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Spin torque nano-oscillators (STNO) generate high frequency output voltage signals upon injection of a spin polarized DC current into a magnetic tunnel junction. For their applications in wireless communications or neuromorphic computing schemes, a central aspect is the control of their phase dynamics via injection locking to an external rf signal [1, 2]. Many interesting, non-linear effects may arise at higher excitation levels that manifest themselves in the Arnold tongue [3, 4]. For the in-plane magnetized STNO (in-plane magnetized polarizer and free layer) [2], we provide here analytical expressions for (i) the asymmetric Arnold tongue boundaries, arising from the dependence of the forcing on the STNO power; (ii) the Takens-Bogdanov bifurcation threshold [4] which leads to a reduced locking range and (iii) the hysteresis boundaries [3,4]. These effects are discussed within a common model for ‘pure’ rf signals arising from either a spin polarized current or a magnetic field. ‘Pure’ refers to a well-defined direction &lt;i&gt;u&lt;/i&gt;&lt;sub&gt;i&lt;/sub&gt; (i=x,y,z ; &lt;i&gt;u&lt;/i&gt;&lt;i&gt;&lt;sub&gt;x&lt;/sub&gt;&lt;/i&gt; the STNO easy axis) of the spin polarization or a magnetic field [2]. In a further study we also investigate synchronization at harmonic order n=1 (rf signal frequency close to the STNO frequency) for mixed spin polarization with x- and y- components. The important and unexpected finding here is that the STNO phase does not follow an arcsine (Adler-type) behavior [5] within the locking range, but shows a discrete jump that can be as large as Pi/2. Our analysis shows that the phase jump results from a competition between locking to the &lt;i&gt;u&lt;/i&gt;&lt;sub&gt;x&lt;/sub&gt;- and &lt;i&gt;u&lt;/i&gt;&lt;sub&gt;y&lt;/sub&gt;-components of the rf signal, where locking to the &lt;i&gt;u&lt;/i&gt;&lt;sub&gt;x&lt;/sub&gt;-component at n=1 is enabled through the fact that the free running oscillation power is non-constant with frequency components at f. This demonstrates that, for mixed rf signals, injection locking can be more complex, and that the results can be of interest for applications such as the phase jump for phase shift keying concepts [6].&lt;br/&gt;MIG was supported by the French Space Agency (CNES) and the CEA NUMERICS program, which has received funding from the European Union&#39;s Horizon 2020 program (No 800945). Partial support was provided by the French National Research Agency (ANR-20-CE24-0002).

2022 Joint MMM-INTERMAG

Arnold tongue of injection locked spin torque nano-oscillators under high and mixed driving signals

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INTERMAG 2021

Spin torque nano-oscillators (STNO) generate high frequency output voltage signals upon injection of a spin polarized DC current into a magnetic tunnel junction. For their applications in wireless communications or neuromorphic computing schemes, a central aspect is the control of their phase dynamics via injection locking to an external rf signal [1, 2]. Many interesting, non-linear effects may arise at higher excitation levels that manifest themselves in the Arnold tongue [3, 4]. For the in-plane magnetized STNO (in-plane magnetized polarizer and free layer) [2], we provide here analytical expressions for (i) the asymmetric Arnold tongue boundaries, arising from the dependence of the forcing on the STNO power; (ii) the Takens-Bogdanov bifurcation threshold [4] which leads to a reduced locking range and (iii) the hysteresis boundaries [3,4]. These effects are discussed within a common model for ‘pure’ rf signals arising from either a spin polarized current or a magnetic field. ‘Pure’ refers to a well-defined direction ui (i=x,y,z ; ux the STNO easy axis) of the spin polarization or a magnetic field [2]. In a further study we also investigate synchronization at harmonic order n=1 (rf signal frequency close to the STNO frequency) for mixed spin polarization with x- and y- components. The important and unexpected finding here is that the STNO phase does not follow an arcsine (Adler-type) behavior [5] within the locking range, but shows a discrete jump that can be as large as Pi/2. Our analysis shows that the phase jump results from a competition between locking to the ux- and uy-components of the rf signal, where locking to the ux-component at n=1 is enabled through the fact that the free running oscillation power is non-constant with frequency components at f. This demonstrates that, for mixed rf signals, injection locking can be more complex, and that the results can be of interest for applications such as the phase jump for phase shift keying concepts [6]. MIG was supported by the French Space Agency (CNES) and the CEA NUMERICS program, which has received funding from the European Union's Horizon 2020 program (No 800945). Partial support was provided by the French National Research Agency (ANR-20-CE24-0002).

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The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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Magnetic antiskyrmions (ASk) are topological magnetic textures that can be stabilized in ultrathin disks by anisotropic Dzyaloshinskii-Moriya interaction (DMI)1. In addition to their recent experimental observation2,3, they have attracted a lot of attention and their dynamic behaviour has been studied in thin nanowires1 and confined nanodisks4. To further our understanding we investigated the resonant modes of magnetic antiskyrmions in antiferromagnetic-exchange coupled thin disks, i.e. in a bilayer SAF, for different microwave excitations. Their response is, as expected, similarly rich to the dynamic behaviour of skyrmions in SAF5, and the spectra revealed several gyrotropic and breathing modes for in and out of plane excitation, respectively. By analysing the trajectories of the antiskyrmion guiding centre in the two disks as well as the spatial distribution of the spectral power we have identified a variety of coupled modes in such a system. Furthermore, we show that these modes can be controlled by tuning the strength of the interlayer exchange coupling between the two disks.

Antiskyrmion excitation modes in antiferromagnetic-exchange coupled disks

Magnetic Hopfions are three-dimensional topological solitons which are considered as a promising candidate for future spintronics devices and non-volatile magnetic storage. These 3D spin textures are found to be stabilized in frustrated, chiral magnets or materials hosting target skyrmions [1-3]. In this work, we have investigated the stability of Hopfions and their spin-wave behavior in individual nanostructures of thin films of Co/Pt with interfacial DMI. We have performed micro-magnetic simulations to understand in details the process of nucleation and annihilation of Hopfions in nanostrutcures in presence of an external magnetic field. Our results demonstrate a narrow region in the hysteresis loop where Hopfions are stabilized (Fig.1). The spin-wave dynamics in this topological structures were studied by perturbing the system with an external transient square field pulse of duration 60 ps and amplitude of 3 mT applied along the x-axis (in the plane of the hopfion). We identified a fundamental spin-wave mode which is excited at about 2 GHz. The nature of this mode is investigated by carrying out fast fourier transform (FFT) at each cell of the discretized geometry. The profile of the mode is investigated by analyzing the image created by plotting the normalized mode-power as well as the phase at the corresponding locations of the nanostructures. Our results suggest that the mode exhibits maximum power in the region of the Hopfions where spins have strong z-components. Furtermore, the excited mode appears to be independent of the external magnetic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/e34473752509d96cae2821cba32de829.jpg)

Fig. 1. (a) Field dependece of the normalized magnetization in Co/Pt nanodisc. Inset shows the spin configuration of hopifion stabilized in the highlighted region of the magnetization vs field curve, (b) Spin wave spectra calculated in individual hopfion stabilized in Co/Pt nanodisk exhibiting interfacial DMI. Inset shows the power profile of the excited fundamental mode at 2.03 GHz.

Magntic Hopfions in Co/Pt nanostructures and their spin-wave behavior

Sequences of geometrically confined magnetic domain walls (DWs) in micrometer long, flat wires currently hold good promises as a possible system for designing new magnetic memories cell devices. Due to the strong shape anisotropy, narrow and flat rectangular wires (FRW), made of soft ferromagnetic (FM) materials, are prone to have the magnetization oriented along the wire edge. Such wires may turn into possible platforms to hold sequences of DWs, provided a periodic sequence of pinning centers might be created along the wire. A bilayer consisting of an FRW exchange coupled to a two-sublattices vicinal antiferromagnetic (AF) substrate fulfills the main requirements to construct a periodic sequence of pinned DWs.1,2 This may lead to a corresponding arrangement in the FM film, which may turn structured with alternating domains separated DWs centered at the step’s edges. We presently report a theoretical study of spin-polarized current domain wall (DW) release, considering a vicinal AF substrate with a small miscut angle so that there is no interaction between adjacent DWs. We consider a DW separating two opposite domains in a FM (Py or Fe) nanostrip coupled to a two-sublattice uniaxial AF substrate. In the absence of electric current, the DW center is on the interface step defect. We consider a spin-polarized current applied along the easy axis AF. The spin-transfer torque is restricted to the DW region. Our results indicate that the electric current density threshold value (J*) to detach the interface pinned DWs increases as the interface exchange energy increases. Furthermore, J* is smaller for Py/AF nanostrips than for Fe/AF nanostrips with the same geometrical dimensions. For instance, for an interface field value of 200 Oe, considering a Py nanostrip consisting of 125 nm long, 25 nm wide, and 10 nm thick domains separating the DWs, we have found that, for small values of electric current density, the DW center moves from the interface step defect, contrary to the electric current direction, for around 27.5 nm, until J*=29x107 A/cm2. The corresponding values for an Fe nanostrip with the same dimensions and interface field strength are 15 nm and J*=34x107 A/cm2.

Domain wall depinning from FM/AF interface defects by spin-polarized current

Magnetic nanowires (MNWs) are increasingly important for applications in memory [1], biolabels [2], and nanowarming of cryopreserved organs [3]. Object Oriented Micromagnetic Framework (OOMMF) was used to visualize moments inside MNWs. The magnetic properties of MNWs were investigated as a function of nanowire shape. Experimental measurements of magnetic nanowires (MNW) often reveal features that are difficult to explain due to statistical shearing of features and interaction fields in large arrays [4]. Measurements of individual nanowires can be used to remove interaction field effects [5], and other measurements such as first-order reversal curves (FORC) [6] and the projection method [7] often attempt to statistically separate bulk effects. For nickel MNWs with fixed length of 3 μm, the coercivity and the remanence are negatively correlated to the diameter of the wire (Fig.1). When the diameter is increased to 200 nm, a ‘wasp waist’ appeared in the hysteresis loop, Fig.2. (solid line) The reversal process is three-folded. First, since there is a large area for curling in 200nm-diameter nanowires, the magnetic moments at the tips of the MNW fall from pointing along z-axis to aligning in the xy-plane and in the form of vortex. Next, the outer shell of moments turns from in-plane to the negative z-axis. Finally, the axial core of the MNW switches and the wire is fully saturated in the negative z-direction. Interestingly, at Hc , the two vortex walls converged in the center of the MNW and a three-dimensional vortex(hedgehog) [8] was observed using a 2D cross-sectional view, Fig.2 (b). Next, when flat-tipped cylinders were replaced with ellipsoids so that the tips of wire had infinitesimally small areas where vortex was no longer supported, the coercivity of the nanowires significantly increased (Fig.1 and 2, dashed line)

![](https://assets.underline.io/uploads/markdown_image/1/image/e49db041cb0a1b0259e8e6a56d35c4b1.jpg)

Fig.1 Hysteresis loops for nickel MNWs with diameter of 30 nm(black), 50 nm(red) and 100 nm(blue) in cylinder shape (solid line) and ellipsoid shape (dashed line).

![](https://assets.underline.io/uploads/markdown_image/1/image/c86cfaf97d6b8d26bb6cd72e78d6f7a2.jpg)

Fig.2 (a) Hysteresis loop for 200 nm nickel MNWs with cylinder shape (solid line) and ellipsoid shape (dashed line). (b) The corresponding 2D and 3D moment map

Magnetic Properties Affected by the Size and Shape of Magnetic Nanowires

The antiferromagnetic (AFM) medium is an ideal candidate for emitting terahertz (THz) electromagnetic (EM) waves, which have extraordinary potential application in information technology and biological medicine [1–4]. In this work, we report the theoretical and numerical investigation about the emission of terahertz (THz) spin waves from a moving AFM DW under the magnetic anisotropy energy gradient (dEa/dx). The DW is accelerated by dEa/dx that can be generated using the technique displayed in Fig. 1: A DC voltage applied across the wedge-shaped insulating layer generates dEa/dx, which triggers the AFM DW to move towards the end with a lower magnetic anisotropy energy [5]. Based on an atomistic spin model, we have found that the DW velocity exceeds the relativistic limit and keeps increasing, which is accompanied by the broadening of the DW width. This is different from a traditional relativistic AFM DW motion with an upper limit of the DW velocity and Lorentz contraction [6]. The DW acceleration effectuates the DW precession. When the frequency of the DW precession is greater than the frequency gap of the spin wave propagating in the AFM medium, a continuous THz spin wave is resonantly emitted. The frequency of the spin wave keeps increasing with the decrease of the anisotropy energy constant, which seems to be contrary to the AFM spin wave obeying the relativistic-like dynamics. Our results demonstrate that obeying the relativistic dynamics is not necessary for the spin-wave emission from a moving AFM DW.

![](https://assets.underline.io/uploads/markdown_image/1/image/b3d5ff6329dec082c7c8fd30cb4693b4.jpg)

Fig.1. Schematic of calculation model: emission of THz spin wave from a moving DW triggered by a voltage-induced gradient of magnetic anisotropy energy (the emission of the spin wave is accompanied by an increase of the DW width due to the reduction of the magnetic anisotropy energy).

Resonant spin-wave emission from the moving AFM DW without Lorentz invariance

Magnetic domain walls (DWs), as a novel information carrier, provide a reliable technique for modern information processing. In previous study, to control the movement of magnetic DWs, different external excitations, such as magnetic field, spin transfer torque, spin-orbit torque, electric field, and spin waves, etc., have been used as the driving force for the DW motion. In this work, we combine electromechanical and micromagnetic simulations to explore a promising way to manipulate the perpendicular magnetic anisotropy (PMA) DW motion in a given direction using voltage-controlled gradient strain. In particular, we show the voltage induced gradient strain driven the DW motion more complicated than simple uniform linear motion, especially for DW motion in a finite length nano-stripe. As shown schematically in Fig. 1(a), the proposed device consists of a ferromagnetic (CoFeB)/heavy metal (FM/HM) nanostrip deposited on a piezoelectric substrate. The initial spin configurations with different i-DMI coefficient are obtained by relax the DW along the y direction. Fig. 1(b) shows the snapshots of magnetization with various i-DMI coefficient when simulated 10 ns after the strain gradient applied. As shown in Fig. 1(b), the direction of the DW will parallel to the x direction due to the i-DMI and form a Neel type DW, while a Bloch type DW formed without i-DMI. From the micromagnetic simulation results shown in Fig. 1(b), along with the DW displacement, DWs are found to tilt away from the y-axis, which agrees with the experimental observations. From these magnetization images, we can also directly see an increment of DW tilting angel as the i-DMI constant increase. These results indicates that strain gradient induced PMA DW motion can be used to develop next generation artificial intelligent memory and logic devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/66bfbae9388ee43210a60dde3c0ce6ad.jpg)

Fig. 1 (a) Schematic view of the investigated strain driven domain wall in a nanostripe with voltage-controlled strain gradient. (b) Snapshots of the magnetization pattern with different i-DMI coefficient.

Dynamics of Domain Wall Induced by the Voltage-controlled Strain Field Gradients

Spin-based neuromorphic computing (NC) is investigated as a power-efficient enabler for artificial intelligence [1-3]. Hence, magnetic tunnel junction (MTJ) based domain wall devices are investigated as synthetic synapses and neurons to mimic their biological counterparts [2-3]. While a biological neuron resets itself after firing, such a function has not been demonstrated experimentally in spintronic systems [3]. The self-reset feature of neurons reduces the power consumption and the complexity of NC architecture. In this work, we present a domain wall (DW) based artificial neuron that automatically resets without applying a reset pulse. At first, we simulated a synthetic antiferromagnetic (SAF) DW device where the free layer (CoFeB) is driven by spin-transfer-torque (STT) to achieve integrate function. Leakage function was achieved when the antiferromagnetic coupling (AFC) between the free and pinned layers pushes the DW towards the starting position automatically (Fig. 1). The left quarter region of the free layer is ferromagnetically coupled (FC) to the pinned layer so that as DW returns (during reset), it is stabilized at the initial position (Fig. 2(a) and Fig. 2(b)). Experimentally, the integrate and leakage functions are also achieved as illustrated in Fig. 2(d) and Fig. 2(e), suggesting that an SAF-based neuron is viable for NC.

![](https://assets.underline.io/uploads/markdown_image/1/image/4f911e30d06059db93a1e69c895660a8.jpg)

Fig.1 Proposed design of the artificial neuron. Domain wall is moved forward by STT and (after firing), reset backward by SAF coupling.


![](https://assets.underline.io/uploads/markdown_image/1/image/c297c876d46646d24593590c9f897b4d.jpg)

Fig. 2 (a and b) DW position (free layer) vs time graphs for different AFC strength in the simulations. (c, d and e) Kerr images of the fabricated SAF neuron device with magnetic field in the -z direction. (c) No current applied and device is in antiparallel configuration, (d) current is applied, pushing the DW and achieving the integrate function, and (e) current is removed and the device automatically resets, indicating the leakage function.

Synthetic antiferromagnetic domain wall device as artificial neuron for neuromorphic computing

Domain wall (DW) dynamics has been intensively studied as the base for novel spintronic devices [1-2]. These dynamics have attracted special attention in ultrathin ferromagnetic strips that show high perpendicular magnetic anisotropy (PMA) since the formation of ‘up’ or ‘down’ domains, separated by DWs, can be exploited to store bits. One of the challenges is the initial nucleation of the domains in sequence, typically performed by an Oersted magnetic field from an attached microwave ‘bit line’ orthogonal to the strip [3]. However, fabrication constraints due to the size of the bit line sets a lower bound for the bits width and, more importantly, the Oersted field that nucleates a DW can annihilate a shifted DW in the strip [3]. Solutions based on a double bit line have been proposed [3], but these still require injecting external currents and a more intricate and multilayer design. In this work, we propose a different approach based on a fluctuating DW as the source. The DW is pinned at one end at the junction of a Y-shaped strip (see red solid circles, Fig. 1) as the DW fluctuates across the strip width (see Fig. 1(b)-(d)). The pinned DW can be originally nucleated by using a bit line. Then, it is forced to oscillate by applying non-localized, pulsed currents (JHM) in the heavy metal (HM) in a controlled manner. Such oscillation injects subsequent DWs that travel along the strip. We characterize the obtained ‘domain sequence’ as a function of duration (d) and periodicity (T) of the current pulses. Particular combinations result in ‘coherent’ sequences of domains (see red dashed circles in Fig. 1(e)-(f)) and/or domain reversal, which may be useful to develop DW-based applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/8e6fb080c602a3bfb788123271a304d5.jpg)

Fig.1. Nucleation of two domains (see red dashed circles in (f)) in the Y-shaped ferromagnetic nanostrip. Black and white are ‘down’ and ‘up’ domains, respectively (see green symbols), and grey represents the HM layer beneath the strip. A pulsed current (JHM) is applied in the x-direction over the entire HM. (a)-(d) Snapshots from a full fluctuation of the DW during a pulse of 0.6 ns duration, and (e) after a stabilisation period (JHM=0) of the nucleated domain (~90 nm width) at 1.4 ns. (f) Results after two pulses of period 1.4 ns.

Fast and Controlled Sequential Nucleation of Narrow Domains in a Ferromagnetic Nano-strip with Non-localised Electrical Currents

One of the tools to move a skyrmion is an injected current[1,2], which gives rise to the risk of producing excessive heat. On the other hand, spin-waves (SW) are a promising way to drive skyrmions without producing such high temperatures and hence affecting the skyrmionic device's performance. Most recent work on SW-driven skyrmion motion is based on micromagnetic simulations on single-layers with either small magnetic track width (W), high SW excitation rf field (Arf), or low Gilbert damping constant α[3,4]. However, to complement the realization of skyrmion motion experimentally, simulations should meet experimentally achievable conditions at the same time: W ≥100nm, Arf ≤100mT and α &gt;0.05. The dataset and characteristic analysis we present concern SW driven skyrmion simulations by OOMMF[5] with fixed parameters, D=1.5 mJ/m2, MS=1004 KA/m, A=3.6 pJ/m, Ku=1 MJ/m3. We varied the α=0.02-0.06, W=40-100 nm and Arf=100-800 mT. The size of the relaxed skyrmion is from 20nm to 23nm when W increases from 40nm to 100nm, so there is more than 5 nm between the skyrmion to each sample edge at initial stage. Fig.1 shows the skyrmion displacement (dsky) with time along the SW propagation direction for various W is shown. When W &gt;50 nm, the skyrmion will initially move opposite to the SW propagation direction before moving forward at longer times. Fig.2 shows that with larger α, the skyrmion will have smaller initial velocity when moving backward. If the leading edge of the skyrmion is regarded as a planar wall on the sample, the domain wall (DW) plane is predicted to rotate when linear momentum transfer between the SW and DW occurs. The angle between the DW wall plane and the out-of-sample direction is defined as DW wall plane angle (ΦDW)[6]. Also, the inset shows that ΦDW of skyrmion leading edge has similar behaviour to the dsky, but with an offset. The skyrmion backward motion gives a possible observation result in the future and guidance about finding a balance between W and magnetic properties to have skyrmions with aiming motion type.

![](https://assets.underline.io/uploads/markdown_image/1/image/20d3a9acd993b9c5372ac498eb39b2f1.jpg)

Fig.1 Time-dependent skyrmion displacement in nm for various W.

![](https://assets.underline.io/uploads/markdown_image/1/image/3a0f9f542c8a882ca49dfb2dc79008fd.jpg)

Fig.2 Skyrmion displacement with time for various α. Inset: comparison between ΦDW and skyrmion displacement when α=0.06.

Transient retrograde motion of spin wave-driven skyrmions in multilayer wave guides

Magnonic devices are based on the wave properties of spin wave (SW), so that one can transport more information in a single device if modes of SWs are freely controllable. SWs in a narrow waveguide have eigenmodes due to standing waves formed in the width direction and the thickness direction [1, 2]. However, excitation of one specific mode of propagating SW has not been realized, yet. Here, we report that high SW mode can be selectively excited by employing SW conversion. We numerically investigated SW propagation in a T-shaped structure using MuMax3 [3]. A 5-nanometer-thick Py film is assumed as the device and a magnetic field of 70 mT is applied along the y-direction (Fig. 1). By applying an rf field to the cells 1 μm away from the junction, Damon-Eshbach spin waves (DESWs) are excited and travel along the x-direction. The DESWs induce a magnetization precession in the junction, which excites SWs in the magnetostatic backward volume wave configuration (BVSWs). First, we varied the SW resonant frequency (f=12, 18 GHz) using a device having w1=500 nm and w2=100 nm. As shown in Fig. 2(a) and (b), the wavelength of the DESW (λDESW) depends on f which is well known [2]. In addition, the BVSWs clearly change their wave front. To extract the mode number, we applied a Fourier transformation (FT) along the y-direction. The FT results indicate that the 1st mode is dominantly excited at 12 GHz but the 2nd mode is dominant at 18GHz (Fig. 2(d) and (e)). We next changed the device to have w2=500 nm and repeated the simulation at 18 GHz (Fig. 2(c)). The result indicates the 12th mode was dominant at this condition (Fig.2 (f)). To further understand the high-mode-excitation, we varied the frequency and plotted the FT peak height of each mode as a function of the ratio of the width to λDESW (Fig.2 (g)). The result implies that modes can be controlled by optimizing a T-shaped waveguide.

![](https://assets.underline.io/uploads/markdown_image/1/image/29fb2eeaedfa09fa59c4d092aa944455.jpg)

Fig.1. Overview of the calculation area. SWs are excited at the orange area.

![](https://assets.underline.io/uploads/markdown_image/1/image/fc18c5114e3e73123e5e447b234cee46.jpg)

Fig.2. (a)-(c) The z-component of the magnetization and (d)-(f) their corresponding FT of BVSWs. (f, w2) is (a, d) (12 GHz, 100 nm), (b, e) (18 GHz, 100 nm), and (c, f) (18 GHz, 500 nm). (g) The peak height of FT of each mode obtained from the device with w2=500 nm.

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Antiskyrmion excitation modes in antiferromagnetic-exchange coupled disks

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