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Enhanced gyration-signal propagation speed in one-dimensional vortex-antivortex lattices and its control by perpendicular bias field

We report on a micromagnetic simulation study of coupled core gyrations in one-dimensional (1D) alternating vortex-antivortex (V-AV) lattices formed in connected soft-magnetic-disk arrays (round-shaped modulated nanostrips). In the V-AV lattices, we found very characteristic standing-wave modes of the coupled gyrations as well as efficiently ultrafast gyration-signal propagation between vortices through the neighboring antivortices, as originating from their combined strong exchange and dipole interactions. Collective core oscillations in the V-AV networks are characterized as unique two-branch magnonic bands that are affected by the polarization ordering between the neighboring vortex and antivortex and controllable by externally applied perpendicular fields each of different field strength and direction. The gyration-signal propagation speed is much faster than that for 1D disk arrays composed only of vortex states, and the propagation speed for the parallel polarization ordering is increased, remarkably, to more than 1 km/sec by application of perpendicular static fields. This work provides a fundamental understanding of the coupled dynamics of topological solitons as well as an additional mechanism for ultrafast gyration-signal propagation; moreover, it offers an efficient means of significant propagation-speed enhancement that is suitable for information carrier applications in continuous nanostrips.

In the meantime, the magnetic antivortex, the topological counterpart of the vortex, has been found to exhibit core gyration and related switching behaviors similar to vortex dynamics. [25][26][27] Periodic vortex and antivortex arrangements have often been found as parts of cross-tie walls. 28,29 Isolated antivortices by contrast, due to their unstable state, possibly can be formed in specially designed geometric confinements. [25][26][27]29 The antivortex also appears together with vortices in dynamic transient states during vortex-core reversals in nanodots [7][8][9][10][11] and magnetization-reversal dynamics, 30 domain-wall motions in nanostrips 31,32 or highly nonlinear chaos dynamics, 33 because the total topological charge is always conserved during such dynamic phenomena. 34,35 Although the dynamics of single antivortices [25][26][27] and their dynamic interactions with vortices 30,36,37 have been reported in earlier studies, the dynamics of coupled vortices and antivortices in round-shaped modulated nanostrips and their coupled gyration propagation have not been well understood in the magnonic band aspect or in terms of gyration-signal propagations through 1D alternating vortex-antivortex (V-AV ) lattices.
In this letter, we report on the fundamental dynamic behaviors of interacting vortices 3 and antivortices in 1D V-AV lattices, specifically in terms of their coupled excitation modes, very unique two-branch band structures, and gyration-signal propagation speed. We found that gyration-signal propagations are much faster (i.e., more than 1 km/s) than those for 1D disk arrays composed only of vortex states, as originating from the strong exchange-dipole interactions between neighboring vortices and antivortices. Also, we discovered that the unique band structure and gyration-propagation speed are controllable according to the strength and direction of perpendicular static fields. This study provides not only fundamental insights into the dynamic interactions between different types of topological solitons but represents the first step toward their implementation in spin-based information signalprocessing devices.
In the present study, we used, for a model system, alternating V-AV lattices in a connected triple-disk structure wherein the diameter of each disk is 2R = 303 nm, the thickness T = 20 nm, and the center-to-center interdistance Dint = 243 nm, for cases of R < Dint < 2R (see Fig. 1). Unlike physically separated disks (i.e., Dint > 2R ), this connected triple-disk structure has two antivortices between neighboring vortices of the same counterclockwise (CCW) chirality in a metastable state. In the initial state, the core magnetizations of the three vortices are upward (called polarization p = +1), and those of the two antivortices are downward (p = -1), resulting in the antiparallel polarization ordering. We numerically calculated the motions of local magnetizations (cell size: 3×3×T nm 3 ) using the OOMMF code, 38 which employs the Landau-Liftshitz-Gilbert equation. 39 We used the typical material parameters of Permalloy (Ni80Fe20, Py): saturation magnetization Ms=8.6×10 5 A/m, exchange stiffness Aex= 1.30×10 -11 J/m, and zero magnetocrystalline anisotropy.
For excitation of all of the coupled modes in such a model system, we applied a static 4 local field of 200 Oe in the + y direction only in the left-end disk, so as to displace the core position to ~54 nm in the -x direction. Upon turning off the local field, the five individual cores of the vortices and antivortices were monitored to extract the trajectories of the coupled core motions under free relaxation. All of the simulation results noted hereafter were obtained up to 200 ns after the field was turned off. respectively, spreading in a wide range from a few hundreds of MHz to the GHz level. Each peak in each frequency spectrum is located at the same corresponding frequency from V1 through V5, indicating that there are five distinct modes in the whole system. Because of the intrinsic damping of each core gyration, those peaks are somewhat broadened and overlapped with the neighboring peaks. There are noticeable differences between the vortex and antivortex-core motions: in antivortex motions, that is, in the AV 2 and AV 4 spectra, the ω3 5 peak disappears, while in the V3 spectrum, the ω2 peak disappears and the ω4 peak becomes very weak. In order to obtain those data with a better spectral resolution, we carried out further simulations with the same model but with an extremely low damping constant, α = 0.0001. The resultant FFT spectra (right column of Fig. 2(b)) show that the five major peaks become sharper. Moreover, for cases of antivortex gyration motions, the higher-frequency ω4 and ω5 modes, which were very weak for α = 0.01, are much stronger for α = 0.0001, due not only to the negligible energy loss but also to the strong exchange interaction between the antivortices and the neighboring vortices. The very small subsidiary peaks in the FFT spectra might be associated with higher harmonics and the nonlinear effect of the complex geometric confinements (e.g., the notch-shaped boundaries) or the discreteness of the core positions due to the finite cell size.
Making inverse FFTs of all of those peaks of each mode with α = 0.0001, we obtained the spatial correlation between the five cores' motion for each mode, as shown in Fig. 3.
The trajectories of the orbiting cores in one cycle period (2π/ω) of gyration starting from about 100 ns are compared for all of the modes, along with the corresponding profiles of the Y components. The Y component profiles contrast markedly between the modes. The shapes of all of the trajectories are eccentric along either the x or y axis, because the strong exchange interaction between the antivortices and the neighboring vortices is additionally employed in such a connected-disk model, with its high asymmetry between the x and y axes. The most noteworthy feature is the fact that the collective core motions show a standing-wave form with a different overall wavelength for a given mode. The orbiting radii of the five cores for a given mode are symmetric with respect to the center of the whole system (i.e., at V3) and are also completely pinned at the imaginary points at both ends, denoted AV 0 and AV6, as reported in Refs. [20,24]. The lower ω1, ω2, and ω3 modes and the higher ω4 and ω5 modes 6 are quite different in terms of their standing-wave forms. For the ω1 mode, all of the cores move in phase, while for the ω2 mode, the core of V3 acts as a node, and for the ω3 mode, the two antivortex cores act as nodes. For the ω4 and ω5 modes, the two antivotices are highly excited relative to the three vortices. Also, the antivortices' core motions are permeated into the bonding axis, thus resulting in higher energy states due to their strong exchange interaction with the neighboring vortices. The difference between the individual modes can be understood by the relative phases between the vortex's and antivortex's effective magnetizations, <M>, induced by their own core shifts and consequently by their dynamic dipolar interaction, as discussed in Ref. 20. The relative phases of the <M> between the neighboring core gyrations determine the mode's average dynamic dipolar energy. However, the strong exchange interaction in such a connected thin-film strip must also be taken into account. Due not only to the unknown potential wells of the antivortices but also the complex asymmetric exchange interaction terms between the vortices and antivortices, it is difficult, in such V-AV lattices, to separately extract the individual contributions of the exchange and dipolar interaction terms to the individual coupled core motions in each mode. A distinct feature of the V-AV coupled lattices is that the strong exchange coupling between the dynamic vortex and antivortex motions dominantly contributes to each mode and differently contributes to the individual modes, as evidenced by the presence of the higher frequency modes that would not be found in vortex-state arrays. 20 Next, on the basis of the above findings, we extended our study to longer 1D chains, as shown in Fig. 4(a). Here we consider two different polarization orderings with the same CCW chirality for all of the individual vortices: antiparallel and parallel orderings represented by (pV, pAV) = (+1, -1) and (pV, pAV) = (+1, +1), respectively. For such V-AV finite lattices, the thickness was set to 40 nm to make them more stable than in a thinner strip 7 and we applied a static local field of 200 Oe in the -y direction only in the left-end disk. The other simulation conditions were the same as those for the earlier model shown in Fig. 1.
From the FFTs of the x components of all of the 25 core positions, we obtained dispersion curves in the reduced zone scheme for the antiparallel and parallel polarization orderings, as shown in Fig. 4(b). Here, we assume that the interdistance between the neighboring vortices and antivortices is equal to the average value, int  In fact, this connected-disk system is analogous to a nanostrip with round-shaped width modulations. Thus, from the technological perspective, as with domain-wall motions in a given nanostrip, gyration-signal propagation through alternating V-AV lattices in a planarpatterned nanostrip can be used as an information carrier. In order to estimate the gyrationsignal propagation speeds, we plotted the displacements of the individual cores from their own center positions in the whole system with time, as seen in Fig. 4(c). This figure clearly indicates that the excited gyrations propagate well through the entire connected-disk 9 nanostrips. The two slopes noted by the white lines compare the substantial propagation speed difference between the parallel and antiparallel polarization orderings. The speed for the parallel (antiparallel) polarization ordering is 0.83 (0.64) km/s, representing a 30% increase relative to the antiparallel ordering. It is known that in the case of dipolar-coupled arrays composed only of vortex-state disks, the gyration propagation for the antiparallel polarization ordering is faster than that for the parallel polarization ordering, owing to the stronger dynamic dipolar interaction, as determined by the relative rotation senses of the two core gyrations. 18,20 However, for the cases of V-AV lattices the faster propagation speed for the parallel ordering than for the antiparallel ordering cannot be explained only in terms of the dynamic dipolar interactions between the vortices and antivortices. Although we could not understand why the exchange interaction affects the faster gyration propagation for the parallel polarization ordering, we could confidently conclude that the exchange interaction between the vortices and antivortices gives rise to a faster propagation speed for either polarization ordering in the V-AV lattices than in physically separated disk arrays of only vortex states. We also note that the gyration-signal propagation through the antivorticies is comparable to or faster than the domain-wall speeds in nanostrips. 42 This is another antivortex-mediated mechanism for faster gyration-signal propagation in continuous nanostrips.
Next, in order to control the observed two-branch band structure and the gyration- Owing to the combined strong exchange and dipolar coupling between the neighboring vortices and antivortices, the vortex and antivortex's roles would not be discriminated separately in those band structures within most k ranges.
In addition, we also estimated the gyration-signal propagation speeds versus Hz for both polarization ordering cases. Figure 6 to the following several advantages of this signal-propagation mechanism and its applicability to any potential spin-based signal-processing devices: 1) such V-AV lattices can be made of simple, single-material, round-shaped modulated patterned thin-film nanostrips; 2) the parallel polarization ordering between neighboring vortices and antivortices is readily available by application of a strong magnetic field perpendicularly to the strip plane; 3) such ultrafast gyration-propagation speed is caused by the combined strong exchange-dipolar coupling between the neighboring vortices and antivortex, and accordingly the propagation speed is higher than 1 km/sec and controllable by application of Hz.
In summary, we studied the dynamics of coupling between vortices and antivortices in their alternating 1D periodic arrays. Standing-wave discrete modes and their dispersion relations were found to have characteristics of two branches: lower and higher bands. These lower and higher bands are strongly influenced by the polarization ordering between the neighboring vortex and antivortex, and thereby are controllable. We found faster gyrationsignal propagations in continuous nanostripes composed of alternating vortex and antivortex lattices for the parallel polarization ordering than for the antiparallel one; both are much faster than only-vortex-state arrays. Also, the band structures and gyration-propagation speed are markedly variable by means of the perpendicular bias field. This work provides an additional mechanism of ultrafast gyration-signal propagation via exchange-dipole interaction, and offers, moreover, a fundamental understanding of vortex and antivortex coupled dynamics in one-dimensional magnonic crystals.