Effective dynamics for ferromagnetic thin films: A rigorous justification

Robert V. Kohn; Valeriy V. Slastikov

doi:10.1098/rspa.2004.1342

Effective dynamics for ferromagnetic thin films: A rigorous justification

Robert V. Kohn, Valeriy V. Slastikov

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In a thin-film ferromagnet, the leading-order behaviour of the magnetostatic energy is a strong shape anisotropy, penalizing the out-of-plane component of the magnetization distribution. We study the thin-film limit of Landau-Lifshitz Gilbert dynamics, when the magnetostatic term is replaced by this local approximation. The limiting two-dimensional effective equation is overdamped, i.e. it has no precession term. Moreover, if the damping coefficient of three-dimensional micromagnetics is α, then the damping coefficient of the two-dimensional effective equation is α + 1/α; thus reducing the damping in three dimensions can actually increase the damping of the effective equation. This result was previously shown by García-Cervera and E using asymptotic analysis: our contribution is a mathematically rigorous justification.

Original language	English (US)
Pages (from-to)	143-154
Number of pages	12
Journal	Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume	461
Issue number	2053
DOIs	https://doi.org/10.1098/rspa.2004.1342
State	Published - Jan 8 2005

Keywords

Effective dynamics
Micromagnetics: thin film

ASJC Scopus subject areas

Mathematics(all)
Engineering(all)
Physics and Astronomy(all)

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10.1098/rspa.2004.1342

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abstract = "In a thin-film ferromagnet, the leading-order behaviour of the magnetostatic energy is a strong shape anisotropy, penalizing the out-of-plane component of the magnetization distribution. We study the thin-film limit of Landau-Lifshitz Gilbert dynamics, when the magnetostatic term is replaced by this local approximation. The limiting two-dimensional effective equation is overdamped, i.e. it has no precession term. Moreover, if the damping coefficient of three-dimensional micromagnetics is α, then the damping coefficient of the two-dimensional effective equation is α + 1/α; thus reducing the damping in three dimensions can actually increase the damping of the effective equation. This result was previously shown by Garc{\'i}a-Cervera and E using asymptotic analysis: our contribution is a mathematically rigorous justification.",

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T1 - Effective dynamics for ferromagnetic thin films

T2 - A rigorous justification

AU - Kohn, Robert V.

AU - Slastikov, Valeriy V.

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N2 - In a thin-film ferromagnet, the leading-order behaviour of the magnetostatic energy is a strong shape anisotropy, penalizing the out-of-plane component of the magnetization distribution. We study the thin-film limit of Landau-Lifshitz Gilbert dynamics, when the magnetostatic term is replaced by this local approximation. The limiting two-dimensional effective equation is overdamped, i.e. it has no precession term. Moreover, if the damping coefficient of three-dimensional micromagnetics is α, then the damping coefficient of the two-dimensional effective equation is α + 1/α; thus reducing the damping in three dimensions can actually increase the damping of the effective equation. This result was previously shown by García-Cervera and E using asymptotic analysis: our contribution is a mathematically rigorous justification.

AB - In a thin-film ferromagnet, the leading-order behaviour of the magnetostatic energy is a strong shape anisotropy, penalizing the out-of-plane component of the magnetization distribution. We study the thin-film limit of Landau-Lifshitz Gilbert dynamics, when the magnetostatic term is replaced by this local approximation. The limiting two-dimensional effective equation is overdamped, i.e. it has no precession term. Moreover, if the damping coefficient of three-dimensional micromagnetics is α, then the damping coefficient of the two-dimensional effective equation is α + 1/α; thus reducing the damping in three dimensions can actually increase the damping of the effective equation. This result was previously shown by García-Cervera and E using asymptotic analysis: our contribution is a mathematically rigorous justification.

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Effective dynamics for ferromagnetic thin films: A rigorous justification

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Keywords

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