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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].&lt;br/&gt;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).&lt;br/&gt;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 &lt;i&gt;x-&lt;/i&gt;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.

2022 Joint MMM-INTERMAG

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

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.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

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This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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.

poster

Unlike traditional memories such as SRAM and DRAM, STT-MRAM has many advantages, especially non-volatility, which has attracted the attention of many researchers in recent years. Nowadays, MRAM is one of the most competitive candidates for next-generation memory. However, spintronic devices require large switching current to achieve fast switching speed, which leads to more power consumption and heat fluxes of MRAM. Power budget and thermal issues limit the 3D stacking of MRAM. Therefore, it is necessary to establish a thermal model of the power-related 3D stacked MRAM in the early stage of design. Since in the duality between thermal and electrical phenomena [1], heat flow is analogous to current, temperature is analogous to voltage, thermal conductance is analogous to electrical conductance, and thermal capacity is analogous to electrical capacitance, a RC thermal model can be established, which can avoid using computationally intensive and time-consuming detailed finite-element model [2]. Because 3D stacked MRAM faces cooling problems, it is not only necessary to consider the circuit-level memory layer when analyzing the thermal behavior, but also to take the heat sink responsible for heat dissipation into consideration from the perspective of package level and model together. The content of this paper is organized as follows. Section I is the introduction,which introduces 3D stacking technology and puts forward the thermal issues of 3D memory. Section II presents various thermal modeling approaches. Section III proposes a power and thermal model for 3D-stacked MRAM as shown in Fig. 1, which also including the simplified memory circuit. In Section IV, the effects of the number of stacked layers, the size of dies, power density, and heat dissipation capacity on temperature are evaluated based on the established RC thermal model. Section IV shows the conclusions. It is demonstrated that this model facilitates the communication between the circuit designer responsible for power consumption and the package designer responsible for heat dissipation in the early stage of MRAM design.

![](https://assets.underline.io/uploads/markdown_image/1/image/c2c6a421afbb60a422fb189be0aae3db.jpg)

A RC Thermal Model of 3D-Stacked Stand-alone MRAM

A RC Thermal Model of 3D-Stacked Stand-alone MRAM

Current-induced domain wall motion (CIDWM) shows the promising prospect in low-power, high-density, and high-speed circuits application. Recent researches have demonstrated that the MTJ device based on CIDWM has great potential in mimicking a non-volatile artificial neuron and synapse [1, 2]. However, low speed and high-power consumption limit the further development of traditional CIDWM driven by spin transfer torque (STT). By combining the high-speed, low-power advantages of spin orbit torque (SOT) MTJ and the ability of interfacial Dzyaloshinskii–Moriya interaction (DMI) to stabilize the Néel–type domain wall, SOT and DMI-based domain wall (DW) MTJ can achieve low threshold current and high propagation speed. Currently, the corresponding device model is highly required for the circuit design in the field of the artificial neuron and synapse simulation. In the paper, a compact model for CIDWM-MTJ based on SOT and DMI is presented. It integrates SOT mechanism for magnetization reversal and domain wall nucleation, CIDWM behaviors, and tunnel resistance theory of MTJ nanopillar. The micromagnetic simulation and the temporal evolution of the DW position is carried out by solving the LLG equation. Finally, a hybrid MTJ/CMOS circuit is simulated to verify the developed model. The presented model combines the CIDWM dynamics and MTJ magnetic dynamics, showing potential application for the simulation of all-spin artificial neural network. Fig. 1 shows the three terminal device structure for SOT and DMI-based CIDWM-MTJ. Iwrite flows between T2 and T3, generating SOT. DW position in the free layer (FL) is propagated by the SOT and the DMI at the HM/FL interface.

![](https://assets.underline.io/uploads/markdown_image/1/image/aa878dc9bb9464796500bbee572ec33b.jpg)

Fig. 1 Three terminal device structure for SOT and DMI-based CIDWM-MTJ.

Compact Model of Domain Wall MTJ Device Driven by Spin Orbit Torque and Dzyaloshinskii–Moriya Interaction

Spin torque switching mechanisms and switching speed of perpendicular magnetic tunnel junctions(pMTJs) free layers are of significant interest for industrial applications and understand the fundamental physics of magnetic material’s response to a spin-polarized voltage across it. A voltage pulse, with an amplitude above the critical voltage Vc0[1], is applied across the junction to switch the magnetization of the pMTJ free layer. The pulse width must be long enough for guaranteed switching and short enough to avoid unnecessary energy consumption and unwanted heating. We have simulated zero-temperature domain wall mediated spin-torque switching of disk-shaped pMTJs with CoFeB free layer[2] to study the optimum pulse width for successful spin-torque switching. Our data show that 1) below a minimum pulse width(tmin), spin-torque switching does not occur, 2) above a maximum pulse width(tmax) switching is deterministic, and 3) for tmin&lt;tpulse&lt;tmax, final magnetization state depends on the pulse width (see Figure 1). Previous research has shown that the domain wall structure and position at the beginning of simulation affect the switching dynamics and the final magnetization state[3]. Since the exchange interaction sets the magnetic structure of the domain wall, we varied the exchange interaction strength to investigate its effect on the final magnetization state. Our results confirm that exchange interaction impacts the pulse width required for switching (Figure 2). We will present our results on the effect of the exchange constant of the free layer on pulse width required for switching and discuss the implications of such effects in pMTJ devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/508e7d1271f47716bb6d7de92f1feb71.jpg)

Figure 1: Time evolution of spatially averaged magnetization along the axis perpendicular to a 30nm pMTJ disk when applied pulse voltage is 1.6Vc0. Blue(red) represent (un)successful switching.

![](https://assets.underline.io/uploads/markdown_image/1/image/78ffc3ac1f2d1e61ffbe30330dcfd426.jpg)

Figure 2: The free layer exchange constant and pulse duration(tpulse) determine the final magnetization state of the junction. When exchange interaction is weaker, longer pulses are needed for switching. Blue(red) represent (un)successful switching.

Nontrivial Relationship Between Current Pulse Width and Spin Torque Switching of Perpendicular Magnetic Tunnel Junctions

We have designed hybrid memory for magnetic random-access memory (MRAM) cells, consisting of low Curie temperature (Tc) CoPd/Pd multilayers (ML) and high Tc Co/Pd ML, and confirmed the thermally-assisted (TA) spin-transfer-torque (STT) switching [1-3]. In this study, we fabricated samples with different high Tc Co/Pd interface layer as [Co(x) /Pd(1.6-x)]1 ML / [CoPd/ Pd]2 ML (x = 0.4, 0.6, 0.8 nm), and investigated the temperature dependence of intrinsic critical current density Jc0 and thermal stability factor Δ. We also measured the magnetization dynamics of the hybrid memory layer by using the time-resolved magneto-optical Kerr effect (TRMOKE), and discussed the effect of the high TC interface layer on TA-STT switching. Current-perpendicular-to-plane type giant magneto-resistance (CPP-GMR) structures with the hybrid memory and Co/Pt reference layers were fabricated to investigate Jc0 and Δ, which are estimated from the pulse width dependence of the STT switching current density. The samples without a reference layer were also fabricated to estimate the damping constant α of the hybrid memory layer by the TRMOKE measurements. Figure 1 shows the temperature dependence of Jc0 of the hybrid memory layer with various of Co thickness x in the interface high TC layer. Jc0 decreased by elevating the temperature, and Jc0 of x = 0.8 drastically decreased compared to the other samples. Figure 2 shows the dependence of α on the Co thickness of the high TC layer. α of hybrid layers decreased with increasing x, and a drastic decrease was confirmed when x &gt; 0.6 nm. The decrease of α suggests the increase of exchange coupling between high Tc and low Tc MLs. The exchange coupling between high Tc and low Tc MLs is considered to have significant influence on the Jc0 at room temperature and its temperature dependence as shown in Fig. 1. MuMax3 simulation also supports the increase of Jc0 by reducing the exchange coupling between high Tc and low Tc MLs, which will be discussed in the presentation.

![](https://assets.underline.io/uploads/markdown_image/1/image/94ec25d1aa1e302edbac1b1c56b558b5.jpg)

Fig. 1 Temperature dependence of Jc0 of the hybrid memory layer of [Co(x)/Pd(1.6-x)]1 / [CoPd/Pd]2.

![](https://assets.underline.io/uploads/markdown_image/1/image/dfa4b70bda8ecc400ddff06cd2c2bfe4.jpg)

Fig. 2 Dependence of damping constant α of the hybrid memory layer on the thickness of Co x in high TC Co/Pd layer.

Effect of high Tc interface layer in high Tc/ low Tc hybrid memory layer on thermally assisted spin-transfer-torque switching

Enhancement of switching perpendicular magnetic tunnel junctions (p-MTJs) with an electric-field has been a milestone for realizing ultra-low energy memory and computing devices. To integrate with current STT-MTJ and SOT-MTJ devices the typical linear fJ/Vm range voltage controlled magnetic anisotropy (VCMA) needs to be enhanced through stack engineering. We have investigated possible non-linear voltage controlled magnetic anisotropy (VCMA) in heavily electron depleted devices, where the magnetic anisotropy could be reduced with positive and negative voltages, differing with the typical linear VCMA voltage behavior. A possible bidirectional and 1.1 pJ/Vm VCMA effect can be realized based on theoretical predictions with heavily electron depleted Fe/MgO interfaces. To improve upon existing VCMA technology we have proposed inserting high work function materials under the magnetic layer which will deplete electrons from the magnetic layer biasing the gating window into the electron-depleted regime where the pJ/Vm and bidirectional VCMA effect was predicted. We have demonstrated tunable control of the under-layer’s work function by utilizing Ta/Pd(x)/Ta trilayers. By modulating the Pd thickness from 0-20 nm we have observed a tunable change in the Ta layer’s work function from 4.3eV to 4.8eV. To investigate the extent of the electron depletion as a function of the Pd thickness in the trilayer we have performed DFT calculations on supercells of Ta/Pd (X)/Ta /CoFe /MgO. We are working to improve the quality of our gate dielectric materials to avoid hysteretic ionic effects allowing for a reliable VCMA measurement of the Ta/Pd/Ta samples.

![](https://assets.underline.io/uploads/markdown_image/1/image/a8360ad2cb5f5ae617004083260b5005.jpg)

Enhancement of Voltage Controlled Magnetic Anisotropy (VCMA) Through Electron Depletion

Enhancement of Voltage Controlled Magnetic Anisotropy (VCMA) Through Electron Depletion

Controlling and switching magnetization by the process of current-induced torque has a considerable advantage in low power consumption for a memory device and spin logic device applications. For a heavy metal (HM) / ferromagnet(FM) / heavy metal (HM) based perpendicular magnetic anisotropy system with broken normal space inversion symmetry (SIA), current-induced magnetization reversal (CIMR) takes place using spin-orbit torque (SOT) with the assistance of an in-plane (IP) magnetic field. For a similar HM/FM/HM-based PMA system having broken lateral symmetry, also known as the quasi-PMA system, the CIMR occurs without the assistance of the IP field. The lateral SIA [1-4] helps in the CIMR process by producing an effective out-of-plane (OOP) field which does not depend on the direction of the IP field and initial magnetization. Thus, the quasi-PMA system has a massive significance for magnetic field-free CIMR. Pt/CoFeB/Pt based quasi-PMA heterostructure has been chosen for CIMR study with the optimal bottom layer Pt thickness for which the effect of lateral SIA is maximum and consists of a ferromagnetic layer with in-plane magnetic anisotropy (IMA) at the top layer of the heterostructure. The IMA ferromagnetic layer [5] contributes to the CIMR process by producing the stray field. Thus, by designing a heterostructure to combine the SOT originated by spin-Hall effect, the stray field by IMA ferromagnetic layer, and the lateral SIA effect, we have enhanced the SOT efficiency of the system to reduce the critical current density (Jc) for CIMR both under the absence and the presence of the different IP fields. The switching phase diagram (SPD) for this device has been shown in Fig. 1. We have achieved Jc as low as (8.65 ± 0.20) x 1010 A/m2 and (3.35 ± 0.17) x 1010 A/m2 at zero bias and 60 mT, respectively. The device has massive potential in low-power magnetic data storage application.

![](https://assets.underline.io/uploads/markdown_image/1/image/d095b538536a799fbce2047bef1a8a64.jpg)

Fig.1. Current-induced switching phase diagram for the device Ta(2)/Pt(3)/CoFeB(0.6)/Pt(0.5)/Ta(1.5)/CoFeB(2.5)/Pt(1)

Improving the spin-orbit torque efficiency in Pt/CoFeB/Pt based perpendicularly magnetized system to decrease the critical current density for current-induced magnetization switching

Application of electric fields (E-fields) on perpendicular magnetic tunnel junctions (p-MTJs) is a promising solution for improving energy efficiency in spintronics based memory and logic systems [1]. Many studies on E-field effects on p-MTJs have focused on voltage-controlled magnetic anisotropy (VCMA) of the p-MTJs free layer to reduce the energy required for p-MTJs to switch magnetization states [2]. In this approach, VCMA temporarily decreases the energy barrier separating the two magnetization states and bidirectional switching is achieved using either an external magnetic field or a spin-polarized current. Another approach is to utilize voltage controlled exchange coupling (VCEC) between two ferromagnetic layers in synthetic antiferromagnetic (SAF) structures. Recently, bidirectional switching through VCEC has been demonstrated on p-MTJs with bottom dual-SAF free layers [3]. Benchmarking on MRAM technologies has shown VCEC-MRAM to have much lower energy dissipation than spin-orbit torque (SOT) MRAM [4]. However, SOT-MRAM has faster write speeds since SOT allows for ultra-fast (≤ 400ps) switching [5]. In this work, we explore the potential for SOT-assisted VCEC switching on p-MTJs with bottom dual-SAF free layers fabricated on bilayer spin hall channels. Fig. 1 shows that the SOT current can tune the coercivity of the p-MTJ, which illustrates that SOT can assist in VCEC switching. Fig. 2 shows that a voltage applied to the MTJ causes either a positive or negative shift in its stray field (center of the field switching curve), which indicates that voltage influences interlayer exchange coupling in the SAF free layer, thus demonstrating VCEC. Since the coercivity of the p-MTJs can be tuned with both VCEC and SOT, dual-biasing of SOT+VCEC can possibly generate stochastic signals for probabilistic-bits. By utilizing the effects of both VCEC and SOT, we introduce a potential solution for low-energy and ultra-fast MRAM technology as well as more novel computing paradigms such as probabilistic spin logic.

![](https://assets.underline.io/uploads/markdown_image/1/image/c46c24b4bba3f3242ec62765e893ac77.jpg)

Fig. 1. Anomalous hall resistance versus field measurements at SOT currents ranging from 1mA to 2.25mA.

![](https://assets.underline.io/uploads/markdown_image/1/image/98c15c49ed5dd8b81e43a340598f2316.jpg)

Fig. 2. Tunneling magneto-resistance (TMR) vs. field measurements at VCEC voltages of +1.1V and -1.1V.

Spin-orbit torque assisted switching with voltage controlled exchange coupling

Magnetic tunnel junctions have been implemented in a broad range of applications[1] due to high sensitivity, excellent reliability and low power consumption. This work discusses the implementation of a vertical packed strategy[2] applied to magnetic tunnel junctions. A multi- TMR device was designed to bring higher signal with reduced noise and compact footprint by increasing the number of sensors vertically within a single sensor[fig.1] or an arrays of sensors per level. A CoFeB/MgO/CoFeB based magnetic tunnel junction was selected and the microfabrication process was established to achieve a functional multi-level TMR device, using a 300 nm thick SiN layer to isolate consecutive levels. Then a common electrical contact is created to connect all levels in a parallel configuration to reduce the resistance. The sensor stack presents an initial RxA of 60 kΩ.µm2 and a TMR=120% deposited by a N3600/N8800 tool at INESC MN. Here we show results for a three-level device microfabricated with a total of 600 sensors within an area of 0.12 mm2. The impact on the sensor transfer curve of topography and misalignments across levels are discussed. Although accumulated roughness reduces the TMR signals, thus, the sensitivity, the magnetostatic coupling is negligible due to optimized level separation, which successfully mitigates the stray fields. Furthermore, a magnetic field of 71 nT/Hz1/2 at 35 Hz was measured on unshielded environment [fig.2]. This value is obtained from converting the voltage output into field units using a sensitivity of 3V/T. Also, the electrical robustness of the multi-TMR devices was evaluated through the breakdown voltage showing similar robustness compared with the arrays in-plane.This work shows an ambitious architecture design for TMR sensor integration in applications requiring small size and low magnetic field detection levels. Comparing with series of sensors achieving 20 nT/Hz1/2 at 1 Hz on 10mm2[3] or 115 pT/Hz1/2 at 200 Hz occupying an area of 4.5x5.5 mm2[4], the proposed geometry is at least 2 orders of magnitude more compact.

![](https://assets.underline.io/uploads/markdown_image/1/image/6c9bba3f2d0ab6cfa2916b7f4e455072.jpg)

Scheme of vertically packed array (arrows represent current flow).

![](https://assets.underline.io/uploads/markdown_image/1/image/4552efa2d071d8bb23eac98f05a1196d.jpg)

Output of device subjected to variable current amplitude applied to the external coil at 35 Hz.

Multi-TMR device with a compact footprint for low magnetic field detection at room temperature

Rare earth iron garnets (REIGs) are magnetic insulators that can be grown as nanometer thick films with perpendicular magnetic anisotropy (PMA), making them attractive for studies on ultrafast spin wave dynamics. PMA has been achieved in epitaxial films of several REIG systems on garnet substrates due to magnetoelastic anisotropy from lattice mismatch strain1-2. Polycrystalline REIG films as thin as 22nm can also be grown on silicon with PMA due to thermal expansion mismatch between the film and substrate following a high temperature anneal3. However, it is of interest to develop ultrathin polycrystalline garnet on silicon with PMA for spintronics devices integrated with CMOS technology. Films below ~20nm thickness crystallize poorly and require a garnet underlayer to provide lattice templating that promotes crystallization. Previous research shows this templating effect still holds when a Pt layer is present between the garnet film and the substrate4. Here we report on the growth and properties of 10nm thin polycrystalline dysprosium iron garnet (DyIG) and dysprosium yttrium iron garnet (Y:DyIG) films on Si (100) substates using an underlayer of a non-magnetic garnet, Gd3Ga5O12, coated with ~1.5nm of Pt to prevent interdiffusion. This enabled the growth of ultrathin polycrystalline DyIG films directly on Si with bulk-like magnetization and PMA. Films were grown by pulsed laser deposition and later crystallized by a rapid thermal anneal (750 °C/5 min). Thermal mismatch creates a tensile strain in the films, which combined with the negative magnetostriction coefficients of DyIG overcomes the shape anisotropy and results in PMA. X-ray diffraction shows single phase pure garnet and atomic force microscopy of the films indicates sub-nanometer surface roughness. Vibrating sample magnetometry (VSM) measurements of a 10nm DyIG/Pt/GGG/Si film showed PMA with a saturation magnetization of 15 emu/cc and a coercivity of ~300 Oe. The spin Hall magnetoresistance and spin mixing conductance of Pt/DyIG/Pt/GGG/Si will be compared with that of Pt/REIG single crystal films.

![](https://assets.underline.io/uploads/markdown_image/1/image/d9868548b1779ee257badc65ac17c2d0.jpg)

In plane and out of plane VSM hysteresis loops of 10nm DyIG/Pt/GGG/Si film with PMA.

![](https://assets.underline.io/uploads/markdown_image/1/image/14b77bc84f7b6309f7c23e005e96455a.jpg)

Grazing incidence X-ray diffraction of 10nm/Pt/GGG/Si film.

Integration of 10 nm thick Dysprosium Iron Garnet Films on Silicon

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.

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

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