG. SHG anisotropy m...
Figure 11.7 Interplay of crystallographic structures during the growth of Bi...
Figure 11.8 Spatially resolved SHG image of polar domains in a strained SrMn
Figure 11.9 Spatially resolved EFM image of polar domains in a strained SrMn
Figure 11.10 Ising–Néel‐type domain walls in PbZr...
Figure 11.11 Interface conduction for LaAl films grown on SrTi. (a) Depend...
Figure 11.12 Electronic states around the Fermi energy of STO in the LAOSTO...
Figure 11.13 Characterisation of the LAOSTO interface by SHG. (a, b) Spectr...
Figure 11.14 Ultrafast processes at the LAOSTO interface. (a) Time dependen...
Figure 11.15 Terminology for the orientation of tetragonal ferroelectric thi...
Figure 11.16 Ferroelectric poling dynamics of BaTi films. (a) SHG anisotrop...
Figure 11.17 Conversion of...
Figure 11.18 The concept of in situ SHG. Sketch of simultaneous in situ moni...
Figure 11.19 ISHG probing of the emergence of ferromagnetic order in Ni film...
Figure 11.20 Experimental realisation of ISHG. Exemplary set‐up with a PLD c...
Figure 11.21 ISHG tracking of the emergence of ferroelectricity during depos...
Figure 11.22 Tracking ferroelectricity in multi‐layer heterostructures by IS...
Figure 11.23 Symmetry engineering of a hexagonal TbMn film. (a) RHEED inten...
Figure 11.24 Polarisation breakdown in BaTi. (a–c) ISHG intensity in depend...
Figure 11.25 Coupling between the interfaces of an ultrathin ferroelectric f...
Chapter 12
Figure 12.1 Evidence for symmetry‐breaking loop currents in SrIr. (a) SHG ...
Figure 12.2 Evidence for broken inversion symmetry in the pseudogap region o...
Figure 12.3 SHG on quasicrystalline structures. (a) Plasmonic quasicrystalli...
Figure 12.4 Band crossing and spin–momentum locking in a topological insulat...
Guide
Cover
Table of Contents
Title Page
Copyright
Dedication
Acknowledgements
Begin Reading
References
Index
End User License Agreement
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Nonlinear Optics on Ferroic Materials
Manfred Fiebig
Author
Prof. Manfred Fiebig
ETH Zurich
Department of Materials
Vladimir‐Prelog‐Weg 4
8093 Zurich
Switzerland
Cover Image: © Jannis Lehmann and Martin Sarott, based on input from Martin Lilienblum and Yevheniia Kholina
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For Gertrud –
The most important
chapter of this story
Preface
This book brings three fields of physics together, namely symmetry, spontaneous long‐range (‘ferroic’) order, and nonlinear (in its response to the light field) laser optics. Nonlinear optics on ferroic materials is a subject that came to life in the early 1990s. Since then, it has been developing with remarkable success. Despite this interest, there appears to be no comprehensive presentation of the topic in book format. In fact, even review articles on the subject are scarce and generally focus on a single type of ferroic order or other selected aspects. Writing a book on nonlinear optics on ferroic materials therefore seems to be both timely and urgent.
As an introductory work, it aims at as wide a circle of readers as possible. This is a necessity because with symmetry, ferroic order, and laser optics, the book combines three vastly different areas, and the fewest readers will have a background in all of them. A prerequisite of this book is therefore the creation of a common perspective on the three subjects and, thus, of a common terminology to present them. This may tempt specialists in the respective fields to grumble about the inappropriate choice of language. The reward, however, is mutual understanding with the prospect of building bridges and joining forces in solving a timely problem. Given the current world situation, this seems to be more important than ever.
The introductions on symmetry, ferroic order, and nonlinear optics in Chapter 1 are written such that graduate students in the natural or engineering sciences should be able to follow and encourage them to read on. The book should also be quite useful for those who are looking for a topic for their doctoral thesis or the academic career that builds on it. Its approach is interdisciplinary, it points out numerous blank spots in the field, and research on these has a good chance of leading to novel, important insights.
Most of all, I would like to have readers fun in reading this work. I seriously hope that they feel some of the excitement I experienced while working in the field of nonlinear optics on ferroic materials and also in writing this book. Time to begin! So: Yeah – Is everything in place?
ZürichSeptember 2023
Manfred Fiebig
Acknowledgements
(After a near‐fatal attack)
C/E JOHANN: Electric motor fixed. Main pump fixed. Water can be pumped into tanks, then blown out with compressed air. Compass ready. Sonar ready.
LT. WERNER: Good, chief! Good. Good. — Take a rest now.
LT. WERNER: You just have to have good people. Good people!
The Boat, Wolfgang Petersen, Germany (1981)
This work would not have been possible without invaluable contributions from many different sides. First of all, I would like to thank my mentors who have influenced me both scientifically and through their person as well as with the attitude according to which they treat both research and researchers. They are Dietmar Fröhlich, who opened up the fascinating worlds of nonlinear optics and lasers to me, Roman Pisarev, who taught me about the wonders of magnetism and time reversal, Kenjiro Miyano, who was very tolerant of the peculiarities of a young strong‐willed postdoc buzzing around his labs, and finally Thomas Elsässer, who gave me unimagined freedom in letting me develop my scientific ways under his roof. I also appreciate occasional kicks by Burkard Hillebrands, Yoshinori Tokura, Hans Schmid, and Ramamoorthy Ramesh in the right direction. I thank the institutions that hosted me over the years: The Technical University of Dortmund, The University of Tokyo – Todai, the Max Born Institute, The Rhenish Friedrich Wilhelm University of Bonn, and the Swiss Federal Institute of Technology in Zurich – ETH Zurich. I furthermore acknowledge financial support by the German Research Council (DFG), the Japan Science and Technology Corporation (JST), the Swiss National Science Foundation (SNSF), and the European Research Council (ERC).
Science can only be truly successful and enjoyable if the people working together on a daily basis are not just colleagues but, at least to some extent, friends. Here, I would like to thank Nicola Spalding in particular. For 20 years now, I have enjoyed our tossing around of more or less crazy ideas (with replaying the Big Bang and teleportation certainly on the ‘more’ side). Discussions with Nicola helped to stay grounded, and I particularly enjoyed our occasional ‘Department (Anti‐) Leadership Retreats’ into the Alps. I wholeheartedly thank Dennis Meier for all the fun we had in exploring multiferroics. His optimistic, cooperative approach to science and scientists and his worldly‐wise view of practical science are ever so refreshing. Furthermore, I thank Takuya Satoh for 25 years of collegiality and friendship. At some point we wondered: ‘Why not plan our careers together?’, and that actually worked out, despite mostly staying on opposite sides of the globe. I also thank Hans Kroha for bringing up some of the most intricate and exciting physics problems and the enjoyable collaboration on these. A theory that is too complex for Hans to explain in understandable words does not exist. Finally, I thank Kathrin Dörr for numerous personal discussions about the world of science and the life around it.
Most of my own findings in this book exist only because I have had the privilege of heading a group of really fantastic people over the years. Here, Thomas Lottermoser has accompanied me for the longest time. With his calm and thoughtful manner, he provided the necessary counterpoint to my rather spontaneous and impulsive ideas. His knowledge of human nature was invaluable in the search for new team members. Morgan Trassin, on the other hand, represents the scientific counterpart of my activities. He understands ferroelectrics and taught me to use the word ‘device’ without getting a hiccup. In short, he complements me in those areas of research where I am not very gifted, and I greatly enjoy our daily banter about it. I am indebted to Dennis Meier, Mads Weber, and Shovon Pal, who entered the group on a postdoctoral position but were doing so much more than just great research. As Junior Group Leaders they left their unmistakable mark on the ‘Laboratory for Multifunctional Ferroic Materials’. I cannot thank enough the team of wonderful doctoral and postdoctoral researchers I have been able to work with over the years. They are at the forefront of experiments and simulations that define my own success. More than anything, I enjoy the daily encounters with the FERROICS: a blend of individuals from 15 or so countries from all over the world. I admire their courage to make the challenging leap into a sometimes completely foreign environment, motivated by the urge to follow their talent and their cultural curiosity. They are the best proof that it is possible to live in peace with one another if only one wants to, and how enriching and productive this is for everybody.
As for the practical side of preparing this book, I am grateful to Hans Kroha, Thomas Lottermoser, Arkadiy Simonov, Morgan Trassin, and Thomas Weber for critical reading of the manuscript. I thank Joohee Bang, Lea Forster, Michael Giger, Marcela Giraldo, Elzbieta Gradauskaite, Yevheniia Kholina, Jannis Lehmann, Jingwen Li, Shovon Pal, Martin Sarott, and Arkadiy Simonov for their help in preparing the figures and Yevheniia Kholina, Jannis Lehmann, Shovon Pal, and Martin Sarott for their contributions to the cover design. And none of the experiments would have worked without the excellent technical and administrative personnel in our lab, lately in the Department of Materials and at ETH at large, whom I sincerely thank for their continuous support.
I would also like to acknowledge a number of locations for providing the recreative environment and counterbalance after hours at the institute. These are Tokyo, Berlin (yes, Morgan!), and Zurich with all its great cinemas, as well as, in an odd way, Boston. I must have spent months in the latter, always around the end of November, as attendee of the incessantly productive meetings of the Materials Research Society. I secretly wonder if there is a Boston without cold, darkness, and Christmas carols. In addition, I thank Dieter Kosslick and his Berlinale. This fantastic and totally unpretentious film festival offers fascinating insights into the world, and at least one hire in my group is in part due to curiosity about a country that arose in me because it frequently produces the best festival films.
My greatest thanks go to my friends and family. Here, I thank Dirk and Kerstin, dear friends since the first semester in Dortmund. They took care that I have that other knowledge that is indispensable for becoming a respected physicist: ‘The Hitchhiker's Guide to the Galaxy’ and the ‘Tales of Pirx the Pilot’, to name just two. And thanks for your bits and bytes when booking the Berlinale tickets! I cannot thank my parents enough who supported me unconditionally in my choice of career and encouraged me in it, especially in difficult times. Unfortunately, my father is no longer with us to hold the printed result of this support in his hands, but I hope that my mother will enjoy it for herself and in his place. I also thank my sister and her family. They support me by their interest in the wonders of nature. It has encouraged me to always be able to explain what I do to lay people so they can get an idea of what I spend taxpayers' money on. I wrote Chapter 1 for Elke, Frank, and Lili‐Marie in particular. Last but not least, I thank Gertrud, the most important person in my life. She had to endure not only my curses about things not working but also all the relocations that sometimes went to places that were only among the second best. It is a great fortune that Gertrud pursues a profession that made these moves possible for her at all. Still, she always had to give up something and leave dear things behind when we went to a new place. How can I ever make up for that? I especially thank Gertrud for our two and a half years of Japan adventures and for her optimism and good humour, which I cannot muster so easily, and certainly not before 10 in the morning. For the time at your side, I thank you from the bottom of my heart.
Manfred Fiebig
1A Preview of the Subject of the Book
MARIANNE: Forgive me, I'd hate to be in your place.
HÉLOÏISE: We are in the same place. Exactly the same place. Come here. Come. Step closer. Look. If you look at me, who do I look at?
Portrait of a Lady on Fire, Céline Sciamma, France (2019)
1.1 Symmetry Considerations
Almost everyone can relate to the concept of symmetry. People usually associate symmetry with something that looks the same on the right‐ and left‐hand side of some centre. Often, symmetry is associated with beauty, whereas asymmetry is considered as unpleasant. On the other hand, asymmetry may be used to create tension and make an object or image appear as interesting where symmetry might convey an impression of dullness. Most people also have an intuitive understanding of the consequences of symmetry. Imagine a picture of a symmetric object, say, of a human body, in which the right half is the (approximate) mirror image of the left half. Even someone who is not an orthopaedic will assume that the arrangement of bones and muscles in foot on the left is a mirror image of the arrangement of bones and muscles in the foot on the right. Here, a correct transfer from the symmetry of the larger object on the symmetry of its hidden components is made. Symmetry obviously allows us to make conclusions about the structure of objects even if we do not understand their composition and functionality in detail.
This principle can be extended to impressive lengths. Imagine an intelligent alien life form that is presented with pictures taken on Earth, as in Figure 1.1, showing a tree and a cow from above. Those aliens may have no idea what these objects represent. They will notice, however, that the tree thing looks roughly the same in all directions. So, whatever that object represents, it is probably rooted to the ground because if it were consciously mobile, it would most likely have a sort of front end in the direction in which it moves in order to detect what lies ahead. Because of this particular purpose, this front end is expected to look different from the rest of its body.
Figure 1.1 The potential of symmetry analysis. (a) Deciduous tree from above. There is no direction here that stands out above another, see the double arrows. In consequence, rotation around the centre does not change the general appearance of the tree. With the lack of a built‐in direction it is expected that trees do not move but rather stay rooted to the ground. (b) Cow from above. With respect to the long axis there is a clear difference between the two ends of the cow. It can be interpreted as a direction built into the animal (arrow), which eventually indicates its direction of motion. With respect to its sides, there is no direction that would distinguish left from right, so no motion along this axis is expected, as indicated by the double arrow.
This is the very concept applying to the cow thing. It does not look the same in all directions. Specifically, with respect to its long axis, one end is different from its opposite. Presumably, this therefore indicates the direction of motion of the object. A sentient mobile being needs to see what lies ahead but not what lies behind, so that the two ends will look different. In contrast, the long sides of the cow thing look about the same. There is no preferred direction here, and so, these sides will not be related to the direction of the movement.
Hence, from the rotational symmetry of the tree and its absence in the case of the cow, the alien life form concludes that not only the latter is consciously mobile but also the former is rooted to the ground. The aliens can also suspect that the cow represents the more intelligent form of life as it controls its direction of motion. That is a lot of knowledge about two systems whose meaning and inner structure are completely unknown to its observer, and all of it is derived from symmetry. Furthermore, it is the absence rather than the presence of the symmetry that tells something about the structure and function of the associated object, here exemplified by the directional structure of the cow that gives away its conscious mobility by breaking the rotational symmetry exhibited by the immobile tree.
In the context of this book, we deal with materials whose atomic structure and electronic interactions we often do not know in detail. Therefore, symmetry is our most powerful tool in extracting the structure and function of these materials, very much in the same way as we have done with Figure 1.1. We consequently employ experimental methods that are strongly rooted in symmetry for our investigations. Two of the symmetry operations we consider have already been mentioned, namely rotations and mirror operations. Both describe the reorientation of an object in the three‐dimensional space we are living in, but they do not change the shape of the object, for example by stretching it. Accordingly, we only consider symmetry operations that preserve the length of an object in each direction. These are translations from one point in space to another, rotations around a certain axis, and mirror operations on a designated point or plane. Reversing the direction of the passage of time also does not change the length of an object and is therefore considered. This might seem odd since the direction of time cannot be changed. Time reversal makes sense, however, when we discuss an electric current flowing from location A to location B. Reversing the direction of time converts it into a current flowing from B to A, and considering if a material remains unchanged or not under such an electric‐current reversal is not unphysical. In fact, it will turn out that time‐reversal symmetry is crucial for describing magnetically ordered systems because magnetic fields are classically generated by electric currents.
1.2 Ferroic Materials
Almost everyone can also relate to magnetism. It is exhibited by certain objects called magnets that attract iron, which is useful because it makes postcards stick to the fridge. For the attentive observer, magnetism can be found in almost all areas of daily life. Electric motors, current generators, sensors, computer hard disks, and compasses are among the objects usually associated with it. In fact, in a typical household, hundreds of magnets can be found, with more than 100 built into a car alone. The fact that magnetism has been with humankind for at least 2500 years makes us forget that it is one of the most mysterious phenomena of nature. It acts without carrier medium across space, a concept captured, yet not explained, by the introduction of a magnetic field. Magnetic fields are generated by electric currents, but no such current is found in a rod magnet. Instead, we had to introduce the notion of a quantum‐mechanical spin as its source, but again, this mostly represents a description rather than a true explanation. Few people realise that with a magnet for less than a euro, they have quantum mechanics in its purest form in their hands.
Readers may remember from school that matter is made up of atoms that are themselves small magnets as depicted in Figure 1.2a. If all these point in the same direction, the very small fields of a very large number of atomic magnets, called magnetic moments, add up to yield the characteristic magnetic field surrounding a magnet, as sketched in Figure 1.2b,d. This picture already leads to one of the most important properties of a magnet. It represents a form of order in a material that is not enforced by some external influence but arises spontaneously below a certain temperature. There is field that can act on a magnet and orient it in a certain direction, such as Earth's magnetic field in the case of a compass needle.
Eventually, it turned out that magnetism is only one of several forms of order that are associated with a surrounding field and arise spontaneously in a material below a certain temperature. The generic term ferroic was introduced to tag these. It refers to the magnetic order of iron, but as a prefix it indicates ordered states as described above in any type of material, even if iron is not involved. Along with the introduction of this prefix, the somewhat unspecific term ‘magnetic’ was replaced by ferromagnetic in order to distinguish it from other forms of magnetic order. Note that in line with what we have just said, nickel also counts as ferromagnet rather than being denominated as ‘niccolomagnet’. Chemical elements that are ferromagnetic at room temperature are iron, cobalt, and nickel, and certain rare metals are coming close.
Almost exactly a century ago, it was recognised that matter can spontaneously order itself electrically. A crude analogy to the atomic magnets mentioned above would be that of minuscule batteries, formed, for example by a pair of atoms of which one is positively and one is negatively charged. If all these pairs, called electric dipoles, spontaneously point in the same direction, we have a material that is electrically ordered and surrounded by an electric field as shown in Figure 1.2e that can attract charged particles. A material of this type would be called ferroelectric. Although few people are aware of this property, ferroelectrics play a not inconsiderable role in our daily lives. Sonar and certain loudspeakers, buzzers, or sensors are based on ferroelectrics, and even computer components based on ferroelectric rather than ferromagnetic memory are in operation.
Finally, about half a century ago it was found that certain materials can deform spontaneously, which can be associated with a mechanical strain field. This property is denominated as ferroelastic, and it concludes the set of currently fully established forms of ferroic order that nature can display.
Figure 1.2 Magnetic and electric order of materials. (a) Atoms in a crystal representing minuscule magnets. In most materials these so‐called magnetic moments point in random directions so that the total magnetic fields cancel out. (b) Spontaneous order may occur, where all the atomic magnets point in the same direction. The magnetic moments of such a ferromagnet add up to reveal its characteristic magnetic field. (c) Spontaneous alternating arrangement of the magnetic moments still represents an ordered state, yet without a magnetic field because of the cancellation for the oppositely oriented magnetic moments. A material of this type is denominated as antiferromagnet. (d) Magnetic field surrounding the ordered magnetic moments of a ferromagnet. (e) Electric field surrounding the ordered electric dipoles of a ferroelectric. The lines in (d) and (e) indicate the direction of the field. Despite the very similar field distribution, the origins of the ferromagnetic and ferroelectric orders are quite different.
In addition to the three types of ferroics we have just mentioned, there are a number of variants, which also play important roles in science and technology. Foremost, there are materials where the atomic magnetic moments are ordered in an alternating fashion. If for a specific atomic magnet the north pole is pointing up, it is the south pole for the next atom, then again the north pole, and so forth, see Figure 1.2c. This is a form of spontaneous order as stringent as in the case of ferromagnetism, but because half of the magnetic moments point in one and the other half in the opposite direction, there is no resulting magnetic field that would surround such an object. A material exhibiting this kind of order is denominated as antiferromagnetic. Metallic chromium and manganese are well‐known materials that are antiferromagnets at room temperature. In terms of technological applications, a fieldless magnet appears to be quite useless because it is not much different from materials that are not ordered in the first place. This is not true, however. A ferromagnet brought into contact with an antiferromagnet may sense the order of the latter, and this influence can be used to improve the technological performance of the ferromagnet. This principle is used in the read‐write heads of computer hard disks. Furthermore, antiferromagnetism is closely related to superconductivity, the lossless, and thus energy‐saving and waste‐heat‐avoiding flow of an electric current.
Because of the absence of a magnetic field, there is no general agreement on which forms of magnetic order should be counted as antiferromagnetism and which ones should not. This ambiguity is quite astonishing considering how intensively and how long the magnetic properties of matter have been studied. When it comes to the antiferroic equivalents for electric and elastic order, the situation is even worse. A spontaneous alternating arrangement of electric dipoles might be called antiferroelectric, but whereas the ferro‐ and antiferromagnetism are often associated with opposite signs of the same quantum‐mechanical interaction, such a connection cannot be drawn in the case of antiferroelectricity. For this reason, the definition of an antiferroelectric is not only even more ambiguous than that of an antiferromagnet, but some scientists even doubt whether introducing the concept of antiferroelectricity makes sense at all. The situation is not better in relation to antiferroelastic materials.
We thus find ourselves in a rather unexpected situation. Ferromagnetism has been known to humankind for millennia, is known to almost everyone, is of enormous technical importance, and is well researched. The concept of ferroic order at large, however, is not very well defined in certain important aspects. In fact, the first proposition of an overarching concept for characterising it was only made in 1970 [1]. That approach was largely based on the symmetry change that occurs when the ferroic state is formed. A more comprehensive concept that included not only symmetry but also a number of phenomenological properties from physics and materials science was only introduced in the year 2000 [2]. What unites these two approaches is that they are based on the involvement of a very large number of atoms. Interactions on the level of the individual atoms that drive the spontaneous order are not part of the definition of a ferroic state. In the case of ferromagnetism, this is often forgotten. It is usually associated with a specific quantum‐mechanical correlation between atoms, but there are manifestations of ferromagnetic order that are driven by other interactions.
As we have seen, the research field of ferroic materials is still in the midst of development, with a number of construction sites at key points. Questions of major interest are:
Are there forms of ferroic order other than ferromagnetism, ferroelectricity, and ferroelasticity, the three established manifestations?
What happens if more than one type of ferroic order is present in the same material, a constellation we denominate as
multiferroic
?
How can we remove the existing ambiguities surrounding the concept of ferroic order?
In this book, we address these issues and propose new concepts, methods, and materials that we hope will advance the field of ferroics in some of its central aspects.
1.3 Laser Optics
Finally, almost everyone knows lasers and can associate the term optics with something involving light. In the combination of the two terms, people would generally imagine a source of intense light, where the latter is sent through transparent media such as microscopes and camera lenses, possibly in order to obtain a particularly bright image of an object illuminated with the laser radiation. In fact, this is exactly what we are planning to do here. Humans are ocular animals; the majority of information is received through the eye. Using a laser instead of a light bulb or the sun also permits us to see hitherto inaccessible aspects of an object because lasers represent not only a very bright but also a very clean source of light.
As in the case of (ferro‐)magnets, there are some very surprising aspects about lasers that are not known to the majority of people. Similar to magnets, lasers are very quantum‐mechanical objects. Coercing a material into emitting an intense, directed light beam can only be understood by resorting to the odd world of atoms where objects can appear as both a particle and a wave. A simple laser pointer can be bought and used by everyone and costs less than 10 euros, which makes us forget that it took until about 1960 to bring physics and technology together and demonstrate laser emission for the first time with a device as sketched in Figure 1.3a [3]. Furthermore, even though lasers are considered as an extremely intense source of light, capable of damaging the eye, they are in fact not very powerful. Some of the most intense laboratory lasers emit light of no more than about 10 W. The weakest vintage light bulbs used in households emit at least 25 W, and even LED light bulbs of 10 W are not particularly bright.
Figure 1.3 Nonlinear optical processes with lasers. (a) Design of the ruby laser used for the first demonstration of laser emission in 1960 [3]. A ruby crystal is optically excited with light from a flash lamp and driven to emit directed visible and very ‘clean’ deep red light with the help of two parallel mirrors. (b) First demonstration of a nonlinear optical process in the visible range. A quartz crystal is irradiated with the deep red light from a ruby laser (wavelength of 694 nm) to produce ultraviolet light at half the wavelength. The numbers indicate . The light was detected with a chemical film that is blackened by the incident laser light (big blotch) and the much weaker emission at half the wavelength (arrow). It is a curiosity of this landmark publication that the actual data point (arrow) is not visible. It was erased by the journal staff as an alleged dust particle when the figure was processed for publication.
Source: (a) Reproduced with permission from Yadav [4]. (b) Reproduced with permission from Franken et al. [5]/with permission of American Physical Society.
The exceptional intensity associated with laser light comes about in two ways. First, the laser light is highly directed, whereas a light bulb emits its radiation in all directions. At the same emitted power, a laser beam of 4 mm diameter at a distance of 1 m from the laser is a million times as intense with respect to the area it illuminates than a light bulb. Second, lasers often emit light pulses rather than continuous radiation. Hence, the emission is ‘compressed’ into a very short time bracket, whereas the laser is ‘off’ during the rest of the time. While the emission is taking place, it is therefore much stronger than if the emission was occurring in a continuous way. If the two types of pulsed lasers we consider in the context of this book were operated all the year round without interruption, they would only emit light for an integrated time of 10 s and 10 ms, respectively.
Apart from its unsurpassed intensity, laser light is also very clean in the sense that it has a very well‐defined colour and thus wavelength, unlike light bulbs or the sun, which emit a broad distribution of wavelengths interpreted by the eye as white. In addition, the laser light wave forms a very even wave pattern, such as in the case of a rock thrown into a calm lake as opposed to wind rippling its surface.
With the laser, we thus have an extremely intense and uniform light source that helps us to detect optical processes that are normally too weak to be observed. These are foremost processes, where the colour of the light changes when it interacts with a material. Typically, an object that is illuminated with light at a specific colour (as opposed to the white light emitted by the sun or a light bulb) scatters back light at exactly this wavelength. We can picture this scattering process in the way that the atoms of the material absorb an optical quantum, called photon, from the light field and emit it again after a while. The photon and its energy do not change in the process so that the colour of the light remains the same.
If the radiation is very intense, as when using a laser, it is possible that an atom absorbs two photons at once because they are so densely distributed. In the subsequent re‐emission, however, only a single photon is typically generated, which then carries the energy of both of the two ingoing photons. The higher energy corresponds to a shorter wavelength and, hence, to a change in colour. Thus, an object illuminated with deep red laser light emits a little bit of deep blue light as well. The part of optics dealing with wavelength‐shifting processes of this type is denominated as nonlinear optics.
It is quite striking how closely the foundation of the field of nonlinear optics is tied to the invention of the laser as the intense light source permitting us to detect nonlinear optical processes. The laser was introduced in 1960 [3], and the first report of a nonlinear optical process was published in 1961 [5], see Figure 1.3b. This rapid succession was possible because the theory for the simultaneous absorption of two photons had already been existing for 30 years, and only the appropriate tool for visualising it was missing [6]. By now, nonlinear optical processes have become very important in studying the structure and properties of materials. Since more photons and more wavelengths than in a conventional optical process are involved, nonlinear optics opens up access to a larger reservoir of information about a material. In addition, it allows researchers to literally ‘see’ this information, for example when taking photos of a sample using the light generated in a nonlinear optical process.
1.4 Creating the Trinity
In Sections 1.1–1.3, we have introduced three seemingly unrelated subjects. Symmetry has a proximity to mathematics, ferroic order refers to materials, and nonlinear laser optics deals with electromagnetic radiation fields. In the following discussion, we will see that these three so very different subjects are in fact perfectly made for one another. As we have explained, symmetry is a tool that enables us to make rather specific statements about systems whose inner structure and functioning are unknown to us. This makes it perfect for characterising and analysing ferroic systems because ferroic order is defined at the macroscopic level, that is disregarding the inner structure. In particular, for some of the lesser studied manifestations of ferroic order, we do not know the microscopic, atomic origin of the transition to spontaneous order.
All types of ferroic order break symmetries by definition, which may help us to develop a concept for ferroic order at large [1] and search for materials exhibiting novel types of ferroic order. Symmetry, or rather its loss, is also particularly well suited to describe materials exhibiting more than one type of ferroic order in the same phase.
In summary, symmetry can help us to find novel types of ferroic order, to explore systems with multiple manifestations of ferroic order, and to find overarching criteria helping us to overcome the existing ambiguities surrounding the concept of ferroic order.
While symmetry is our conceptual approach to exploring ferroic states of matter, nonlinear laser optics is the practical way to probe it. Just like matter, light as an electromagnetic radiation field has its characteristic symmetries. For example, an oscillating electric field as simple representation of a light wave would not look different if it is mirrored on the plane in which the oscillation occurs. This mirror operation thus is a symmetry operation with respect to the light field. In contrast, time reversal is not a symmetry operation because it would reverse the direction in which the light is propagating. One can therefore assume that light can address and thus probe a specific type of ferroic order if the symmetry of the light field is compatible with the symmetry of the ferroic state. The symmetry of the light field is controlled by setting its polarisation and direction of propagation. This makes polarisation‐dependent optical spectroscopy the perfect tool for investigating ferroic materials because for both the experimental tool and the system to which it is applied, symmetry is the common ground.
The particular advantage of optical experiments is that we can expand from linear optics involving a single light field towards nonlinear optics, where multiple light fields are brought in connection, as described above. By combining the symmetries of these light fields in the appropriate way, the very specific symmetry configurations of a ferroic state can be addressed with high selectivity. This can even be used to the extent that in systems featuring multiple types of ferroic order, the respective ferroic states can be addressed selectively by different nonlinear optical experimental configurations. Specifically, in a multiferroic exhibiting magnetic and electric order at the same time, the coexistence and interaction of the two forms of order can thus be investigated. No other experimental technique permits this to the extent nonlinear optics does.
As we see, the combination of symmetry, ferroic order, and nonlinear optics with lasers can give us unprecedented access to one of the most fascinating classes of materials. The nonlinear optical properties of ferroics have been investigated since the invention of the laser. From then on, the field has been developing with remarkable success. Despite several decades of research, however, there is only a relatively small number of review articles on this subject, and these articles are mostly focused on selected aspects. In particular, there appears to be no monograph presenting a comprehensive view on nonlinear optics applied to ferroic materials. It is the purpose of the work at hand to change this.
1.5 Structure of this Book
The Part I of this book is devoted to the basics and presents self‐contained introductions on symmetry, ferroic order, and nonlinear optics. Rather than summarising earlier literature on these well‐covered fields, we focus on those aspects that are little considered in the existing literature or that are relevant in bringing the three subjects together. This part concludes with an intuitive example uniting the introductions on symmetry, ferroic order, and nonlinear optics in a single model compound. This example provides a first glimpse at the extraordinary power of applying nonlinear optics to the study of ferroic materials and reveals several properties in our model compound that are inaccessible with other characterisation techniques.
The Part II of this book makes the transition from basics to materials
