One-dimensional in-plane edge domain walls in ultrathin ferromagnetic films

Abstract

We study existence and properties of one-dimensional edge domain walls in ultrathin ferromagnetic films with uniaxial in-plane magnetic anisotropy. In these materials, the magnetization vector is constrained to lie entirely in the film plane, with the preferred directions dictated by the magnetocrystalline easy axis. We consider magnetization profiles in the vicinity of a straight film edge oriented at an arbitrary angle with respect to the easy axis. To minimize the micromagnetic energy, these profiles form transition layers in which the magnetization vector rotates away from the direction of the easy axis to align with the film edge. We prove existence of edge domain walls as minimizers of the appropriate one-dimensional micromagnetic energy functional and show that they are classical solutions of the associated Euler-Lagrange equation with Dirichlet boundary condition at the edge. We also perform a numerical study of these one-dimensional domain walls and uncover further properties of these domain wall profiles.

Edge Domain Walls in Ultrathin Exchange-Biased Films

“…One would naturally expect that the above model can be easily modified to describe the finite sample case by restricting the domains of integration to D. However, this is not the case as such a model would miss the contribution of the edge charges to the magnetostatic energy [23]. On the other hand, a simple extension of the magnetization m from D to the whole of R 2 by zero and treating ∇·m distributionally would not work in general, as in this case the magnetostatic energy becomes infinite unless the magnetization is tangential to the boundary ∂D of the sample (for further discussion see [27]). This is due to the fact that a discontinuity in the normal component of the magnetization at the sample edge produces a divergent contribution to the magnetostatic energy.…”

“…Proof. For the proof of (5.9), we refer to the Appendix in [27]. To obtain (5.10), we first note that the minimum in the right-hand side of (5.10) is attained.…”

“…For instance, the nanostructure edges often determine the mechanism of the magnetization reversal process [14,21,34]. However, despite the importance of edge effects there exist just a handful of rigorous analytical studies characterizing the magnetization behavior near the film edges [23,25,27,31,35].…”

“…The latter is ensured by the choice of the reconstruction of the magnetization vector from the average of its component in the direction normal to the edge. We note that this argument crucially uses the specific form of the exchange bias energy and does not apply in the case of the uniaxial anisotropy considered by us in [27]. Once the one-dimensional nature of the minimizers has been established, the derivation of the Euler-Lagrange equation and the regularity still requires a delicate analysis due to the fact that nonlocality remains intertwined with the rest of the terms, producing an integro-differential equation.…”

Transverse domain walls in thin ferromagnetic strips

“…The mathematical understanding of domain wall profiles in ferromagnets rests on the micromagnetic modeling framework, whereby the magnetization configurations representing these profiles are viewed as local or global minimizers of the micromagnetic energy functional [29,42]. This framework has been successfully used to characterize a great variety of domain walls and other magnetization configurations (for an overview, see [15]; for some more recent developments, see [12,19,30,31,35,46,47,53]). However, head-to-head domain walls pose a fundamental challenge to micromagnetic modeling and analysis, since these magnetization configurations carry a non-zero magnetic charge, which may lead to divergence of the wall energy in infinite samples due to singular behaviors of the stray field [47].…”