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Topology and Magnetic Domain Walls

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Topology in Magnetism

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

Abstract

This chapter presents the various connections between topology and magnetic domain walls. To begin with, we expose what topology tells us about magnetic domain walls, answering questions like: may domain walls be topological defects?, are there topologically different classes of domain walls? We then turn to dynamics, explaining the profound link between topology and the dynamics of magnetic textures, domain walls here. Experimental aspects are reviewed in the next sections, constructed according to material and sample types, with a special role played by bubble garnet films where a number of fundamental concepts were introduced. The authors try to provide a unified view of the vast literature on the subject, that spreads over four decades and different research thematics.

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Notes

  1. 1.

    An example of a topologically non-trivial space is the Möbius strip [1].

  2. 2.

    Other values of p and \(p'\) are given in mathematical textbooks, e.g. [7] Sect. 2.4.1.

  3. 3.

    The mathematical objects adapted to this case are called relative homotopy groups. See e.g. [7] for a reference textbook. As the spaces that are considered are easily visualized, we can avoid the mathematical machinery here.

  4. 4.

    If two paths existed that use the two sides of \(S^1\), they would form a loop that encloses \(S^1\) once, so that a topological defect of \(n=2\) would exist, a vortex point in \(d=2\) or a vortex line in \(d=3\).

  5. 5.

    Whereas for domain walls a rich taxonomy has developed, with Bloch, Néel, asymmetric Bloch, asymmetric Néel, cross-tie walls, followed recently by transverse, vortex, asymmetric transverse, Bloch point walls etc., such is not the case of lines, that are nearly always called Bloch lines. Feldtkeller [8] proposed a denomination logics for lines, based on which type of wall they display in their core, but this has not been adopted by the whole community.

  6. 6.

    This can be figured out simply by remarking that, at the VBL core one has \(\partial \mathbf {m} / \partial x\) along \(+x\) and \(\partial \mathbf {m} / \partial y\) along \(-z\), so that \(\partial \mathbf {m} / \partial x \times \partial \mathbf {m} / \partial y\) is along \(+y\), parallel resp. antiparallel to the local magnetization for cases E resp. F.

  7. 7.

    The functional derivative differs from the partial derivative as soon as \(\mathscr {E}\) contains terms involving gradients of \(\mathbf {m}\). In the energy differential, the terms involving the gradients of the variation of \(\mathbf {m}\) have to be integrated by parts to be transformed into terms involving only the variations of \(\mathbf {m}\). This integration also produces surface terms, that contribute to the micromagnetic boundary conditions. See [2] for details.

  8. 8.

    Note that, in general, velocity is not linear with field as the moving domain wall structure differs from the rest structure, so that the Thiele domain wall width is not a constant.

  9. 9.

    This is not exactly so, though, when \(\Delta _\mathrm {T}\) varies much with field.

  10. 10.

    It may prove more pedagogical to input all external actions in \(\mathbf {T}\) rather than in \(\mathbf {H}_\mathrm {eff}\); then \(\mathbf {F}\) is zero as the self-energy of the texture \(\mathbf {m}_0\) should be independent of position (assuming a uniform medium, infinitely extended). See Sect. 2.4 for the special case of an infinite extension in one dimension only.

  11. 11.

    This domain wall width is implicitly defined in [30] (Sect. 2.4) when applying the Thiele equation to the dynamics of a simple domain wall. The explicit definition of the Thiele domain wall width occured later [32].

  12. 12.

    We stick to the historical formulation that refers to the domain wall energy density \(\sigma \), but in fact it is the total energy \(E=\int {\mathscr {E} d^3r}=\int {\mathscr {E} dqdS}=\int {\sigma dS}\). Indeed, expressing the effect of an easy axis field by \(\delta \sigma / \delta q=2 \mu _0 M_\mathrm {s} H\), which means a position-dependent domain wall energy, is physically artificial.

  13. 13.

    Soft magnetic films often display a small in-plane uniaxial anisotropy.

  14. 14.

    By convention for bubbles, the bias field is up so that the bubble core magnetization is down. Thus topological and winding numbers are in one-to-one correspondence (in fact, they are simply opposite, as the core is down-magnetized).

  15. 15.

    The two VBLs indeed have same core magnetization, namely that of the HBL, same magnetization gradient normal to the domain wall and opposite magnetization gradients along the domain wall, hence opposite vertical gyrovector components.

  16. 16.

    It is thus meaningless to try to call these walls Bloch or Néel, as the latter have no volume charge resp. a dipolar volume charge.

  17. 17.

    This is not exact, actually, as magnetization tilts slightly out-of-plane close to the half antivortex and half hedgehog vortex. The surface thus covered on the sphere is, however, not topologically protected.

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Acknowledgements

The authors would like to acknowledge Maurice Kléman, formerly at LPS, for being an inspiration and, by his pionneering works, a source for the content of this chapter. They thank Jean-Yves Chauleau for providing MFM images of domain walls in Ni-Fe nanostrips, borrowed from his Ph.D. thesis [85]. The critical reading of the manuscript by O. Fruchart is gratefully acknowledged.

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  1. Laboratoire de Physique des Solides, Université Paris-Sud, CNRS UMR 8502, Bâtiment 510, Centre universitaire, 91405, Orsay, France

    André Thiaville & Jacques Miltat

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  1. André Thiaville
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Correspondence to André Thiaville .

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  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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Thiaville, A., Miltat, J. (2018). Topology and Magnetic Domain Walls. 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_2

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