Boundary vortices in thin magnetic films

Kurzke, Matthias

doi:10.1007/s00526-005-0331-z

Cited by 51 publications

(99 citation statements)

References 10 publications

(15 reference statements)

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“…When the nonlinearity f is given by f (u) = sin(cu) for some constant c, problem (1.1) in a half-plane is called the Peierls-Nabarro problem, and it appears as a model of dislocations in crystals (see [21,36]). The Peierls-Nabarro problem is also central to the analysis of boundary vortices in the paper [28], which studies a model for soft thin films in micromagnetism recently derived by Kohn and Slastikov [26] (see also [27]). …”

Section: Introductionmentioning

confidence: 99%

Layer solutions in a half‐space for boundary reactions

Cabré

Solà-Morales

2005

Comm Pure Appl Math

248

362

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Section: Introductionmentioning

confidence: 99%

Layer solutions in a half‐space for boundary reactions

Cabré

Solà-Morales

2005

Comm Pure Appl Math

248

362

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“…Apart from a few special cases (e.g. an ellipsoidal particle), finding configurationsm(r) of lowest energy is a difficult computational problem.Analytical treatment is nonetheless possible in a thinfilm limit [10,11] defined for a strip of width w and thickness t aswhere λ = A/µ 0 M 2 is a magnetic length scale (5 nm in permalloy). Taking this limit yields three simplifications: (a) magnetization lies in the plane of the film, m = (cos θ, sin θ, 0); (b) it depends on the in-plane coordinates x and y, but not on z; (c) the magnetic energy becomes a local functional of magnetization [10,11]:…”

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confidence: 99%

“…The above solutions can be easily adopted to the cases when the ratio Λ/w is small but finite. The singular cores of the halfvortices reside outside the film, the distance Λ away from the edge [10].…”

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Fractional Vortices and Composite Domain Walls in Flat Nanomagnets

2005

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We provide a simple explanation of complex magnetic patterns observed in ferromagnetic nanostructures. To this end we identify elementary topological defects in the field of magnetization: ordinary vortices in the bulk and vortices with half-integer winding numbers confined to the edge. Domain walls found in experiments and numerical simulations in strips and rings are composite objects containing two or more of the elementary defects.Topological defects [1,2] greatly influence the properties of materials by catalyzing or inhibiting the switching between differently ordered states. In ferromagnetic nanoparticles of various shapes (e.g. strips [3] and rings [4]), the switching process involves creation, propagation, and annihilation of domain walls with complex internal structure [5]. Here we show that these domain walls are composite objects made of two or more elementary defects: vortices with integer winding numbers (n = ±1) and edge defects with fractional winding numbers (n = ±1/2). The simplest domain walls are composed of two edge defects with opposite winding numbers. Creation and annihilation of the defects is constrained by conservation of a topological charge. This framework provides a basic understanding of the complex switching processes observed in ferromagnetic nanoparticles.In ferromagnets the competition between exchange and magnetic dipolar energies creates nonuniform patterns of magnetization in the ground state. Whereas the exchange energy favors a state with uniform magnetization, magnetic dipolar interactions align the vector of magnetization with the surface. In a large magnet a compromise is reached by the formation of uniformly magnetized domains separated by domain walls. In a nanomagnet magnetization varies continuously forming intricate yet highly reproducible textures, which include domain walls and vortices [5,6,7]. Numerical simulations [8, 9] reveal a rich internal structure and complex dynamics of these objects. For example, collisions of two domain walls can have drastically different outcomes: complete annihilation or formation of other stable textures [7]. These puzzling phenomena call for a theoretical explanation.An elementary picture of topological defects in nanomagnets with a planar geometry is presented in this Letter. It is suggested that the elementary defects are vortices with integer winding numbers and edge defects with half-integer winding numbers. All of the intricate textures, including the domain walls, are composite objects made of two or more elementary defects.For simplicity, we consider a ferromagnet without intrinsic anisotropy, which is a good approximation for permalloy. The magnetic energy consists of two parts: the exchange contribution A |∇m| d 3 r, wherê m = M/|M| is the unit vector pointing in the direction of magnetization M, and the magnetostatic energy (µ 0 /2) |H| 2 d 3 r. The magnetic field H is related to the magnetization through Maxwell's equations, ∇ × H = 0 and ∇ · (H + M) = 0. Apart from a few special cases (e.g. an ellipsoidal particle), f...

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“…The research presented in this article was carried out as part of my thesis [6] under the supervision of Prof. Stefan Müller, and I am thankful for his many helpful suggestions. During this research, I was supported by the DFG, first through the Graduiertenkolleg at the University of Leipzig, then through Priority Program 1095, and I want to express my gratitude for the support.…”

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confidence: 99%

A nonlocal singular perturbation problem with periodic well potential

Kurzke

2005

ESAIM: COCV

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Abstract. For a one-dimensional nonlocal nonconvex singular perturbation problem with a noncoercive periodic well potential, we prove a Γ-convergence theorem and show compactness up to translation in all L p and the optimal Orlicz space for sequences of bounded energy. This generalizes work of Alberti, Bouchitté and Seppecher (1994) for the coercive two-well case. The theorem has applications to a certain thin-film limit of the micromagnetic energy.Mathematics Subject Classification. 49J45.

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Boundary vortices in thin magnetic films

Cited by 51 publications

References 10 publications

Layer solutions in a half‐space for boundary reactions

Layer solutions in a half‐space for boundary reactions

Fractional Vortices and Composite Domain Walls in Flat Nanomagnets

A nonlocal singular perturbation problem with periodic well potential

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