Formation and coarsening of the concertina magnetization pattern in elongated thin-film elements

Jutta Steiner, Rudolf Schäfer, Holm Wieczoreck, Jeffrey McCord, and Felix Otto

Phys. Rev. B 85, 104407 – Published 12 March 2012

Abstract

The concertina is a magnetization pattern in elongated thin-film elements of a soft ferromagnetic material. It is a ubiquitous domain pattern that occurs in the process of magnetization reversal in the direction of the long axis of the small element. Van den Berg and Vatvani [IEEE Trans. Magn. 18, 880 (1982)] argued that this pattern grows out of the flux-closure domains at the sample's tips as the external field is reduced. Based on experimental observations and theory, we argue that in sufficiently elongated thin-film elements the concertina pattern rather bifurcates from an oscillatory buckling mode. Typical sample widths and thicknesses are of the order of 10-100 $μ$ m and of the order of 10-150 nm, respectively. Using a reduced model that is derived by asymptotic analysis from the micromagnetic energy and that is also investigated by means of numerical simulation, we quantitatively predict the average period of the concertina pattern and qualitatively predict its hysteresis. In particular, we argue that the experimentally observed coarsening of the concertina pattern is due to secondary bifurcations related to an Eckhaus instability. We also link the concertina pattern to the magnetization ripple and discuss the effect of a weak (crystalline or induced) anisotropy.

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Received 5 July 2011

DOI:https://doi.org/10.1103/PhysRevB.85.104407

Authors & Affiliations

Jutta Steiner

Centre de Mathémathiques Appliqueées, École Polytechnique, 91128 Palaiseau Cedex, France

Rudolf Schäfer and Holm Wieczoreck

Leibniz Institute for Solid State and Materials Research Dresden (IFW Dresden), Institute for Metallic Materials, Helmholtzstraße 20, 01069 Dresden, Germany

Jeffrey McCord

Institute for Materials Science, Technical Faculty of CAU Kiel, Kaiserstraße 2, 24143 Kiel, Germany

Felix Otto

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

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Vol. 85, Iss. 10 — 1 March 2012

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Images

Figure 1
Concertina in a very elongated (length 2 mm) sample of width $50 μ$ m and thickness 50 nm (left) and in a sample of width $35 μ$ m, thickness 40 nm, and moderate length $110 μ$ m (right). The left image shows only the center of the very elongated sample, i.e., less than $10 %$ of the whole sample length. The right image shows an entire sample with its closure domains at the tips. As indicated by the blue arrows, the gray scales encode the transverse component of the magnetization in the domains. Notice that the gray scales in the different pictures are not comparable. In particular the legend in the center has to be understood qualitatively.Reuse & Permissions
Figure 2
Formation of the concertina pattern in the experiment: The pictures show a section near the center of the elongated thin-film element. The oscillatory instability grows into a domain-wall pattern which coarsens several times as the strength of the destabilizing field decreases (from left to right). The two upper series show a sample of 30 nm thickness of low anisotropy. The two lower series show a sample of 30 nm thickness of higher anisotropy. The widths are 30 and $50 μ$ m, respectively.Reuse & Permissions
Figure 3
We assume an idealized elongated sample that is infinitely extended or periodic in the direction of the long axis $x_{1}$ . It is of width $ℓ$ and thickness $t$ . The uniform saturation magnetization $m^{*} \equiv (1, 0, 0)$ is sketched together with the homogeneous external saturation field $H_{ext}$ that is applied along the long axis of the sample.Reuse & Permissions
Figure 4
The scale of the external field $h_{ext}$ .Reuse & Permissions
Figure 5
The creation of a triplet (a)–(c) out of the initial doublet as shown and sketched by van den Berg and Vatvani; the sketch is duplicated and the images replicated from Figs. 8 and 9 in Ref. 1. (Reprinted with permission from Ref. 1. Copyright 1981, American Institute of Physics.) The doublet in (a) was created by applying a uniform magnetic field which pushed the central wall of an initial Landau state toward the long edge, where it broke up. The arrows indicate the direction of the magnetization $(m_{1}, m_{2})$ . The experimental images (d) and (e) show the buildup process of the concertina via the repeated generation of triplets. The field-penetrated region is hatched.Reuse & Permissions
Figure 6
Concertina in Permalloy samples of width $ℓ = 100 μ$ m and thickness $t = 30$ nm (left), 80 nm (center), and 300 nm (right). Obviously, the average period of the pattern is a decreasing function of the thickness $t$ .Reuse & Permissions
Figure 7
Coherent-rotation perturbation with generated surface charges that act like two oppositely charged wires at distances much larger than $t$ (left). The right image sketches the stray field generated by two oppositely charged wires in the $x_{2} x_{3}$ plane.Reuse & Permissions
Figure 8
Buckling mode with generated density of volume charges, which act like surface charges on a plate at distances much larger than $t$ , and the generated stray field. For reasons of a clear presentation the field is drawn only in the region above the sample (left). The right image shows an $x_{2} x_{3}$ section of the stray field generated by the surface-charge density on the plate (independent of $x_{1}$ ) that changes sign over distance $ℓ$ with respect to $x_{2}$ . The stray field extends roughly a distance $ℓ$ from the sample into space.Reuse & Permissions
Figure 9
Oscillatory buckling mode and generated surface charges and stray field—for reasons of clarity drawn only in the region above the sample. Compared to Fig. 8, the opposite charges are now at closer distance $w^{*}$ , so that the stray field extends roughly only a distance $w^{*}$ into space. Moreover, the dominant components of the stray field are $h_{1}$ and $h_{3}$ .Reuse & Permissions
Figure 10
Phase diagram for the four regimes for nucleation. The boundaries of the different regimes are obtained from Table by representing $ℓ / d$ as a function of $t / d$ at the regime boundaries; for example, the boundary between regimes III and IV is then given by the graph $ℓ / d = {(t / d)}^{2}$ . Inside each region the corresponding mode is the first mode to become unstable, i.e., the corresponding strength of the critical field of the mode is smaller than the strength for the other three modes. The shaded region corresponds to samples whose thickness $t$ is larger than their width $ℓ$ .Reuse & Permissions
Figure 11
The theoretical period of the unstable mode is in good agreement with the measurements: The upper image shows the ratio of the experimentally observed period and the period of the unstable mode. The white patches correspond to broken or defect-ridden samples. The lower displays the ratio of the period $w^{*}$ and the smallest period at all, i.e., $w^{*} (ℓ = 50 μ m, t = 150 nm)$ . Both images share the same color map (grayscale).Reuse & Permissions
Figure 12
Energy landscape close to the bifurcation. The loss of stability at the critical field leads to a first-order phase transition—on a large scale, however, the energy is coercive.Reuse & Permissions
Figure 13
Unstable mode ${{\hat{m}}_{2}^{*}}$ and additional curvature correction ${A {\hat{m}}_{2}^{*} + A^{2} {\hat{m}}_{2}^{* *}}$ , $A = 0.1$ , with generated charges on the domain $[0, {\hat{w}}^{*}) \times (0, 1)$ . The values are normalized by the maximum so that the grayscale reaches from $- 1$ to 1.Reuse & Permissions
Figure 14
Numerical simulations: The ${\hat{w}}^{*}$ -periodic branch close to the bifurcation and the pattern at the indicated fields (increasing average amplitude ${〈 {\hat{m}}_{2}^{2} 〉}^{1 / 2}$ from left to right). By ${〈 {\hat{m}}_{2}^{2} 〉}^{1 / 2}$ we denote the spatial root mean square of ${\hat{m}}_{2}$ , i.e., the amplitude of the average magnetization. The grayscales encode the $m_{2}$ component but are not comparable. The whole spectrum is considered so that the structure of the pattern can best be resolved; bright regions correspond to ${\hat{m}}_{2} < 0$ , dark regions to ${\hat{m}}_{2} > 0$ . The computational domain is $\hat{Ω} = [0, \hat{w}) \times [0, 1]$ .Reuse & Permissions
Figure 15
Numerical simulations: The ${\hat{w}}^{*}$ -periodic concertina pattern exhibits a clear scale separation as ${\hat{h}}_{ext}$ increases; ${\hat{h}}_{ext} = 6.84, 23.2, 40.1, 57.3$ from left to right. The grayscales linearly encode the ${\hat{m}}_{2}$ component and are comparable. The computational domain is $\hat{Ω} = [0, \hat{w}) \times [0, 1]$ .Reuse & Permissions
Figure 16
Domain theory: The mesoscopic charge-free ansatz function.Reuse & Permissions
Figure 17
Domain theory and numerical simulations: The domain-theoretical prediction for the rescaled optimal amplitude (dashed) and the computed amplitude based on the reduced model (solid); corresponding values on the left axis. For the reduced model we display the amplitude, i.e., the maximal value which is attained in the quadrangular domain. The right axis indicates the value of the relative error (dot dashed).Reuse & Permissions
Figure 18
Domain theory: Generalized tilted ansatz function.Reuse & Permissions
Figure 19
Numerical simulations: The optimal period of the concertina pattern as a function of the external field computed on the basis of the reduced model.Reuse & Permissions
Figure 20
Concavity of the minimal energy per period implies an instability under modulation of the wavelength.Reuse & Permissions
Figure 21
Numerical simulations: Comparison of the optimal and marginally stable periods of the concertina pattern as a function of the external field, both computed on the basis of the reduced model. In the region below the red (lower) curve the minimal energy per period is concave and thus a concertina of that period is unstable and coarsens.Reuse & Permissions
Figure 22
Numerical simulations and domain theory: The optimal and marginally stable periods computed on the basis of the reduced model (dashed) match the predictions on the basis of domain theory in the regime ${\hat{h}}_{ext} ≫ 1$ .Reuse & Permissions
Figure 23
Bifurcation analysis: The optimal and marginally stable periods as functions of the external field obtained on the basis of the extended bifurcation analysis.Reuse & Permissions
Figure 24
Numerical simulations and bifurcation analysis: The prediction on the basis of the reduced model (dashed) matches the prediction on the basis of the extended bifurcation analysis for a near-degenerate value of $\hat{Q} = 0.0295$ close to ${\hat{Q}}^{*} \approx 0.03$ ; cf. Sec. 6.Reuse & Permissions
Figure 25
Numerical simulations: The appearance of the secondary instability under $N {\hat{w}}^{*}$ -periodic perturbations as a function of $N$ ; the markers indicate the occurrence of the instability for $N = 2, \dots, 8$ (right to left). The critical field for $N = 8$ is $5.602$ .Reuse & Permissions
Figure 26
Numerical simulations. Bifurcation diagram for $4 {\hat{w}}^{*}$ perturbations: The bifurcation branches that connect the ${\hat{w}}^{*}$ -periodic (blue solid) and the $\frac{4}{3} {\hat{w}}^{*}$ -periodic (orange dashed) branches. The magnetization patterns at the indicated fields are shown in Fig. 27.Reuse & Permissions
Figure 27
Numerical simulations: Reflectional-symmetric with respect to center wall (top two rows) and rotational-symmetric with respect to the midpoint of the white face (bottom two rows) magnetization patterns on the unstable bifurcation branch connecting the ${\hat{w}}^{*}$ -periodic and the $\frac{4}{3} {\hat{w}}^{*}$ -periodic branches. The central fold collapses (top); the white face disappears and two adjacent black faces merge (bottom). The computational domain is $\hat{Ω} = [0, 4 \hat{w}) \times [0, 1]$ .Reuse & Permissions
Figure 28
Numerical simulations: The coarsened concertina pattern degenerates as the external field is reduced. The numerical simulations confirm the prediction based on domain theory: The pattern degenerates at the turning point of the branch. The computational domain is $\hat{Ω} = [0, 5 \hat{w}) \times [0, 1]$ .Reuse & Permissions
Figure 29
Numerical simulations: The marginally stable (bottom curves), optimal (middle curves), and maximal period (top curves) of the concertina pattern as functions of the external field ${\hat{h}}_{ext}$ . The dashed and solid curves depict the results on the basis of the reduced model and on the basis of domain theory, respectively.Reuse & Permissions
Figure 30
Numerical simulations: The hysteresis loop. As we increase the field, the concertina pattern coarsens if the period is smaller than the stable period. As we decrease the field starting from a coarsened concertina the pattern degenerates as we reach the turning point of the branch. The pattern refines toward the optimal period until it finally disappears. The computational domain is $\hat{Ω} = [0, 4 \hat{w}) \times [0, 1]$ .Reuse & Permissions
Figure 31
Experiment: The hysteresis cycles of a Permalloy sample of 30 nm thickness and $50 μ$ m width. The upper row shows the pattern as the external field increases (from left to right); the lower row shows the pattern as the external field decreases (from right to left).Reuse & Permissions
Figure 32
Tangent predictor-corrector continuation method.Reuse & Permissions
Figure 33
Experiment and numerical simulations: The coarsening of the concertina pattern in a Permalloy sample (top row) of 30 nm thickness and $70 μ$ m width compared to the numerical simulations (bottom row). A ripplelike structure grows into the concertina pattern. Within the numerical simulations we iteratively increment the external field and minimize the energy. The computational domain is of period $6 {\hat{w}}^{*}$ . The numerical images are scaled according to (12) to compare them to the experiment. The numerical images hence display 1.8 times the unit cell; the numerical images therefore appear to be more uniform than the experimental concertina.Reuse & Permissions
Figure 34
The order of the different effects of anisotropy.Reuse & Permissions
Figure 35
Numerical simulations: Transition from sub- to supercritical bifurcation as the strength of the transverse anisotropy increases. For $Q = 0.03 \approx Q^{*}$ the bifurcation degenerates.Reuse & Permissions
Figure 36
Numerical simulations: Loss of the turning point as the strength of the longitudinal anisotropy increases.Reuse & Permissions
Figure 37
Experiment: Permalloy samples of width 60– $150 μ$ m of high transverse anisotropy and at the end of the coarsening process. The $s i x$ samples on the right are of thickness 30 nm, the six on the right of thickness 50 nm. The period of the pattern appears to be independent of the width of the samples, in agreement with our theoretical prediction as an effect of anisotropy.Reuse & Permissions
Figure 38
Scaling behavior of the optimal and marginally stable periods and the amplitude of the transverse component in the regime $t ℓ^{- 1} ≪ Q ≪ d^{- 2 / 3} ℓ^{- 2 / 3} t^{4 / 3}$ .Reuse & Permissions
Figure 39
Continuous transition from the concertina pattern via the Landau state to the reversed concertina. Note that the total length of the walls and the Zeeman energy do not change while the anisotropy energy is smaller in the case of the Landau state.Reuse & Permissions
Figure 40
Experiment: Hysteresis of a CoFeB sample of 60 nm thickness and $30 μ$ m width. Following the coarsening we observe a transition to a Landau state at zero external field, which turns into a concertina that degenerates and refines and finally disappears.Reuse & Permissions

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