Herein, we report an exotic domain-wall dynamics showing double Walker breakdowns in magnetic multilayer films composed of two magnetic layers. Such multiple Walker breakdowns are attributed to the internal magnetic dipole field, which is antisymmetric on the domain walls of the lower and upper magnetic layers. A micromagnetic simulation shows four phases of the domain-wall dynamics, which result in a phase diagram with the phase boundaries of the double Walker breakdown fields. Such double Walker breakdowns lead to two minima in the variation of the domain-wall velocity, as often observed experimentally.
Research Article
Multiple Walker Breakdowns in Magnetic Multilayers
https://doi.org/10.21203/rs.3.rs-775358/v1
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Multiple Walker breakdowns
magnetic multilayers
films
internal magnetic dipole field
antisymmetric
domain-wall velocity
Recently, magnetic thin films have drawn great technological attention because of their prospects for use in next-generation memory and logic devices [1, 2]. The data bits in these devices are stored in their magnetization states and/or ordering structures such as magnetic domains and domain walls (DWs). The device operation is therefore performed by switching between the magnetization states as done for the magnetic random-access memory and/or the displacement of magnetic structures as done for the magnetic racetrack memory [3, 4]. These magnetization dynamics and their characteristics are determined by magnetic phenomena such as perpendicular magnetic anisotropy, Dzyaloshinskii–Moriya interaction, and the spin-orbit coupling effect [5–9]. As these phenomena are generated at interfaces adjacent to magnetic layers, much effort has been devoted to controlling the interface properties in magnetic thin films and multilayered structures [10].
For multilayered structures with multiple magnetic layers, the magnetic properties of each magnetic layer are largely determined by their own interfaces. Because ensuring the homogeneity among all interfaces is difficult, it is natural that each magnetic layer has magnetic properties different from the others [11, 12]. Static magnetic properties, such as the stable DW configuration, have been reported to significantly affect the dynamics of the DW motion [13]. For example, a stable DW configuration determines the direction of DW motion driven by the spin-orbit torque. Therefore, it is essential to understand the layer-resolved DW configurations and their roles in DW dynamics in magnetic multilayered structures [14, 15]. In this study, we prepared magnetic multilayered structures composed of two magnetic layers and then investigated the DW dynamics in these structures using a magneto-optical Kerr effect microscope. Interestingly, the magnetic multilayered structures exhibit an exotic coupled behavior of the DW dynamics. A micromagnetic study was carried out to explain the exotic coupled behavior within the context of multiple Walker breakdowns of DWs in each magnetic layer.
Sample preparations and DW velocity measurements. For this study, we prepared two types of magnetic films with different numbers of magnetic layers. Here, we denote Samples I and II for films composed of a single magnetic layer and double magnetic layers, respectively. The detailed layered structures comprise 5-nm Ta/2.5-nm Pt/0.3-nm Co/1.5-nm Pt for Sample I and 5-nm Ta/2.5-nm Pt/0.3-nm Co/0.3-nm Pt/0.3-nm Co/1.5-nm Pt for Sample II, as illustrated in Figs. 1(a) and (b), respectively. The films were deposited on Si wafers with a 100-nm SiO2 layer using dc magnetron sputtering.
Both films exhibited strong perpendicular magnetic anisotropy and clear magnetic domain patterns. Figures 1(c) and (d) show the magnetic domain images of Samples I and II, respectively. Each image is overlaid with several snapshots taken at a specified time interval under the application of the out-of-plane magnetic field, 
(red symbol), together with the in-plane magnetic field, 
(blue arrow). The brightest circle in the middle represents the initial circular magnetic domain, which exhibits gradual expansion over time, as shown by the gradual change in image contrast.
Micromagnetic simulation. we employed the object-oriented micromagnetic framework (oommf) code in three-dimensional geometry. The simulation geometry was set as a lateral wire structure onto the film of two ferromagnetic (FM) layers separated by a nonmagnetic (NM) spacer. The lateral wire structure has a length of 2,000 nm and a width of 250 nm. All FM and NM layers have an identical thickness of 0.4 nm. The mesh size was chosen to be 1 nm along the length, 250 nm along the width, and 0.4 nm along the thickness. Notably, the mesh size along the width is set to be identical to the wire width for a direct comparison with an analytic model of one-dimensional DWs and for a fast simulation speed by reducing the number of meshes. The mesh size along the length was set to be sufficiently small to describe the internal DW structures. The magnetic parameters are chosen as the typical values of Co/Pt multilayers, as given by the exchange stiffness of 5 × 10-12 J/m, saturation magnetization of 1.4 × 106 A/m, and surface magnetic anisotropy of 1.93 × 106 J/m3 [16]. For fast relaxation to the ground state, the damping constant was set at α = 0.5.
Double minimum of DW velocity about 𝑯𝒙 The DW displacements are then measured along the direction (yellow arrow) parallel to 𝐻𝑥, providing the DW speed, 𝑣DW, in the direction of 𝐻𝑥. Figures 2(a) and (b) plot the measured 𝑣DW onto the 2-dimensional (2D) coordinate plane with respect to 𝐻𝑥 (abscissa) and 𝐻𝑧 (ordinate) for Samples I and II, respectively. Sample I exhibits a typical symmetrical 𝑣DW variation with respect to 𝐻𝑥. The line profile along the dashed red line in Fig. 2(a) is plotted in Fig. 2(b), clearly confirming the symmetrical variation with a single minimum. Such symmetrical variation is known to be caused by the transition of the stable DW configuration under the influence of 𝐻𝑥 [13].
Interestingly, Sample II shows an exotic variation of 𝑣DW with two minima, as shown in the 2D plot of Fig. 2(c) and the line profile in Fig. 2(d). Such an exotic variation goes well beyond the model based on a single DW configuration. To check whether the exotic variation with two minima is ascribed to the two magnetic layers as in Sample II, the coupled configuration between the DWs in the two magnetic layers was investigated by considering micromagnetism.
Multiple walker breakdown. The stationary DW configurations at 𝐻𝑧 = 0 were first examined under application of 𝐻𝑥 over the range from 0 to 120 mT. Figures 3(a)–(e) show the cross-sectional view of the stable DW configurations in the stationary state for different 𝐻𝑥 values of (a) 0, (b) 30, (c) 60, (d) 90, and (e) 120 mT, respectively. The arrow inside each mesh shows the direction of the magnetization. The color of each arrow corresponds to the 𝑥 component of the magnetization (red for +𝑥 direction and blue for −𝑥 direction), whereas the color inside each mesh corresponds to the z component of the magnetization (red for +𝑧 direction and blue for −𝑧 direction).
When 𝐻𝑥 = 0, as shown in Fig. 3(a), the dipolar magnetic field from the lower magnetic layer generates a local magnetic field, 𝐻dip. For the present geometry, such 𝐻dip has a net in-plane component along the +𝑥 direction (upper gray arrow) at the position of the DW in the upper magnetic layer. Thus, the upper DW becomes stabilized as the Néel-type configuration with internal magnetization along the +𝑥 direction. Inversely, the dipolar magnetic field from the upper magnetic layer generates a net 𝐻dip along the −𝑥 direction (lower gray arrow) at the lower DW, thus resulting in the opposite Néel-type configuration along the −𝑥 direction.
Consequently, an antiparallel alignment of the internal magnetization appears between the upper and lower DWs, as shown in Fig. 3(a).
Under the application of 𝐻𝑥 (blue arrows), the total in-plane magnetic field, 𝐻tot (= 𝐻𝑥 + 𝐻dip) becomes unbalanced at positions between the upper and lower DWs. For the present geometry, an increase in 𝐻𝑥 causes the decrease of 𝐻tot at a lower DW, as shown by the lower gray arrows in Figs. 3(b) and (c). When 𝐻tot vanishes, the DW becomes a Bloch-type configuration, as shown in Fig. 3(d). A further increase in 𝐻𝑥 results in the parallel alignment of the internal magnetization between the upper and lower DWs, as shown in Fig. 3(e). To examine the effect of these different DW configurations, the DW speed, 𝑣DW is calculated under the application of 𝐻𝑧 . Figures 3(f)–(j) plot 𝑣DW with respect to 𝐻𝑧 for different 𝐻𝑥 values. When 𝐻𝑥 = 0, as shown in Fig. 3(f), 𝑣DW exhibits an abrupt breakdown at a certain strength of 𝐻𝑧 . This breakdown is well known as the Walker breakdown, which is caused by the precession of the internal magnetization inside the DWs.
Phases of DW during DW motion. Figure 4(a) shows a 2D plot of 𝑣DW with respect to 𝐻𝑥 (abscissa) and 𝐻𝑧 (ordinate). It is clear from the plot that several different phases are separated by the phase boundaries. The phase boundaries are attributed to the Walker breakdown, below which a steady-state motion with a fixed magnetization angle appears, above which a precessional motion occurs. For better visualization, Figure 4(b) illustrates the four phases: phase SS of steady-state motions (both DWs), phase PS of precessional (upper DW) and steady-state (lower DW) motions, phase SP of steady-state (upper DW) and precessional (lower DW) motions, and phase PP of precessional motions (both DWs). The phase boundaries indicate the Walker breakdown fields, 𝐻W u (red symbols) and 𝐻W 𝑙 (black symbols), for the upper and lower DWs, respectively.
Inside the phase diagram, the line profiles (horizontal dashed lines) under a fixed 𝐻𝑧 provide the 𝑣DW variation with respect to 𝐻𝑥 , similar to the experimental measurement procedure, shown in Fig. 2(d).
DW velocity by walker breakdown. Figures 5(a)–(c) plot the 𝑣DW variation with respect to 𝐻𝑥 under several fixed 𝐻𝑧 values. For a weak 𝐻𝑧 , the line profile along the first horizontal dashed line in Fig. 4(b) stays inside phase SS for both DWs. The 𝑣DW variation almost exhibits a plateau in the middle with respect to 𝐻𝑥, as shown in Fig. 5(a). However, a closer observation showed two separated minima with small amplitudes. These separated minima become obvious for an intermediate 𝐻𝑧 , where the line profile along the second horizontal, dashed line in Fig. 4(b) crosses phases PS, SS, and SP. Because 𝑣DW in phases PS and SP is largely reduced owing to the Walker breakdown, the 𝑣DW variation naturally exhibits two large minima inside these phases, as shown in Fig. 5(b). Finally, for a strong 𝐻𝑧 , the line profiles along the third horizontal dashed line in Fig. 4(b) cross phases PS, PP, and SP. Because the 𝑣DW of phase PP is slower than those of the other phases owing to the Walker breakdowns for both DWs, its 𝑣DW variation exhibits a single minimum at the middle, that is, inside phase PP, as shown in Fig. 5(c). Notably, the experimental value of 𝐻𝑧 in Fig. 2(b) ranges between those values shown in Fig. 5(a) and (b). Therefore, the double Walker breakdowns for each DW of the two magnetic layers can explain the experimental observation of the exotic 𝑣DW variation.
The magnetic field for the breakdown is denoted as the Walker breakdown field, 𝐻W.
In addition, when 𝐻𝑥 = 0, both the upper and lower DWs have the same |𝐻tot| value, resulting in identical strengths of 𝐻W. Therefore, both DWs exhibit a single Walker breakdown simultaneously, as shown in Fig. 3(f). However, under the application of 𝐻𝑥, the upper and lower DWs have different values of |𝐻tot|, resulting in different strengths of 𝐻W. This could possibly result in two separate Walker breakdowns. As 𝐻𝑥 increases, such a separation becomes more obvious and dominant, as shown in Figs. 3(g)–(j). In summary, we investigated the phases of the DW motion among two magnetic layers. The micromagnetic simulation results reveal that each magnetic layer has a different internal dipolar magnetic field, and thus the DW in each magnetic layer experiences different magnetic fields. Owing to the different Walker breakdown fields for each DW, the 𝑣DW variation exhibits two minima with respect to 𝐻𝑥 . The presented simulation results satisfactorily explain the experimental observation of the two minima in the 𝑣DW variation.
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Acknowledgements
Authors thank Soong-Geun Je for inspirational discussions. This work was supported by Samsung Electronics Co., Ltd, the Samsung Science and Technology Foundation (SSTF-BA1802-07), and National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT (MSIT) (2015M3D1A1070465 and 2020R1A5A1016518).
Author Contributions
J.M. and D.-Y.K. designed this study. J.M. and S.-H.L. fabricated samples. J.M. and K.K. performed experiments. J.M. and J.Y. performed simulations. J.M., J.Y. and S.-B.C analyzed the data and wrote the manuscript. All authors discussed the results and commented on the manuscript. J.M. and J.Y. contributed equally to this work
Competing interests
The authors declare no competing interests.
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No competing interests reported.