MICROMAGNETICS OF THIN FILMS IN THE PRESENCE OF DZYALOSHINSKII–MORIYA INTERACTION
ELISA DAVOLI, GIOVANNI DI FRATTA, DIRK PRAETORIUS, MICHELE RUGGERI
Abstract. In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii–Moriya interaction (DMI). Our analysis includes both a stationary Γ-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional model with the two-dimensional thin-film limit.
. Introduction
.. Chiral effects in micromagnetics. Due to an increasing interest in spintronics applica- tions, magnetic skyrmions are currently the subject of intense research activity, spanning from mathematics to physics and materials science. Although these chiral structures can emerge in different experimental settings, our mathematical analysis is centered on thin films derived from bulk materials without inversion symmetry, where the Dzyaloshinskii–Moriya interaction (DMI) can twist the otherwise ferromagnetic spin arrangement (cf. []). In this paper, we identify a limiting model for micromagnetic thin films in the presence of bulk DMI. Our investigations focus on the limiting behavior of the observable magnetization states as the thickness parameter tends to zero. The analysis includes both a stationary Γ-convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert (LLG) equation. In particular, the derived limiting model unveils some physics. Indeed, quite unexpectedly, we find that part of the DMI behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy originating from the magnetostatic self-energy. Our analytical derivation is complemented by a numerical approximation of the limiting solution via a projection-free tangent plane scheme, and the results of numerical simulations are discussed. Before stating our main results, we set up the physical framework and introduce some notation. In the variational theory of micromagnetism (cf. [, , ]), the observable states of a rigid ferromagnetic body occupying a region Ω ⊆ R3 are described by its magnetization M, a vector field verifying the so-called saturation constraint: There exists a material-dependent positive constant Ms such that |M| = Ms in Ω. The saturation magnetization Ms := Ms(T ) depends on the temperature T , but vanishes above the so-called Curie temperature Tc which is characteristic of each crystal type. However, when the ferromagnet is at a fixed temperature well below Tc, the value of the saturation magnetization can be considered constant in Ω. Therefore, we can express
Mathematics Subject Classification. S , C , Q, D , M, M . Key words and phrases. Micromagnetics, Landau–Lifshitz–Gilbert Equation, Magnetic thin films, Dzyaloshinskii– Moriya interaction, Γ-Convergence, Finite elements. MICROMAGNETICS OF THIN FILMS IN THE PRESENCE OF DZYALOSHINSKII–MORIYA INTERACTION
2 the magnetization in the form M := Msm, where m :Ω → S is a vector field taking values in the unit sphere S2 of R3. Although the modulus of m is constant in space, in general, it is not the case for its direc- tion. In single-crystal ferromagnets (cf. [, , ]), the observable magnetization states can be described as the local minimizers of the micromagnetic energy functional which, after a suitable nondimensionalization, reads as
1 2 1 Gsym,Ω(m) := |∇m| dx + ϕan(m) dx − hd[mχΩ] · m dx − ha · m dx (.) 2 ˆΩ ˆΩ 2 ˆΩ ˆΩ =:EΩ(m) =:AΩ(m) =:WΩ(m) =:ZΩ(m)
1 2 with m ∈ H (Ω, S ), and where mχΩ denotes the extension of m by zero to the whole space outside Ω.
The exchange energy EΩ(m) penalizes nonuniformities in the orientation of the magnetization. The magnetocrystalline anisotropy energy AΩ(m) accounts for the existence of preferred directions 2 of the magnetization. In general, ϕan : S → R+ is assumed to be a nonnegative Lipschitz continuous function that vanishes only on a distinguished set of directions known as easy axes. The quantity WΩ(m) represents the magnetostatic self-energy and describes the energy due to the 2 3 3 stray field hd[mχΩ] generated by mχΩ ∈ L (R , R ). The stray field can be characterized as the unique solution in L2(R3, R3) of the Maxwell–Ampère equations of magnetostatics [, , ]: div b[mχΩ] = 0, curl hd[mχΩ] = 0, (.) b[mχΩ] = µ0(hd[mχΩ] + mχΩ),
where b[mχΩ] denotes the magnetic flux density, and µ0 is the vacuum permeability. The linear operator hd : mχΩ 7→ hd[mχΩ] is then a bounded, nonlocal, and negative-definite operator which satisfies the following energy bounds:
2 2 0 6 |hd[mχΩ]| dx = − hd[mχΩ] · m dx 6 kmkL2(Ω). (.) ˆR3 ˆΩ
Finally, the term ZΩ(m) is the Zeeman energy and models the tendency of the specimen to have its magnetization aligned with the (externally) applied field ha. Overall, the variational analysis of (.) arises as a nonconvex and nonlocal problem.
In this work, other than the classical energy terms in Gsym,Ω, we consider the possible lack of centrosymmetry in the crystal lattice structure of the ferromagnet. Generally speaking, this can be done by superimposing to the energy density in (.) the contribution from suitable Lifshitz invariants of the chirality tensor ∇m × m. Here, we are interested in the bulk DMI contributions, whose energy density is given by the trace of the chirality tensor. Precisely, for every m ∈ H1(Ω, S2), we define the functional
DΩ(m) := κ curl m · m dx. (.) ˆΩ The normalized constant κ ∈ R is the so-called DMI constant, and its sign determines the chirality of the system. The full micromagnetic energy functional we are interested in is then Gsym,Ω(m) + DΩ(m). Since one of our main aims concerns the derivation via Γ-convergence of a D model, to streamline the presentation of the results, we will neglect the energy terms AΩ(m) and ZΩ(m) because in this dimension reduction context they act as Γ-continuous perturbations [] and their MICROMAGNETICS OF THIN FILMS IN THE PRESENCE OF DZYALOSHINSKII–MORIYA INTERACTION
Γ-continuous limit is straightforward to compute. Summarizing, for every m ∈ H1(Ω, S2), we consider the micromagnetic energy functional
1 2 1 GΩ(m) := |∇m| dx + κ curl m · m dx− hd[mχΩ] · m dx. (.) 2 ˆΩ ˆΩ 2 ˆΩ
1 2 The existence of at least one minimizer of GΩ(m) in H (Ω, S ) easily follows by the direct method of the calculus of variations. Indeed, although the bulk DMI energy DΩ is, a priori, neither positive nor negative, it is linear in the partial derivatives and can be controlled by the exchange energy. P3 Namely, by the identity curl m = i=1 ei × ∂im, one obtains
1 3 1 3 E (m)+D (m) = X |d m|2 dx − κ2 X |e × m|2 dx Ω Ω 2 ˆ i 2 ˆ i i=1 Ω i=1 Ω 1 3 = X |d m|2 dx − κ2|Ω|, (.) 2 ˆ i i=1 Ω
for every m ∈ H1(Ω, S2), where, for i = 1, 2, 3, the quantities
dim := ∂im − κ(ei × m) (.)
represent the so-called helical derivatives [ ,]. The name is motivated by the fact that dim = 0 for helical fields m which perform a rotation of constant frequency κ perpendicular to the ei axis, that is, counter-clockwise or clockwise if the sign of κ is positive or negative, respectively. Note that they can also be regarded as a variant of covariant derivatives emerging in the context of gauged sigma models [].
.. State of the art. The analysis of micromagnetic thin films is a subject with a long history. It dates back to the seminal papers [, ], where the authors show that in planar thin films, the effect of the demagnetizing field operator drops down to an easy-surface anisotropy term. Landau states in thin ferromagnetic films are investigated in []. A complete reduced theory for thin films micromagnetics has been established in []. Recently, in [] the authors have considered static dimension reduction under possible degeneracy of the material coefficients. A thorough analysis of soft ferromagnetic films has been the subject of [, , ] (see also [] for the case of nonuniformly extruded thin shells). The associated thin-film dynamics has been analyzed in [,,]. In [ ,], the authors characterize Neel walls in ferromagnetic thin films. The various regimes arising as effect of the presence of external fields are the focus of [, ]. The study of domain walls has been undertaken in [ , ,,] (see also [ ] for the case of D walls under fourfold anisotropy and [] for ultrathin films), whereas that of boundary vortices is carried out in [, , , ]. The effects of periodic surface roughness have been studied in []. In [ ], the author examines second-order phase transitions with dipolar interactions. We also refer to [, Section .] and the references therein for a review on the mathematics of magnetic planar thin films. Magnetic curved thin films have been the object of extensive investigations in recent times, because of their capability to induce an effective antisymmetric interaction even in the absence of DMI (see, e.g., [, , ] and the topical review []). We finally mention [] and [] for the magnetoelastic case. Due to the nonlinear, nonconvex, and nonlocal nature of the problem, the computation of minimizers of the micromagnetic energy and the numerical approximation of the LLG equation are challenging tasks. In the last twenty years, these problems have been the subject of several mathematical studies; see, e.g., the monograph [], the review articles [, ], the papers [, , MICROMAGNETICS OF THIN FILMS IN THE PRESENCE OF DZYALOSHINSKII–MORIYA INTERACTION
, , , ], and the references therein. As far as the numerical analysis of the LLG equation in the presence of DMI is concerned, we refer to the recent work [].
.. Contributions of the present work. For ε > 0, consider the thin-film domain Ωε := 2 ω × (0, ε), where ω is a bounded domain in R having Lipschitz boundary, and assume that Ωε is a thin specimen of a ferromagnetic body. As it is customary in dimension reduction, we introduce 2 2 the family of diffeomorphisms Φε :(σ, s) ∈ R × R → (σ, εs) ∈ R × R. We denote by ΦM,ε the 1 2 restriction of Φε to the set M := ω × I, I = (0, 1). To every mε ∈ H (Ωε, S ), we associate a 1 2 map u ∈ H (M, S ) given by u := mε ◦ ΦM,ε. Hereinafter, we use the convention that when we talk about weak convergence in H1(M, S2) we refer to the topology induced in H1(M, S2) by the weak topology of H1(M, R3); a similar remark is understood for strong convergence in H1(M, S2). Note that, owing to the Rellich theorem, H1(M, S2) is a weakly closed subset of H1(M, R3). The first result of this paper concerns the variational characterization of the asymptotic behavior
of a rescaling of the sequence GΩε in the limit for ε → 0. This amounts to the identification of the Γ-limit of the family of energy functionals defined by
1 2 1 2 Fε(u) := |∇εu| dx + κ curl εu · u dx + |hε[uχM]| dx, (.) 2 ˆM ˆM 2 ˆR3
1 2 −1 for every u ∈ H (M, S ), where hε[uχM] := hd[mεχΩε ] ◦ Φε = hd[(uχM) ◦ ΦM,ε] ◦ Φε, ∇ε is −1 T the ε-rescaled gradient operator given by ∇ε = (∂1, ∂2, ε ∂s) and curlε u = ∇ε × u. Note that 1 Fε(u) = ε GΩε (mε). We show that the asymptotic behavior of minimizers of Fε is encoded in the minimizers of an effective functional with a complete local character. Precisely, our first result is stated in the following theorem.
1 2 Theorem .. The family of functionals (Fε)ε>0 is equicoercive in the weak topology of H (M, S ) 1 2 and Γ-converges, weakly in H (M, S ), to the energy functional F0 defined by 2 2 1 2 1 + κ 2 κ F0(u) := |∇ωu(σ)| dσ + κ curl ωu · u dσ + (u(σ) · e3) dσ − |ω|, (. ) 2 ˆω ˆω 2 ˆω 2
1 2 if u ∈ H (M, S ) is independent of the s-variable, and F0(u) = +∞ otherwise. In the expression above, ∇ω := (∂1, ∂2) and curl ω := e1 ×∂1 +e2 ×∂2, denote the corresponding differential operators meant with respect to the tangential variables (x1, x2) ∈ ω. In particular, if for every ε the map 1 2 uε ∈ H (M, S ) is a minimizer of Fε, then, upon possible extraction of a subsequence, (uε)ε>0 1 2 converges strongly in H (M, S ) to a minimizer of F0.
Remark .. Note that the definition of curl ω = e1 × ∂1 + e2 × ∂2 in Theorem . is formally consistent with the D curl if ∂3u = 0. Remark .. We stress that part of the DMI energy in (.) contributes, in the limiting en- ergy (. ), to an increase in the shape anisotropy of the thin film through the energy density 2 2 κ (u(σ) · e3) . Also, we observe that in contrast to the classical setting κ = 0 (cf. [, ]), where constant in-plane magnetizations are the minimizers of F0 (and higher-order terms in the asymp- totic expansions are needed to gather more information on the direction of the magnetization), here the presence of the DMI makes the reduced energy functional F0 nontrivial.
The proof strategy essentially relies on the notion of Γ-convergence (see []). The equicoercivity of the energy functionals is established in Proposition . and is based on the observation in (.). In Propositions . and ., respectively, we prove that F0 provides a lower bound for the asymptotic behavior of the energies (Fε)ε>0 and show that this lower bound is indeed optimal. MICROMAGNETICS OF THIN FILMS IN THE PRESENCE OF DZYALOSHINSKII–MORIYA INTERACTION
Our second contribution concerns the derivation of a thin-film model for the LLG equation which describes the magnetization dynamics in small ferromagnetic samples. In our context, the LLG equation reads (in strong form) as
ε ∂tu = −u × h [u] + αu × ∂tu in M × R+, eff ∂nu = −κu × n on ∂M × R+, (. ) u(0) = u0 in M,
where α > 0 is a dimensionless damping factor, while u0 is the magnetization at time t = 0. The ε dynamics is driven by the effective field heff [u] := −∂uFε which is defined as the opposite of the first-order variation of the energy Fε. Our second result concerns families of weak solutions of the LLG equation (we refer to Section for the precise definition) and reads as follows. 0 0 1 2 Theorem .. Let u (σ, s) := u (σ) ∈ H (ω, S ) and, for every ε > 0, let uε be a weak solution 0 ∞ 1 2 of (. ) with initial datum u . Then, there exists a magnetization u0 ∈ L (R+; H (ω, S )) 1 2 such that u0 ∈ H (ω × (0,T ), S )) for every T > 0 and, up to the extraction of a nonrelabeled subsequence, ∗ ∞ 1 2 uε * u0 weakly in L (R+; H (M, S )). The limit magnetization u0 satisfies in the weak sense the boundary value problem
0 ∂tu0 = −u0 × h [u0] + αu0 × ∂tu0 in ω × R+, eff ∂nu0 = −κu0 × n on ∂ω × R+, (.) 0 u0(0) = u in ω,
0 2 where the limiting effective field heff [u0] := ∆ωu0 −2κcurl ωu0 −(1+κ )(e3 ⊗e3)u0 coincides with the opposite of the first-order variation of the Γ-limit energy F0 (cf. (. )). Moreover, for almost every T > 0, the following energy inequality holds
T 2 0 F0(u0(T )) + α k∂tu0kL2(ω,R3) dt 6 F0(u ). (.) ˆ0
The proof of this second result is based on the combination of some a priori energy estimates with [, Lemma .] (see also [, Lemma ] for a more general statement), and on the application of the Aubin–Lions lemma. Our third and last result concerns the numerical approximation of (. ) and (.). We propose an improved version of the projection-free finite element tangent plane scheme introduced in [] (see Algorithm below). The main novelty consists in an implicit-explicit approach for treating the effective field contributions, which is designed in such a way that the discrete energy law satisfied by the approximations mimics the dissipative energy law of the continuous problem (Proposition .). Moreover, suitable time reconstructions built using the approximations generated by the algorithm converge weakly and unconditionally towards a weak solution of the problem (Theorem .). Here, the word ‘unconditionally’ refers to the fact that the convergence analysis of Algorithm does not require any coupling condition between the spatial mesh size and the time-step size. Besides its own mathematical interest, the derivation of effective thin-film micromagnetic models also has relevant practical implications. The capability to perform reliable micromagnetic simu- lations using a D model is very favorable in terms of computational cost. On the one hand, the complexity of the simulation is clearly reduced by the dimension reduction (from D to D). On the other hand, since in the thin-film limit the nonlocal magnetostatic interaction reduces to a local shape anisotropy, one can save the cost of solving the magnetostatic Maxwell equations (.) MICROMAGNETICS OF THIN FILMS IN THE PRESENCE OF DZYALOSHINSKII–MORIYA INTERACTION
(which usually involves linear algebra operations with large fully populated matrices). This has a significant impact on the overall computational cost, as, for practically relevant problem sizes, the computation of the magnetostatic interaction is usually the most time-consuming part of a micromagnetic simulation []. In the last section of the present work, presenting the results of two numerical experiments, we validate our theoretical findings. At the same time, we show the effectivity of the proposed numerical scheme and give a flavor of the computational benefits of our approach.
.. Outline. The paper is organized as follows. In Section , we prove the static Γ-convergence result of Theorem .. In Section , we recall the definition of weak solutions of the LLG equation, and provide a convergence analysis from the D setting to the reduced limiting model. Section is devoted to the description and the analysis of the approximation scheme, and Section to the presentation of the results of numerical simulations.
. Gamma-convergence: Proof of Theorem .
To prove Theorem ., we first show that the family (Fε)ε∈I is equicoercive in the weak topol- ogy of H1(M, S2). This step assures the validity of the fundamental theorem of Γ-convergence concerning the variational convergence of minimum problems (cf. [, ]).
Proposition .. There exists a (nonempty) weakly compact set K ⊂ H1(M, S2) such that infH1(M,S2) Fε = infK Fε for every ε ∈ I.
Proof. Owing to (.), the following identity holds true for every u ∈ H1(M, S2):
2 1 2 1 2 Fε(u) + κ |M| = |Dεu| dx + |hε[uχM]| dx, (.) 2 ˆM 2 ˆR3
where Dε is the ‘Jacobian’ matrix of the ε-rescaled helical derivatives