ABSTRACT Title of Dissertation: MAGNETIC AND TOROIDAL SYMMETRY OF LITHIUM TRANSITION METAL ORTHOPHOSPHATES Stephanie Gnewuch Doctor of Philosophy, 2024 Dissertation Directed by: Professor Efrain Rodriguez Department of Chemistry and Biochemistry LiCoPO4 is the foremost candidate material for a novel type of ferroic ordering called ferrotoroidicity. In this work, the synthesis of polycrystalline sample of LiCoPO4 is discussed, along with the structural analog LiMnPO4. Their magnetic susceptibility and magnetic structure were determined and analyzed and found to be consistent with previous reports on single crystal materials. This work also provides a thorough introduction to ferrotoroidicity, a history of its theoretical development, and a summary of the most studied candidate materials. The work then presents a detailed methodology for determining the toroidal structure which would result for the magnetic structure in candidate ferrotoroidal materials. The model provides a method for determining how many toroidal moments would be present, where they would be located within the unit cell, and along which crystallographic direction they would be oriented. Detailed examples for determining the magnetic structure are provided for LiCoPO4 and analogous structures with the olivine structure type, as well as several structures with the pyroxene structure type. The results demonstrate a method for understanding ferrotoroidal arrangements, anti-ferrotoroidal arrangements and non-toroidal structures. MAGNETIC AND TOROIDAL SYMMETRY OF LITHIUM TRANSITION METAL ORTHOPHOSPHATES by Stephanie Gnewuch Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2024 Advisory Committee: Dr. Efrain Rodriguez, Chair/Advisor Dr. Lourdes Salamanca-Riba, Dean’s Representative Dr. William Ratcliff Dr. Mercedes Taylor Dr. Andrei Vedernikov Preface Since this document is a doctoral dissertation, it is important to explain the delegation of labor for the synthesis work between my teammates. I was the first group member to start on this project on ferrotoroidal materials, in the summer of 2016. The first objective was to synthesize large single crystals of candidate olivine materials, since they would be needed for detailed physical property measurements and neutron experiments under applied fields. My first synthetic approach was using hydrothermal methods. The rationale was that hydrothermal preparation would be less likely to produce metal oxide impurities which would interfere with magnetization measurements. Unfortunately, large single crystal growth proved impossible due to the low solubility of transitional metal phosphates in aqueous media up to 220◦C, which was the maximum temperature rating for the acid digestion vessels used for the synthesis. However, my initial synthetic attempts ultimately proved successful at producing polycrystalline LiCoPO4, LiFePO4, LiMnPO4, and various solid solutions of cobalt, iron, and manganese on the metal site. However, a disadvantage of the hydrothermal method was the size of the vessel and solubility of the starting reagents limited how much polycrystalline sample could be produced at one time. Another disadvantage of the hydrothermal method was it is not possible to produce LiNiPO4. All attempts at hydrothermal synthesis of LiNiPO4 produced apatite, which is a nickel phosphate mineral.[1] This is because nickel phosphate is very insoluble under the hydrothermal ii conditions possible using acid digestion vessels. The hydrothermal synthesis methods will be discussed in detail in §5.4. To address the disadvantages of the hydrothermal method, by the summer of 2017 I was also using solid state methods to produce olivine materials. The primary advantage of solid state methods is larger quantities can be synthesized at one time. It is also possible to synthesize LiNiPO4 and solid solutions containing nickel. However, it is vital to perform the synthesis under inert atmosphere (argon flow gas) to avoid oxidation of the transition metals. It is also important to carefully control the heating profile of the synthesis. The solid sate synthesis methods will be discussed in detail in §5.3. By the Spring of 2017 I also had been working with an undergraduate student researcher, Noah Bender. The project I had him work on was delithiating LiFePO4 to ultimately synthesize NaFePO4. The project was inspired by the detailed work of Maxim Avdeev.[2] Avdeev prepared LiFePO4 via a solid state technique and then delithiated it in a solution of nitronium tetrafluoroborate (NO2BF4) in acetonitrile. Gwenaëlle Rousse, who performed a detailed neutron diffraction study of LiFePO4 and FePO4, used a similar delithiation process.[3] Avdeev then sodiated the FePO4 powder by stirring with sodium iodide (NaI) in acetonitrile with gentle heating (60◦C, 48 hr) to form NaFePO4 with the olivine structure type. Avdeev and his colleagues then performed magnetic susceptibility and neutron diffraction experiments and found it had the same magnetic structure as LiFePO4. Therefore NaFePO4 was a desirable ferrotoroidal candidate, if we could ultimately grow large single crystals of it. By the end of the Fall of 2017 Noah had successfully been able to delithiate LiFePO4 powder using a peracetic acid solution method.[4, 5] In the Spring of 2017 Jacob Tosado also joined this research project. His role was building a prototype spherical neutron polarimetry device at the NIST Center for Neutron Research. The iii objective was to use spherical neutron polarimetry to analyze large single crystals of olivine materials to probe for evidence of ferrotoroidicity. He continued to work on the project in Maryland until the end of Spring 2021. By the summer of 2017 I had been successful at producing single crystalline samples of three end members LiMPO4 (M = Mn, Fe, Co) using a LiCl salt flux under argon atmosphere. To avoid oxidation, it had been suggested to me by my colleague Xiuquan Zhou to use graphite crucibles. There were a few small crucibles in storage leftover from a project by Chris Borg who had graduated in 2015. The crucibles were sized to fit inside the quartz synthesis tubes used ubiquitously in the Rodriguez research group. The crucibles were approximately 1 cm in diameter and about 1.5 cm tall. They proved a very successful growth container, but were easily damaged and could not be reused more than once or twice. The heating conditions would make the crucibles brittle, and the lid would tightly seal to the crucible bottom. Therefore opening it would damage the lid and crucible bottom. The crucibles were also quite small, which limited the surface area for crystal growth. Sometime in 2017 I discovered a very detailed research paper on flux growth of LiFePO4 by Yuri Jensen from the Peter Khalifah research group.[6] In it, the authors also concluded that graphite was an ideal crucible material since they are inexpensive yet effective. This confirmed my conclusion that graphite crucibles were a viable alternative to platinum crucibles and worthy of being pursued further. Indeed, in 2019 Rasmus Toft-Petersen, David Vaknin and their colleagues who had used platinum crucibles, discovered their olivine single crystals were impure.[7] They reused the platinum crucibles for different growths and inadvertently cross-contaminated their samples. The Janssen paper also used iron(II) oxalate dihydrate as the iron precursor for the solid state synthesis of LiFePO4, which I also found to be suitable. The organic oxalate ligand helps iv prevent oxidation of the iron ions. (The Avdeev 2013 paper used iron(II) oxalate as well.)[2] I advanced to candidacy at the end of the Fall semester in 2017, officially advancing January 2018. By the summer of 2018 I had decided to purchase large rectangular graphite crucibles with thick sides for the single crystal growth. The thick sides enabled them to be reused, and the large volume inside the crucibles provided a large surface area for nucleation and growth of large crystals. The lids were secured to the tops with inert metal wire. There was a large spool of nickel wire in the lab leftover from some other project which proved suitable for the task due to its high melting point. I believe it may have been specifically nichrome wire (a nickel-chromium alloy). I also knew by that time that in a LiFexMn1−xPO4 solid solution, the iron could be selectively oxidized to form ferrisicklerite (Li1−xFexMn1−xPO4). I learned this from an article I had found authored by a team of geologists.[8] The Spring of 2018 Timothy Diethrich had joined the Rodriguez research group and the Summer of 2018 I taught him my methods for olivine synthesis and crystal growth. I had also shared with him an article on ferrisicklerite summer.[8] By the end of the Fall of 2018 Tim had mastered single crystal growth for the entire series of LiFexMn1−xPO4. Tim and I continued to brainstorm together ways to optimize the solid state preparation of LiFexMn1−xPO4. Over the Summer of 2018 I began working on the toroidal theory presented in the previous chapters of this dissertation. I also completed the neutron powder diffraction experiments presented in Chp. 5. Therefore my objectives shifted to focusing on ferrotorodial theory, and on detailed study of the magnetic structure of the LiCoPO4 and LiMnPO4 end members. In an effort to delegate work so I could devote more time to toroidal theory, Tim took the lead on the LiFexMn1−xPO4 preparation and single crystal growth. He also eventually took the v lead on the delithiation work as Noah prepared to graduate in the Spring of 2019. Tim also began working on synthesizing and studying the magnetism in thiophosphate materials in the Spring of 2019. I continued to collaborate with Tim on the LiFexMn1−xPO4 series, particularly in determining which compositions would have the suitable symmetry for ferrotoroidicity. Tim graduated in the Fall of 2022. I also have powder diffraction and magnetization data for various solid solutions, but since Tim Diethrich took the lead on the project in 2019, I wanted there to be a clear separation in our dissertations of whose work was whose. In Tim’s dissertation and publicatons, all of the samples he synthesized himself. Likewise, all of the data he collected himself. His work was greatly informed and aided by my preliminary efforts, but ultimately his own. In addition, I do not intend at this point in time to publish any of my work on solid solutions of olivine materials, particularly LiMnxCo1−xPO4 and LiMnxFe1−xPO4. There are already many published structural and magnetization studies on those materials, since they are well-known cathode materials for batteries. The magnetic studies of the LiFexMn1−xPO4 delithiated samples, however, were novel enough for Tim to publish. We have struggled to delithiate samples containing cobalt. Indeed, Ehrenberg and his colleagues could only perform neutron powder diffraction on partially delithiated LixCoPO4, and discussed challenges with the delithiated form converting to amorphous powder.[9] In addition, MnPO4 is unstable due to Jahn-Teller distortion.[10, 11] Published Articles resulting from this work: 1. Gnewuch, S.; Rodriguez, E. E. The Fourth Ferroic Order: Current Status on Ferrotoroidic Materials. J Solid State Chem 2019, 271, 175–190. 2. Gnewuch, S.; Rodriguez, E. E. Distinguishing the Intrinsic Antiferromagnetism in Polycrystalline vi LiCoPO4 and LiMnPO4 Olivines. Inorg Chem 2020, 59, 5883-5895. 3. Diethrich, T. J.; Gnewuch, S.; Dold, K. G.; Taddei, K. M.; Rodriguez, E. E. Tuning Magnetic Symmetry and Properties in the Olivine Series Li1–xFexMn1–xPO4 through Selective Delithiation. Chem. Mater. 2022, 34, 5039–5053. Unpublished Work contained in this dissertation: • Chp. 2 contains the contents of Gnewuch 2019, but is completely re-organized and a great deal of content was added. • Chp. 3 is yet-unpublished work, except for §3.5, which presents my contribution to Diethrich 2022. • Chp. 4 is entirely yet-unpublished work. • Chp. 5 is the contents of Gnewuch 2020. vii Dedication To my Mother Mary. Thank you for all the prayers and support. viii Acknowledgments This has been an amazing journey through graduate school to get to this point. There are many people I would like to personally thank for their support along the way: I would first like to thank my advisor, Dr. Efrain Rodriguez. I came to graduate school to pursue additional study in crystallography,and it’s been an exciting opportunity learning powder diffraction and neutron scattering in his research group. Not only has he guided my studies and research, but has also encouraged and supported my growth personally. I am forever indebted to his unwavering faith in my intellectual potential, and support to see things through to the end. I would like to especially thank my colleagues Dr. Jacob Tosado and Dr. Timothy Diethrich who collaborated on experiments and spent countless hours planning and brainstorming in our combined efforts to support this larger research project investigating potential ferrotoroidic materials with neutron diffraction experiments. Our camaraderie and mutual support made this journey scientifically fruitful and personally fulfilling, as we celebrated each other’s successes and supported each other through the many challenges we each faced. I would also like to thank the undergraduate researchers Noah Bender and Kaitlyn Dold, who supported synthesis and crystal growth efforts on this project. I would also like to thank and acknowledge the U.S. Department of Energy Grant Number: DE-SC0016434 for the funding which made this research possible. I would like to thank the other graduate students and post-doctoral associates in the Rodriguez ix group who shared equipment, resources, and experience which made day-to-day research smooth and productive. In particular: Dr. Amber Larson, Dr. Daniel Taylor, Dr. Xiuquan Zhou, Dr. Rishvi Jayathilake, Dr. Austin Virtue, Dr. Brandon Wilfong, Dr. Ryan Stadel, Dr. Lahari Balisetty, Dr. Tianyu Li, Dr. Matthew Leonard, Huafei Zheng, Justin Yu, and Chris Borg. Dr. Brandon Wilfong deserves an additional acknowledgment for sharing expertise performing the SQUID experiments, as well as the post-doctoral associates and research scientists at the Maryland Quantum Materials Center Dr. Shanta Saha, Dr. Halyna Hodovanets, and Dr. Joshua Higgins, who assisted with instrument training and troubleshooting. The other facility at the University of Maryland which was invaluable in my research was the X-Ray Crystallography Center. A special thank-you goes to Dr. Peter Zavalij for his advice and expertise performing countless laboratory X-Ray diffraction experiments over my time in the program for preliminary sample analysis and alignment. The data collected from neutron and synchrotron beamlines was crucial for this research. I would like to thank the beamline staff at BT1 at the NIST Center for Neutron Research (NCNR) for their assistance performing experiments, including Dr. Craig Brown, Dr. Hui Wu, and Dr. Juscelino Leo. I would also like to thank Dr. Nicholas Butch for assistance with sample alignment on the Laue diffractometer at the NCNR on several occasions. I would not have been able to perform this work without the training I received in X-ray and neutron scattering and magnetic structure analysis I received from the numerous scientists who organized and participated in the following summer schools: 2018 and 2021 School on Representational Analysis and Magnetic Structures, University of Maryland, College Park; 20th National School on Neutron and X-Ray Scattering, Argonne National Laboratory and Oak Ridge National Laboratory; International School of Crystallography 53rd Course: Magnetic Crystallography, x Erice, Italy. I am grateful for their tireless efforts to educate and inspire the next generation of scientists. I would like to thank my candidacy committee members for their crucial advice at a critical time in my program: Dr. Nicholas Butch, Dr. Bryan Eichhorn, Dr. Lawrence Sita, and Dr. Andrei Vedernikov. Similarly, I would like to thank my dissertation committee members: Dr. William Ratcliff, Dr. Lourdes Salamanca-Riba, Dr. Mercedes Taylor, and Dr. Andrei Vedernikov. Of course, I would like to thank my parents, brothers, and friends for the love, prayers, and support they gave, getting me through this long journey. Lastly, I’d like to thank my undergraduate research mentor from Otterbein University Dr. Dean Johnston for introducing me to crystallography, and encouraging me to pursue further studies. Likewise, I’d like to thank my undergraduate research mentor from the NIST Summer Undergraduate Research Fellowship program Dr. Zeric Hulvey. It was he who introduced me to neutron diffraction at the NCNR and the software programs for analyzing data from BT1. Without their inspiration, I never would have known these opportunities existed, to have embarked on this memorable journey. xi Table of Contents Preface ii Dedication viii Acknowledgements ix Table of Contents xii List of Tables xiv List of Figures xvi Chapter 1: General Introduction 1 Chapter 2: Introduction to Ferrotoroidal Theory 4 2.1 Early Theory of Toroidal Moments in Crystalline Solids . . . . . . . . . . . . . . 5 2.2 Experimental Evidence for Ferrotoroidicity, and Modern Descriptions of Toroidal Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Toroidal Point Groups Based upon the Limiting Group Symmetry . . . . . . . . 29 2.4 Subsequent Neutron Scattering Studies of LiMPO4 (M = Fe, Co, Ni, Mn) . . . . 37 2.5 Challenges Proposing Other Ferrotoroidic Candidates . . . . . . . . . . . . . . . 43 2.6 Magnetoelectric Pyroxenes as Candidate Ferrotoroidic Materials . . . . . . . . . 47 2.7 Evidence for Hysteric Poling of Toroidal Domains . . . . . . . . . . . . . . . . . 62 2.8 Most Recent Work on Ferrotoroidic Candidates . . . . . . . . . . . . . . . . . . 70 2.9 Most Recent Theoretical Developments related to Ferrotoroidic Order . . . . . . 71 Chapter 3: A Magnetic Symmetry-Based Model for a Ferrotoroidal Lattice 76 3.1 Introduction and Motivation for Modeling Toroidal Moments Using the Conventions of Magnetic Space Group Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Space-Time Inversion Symmetry ( 1̄′ ) . . . . . . . . . . . . . . . . . . . . . . . 83 3.3 Behavior of Axio-Polar Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4 Ferrotoroidal and Antiferrotoroidal, and Non-Ferrotoroidal point groups . . . . . 94 3.5 Determining a Magnetic Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.6 Ferrotoroidal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Chapter 4: Toroidal Structure in Antiferromagnetic Materials with Olivine and Pyroxene Structure Types 120 xii 4.1 Toroidal Structure in the Olivine Structure Type . . . . . . . . . . . . . . . . . . 120 4.2 Example defining tγ and Tnet using Γ − 2 . . . . . . . . . . . . . . . . . . . . . . . 128 4.3 Determining tγ and Tnet using Γ − 4 . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.4 Determining tγ and Tnet using Γ − 1 . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.5 Determining tγ and Tnet using Γ + 2 . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.6 Determining tγ and Tnet using individual basis vectors from Γ − 1 and Γ + 1 . . . . . 137 4.7 Determining tγ and Tnet using individual basis vectors from Γ − 2 and Γ − 4 . . . . . 139 4.8 Comparison of the Magnetic and Toroidal Structures in Various Transition Metal Orthophosphate Materials with the Olivine Structure Type . . . . . . . . . . . . 141 4.9 Toroidal Structure in the Pyroxene Structure Type . . . . . . . . . . . . . . . . . 150 4.10 Example defining tγ and Tnet using Γ − 1 . . . . . . . . . . . . . . . . . . . . . . . 155 4.11 Example defining tγ and Tnet using Γ − 2 . . . . . . . . . . . . . . . . . . . . . . . 157 4.12 Comparison of the Magnetic and Toroidal Structures in Various Antiferromagnetic Materials with the Pyroxene Structure Type . . . . . . . . . . . . . . . . . . . . 159 4.13 General Summary and Discussion of the Methodology . . . . . . . . . . . . . . 168 Chapter 5: Magnetism and Magnetic Structure of LiCoPO4 and LiMnPO4 171 5.1 Olivine Structure Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 5.2 Experimental: General Goals of Synthesis Methodology . . . . . . . . . . . . . 176 5.3 Experimental: Hydrothermal Synthesis Methodology . . . . . . . . . . . . . . . 178 5.4 Experimental: Solid State Synthesis Methodology . . . . . . . . . . . . . . . . . 180 5.5 Experimental: Preliminary Phase Identification with Laboratory Powder X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.6 Magnetic Susceptibility Measurements . . . . . . . . . . . . . . . . . . . . . . . 184 5.7 Neutron Powder Diffraction Experiments of Solid State Samples . . . . . . . . . 193 5.8 Magnetic Symmetry Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 5.9 General Discussion of the Magnetic Susceptibility for LiMnPO4 . . . . . . . . . 209 5.10 General Discussion of the Magnetic Susceptibility for LiCoPO4 . . . . . . . . . 212 5.11 General Discussion of the Magnetic Structure . . . . . . . . . . . . . . . . . . . 213 5.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Chapter 6: Conclusions and Future Directions 217 Bibliography 220 xiii List of Tables 2.1 Irreducible Representations of the Point Group 1̄1′ . . . . . . . . . . . . . . . . . 5 2.2 Magnetic Symmetry Associated with the Irreducible Representations of the Point Group Pnma1′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 The 31 magnetic point groups that allow for ferrotoroidicity . . . . . . . . . . . . 32 2.4 The 19 magnetic point groups which permit only a linear magneto-electric effect . 33 2.5 The 31 magnetic point groups allowing a toroidal moment along one, two, or three directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 The limiting groups of primary ferroics . . . . . . . . . . . . . . . . . . . . . . . 37 2.7 Irreducible representations of the point group P21/c1′ and associated magnetic symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1 Behavior of Vector Types under Inversion Symmetries . . . . . . . . . . . . . . . 90 3.2 Comparison of the Transformations of Axial and Axio-polar Vectors under Four Magnetic Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.3 Classifications of the Point Groups Labeled “Other” in Tab. 3.4 . . . . . . . . . 102 3.4 Ferrotoroidal and Anti-ferrotoroidal Point Groups. . . . . . . . . . . . . . . . . 103 3.5 Symmetry parameters of the toroidal moment in centrosymmetric subgroups of mmm1′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 3.6 Symmetry parameters of the toroidal moment in centrosymmetric subgroups of 2/m1′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.7 Magnetic symmetry associated with the eight irreducible representations of the parent paramagnetic space group Pnma1′ . . . . . . . . . . . . . . . . . . . . . . 108 3.8 Character table for the irreducible representations of Pnma1′ . . . . . . . . . . . 112 3.9 Magnetic symmetry for various LiFexMn1−xPO4 and Li1−xFexMn1−xPO4 compositions113 4.1 Wyckoff letters in the crystallographic space group Pnma′, along with their multiplicities, site symmetries and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.2 Definitions of the rβ α vectors centered about the 4a and 4b sites for the toroidal moments in the olivine structure with the space group Pnma . . . . . . . . . . . 125 4.3 Magnetic symmetry associated with the eight irreducible representations of the parent paramagnetic space group Pnma1′ . . . . . . . . . . . . . . . . . . . . . . 128 4.4 Definitions of the mα vectors for the spin modes Cy of Γ − 2 . . . . . . . . . . . . . 129 4.5 Definitions of the mα vectors for the spin modes Ax +Cz of Γ − 4 . . . . . . . . . . 133 4.6 Definitions of the mα vectors for the spin modes Cx +Az of Γ − 1 . . . . . . . . . . 135 4.7 Definitions of the mα vectors for the spin modes Gx +Fz of Γ + 2 . . . . . . . . . . 136 xiv 4.8 Definitions of the mα vectors for the Gy spin mode of Γ + 1 and the Cx spin mode of Γ − 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.9 Definitions of the mα vectors for the Cy spin mode of Γ − 2 and the Cz spin mode of Γ − 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.10 Summary of the magnetic symmetry and refined magnetic structure in various olivine analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.11 Summary of the toroidal order in various olivine analogs . . . . . . . . . . . . . 147 4.12 Wyckoff letters in the crystallographic space groups P2′1/c and P21/c′, along with their multiplicities, site symmetries and coordinates . . . . . . . . . . . . . 151 4.13 Definitions of the rβ α vectors centered about the 2a, 2b, 2c, and 2d sites for the toroidal moments in the pyroxene structure with the space group P21/c . . . . . . 153 4.14 Magnetic symmetry associated with the four irreducible representations of the parent paramagnetic space group P21/c1′ . . . . . . . . . . . . . . . . . . . . . 154 4.15 Definitions of the mα vectors for the spin modes Ax +Cy +Az of Γ − 1 . . . . . . . 155 4.16 Definitions of the mα vectors for the spin modes Cx +Ay +Cz of Γ − 2 . . . . . . . 158 4.17 Summary of the magnetic symmetry and refined magnetic structure in various pyroxene analogs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.18 Summary of the toroidal order in various pyroxene analogs . . . . . . . . . . . . 166 5.1 Summary of the Curie-Weiss fitting parameters for LiCoPO4 and LiMnPO4 . . . 189 5.2 Refined structural models for LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . 198 5.3 Refined structural models for LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . 200 5.4 Crystallographic parameters of LiCoPO4 and LiMnPO4 . . . . . . . . . . . . . . 201 5.5 Character table for the irreducible representations of Pnma1′ . . . . . . . . . . . 205 5.6 Magnetic symmetry associated with the eight irreducible representations of the parent paramagnetic space group Pnma1′. . . . . . . . . . . . . . . . . . . . . . 206 5.7 Magnetic model and parameters deduced from Rietveld refinement of neutron powder diffraction patterns taken at 10 K. . . . . . . . . . . . . . . . . . . . . . 207 5.8 Theoretical magnetic moment values for LiCoPO4 and LiMnPO4 . . . . . . . . . 209 xv List of Figures 2.1 Structural models of a toroidal dipole moment proposed by Dubovik, Tugushev, and their colleagues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Temperature-dependence of the off-diagonal α32 and α23 components in the magnetoelectric tensor of the m′m2′ phases of various boracites . . . . . . . . . . . . . . . . . . . 9 2.3 The temperature dependence of the αxy and αyx components of the magnetoelectric tensor in LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Butterfly loops measured in LiCoPO4 near the Néel temperature just below 22 K . 13 2.5 Magnetic Structure of LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 The SHG measurements of LiCoPO4 presented by Van Aken . . . . . . . . . . . 20 2.7 Model of the ring of spins in LiCoPO4 leading to the toroidal moment . . . . . . 22 2.8 Adaptation of the model for producing net toroidization in a crystal lattice . . . . 24 2.9 Model of antiferrotoroidic order in YMnO3 proposed by Spaldin, Fiebig, and Mostovoy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.10 Simplified version of Ascher’s 1974 Venn diagram . . . . . . . . . . . . . . . . 31 2.11 The antiferromagnetic magnetization around the Co magnetic ions in the (101) plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.12 The spherical neutron polarimetry results for MnPS3 . . . . . . . . . . . . . . . 46 2.13 The structure of clinopyroxene and orthopyroxene . . . . . . . . . . . . . . . . . 48 2.14 Model for toroidal moments proposed by Baum et al. . . . . . . . . . . . . . . . 50 2.15 Temperature dependence of the electric polarization under applied magnetic field in LiFeSi2O6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.16 Possible magnetic order for a system with P21/c structural symmetry . . . . . . . 53 2.17 Model for toroidal moments in LiFeSi2O6 proposed by Tolédano et al. . . . . . . 54 2.18 Magnetic structure of LiFeSi2O6 and LiFeGe2O6 . . . . . . . . . . . . . . . . . 57 2.19 Temperature dependence of the magnetic field-induced pyroelectric current, polarization, and induced electric polarization under various magnetoelectric annealing conditions for CaMnGe2O6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.20 Proposed ferrotoroidic and antiferrotoroidic domains in LiCoPO4 . . . . . . . . . 63 2.21 The temperature dependence of the SHG intensity in the (100) plane of Li(Ni0.80Fe0.20)PO4 and calculated temperature dependence of the spin angle . . . . . . . . . . . . . 65 2.22 Hysteretic poling of LiCoPO4 (100) by magnetic and electric fields . . . . . . . . 66 2.23 The toroidal and non-toroidal contributions to the SHG intensity of LiCoPO4 (001) under magnetic and electric fields . . . . . . . . . . . . . . . . . . . . . . 67 2.24 Model presented by Zimmermann, Meier, and Fiebig for the “ring” of spins leading to a net toroidal moment Ty . . . . . . . . . . . . . . . . . . . . . . . . 68 xvi 2.25 Model presented by Zimmermann, Meier and Fiebig for the “ring” of spins leading to a net toroidal component +Tz in one domain, and −Tz in the opposite orientational domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.26 Van Aken’s diagram comparing the behavior of the four kinds of ferroic order under time inversion (1′), spatial inversion (1̄) symmetries . . . . . . . . . . . . . 71 2.27 Gopalan’s diagram comparing the behavior of axial and polar vectors under time inversion (1′), spatial inversion (1̄), and rotation inversion (1Φ) symmetries . . . . 72 2.28 Cástan’s diagram comparing the behavior of vectorial ferroic properties under spatial and time inversion symmetries . . . . . . . . . . . . . . . . . . . . . . . 74 2.29 Cheong’s diagram comparing the behavior of vectorial ferroic properties under spatial and time inversion symmetries . . . . . . . . . . . . . . . . . . . . . . . 75 3.1 Depiction of an infinite rotation axis perpendicular to the plane of a circle . . . . 85 3.2 Simple depiction of a ring of two moments (spin A and spin B) within a two- dimensional lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.3 Transformation of axial vectors aligned parallel to the 2-fold rotation axis in four magnetic point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4 Transformation of axio-polar vectors aligned parallel to the 2-fold rotation axis in four magnetic point groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.5 Group-subgroup trees where the maximal supergroups are 6/m′mm and 4/m′mm . 99 3.6 Group-Subgroup trees where the maximal supergroups are 6/m′m′m′ and 4/m′m′m′100 3.7 Group-Subgroup trees where the maximal supergroups are 6′/mmm′, 4′/m′m′m, and 6′/m′mm′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.8 Subgroup Diagram for Pnma, Pn′m′a′, Pnma′, and Pnm′a . . . . . . . . . . . . . 111 3.9 Depictions of five one-dimensional unit cells with different origin symmetries . . 115 3.10 Toroidal lattice within a one-dimensional antiferromagnetic lattice . . . . . . . . 119 4.1 Magnetic Structure of LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4.2 Ferrotoroidal Structure of LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.3 Antiferrotoroidal Structure of LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . 149 4.4 Magnetic Structure of LiFeSi2O6 . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.5 Ferrotoroidal Structure of LiFeSi2O6 . . . . . . . . . . . . . . . . . . . . . . . . 167 5.1 Delithiated LiFePO4 sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.2 Preliminary refinements from X-ray powder diffraction data of hydrothermal and solid state LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 5.3 Preliminary refinements from X-ray powder diffraction data of hydrothermal and solid state LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.4 Temperature-dependent magnetic susceptibility of hydrothermal LiCoPO4 and LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.5 Temperature-dependent magnetic susceptibility of hydrothermal LiMnPO4 . . . . 185 5.6 Temperature-dependent magnetic susceptibility of solid state LiCoPO4 and LiMnPO4187 5.7 Inverse susceptibility of solid state and hydrothermal LiCoPO4 and LiMnPO4 . . 188 5.8 Field-dependent magnetic susceptibility of solid state and hydrothermal LiCoPO4 and LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 xvii 5.9 Field-dependent magnetic susceptibility of hydrothermal LiMnPO4 at 2 K . . . . 192 5.10 Field-dependent magnetic susceptibility of (top) solid state LiCoPO4 . . . . . . . 193 5.11 Indexed neutron powder diffraction patterns for LiCoPO4 data taken at 10 K . . . 195 5.12 Indexed neutron powder diffraction patterns for LiMnPO4 data taken at 10 K . . . 196 5.13 Refinements of the nuclear and magnetic structures with neutron powder diffraction data of LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 5.14 Refinements of the nuclear and magnetic structures with neutron powder diffraction data of LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 5.15 Temperature dependence of the integrated intensity of the (012) reflection of LiCoPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.16 Temperature dependence of the integrated intensity of the (010) reflection of LiMnPO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.17 Magnetic structure model of LiCoPO4 and LiMnPO4 . . . . . . . . . . . . . . . 214 xviii Chapter 1: General Introduction So often in crystallography, a lower symmetry can be wrongly assigned to a crystal structure because the higher symmetry is overlooked. In many cases it might be undetected due to instrumental limitations. In other instances the additional symmetry is simply not considered. But to uncover hidden symmetries, it is first necessary to have the mathematical language to describe them. Such is the case with ferrotoroidal materials (also called ferrotoroidic materials). The primary objective of my doctoral work was to develop the language for describing this fourth category of primary ferroic materials. There are three well-established primary ferroic orders: ferroelectric polarization, ferromagnetic magnetization, and ferroelastic strain.[12] Each are classified by an order parameter which describes the long-range ordering which occurs due to loss of symmetry elements as the material undergoes a phase transition from its higher symmetry para- ferroic phase to its lower symmetry ordered ferroic phase. Several researchers have long known that certain magnetoelectric materials which display a liner magnetoelectric effect fulfill many of the criteria for ferrotoroidal materials. The first chapter of this document recounts the historical development of the theory behind ferrotoroidal materials. The general consensus is toroidal moments would arise from a “ring” of magnetic spins, oriented head-to-tail. There also have been many attempts at defining which magnetic point groups would allow ferrotoroidal order. The general consensus is the magnetic point group 1 should also permit a linear magneto-electric effect. The connection to the magneto-electric effect is strengthened by the assertion that crossed magnetic and electric fields should be able to control ferrotoroidal domains in a material. Chapter 2 also discusses the various materials which have been or are considered candidate ferrotoroidal materials. The most notable of them are LiCoPO4 with the olivine structure type, and LiFeSi2O6 with the pyroxene structure type. Both materials have a number of compositional analogs which are also candidate materials. The chapter discusses all of the magnetism-related research on these candidate materials, and the arguments which have been put forth for them being ferrotoroidal candidates. Upon reviewing the literature, I identified a number of unanswered questions: • Where will the toroidal moments be located? • How will the toroidal moments be oriented? • How can a “ring of spins” be defined systematically in a magnetic material? • What is the symmetry of a “ring of spins”? • How can the structure and composition of a material be altered to maximize the net toroidal moment in a material? I recognized they all could be answered by developing a model for understanding the toroidal structure in a material which would result from the complementary magnetic structure. Since the only symmetry criteria for ferrotoroidic materials are based upon their magnetic point symmetry, there had been no consistent method for defining where the toroidal moments would order within a material. Since the magnetic moments have defined, discrete locations within a material, I argue in Chapter 3 that there should also be defined, discrete locations for the 2 toroidal moments. I also argue that they must occur at sites with 1̄′ symmetry, since this is the defining symmetry of any “ring of spins.” In Chapter 4, I present thorough example calculations for determining the possible toroidal structures in various materials with the pyroxene and olivine structure types. In doing so, I demonstrate how toroidal structures could be classified as ferrotoroidal, antiferrotoroidal, non- toroidal, and ferritoroidal. I also provide example calculations for determining the magnitude and directions of individual toroidal moments within the unit cell, and the net toroidal moment per unit cell. Of course, the ultimate goal of this research is to test candidate materials. And my ultimate goal as a chemist is to create candidate materials which maximize the net torodial moment in a material. Chapter 5 presents my work on two end members in the olivine series: LiCoPO4 and LiMnPO4. I used two synthetic methods to prepare polycrystalline samples. The first was hydrothermal methods, which are less likely to create metal oxide impurities. The second was solid state methods, which can create larger quantities of product. I performed detailed temperature- and field-dependent magnetization measurements on the materials created by the two different methods, to compare their signal. I then performed neutron powder diffraction experiments, and determined their magnetic structures through symmetry analysis. I used the structural information of LiCoPO4, LiMnPO4, and other reported structures in my ferrotoroidal models in Chapter 4. In doing so, I demonstrate this method of modeling toroidal order can be applied to different materials, and used to predict optimal materials for further study. 3 Chapter 2: Introduction to Ferrotoroidal Theory This chapter contains the contents of Gnewuch 2019, but is completely re-organized and a great deal of content was added. Gnewuch, S.; Rodriguez, E. E. The Fourth Ferroic Order: Current Status on Ferrotoroidic Materials. J Solid State Chem 2019, 271, 175–190. This chapter begins (§2.1) with a description of the development of the classification criteria of ferrotoroidic materials based upon their magnetic ordering. The second section (§2.2) discusses the major reported experiments investigating certain materials with the olivine structure type for evidence of ferrotoroidic ordering, leading up to a landmark study by Van Aken et al. in 2007 on olivine material LiCoPO4. Those experiments motivated subsequent work by a number of experimentalists in neutron scattering, as well as theorists. The symmetry requirements permitting ferrotoroidic order were revisited during that time period, and §2.3 summarizes the modern understanding of the symmetry requirements for ferrotoroidic order. A summary of the neutron scattering studies on candidate materials with the olivine structure type are presented in §2.4. Other candidate materials are discussed in §2.5 and §2.6, particularly those with the pyroxene structure type based upon work on LiFeSi2O6. Another noteworthy study in 2014 provided evidence for the ferroic nature of the magnetoelectric response in LiCoPO4, which is discussed in §2.7. The chapter concludes with a summary of the developments in theory (§2.9) and proposed 4 Table 2.1: Irreducible Representations of the Point Group 1̄1′. The representation labels are based upon the scheme used in Ref. [14]; the classification criteria are based upon the scheme used in Ref. [16]. 1̄1′ 1 1̄ 1′ 1̄′ Classification Γ1 1 1 1 1 Strain Γ1̄ 1 1 -1 -1 Magnetization Γ11′ 1 -1 1 -1 Polarization Γ1̄′ 1 -1 -1 1 Toroidization ferrotoroidic candidates since that time (§2.8). 2.1: Early Theory of Toroidal Moments in Crystalline Solids Edgar Ascher is considered the first to postulate that if there are symmetries which permit net magnetization, net electric polarization, and the null case of strain, there must be a fourth category which describes yet another type of ordered “polarization.”[13] His argument is based upon how the symmetry elements of the point group 1̄1′ can be reduced into four irreducible representations, given in Table 2.1. According to this scheme, magnetic moments are invariant under spatial inversion (1̄), polar moments are invariant under time inversion (1′), and a third kind of moment is invariant under space-time inversion (1̄′). Throughout the 1960’s and 70’s Ascher and his colleague Janner postulated this third category could describe ordered current density, based upon the point group symmetry materials display under electromagnetic fields.[13, 14, 15] The papers published over this time period categorize the magnetic (i.e., Shubnikov) point groups based upon which space and time inversion symmetry elements they contain. Initially they classified the fourth category, with the representation Γ1̄′ as “spontaneous current” and postulated it was related to superconductivity. Ascher abandoned that theory within a couple years, but the classification scheme remained plausible. 5 a) b) Figure 2.1: Structural models of a toroidal dipole moment proposed by Dubovik, Tugushev, and their colleagues.[17, 18, 19] a) A current loop arranged as a torus. The toroidal moment labeled “T” is indicated along the rotation axis with an upwards-pointing arrow. Fig. 2.1a is reproduced from Fig. 1 in Ref. 17. b) Local moments labeled “Πi” are precessing on a circle. Their centrosymmetric arrangement gives rise to the toroidal dipole moment “TΠ.” Fig. 2.1b is reproduced from Fig. 1c in Ref. 18. Independently from Ascher, V. M. Dubovik and his colleague considered the electrodynamic conditions required for toroidal moments to arise from current density around magnetic dipoles.[17] They concluded a toroidal dipole would occur along the rotation axis perpendicular to electric current rings arranged as a torus. Fig. 2.1a is a diagram of this arrangement. Dubovik continued developing that work with V. V. Tugushev and several other colleagues, and like Ascher also considered the multipole moments which would be generated from the charge density of the arrangement.[18, 19] Ascher then classified the moments according to their symmetry transformations under space and time inversion operations, and called these the magnetic dipole moment M, the charge dipole moment P, and the toroidal dipole moment T. Dubovik and Tugushev recognized the simplest scenario to produce a toroidal dipole moment would be an antiferromagnetic arrangement of magnetic spins arranged in a ring head-to-tail, which would approximate a torus. Fig. 2.1b presents their idealized models for this arrangement. Mathematically, the expression for the magnetic toroidal moment T from the localized 6 current density distribution j(r) centered about an origin is written as T = 1 10c ∫ [r(r · j)−2r2j]d3r, (2.1) where c is the speed of light in a vacuum, and r is the radius of the ring centered on the origin.[19] A detailed discussion of the physical manifestation of the toroidal dipole moment is outside the scope of this work. However, it is important to note the interaction energy between a toroidal moment T and homogenous magnetic field B is minimized when the toroidal moment is parallel to the curl of the magnetic field. Mathematically, the contribution to the free energy density FT can be expressed as:[19] ∆FT ∼ T · (∇×B). (2.2) Therefore the toroidal moment would be perpendicular to a ring of magnetic moments in an electro-magnetic system. In addition, there is also a contribution to the free energy density from crossed electric E and magnetic B fields:[19] ∆FT ∼ T · (E×B). (2.3) Therefore materials which would permit a net toroidal moment T would also by symmetry permit a linear magnetoelectric effect. Specifically, the signal from a net toroidal dipole in the material would appear as antisymmetric components in the magnetoelectric tensor. This connection between the linear magnetoelectric effect and toroidal moments is the reason the magnetoelectric effect is studied intensely in candidate ferrotoroidic materials. Later in the paper,[19] Dubovik and Tugushev considered how a local magnetic moment µloc would manifest in a system of magnetic moments m at points arranged in a ring with radius r: µloc(r) = ∑ i miδ (r− ri). (2.4) This is the same mathematical description used for electric polarization. The net toroidal dipole moment T from such a ring of local magnetic moments is then: 7 T(µloc) = 1 2 ∫ r×µ⊥d3r. (2.5) Around the same time, Tugushev and Krotov [20] tabulated all the symmetry invariants which would permit toroidal ordering for an antiferromagnetic arrangements of spins, and organized them according to crystal system. This method of describing a magnetic structure involves defining an antiferromagnetic material as a superposition of two anti-aligned ferromagnetic lattices. The symmetry invariants describe the directions in which the symmetry allows a net alignment of spins. Since symmetry elements are described by the invariance of a point or vector (or other entity) under symmetry operations, these vectors are called the symmetry invariants. They further classified the invariants according to the spatial parity and point group for each of the 230 space group types. The object of this scheme is to predict if a particular antiferromagnetic spin orientation would permit toroidal ordering. They only considered collinear antiferromagnetic structures in which the magnetic cell coincides with the structural unit cell, since the resulting toroidal moment arrangement would be uniform. They also assumed that materials in which either the magnetic spins were canted, or the magnetic and structural unit cells did not coincide, would result in an anti-toroidal arrangement. The results revealed that in all cases in which toroidal ordering is permitted, the toroidal vector invariants were always accompanied by antiferromagnetic vector invariants. One would conclude from this, that while the magnetic ordering is distinct from the toroidal ordering, a magnetically ordered system of the appropriate symmetry is a necessary condition for it to occur. They cannot occur independently from one another. Gorbatsevich, Kopaev, Ginzburg, and their colleagues continued to develop the theoretical basis for ordered toroidal moments.[21, 22, 23] Like Ascher initially, they were inspired by the “current loops” which would produce a toroidal moment. They were initially convinced the 8 a) b) c) Figure 2.2: Temperature-dependence of the off-diagonal α32 and α23 components in the magnetoelectric tensor of the m′m2′ phases of the following boracites: a) Co3B7O13Br, b) Co3B7O13I, and c) Ni3B7O13Cl. Reproduced from Figure 1 in Ref. 24. An identical figure is presented in Ref. 25. “loops” were related to superdiamagnetism. Ginzburg realized the connection that the point groups which would permit toroidal moments, also permit the linear magnetoelectric effect.[21] Gorbatsevich and Kopaev’s 1994 paper[22] cites Ascher’s classification scheme,[15] and also reviews the current understanding of how toroidal moments are related to the electric and magnetic properties. While both the spin and orbital contributions from ordered magnetic moments can contribute to a toroidal moment, the authors began to consider how a configuration could occur in a crystalline material. Considering only the contribution from the magnetic spins, a simple configuration would be one or more pairs of anti-aligned magnetic moments. This type of compensated configuration of magnetic moments is always classified as antiferromagnetic. A few years later, Sannikov reported that in several boracites, an unexpected discontinuity was observed in certain off-diagonal components of the temperature-dependent magnetoelectric susceptibility.[24, 25] The measurements are presented in Fig. 2.2. (Sannikov published identical 9 figures in both Ref. 24 and 25.) Boracites have the general formula M3B7O13X where M is a divalent cation and X is a halide anion. Sannikov was studying the Co-Br, Co-I, and Ni-Cl boracites in particular. In the papers, he presented detailed phenomenological arguments for why a toroidal dipole moment can be regarded as a primary order parameter. He explained the sharp peak observed in the off-diagonal components of the temperature-dependent magnetoelectric susceptibility are indicative of a phase transition. He proposed this phase transition would be to an ordered ferrotoroidic state, and the observed discontinuity could be considered a characteristic signal of a ferrotoroidic phase transition. This work was the experimental evidence establishing the connection between the magnetoelectric effect, and ferrotoroidic order. He published additional examples of boracites exhibiting this characteristic signal in subsequent years.[26, 27] As a note, his 2007 paper “Ferrotoroics” serves as a concise summary of the phenomenological theory he developed.[28] He was the sole author on all these papers. Around this same time, Popov and a number of co-authors published a paper reporting measurements of the linear magnetoelectric effect in Ga2−xFexO3, and present analysis arguing the materials can produce a net toroidal moment.[29] They cite work by Gorbatsevish and Dubovik when defining the toroidal moments per unit cell. First, they considered Eq. 2.1 (mentioned previously) for the toroidal moments. Similarly, the net magnetic moment M is: M = 1 2c ∫ r× j(r)d3r. (2.6) Since not stated previously, the current density is the curl of the spin density of localized spins, which is expressed mathematically as: j(r) = c∇×S(r). (2.7) In the case of toroidal moments, the spin density has a divergence of zero because it is arranged in a ring. The ring of spins will then produce current density perpendicular to it. The magnetic 10 ions in the unit cell of any magnetic crystalline material are in discrete locations. Therefore the net magnetic moment per unit cell due to the spins of all the magnetic moments S (with units of µB) can be determined by a summation: M = 2µB ∑ α Sα . (2.8) This expression is simpler to compute than integrating over the unit cell. In the same way, Popov and his colleagues re-wrote Eq. 2.5 as a summation of discrete toroidal moments in a unit cell. Their final expression for the net toroidal moment per unit cell is: T = µB 2 ∑ α rα ×Sα . (2.9) They defined the rα vector as the “radius” vector with the origin at the center of the unit cell, and the terminus as the position of magnetic atom α with spin S. Popov and his colleagues go on to explain that in Ga2−xFexO3, there is a displacive phase transition so that the non-centrosymmetric mm2 structure transforms to the centrosymmetric mmm structure. Therefore, they defined the rα vectors for the initial structure and the distorted structure, and calculated the toroidization from the difference between the structures. Popov and several colleagues published a similar (shorter) paper three years later on BiFePO3.[30] In it, they explained the asymmetry in their magnetoelectric measurements arose from toroidal moments, based upon the symmetry invariants of the antiferromagnetic structure.1 Independently from the Russian scientists, Swiss scientists Schmid, Rivera, and their colleagues were studying the magnetoelectric susceptibility in various materials, including various boracites and LiCoPO4 (among others). In the articles on LiCoPO4,[31, 33] Rivera and Schmid study the 1A brief note: The works mentioned thus far (besides Edgar Ascher’s paper) were published by scientists from the former Soviet Union, and therefore much of their early work was originally published in Russian. As a result, there is not a consistent term used when referring to “materials with toroidal moments.” They are sometimes named “ferrotoroics” or simply materials with “toroid” or “toroidal” moments. The discrepancies complicate literature searches. I only consulted the work which has been translated into English. 11 a) b) Figure 2.3: The temperature dependence of the αxy and αyx components of the magnetoelectric tensor in LiCoPO4. The components are highly asymmetric, with the αyx component approximately two times greater than the αxy component. a) Reproduced from Fig. 17 in Ref. 31. b) The data was republished with an improved fitting model as Fig. 5 in Ref. 32. temperature dependent magnetoelectric susceptibility, and observe the component αxy is about half the value of αyx. Their measurements are presented in Fig. 2.3a. They also observe “butterfly loops,” near the transition temperature, which is just below 22 K. Fig. 2.4 presents their measurements. The “V” shape is due to reversing the antiferromagnetic domains in the material, as the magnetic field is reversed. However, the asymmetric shape to the features are what give them the name “butterfly” loops. Rivera was surprised to find such a feature in an antiferromagnet because it is typically only observed in ferromagnetic or ferrimagnetic (weakly ferromagnetic) crystals. The magnetoelectric tensor measurements only detected αxy and αyx terms, consistent with the point group mmm′ and not a lower symmetry. Rivera and Schmid collaborated with Kornev and several other colleagues to more thoroughly analyze this effect in both LiCoPO4 and LiNiPO4. Their paper published in 2000 includes an improved fitting model for the temperature dependence of the αxy and αyx susceptibility, shown 12 Figure 2.4: Butterfly loops measured in LiCoPO4 near the Néel temperature just below 22 K. The electrodes for measuring the charge were applied on the (100)-cut crystal face. The magnetic field is applied parallel to the y axis, which is parallel to the magnetic spins. Reproduced from Fig. 11 in Ref. 31. in Fig.2.3. The model is given by the equation α = a(TN −T )1/2 1−b(TN −T ) (2.10) where TN is the antiferromagnetic transition temperature. The fitted parameter values for (αxy,αyx) are a = (−6.881,−14.72) and b = (−0.03,−0.561). They also present a number of additional magnetoelectric measurements under a variety of fields and temperatures. Kornev and the others then present a phenomenological model for understanding the butterfly loops and other behavior in these materials.[32] They consider the magnetic symmetry according to the formalism assigning components of spin density waves associated with the different possible irreducible representations for describing the symmetry breaking that occurs during the phase transition. These are described by a net vector, which is the sum of the contributions from 13 Table 2.2: Magnetic Symmetry Associated with the Irreducible Representations of the Point Group Pnma1′. Adapted (and corrected) from Tab. 1 in Ref. 32. Γa M.S.G.b M.P.G.c αi j ̸= 0d Basise Γ + 1 Pnma mmm none L1 y Γ − 1 Pn′m′a′ m′m′m′ αxx, αyy, αzz L1 y, L2 x, L3 z Γ + 2 Pn′m′a m′m′m none Mz, L1 x Γ − 2 Pnma′ mmm′ αxy, αyx Tz, L2 y Γ + 3 Pnm′a′ mm′m′ none Mx, L1 z Γ − 3 Pn′ma m′mm αyz, αzy Tx, L3 y Γ + 4 Pn′ma′ m′mm′ none My Γ − 4 Pnm′a mm′m αxz, αzx Ty, L2 z, L3 x a Irreducible Representation, b Magnetic Space Group, c Magnetic Point Group, d Non-zero terms of the magnetoelectric susceptibility tensor αi j, e Basis functions where the antiferromagnetic basis functions are L1 = s1 + s2 − s3 − s4, L2 = s1 − s2 + s3 − s4, and L3 = s1 − s2 − s3 + s4, the ferromagnetic M = s1 + s2 + s3 + s4, and the ferrotoroidic T. each of the spins. For example, a ferromagnetic configuration is notated M = s1 + s2 + s3 + s4. Antiferromagnetic configurations are noted as L2 = s1 − s2 + s3 − s4, etc. L1 and L3 are included in the caption of Tab. 2.2, which is an adaptation of Table 1 in their paper.[32] In their table presenting the irreducible representations of the magnetic structure Pnma1′, they indicate that three invariants are associated with T which they call “toroidal momentum.” In their discussion of the antiferromagnetic phase transition Pnma1′ → Pnma′ for LiCoPO4, they note the invariant L2 y is also associated with a nonzero Tz value (i.e., it is spontaneous). After two brief sentences they conclude, “We shall not consider further this invariant.” In their conclusions they suggest the butterfly loops could be due to an incommensurate modulation of the spins, but acknowledge a good deal more experimental and theoretical work is needed for a more conclusive analysis. In 1999, Schmid published an article titled “On the Possibility of Ferromagnetic, Antiferromagnetic, Ferroelectric, and Ferroelastic Domain Reorientations in Magnetic and Electric Fields.”[34] This 14 paper is the first attempt to expand Aizu’s classification of ferroic materials.[35] The work does not have a separate classification of “ferrotoroidal” materials, but includes a lengthy table of materials and their ferroic classifications. In particular, it notes a number of compounds as “ferrotoroidal,” including a number of boracites, LiCoPO4, and Ga2−xFexO3. Schmid cites Popov,[29] but primarily cites work on boracites which were published in the journal Ferroelectrics. It is unclear if he was familiar at that time with any of the work which had been performed across the iron curtain. Schmid wrote an extensive review of magnetoelectric materials in 2000.[36] In it, he listed a lengthy table of 34 known magnetoelectrics by their magnetic point group, ordering temperature, and maximum magnetoelectric coefficient value. LiCoPO4 had the sixth-highest magnetoelectric coefficient in that table. In addition, since its point symmetry was mmm′ and it had an orthorhombic crystal lattice, the a, b, and c axes would be inequivalent. This can be helpful when performing various kinds of physical measurements because it is easier to unambiguously align the sample and assign the axes. In addition, it had been known for several decades that the entire series of LiMPO4 (M = Fe, Co, Mn, Ni) structural analogs could be grown from an inexpensive LiCl salt flux. Michel Mercier had studied this series in the mid-1960’s through the early 1970’s as part of his doctoral dissertation at the University of Grenoble with Néel, Bertaut, and others.[37, 38, 39, 40, 41, 42, 43] He had been able to grow large, quality single crystals several millimeters in size which certainly made crystal alignment much easier for performing the magnetoelectric measurements. The crystals are also indefinitely stable under ambient conditions. They do not need to be stored under inert atmosphere. Rivera and Schmid’s earliest reported work on LiCoPO4 cited Mercier,[38] so they were undoubtedly aware this was an ideal system for detailed study for practical reasons. 15 Besides the practical advantages for physical property measurements, the LiMPO4 (M = Fe, Mn, Ni, Co) series is also ideal because of its structural chemistry. They belong to the general class of metal orthophosphates with the olivine-structure type with the space group Pnma (No. 62). The particular structure type for LiMPO4 is triphylite because the divalent transition metal ion is located on the 4c Wyckoff position and the monovalent lithium ions are located on 4a site. (If the situation is reversed with the transition metal on the 4a site, then the structure is named maricite.) In triphylite, the PO3− 4 units bridge the two layers of ions, so that the terminal oxygen atoms on the PO3− 4 ions form a distorted octahedral coordination environment around the Co ions. The structure of LiCoPO4 is depicted in Fig. 2.5. It is also trivial to prepare solid solutions of triphylite, in which combinations of Fe2+, Mn2+, Ni2+, and Co2+ are substituted on the 4c site, using stoichiometric quantities of starting reagents. Therefore, the variety of different magnetically-active ions which can be used to prepare structurally analogous materials makes this series rich for study, since the magnitude of the magnetic moment and even magnetic ordering can be rationally tuned by substitution on the 4c site. Schmid subsequently wrote a more developed article on ferrotoroidics in the conference proceedings of the International Symposium on Ferroic Domains and Mesoscopic Structures which took place in Nanjing, China in 2000.[45] In this work, he presents Ascher’s table as the starting point for rationalizing how ferrotoroidics could be considered a fourth category of ferroic materials. Schmid proposes that materials in this fourth category should exhibit the same hallmarks of other ferroics, including domain structure and hysteretic poling under applied fields. Because of this connection to other ferroics, he suggests standardizing the name to “ferrotoroidic” to maintain the “-oic” ending. He presents an extensive table re-defining primary, secondary, and tertiary ferroics according to those “driving fields.” In this work, he cites multiple papers by 16 a) b) Figure 2.5: The magnetic structure of LiCoPO4. Cobalt ions are shown as blue spheres inside distorted octahedra. Phosphorus atoms are shown as grey spheres inside grey tetrahedra. Oxygen atoms are red spheres, located at the vertices of the polyhedra. The Li ions are omitted for clarity. The magnetic moments on the cobalt ions are indicated as red arrows. The depicted magnetic structure is Pnma′, in which the magnetic moments are aligned along the b axis. a) View along the c axis, [001] direction. b) View along the a axis, [100] direction. Reproduced from Gnewuch & Rodriguez (2019).[44] Sannikov, Gorbatsevich, Ginzburg, Dubovik, and Popov.[19, 21, 22, 25, 29] (I cite the articles I was able to access.) Schmid published a similar review article from the proceedings of the 5th Magnetoelectric Interaction Phenomena in Crystals (MEIPIC-5) conference in 2004.[46] Around this time Schmid, his colleague Rivera, and two colleagues from Ames Labs, Vaknin and Zarestky, were intensely studying the magnetic ordering and magnetoelectric effect in LiCoPO4, LiNiPO4 and other olivine analogs.[47, 48, 49, 50, 51] Also around this time, Manfred Fiebig began studying magnetoelectric phenomena and began collaborating with Rivera and Schmid, continuing to explore the connection to ferrotoroidicity in magnetoelectric materials.[52] While the magnetic spins in LiCoPO4 are aligned in an anti-ferromagnetic arrangement along the b axis, they are aligned along the c axis in LiNiPO4. This had been known since Santoro, Segal, and Newman’s early neutron diffraction experiments,[53] and Mercier’s magnetoelectric 17 experiments.[38] Although, even Mercier noted the direction of the spins in LiNiPO4 as “c?” with a question mark in the table of his paper, and ended with a comment he was not certain of the magnetic structure.[38] In 2000 Kornev, Rivera, Schmid, and their co-authors observed unexpected “butterly loops” in his magnetoelectric measurements for both LiCoPO4 and LiNiPO4.[32, 54] This type of feature is seen in materials with a weak ferromagnetic signal, but the magnetic point groups mmm′ for LiCoPO4 and mm′m for LiNiPO4 associated with their established ordering should not allow a ferromagnetic signal. From 2001-2003 Kharchenko et al. followed up with detailed temperature and field-dependent magnetic susceptibility measurements of LiCoPO4 and LiNiPO4 single crystals from Schmid.[55, 56, 57] They all confirmed a very weak ferromagnetic signal. In addition, Kharchencko proposed that while the ground state magnetic ordering is commensurate in LiNiPO4, as it is heated just above its magnetic transition temperature it undergoes a phase transition to an incommensurate phase. In 2004 Vaknin and Zarestky collaborated with Rivera and Schmid, and confirmed through neutron diffraction studies the ordering was indeed incommensurate above about 21 K until around 36 K — above which the material behaves like a paramagnet.[49, 50, 51] They did not observe an incommensurate transition in LiCoPO4. This made LiCoPO4 a more ideal model candidate for ferrotoroidicity, since the magnetic ordering was less complex than in LiNiPO4. 2.2: Experimental Evidence for Ferrotoroidicity, and Modern Descriptions of Toroidal Order Interest in ferrotoroidicity was re-ignited around 2007 when Van Aken, Rivera, Schmid, and Fiebig published an article in Nature presenting evidence for ferrotoroidic domains in LiCoPO4.[58] 18 Their work used second harmonic generation (SHG) spectroscopy to visualize the domains in large single crystals. Figure 2.6 are a reproduction of their results. In this technique, electromagnetic radiation of a particular frequency is directed to the sample, so that a polarization is induced and radiation is emitted at double the incident frequency. This can be expressed as P(2ω) = ε0χ̂E(ω)E(ω) where P is the non-linear polarization of the emitted light, ε0 is the permittivity of free space, E(ω) is the electric field component of the incident light of a given frequency ω , and χ̂ the second-order susceptibility.[58] Like all physical property tensors, the components of χ̂ are restricted by the point symmetry of the crystal. Crystal domains are readily imaged using this technique, as their different orientations will cause some domains to appear brighter or darker than others. It should be noted, the researchers have specially built instrumentation for performing these measurements under cryogenic temperatures. With this, they were able to measure the SHG intensity as a function of temperature. They subsequently published a more detailed article on their measurements in Physical Review Letters.[59] The prevailing assumption by that time was that only a change in toroidization would be observable, and the change would be associated with some slight change in the structure of the material. For example, in the work by Popov et al. explained a few paragraphs ago,[30] the toroidal moment was calculated using the difference between the centrosymmetric and non- centrosymmetric structures. However, it is also possible to have small displacements of atoms which preserve the underlying space group symmetry. For example, in LiCoPO4 with the space group Pnma, the Co2+ ion lies on the 4c Wyckoff position. The coordinates for atoms lying on this position are given by (1 4 + ε , 1 4 , −δ ) where the ε and δ describe the displacements of the atoms.[58] The other three sites are generated from the first. In addition, the single crystal neutron diffraction experiments of LiCoPO4 by Vaknin, Zarestky, and Schmid had revealed the 19 spins on the Co ions were slightly canted away from the y axis by a small angle (4.6◦).[48] In addition to the canting, Vaknin and his colleagues also referenced a 2001 article by Kharchenko, Schmid, and others. In it, they reported a weak ferromagnetic signal along the y axis from field- Figure 2.6: The SHG measurements of LiCoPO4 presented by Van Aken et al. (2007). a) The image obtained from light corresponding to the χxyy component of the SHG susceptibility tensor. b) An illustration of the domain structure. c) The image obtained from the interference of light from the χyyz and χzyy components. The new domains which result are outlined in red. They correspond to the walls of the ferrotoroidic domains. The inset is evidence the domain wall moved upon being cooled below the antiferromagnetic transition temperature. Originally published in Ref. [58] and reproduced in the review article by Gnewuch & Rodriguez (2009).[44] 20 and temperature-dependent magnetization measurements of a single crystal of LiCoPO4. The combination of these two subtleties lowers the magnetic space group symmetry from Pnma′ to P2′111 (or P12′11 depending upon the chosen axis), which corresponds to lowering the point group symmetry from mmm′ to 2′. Van Aken and his colleagues suggested the possible mechanism for a toroidal state in LiCoPO4 is due to these subtle changes in the magnetic structure.[58] The reduction in the point symmetry from mmm′ to 2′ lowers the number of symmetry operations from eight to two, which would result in four different domain states. They argued that two of the domains could be considered antiferromagnetic, and the other two ferrotoroidic domains. A year after the paper was published, Schmid (one of the co-authors) argues all four domains are ferrotoroidic.[60] His argument is based upon Litvin’s work published shortly after the Nature article.[61] Litvin extended Aizu’s ferroic classification scheme to include ferrotoroidic order, and to determine the number of distinguishable domains permitted in a ferrotoroidic material. His result was that the permitted number of domain states does not change. It is still determined by the magnetic domain states. Therefore Schmid argues that the antiferromagnetic domains should be considered ferrotoroidic domains. He considers them two different descriptions for the same physical state. The Nature article by Van Aken et al. also made a first attempt at modeling a “ring of spins” in the crystal structure. Figure 2.7 is a reproduction of the figure they presented in their paper.[58] Since T ∝ ∑α rα ×Sα they defined the center of the unit cell as the origin of the ring of spins, and the r vectors were then the vectors pointing from the center to each of the four spins.[58] Due to the inherent symmetry of the unit cell, this meant that the r vectors had two different magnitudes, so that r1,3 > r2,4. The assumption then was that the r1 and r3 vectors would give a greater contribution than the other two. In addition, the r1 and r3 vectors could be 21 a) b) Figure 2.7: Model of the ring of spins in LiCoPO4 leading to the toroidal moment, as presented in Van Aken et al. (2007). Reproduced from Figure 3 in Ref. 58. considered to form a clock-wise ring, while the r2 and r4 vectors would give an anti-clockwise ring. Therefore the contributions from each would be opposite and partially cancel with one another. Consequently, one would only observe a net toroidal component Ty. The slight canting of the spins would slightly change the magnitude of T since it is defined by rα ×Sα . The major significance of the 2007 Van Aken article was re-igniting interest in toroidal materials over the next several years. Several theorists, notably Eerenstein, and also Ederer and Spaldin were exploring how ferrotoroidic materials would be classified within the established fields of ferroics, multiferroics, and magnetoelectrics.[62, 63, 64] Ederer and Spaldin in particular were interested in developing a clearer model for how toroidal moments could occur in a crystalline solid, with a periodic boundary. In their 2007 article,[63] they began by considering a local distribution of magnetic moments in an ordered antiferromagnetic material, described by Popov et al.[29] according to Eq. 2.4, 2.5, and 2.9. Ederer presented a slightly simpler form of Eq. 2.9: t = 1 2 ∑ α rα ×mα . (2.11) Since mα = gµBSα , the units of µB are implied by treating the spins as individual magnetic moments on magnetic ions. Using this same notation, the net toroidal moment per unit volume is T = 1 2V ∑ i ri ×mi, (2.12) 22 where V is the volume of the unit cell and the index i indicates the sum of all [rα ×mα ] in the material. Popov and his colleagues had performed similar analyses with the [rα ×Sα ] vectors in Ga2−xFexO3, giving units of a net toroidal moment per unit cell as µBÅ. Ederer and Spaldin report the units as µBÅ−2 for the spontaneous toroidization and as µBÅ per unit cell. They give examples calculating the net toroidal moment for LiCoPO4, Ga2−xFexO3, and BiFeO3. They calculated the toroidal moment per unit cell for LiCoPO4 to be 1.75 µBÅ. Ederer and Spaldin also discuss choice of origin in some detail in their 2007 article.[63] They also point out that only the “fully compensated” components in an antiferromagnetic configuration would contribute to a toroidal moment, and the toroidal moment would be independent of the chosen origin in this case. Figure 2.8b is a graphical depiction of how Spaldin et al. proposed the displacements of the magnetic moments can lead to a toroidization in the lattice. Their argument is based upon how net polarization occurs in a lattice due to ion displacements, which is depicted in Fig. 2.8a. Spaldin published similar figures with co-authors in a few articles published since 2007 as well. See Fig. 3 & 5 in Ref. 63, Fig. 4 in Ref. 65, and Fig. 6 in Ref. 66. They did warn that if a structural distortion accompanied the toroidal origin, it is important to maintain the same origin in both structures. The contribution to the toroidal moment described according to Eq. 2.11 is due to an antiferromagnetic configuration of moments. If the moments are not fully compensated so that there is a ferromagnetic component (like in a ferrimagnet) then that contribution can be described according to t̃ = 1 2 R×m, (2.13) where R is the average position of the magnetic moments. While t is independent of the origin, t̃ is dependent. They went on to argue that since the definition of ferrotoroidic ordering requires breaking both time and spatial inversion symmetry, in antiferromagnetic configurations this could 23 a.) ++ + ++ + – – – – – – b.) 2a a a (1–d)a (1+d)a Figure 2.8: Adaptation of the model for producing net toroidization in a crystal lattice. a) Rectangle indicates one unit cell with length 2a of two oppositely charged particles, separated by a distance a. As the charges move closer to one another by a displacement d they produce a net polarization. b) The charged particles are replaced with magnetic moments of opposite spin. This is the simplest depiction of an antiferromagnetic arrangement. As the moments are distorted from their original location, the net change in the arrangement would be described as the net toroidization. Reproduced from Gnewuch & Rodriguez (2019). occur from changes in the orientation or positions of the magnetic moments. Since a ferro(i)magnet always breaks time inversion symmetry, then in order to also break spatial inversion symmetry, the positions of the magnetic moments must not be related by spatial inversion. In other words, their arrangement must be noncentrosymmetric. By this reasoning, they argued simply changing the orientations of the magnetic moments in an antiferromagnetic arrangement would be sufficient to break both spatial and time inversion symmetry. Ederer and Spaldin concluded by warning that if any structural displacements accompany the magnetic ordering, care must be taken to ensure a consistent origin choice. For this reason, they advocated choosing the center of the unit cell as the origin for defining the r vectors when calculating rα ×mα . 24 As a brief note, Ederer published an article using similar analysis in 2009.[64] (He was the sole author.) In it, he compares the structures of BiFeO3 and FeTiO3, elaborating on the analysis for BiFeO3 presented in his previous work.[63] He argues the slight rotations of the oxygen octahedra in the paraelectric R3̄c phase of FeTiO3 lowers the magnetic symmetry and produces toroidal moments in the material. He then goes on to argue the antiferromagnetic order parameter can be used to explain the magnetoelectric tensor in the material. He emphasizes both the position and orientation of the magnetic moments need to be analyzed together when determining how the antiferromagnetic configuration can produce toroidal moments. He also argues that if a toroidal material has a spontaneous non-zero toroidal moment, a non-toroidal material would correspond to a “centrosymmetric” ensemble of toroidal moments. However, he cautions against using Eq. 2.11 in a single unit cell to conclude t = 0, and that the material is therefore non-toroidal. As a note, if one does this with BiFeO3, the toroidal moments produced are in a centrosymmetric arrangement and therefore cancel with one another, leading to t = 0. Separately from Ederer, Spaldin continued to collaborate with a variety of colleagues on theoretical considerations for ferrotoroidicity. In 2008 she co-authored an extensive review paper with Fiebig and Mostovoy exploring toroidal moments and the connection to the magnetoelectric effect.[65] They clarified that the toroidal moment would be considered a vector order parameter analogous to magnetic and electric dipoles. Long-range ordering of the toroidal moments should occur spontaneously below a certain critical temperature. The ordering should be able to be described according to a single order parameter. Orientational domains would be expected to spontaneously occur below the critical temperature. Like other ferroics, there should be a vector (or tensor) field which could control the domains. (In the following years, Litvin published three papers tabulating all the tensors which would distinguish ferrotoroidic domains.)[16, 67, 68] 25 Since different domains have different orientations of the order parameter, applying a field would change the energy between the states so that under a strong enough field they could be reoriented. To be considered a primary ferroic, the switching would need to occur according to only one field. The paper by Spaldin, Fiebig, and Mostovoy also further develops the concept of a toroidal lattice.[65] They were particularly concerned with how to achieve simultaneously breaking spatial and time inversion symmetry. They assume there would need to be a structural distortion of some kind, so that the magnetic ions would shift. This would result in a change in the toroidization, which would be the measurable quantity. In ferroelectric transitions, structural distortions break the spatial inversion within the cell, leading to a polarization in the cell. Only changes in polarization are measurable, so it seemed reasonable toroidization would be the same. In addition, the toroidization in a lattice would not be a single value, because it would have a different value for the translation of every lattice vector R. If V is the volume of the primitive cell, then the change in the toroidization would be: ∆T = gµB 2V R×S, (2.14) where g is the gyromagnetic ratio. The authors call this the “toroidization increment” of the “toroidization lattice.” The assumption would be that the toroidization increments would be defined by the shifts in the ions in the lattice to form antiferromagnetic dimers (or pairs of anti- aligned moments). The shifts would be relative to a high-temperature non-toroidized reference structure (presumably the paramagnetic state), which would be straightforward to calculate as long as a consistent origin is chosen. One mechanism Spaldin et al. proposes for toroidization is spin-lattice coupling due to superexchange interactions.[65] This kind of distortion could break both spatial and time inversion symmetries simultaneously. The strength of superexchange interactions depends on the 26 metal cation–oxygen anion–metal cation bond angle, where the metal is a magnetic ion. These interactions are described by the Anderson-Kanamori-Goodenough rules.[69] For example, if the angle is close to 90◦ the magnetic moments tend to align in a ferromagnetic arrangement; if the angle is closer to 180◦ the moments align in an antiferromagnetic arrangement. The rules depend upon the particular cation (metal and oxidation state) because it depends upon the degree of orbital overlap (covalency) between the oxygen atoms and the metal cations. If the involved atoms shift due to an applied field or phase transition, the exchange interactions would therefore be impacted as well. The other possible mechanism the authors suggest would be due to purely electronic interactions with no atomic distortions. The change in the spin states would cause virtual excited states in the metal cation–oxygen anion–metal cation bonds. The changes in the electronic structure would cause a dipole moment to form as the electrons are redistributed, creating oxygen holes as the electrons move closer to the metal cations. The paper by Spaldin, Fiebig, and Mostovoy [65] also discusses several limitations to using the linear magnetoelectric effect to study potential ferrotoroidic materials. While the magnetic multipoles of torus-like spin arrangements would produce toroidal moments, and while toroidal materials would be expected to display a linear magnetoelectric effect, they argue it is not possible to assign specific components of the magnetoelectric tensor to contributions from toroidal moments alone. In other words, toroidal moments would indeed give an off-diagonal magnetoelectric response, but there can be other causes for anti-symmetric off-diagonal terms. As an example, they cited the conical spiral ordering in ZnCr2Se4 where the rotation axis is parallel to the magnetic propagation vector.[70, 71] That configuration would result in the antisymmetric off-diagonal components αxy = −αyx in the magnetoelectric tensor. They also propose general criteria for good magnetoelectric materials. For example, the upper limit of the magnetoelectric 27 tensor is proportional to the dielectric susceptibility χe ii and magnetic susceptibility χm j j so that αi j ≤ √ χe iiχ m j j. (2.15) They point out that magnetic geometrically frustrated systems often have large magnetic susceptibilities, which would make them good materials to consider. Magnetic frustration can also break time inversion symmetry. To conclude, they emphasize that while the magnetoelectric effect is useful for studying ferrotoroidic materials, the measurements are indirect. The anti-symmetric off- diagonal αi j =−α ji tensor components are not unique to ferrotoroidic materials. Spaldin, Fiebig, and Mostovoy [65] also clarify that a ferrotoroidic would be considered a primary and not a secondary ferroic. The free energy term of a ferromagnetoelectric is F = −αi jEiH j. It is a secondary ferroic because both the electric and magnetic field are required to switch the domains. It is, by definition, always observed in materials which exhibit a linear magnetoelectric effect. The authors argue that even though crossed electric and magnetic fields (E ×H) are used to approximate a toroidal field in experimental measurements, it is only the toroidal field which controls the order parameter in a ferrotoroidic. Therefore, ferrotoroidic materials can be considered primary ferroics. It should be mentioned here, that a linear magnetoelectric effect is not unique to ferrotoroidic materials. However, ferrotoroidic materials would exhibit a linear magnetoelectric effect with anti-symmetric off-diagonal components. It is a necessary but not sufficient condition for ferrotoroidicity. The challenge is that from a point symmetry perspective you can only predict which tensor terms of a physical property are allowed or disallowed by symmetry. The actual physical response can only be determined experimentally. Finally, Spaldin, Fiebig, and Mostovoy[65] consider the possibility of antiferrotoroidic order. They explain that such a system would need to break the same symmetries as ferrotoroidic order, but not lead to coupling with an external field. In other words, it must break both spatial and 28 Figure 2.9: Model of antiferrotoroidic order in YMnO3 proposed by Spaldin, Fiebig, and Mostovoy (2008). The figure on the left is simplified depiction of the arrangement of Mn3+ spins. The open circles indicate the spins along z = 0 and the filled circles indicate the spins translated to z = c/2. The three sublattices are indicated by ovals and the toroidal moment for each is labeled with large white arrows. The smaller figure on the right emphasizes the toroidal moments for the sublattices would cancel, but lead to a net toroidal moment along the z axis. Reproduced from Fig. 8 in Ref. 60. time inversion symmetry but yet lead to a net toroidal moment of zero. They suggest the domains in YMnO3 could be considered antiferrotoroidic, but do not suggest a possible microscopic mechanism for how such a state could occur. Their model is presented in Fig. 2.9. They propose there are three different types of ferrotoroidic sublattices present, represented by large arrows in the x and y plane. While the net toroidal moments of the three different orientational sublattices would cancel one another, their arrangement would produce a net toroidal moment along the z direction. 2.3: Toroidal Point Groups Based upon the Limiting Group Symmetry Around the same time Spaldin, Fiebig, and Mostovoy were writing their article, Schmid also wrote an extensive review article of ferrotoroidic materials.[60] Schmid was particularly interested in the point symmetry which would permit ferrotoroidic ordering, and how to unambiguously classify a ferrotoroidic material as a primary ferroic. In crystallography the term “point group” 29 can be used in several different contexts: molecular or site symmetry, and the point symmetry of the space group. The point symmetry of a macroscopic crystal represents the vectors normal to the crystal faces and is useful for understanding the macroscopic physical properties. However, molecular or site symmetry describes the symmetry around a point in the crystal structure. Since the early work on toroidal materials was investigating the physical properties, the exploration of the symmetry was restricted to point symmetry of the macroscopic crystal. Therefore the assumption when considering candidate materials is the point symmetry of the space group must be included in the list of point groups which permit toroidal ordering. Ascher’s second article [15] included Venn Diagrams of the different point groups which would be considered ferromagnetic, ferroelectric, and the “third” category (what he called ferrokinetic or ferroconductive). Throughout the rest of the 20th century most every research article on “ferrotoroidal” order included some discussion of permitted point groups. However, there was never any consensus on a list. Schmid presented a Venn Diagram adapted from Ascher’s work.[13, 15, 60] Figure 2.10 is a simplified verison. The symmetric categorization seemed compelling, with 19 point groups permitting one type of ordering for each category, four point groups permitting two types of ordering for each category, and nine point groups permitting all three types of ordering. Schmid reviewed the work by several researchers who had considered the permitted tensor terms of magnetoelectric materials.[60] For example, Grimmer considered which tensors would be permitted under 1̄′ inversion,[72] based upon Ascher’s 1966 table (Tab. 2.1). These were tabulated by point group. Schmid and his colleague Rivera had long considered the different tensors of magnetoelectric materials. The work presented in his 2008 paper includes a number of tables organized by point group symmetry which were modifications of his earlier work 30 T 14 MPT 9 P 14 M 14 MP 4 PT 4 MT 4 Figure 2.10: Simplified version of Ascher’s 1974 Venn diagram showing the relationships between the the point groups associated with net magnetization M, polarization P, and toroidization T.[15] A modified version of this diagram was presented in Schmid (2008).[60] The premise is that depending upon the point symmetry, one, two, or three types of ordering can be permitted in the same phase for each category. In a primary ferroic, only one type of ordering should be allowed. Reproduced from Gnewuch & Rodriguez (2019).[44] published in 1973, 1994, and 2000.[36, 73, 74] The 1994 paper meticulously compares the point groups from tables by half a dozen contemporaries studying magnetic materials including Birss, Cracknell, and Siratori. It helpfully clarifies conversion of SI units in magnetoelectric measurements. It also clarifies how the chosen setting for the point group impacts the permitted tensor terms. In his 2008 paper, Schmid extends his original work on the classifications of ferroic order (primary, secondary, teriary, etc.) to include ferrotoroidic ordering, and emphasizes that multiple types of ordering can co-exist with one another. For example, he explains that certain point groups can permit both ferrotoroidic and antiferromagnetic order. Table 2.3 summarizes the permitted point groups Schmid determined. Since Schmid was working closely with the experimentalists Fiebig, Van Aken, and Rivera, he was particularly concerned with understanding how the magnetic point symmetry would 31 Table 2.3: The 31 magnetic point groups that allow for ferrotoroidicity, classified according to Table 7 in Ref. 60. Order allowed Magnetic point groups Ferrotoroidic 1̄′, 2/m′, 2′/m, m′mm, and antiferromagnetic 4̄′,4/m′, 4/m′mm, 4̄′m2′, 3̄′, 3̄′m, 6̄′, 6/m′, 6/m′mm, 6̄′m2′ Ferrotoroidic 22′2′, 42′2′, 32′, 62′2′ and ferromagnetic Ferrotoroidic mm2, 4mm, 3m, 6mm and ferroelectric All three 1, 2, 3, 4, 6, ferroic orders m, 2′, m′, mm′2′ impact the physical property measurements. This would influence how they were classifying domains in the SHG (second harmonic generation) studies. For example, I mentioned previously that in the 2007 Van Aken article, they claimed there were two antiferromagnetic and two ferrotoroidic domains. By Schmid’s 2008 paper,[60] he is convinced all four domains are both antiferromagnetic (weakly ferromagnetic) and ferrotoroidic. The next experimental challenge to address would be controlling the domains with applied electric and magnetic fields. To do this systematically, it is first necessary to determine how ferrotoroidics are classified compared to other kinds of materials with magnetic and electric ordering, such as magnetoelectrics, multiferroics, etc. In Table 2 of that paper,[60] Schmid organizes all of the 122 Heesch-Shubnikov magnetic point groups by their type of magnetic and/or electric ordering, and if an “invariant velocity vector Vs” is permitted or not. Schmid explains the Vs vector is that “third” type of vector Ascher considered in 1974.[15] Schmid believes this is appropriately called “net toroidization,” analogous to “magnetization” and “electric polarization.” Of the 122 magnetic point groups, 19 permit only a linear magnetoelectric effect and not any higher ordering. (For example, ferromagnetobielectrics are controlled by 32 Table 2.4: The 19 magnetic point groups which permit only a linear magneto-electric effect,a sorted by whether they permit a spontaneous toroidal moment Ts. The information is adapted from Table 2 in Ref. 60. Ts not permitted Permits Ts 1̄′, 2/m′, 2′/m m′m′m′ mmm′ 4′/m′, 4′/m′m′m, 4/m′m′m′ 4/m′, 4/m′mm 3̄′m′ 3̄′, 3̄′m 6/m′m′m′ 6/m′, 6/m′mm 432, m′3, m′3m′ ai.e. these point groups do not permit any higher order effects such as a linear magnetobielectric effect one electric and two magnetic fields so that the free energy F is proportional to three terms: F ∝ ∆γi jkHiE jEk.) They are listed in Tab. 2.4. Schmid’s 2008 article [60] cited Litvin’s early 2008 article [61] but was published before Litvin’s series of articles mentioned previously.[16, 67, 68] Litvin’s concern is focused on the consistent mathematical treatment of ferrotoroidics with other symmetry classification methods for primary ferroics.[61] Table 2.5 includes all the toroidal point groups and the forms of the toroidal tensor Litvin tabulated in Ref. 67. It is worthy to note that one of the benefits of investigating olivine LiCoPO4 as a candidate ferrotoroidic material is that the space group Pnma′ with the point group mmm′ would have a toroidal moment along only one direction (0,0,T3). Note: m′mm would have the net toroidal moment (T1,0,0). If the magnetic space group is lower and actually P2′111 with a point group 2′y it would be permitted along two directions (T1,0,T3). Note: 2′z would have the net toroidal moment (T1,T2,0). Depending upon which symmetry elements in the point group are primed will impact the final allowed directions of the toroidal moment. A Note: A modern resource (currently being improved and expanded) to determine the 33 Table 2.5: The 31 magnetic point groups allowing a toroidal moment along one, two, or three directions, as adapted from Tables 2 and 3 in Ref. 67. Magnetic Point Form of the Toroidal Magnetic Point Form of the Toroidal Group (MPG) Physical Property Tensor Group (MPG) Physical Property Tensor 1 (T1,T2,T3) 4zmxmxy (0,0,T3) 1̄′ (T1,T2,T3) 4̄′z2 ′ xmxy (0,0,T3) 2z (0,0,T3) 4z/m′ zmxmxy (0,0,T3) 2′z (T1,T2,0) 3z (0,0,T3) mz (T1,T2,0) 3̄′z (0,0,T3) m′ z (0,0,T3) 3z2′x (0,0,T3) 2′z/mz (T1,T2,0) 3zmx (0,0,T3) 2z/m′ z (0,0,T3) 3̄′zmx (0,0,T3) 2′x2′y2z (0,0,T3) 6z (0,0,T3) mxmy2z (0,0,T3) 6̄′z (0,0,T3) m′ xmx2′z (T1,0,0) 6z/m′ z (0,0,T3) m′ xmymz (T1,0,0) 6z2′x2′1 (0,0,T3) 4z (0,0,T3) 6zmzm1 (0,0,T3) 4̄′z (0,0,T3) 6̄′zmx2′1 (0,0,T3) 4z/m′ z (0,0,T3) 6z/m′ zmxm1 (0,0,T3) 4z2′x2′xy (0,0,T3) tensor properties based upon the magnetic point group is the program MTENSOR.[75] It is available from the Bilbao Crystallographic Server at https://cryst.ehu.es, under the programs for “Magnetic Symmetry and Applications.” The server webpage directs you to input the appropriate magnetic point group from a list of options, and then the desired physical property tensor. For toroidal moments, this is the “Polar Toroidal Moment Ti.” The server then runs the program to calculate the tensor, and displays the computed result on a new webpage. In the sidebar the authors of MTENSOR explain they define an “Axial Toroidal Moment Ai” as an axial tensor invariant under time inversion symmetry, and a “Polar Toroidal Moment Ti” as variant under time inversion symmetry. Neither are an optimal label, since the symmetry of toroidal moments are best described by “axio-polar” vectors, which will be discussed later in §3.3. However, “axio- 34 polar” vectors are in fact variant under time-inversion (1′). Therefore the method used by the authors of MTENSOR computes a logically consistent result for the “Polar Toroidal Moments.” As will be explained in §3.3 & 3.6, space-time inversion (1̄′) only has meaning in geometric space because inversion symmetry of any kind (i.e. 1̄ or 1̄′) is defined about a point of inversion (i.e. inversion center). It is therefore difficult to fully appreciate how space-time inversion (1̄′) is distinct from space (1̄) and time (1′) inversion in terms of point symmetry alone since they can appear to produce identical results. In geometric space it is clear they in fact do not. Around the same time as Schmid, Rivera wrote his own review article [76] on magnetoelectric materials. In it, Schmid requested Rivera include an updated form of the table of tensors for the linear magnetoelectric effect in the appendix of his 2008 article.[60] The updated table includes all of the different settings for the monoclinic point groups. Rivera discusses Schmid’s 2008 article extensively. He cautions that (Ei ×Hi) cannot be approximated as a vector S in order to neatly claim the density of free energy can be written according to the vector components (E,H,S,σ). This is because when (E ×H) is substituted into the free energy expressions for S, the results are the same as simply considering the free energy in terms of (E,H,σ). This is because S is not a new independent variable. Schmid justified the redundancy by stating TiSi is a special case of the linear magnetoelectric effect.[60] In other words, Schmid treats the fact that Ti(Ei ×Hi) = TiSi as evidence that ferrotoroidic order can coexist with other types of ordering, and therefore other physical properties. They are different mathematical expressions for the same physical phenomena. Schmid’s 2008 paper also discusses the connection between Ascher’s 1966 Table [13] and the limiting groups for types of ferroic ordering, citing Dubovik’s 1990 article.[19] However, it is in subsequent years that the theoreticians Saxena and Lookman from Los Alamos National 35 Laboratory, and Castán and Planes from the University of Barcelona, deeply explore the limiting groups and their implications for ferrotoroidic order.[77, 78, 79, 80] Saxena was interested in the underlying symmetry of order parameters.[77, 78] All ferroic ordering can be described according to an order parameter. This is a vector with a certain symmetry which describes the net order in the material. For example, electric polarization can be described with a polar vector; magnetization can be described with an axial vector. The symmetry of a polar vector can be considered like that of a stationary cone, the symmetry of an axial vector like that of a rotating cylinder, and the symmetry of uniaxial stress as a stationary cylinder. Saxena proposed that the symmetry of an “axio-polar” vector like in ferrotoroidic materials would be that of a rotating cone. By considering the symmetries of these geometric entities, Sexena deduced the limiting groups for each of them,