2-d stability of the Néel wall

Abstract

We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls.We prove stability of the 1-d transition layer under 2-d perturbations. This amounts to the investigation of the following singularly perturbed energy functional:The topological structure of this two-dimensional problem allows us to use a duality argument to infer the optimal lower bound. The lower bound relies on an -perturbation of the following logarithmically failing interpolation inequality |∇ 1/2 φ| 2 dx sup |φ| |∇φ| dx.

One-dimensional Néel walls under applied external fields

“…Using a further one-dimensional thin film reduction of the micromagnetic energy introduced in [17], Capella, Melcher and Otto outlined the proof of uniqueness of the Néel wall profile and its linearized stability with respect to one-dimensional perturbations [23]. Stability of geometrically constrained one-dimensional Néel walls with respect to large two-dimensional perturbations in soft materials was demonstrated asymptotically in [24]. More recently, Γ-convergence studies of the one-dimensional wall energy in the limit of very soft films and in the presence of an applied in-plane field normal to the easy axis were undertaken in [25,26], and a rigorous derivation of the effective magnetization dynamics driven by the reduced thin film energy introduced in [23] from the full three-dimensional Landau-Lifshitz-Gilbert equation was presented in [27].…”

A compactness result in thin-film micromagnetics and the optimality of the Néel wall

“…Following [5,7], the setting is determined by our goal to prove the optimality of Néel walls under 2-d variations. Let = (−1, 1) × R be a 2-d sheet (the cross section of the thin ferromagnetic sample) (see where m 1,∞ ∈ [0, 1) is some fixed number and we use the shorthand notation x = (x 1 , x 2 ).…”

“…In fact, ε is a non-dimensional quantity formed from three length scales: a material length scale, the film thickness and the film width (see [5,7]). The first term in (5) comes from the exchange energy (in fact, it is smaller than the usual exchange energy term represented by the Dirichlet integral of m ), and the energy of the stray field is also called the magnetostatic energy. Notice that the stray field h can be minimized out.…”

Control and stabilization of steady-states in a finite-length ferromagnetic nanowire

“…Now, multiplying (14) by sin ω and (15) by cos ω and adding these two equalities, it follows that θ = sin θ cos θ.…”

“…The set of steady-states of (1) is known to be very rich in the sense that it contains a number of diverse pattern configurations, such as Bloch or Néel walls (see [14,18]). This diversity could be used in magnetic storage technologies in order to encode information or to perform logic operations (see [2,5,16,28]).…”