Uniqueness of one-dimensional Néel wall profiles

Abstract

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin-film micromagnetic model, we analyse the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the Néel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behaviour at infinity. Thus, we establish uniqueness of the one-dimensional monotone Néel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.

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

“…These features have been predicted theoretically, using micromagnetic treatments [16,19,22,36,39], and verified experimentally [5]. Recent rigorous mathematical studies of Néel walls confirmed these predictions and provided more refined information about the profile of the Néel wall, including uniqueness, regularity, monotonicity, symmetry, stability and precise rate of decay [9,10,12,20,33,38].…”

“…Hence θ ∈ L 2 (R + ), which by Sobolev embedding [8,Theorem 8.8] implies that θ ∈ C 1 (R + ) and θ ∈ L ∞ (R + ). Focusing now on the nonlocal term, let u(x) be defined by (38), extended, as usual, by zero to x < 0, and let h(x) denote the integral in (73), with x ∈ R. Note that by chain rule we have u ∈ C 1 (R + ), and u (x) experiences a jump discontinuity at x = 0 whenever u (0 + ) = 0. Also, by weak chain rule [8,Corollary 8.11] we have u ∈ L 2 (R + ).…”

Separation of domain walls with nonlocal interaction and their renormalised energy by $Γ$-convergence in thin ferromagnetic films

“…where Ḣ1/2 (R) is the homogeneous Sobolev space of fractional order 1/2 and we write x 1 for the independent variable for reasons that will become clear later. For details about the background of this model, and how it is derived from the full micromagnetic energy, we refer to previous work [8,9,30,31,7,21,16,5,29,32,18,19,20]. We note, however, that the first term on the right hand side in (1) is called exchange energy and the second term is called stray field energy or magnetostatic energy in the theory of micromagnetics.…”

One-dimensional domain walls in thin ferromagnetic films with fourfold anisotropy

“…Remark 1. It should be possible to use the arguments of [31] to prove that the minimizers in theorem 1 are the unique monotone 90 • wall profiles, i.e., that the minimizers are the unique, up to translations, monotone critical points of the energy E −π/4 in A π/2 .…”

“…[1,25]). More recent micromagnetic studies have led to a good present understanding of the Neel wall's internal structure [1,4,[26][27][28][29][30][31], the main features of which (sharp inner core with slowly decaying tails) have been verified experimentally [32][33][34].…”