Micromagnetics of curved thin films

In this paper, we aim at a reduced 2d-model describing the observable states of the magnetization in curved thin films. Under some technical assumptions on the geometry of the thin-film, it is well-known that the demagnetizing field behaves like the projection of the magnetization on the normal to the thin film. We remove these assumptions and show that the result holds for a broader class of surfaces; in particular, for compact surfaces. We treat both the stationary case, governed by the micromagnetic energy functional, and the time-dependent case driven by the Landau-Lifshitz-Gilbert equation.


Introduction and Physical Motivations
According to the Landau-Lifshitz theory of ferromagnetism (cf. [6,35,9,8,25,29]), the observable states of a rigid ferromagnetic body, occupying a region Ω ⊆ R 3 , are described by its magnetization M : Ω → R 3 , a vector field verifying the fundamental constraint of micromagnetism: there exists a material-dependent positive constant M s ∈ R + such that |M | = M s in Ω. In general, the spontaneous magnetization M s := M s (T ) depends on the temperature T and vanishes above a critical value T c , characteristic of each crystal type, known as the Curie temperature. Assuming the specimen at a fixed temperature well below T c , the value of M s can be considered constant in Ω. Therefore, the magnetization can be expressed in the form M := M s m where m : Ω → S 2 is a vector field taking values in the unit sphere S 2 of R 3 .
Although the modulus of m is constant in space, in general, it is not the case for its direction. For single crystal ferromagnets (cf. [1,3]), the observable magnetization states correspond to the local minimizers of the micromagnetic energy functional which, after a suitable normalization, reads as (cf. [35, p. 22] or [25, p. 138 with m ∈ H 1 (Ω, S 2 ) and mχ Ω the extension of m by zero outside Ω.
The variational problem (1) is non-convex, non-local, and contains multiple length scales.
The first term, E (m), is the exchange energy and penalizes nonuniformities in the orientation of the magnetization. It summarizes the effects of short-range exchange interactions among neighbor spins. The magnetocrystalline anisotropy energy, A (m), models the existence of preferred directions of the magnetization; the energy density φ an : S 2 → R + is a non-negative function that vanishes only on a finite set of directions, the so-called easy axes. The quantity W (m) represents the magnetostatic self-energy and describes the energy due to the demagnetizing field (stray field) h d [mχ Ω ] generated by mχ Ω ∈ L 2 (R 3 , R 3 ). The operator h d : m → h d [m] is, for every m ∈ L 2 (R 3 , R 3 ), the unique solution in L 2 (R 3 , R 3 ) of the Maxwell-Ampère equations of magnetostatics [9,27,16,34]: where b[m] denotes the magnetic flux density and µ 0 is the magnetic permeability of the vacuum. Finally, the term Z (m) is the Zeeman energy and models the tendency of the specimen to have its magnetization aligned with the (externally) applied field h a . The applied field is assumed to be unaffected by variations of m.
The dynamics of the magnetization is described by the Landau-Lifshitz-Gilbert (LLG) equation [29,20] ∂ m ∂t The LLG equation is driven by the effective field h eff [m] := −∂ m G Ω (m) and includes both conservative precessional and dissipative contributions; the constant α is the Gilbert damping constant.
The four terms in the energy functional (1) take into account effects originating from different spatial scales, such as short-range exchange forces and long-range magnetostatic interactions. The competition among the four contributions in (1) explains most of the striking pictures of the magnetization observable in ferromagnetic materials [25]; in particular, the domain structure already suggested in 1906 by Weiss [43], i.e., regions of uniform or slowly varying magnetization (magnetic domains) separated by thin transition layers (domain walls). Depending on the relations among the material and geometric parameters of the ferromagnetic particle, various asymptotic regimes arise, and their investigation can be efficiently carried out by the dimension reduction techniques of calculus of variations (see, e.g., [4,12,14,13,22,28,26,37]).

State of the art.
In the last decade, magnetic systems with the shape of a curved thin film have been subject to extensive experimental and theoretical research (nanotubes [23,30], 3d helices [40,31], thin spherical shells [36,39,17,32]). The embedding of two-dimensional structures in the three-dimensional space permits to alter the magnetic properties of the system by tailoring its local curvature. Even in the case of an anisotropic crystal, curved geometries can induce an effective antisymmetric interaction and, therefore, the formation of magnetic skyrmions or other topological defects. In addition to fundamental reasons, the wide range of magnetic properties emerging in this way makes these structures well-suited for technological applications, e.g., racetrack memory devices, spin-wave filters, and magnetoencephalography (see the topical review [41]).
Under technical (and strong) assumptions on the geometry of the thin-film, it is wellknown that the demagnetizing field behaves like the projection of the magnetization on the normal to the thin film. The first mathematical justification of this observation in micromagnetics emerged from the work of Gioia and James [22]; they showed that in planar thin films the effect of the demagnetizing field operator is represented by an easy-surface anisotropy term. A generalization of this result to the case of curved thin films is given in [10], although under the technical assumption that the thin geometry is generated by a surface diffeomorphic to the closed unit disk of R 2 . Also, in [38], a Γ-convergence analysis is performed on pillow-like shells, i.e., on shells of small thickness ε > 0 having the form with ω ⊆ R 2 and γ 1 , γ 2 functions vanishing on the boundary of ω. All these investigations, having a local character, do not cover significant scenarios like the one of a spherical thin film [36,39,17]. Lack of mathematical models in this context motivated our work [15], which addressed the question but under a convexity assumption on the surface generating the thin film.
Finally, in [16] we established three distinct variational principles for the magnetostatic self-energy that, through the explicit construction of suitable families of scalar and vector potential, permit to circumvent the technical difficulties in [10] and [15], at least in the stationary case. Indeed, it is essential to remark that the approach in [16], dealing with energy estimates rather than with the weak and strong convergence of the family of demagnetizing field operators (as we do in Lemma 1), is not suitable for the analysis of the time-dependent case.

Contributions of present work.
The main aim of this paper is to overcome the technical assumptions on the geometry of the thin-film required in [10] and [15]. We extend the validity of the reduced 2d models in [10] to the general framework of smooth (C 2 is sufficient) and bounded orientable surfaces in R 3 . (In particular, our results cover the class of compact surfaces.) We consider both the stationary case, which is governed by the micromagnetic energy functional, and the time-dependent case driven by the Landau-Lifshitz-Gilbert equation (the methods in [16] can not treat this case).
Note that, the extension to surfaces that do not admit (like in [10]) a single chart atlas, is not achieved by a local-to-global gluing argument. In fact, the presence of the nonlocal demagnetizing field operator h d (cf. (2)) makes this approach problematic. To have a rigorously justified micromagnetic model of curved thin films, the asymptotic analysis must take into account the global geometry of the surface from the very beginning; this requires some work. In that respect, our primary tool is Lemma 1 which concerns the weak limit of a family of demagnetizing field operators defined on curved thin films; we put some effort to make the proof of Lemma 1 short and direct, so that the central geometric scaling involved should appear evident (cf. Figure 1).
We want to emphasize that, although the results in themselves are not surprising, they give a definitive answer to the long-standing question of whether the Gioia-James regime still hold for surfaces that, like the sphere, cannot be covered by a single chart. Our results give an affirmative answer to this question and, therefore, put on solid ground the 2d models currently used for the analysis and simulation of magnetism in curved geometries. Indeed, it is important to stress that although several geometric regimes can be investigated (see, e.g., the survey article [26]), the Gioia-James regime is the most interesting one for the study of magnetic skyrmions in curved geometries [41]. In this regard, the primary motivation for our paper comes from the physics community; as remarked in [19]: being aware that these (dimension reduction) results were obtained for plane films, we assume that the same arguments are valid for smoothly curved shells.

Outline.
The paper is organized as follows. In Section 2, we state the main results: Lemma 1 which concerns the limiting behavior of the demagnetizing field in curved thin films; Theorem 1, which deals with the curved thin-film limit in the stationary case; and Theorem 2 that accounts for the time-dependent setting. The proof of Theorem 1 is given in Section 3; the analysis of the limiting LLG equation is in Section 4.

Statement of main results
In the following, S will always denote a bounded, orientable, and smooth (C 2 ) surface in R 3 . The normal field associated with the choice of orientation for S will be denoted by n : S → S 2 . We stress that although in the following we will always refer to surfaces without boundary, the distinction between surfaces with or without boundary is here irrelevant; indeed, contributions due to the presence of a boundary can appear only at higher-order terms in the expansion of the demagnetizing field.
Remark 2.1. The range of surfaces included by our analysis is broad. Indeed, any compact and smooth surface is orientable and admits a tubular neighborhood (of uniform thickness) [18,Prop. 1,p. 113]. In particular, our analysis holds for bounded convex surface (e.g., planar surfaces, the sphere, the ellipsoid) as well as non-convex ones (e.g., the torus). Also, it covers the class of bounded surfaces that are diffeomorphic to an open subset of a compact surface (e.g., the finite cylinder or the graph of a C 2 function).
For every ξ ∈ S the symbols τ 1 (ξ), τ 2 (ξ) are used to denote an orthonormal basis of T ξ S made by its principal directions, i.e., an orthonormal basis consisting of eigenvectors of the shape operator of S (cf. [18, p. 146]). We then write κ 1 (ξ), κ 2 (ξ) for the principal curvatures at ξ ∈ S. Note that, for any x ∈ Ω δ the trihedron constitutes an orthonormal basis of T π(x) Ω δ that depends only on S. Also, we denote by √ g ε the metric factor which relates the volume form on Ω ε to the volume form on M, by h 1,ε , h 2,ε the metric coefficients which link the gradient on Ω ε to the gradient on M. A direct computation shows that (cf., e.g., [33, p. 69 where H(ξ) and G(ξ) are, respectively, the mean and Gaussian curvature at ξ ∈ S. In what follows, we shall always assume that the thickness δ is sufficiently small so that for some positive constant c M , and every ε ∈ I δ + .
Also, we shall denote by H 1 (M, R 3 ) the Sobolev space of vector-valued functions defined on M (see [2]) endowed with the norm Here, ∇ ξ u is the tangential gradient of u on S. Finally, we write H 1 (M, S 2 ) for the subset of Next, we recall that (cf. [16] Therefore, the demagnetizing field can be described as the map which to every magnetization m ∈ L 2 (R 3 , R 3 ) associates the distributional gradient of the unique solution of (11) in defines a selfadjoint and positive-definite bounded linear operator from L 2 (R 3 , R 3 ) into itself: for every m ∈ L 2 (Ω, R 3 ). Moreover, for any ε ∈ I δ + , the demagnetizing field h d [mχ Ωε ] satisfies the weak form of the Maxwell-Ampère equations (cf. [16]) The anisotropy energy density φ an : S 2 → R + , which does not depend on ε ∈ I δ + , is assumed to be a non-negative Lipschitz continuous function that vanishes only along the easy directions. The hypotheses on φ an are sufficiently general to embrace all classes of crystal anisotropies arising in applications (e.g., uniaxial, triaxial, and cubic, where the anisotropy density is a polynomial on S 2 ). Finally, the applied field h a is assumed to be Lipschitz continuous.
The crucial tool for our dimension reduction analysis, both in the stationary and timedependent case, is contained in the next result, whose proof will be given in Section 3.5.

The stationary case. The micromagnetic energy functional on curved thin films.
To avoid uninformative results (i.e., to avoid that the Γ-limit reduces to the null functional), we consider a rescaled version of the energy functional (1). Precisely, let Ω δ be a bounded open set generated as a tubular neighborhood of a smooth orientable surface S. For any ε ∈ I δ + := (0, δ) we denote by G ε the micromagnetic energy functional defined on where mχ Ωε is the extension by zero of m to the whole space outside Ω ε . The existence for any ε ∈ I δ + of at least a minimizer for G ε in H 1 (Ω ε , S 2 ) is a straightforward application of the direct method of the calculus of variations (cf. [42]).
We are interested in the asymptotic behavior of the family of minimizers of (G ε ) ε∈I δ + as ε → 0.
Let us introduce the following functionals defined on H 1 (M, S 2 ), which can be thought as the pull-back of E ε , W ε , A ε and Z ε on the product manifold M := S × I, I := (−1, 1): • The exchange energy on M reads as • The magnetostatic self-energy on M is defined by Here, the symbol h ε [u] ∈ L 2 (M, R 3 ) stands for the demagnetizing filed on M: The family of diffeomorphisms (ψ ε ) ε∈I δ + is the one given in (4). • The anisotropy and interaction energies on M, respectively given by For every ε ∈ I δ + , we have used the symbol h ε a for the expression of h a on M which, for every (ξ, s) ∈ M, is defined by h ε a (ξ, s) := h a (ψ ε (ξ, s)). Note that, in the new coordinate system, the applied field depends upon ε ∈ I δ + .
The main result for the stationary case is stated in the next result.
By this, we mean that a configuration m ε ∈ H 1 (Ω ε , The family (F ε ) ε∈I δ + is equicoercive in the weak topology of H 1 (M, S 2 ) and the Γ-limit F 0 := Γ-lim ε→0 F ε is given by if ∂ s u = 0 and +∞ otherwise. Therefore, and if (u ε ) ε∈I δ + is a minimizing family for (F ε ) ε∈I δ + , there exists a subsequence of (u ε ) ε∈I δ+ which weakly converges in H 1 (M, S 2 ) to a minimum point of F 0 .
The proof of Theorem 1 is given in several steps. In Subsection 3.1 we show that for any ε ∈ I δ+ and any m ε ∈ H 1 (Ω ε , S 2 ) the equality G ε (m ε ) = F ε (m • ψ ε ) holds, where ψ ε stands for the diffeomorphism of M onto Ω ε given by ψ ε (ξ, s) := ξ + εsn(ξ). In Subsection 3.2 we show that the family (F ε ) ε∈I δ + is equicoercive for the weak topology of H 1 (M, S 2 ). Finally, the complete characterization of the Γ-limit F 0 is the object of Sections 3.3-3.5.
This means that, with respect to the (topological) product space I δ + × H 1 (M, S 2 ), the following relations hold Hence, the theorem on the sum of Γ-limits holds (cf. [11,Prop. 6.20, p. 62]), namely: For this reason, in the sequel, we shall only focus on the family (E ε M + W ε M ) ε∈I δ + .
In micromagnetics, −dG Ω (m) is referred to as the effective field and usually denoted by h eff [m]. The stationary condition (30), which in the strong form reads as represents the so-called Brown's static equations. In what follows, to shorten the length of the formulas, we will neglect both the magnetocrystalline anisotropy and the Zeeman energy.
As already pointed out in Remark 2.3, these terms do not alter the mathematical analysis of the question. Then, a standard computation shows that for every φ ∈ H 1 (Ω, R 3 ) one has  [29,21], which model the magnetization dynamics as a dissipative precessional dynamics driven by the effective field. In dimensionless form, the LLG reads as where α > 0 is the Gilbert damping constant, and m * is the magnetization state at time t = 0.

Definition 1.
Let Ω be an open set of R 3 and m * ∈ H 1 (Ω, S 2 ). We say that the vector field m : R + × Ω → S 2 is a (global) weak solution to the LLG, if for every T > 0 the following conditions are fulfilled: iv. With G Ω the energy functional (1), the following energy inequality holds: The existence of at least a global weak solution of (33) dates back to [5]. Therefore, in our setting, for any ε > 0, and any initial datum m * ε ∈ H 1 (Ω ε , S 2 ), there exists at least a global weak solution m ε of the LLG equation on Ω ε with initial datum m * ε . As in the statement of the Γ-convergence result, it is convenient to rewrite the LLG as an equation on the product manifold M = S × I. To this end, for any u ε ∈ H 1 (M, S 2 ), we introduce the effective field on M, defined for every φ ∈ H 1 (M, where, to shorten notation, we set This leads to the Definition 2 below, which gives an equivalent notion of (global) weak solution for configurations evolving in time on M. In what follows, for every T ∈ R + we set M T := (0, T ) × M. Also, with a slight abuse of notation we set and Definition 2. Suppose that u * (ξ, s) := u * (ξ)χ I (s) with u * (ξ) ∈ H 1 (S, S 2 ). We say that the vector field u ε : R + × M → S 2 is a (global) weak solution of the LLG equation if for every T > 0 the following conditions are fulfilled: iv. The following energy inequality holds, where F ε represents the energy functional (24): Remark 2.4. The equivalence between the two definitions has to be understood in the following sense. If m ε is a global weak solution in the sense of Definition 1 (with initial datum m * ε ), then u ε (t) := m ε (t) • ψ ε is a global weak solution in the sense of Definition 2 (with initial datum u * ε := m * • ψ ε ) where, as before, For curved thin films, our main result reads as follows.

The limit vector field u 0 satisfies the LLG equation on S:
Moreover, u 0 (0) = u * and the following energy inequality holds for all T > 0 with F 0 given by (26).
The proof of Theorem 2 is given in Section 4, and it is based on the characterization of the demagnetizing field operator on curved thin films (cf. Lemma 1).

Compactness.
We now show that the family (F ε ) ε∈I δ + is equicoercive in the weak topology of H 1 (M, S 2 ). This means, by definition (see [7]), that there exists a nonempty and weakly compact set The equicoercivity of (F ε ) ε∈I δ+ will assure that we can rely on the fundamental theorem of Γ-convergence concerning the variational convergence of minimum problems (cf. [7,11]).
Since A ε M and Z ε M are uniformly bounded terms (with respect to ε ∈ I δ + ), it is sufficient to show the equicoercivity of the family V ε := (E ε M + W ε M ) ε∈I δ + . To this end, we observe that for any constant in space v ∈ H 1 (M, S 2 ) we have Taking into account (12) and that √ g ε is bounded on M, uniformly with respect to ε ∈ I δ + , we end up with for a suitable positive constant κ M depending only on M. Therefore, for every ε ∈ I δ + , the minimizers of (V ε ) ε∈I δ + are in K(M, Also, since the principal curvatures κ 1 , κ 2 are bounded in S, whenever the radius δ ∈ R + of the tubular neighborhood Ω δ is sufficiently small, we have inf (ξ,s)∈M h i,ε (ξ, s) c M for every ε ∈ I δ + (cf. (9)). Therefore, since W ε M is always nonnegative because of (12), we get and therefore if u ∈ K(M, S 2 ) then In other words, the set K(M, To prove that H 1 b (M, S 2 ) is weakly compact it is sufficient to show that the set H 1 b (M, S 2 ) is weakly closed. To this end, we note that if (u n ) n∈N is a sequence in H 1 b (M, S 2 ) such that u n ⇀ u 0 weakly in H 1 (M, R 3 ), due to the Rellich-Kondrachov theorem, u n → u 0 strongly in L 2 (M, R 3 ), and therefore, up to the extraction of a subsequence, 1 ≡ |u n | → |u 0 | a.e. in M. Thus u 0 (ξ, s) ∈ S 2 for a.e. (ξ, s) ∈ M and this concludes the proof.

The identification of the Γ-limit.
In this section, we compute F 0 := Γ-lim ε→0 F ε . As pointed out in Remark 2.3, it is sufficient to focus on the Γ-convergence of the family We set V 0 := E 0 + W 0 with E 0 and W 0 given by (25). Note that, as a consequence of (12), V ε (u) 0 for any u ∈ H 1 (M, S 2 ).
Therefore, for the identification of the Γ-limit of (V ε ) ε∈I δ it is sufficient to restrict the analysis to the families of H 1 (M, S 2 ) functions which weakly converge to an element u 0 ∈ H 1 (M, S 2 ) which is 0-homogeneous, i.e., of the form for someũ 0 ∈ H 1 (S, S 2 ), i.e., not depending on the s variable. In the following, with a slight abuse of notation, we shall write u 0 (ξ) instead ofũ 0 (ξ).

The
Proof. We start by addressing the Γ-lim inf inequality for (E ε M ) ε∈I δ + . Taking into account the lower semicontinuity of the norm, for any u ε ⇀ u 0 in H 1 (M, S 2 ), with u 0 of the type (57), we get In deriving the last equality, we used (56) and denoted by ∇ ξ u ε (ξ, s) the tangential gradient of u ε on S whose norm, with respect to an orthonormal basis (τ 1 (ξ), τ 2 (ξ)) of T ξ S, can be expressed as |∇ ξ u ε (ξ, s) Next, from (9), there exists a strictly positive real-valued function γ : I δ + → R + such that, at least in a neighborhood of 0 ∈ R, the following estimate holds: and, therefore, Substituting (68) into (62) we get that if u ε ⇀ u 0 weakly in H 1 (M, S 2 ) and lim inf ε→0 V ε (u ε ) < +∞, then the following estimate holds We now address the existence of a recovery sequence. To this end, it is sufficient to note that for every This concludes the proof.

3.5.
Proof of Lemma 1 and the Γ-limit of the family (W ε M ) ε∈I δ+ . This section is devoted to the identification of the Γ-limit of the family (W ε M ) ε∈I δ + of magnetostatic self-energies on M. We will first prove Lemma 1.
3.6. The Γ-limit of the family (F ε ) ε∈I δ . We complete the proof of Theorem 1 by showing that F 0 is given by (26). As pointed out in Remark 2.3, it is sufficient to show that Γ-lim V ε = E 0 + W 0 . We note that if u ε ⇀ u in H 1 (M, S 2 ) and ∂ s u = 0 then lim inf ε→0 V ε (u ε ) = +∞. Therefore, the Γ-lim inf inequality is trivially satisfied. On the other hand, if ∂ s u = 0, then from Proposition 1 and Proposition 2 we get Finally, for any u 0 ∈ H 1 (M, S 2 ) such that ∂ s u 0 = 0, the constant (with respect to the index ε) family (u ε ) ε∈I δ + = (u 0 ) ε∈I δ + is a recovery sequence. This completes the proof of Theorem 1.

The time-dependent case: proof of Theorem 2
Let now (u ε ) ε∈I δ + be a sequence of weak solutions of the LLG, in the sense of Definition 2, and suppose that u ε (0) = u * for some u * (ξ) ∈ H 1 (S, S 2 ). Since u * does not depend on the s-variable, from (12), we get the existence of a positive constant κ M (u * ), depending only on the volume of M and on ∇ ξ u * 2 L 2 (S) , such that, for every ε > 0 F ε (u * ) κ M (u * ) .
Taking the limit for ε → 0 of the expression of the effective field (36) we get that for every φ ∈ H 1 (M T , R 3 ) which does not depend on the s-variable, Therefore, passing to the limit for ε → 0 in the family of LLG equation (40), we end up with the weak formulation of the limit LLG equation which reads, for every φ ∈ H 1 (S T , R 3 ), as In particular, in the smooth setting, integrating by parts in (103), we get the strong form of the limiting LLG equations that read as where −h 0 eff [u 0 ] := −∆ ξ u 0 + (n ⊗ n)u 0 . Finally, the energy inequality (44) is a direct consequence of the energy inequalities satisfied by the family of solutions (u ε ) ε∈I δ + : Indeed, by the lower semicontinuity of the norms we have F 0 (u 0 (T )) + and the right-hand side coincides with F 0 (u * ) because, as shown in Section 3.6, u * is a recovery sequence for the Γ-limit F 0 . This completes the proof.

Acknowledgment
The author acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65) and of the Vienna Science and Technology Fund (WWTF) through the research project Thermally controlled magnetization dynamics (Grant MA14-44).