A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types

A reduced model for domain walls in soft ferromagnetic films at the cross-over from symmetric to asymmetric wall types.

Lukas Döring^∗ Radu Ignat^† Felix Otto^∗ September 3, 2013

Abstract

We study the Landau-Lifshitz model for the energy of multi-scale transition layers – called “domain walls” – in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors m^±α ∈ S² that differ by an angle 2α. Assuming trans- lation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parame- ter regime confirms the experimental, numerical and physical predictions: The minimal energy splits into a contribution from an asymmetric, divergence-free core which per- forms a partial rotation inS² by an angle2θ, and a contribution from two symmetric, logarithmically decaying tails, each of which completes the rotation from angleθtoαin S¹. The angleθis chosen such that the total energy is minimal. The contribution from the symmetric tails is known explicitly, while the contribution from the asymmetric core is analyzed in [7].

Our reduced model is the starting point for the analysis of a bifurcation phenomenon from symmetric to asymmetric domain walls. Moreover, it allows for capturing asym- metric domain walls including their extended tails (which were previously inaccessible to brute-force numerical simulation).

Keywords:Γ-convergence, concentration-compactness, transition layer, bifurcation, micromagnet- ics.

MSC:49S05, 49J45, 78A30, 35B32, 35B36

∗Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany (email:

Lukas.Doering@mis.mpg.de, Felix.Otto@mis.mpg.de)

†Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse, France (email: Radu.Ignat@math.univ-toulouse.fr)

1 Introduction

1.1 Model

We consider the following model: The magnetization is described by a unit-length vector field

m= (m₁, m₂, m₃) : Ω→S², where the two-dimensional domain

Ω =R×(−1,1)

corresponds to a cross-section of the sample that is parallel to thex₁x₃-plane. The following

“boundary conditions atx₁ =±∞” are imposed so that a transition from the angle −α to α∈(0,^π₂]is generated and a domain wall forms parallel to thex₂x₃-plane (see Figure 1):

m(±∞,·) =m^±_α := (cosα,±sinα,0), (1) with the convention:

f(±∞,·) =a_± ⇐⇒

Z

Ω+

|f−a₊|²dx+ Z

Ω−

|f −a₋|²dx <∞, (2) where Ω₊ = Ω∩ {x₁ ≥ 0} and Ω₋ = Ω∩ {x₁ ≤ 0}. Throughout the paper, we use the variables x = (x₁, x₃) ∈ Ω together with the differential operator ∇ = (∂_x1, ∂_x3), and we denote bym^′ = (m1, m₃)the projection of m on the x₁x₃-plane.

Ω

x1

x2

x3

Figure 1: The cross-sectionΩin a ferromagnetic sample on a mesoscopic level.

We focus on the following micromagnetic energy functional depending on a small parame- terη:¹

E_η(m) = Z

Ω|∇m|²dx+λln¹_η Z

R²|h(m)|²dx+η Z

Ω

(m₁−cosα)²+m²₃dx, η∈(0,1), (3) subject to the boundary conditions (1), whereλ >0is a fixed constant andh=h(m) :R² → R² stands for the uniqueL² stray-field restricted to thex₁x₃-plane that is generated by the

1We refer to Section 2 for more information onEη and the parametersηandλ.

static Maxwell equations:²

(∇ ·(h+m^′1_Ω) = 0 inD^′(R²),

∇ ×h= 0 inD^′(R²). (4) The first term of (3) is called the “exchange energy”, favoring a constant magnetization. The second term (called “stray-field energy”) can be written as the H˙⁻¹(R²)-norm of the 2D divergence ofm^′ (wherem is always extended by0 outside ofΩ):

Z

R²|h(m)|²dx=k∇·(m^′1_Ω)k²H˙⁻¹(R²):= sup Z

Ω

m^′· ∇v dx

v∈C_c^∞(R²),k∇vkL²(R2)≤1

. The last term in (3) (a combination of material anisotropy and external magnetic field) forces the magnetization to favor the “easy axis” m^±_α and serves as confining mechanism for the tails of the transition layer. We refer to Section 2 for more physical details about this model.

We are interested in the asymptotic behavior of minimizers m_η of E_η with the boundary condition (1) as η ↓ 0. The main feature of this variational principle is the non-convex constraint on the magnetization (|m_η|= 1) and the non-local structure of the energy (due to the stray field h(m_η)). The competition between the three terms of the energy together with the boundary constraint (1) induces an optimal transition layer that exhibits two length scales (cf. Figure 3):

• an asymmetric core of size |x₁| . 1

(up to a logarithmic scale in η) where the magnetizationm_η is asymptotically divergence-free (so, generating no stray field) and hence the leading order term inE_η is given by the exchange energy; in this region,m_η describes a transition path onS² between the two directions m^±_θ determined by some angleθ.

• two symmetric tails of size 1 . |x₁| . ¹_η

(up to a logarithmic scale in η) where m_η asymptotically behaves as a symmetric Néel wall: a one-dimensional (i.e., m_η = m_η(x₁)) rotation on S¹ := S¹× {0} ⊂ S² between the angles θ and α (on the left and right sides of the core). Here, the formation of the wall profile is driven by the stray-field energy that induces a logarithmic decay ofm_1,η on these two tails.

The constantλ >0 and the wall angle α play a crucial role in the behavior of a minimizer m_η. In fact, for eitherα ≪1, or α ∈(0,^π₂] arbitrary but λ small, a minimizer is expected to be asymptotically symmetric (i.e., m_η =m_η(x₁)) as η ↓0. However, for sufficiently large λ, there exists a critical wall angle α^∗ where a bifurcation occurs: It becomes favorable to nucleate an asymmetric domain wall in the core of the transition layer.

In [10, Section 3.6.4 (E)], Hubert and Schäfer state:

2Existence and uniqueness of the stray field are a direct consequence of the Riesz representation theorem in the Hilbert spaceV =n

v∈L²_loc(R²)

∇v∈L²,⁻R

B(0,1)v dx= 0o

endowed with the normk∇vkL2: Indeed, by (1), the functionalv7→R

Ω m^′−(cosα,0)

· ∇v dxis linear continuous onV so that there exists a unique solution h =−∇u withu ∈V of (4) written in the weak form R

R²∇u· ∇v dx= R

Ωm^′· ∇v dxfor every v∈C_c^∞(R²).

“The magnetization of an asymmetric Néel wall points in the same direction at both surfaces, which is [. . . ] favourable for an applied field along this direction.

This property is also the reason why the wall can gain some energy by splitting off an extended tail, reducing the core energy in the field generated by the tail.

[. . . ] The tail part of the wall profile increases in relative importance with an applied field, so that less of the vortex structure becomes visible with decreasing wall angle. At a critical value of the applied field the asymmetric disappears in favour of a symmetric Néel wall structure.”

To justify this physical prediction, we will establish the asymptotic behavior of {E_η}η↓0

through the method ofΓ-convergence. The limiting reduced model does then show that the minimal energy splits into the separate contributions from the symmetric and asymmetric regions of the transition layer. This makes it possible to infer information on the size of the regions and the conjectured bifurcation from symmetric to asymmetric walls. For details, we refer to the end of Section 1.2.

1.2 Results

Let α ∈ (0,^π₂] and η ∈ (0,1). Observe that for m: Ω → S², finite energy E_η(m) < ∞ is equivalent to m ∈ H˙¹(Ω,S²) and m^′(±∞,·) ⁽²⁾= (cosα,0) (which in particular implies

|m₂|(±∞,·) = sinα, see Lemma 3). In the following we focus on the set of magnetizations of wall angle α∈(0,^π₂]with a transition imposed by (1):

X^α:=n

m∈H˙¹(Ω,S²)

m(±∞,·) =m^±_αo

. (5)

Our main result consists in provingΓ-convergence of {E_η}η↓0, defined onX^α ⊂H˙¹(Ω,S²), in the weak H˙¹-topology to the Γ-limit functional

E₀(m) = Z

Ω|∇m|²dx+ 2π λ cosθ_m−cosα2

, (6)

which is defined on a spaceX₀ ⊂H˙¹(Ω,S²):

In order to give the definitions of X₀ (see (8)) and the angleθ_m associated to m∈X₀ (see (7)), we need some preliminary remarks. First, due to the logarithmic penalization of the stray field in (3) asη↓0, limiting configurations of a family {m_η}η↓0 of uniformly bounded energy E_η(m_η) ≤ C (e.g., minimizers of E_η) are stray-field free. Second, note that for any m ∈ H˙¹(Ω,S²) with ∇ ·(m^′1_Ω) = 0in D^′(R²) (i.e., ∇ ·m^′ = 0 in Ω and m₃ = 0 on ∂Ω) there is a unique constant angleθ_m ∈[0, π]such that

¯

m₁(x₁) :=− Z 1

−1

m₁(x₁, x₃)dx₃ = cosθ_m for allx₁∈R. (7) Observe that such vector fields have the property m^′(±∞,·) = (cosθ_m,0) in the sense of (2) (see (30) and (31) below) and moreover, |m₂|(±∞,·) = sinθ_m (see Lemma 3 if θ_m ∈ (0, π), and Remark 1 below if θ_m ∈ {0, π}). We define X₀ as the non-empty (see

Appendix) set of such configurations m that additionally change sign as|x₁| → ∞, namely m₂(±∞,·) =±sinθ_m in the sense of (2):

X₀ :=n

m∈H˙¹(Ω,S²)

∇ ·m^′= 0 inΩ, m₃ = 0 on ∂Ω, m(±∞,·) =m^±_θ

m

o

. (8) Note, however, that due to vanishing control of the anisotropy energy as η ↓ 0, a limiting configurationm in general satisfies (1) for an angle θ_m that differs fromα.

Remark 1. Observe that if θ_m ∈ {0, π} for m ∈H˙¹(Ω,S²) with∇ ·(m^′1_Ω) = 0 in D^′(R²) – in particular if m∈X₀ –, we have m∈ {±e₁}: Indeed, since |m¯₁| ≡1 in R and |m|= 1 in Ω, we deduce |m₁| ≡1 andm₂ ≡m₃≡0 in Ω.

We further remark that the first term in the Γ-limit energy (6) accounts for the exchange energy of the asymmetric core of a transition layerm_η asη↓0, while the second term inE₀ accounts for the contribution coming from stray field/anisotropy energy through extended (symmetric) tails of the wall configurations at positiveη.

OurΓ-convergence result is established in three steps. We start with compactness results. The main difficulty comes from the boundary conditions (1), which are in general not carried over by weak limits of magnetization configurations with uniformly bounded exchange energy.

However, since the energy E_η is invariant under translations in x₁-direction, and due to the constraint (1) in X^α, a change of sign in m₂ can be preserved as η ↓ 0 by a suitable translation inx₁.

Proposition 1 (Compactness). Let α ∈ (0,^π₂]. The following convergence results hold up to a subsequence and translations in the x₁-variable:

(i) Let {m_η}η↓0 ⊂ X^α with uniformly bounded energy, i.e., sup_η↓0E_η(m_η) < ∞. Then m_η −⇀ m weakly in H˙¹(Ω) for somem∈X₀.

(ii) Let {m_k}k↑∞ ⊂ X^α with uniformly bounded energy E_η for η ∈ (0,1) fixed, i.e., sup_kE_η(m_k)<∞. Then m_k−⇀ m weakly in H˙¹(Ω)for some m∈X^α. Moreover, the corresponding stray fields{h(m_k)}k↑∞ converge weakly inL²(R²), i.e.,h(m_k)−⇀ h(m) in L²(R²).

(iii) Let{m_k}k↑∞⊂X₀ with uniformly bounded exchange energy, i.e.,sup_kR

Ω|∇m_k|²dx <

∞, such that the angles θ_k := θ_mk associated to m_k in (7) satisfy θ_k ∈ [0, π]. Then θ_k →θ for some angle θ∈[0, π]andm_k −⇀ mweakly inH˙¹(Ω) for somem∈X₀ with θ_m =θ (i.e., m∈X₀∩X^θ).

The main ingredient in Proposition 1 is the following concentration-compactness type lemma related to the change of sign at ±∞:

Lemma 1. Let u_k:R→R,k∈N, be continuous and satisfy the following conditions:

lim sup

k↑∞

Z

R|_ds^du_k(s)|²ds <∞, (9) lim sup

s↓−∞

u_k(s)<0 and lim inf

s↑∞ u_k(s)>0 for everyk∈N, (10)

where we denote by _ds^du_k the distributional derivative of the function u_k.

Then for eachk∈N, there exists a zero z_k of u_k and a limitu∈H˙¹(R)such that u(0) = 0, u_k(·+z_k)→u locally uniformly inR and weakly inH˙¹(R) for a subsequence and

lim sup

s↓−∞

u(s)≤0 as well as lim inf

s↑∞ u(s)≥0. (11)

The second step consists in proving the following lower bound:

Theorem 1 (Lower bound). Let α ∈ (0,^π₂]. For m ∈ X₀ and any family {m_η}η↓0 ⊂ X^α withm_η −⇀ m inH˙¹(Ω)as η↓0, the following lower bound holds:

lim inf

η↓0 E_η(m_η)≥E₀(m). (12)

The last step consists in constructing recovery sequences for limiting configurations:

Theorem 2 (Upper bound). For α ∈ (0,^π₂] and every m ∈ X₀ there exists a family {m_η}η↓0 ⊂X^α withm_η →m strongly in H˙¹(Ω) and

lim sup

η↓0

E_η(m_η)≤E₀(m). (13)

As a consequence, one deduces the asymptotic behavior of the minimal energyE_η over the spaceX^α asη ↓0.

Corollary 1. For α ∈(0,^π₂]and θ∈[0, π]we define E_asym(θ) = min

m∈X0

θm=θ

Z

Ω|∇m|²dx and

E_sym(α−θ) = 2π cosθ−cosα2

. Then it holds

limη↓0 min

mη∈X^αE_η(mη) = min

m∈X0

E₀(m) = min

θ∈[0,π]

E_asym(θ) +λ E_sym(α−θ)

. (14) In fact, the optimal angle θ is attained in [0,^π₂]. Moreover, every minimizing sequence {m_η}η↓0 ⊂ X^α of {E_η}η↓0 in the sense of E_η(m_η) → min_X0E₀ is relatively compact in the strongH˙¹(Ω)-topology, up to translations inx₁, with accumulation points in X₀. One benefit of (14) is splitting the problem of determining the optimal transition layer into two more feasible ones: First, the energy of asymmetric walls (i.e. walls of small width) has to be determined (at the expense of an additional constraint on∇ ·m^′). Afterwards, a one-dimensional minimization procedure is sufficient to determine the structure of the wall profile. Direct numerical simulation of (3) has been a difficult endeavor (see [17] and also [10, Section 3.6.4 (E)]).

1.3 Outlook

In the following we briefly discuss an application of our reduced model to the cross-over from symmetric to asymmetric Néel wall and point out further interesting (topological) questions and open problems associated with the energy of asymmetric domain walls.

Bifurcation. The previous result represents the starting point in the analysis of the bifur- cation phenomenon (from symmetric to asymmetric walls) in terms of the wall angleα (see also [7]). We will prove that there is a supercritical (pitchfork) bifurcation (cf. Figure 2): This means that for small anglesα ≪1, the optimal transition layerm_η of E_η is asymptotically symmetric (the symmetric Néel wall); beyond a critical angle α^∗, the symmetric wall is no longer stable, whereas the asymmetric wall is. To understand the type of the bifurcation, by (14), we need to compute the asymptotic expansion of the asymmetric energy up to order θ⁴ asθ→0(since the symmetric part of the energy is quartic for small anglesθ, α≪1, i.e., E_sym(α−θ).α⁴).³ In fact, we show (see [7]):

E_asym(θ) = 4πθ²+³⁰⁴₁₀₅πθ⁴+o(θ⁴) as θ↓0. (15) This allows us to heuristically determine a critical angle α^∗ at which the symmetric Néel wall loses stability and an asymmetric core is generated. Moreover, a new path of stable critical points with increasing inner wall angleθbranches off ofθ= 0(see Figure 2). Indeed, for smallα, combining with (15), the RHS of (14) as function of θ ∈[0, α]has the unique critical point θ= 0 if α≤α^∗ where the bifurcation angle α^∗ is given by

α^∗ = arccos 1−²_λ

+o(1), as α→0.

(Observe that α^∗ ∈ [0,^π₂] provided λ ≥ 2; therefore, the bifurcation appears only if λ is large enough.) Forα > α^∗, the symmetric wall becomes unstable under symmetry-breaking perturbations and the optimal splitting angleθbecomes positive; hence, the asymmetric wall becomes favored by the system. Moreover, the second variation of the RHS of (14) along the branch of positive splitting angles is positive so that the bifurcation from symmetric to asymmetric wall is supercritical.

Topological degree and vortex singularity. We now discuss topological properties of stray-field free magnetization configurations: In fact, if m∈X₀ satisfies (1) for some angle θ∈(0,^π₂], denoting the “extended” boundary ofΩ

Bdry:=∂Ω∪

{±∞} ×[−1,1]

∼=S¹, (16)

then one can define the following winding number of m on Bdry: due to m₃ = 0 on ∂Ω as well asm₃(±∞,·) = 0 (so, (m₁, m₂) : Bdry→S¹), one obtains (by the homeomorphism (16)) a mapm˜ ∈H¹²(S¹,S¹) to which a topological degree can be associated (see, e.g., [4]).

3Observe that for givenα∈(0,^π₂]the optimal wall angleθα= argmin (Easym(θ) +λEsym(α−θ))∈[0,^π₂] satisfies the estimateθα.α². Indeed, by comparison withθ= 0we haveEasym(θα) + 2πλ(cosθα−cosα)²≤ 2πλ(1−cosα)². Omitting Easym(θα) we first obtainθα →0 as α↓ 0, so that by (15) one deduces that 2θ²α/λ(1−cosα)² for small α >0. From here, the desired estimate follows.

stable unstable θ

α^∗ α 0

Figure 2: Bifurcation diagram for the angle θ of the asymmetric core, depending on the global wall angleα.

In particular, in the case of smoothm˜:S¹ →S¹, the topological degree (also called winding number) ofm˜ is defined as follows:

deg( ˜m) := 1 2π

Z

S¹

det( ˜m, ∂_θm)˜ dH¹ where ∂_θm˜ is the angular derivative of m.˜

We will show the following relation between the winding number of m ∈ X₀ on Bdry and topological singularities of (m₁, m₃) inside Ω: the non-vanishing topological degree of (m1, m₂) : Bdry → S¹ generates vortex singularities of (m1, m₃) as illustrated in Figure 3. By vortex singularity of v := (m₁, m₃), we understand a zero of v carrying a non-zero topological degree. In general, this is implied by the existence of a smooth cycle (i.e., closed curve) γ ⊂Ωsuch that|v|>0on γ and deg(_|^v_v|, γ)6= 0; the vector field v then vanishes in the domain bounded by γ.

Lemma 2. Let m∈X₀ (i.e. m∈H˙¹(Ω,S²) with ∇ ·(m^′1_Ω) = 0 in D^′(R²)) such that (1) holds for some angle θ∈(0,^π₂]. Suppose that(m1, m₂) :Bdry→S¹ has a non-zero winding number onBdry. Then there exists a vortex singularity of(m₁, m₃)inΩcarrying a non-zero topological degree.

Motivated by Lemma 2, let us introduce the set

L^θ={m∈X₀∩X^θ : degm= 1}

for a fixed angle θ ∈ (0,^π₂]. First of all, we have that L^θ 6= ∅ (see Appendix).⁴ Since X₀ =∪θ∈[0,π] X₀ ∩X^θ

, the relationL^θ 6=∅ obviously implies that X₀∩X^θ 6=∅ for every θ ∈(0, π) which is essential in our reduced model given by the Γ-convergence program. A natural question concerns the closure (in the weak H˙¹(Ω)-topology) of the set L^θ. This is

4Naturally, one can address a similar question by imposing an arbitrary winding numbern. For the case n= 0, we analyze this problem in [7] which is typical for asymmetric Néel walls; in particular, for small angles θ, we construct an elementm∈X0∩X^θ withdegm= 0and asymptotically minimal energy. Moreover, given anym∈X0∩X^θwith finite energy, one can use a reflection and rescaling argument to define a finite-energy magnetization onΩwith degree0(see Remark 5 (iii) ).

important in order to define the (limit) asymmetric Bloch wall by minimizing the exchange energy onL^θ.⁵

Open problem 1. Is the following infimum inf

m∈L^θ

Z

Ω|∇m|²dx attained for every angleθ∈(0,^π₂]?

1.4 Structure of the paper

This paper is organized as follows: In Section 2, we explain the relation of (3) to the full Landau-Lifshitz energy, as well as the physical background of our analysis.

In Section 3, we prove the compactness results in Lemma 1 and Proposition 1, which in particular yield existence of minimizers ofE_η,E_asym(θ) and E₀.

Section 4 contains the proofs of the lower and upper bound (Theorems 1 and 2) of our Γ-convergence result and also, the proof of Corollary 1.

In the Appendix, finally, we show that the set X₀∩X^θ is non-empty for any given angle θ∈(0,^π₂]. To this end, we construct an admissible configuration inE_asym(θ)with non-zero topological degree on the boundary ofΩ(i.e., of asymmetric Bloch-wall type). Moreover, we prove Lemma 2.

2 Physical background

In this section, we denote by ∇ = (∂_x1, ∂_x2, ∂_x3) the full gradient of functions depending on x = (x₁, x₂, x₃). Recall that the prime ^′ denotes the projection on the x₁x₃-plane, i.e.

∇^′ = (∂x1, ∂_x3),x^′ = (x1, x₃).

Micromagnetics. Let ω ⊂ R³ represent a ferromagnetic sample whose magnetization is described by the unit-length vector-field m: ω → S². Assume that the sample exhibits a uniaxial anisotropy with e₂ = (0,1,0) as “easy axis”, i.e. favored direction of m. The well-accepted micromagnetic model (see e.g. [5, 10]) states that in its ground state the magnetization minimizes the Landau-Lifshitz energy:

E^3D(m) =d² Z

ω|∇m|²dx+ Z

R³|h(m)|²dx+Q Z

ω

m²₁+m²₃dx−2 Z

ω

h_ext·m dx. (17) Here, the exchange length d is a material parameter that determines the strength of the exchange interaction of quantum mechanical origin, relative to the strength of the stray field h=h(m). The stray field is the gradient field h=−∇u that is (uniquely) generated by the distributional divergence ∇ ·(m1_ω) via Maxwell’s equation

∇ ·(h+m1_ω) = 0 inD^′(R³). (18)

5This question is related to the theory of Ginzburg-Landau minimizers with prescribed degree (see e.g.

Berlyand and Mironescu [2]).

The non-dimensional quality factor Q >0 is a material constant that measures the relative strength of the energy contribution coming from misalignment ofmwithe₂.⁶The last term, called Zeeman energy, favors alignment of mwith an external magnetic field h_ext:ω→R³. Derivation of our model. We assume the magnetic sample to be a thin film, infinitely extended in the (x₁x₂)-plane, i.e. ω =R²×(−t, t), where two magnetic domains of almost constant magnetizationm≈m^±_α have formed for±x₁≫t. Physically, such a configuration is stabilized by the combination of uniaxial anisotropy and suitably chosen external field h_ext = Qcosαe₁. Moreover, we assume that m and hence, the stray field h = (h1,0, h3) are independent of the x₂-variable so that (17) formally reduces to integrating the energy density (per unit length inx₂-direction):

E^2D(m) =d² Z

ω^′|∇^′m|²dx^′+ Z

R2|h^′|²dx^′+Q Z

ω^′

(m1−cosα)²+m²₃dx^′

where ω^′ = R×(−t, t) and h^′ = h^′(m) = −∇^′u satisfies (4) driven by the 2D divergence of m^′1_ω′. Recall that the prime ^′ here denotes a projection onto the coordinate directions (x₁, x₃) transversal to the wall plane. After non-dimensionalization of length with the film thickness t, i.e., setting x˜^′ = ^x_t^′,ω˜^′ = ^ω_t^′, m(˜˜ x^′) = m(x^′), u(˜˜ x^′) = ^u(x_t^′), the above specific energy (per unit length inx₂) is given by

E˜^2D( ˜m) =d² Z

˜

ω^′|∇˜^′m˜|²d˜x^′+t² Z

R²

|∇˜^′u˜|²d˜x^′+Qt² Z

˜ ω^′

( ˜m₁−cosα)²+ ˜m²₃d˜x^′, (19) where the differential operator∇˜^′ refers to the variablesx˜^′= (˜x₁,x˜₃) andu˜:R² →Ris the 2Dstray-field potential given by

∆˜^′u˜= ˜∇^′·( ˜m1_ω˜′) in D^′(R²).

Throughout the section,we omit˜and ^′.

Symmetric walls. In the regime of very thin films (i.e. for a sufficiently small ratio of film thicknesstto exchange length d, see below for the precise regime), the symmetric Néel wall m is the favorable transition layer: It is characterized by a reflection symmetry w.r.t.

the midplane x₃ = 0, see 3) below. In fact, to leading order in _d^t, it is independent of the thickness variablex₃, i.e. m=m(x1), and in-plane, i.e.m₃= 0. The symmetric Néel wall is a two length-scale object with a core of sizew_core=O(^d_t²)and two logarithmically decaying tails w_core . |x₁| . w_tail = O(_Q^t) (see e.g. Melcher [15, 16]). It is invariant w.r.t. all the symmetries of the variational problem (besides translation invariance):

1) x₁ → −x₁,x₃ → −x₃,m₂ → −m₂; 2) x₁ → −x₁,m₃ → −m₃,m₂ → −m₂; 3) x₃ → −x₃,m₃ → −m₃;

4) Id.

6A typical, experimentally accessible, soft ferromagnetic material is Permalloy, for whichd ≈5nm and Q= 2.5·10⁻⁴.

The specific energy of a Néel wall of angleα= ^π₂ is given by E^2D(symmetric Néel wall) =O(t² 1

ln_w^wtail

core

) =O(t² 1 ln_d^t2²Q

)

(see e.g. [19, 5]). For a symmetric Néel wall of angle α < ^π₂, the energy is asymptotically quartic inα as it is proportional to(1−cosα)² (see e.g. [11]).

Asymmetric walls. For thicker films, the optimal transition layer has an asymmetric core, where the symmetry 3) is broken (see e.g. [8, 9]). The main feature of this asymmetric core is that it is approximately stray-field free. Hence to leading order, the asymmetric core is given by a smooth transition layer mthat satisfies (1) and

m:ω →S²,∇ ·m^′= 0 in ω and m₃= 0 on ∂ω. (20) Observe that (m₁, m₂) :∂ω → S¹ since m₃ vanishes on ∂ω, so that one can define a topo- logical degree of (m₁, m₂) on ∂ω (where ∂ω is the closed “infinite” curve R × {±1}

∪ {±∞} ×[−1,1]

). The physical experiments, numerics and constructions predict two types of asymmetric walls, differing in their symmetries and the degree of (m₁, m₂) on∂ω:

(i) For small wall angles α, the system prefers the so-called asymmetric Néel wall. Its main features are the conservation of symmetries 1) and 4) and a vanishing degree of (m₁, m₂) on ∂ω (see Figure 3). Due to symmetry 1), the m₂ component of an asymmetric Néel wall vanishes on a curve that is symmetric with respect to the center of the wall (by x → −x). Moreover, the phase of (m₁, m₂) is not monotone at the surface|x₃|= 1.

(ii) For large wall anglesα, the system prefers the so-called asymmetric Bloch wall. These walls only have the trivial symmetry 4). Another difference is the non-vanishing topo- logical degree on∂ω (i.e.,deg (m₁, m₂), ∂ω

=±1). Therefore, a vortex is nucleated in the wall core, and the curve of zeros of m₂ is no longer symmetric with respect the center of the wall (see Figure 3). Moreover, the phase of (m₁, m₂) is expected to be monotone at the surface|x₃|= 1.

The asymmetric wall has a single length scalew_core∼tand the specific energy comes from the exchange energy (see e.g. [19, 5]). It is of the order

E^2D(asymmetric wall) =O(d²).

For small wall angles, the energy of the optimal asymmetric wall is asymptotically quadratic inα (see [5]).

Regime. We focus on the challenging regime of soft materials of thickness t close to the exchange lengthd(up to a logarithm), where we expect the cross-over in the energy scaling of symmetric walls and asymmetric walls (see [19]):

Q≪1 and ln_Q¹ ∼(_d^t)².

7The magnetization was obtained by numerically solving the Euler-Lagrange equation corresponding to E (θ). To this end, a Newton method with suitable initial data was employed.

-1 -0.5 0 0.5 1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x3

x₁

-1 -0.5 0 0.5 1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

x3

x₁

Figure 3: Asymmetric Néel wall (on the left) and asymmetric Bloch wall (on the right).

Numerics.⁷

Rescaling the energy (19) by d² and setting

η :=Q_d^t2₂ ≪1 and λ:= ^t2

d²ln_η¹ >0,

thenλ=O(1)is a tuning parameter in the system, and the rescaled energy, which is to be minimized, takes the form of energy E_η given in (3) under the constraint

m: Ω =R×(−1,1)→S², m(±∞,·) =m^±_α,

h=−∇u:R² →R², ∇ ·(h+m^′1_Ω) = 0 inD^′(R²).

Observe that the parameter λ measures the film thickness t relative to the film thickness dln¹² _Q¹ characteristic to the cross-over. The limit η ↓0 corresponds to a limit of vanishing strength of anisotropy, while at the same time the relative film thickness _d^t increases in order to remain in the critical regime of the cross-over.

Other microstructures in micromagnetics. In other asymptotic regimes, different pat- tern formation is observed. Let us briefly mention three other microstructures that were recently studied: the concertina pattern, the cross-tie wall and a zigzag pattern.

Concertina pattern. In a series of papers ([20, 23] among others) the formation and hysteresis of the concertina pattern in thin, sufficiently elongated ferromagnetic samples were studied.

While in this case the transition layers between domains of constant magnetization are symmetric Néel walls, the program carried out for the concertina (a mixture of theoretical and numerical analysis, and comparison to experiments) serves as motivation for our work on the energy of domain walls in moderately thin films. Moreover, we hope that our analysis of the wall energy is helpful for studying a different route to the formation of the concertina pattern in not too elongated samples as proposed in [24], see also [6].

Cross-tie wall. An interesting transition layer observed in physical experiments is the cross-tie wall (see [10, Section 3.6.4]). It was rigorously studied in a reduced2Dmodel (by assuming vertical invariance of the magnetization) where a forcing term amounts to strong planar anisotropy that dominates the stray-field energy (see [1, 21, 22]). For small wall angles θ∈(0,^π₄], the optimal transition layer is given by the symmetric Néel wall; for larger angles

θ > ^π₄, the domain wall has a two-dimensional profile consisting in a mixture of vortices and Néel walls. The energetic cost of a transition in this2Dmodel is proportional tosinθ−θcosθ, so it is cubic inθ as θ→ 0. This is due to the scaling of the stray-field energy (because of the thickness invariance assumption), which makes this reduced 2D model seem artificial.

In the physics literature, it is known that for the full 3D model and large wall angles the cross-tie wall may also be favored over the asymmetric Bloch wall. We hope that our more realistic wall-energy density confirms and helps to quantify this issue.

A zigzag pattern. In thick films, zigzag walls also occur. This pattern has been studied by Moser [18] in a3Dmodel with a uniaxial anisotropy in an external magnetic field perpendic- ular to the “easy axis” (rather similar to our model). In fact, zigzag walls are to be expected there; however, this question is still open since the upper bound given for the limiting wall energy through a zigzag construction does not match the lower bound. Recently, in a reduced 2D model, Ignat and Moser [13] succeeded to rigorously prove the optimality of the zigzag pattern (for small wall angles). This was due to the improvement of the lower bound based on an entropy method (coming from scalar conservation laws). Remarkably, the function sinθ−θcosθplays an important role for the limiting energy density in that context as well as for the cross-tie wall.

3 Compactness and existence of minimizers

In this section we prove compactness results for sequences {m_k}k↑∞ of magnetizations of bounded exchange energy. As an application we will derive existence of minimizers of E_η (for some fixed η ∈ (0,1)) and E_asym(θ) subject to a prescribed wall angle θ ∈ (0, π), and show that the optimal angle inE₀ is attained (cf. (14)).

All these statements are rather straightforward up to one point: The condition of sign-change

±m₂(±∞,·)≥0

can in general not be recovered in the limit as shown in Figure 4.

¯ m2,η

x1

Figure 4: Thex₃-averagem¯_2,ηof them₂-component. The arrow←→denotes that the length of the corresponding interval grows to +∞ asη ↓ 0. Then the limit m¯₂ (as η ↓ 0) has the same sign at+∞ and−∞.

However, we will show that one can always choose zeros x_1,η of m¯_2,η in such a way that m_η(·+x_1,η,·) has the correct change of sign in the limit η↓0.

In the sequel we denote by C > 0 a universal, generic constant, whose value may change from line to line, unless otherwise stated.

3.1 Compactness

We start by proving the1D concentration-compactness result stated in Lemma 1.

Proof of Lemma 1: Due to (10), the set Z_k := {z ∈ R

u_k(z) = 0} of zeros of u_k is non- empty, and up to a translation in x₁-direction we may assume u_k(0) = 0for allk∈N. Step 1: For every sequence {z_k ∈ Z_k}k↑∞ there exist a subsequence Λ ⊂ N and a limit u:R →R such that u_k(·+z_k)→ u locally uniformly for k↑ ∞, k∈Λ. Moreover, we have the bound

Z

R|_ds^du|²ds≤lim inf

k↑∞

k∈Λ

Z

R|_ds^du_k|²ds <∞. Indeed, by Cauchy-Schwarz’s inequality, we have fort6= ˜tthat

|u_k(t)−u_k(˜t)|²

|t−˜t| = Rt

˜t d

dsu_kds2

|t−t˜| ≤ Z

R|_ds^du_k|²ds;

thus, by (9), we deduce that{u_k(·+z_k)}k↑∞ is uniformly Hölder continuous with exponent

1

2. In particular, since u_k(z_k) = 0, we also have that {u_k(·+z_k)}k↑∞ are locally uniformly bounded. Hence, the Arzelà-Ascoli compactness theorem yields uniform convergence on each compact interval [−n, n],n∈N, up to a subsequence. By a diagonal argument, one finds a subsequence Λ⊂N and a continuous limitu:R→R such that

u_k(·+z_k)→ulocally uniformly for k↑ ∞,k∈Λ.

Moreover, theL²(R)-estimate on _ds^dufollows from weak convergence inL² of _ds^du_kand weak lower-semicontinuity of theL² norm.

Step 2:Inductive construction of zeros. Assume by contradiction that for every sequence {z_k ∈ Z_k}k↑∞, no accumulation point u (w.r.t. to locally uniform convergence) of the se- quence{u_k(·+z_k)}k↑∞satisfies (11). We will show by an iterative construction that one can select a subsequence of{u_k}k↑∞ such that each termu_k has asymptotically infinitely many zeros (i.e., #Z_k → ∞ask↑ ∞) with large distances in-between.

More precisely, we prove that for everyl∈Nthere exist a limitu^l∈H˙¹(R)and subsequences Λ_l ⊂Λ_l−1 ⊂. . .⊂Λ₁ ⊂N, such that for allk∈Λ_l there exists an additional zero z^l_k∈Z_k ofu_k with the properties:

1≤mini6=j≤l|zⁱ_k−z_k^j| → ∞ and u_k(·+z_k^l)→u^l locally uniformly, as k↑ ∞, k∈Λ_l. In Step 3, we finally show that this construction implies thatu≡0is one of the accumulation points of {u_k(·+z_k)}k↑∞ for z_k ∈Z_k a diagonal sequence of these z_k^l, i.e., (11) is satisfied, in contradiction to our assumption.

At levell= 1, we choose the zeroz_k¹ = 0ofu_kfor every k∈N. Then by Step 1, there exists a subsequenceΛ₁⊂N and a limit u¹ ∈H˙¹(R) such that

u_k(·+z_k¹)→u¹ locally uniformly for k↑ ∞,k∈Λ₁.

By assumption,u¹ does not satisfy (11). Hence, there existsε₁ >0such that for everys >0 we can finds₁> s such that

u¹(s₁)≤ −ε₁<0 or u¹(−s₁)≥ε₁>0.

By uniform convergence, we also deduce that for every s >0 there exists an index k_s ∈Λ₁ such that

sup

[−s1,s1]|u_k(·+z_k¹)−u¹| ≤ ^ε₂¹ for k≥k_s, k∈Λ₁, which in particular implies that

u_k(s₁+z_k¹)<0 or u_k(−s₁+z¹_k)>0, for k≥k_s, k∈Λ₁. (21) At levell= 2, we proceed as follows: By the construction at levell= 1, for everys:=n∈N we choose as above s₁ ≥ n and k := k_n ∈ Λ1 (here, {k_n}n↑∞ is to be chosen increasing).

We also know thatu_k satisfies (11) which implies by (21) thatu_kchanges sign at the left of

−s₁+z_k¹ or at the right ofs₁+z¹_k. Choosez_k²∈Z_k as this new zero ofu_k. Sincez_k¹

n = 0, we have

|z¹_kn−z_k²_n| → ∞ asn↑ ∞.

Let Λ˜2 = {k_n|n∈N} ⊂ Λ1 be the sequence of these indices. By Step 1, there exist a subsequence Λ₂ ⊂Λ˜₂ and a limit u² ∈H˙¹(R) such that

u_k(·+z_k²)→u² locally uniformly for k↑ ∞,k∈Λ2.

We now show the general construction, i.e. how one obtains the(l+ 1)^th set of zeros from the construction after thel^th step. Indeed, suppose the functions u¹, . . . , u^l, the sequences Λ_l ⊂ . . . ⊂ Λ₁ ⊂ N and the zeros z¹_k, . . . , z_k^l of u_k for every k ∈ Λ_l have already been constructed. We now construct u^l+1, Λ_l+1 and z_k^l+1 for k ∈ Λ_l+1: By assumption, none of the limits u^j,1 ≤j ≤l, satisfies (11). Hence, there exists ε_l >0 such that for every s >0 we can finds₁, . . . , s_l≥s with the property:

u^j(s_j)≤ −ε_l<0 or u^j(−s_j)≥ε_l>0

for every 1≤j≤l.

By uniform convergence, we also deduce that for every s >0 there exists an index k_s ∈Λ_l such that for every 1≤j≤l and everyk≥k_s withk∈Λ_l:

sup

[−sj,sj]|u_k(·+z_k^j)−u^j| ≤ ^ε₂^l and min

1≤i6=j≤l|z_kⁱ −z_k^j| ≥4 max

1≤j≤ls_j.

In particular, for every s := n ∈ N we choose as above s₁, . . . , s_l ≥ n and k := k_n ∈ Λ_l (again, {k_n}n↑∞ is to be chosen increasing). Then we deduce that for all 1 ≤ j ≤ l and k∈Λ_l:

u_k(s_j+z_k^j)<0 or u_k(−s_j+z_k^j)>0, and thel intervals{I_j := [z^j_k−s_j, z_k^j +s_j]}1≤j≤l are disjoint.

Since u_k satisfies (11), there exists a new zero z_k^l+1 ∈ Z_k \Sl

j=1I_j of u_k. Indeed, let us assume (after a rearrangement) that these intervals are ordered I₁ < I₂ <· · ·< I_l. If there is no zero to the left of I₁ (i.e., on(−∞, z¹_k−s₁]) and in-between these l intervals (i.e., on Sl−1

j=1[z_k^j+s_j, z_k^j+1−s_j+1]), thenu_k must have a negative sign at the right endpoint of each intervalI_j (i.e.,u_k(z^j_k+s_j)<0) with1≤j≤l. In particular, there must be a zero of u_k at the right ofI_l, that we call z_k^l+1.

SetΛ˜_l+1 ={k_n|n∈N} ⊂Λ_l. Then

1min≤j≤l|z_k^j −z_k^l+1| → ∞ ask↑ ∞, k∈Λ˜_l+1. Finally, by Step 1, there existΛ_l+1⊂Λ˜_l+1 and u^l+1 such that

u_k(·+z^l+1_k )→u^l+1 locally uniformly inR ask↑ ∞, k ∈Λ_l+1, which finishes the construction at the level l+ 1.

Step 3:Construction of vanishing diagonal sequence.We prove that the assumption in Step 2 (i.e. the assumption that no accumulation point of a sequence of translates of {u_k}k↑∞

satisfies (11)) leads to a contradiction:

Consider the construction done in Step 2. The sequence {u^l}l↑∞ is uniformly bounded in H˙¹(R). Hence, as in Step 1, there is a subsequenceΛ⊂N and a functionusuch thatu^l→u locally uniformly forl↑ ∞,l∈Λ. In the following, we prove that u≡0 onR (in particular (11) is satisfied). Indeed, we first observe that 0 = u^l(0) → u(0) as l ↑ ∞, l ∈ Λ; thus, u(0) = 0. Let nowa >0and we want to prove thatu(a) = 0. For that, letl∈Λand k∈Λ_l. Then for1≤j≤l,

|u_k(a+z_k^j)|²

a = |u_k(a+z^j_k)−u_k(z^j_k)|²

a ≤

Z a+z^j_k

z_k^j |_ds^du_k|²ds.

Fork=k(a)∈Λ_l sufficiently large, the intervals{[z_k^j, a+z^j_k]}1≤j≤l are disjoint and we have X

1≤j≤l

|u_k(a+z_k^j)|²

a ≤ X

1≤j≤l

Z a+z_k^j

z_k^j |_ds^du_k|²ds≤ Z

R|_ds^du_k|²ds.

Lettingk↑ ∞,k∈Λ_l, it follows X

1≤j≤l

|u^j(a)|²

a ≤lim sup

k↑∞

Z

R|_ds^du_k|²ds <∞.