Magnetic Domains in Thin Ferromagnetic Films with Strong Perpendicular Anisotropy

Knüpfer, Hans; Muratov, Cyrill B.; Nolte, Florian

doi:10.1007/s00205-018-1332-3

Cited by 26 publications

(54 citation statements)

References 59 publications

(77 reference statements)

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“…Finally, we consider a regime in which simultaneously the film thickness goes to zero, while the film's lateral dimension goes to infinity with a suitable rate that is exponential in the film thickness. We note that these types of scalings were previously discussed in the physics literature [20,35] and have been recently treated by Knüpfer, Muratov and Nolte within the framework of micromagnetics [22]. In this regime, after a rescaling that fixes the domain in the plane we prove a Γ-convergence result for the reduced energy in Theorem 2.4.…”

Section: Introductionsupporting

confidence: 56%

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A universal thin film model for Ginzburg–Landau energy with dipolar interaction

Muratov

2019

Calc. Var.

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We present an analytical treatment of a three-dimensional variational model of a system that exhibits a second-order phase transition in the presence of dipolar interactions. Within the framework of Ginzburg-Landau theory, we concentrate on the case in which the domain occupied by the sample has the shape of a flat thin film and obtain a reduced two-dimensional, non-local variational model that describes the energetics of the system in terms of the order parameter averages across the film thickness. Namely, we show that the reduced two-dimensional model is in a certain sense asymptotically equivalent to the original three-dimensional model for small film thicknesses. Using this asymptotic equivalence, we analyze two different thin film limits for the full three-dimensional model via the methods of Γ-convergence applied to the reduced twodimensional model. In the first regime, in which the film thickness vanishes while all other parameters remain fixed, we recover the local two-dimensional Ginzburg-Landau model. On the other hand, when the film thickness vanishes while the sample's lateral dimensions diverge at the right rate, we show that the system exhibits a transition from homogeneous to spatially modulated global energy minimizers. We identify a sharp threshold for this transition.

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Section: Introductionsupporting

confidence: 56%

“…We now specify the scaling of δ ε with ε for which modulated patterns emerge. This scaling has been recently identified in [22] in the studies of a closely related model from micromagnetism. For λ > 0 fixed, we set δ ε = λ γ| ln ε| , (2.18) and consider the limit behavior of the energies in (2.13) and (2.14) as ε → 0.…”

Section: )mentioning

confidence: 55%

A universal thin film model for Ginzburg–Landau energy with dipolar interaction

Muratov

2019

Calc. Var.

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“…The first instance of this idea is contained in the following lower bound for the energy in terms of the perimeter, adapted from Knüpfer, Muratov and Nolte [37]. Not only is it the first step to obtain uniform regularity estimates and the lower bound for the Γ-limit E λ,δ Γ → (1 − λ)P , it will in Section 6 also be the starting point in proving the Γ-lim inf inequality of the Γ-convergence in the critical case λ = 1.…”

Section: The Subcritical Case λ <mentioning

confidence: 99%

A Nonlocal Isoperimetric Problem with Dipolar Repulsion

Muratov

Simon

2019

Commun. Math. Phys.

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We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order Γ-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.1 |x| 3 . This is the interaction experienced at large distances by spins lying on the plane and oriented perpendicularly to it. Just like the classical Gamow's model, the model under consideration is relevant to the multidomain patterns observed in perpendicularly magnetized thin film ferromagnets [30], as well as ferroelectric films [57], Langmuir monolayers [5] and ferrofluid films subject to a strong perpendicular applied field [31,54] (for an experimental realization involving a magnetic garnet film, see Figure 1). Note, however, that setting G(x) = 1 |x| 3 would result in an ill-defined problem because of the divergence of the integral defining the nonlocal contribution at short scales. Furthermore, redefining the energy in the spirit of nonlocal minimal surfaces [11] to reduce the integral to that over Ω and Ω c , up to an additive constant, would not help, either, since the singularity of the kernel is still too strong. Therefore, a genuine regularization at short scale δ > 0 is necessary to make sense of the energy in (1.1) with this kind of kernel. This is a novel feature of the considered nonlocal isoperimetric problem compared to those studied previously.More specifically, we study the following nonlocal isoperimetric problem, as proposed by . We wish to minimize E(Ω) := αP (Ω) + β 2ˆΩˆΩ g δ (|x − y|) |x − y| 3 dx dy, (1.2) among all finite perimeter sets Ω ⊂ A m of fixed mass m > 0, i.e., with A m := {Ω ⊂ R 2 : P (Ω) < ∞, |Ω| = m}. (1.3) 3 Here, α, β > 0 are fixed parameters, P (Ω) is the perimeter of a measurable set Ω in the sense of De Giorgi [4]: P (Ω) := sup ˆΩ ∇ · φ dx : φ ∈ C 1 c (R 2 ; R 2 ), |φ| ≤ 1 , (1.4)

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“…for some λ > 0 playing the role of the renormalized effective film thickness. Notice that this scaling has recently appeared in a different context in the studies of thin ferromagnetic films with perpendicular magnetic anisotropy [22]. At the same time, according to (2.10) the leading order contribution to the magnetostatic energy of the edge charges for the optimal choice of the edge domain wall width L ε = O(1) turns out to be the same as the energy of the monodomain state.…”

Section: Thin Film Regimementioning

confidence: 87%

Edge Domain Walls in Ultrathin Exchange-Biased Films

2020

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We present an analysis of edge domain walls in exchange-biased ferromagnetic films appearing as a result of a competition between the stray field at the film edges and the exchange bias field in the bulk. We introduce an effective two-dimensional micromagnetic energy that governs the magnetization behavior in exchange-biased materials and investigate its energy minimizers in the strip geometry. In a periodic setting, we provide a complete characterization of global energy minimizers corresponding to edge domain walls. In particular, we show that energy minimizers are one-dimensional and do not exhibit winding. We then consider a particular thin film regime for large samples and relatively strong exchange bias and derive a simple and comprehensive algebraic model describing the limiting magnetization behavior in the interior and at the boundary of the sample. Finally, we demonstrate that the asymptotic results obtained in the periodic setting remain true in the case of finite rectangular samples.

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Magnetic Domains in Thin Ferromagnetic Films with Strong Perpendicular Anisotropy

Cited by 26 publications

References 59 publications

A universal thin film model for Ginzburg–Landau energy with dipolar interaction

A universal thin film model for Ginzburg–Landau energy with dipolar interaction

A Nonlocal Isoperimetric Problem with Dipolar Repulsion

Edge Domain Walls in Ultrathin Exchange-Biased Films

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