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Solitons in Real Space: Domain Walls, Vortices, Hedgehogs, and Skyrmions

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Topology in Magnetism

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 192))

Abstract

Recent years have seen tremendous progress in the understanding of topological phenomena in magnetism, in particular at the nanoscale. In this overview, we consider smooth topological textures such as smooth domain walls, meron or vortices, and most importantly skyrmions. These structures derive their topological stability from the fact that they cannot be undone without violating the continuity of the magnetization field, similar to a knot in a rope. Owing to their topological stability, domain walls and skyrmions are prominent candidates in racetrack-type memories introduced by Parkin and co-workers. These smooth textures should be contrasted with singular topological point defects where the magnetization field is forced to vanish in a submanifold. Such point defects include Ising domain walls, vortices of easy-plane spins, and 3D Bloch points, ‘hedgehogs’, or ‘monopoles’. As domain walls, vortices, and skyrmions including their dynamical versions will be discussed in detail in later chapters by Thiaville and Miltat, Behncke and Meier, Chen, Bauer et al., and Åkerman, we give analytical arguments how domain walls emerge in quasi 1D nanowires, how magnetization reverses via nucleation, and why skyrmions exist in thin films. A variational ansatz for skyrmions that is derived from an exact \(2\pi \) domain wall profile provides an excellent approximation to numerical and experimental observations in films that include Dzyaloshinskii-Moriya interaction (DMI) and dipolar interactions. In systems of vanishing DMI, the two helical states of a skyrmion are degenerate, and switching between the two helicities occurs in a topologically allowed fashion. This mechanism is closely related to domain wall nucleation in nanowires. Finally we show that dynamical skyrmions may be regarded as 2D siblings of domain wall breathers, and can be described by the same variational ansatz inspired from \(2\pi \) domain walls as static skyrmions in thin films.

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Notes

  1. 1.

    Here we refer to micromagnetics as a formalism based on a continuous magnetization field defined in continuous space. In recent years it has become common to use the term micromagnetics to exclusively describe numerical approaches. However, this is too narrow a terminology and potentially misleading as the formalism as originally set out by Brown [23] was an analytical continuum theory and did not refer to numerical methods. In contrast, numerical methods involve a discrete mesh whose scale is usually considerably larger than the physical lattice.

  2. 2.

    If every map from \(S^1\) into a space X is null-homotopic, then X is called ‘simply connected’ (e.g., \(\mathbb {R}^2\)).

  3. 3.

    Specifically, we have \({\varOmega }= * r dr \equiv {1 \over n!} \varepsilon _{i_0 i_1 \dots i_n} x^{i_0} dx^{i_1} \wedge \dots \wedge dx^{i_n}\). This differential n-form satisfies \(d {\varOmega }= \omega \) where \(\omega \) is the standard volume element on \({\mathbb R}^{n+1}\). Correspondingly, \( \int _N {\varOmega }= \int _{B^{n+1}} d {\varOmega }= (n+1) \int _{B^{n+1}} \omega = (n+1) V_{n+1}\), where in the first step we made use of Stokes’ theorem and \(B^{n+1}\) denotes the unit ball in \({\mathbb R}^{n+1}\) with \(\partial B^{n+1} = S^n\) and \(V_{n+1}\) its volume.

  4. 4.

    For a nanowire with effective easy-axis along the wire (x-axis), as is relevant for data storage in perpendicular hard disk media, the field is applied along wire and the effective anisotropies are given by \(-K_{e,eff} m_x^2 +K_{h,eff} m_z^2\) with \(K_{e,eff} = K_{e,cryst} + (\mu _0/2) M_0^2 (N_x-N_y)\) and correspondingly for \(K_{h,eff} \). The nucleus solution is then obtained via the replacement \(\theta \rightarrow \phi \). For a discussion of effective anisotropies for different sample shapes, cf. [19].

  5. 5.

    It is amusing to note that Bloch in his paper actually acknowledges Heisenberg for the solution of the corresponding differential equation, so perhaps it should be more appropriately named the ‘Heisenberg-Bloch’ wall. Also, note that in the sequel we usually do not distinguish between Bloch and Néel walls for the quasi 1D situation.

  6. 6.

    With \(\partial _x \theta _{QC} = C \, \mathrm{sech}\, x\) and \(\cos \theta _{QC} = - \tanh (Qx)\) we immediately verify that \((1/\pi )\int _{-\infty }^\infty \! dx \; \partial _x \! \theta _{QC} = C\) and \([ \cos \theta _{QC}(-\infty ) - \cos \theta _{QC}(\infty )]/2 =Q\).

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Acknowledgements

I gratefully acknowledge numerous helpful discussions with J. Åkerman, P. Böni, R.V. Hügli, B. Roessli, and Y. Zhou. This research has been supported by Science Foundation Ireland under 11/PI/1048.

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  1. School of Physics, University College Dublin, Dublin, Ireland

    Hans-Benjamin Braun

  2. School of Theoretical Physics, Dublin Institute of Advanced Studies, Dublin, 4, Ireland

    Hans-Benjamin Braun

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Editors and Affiliations

  1. University of New Hampshire, Durham, NH, USA

    Jiadong Zang

  2. Unité Mixte de Physique CNRS/Thales, Palaiseau, France

    Vincent Cros

  3. Materials Science Division, Argonne National Laboratory, Lemont, IL, USA

    Axel Hoffmann

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Braun, HB. (2018). Solitons in Real Space: Domain Walls, Vortices, Hedgehogs, and Skyrmions. In: Zang, J., Cros, V., Hoffmann, A. (eds) Topology in Magnetism. Springer Series in Solid-State Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-97334-0_1

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