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The Toroidal Moment in Condensed-Matter Physics and Its Relation to the Magnetoelectric Effect Spaldin, Nicola A.; Fiebig, Manfred; Mostovoy, Maxim

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The Toroidal Moment in Condensed-Matter Physics and Its Relation to the Magnetoelectric Effect Spaldin, Nicola A.; Fiebig, Manfred; Mostovoy, Maxim University of Groningen The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect Spaldin, Nicola A.; Fiebig, Manfred; Mostovoy, Maxim Published in: Journal of Physics-Condensed Matter DOI: 10.1088/0953-8984/20/43/434203 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2008 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Spaldin, N. A., Fiebig, M., & Mostovoy, M. (2008). The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect. Journal of Physics-Condensed Matter, 20(43), [434203]. https://doi.org/10.1088/0953-8984/20/43/434203 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 20 (2008) 434203 (15pp) doi:10.1088/0953-8984/20/43/434203 The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect* Nicola A Spaldin1, Manfred Fiebig2 and Maxim Mostovoy3 1 Materials Department, University of California, Santa Barbara, CA 93106-5050, USA 2 HISKP, Universit¨at Bonn, Nussallee 14-16, 53115 Bonn, Germany 3 Zernike Institute for Advanced Materials, University of Groningen, 9747 AG Groningen, The Netherlands E-mail: [email protected], fi[email protected] and [email protected] Received 17 April 2008, in final form 4 June 2008 Published 9 October 2008 Online at stacks.iop.org/JPhysCM/20/434203 Abstract The concept of toroidal moments in condensed-matter physics and their long-range ordering in a so-called ferrotoroidic state is reviewed. We show that ferrotoroidicity as a form of primary ferroic order can be understood both from microscopic (multipole expansion) and macroscopic (symmetry-based expansion of the free energy) points of view. The definition of the local toroidal moment and its transformation properties under the space-inversion and time reversal operations are highlighted and the extension to periodic bulk systems is discussed. Particular attention is paid to the relationship between the toroidal moment and the antisymmetric magnetoelectric effect and to limitations of the magnetoelectric response in ferrotoroidic systems and ferroic materials in general. Experimental access to the ferrotoroidic state by magnetoelectric susceptibility measurements, x-ray diffraction and optical techniques or direct measurement of the bulk toroidization is discussed. We outline the pertinent questions that should be clarified for continued advancement of the field and mention some potential applications of ferrotoroidic materials. 1. Introduction The purpose of this article is to summarize our current understanding of such an elusive type of long-range order: The search for novel states of matter and new types of order ferrotoroidicity. By analogy with the spontaneous alignment is one of the most exciting and fundamental aspects of modern of magnetic dipole moments in ferromagnets, ferrotoroidics condensed-matter physics. Anyon gases in fractional quantum are defined to have a spontaneous alignment of toroidal Hall systems, d-wave electron pairing in high-temperature moments [3–5], of which a classical example is a solenoid superconductors and electron nematic phases in transitional that is bent into a torus [6]. As shown in figure 1 the current metal oxides have recently enriched our understanding of induces a circular magnetic field inside the solenoid, giving rise the complex ways in which strongly correlated matter can to a toroidal moment perpendicular to the magnetic field. The organize itself [1, 2]. Some ordered states escape detection quantum-mechanical equivalent can be generated by certain and remain hidden for a long time, as their nature can only spin orderings, such as head-to-tail arrangements of spins, or be revealed by a purposely designed experimental probe. For by persistent orbital currents. A ferrotoroidic state can then be example, while ferromagnetism has been known to mankind visualized as an array of spin vortices, each of the size of one since time immemorial, antiferromagnetism was discovered unit cell, which is a magnetic analogue of the electronic flux only relatively recently despite the fact that antiferromagnets states discussed in the context of quantum Hall systems and are far more common than ferromagnets in nature. high-temperature superconductors [6, 7]. In this paper we describe the existing state of knowledge * All authors contributed equally to this work. of toroidal moments and toroidal ordering in the solid state. 0953-8984/08/434203+15$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK J. Phys.: Condens. Matter 20 (2008) 434203 NASpaldinet al vice versa. In section 4 the upper bound limiting the magnitude of the magnetoelectric coupling in toroidically ordered and disordered systems is discussed, and the limit imposed by thermodynamic stability is identified. We introduce simple microscopic mechanisms for magnetoelectric behaviour in which the magnitude of the magnetoelectric coefficient is determined by the geometric aspects of the superexchange interaction. In section 5 experimental techniques probing the peculiar symmetry of a ferrotoroidic state are reviewed. The magnetoelectric effect as well as gyrotropic and dichroic x-ray and optical effects are discussed and we show that nonlinear optical techniques are quite powerful in observing ferrotoroidic order and domains. In section 6 various issues related to the future practical merit of ferrotoroidics are highlighted. Ways for measuring and switching a ferrotoroidic state and the potential of ferrotoroidicity for magnetoelectric and data storage devices are discussed. We also speculate on the possible existence of antiferrotoroidic order. A concluding Figure 1. Toroidal moment, magnetoelectric effect and ferrotoroidic section follows in which some pertinent questions to be order. (a) Ring-shaped solenoid with electric-current loops as clarified for the continued advancement of the field are classical example for a toroidal moment. (b) A head-to-tail discussed. Note that Gaussian units are used throughout this arrangement of atomic magnetic moments is a quantum-mechanical article. example of a toroidal moment. The two vortices displayed correspond to opposite toroidal moments. (c) Sketch of a ferrotoroidic domain structure with curls in boxes representing unit 2. Macroscopic and microscopic derivations cells with magnetic vortices as in (c). 2.1. Macroscopic motivation: symmetry expansion The toroidal moment has long been discussed in the context 2.1.1. Ferroic order. A hallmark of any ferroic state is of particle physics (see appendix A) and interest within the formation of domains, which are energetically degenerate the solid state community has recently increased drastically regions of identical structure, that have different orientations of because of its relationship to the magnetoelectric effect and the the ferroic order parameter [8]. An appropriate vector or tensor consequent implications for the active field of multiferroics. field can be used to induce an energy difference between states The toroidal moment constitutes a fundamental building block with different orientations of the order parameter. Such a field both in an expansion of the free energy into contributions can therefore be used to switch domain states, which adopt the orientation with the lowest energy in the presence of the field. of different symmetry as well as in the multipole expansion For example, in ferroelectric materials there is a contribution of the electromagnetic vector potential. Nevertheless, its to the free energy of form −P · E, which causes domains description is not contained in standard textbooks which has sp with spontaneous polarization P parallel to an electric field led to ambiguities and confusion in the field. We hope to rectify sp E to have lowest energy. An electric field of sufficient strength that here. then transforms a sample into a uniformly polarized domain The remainder of this paper is organized as follows. In state with P E since this minimizes the free energy. More section 2 we introduce the concept of ferrotoroidic long-range sp generally, any ferroic state is characterized by a macroscopic order using general macroscopic symmetry considerations property Oˆ corresponding to the order parameter (here P ) based on transformation properties of the order parameter sp that can be switched by an external field Aˆ (here E) because of under space-inversion
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