Interaction Energy of Domain Walls in a Nonlocal Ginzburg–Landau Type Model from Micromagnetics

Abstract

We study a variational model from micromagnetics involving a nonlocal Ginzburg-Landau type energy for S 1 -valued vector fields. These vector fields form domain walls, called Néel walls, that correspond to one-dimensional transitions between two directions within the unit circle S 1 . Due to the nonlocality of the energy, a Néel wall is a two length scale object, comprising a core and two logarithmically decaying tails. Our aim is to determine the energy differences leading to repulsion or attraction between Néel walls. In contrast to the usual Ginzburg-Landau vortices, we obtain a renormalised energy for Néel walls that shows both a tail-tail interaction and a core-tail interaction. This is a novel feature for Ginzburg-Landau type energies that entails attraction between Néel walls of the same sign and repulsion between Néel walls of opposite signs.

Néel walls with prescribed winding number and how a nonlocal term can change the energy landscape

“…Let u = U (m) be the function defined in (7). By Proposition 15, we know that φ is smooth in R. We now use an argument similar to a proof in our previous paper [11,Lemma 12]. As u is harmonic, we calculate, for every R > 0, that…”

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

“…as 0. The result was improved in our previous paper [18]. The first observation is that the asymptotic behaviour of this quantity is more easily expressed in δ = log 1 than in itself because δ is the appropriate size of the core of a Néel wall.…”

“…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.…”

“…The first observation is that the asymptotic behaviour of this quantity is more easily expressed in δ = log 1 than in itself because δ is the appropriate size of the core of a Néel wall. (A Néel wall is a two-scale object, containing a core and two logarithmically decaying tails [30,31,18].) For this reason, we will assume this relationship between and δ throughout the paper and work mostly with δ henceforth.…”

“…Another observation is that the problem behaves similarly to Ginzburg-Landau vortices in some respects. In particular, similarly to the theory of Ginzburg-Landau vortices (see the seminal book of Bethuel-Brezis-Hélein [2]), we can compute a renormalised energy that gives a lot of information about the energy required for a certain collection of Néel walls [18]. It is given by the function 2 1 We include N = 0 here to avoid special treatment of this case later on.…”