# Domain wall

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A domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.[1]

## Magnetism

Domain wall (B) with gradual re-orientation of the magnetic moments between two 180-degree domains (A) and (C)

In magnetism, a domain wall is an interface separating magnetic domains. It is a transition between different magnetic moments and usually undergoes an angular displacement of 90° or 180°. Domain wall is a gradual reorientation of individual moments across a finite distance. The domain wall thickness depends on the anisotropy of the material, but on average spans across around 100–150 atoms.

The energy of a domain wall is simply the difference between the magnetic moments before and after the domain wall was created. This value is usually expressed as energy per unit wall area.

The width of the domain wall varies due to the two opposing energies that create it: the Magnetocrystalline anisotropy energy and the exchange energy (${\displaystyle J_{\mathrm {ex} }}$), both of which tend to be as low as possible so as to be in a more favorable energetic state. The anisotropy energy is lowest when the individual magnetic moments are aligned with the crystal lattice axes thus reducing the width of the domain wall. Whereas the exchange energy is reduced when the magnetic moments are aligned parallel to each other and thus makes the wall thicker, due to the repulsion between them. (Where anti-parallel alignment would bring them closer - working to reduce the wall thickness.) In the end an equilibrium is reached between the two and the domain wall's width is set as such.

An ideal domain wall would be fully independent of position, however, they are not ideal and so get stuck on inclusion sites within the medium, also known as Crystallographic defects. These include missing or different (foreign) atoms, oxides, insulators and even stresses within the crystal. This prevents the formation of domain walls and also inhibits their propagation through the medium. Thus a greater applied magnetic field is required to overcome these sites.

Note that the magnetic domain walls are exact solutions to classical nonlinear equations of magnets (Landau-Lifshitz equation, nonlinear Schrodinger equation and so on).

### Symmetry of multiferroic domain walls

Since domain walls can be considered as thin layers, their symmetry is described by one of the 528 magnetic layer groups.[2][3] To determine the layer's physical properties continuum approximation is used which leads to point-like layer groups.[4] If continuous translation operation is considering as identity then these groups transform to magnetic point groups. It was shown[5] that there are 125 of such groups. It was found that if magnetic point group is pyroelectric and/or pyromagnetic then domain wall carries polarization and/or magnetization respectively.[6] These criteria were derived from the conditions of the appearing of the uniform polarization[7][8] and/or magnetization.[9][10] After their application to any inhomogeneous region they predict existing of even parts in functions of distribution of order parameters. Identification of the remained odd parts of these functions, were formulated[11] based on symmetry transformations which interrelate domains. The symmetry classification of the magnetic domain walls contains 64 magnetic point groups.[12]

Symmetry based predictions of the structure of the multiferroic domain walls have been proven using phenomenology coupling via magnetization[13] and/or polarization[14] spatial derivatives (flexomagnetoelectric[15] effect).

### Depinning of a domain wall

Schematic representation of domain wall unpinning

Non-magnetic inclusions in the volume of a ferromagnetic material, or dislocations in crystallographic structure, can cause "pinning" of the domain walls (see animation). Such pinning sites cause the domain wall to seat in a local energy minimum and external field is required to "unpin" the domain wall from its pinned position. The act of unpinning will cause sudden movement of the domain wall and sudden change of the volume of both neighbouring domains; this causes Barkhausen noise.

## Optics

Recently, a phase-locked dark-dark vector soliton was only observed in fiber lasers of positive dispersion, a phase-locked dark-bright vector soliton was obtained in fiber lasers of either positive or negative dispersion. Numerical simulations confirmed the experimental observations, and further showed that the observed vector solitons are the two types of phase-locked polarization domain-wall solitons theoretically predicted.[16] Another novel type domain wall soliton is the vector dark domain wall consisting of stable localized structures separating the two orthogonal linear polarization eigenstates of the laser emission and dark structure is visible only when the total laser emission is measured.[17]

## String theory

In string theory, a domain wall is a theoretical 2-dimensional singularity. A domain wall is meant to represent an object of codimension one embedded into space (a defect in space localized in one spatial dimension). For example, D8-branes are domain walls in type II string theory. In M-theory, the existence of Horava-Witten domain walls, "ends of the world" that carry an E8 gauge theory, is important for various relations between superstring theory and M-theory.

If domain walls exist, it seems plausible that they will violently emit gravitational waves if two such walls would collide. As the Laser Interferometer Gravitational-Wave Observatory and future observatories of its kind will search for direct evidence of gravitational waves, this phenomenon would be included as well in such searches.

## References

1. S. Weinberg, The Quantum Theory of Fields, Vol. 2. Chap 23, Cambridge University Press (1995).
2. N. N. Neronova & N. V. & Belov, Sov. Phys. - Cryst. 6, 672-678 (1961).
3. D. B. Litvin, Acta Cryst., A55, 963-964 (1999).
4. V. Kopsky, J. Math. Phys. 34, 1548-1576 (1993).
5. J. Privratska, B. Shaparenko, V. Janovec, D. B. Litvin, Ferroelectrics 269 (2002) 39-44.
6. J. Privratska, V. Janovec, Ferroelectrics 222 (1999) 23 - 32.
7. W. B. Walker and R. J. Gooding, Phys. Rev. B 32 (1985) 7408.
8. P. Saint-Grkgoire and V. Janovec, in Lecture Notes on Physics 353, Nonlinear Coherent Structures, in: M. Barthes and J. LCon (Eds.), Spinger-Verlag, Berlin, 1989, p. 117.
9. L. Shuvalov, Sov. Phys. Crystallogr. 4 (1959) 399
10. L. Shuvalov, Modern Crystallography IV : Physical Properties of Crystals, Springer, Berlin, 1988
11. V.G. Bar'yakhtar, V.A. L'vov, D.A. Yablonskiy, JETP Lett. 37, 12, 673-675 (1983)
12. B. M. Tanygin, O. V. Tychko, Physica B: Condensed Matter 404, 21, 4018-4022 (2009)
13. B.M. Tanygin, Journal of Magnetism and Magnetic Materials, Volume 323, Issue 14 (2011) Pages 1899-1902
14. B.M. Tanygin, IOP Conf. Ser.: Mater. Sci. Eng. 15 (2010) 012073.
15. A.P. Pyatakov, A.K. Zvezdin, Eur. Phys. J. B 71 (2009) 419.
16. Han Zhang, D. Y. Tang, L. M. Zhao, X. Wu "Observation of polarization domain wall solitons in weakly birefringent cavity fiber lasers" arXiv:0907.5496v1
17. H. Zhang, D. Y. Tang, L. M. Zhao and R. J. Knize, “Vector dark domain wall solitons in a fiber ring laser’’ OPTICS EXPRESS 18, 4428 (2010). http://www.sciencenet.cn/upload/blog/file/2010/2/201021911159386775.PDF