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2022 | OriginalPaper | Chapter

2. Scientific Background

Author : Dr. Jannis Lehmann

Published in: Toroidal Order in Magnetic Metamaterials

Publisher: Springer International Publishing

Abstract

The following Chapter introduces three main topics of my work: ferroic order, nanomagnetism and metamaterials. First, the concept of ferroic order is presented. A symmetry-based classification is given together with a brief discussion of phase transitions, the emergence of a ferroic order parameter, spontaneous domain formation and the manipulation of an order parameter with a conjugate field. This part closes by unravelling ferrotoroidicity. Second, magnetic properties of sub-micrometre-sized objects made from a ferromagnetic material are discussed. Here, the formation of different kinds of spin structures is explained that serve as building blocks of nanomagnetic arrays. The suppression or—more important here—the support of long-range order in extended magnetostatic-coupled arrays is explained. The third part of this Chapter introduces metamaterials, a class of matter that is assembled on length scales comparable with the wavelength of radiation that interacts with it and that provides design-determined novel material properties and functionalities.

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next chapter Experimental and Computational Methods

Since the spontaneous strain is described by a second-rank tensor instead of a vector, like for the other three primary ferroic states, it appears detached from them [17]. To circumvent this issue ferro-rotational order has been proposed recently as a replacement with a suitable symmetry and a vectorial order parameter [32‐34].

Generally, these exponents are expected to be of universal character and, as such, only dependent on global details of the investigated system like its physical dimension, or the range of the interaction.

For a more detailed study, energy contributions from the magnetic anisotropy (see Sect. 2.2), magnetostriction, strain, crystal defects and grains or annealing procedures have to be taken into account [53‐56].

Note that the so-called anapole moment, first mentioned already in 1958 [79] is a closely related term for the non-radiative mode of an electromagnetic excitation with the same symmetry.

Confusion arises, in this context, due to an assumed identity of a toroidally ordered single-phase ferroic state and a multiferroic state that exhibits a magnetisation perpendicularly aligned to an electric polarisation [156‐158]. The order in these magnetoelectric multiferroics, despite its potentially equivalent macroscopic symmetry, is not in agreement with the classification of the toroidal order as presented here [77], since it is neither magnetically compensated nor requires the same conjugate field for switching [42, 43].

For more realistic shapes such as a ferromagnetic prism, corresponding demagnetisation values have been calculated as well [171].

Not discussed here is the rich variety of magnetic configurations that form between these two variants, which have been observed in discs or polyhedral-shaped magnetic bodies, namely the C state (buckle state), the S state, the triangle state, the flower state, the leaf state or the diamond state [162, 179, 180].

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Title: Scientific Background
Author: Dr. Jannis Lehmann
Publisher: Springer International Publishing
Book: Toroidal Order in Magnetic Metamaterials
Print ISBN: 978-3-030-85494-2

Electronic ISBN: 978-3-030-85495-9

Copyright Year: 2022
DOI: https://doi.org/10.1007/978-3-030-85495-9_2

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