Edge Domain Walls in Ultrathin Exchange-Biased Films

Abstract

We present an analysis of edge domain walls in exchange-biased ferromagnetic films appearing as a result of a competition between the stray field at the film edges and the exchange bias field in the bulk. We introduce an effective two-dimensional micromagnetic energy that governs the magnetization behavior in exchange-biased materials and investigate its energy minimizers in the strip geometry. In a periodic setting, we provide a complete characterization of global energy minimizers corresponding to edge domain walls. In particular, we show that energy minimizers are one-dimensional and do not exhibit winding. We then consider a particular thin film regime for large samples and relatively strong exchange bias and derive a simple and comprehensive algebraic model describing the limiting magnetization behavior in the interior and at the boundary of the sample. Finally, we demonstrate that the asymptotic results obtained in the periodic setting remain true in the case of finite rectangular samples.

Transverse domain walls in thin ferromagnetic strips

“…Notice that in the regime of interest the film thickness reaches an order of only a few atomic layers, making the use of the full three-dimensional micromagnetic energy problematic. As was argued previously, a model that is more appropriate for such ultrathin films is the reduced micromagnetic thin film energy (for a detailed discussion, see [17,46]).…”

“…where γ > 0 is a fixed parameter, which may be obtained from the full three-dimensional micromagnetics via a formal asymptotic reduction and a suitable rescaling of the strip width [17,46]. The conditions on m, which we are going to relax shortly, are needed to ensure convergence of all the integrals in (2.10).…”

“…To this end, we introduce a reduced thin film micromagnetic energy functional that is appropriate for modeling ultrathin ferromagnetic films in which the ferromagnetic layer has thicknesses down to a few atomic layers and, strictly speaking, the macroscopic energy functional in (1.15) is no longer applicable. This two-dimensional reduced thin film energy functional retains the nonlocal character of the micromagnetic energy in (1.15) in the ultrathin film regime and was introduced by us earlier in the studies of exchange biased films [46]. It represents an intermediate level of modeling between the full three-dimensional micromagnetic energy in (1.15) and the two-dimensional thin film limit energy in (1.16).…”

“…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].…”

“…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]. To date, there have been only a handful of micromagnetic studies of such charged domain walls [26,27,36,38,39,46,47].…”