Curvature-stabilized skyrmions with angular momentum

Melcher, Christof; Sakellaris, Zisis N.

doi:10.1007/s11005-019-01188-6

Cited by 11 publications

(4 citation statements)

References 25 publications

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“…If κ −4, the normal vector fields are the only global minimizers of F κ . Also, in [170], it is shown that for κ 0, skyrmionic solutions topologically distinct from the ground state emerge as excited states.…”

Section: Proposition 1 ([76]mentioning

confidence: 97%

“…Their stability, reduced size, and the small current densities sufficient to control them, make magnetic skyrmions extremely attractive for applications in modern spintronics [85]. An in-depth understanding of their rich structure (e.g., chirality, topological charge, stability) leads to challenging problems in a subject area where geometry and continuum mechanics meet topology and analysis, and this has raised interest in magnetic skyrmions also from a mathematical perspective [11,21,57,72,74,76,82,83,115,117,132,156,157,166,169,170,179].…”

Section: Micromagnetics Of Curved Thin Films (By Giovanni DI Frattamentioning

confidence: 99%

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The mathematics of thin structures

Babadjian¹,

Fratta

Fonseca

et al. 2022

Quart. Appl. Math.

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This article offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that domain vanishes. After a short review in Section 1 of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case, giving rise to various well-known models in plate theory (membrane, bending, Von Karmann, etc…), the other sections address various extensions of those initial results. Section 2 adds brittleness and delamination and investigates the brittle membrane regime. Sections 4 and 5 focus on micromagnetics, rather than elasticity, this once again in the membrane regime and discuss magnetic skyrmions and domain walls, respectively. Finally, Section 3 revisits the classical setting in a non-Euclidean setting induced by the presence of a pre-strain in the model.

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Section: Proposition 1 ([76]mentioning

confidence: 97%

Section: Micromagnetics Of Curved Thin Films (By Giovanni DI Frattamentioning

confidence: 99%

The mathematics of thin structures

Babadjian¹,

Fratta

Fonseca

et al. 2022

Quart. Appl. Math.

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“…(2) When α −4, the normal vector fields ±n are the only global minimizers of E. Also, in [28] it is shown that for α −1, skyrmionic solutions topologically distinct from the ground state emerge as excited states.…”

Section: Introductionmentioning

confidence: 99%

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

Fratta¹,

Fiorenza²,

Slastikov³

2021

Preprint

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The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of S 2 -valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of nematic liquid crystals and micromagnetics. We show that minimal configurations are z-invariant and that energy minimizers in the class of weakly axially symmetric competitors are, in fact, axially symmetric. Our main result is a family of sharp Poincaré-type inequality on the circular cylinder, which allows establishing a nearly complete picture of the energy landscape. The presence of symmetry-breaking phenomena is highlighted and discussed. Finally, we provide a complete characterization of in-plane minimizers, which typically appear in numerical simulations for reasons we explain.

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“…Specifically, for the energy of the form of (1.1) consisting of an exchange energy with an additional Skyrme-type higher order term, E ex (m) = A ex ´R2 |∇m| 2 dx + A S ´R2 |∂ 1 m × ∂ 2 m| 2 dx, and a special form of an anisotropy/Zeeman term (see the remark in [27, p. 2]), E a (m) + E Z (m) = K ´R2 |m + e 3 | 4 dx, existence of a skyrmion solution as a minimizer m of E with N (m) = ±1 was proved in [54,49]. Also curvature of the underlying space has been explored as another possible mechanism for ensuring existence of skyrmions [61].…”

Section: Introductionmentioning

confidence: 99%

A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm1$ harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$

Bernand-Mantel,

Muratov,

Simon

2019

Preprint

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We characterize skyrmions in ultrathin ferromagnetic films as local minimizers of a reduced micromagnetic energy appropriate for quasi two-dimensional materials with perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. The minimization is carried out in a suitable class of two-dimensional magnetization configurations that prevents the energy from going to negative infinity, while not imposing any restrictions on the spatial scale of the configuration. We first demonstrate existence of minimizers for an explicit range of the model parameters when the energy is dominated by the exchange energy. We then investigate the conformal limit, in which only the exchange energy survives and identify the asymptotic profiles of the skyrmions as degree 1 harmonic maps from the plane to the sphere, together with their radii, angles and energies. A byproduct of our analysis is a quantitative rigidity result for degree ±1 harmonic maps from the two-dimensional sphere to itself.

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Curvature-stabilized skyrmions with angular momentum

Cited by 11 publications

References 25 publications

The mathematics of thin structures

The mathematics of thin structures

On symmetry of energy minimizing harmonic-type maps on cylindrical surfaces

A quantitative description of skyrmions in ultrathin ferromagnetic films and rigidity of degree $\pm1$ harmonic maps from $\mathbb{R}^2$ to $\mathbb{S}^2$

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