A compactness result for Landau state in thin-film micromagnetics

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www.elsevier.com/locate/anihpc

A compactness result for Landau state in thin-film micromagnetics

Radu Ignat

^a,^∗

, Felix Otto

^b

aLaboratoire de Mathématiques, Université Paris-Sud 11, bât. 425, 91405 Orsay, France bMax Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany

Received 7 September 2010; accepted 2 January 2011 Available online 21 January 2011

Abstract

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parametersεandηand defined over vector fieldsm:Ω⊂R²→S²that are tangent at the boundary∂Ω. We are interested in the behavior of minimizers asε, η→0. They tend to be in-plane away from a region of length scaleε(generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so thatS¹-transition layers of length scaleη(Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state.

Our main result concerns the compactness of vector fields{m_ε,η}ε,η↓0of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg–Landau type problems for the concentration of energy on vortex balls, together with an approximation argument ofS²-vector fields byS¹-vector fields away from the vortex balls.

MSC:primary 49S05; secondary 82D40, 35A15, 35B25

Keywords:Compactness; Singular perturbation; Vortex; Néel wall; Micromagnetics; Ginzburg–Landau energy

1. Introduction

In this paper, we investigate a common pattern of the magnetization in thin ferromagnetic films, called Landau state, that corresponds to the global minimizer of the micromagnetic energy in a certain regime. For that, we focus on a toy problem rather than on the full physical model:

LetΩ⊂R²be a bounded simply-connected domain with aC^1,1boundary corresponding to the horizontal section of a ferromagnetic cylinder of small thickness. Due to the thin film geometry, the variations of the magnetization in the thickness direction are strongly penalized. It motivates us to consider magnetizations that are invariant in the out-of-plane variable, i.e.,

m=(m₁, m₂, m₃):Ω→S²

* Corresponding author.

E-mail addresses:Radu.Ignat@math.u-psud.fr (R. Ignat), otto@mis.mpg.de (F. Otto).

doi:10.1016/j.anihpc.2011.01.001

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and they are tangent to the boundary∂Ω, i.e.,

m·ν=0 on∂Ω, (1)

wherem=(m1, m2)is the in-plane component of the magnetization andνis the normal outer unit vector to∂Ω. We consider the following micromagnetic energy functional:

E_ε,η(m)=

Ω

|∇m|²dx+ 1 ε²

Ω

m²₃dx+1 η

R²

|∇|⁻^1/2

∇ ·m²dx,

whereε andηare two small positive parameters (standing for the size of the vortex core and the Néel wall core, respectively). Here,x=(x1, x2)are the in-plane variables with the differential operator

∇ =(∂_x₁, ∂_x₂).

The first term ofE_ε,η(m)stands for the exchange energy. The second term corresponds to the stray-field energy penalizing the top and bottom surface chargesm₃of the magnetic cylinder, while the last term counts the stray-field energy penalizing the volume charges∇ ·mwhere we will always think of

m≡m1Ω

as being extended by 0 outsideΩ. For more physical details, we refer to Section 3.

Note that the nonlocal term in the energy is given by the homogeneousH˙⁻^1/2-seminorm of the in-plane divergence

∇ ·mthat writes in the Fourier space as:

∇ ·m²_˙

H⁻^1/2(R²)=

R²

|∇|⁻^1/2

∇ ·m²dx:=

R²

1

|ξ|F

∇ ·m²dξ. (2)

Also observe that the boundary condition (1) is necessary so that (2) is finite since

∇ ·m=

∇ ·m 1Ω+

m·ν

1∂Ω inR² (see Proposition 2 in Appendix A).

We are interested in the asymptotic behavior of minimizers of the energyE_ε,ηin the regime ε 1 and η 1.

The main features of this variational model resides in the nonconvex constraint on the magnetization |m| =1 and the nonlocality of the stray-field interaction. The competition of these effects with the quantum mechanical exchange effect leads to a rich pattern formation for the stable states of the magnetization. Generically, a pattern of a stable state consists in large uniformly magnetized regions (magnetic domains) separated by narrow smooth transition layers (wall domains) where the magnetization varies rapidly. The characteristic wall domains observed in thin ferromagnetic films are the Néel walls (corresponding to a one-dimensional in-plane rotation connecting two directions of the magnetization) together with topological defects standing for interior vortices (called Bloch lines) and micromagnetic boundary vortices.

The existence of line singularities at the mesoscopic level of the magnetization in thin films can be explained by the principle of pole avoidance. For this discussion, we first neglect the exchange term inE_ε,η. The stray-field energy will try to enforce in-plane configurations, i.e.,m₃=0 inΩ, together with the divergence-free condition form, i.e.,

∇ ·m=0 inΩ. Together with (1), we arrive at

m=1, ∇ ·m=0 inΩ and m·ν=0 on∂Ω. (3) We notice that the conditions in (3) are too rigid for smooth magnetizationm. This can be seen by writingm= ∇^⊥ψ with the help of a “stream function”ψ. Then up to an additive constant, (3) implies thatψis a solution of the Dirichlet problem for the eikonal equation:

|∇ψ| =1 inΩ and ψ=0 on∂Ω. (4)

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The method of characteristics yields the nonexistence of smooth solutions of (4). But there are many continuous solutions that satisfy (4) away from a set of vanishing Lebesgue measure. One of them is the “viscosity solution”

given by the distance function ψ (x)=dist

x, ∂Ω

that corresponds to the so-called Landau state for the magnetization m. Hence, the boundary conditions (1) are expected to induce line-singularities for solutionsmthat are an idealization of wall domains at the mesoscopic level.

At the microscopic level, they are replaced by smooth transition layers, the Néel walls, where the magnetization varies very quickly on a small length scale η. Note that the normal component of m does not jump across these discontinuity lines (because of (3)); therefore, the normal vector of the mesoscopic wall is determined by the angle between the mesoscopic levels of the magnetization in the adjacent domains (called angle wall). Now, taking into account the contribution of the exchange effect, the energy scaling per unit length of a Néel wall of angle 2θ (with θ∈(0,^π₂]) is given in DeSimone, Kohn, Müller and Otto [7], Ignat and Otto [11] (see also Ignat [8]):

π(1−cosθ )²+o(1)

η|logη| asη→0. (5)

The formation of interior or boundary vortices is explained by the competition between the exchange energy and the penalization of them₃-component for configurations tangent at the boundary. Indeed, there is noS¹-configuration that is of finite exchange energy and satisfies (1). There are only two possible situations: Ifmdoes not vanish on∂Ω, than (1) implies thatmcarries a nonzero topological degree, deg(m, ∂Ω)= ±1. In this case, we expect the nucleation of an interior vortex of core-scaleε. The scaling of the vortex energy is related to the minimal Ginzburg–Landau (GL) energy (see Béthuel, Brezis and Hélein [1]):

min

m∈H¹(Ω,R²) m=ν^⊥on∂Ω

Ω

g_ε m

dx=

2π+o(1)

|logε| asε→0, (6)

where the GL density energy is given in the following:

gε

m

=∇m²+ 1 ε²

1−m²2

. (7)

(Here, we denoteν^⊥=(−ν₂, ν₁).) The second situation consists in having zeros ofm on the boundary. Therefore, we expect that boundary vortices do appear. Roughly speaking, they correspond to “half” of an interior vortex where the vector fieldmis tangent at the boundary; therefore they are different from the micromagnetic boundary vortices analyzed by Kurzke [14] and Moser [16] (see details in Section 3). Remark the importance of the regularity of∂Ω in estimate (6). In fact, if∂Ω has a corner and the boundary conditionm=ν^⊥ on∂Ω in (6) is relaxed to (1), then estimate (6) does not hold anymore, it depends on the angle of the corner (see Proposition 1 and Remark 2). Therefore, at the microscopic level, topological point defects do appear in the Landau state pattern and are induced by (1).

The aim of the paper is to show compactness of magnetizations of energyE_ε,ηclose to the Landau state in order to rigorously justify the limit behavior (3): the delicate issue consists in having the constraint|m| =1 conserved in the limit. For that, we have to evaluate the energetic cost of the Landau state. We expect that the leading order energy of a Landau state is given by the topological point defects and Néel walls. The Landau state configuration consists in several Néel walls and either one interior Bloch line or two “half” Bloch lines placed at the boundary of the sampleΩ. Therefore, by (5) and (6), we expect that the energy of the Landau state has the following order:

2π|logε| + A

η|logη|, (8)

for some positiveA >0 depending on the length and angle of Néel walls.

2. Main results

First of all, we want to rigorously prove the upper bound (8) for the Landau state. Our result gives the exact leading order energy of the Landau state in the case of a domainΩ of a “stadium” shape (see Fig. 1). Note that the Landau state of a stadium consists in a single Néel wall of 180^◦(in our example, the length of the wall is equal to 2, so that A=2π in (8)).

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Fig. 1. Stadium.

Theorem 1.LetΩ=Ω₁∪Ω₂∪Ω₃be the following “stadium” shape domain:

Ω₁=

x=(x₁, x₂)∈R²: x−(1,0)<1, x11 , Ω₂=(−1,1)×(−1,1),

Ω₃=

x=(x₁, x₂)∈R²: x−(−1,0)<1, x1−1 .

In the regimeε η 1, there exists aC¹vector fieldm_ε,η:Ω→S²that satisfies(1)and E_ε,η(m_ε,η)2π|logε| +2π+o(1)

η|logη| asη↓0. (9)

Observe that the vortex energy in the above estimate is relevant only if a vortex costs at least as much as a Néel wall, i.e., _η_|_logη¹ _||logε|(otherwise, the vortex energy would be absorbed by the termo(_η_|_log¹_η_|)). This regimes leads to a sizeεof the vortex core exponentially smaller than the size of the Néel wall coreη(see Remark 1).

Notation.We always denotea bif ^a_b→0; also,abifaCbfor some universal constantC >0.

Now we state our main result on the compactness of theS²-valued magnetizations that have energies near the Lan- dau state. The issue consists in rigorously justifying that the constraint|m| =1 is conserved by the limit configurations asε, η→0. The regime where we prove our result corresponds to the case where a topological defect is energetically more expensive than the Néel wall, that is coherent with the regime where (9) holds.

Theorem 2.Letα∈(0,¹₂)be an arbitrary constant. We consider the following regime between the small parameters ε, η 1:

ε^1/2η, (10)

log|logε| 1

η|logη|. (11)

For eachεandη, we considerC¹vector fieldsmε,η:Ω→S²that satisfy(1)and

E_ε,η(m_ε,η)−2π|logε| 2π α|logε|, (12)

_η_|_logη¹ _|. (13)

Then the family{m_ε,η}ε,η↓0is relatively compact inL¹(Ω, S²)and any accumulation pointm:Ω→S²satisfies m₃=0, m=1 a.e. inΩ and ∇ ·m=0 distributionally inR². (14) The proof of compactness is based on an argument of approximatingS²-valued vector fields byS¹-valued vector fields away from a small defect region. This small region consists in either one interior vortex or two boundary vortices. The detection of this region is done in Theorem 3 and uses some topological methods due to Jerrard [12]

and Sandier [18] for the concentration of the Ginzburg–Landau energy around vortices (see also Lin [15], Sandier and Serfaty [19]). Away from this small region, the energy level onlyallows for line singularities. Therefore, the compactness result forS¹-valued vector fields in [11] applies.

Let us discuss the assumptions (10), (11), (12) and (13). Inequality (13) assures that cutting out the topological defect (one vortex or two boundary vortices), the remaining energy rescaled at the energetic level of Néel walls is uniformly bounded. Inequality (12) together with the choice of α < ¹₂ mean that the energy cannot support three

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“half” interior vortices and is precisely explained in Theorem 3 below. Inequality (11) is imposed due to our method to detect a boundary vortex: it leads to a loss of energy of orderO(log|logε|)with respect to the expected half energy of an interior vortex, i.e.,π|logε| (see Theorem 3 and Proposition 1). This amount of energy could leave room for configurations of Néel walls that may destroy the compactness of|m| =1. Therefore, to avoid this scenario, (11) is imposed. The regime (10) is rather technical: it is needed in the approximation argument ofS²-valued vector fields byS¹-valued vector fields away from the vortex balls. In fact, starting from the values ofmon a square grid of size ε^β, the approximation argument requires zero degree of m on each cell, leading to the conditionβ <1−α(see Lemma 2); furthermore, the conditionε^βηis needed in order that the approximatingS¹-valued vector fields induce a stray field energy of the same order ofm(see (77)). Therefore, (10) can be improved to a larger regime

ε^βη for anyβ <1−α

as presented in the proof (Theorem 2 is stated for the valueβ=1/2 which is the universal choice for everyα <1/2).

However, this slightly improved condition is weaker than the complete regime implied by (12) as explained in the following remark.

Remark 1.Any limit configurationmsatisfies (14). IfΩis a bounded simply-connected domain different than discs, m has at least one ridge (line-singularity) that corresponds to a Néel wall. Therefore, the minimal energy verifies min(1)E_ε,η−2π|logε|_η_|_logη¹ _|. Combining with (12), it follows that

1

η|logη||logε|;

in particular,εe⁻

1

η|logη|, i.e., the core of the vortex is exponentially smaller than the core of the Néel wall. However, in the proof of Theorem 2, this much stronger constraint with respect to (10) is not needed.

We prove the following result of the concentration of Ginzburg–Landau energy around one interior vortex or two boundary vortices for vector fields tangent at the boundary:

Theorem 3.Letα∈(0,¹₂)andΩ⊂R²be a bounded simply-connected domain with aC^1,1boundary. There exists ε₀=ε₀(α, ∂Ω) >0such that for every0< ε < ε₀, ifm:Ω→B²is aC¹vector field that satisfies(1)and

Ω

g_ε m

dx2π(1+α)|logε|, (15)

then there exists either a ballB(x₁^∗, r^∗)⊂Ω(called vortex ball)withr^∗=_|_logε¹_|3 and

B(x^∗₁,r∗)

g_ε m

dx2π logr^∗

ε

−C, (16) or two ballsB(x₂^∗, r^∗)andB(x₃^∗, r^∗)(called boundary vortex balls)withx₂^∗, x₃^∗∈∂Ωand

(B(x^∗₂,r^∗)∪B(x₃^∗,r^∗))∩Ω

g_ε m

dx2π logr^∗

ε

−C, (17)

whereC=C(α, ∂Ω) >0is a constant depending only onαand on the geometry of∂Ω.

The conditionα <1/2 is needed in our proof. In fact, if no topological defect exists in the interior (in which case, condition (1) induces boundary vortices), we perform a mirror-reflection extension ofmoutside the domain. Roughly speaking, the GL energy in the extended domain doubles, i.e., it is of order 2π(2+2α)|logε|and the degree at the new boundary is equal to two; in order to avoid the formation of three interior vortices in the extended region, we should impose 2+2α <3, i.e.,α <1/2.

Notice that the Ginzburg–Landau energy concentration for a boundary vortex in (17) has a cost of orderπ|logε| − Clog|logε|provided that the boundary has regularityC^1,1. We conjecture that the same energetic cost for a boundary

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vortex holds true if the boundary has regularityC^1,β,β∈(0,1). However, if the boundary regularity is onlyC¹, then the energetic cost of a boundary vortex may decrease to(π−_log_|^C_log_ε_|)|logε| whereC >0 is a universal constant.

Proposition 1.We consider in polar coordinates the following C¹ domainΩ = {(r, θ ): r∈(0,₂₀¹), |θ|< γ (r)=

π

2 −_{log log}¹ 1 r

}. For every0< ε <1, there exists aC¹-functionm_ε:Ω∩B_1/200→R²that satisfies(1)on∂Ω∩B_1/200

and

Ω∩B1/200

g_ε m_ε

dx

π− C

log|logε|

|logε|,

whereC >0is some universal positive constant(independent ofε).

The outline of the paper is as follows. In Section 3, we present the physical context of our toy problem. In the next section, we recall two results that we need for the proof of our results: a compactness result forS¹-valued magnetizations and the concentration of the Ginzburg–Landau energy on vortex balls. In Section 5, we prove Theorem 3 and Proposition 1. In Section 6, we give the proof of our main result in Theorem 2. In Section 7, we show the upper bound for the stadium domain stated in Theorem 1. In Appendix A, we prove that (1) is a necessary condition for our configurations to have a finite stray field energy.

3. Physical context

In this section we explain the physical context of our model in thin-film micromagnetics. We consider a ferromagnetic sample of cylinder shape, i.e.

ω=ω×(0, t )

whereω⊂R²is the section of the magnetic sample of lengthandtis the thickness of the cylinder. The microscopic behavior of the magnetic body is described by a three-dimensional unit-length vector fieldm=(m, m₃):ω→S², called magnetization. The observed ground state of the magnetization is a minimizer of the micromagnetic energy that we write here in the absence of anisotropy and external magnetic field:

E^3d(m)=d²

ω

∇, ∂

∂z

m

²dx dz+

R³

∇, ∂

∂z

U (m)

²dx dz. (18)

The parameter d of the material is called exchange length and is of order of nanometers. The stray-field potential U (m):R³→Ris defined by static Maxwell’s equation in the weak sense:

R³

∇, ∂

∂z

U (m)·

∇, ∂

∂z

ζ dx dz=

R³

∇, ∂

∂z

·(m1ω)ζ dx dz, for everyζ ∈C_c^∞ R³

. (19)

Instead of the three length scales,tandd of the physical model, we introduce two dimensionless parameters:

ε:=d

and η:=d² t

(standing for the size of the core of the Bloch line and the Néel wall, respectively).

3.1. Thin-film reduction

We consider the thin-film approximation of the full energy (18) in the following regime:

ε η 1 (20)

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(equivalently,t d ). The assumptiont dimplies that in-plane transitions (Néel walls) are preferred to out-of- plane transitions (asymmetric Bloch walls) between two mesoscopic directions of the magnetization (see Otto [17]).

The hypothesisd assures that constant configurations in general are not global minimizers (see DeSimone [4]).

The main issue is the asymptotic behavior of the energy in the regime of thin films. We first nondimensionalize in length with respect to, i.e.(x,¯ z)¯ =(^x,^z),Ω=^ω,m(¯ x,¯ z)¯ =m(x, z),U (¯ m)(¯ x,¯ z)¯ =¹U (m)(x, z)and then we renormalize the energyE¯^3d(m)¯ =_d¹2tE^3d(m). Omitting the¯, we get

E^3d(m)= η ε²

Ω×(0,^ε_η²)

∇, ∂

∂z

m

²dx dz+ η ε⁴

R³

∇, ∂

∂z

U (m)

²dx dz. (21)

In the regime (20), the penalization of exchange energy enforces the following constraints for the minimizers:

(a) mvaries on length scales^ε_η². (b) m=m(x), i.e.misz-invariant.

With these assumptions, (21) can be approximated by the following reduced energyE^red (see DeSimone, Kohn, Müller and Otto [6], Kohn and Slastikov [13]):

E^red(m)=

Ω

|∇m|²dx

+ 1 ε²

Ω

m²₃dx+|log^ε_η²| 2π η

∂Ω

m·ν2

dH¹+ 1

2η∇ ·m

ac²_˙

H⁻^1/2(R²). (22)

The above formula follows by solving the stray field equation (19) in the regime (20): indeed, forz-invariant con- figurationsm, the Fourier transform in the in-plane variablesx=(x1, x2)turns (19) into a second order ODE in the z-variable that can be solved explicitly (see [13,9]). Then, due to the above assumption (a) and to the regime (20), the stray-field energy asymptotically decomposes into three terms as written in (22): the first term in (22) is penalizing the surface chargesm3on the top and bottom of the cylinder, a second term counts the lateral chargesm·ν in the L²-norm, as well as the third term that penalizes the volume charges(∇ ·m)ac:=(∇ ·m)1Ω as a homogeneous

˙

H⁻^1/2-seminorm. In fact, the last term corresponds to the stray-field energy created by a three-dimensional vector fieldh_ac(m)defined as

hac(m)=

∇, ∂

∂z

Uac(m):R³→R³, that satisfies:

R³

∇, ∂

∂z

Uac(m)·

∇, ∂

∂z

ζ dx dz=

R²

∇ ·m

acζ dx, for allζ∈C_c^∞ R³

.

Then one has

R³

hac(m)²dx dz=1

2(∇ ·m)ac²_˙

H⁻^1/2(R²). (23)

Note that if (1) holds (i.e., no lateral surface charges), then(∇ ·m)ac= ∇ ·(m1Ω)and therefore,hac(m)induces the stray field energy given by (2). In fact, (2) corresponds to the minimal stray field energy in thin films. More precisely, a stray field h=(h, h3)=(h1, h2, h3):R³→R³ is related to the magnetization m:Ω →S² via the following variational formulation:

R²×R

h· ∇ζ+h₃∂ζ

∂z

dx dz=

R²

ζ∇ ·mdx, ∀ζ∈C_c^∞ R³

, (24)

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Fig. 2. Néel wall of angle 2θconfined in[−1,1].

wherezdenotes the out-of-plane variable in the spaceR³. (As before,m≡m1Ω andmsatisfies (1).) Classically, this is,

⎧⎨

⎩

∇ ·h+∂h3

∂z =0 inR³\

R²× {0}

, [h₃] = −∇ ·m onR²× {0},

where[h3]denotes the jump of the out-of-plane component ofhacross the horizontal planeR²× {0}. Then (2) can be expressed as:

R²

|∇|⁻^1/2

∇ ·m²dx=2 min

hwith(24)

R²×R

|h|²dx dz.

Therefore,h_ac(m)is a minimizing stray-field (of vanishing curl) associated with the stray field potentialU_ac(m).

In our regime (20), there are three different structures that typically appear: Néel walls, Bloch lines and micromagnetic boundary vortices. We explain these structures in the following and compare their respective energies. As we already mentioned, a fourth structure, the asymmetric Bloch wall, can appear in thicker films but we do not discuss it here since the asymmetric Bloch wall is more expensive than a Néel wall ift d.

3.2. Néel walls

The Néel wall is a dominant transition layer in thin ferromagnetic films. It is characterized by a one-dimensional in-plane rotation connecting two (opposite) directions of the magnetization. It has two length scales: a small core with fast varying rotation and two logarithmically decaying tails. In order for the Néel wall to exist, the tails are to be contained and we consider here the confining mechanism of the steric interaction with the sample edges. Typically, one may consider wall transitions of the form:

m=(m1, m2):R→S¹ and m(±t )= cosθ

±sinθ

for ±t1, withθ∈(0,^π₂](see Fig. 2), whereas the reduced energy functional is:

E^red(m)=

R

dm dx₁

²dx1+ 1 2η

R

d

dx₁ ^1/2m1

²dx1.

Asη→0, the scale of the Néel core is given by|x₁|w_core=O(η)while the two logarithmic decaying tails scale asw_core|x₁|w_tail=O(1). The energetic cost (by unit length) of a Néel wall is given by

E^red(Néel wall)=O 1

η|logη|

with the exact prefactorπ(1−cosθ )²/2 where 2θis the wall angle (see e.g. [8]).

3.3. Bloch line

A Bloch line is a regularization of a vortex on the microscopic level of the magnetization that becomes out-of-plane at the center. The prototype of a Bloch line is given by a vector field

m:B²→S²

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Fig. 3. Bloch line.

Fig. 4. A micromagnetic boundary vortex.

defined in a circular cross-sectionΩ=B²of a thin film and satisfying:

∇ ·m=0 inB² and m(x)=x^⊥ on∂B². (25)

(For the Bloch line in a thin cylinder, the magnetization is assumed to be invariant in the thickness direction of the film and the word “line” refers to the vertical direction.) Since the magnetization turns in-plane at the boundary of the diskB²(so, deg(m, ∂Ω)=1), a localized region is created, that is the core of the Bloch line of sizeε, where the magnetization becomes perpendicular to the horizontal plane (see Fig. 3). The reduced energy (22) for a configuration (25) writes as:

E^red(m)=

B²

|∇m|²dx+ 1 ε²

B²

m²₃dx.

The Bloch line corresponds to the minimizer of this energy under the constraint (25). Remark that the reduced energy E^redcontrols the Ginzburg–Landau energy, i.e.,

B²

g_ε m

dxE^red(m)

since|∇m|²|∇(m, m₃)|² and(1− |m|²)²=m⁴₃m²₃. Due to the similarity with the Ginzburg–Landau type functional, the Bloch line corresponds to the Ginzburg–Landau vortex and the energetic cost of a Bloch line (per unit-length) is given by (6):

E^red(Bloch line)=O

|logε| with the exact prefactor 2π(see e.g. [9]).

3.4. Micromagnetic boundary vortex

Next we address micromagnetic boundary vortices. A micromagnetic boundary vortex corresponds to an in-plane transition of the magnetization along the boundary from ν^⊥to−ν^⊥(see Fig. 4). The corresponding minimization problem is given by

E^red(m)=

Ω

|∇m|²dx+|log^ε_η²| 2π η

∂Ω

m·ν2

dH¹

within the set of in-plane magnetizationsm:Ω→S¹. The minimizer of this energy is a harmonic vector field with values inS¹driven by a pair of boundary vortices. These have been analyzed in [14,16]. The transition is regularized

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on the length scale of the exchange part of the energy, i.e. the core of the boundary vortex has length of size ^η

|log^ε_η²|. The cost of such a transition is given by

E^red(Micromagnetic boundary vortex)=O log η

|log^ε_η²|

with exact prefactorπ. (Note that the boundary vortices in Theorem 3 correspond in fact to “half” Bloch lines where the vector field is tangent at the boundary, i.e.,m·ν=0 on∂Ω; therefore, their structure is different from the one of micromagnetic boundary vortices, but with the same energetic cost.)

Claim.In the regime(20), then

either E^red(Micromagnetic boundary vortex)E^red(Néel wall) or E^red(Micromagnetic boundary vortex)E^red(Bloch line).

Indeed, assume by contradiction that the above statement fails. Then one has 1

η|logη| log η

|log^ε_η²|

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and log1

ε log η

|log^ε_η²|

. (27)

In the regime(20), one hasε² ε η, therefore(26)turns into 1

η|logη|log log1 ε, while(27)implies that

log1

ε log1 η.

Now it is easy to see the incompatibility between the last two inequalities asε, η→0.

3.5. Our toy problem

The model we presented in the introduction consists in considering configurations without lateral surface charges, i.e., (1) holds true. In this case, our energy functionalE_ε,2η(m)coincides with the reduced thin-film energyE^redsince h_ac(m)induces the stray field energy (23) as in (2). However, (1) would be physical relevant for a global minimizer onlyif boundary vortices were more expensive than both the Néel walls and Bloch line contribution. As explained in the above claim, this assumption is violated in the regime (20). Therefore, our energy functional is not adapted for studying global minimizers in the regime (20), but rather for metastable states that satisfy (1).

Recently, the regime E^red(Micromagnetic boundary vortex) E^red(Néel wall) E^red(Bloch line) was investi- gated in Ignat and Knüpfer [10] for thin films of circular cross-section. It is stated that the global minimal configuration for that geometry is given by a 360^◦-Néel wall that concentrates around a radius so that it becomes a vortex (the Lan- dau state of a disk) at the mesoscopic level.

4. Some preliminaries

The result stated in Theorem 2 is an extension to theS²-valued magnetizations of the following compactness result forS¹-valued magnetizations obtained by the authors in [11]:

(11)

Theorem 4.(See Ignat and Otto [11].) LetBⁿbe the unit ball inRⁿ,n=2,3. For every smallη >0, letm_η:B²→S¹ andh_η=(h_η, h_3,η):B³→R³be related by

B³

h_η· ∇ζ +h_3,η∂ζ

∂z

dx dz=

B²

ζ∇ ·m_ηdx, ∀ζ∈C_c^∞ B³

.

Suppose that

B²

∇ ·m_η²dx+1 η

B³

|h_η|²dx dz C

η|logη|, (28)

for some fixed constant C >0. Then {m_η}η↓0 is relatively compact in L¹(B²) and any accumulation point m:B²→R²satisfies

m=1 a.e. inB² and ∇ ·m=0 distributionally inB².

In the proof of Theorem 3, we will use the following result due to Jerrard [12] for the concentration of the GL energy (7) around vortices (see also Sandier [18], Lin [15]):

Theorem 5.(See Jerrard [12].) Letα∈ [0,1)and d >0be a positive integer. There exists ε0=ε0(d, α) >0such that for every0< ε < ε0, ifm:Ω→R²satisfies the following conditions:

m1

2 on

x∈Ω: dist(x, ∂Ω)r^∗(ε)

for somer^∗(ε)∈ 1

|logε|⁴,1

, (29)

deg

m, ∂Ω=d

and

Ω

g_ε m

dx2π(d+α)|logε|,

then there existnpointsx₁, . . . , x_n∈Ωwithdist(xj, ∂Ω) > r^∗(ε),j=1, . . . , nand positive integersd₁, . . . , d_n>0 such that thenballs{B(x_j, r^∗(ε))}1jnare disjoint,

n j=1

dj=d

and

B(x_j,r^∗(ε))

g_ε m

dx2π dj

logr^∗(ε) ε

−C(d, α), j=1, . . . , n,

whereC(d, α)is a constant only depending ond andα.

In the above theorem,Ω⊂R²is any open bounded set (without any regularity condition imposed for the boundary∂Ω). This is due to hypothesis (29) of having a security region around∂Ω. By degree of aC¹-functionv:C→S¹ defined on a closed curveC⊂R²with the unit tangential vectorτ, we mean the winding number

deg(v,C)= 1 2π

C

det(v, ∂τv) dH¹.

If m:C→R² is a C¹-function with |m|>0 on C, we set deg(m,C):=deg(_|^m_m_|,C). The notion of degree can be extended to continuous vector fields and more generally,VMOvector fields, in particularH^1/2(C, S¹)maps (see Brezis and Nirenberg [2]).

(12)

Fig. 5. Mirror-reflection extension.

5. Proof of Theorem 3 and Proposition 1

First of all, let us define the security region around∂Ω together with some notations that we use in the sequel:

Definition 1. LetΩ is a simply-connected bounded domain of C^1,1 boundary. Thesecurity regionaround∂Ω is the maximal set of points around∂Ω (in the interior and outsideΩ) covered by the normal lines at∂Ω before any crossing occurs. We calldepth of the security regionto be the smallest distance to the boundary∂Ωwhere a crossing of two normal lines occurs and it will be denoted byR(∂Ω).

LetR=R(∂Ω)be the depth of the security region around∂Ω. Forr∈(0, R), we denote the interior subdomain Ω_r⊂Ωat a distancerfrom the boundary, i.e.,

Ω_r=

x∈Ω: dist(x, ∂Ω) > r

and ∂Ω_r=

x∈Ω: dist(x, ∂Ω)=r

(30) be the boundary of this subdomain. Forr∈(−R,0), we write∂Ωr to be the symmetry of∂Ω₋r across the boundary

∂Ω=∂Ω0andΩr ⊃Ω be the extended domain surrounded by∂Ωr.

Letl=H¹(∂Ω)be the length of∂Ω. Setw:[0, l] →∂Ω be aC^1,1arclength parametrization of ∂Ω such that

| ˙w(s)| =1 withw(s)˙ =^dw_ds(s)and letν(s)= ˙w(s)^⊥ be the outer unit normal vector on∂Ω atw(s). Sincew(s)¨ =

d²w

ds²(s)is parallel toν(s)for a.e.s∈ [0, l], we will always write

¨

w(s)= ¨w(s)ν(s)

wherew(s)¨ is the signed length of the vectorw(s)¨ with respect toν(s). Notice that| ¨w(s)|_R(∂Ω)¹ . In the security region around∂Ω, a pointxwrites in the new coordinates as:

x=F (s, t )=w(s)+tν(s), s∈ [0, l], t∈

−R(∂Ω), R(∂Ω)

. (31)

Note that for interior points x ∈Ω, the corresponding normal coordinate t is negative. We define the symmetry transformΦ in the security region around∂Ω:

Φ F (s, t )

=F (s,−t ), s∈ [0, l], t∈

−R(∂Ω), R(∂Ω)

. (32)

A first ingredient that we need in the proof of Theorem 3 is a mirror-reflection extension across the boundary∂Ω. Lemma 1.LetR_∞>0. There existsε₀=ε₀(R_∞) >0such that for every0< ε < ε₀, the following holds:

LetΩbe a simply-connected bounded domain ofC^1,1boundary with the depth of the security regionR(∂Ω)R_∞. Let Φ be the symmetry transform across the boundary∂Ω defined in(32). In the security region, we consider the interior curve

γ=∂Ω 1

|logε|

(see notation(30))andm:Ω→B²is aC¹vector field that satisfies(1), m1/2 onγ and deg

m, γ

=0.

Then there exists an extension vector fieldm˜:Ω₋ 1

|logε|→R²ofm(see Fig.5)into the extended domainΩ₋ 1

|logε| ⊃Ω of boundary

˜

γ=Φ(γ )=∂Ω₋ 1

|logε|

(13)

such that

˜

m≡m inΩ, m˜1/2 onγ˜ and deg

˜ m,γ˜

=2,

˜ γ

g_ε

˜ m(y)

dH¹(y)−

γ

g_ε m(x)

dH¹(x) C

|logε|H¹(∂Ω)+ 1

|logε|

γ

g_ε m(x)

dH¹(x)

(33) and

Φ(W )

gε

m˜(y) dy−

W

gε

m(x) dx

C

H¹(∂Ω)+ 1

|logε|

W

gε

m(x) dx

(34) whereW⊂Ω\Ω 1

|logε| is any open subset ofΩ andC=C(R_∞)is a positive constant depending only onR_∞. Proof. We use the notations introduced at the beginning of this section. We have that| ¨w(s)|_R(∂Ω)¹ _R¹_∞. More- over, differentiating (31), we have that for a.e.s∈ [0, l]andt∈(−R(∂Ω), R(∂Ω)),

DF(s, t )=

αs(t )w(s)˙ ν(s)

and DF⁻¹(s, t )= 1

α_s(t )w(s)˙ ν(s) T

, (35)

where

α_s(t ):=1−tw(s).¨

By (32) and (35), we compute that:

Ss(t ):=DΦ(x)= 2

α_s(t )w(s)˙ ⊗ ˙w(s)−Id for a.e.s∈ [0, l]andt∈

−R(∂Ω), R(∂Ω)

. (36)

The matrixS_s(t )is symmetric and its inverse is given byS_s(t )⁻¹=S_s(−t ). The mirror-reflection extensionm˜ofm is defined as (see Fig. 5):

˜ m

Φ(x)

:=S_s(0)m(x)=2m(x)· ˙w(s)w(s)˙ −m(x) forx∈Ω\Ω_R(∂Ω). (37) (We use thata⊗bc=(b·c)a, for anya, b, c∈R².) Remark that the condition (1) implies that the mirror-reflection extension does not induce jumps at the boundary. Moreover,| ˜m(Φ(x))| = |m(x)|sinceSs(0)=2w(s)˙ ⊗ ˙w(s)−Id is a reflection matrix (i.e., it is symmetric and orthogonal). Therefore,| ˜m|1/2 onγ˜.

The goal is to estimate the energies

Φ(W )g_ε(m˜) dy and

˜

γg_ε(m˜) dH¹. We start by computing the Dirichlet energy of the extensionm˜. For that, we differentiate (37) in the coordinates(s, t ):

D

˜ m

Φ(x)

=S_s(0)Dm(x)DF(s, t )+2

V (s)m(x) 0 , where

V (s):= ˙w(s)⊗ ¨w(s)+ ¨w(s)⊗ ˙w(s). (38)

SinceD(m˜(Φ(x)))=Dm˜(Φ(x))DΦ(x)DF(s, t ), multiplying byDF(s, t )⁻¹S_s(−t ), it implies that Dm˜

Φ(x)(35), (36)

= S_s(0)Dm(x)S_s(−t )+ 2

αs(−t )V (s)m(x)⊗ ˙w(s).

Since Dm˜

Φ(x)T

=S_s(−t )Dm(x)^TS_s(0)+ 2

α_s(−t )w(s)˙ ⊗V (s)m(x), it follows that

Dm˜Φ(x)²=tr Dm˜

Φ(x) Dm˜

Φ(x)T

=tr

S_s(0)Dm(x)S_s(−t )²Dm(x)^TS_s(0)

+ 4

α_s(−t )²V (s)m(x)² + 4

α_s(−t )tr

Ss(0)Dm(x)Ss(−t )w(s)˙ ⊗V (s)m(x)

=I+II+III. (39)

(14)

For the first term in (39), we compute that Ss(−t )²⁽³⁶⁾= −4tw(s)¨

α_s(−t )²w(s)˙ ⊗ ˙w(s)+Id.

Sincetr(SAS⁻¹)=tr(A)andtr(Av⊗Av)= |Av|²|A|²|v|²for any two matricesAandSinR²^×²withSinvertible and any vectorv∈R², we deduce that

I

1+4|t|| ¨w(s)| αs(−t )²

Dm(x)². (40)

For the second term in (39), we have that|V (s)|²⁽³⁸⁾= 2| ˙w(s)⊗ ¨w(s)|²=2| ¨w(s)|²and therefore,

II 4

α_s(−t )²

V (s)²m(x)²8| ¨w(s)|² α_s(−t )²

m(x)². (41)

For the third term in (39), we compute that Ss(−t )w(s)˙ ⁽³⁶⁾= α_s(t )

α_s(−t )w(s)˙ and Ss(0)V (s)⁽³⁶⁾= ¨w(s)

0 1

−1 0

.

Using thattr(Ab⊗c)=c·Abandtr(A)=tr(Ss(0)ASs(0))for any matrixAinR²^×²and any vectorsb, c∈R², we deduce that

III= 4αs(t ) αs(−t )²t r

Dm(x)w(s)˙ ⊗

S_s(0)V (s)m(x)

= −4αs(t )w(s)¨ α_s(−t )² tr

Dm(x)w(s)˙ ⊗m(x)^⊥

= −4αs(t )w(s)¨

α_s(−t )² m(x)^⊥·

Dm(x)w(s)˙ 4αs(t )| ¨w(s)|

α_s(−t )²

m(x)Dm(x). (42)

Since|detDΦ(x)|⁽³⁶⁾= ^α_α^s⁽_s⁻_{(t )}^{t )}, we deduce by (39), (40), (41) and (42), Dm˜Φ(x)²detDΦ(x)α_s(−t )

α_s(t )

1+4|t|| ¨w(s)| αs(−t )²

Dm(x)²+ 8| ¨w(s)|²

α_s(−t )α_s(t )m(x)² +4| ¨w(s)|

αs(−t )m(x)Dm(x). (43)

Therefore, for every open setW⊂Ω\Ω 1

|logε|, we obtain by Young’s inequality,

Φ(W )

Dm˜(y)²dy =

W

detDΦ(x)Dm˜Φ(x)²dx

(43)

W

1+ C

|logε|

Dm(x)²+C|logε|m(x)² dx

1+ C

|logε|

W

Dm(x)²dx+CH¹(∂Ω),

withC=C(R_∞) >0 andεε(R_∞). (We use thatH²(W )_|_logε^C _|H¹(∂Ω).) Also,