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Synthetic antiferromagnets (SAFs) are of interest in a wide variety of spintronic applications owing to their net zero magnetisation and inherent tunability arising from control over complex stacking sequences [1]. They show very fast domain wall (DW) motion in response to spin torques [2]. Here we describe current-driven DW motion in an SAF with stack structure Ta/[Ru/Pt/CoB/Ru/Pt/CoFeB]×5/Ru/Pt which displays both perpendicular magnetic anisotropy (PMA) and the Dzyaloshinskii-Moriya interaction (DMI). The alternating CoB and CoFeB layers had their thicknesses adjusted to give net zero magnetisation (as determined by SQUID-VSM) for applied fields up to ~500 Oe. They were patterned into 2 µm wide wires with Cu leads attached to each end. These were fabricated on Si&lt;sub&gt;3&lt;/sub&gt;N&lt;sub&gt;4&lt;/sub&gt; membranes to permit imaging of DW motion using scanning x-ray transmission microscopy. The distinct chemical composition of the two sublattices means that we could use the element specificity of this technique to observe the CoFeB separately whilst imaging at the Fe L&lt;sub&gt;3&lt;/sub&gt; edge, whilst contrast at the Co L&lt;sub&gt;3&lt;/sub&gt; edge is dominated by the other sublattice. Inverted domain structures observed at these two edges, as shown in Fig. 1, confirm the SAF ordering of the layers. Current pulses drive domain wall motion, as shown in Fig. 2, with velocities up to ~ 20 m/s achieved at current densities as low as 4 × 10&lt;sup&gt;11&lt;/sup&gt; A/m&lt;sup&gt;2&lt;/sup&gt;.

![](https://assets.underline.io/uploads/markdown_image/1/image/d87de30dbef334ff75e4447e90cf2feb.jpg)

Fig. 1: STXM images acquired at zero field the Co and Fe L3 edges, which probe different sublattices of the SAF. A finger-shaped Cu lead is visible on the left. 1 µm scale bar.

![](https://assets.underline.io/uploads/markdown_image/1/image/04b915124bc7866fb80c310fb03a761b.jpg)

Fig. 2: DW velocity against current density measured using 5 ns current pulses.



2022 Joint MMM-INTERMAG

Current-Driven Domain Wall Motion in a Synthetic Antiferromagnet Multilayer

#### Welcome to the 67th Annual Conference on Magnetism and Magnetic Materials, MMM 2022!

The Conference was held from October 31 to November 4 and offers both in-person and on-demand content.

MMM 2022 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 tutorial, and several special sessions. 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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MMM 2022

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

Synthetic antiferromagnets (SAFs) are of interest in a wide variety of spintronic applications owing to their net zero magnetisation and inherent tunability arising from control over complex stacking sequences [1]. They show very fast domain wall (DW) motion in response to spin torques [2]. Here we describe current-driven DW motion in an SAF with stack structure Ta/[Ru/Pt/CoB/Ru/Pt/CoFeB]×5/Ru/Pt which displays both perpendicular magnetic anisotropy (PMA) and the Dzyaloshinskii-Moriya interaction (DMI). The alternating CoB and CoFeB layers had their thicknesses adjusted to give net zero magnetisation (as determined by SQUID-VSM) for applied fields up to ~500 Oe. They were patterned into 2 µm wide wires with Cu leads attached to each end. These were fabricated on Si3N4 membranes to permit imaging of DW motion using scanning x-ray transmission microscopy. The distinct chemical composition of the two sublattices means that we could use the element specificity of this technique to observe the CoFeB separately whilst imaging at the Fe L3 edge, whilst contrast at the Co L3 edge is dominated by the other sublattice. Inverted domain structures observed at these two edges, as shown in Fig. 1, confirm the SAF ordering of the layers. Current pulses drive domain wall motion, as shown in Fig. 2, with velocities up to ~ 20 m/s achieved at current densities as low as 4 × 1011 A/m2.

![](https://assets.underline.io/uploads/markdown_image/1/image/d87de30dbef334ff75e4447e90cf2feb.jpg)

Fig. 1: STXM images acquired at zero field the Co and Fe L3 edges, which probe different sublattices of the SAF. A finger-shaped Cu lead is visible on the left. 1 µm scale bar.

![](https://assets.underline.io/uploads/markdown_image/1/image/04b915124bc7866fb80c310fb03a761b.jpg)

Fig. 2: DW velocity against current density measured using 5 ns current pulses.

technical paper

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

Please register for the Joint MMM-Intermag conference to join the event.

We theoretically study the dynamics of a domain wall in a ferromagnetic nanotube driven by electrons and magnons by investigating the electron-domain-wall and magnon-domain-wall interactions. Due to the geometry of the sample, domain walls are classified by the Skyrmion charge which counts the winding number of magnetic textures on the domain wall [1]. The domain wall with a non-zero Skyrmion charge generates an emergent magnetic field for interacting particles, which exerts the Lorentz force on moving electrons and magnons and thereby deflects their trajectories. This deflection is manifested as the generation of the finite orbital angular momenta of the electrons and magnons that traverse the domain wall [2,3]. For the case of electrons, we can interpret the exchange of orbital angular momenta between electrons and the domain wall as a current-induced torque. From this peculiar torque, the domain wall with the non-zero Skyrmion charge can be driven by an arbitrary small current without the Walker breakdown. For the case of magnons, we obtain the exact solution for the magnon on the Skyrmion-textured domain wall and also their scattering properties with the domain wall with the aid of supersymmetric quantum mechanics (SUSY QM). We also show that there is a critical wavenumber for the total reflection and it is discretized by the Skyrmion charge of the domain wall. Our results show that the orbital angular momenta of magnetic systems and quasi-particles can be intertwined in a curved geometry.

![](https://assets.underline.io/uploads/markdown_image/1/image/52936d7fd5815e871498a79c009adca1.jpg)

Fig. 1. A ferromagnetic nanotube with a domain wall possessing a finite Skyrmion charge. The arrows represent magnetizations of spin-wave excitations. The black curve lines the heads of the arrows. The axis of the nanotube is the z-axis.



Orbital angular momentum of a domain wall and geometrically twisted magnons

We find numerically skyrmionic textures with skyrmion number Q=0 in ferromagnets with the Dzyaloshinskii-Moriya interaction perpendicular anisotropy and the magnetostatic field [1]. The configuration is asymmetric; the largest part has the form of a skyrmion and a smaller part has the form of an antiskyrmion. The texture is topologically trivial (Q=0) and it may be called a chiral droplet. Droplets are stable in an infinite film as well as in disk-shaped magnetic elements. They are stable also in a model without the magnetostatic field [2]. They are found in films for values of the parameters close to the transition from the ferromagnetic to the spiral phase. Chiral droplets are a valuable addition to a short list of skyrmionic textures [3]. The presence of topologically trivial textures (Q=0) along with usual skyrmions (Q=1) in the same magnetic system is rare. It is important because the dynamics of skyrmions is linked to their topology; it depends on the skyrmion number. Under spin-transfer torque (due to an in-plane current), droplets are accelerated in the direction of the current flow and they behave as solitary waves of Newtonian character with no Magnus force effect. This is in stark contrast to the Hall dynamics of the standard Q=1 skyrmion. In the case shown in the figure, the motion is initially complicated, with the droplet making a full π turn while its size is expanded. It is accelerated in the direction of the current and eventually, the velocity acquires a constant value, in very good agreement with the theoretical prediction. The dynamical behavior of a droplet can be compared to that of an axially symmetric skyrmionium [4] in order to show that the global topology rather than the symmetry of the configuration is crucial for skyrmion dynamics. We acknowledge support by the project ThunderSKY, funded by HFRI and GSRT, under Grant No. 871.

![](https://assets.underline.io/uploads/markdown_image/1/image/3fafac4601e2b98489be56ebef970d99.jpg)

The motion of a chiral droplet when an in-plane current is applied. The droplet is represented via the perpendicular component of the magnetization mz (color-coded). In the initial stage, the droplet is following a complicated path, it rotates and its shape adjusts to that of a propagating solitary wave. It eventually moves with no Magnus force effect.

Chiral droplets in ferromagnets, and their current-driven motion

Magnetic skyrmions, which are topologically protected particle-like spin textures, exhibit Brownian motion at ambient temperatures in a ferromagnetic thin film. The skyrmion system can be an ideal platform to design unconventional computers like stochastic/Brownian computers. Therefore, investigation of the Brownian motion of the magnetic skyrmion is now the subject of scientific and technical interest [2-5]. The motion of skyrmions driven by the current [6] and temperature gradient [7] have the chirality depending on the skyrmion number. In this study, we explore the effects of the chiral property of a skyrmion on its Brownian motion experimentally and theoretically. We fabricated the stacking structure as Ta(5)|Co16Fe64B20(1.26)|Ta(0.22)|MgO(1.5)|SiO2(5.0) and observed the position of the skyrmion x(t) = (x(t), y(t)) by using MOKE microscope. Fig. 1 shows the simultaneous velocity-position correlation function &lt;v(t)●x(t)&gt; and &lt;(v(t)×x(t))z&gt;. After enough time has passed, &lt;v(t)●x(t)&gt; and &lt;(v(t)×x(t))z&gt; correspond to the diffusion coefficient Dxx and the angular momentum of the skyrmionic diffusion, respectively, and the latter also has a sizable value. Fig. 2 represents the simulation trajectory in the periodic potential obtained by integrating the Thiele-Langevin eq. [8,9] including the thermal fluctuation and the mass term. The chirality of the Brownian motion is affected by the fluctuated potentials of the samples. This research and development work was supported by ULVAC, Inc., the Ministry of Internal Affairs and Communications, JSPS Grant-in-Aid for Scientific Research (S) Grant Number JP20H05666, Japan and CREST (Non-classical Spin project, JPMJCR20C1) of the Japan Science and Technology Agency.

![](https://assets.underline.io/uploads/markdown_image/1/image/b2fe0462e46a0f673fe46f65f2392e00.jpg)

Fig. 1 velocity-position correlation function and <(v(t)×x(t))z>


![](https://assets.underline.io/uploads/markdown_image/1/image/6aa7d699583b7f1eef889ab6bb6a071e.jpg)

Fig. 2 The simulation result of the motion of the skyrmion in the periodic potential. The red arrows show the direction of the motion.

Chiral Brownian motion of the magnetic skyrmions

Skyrmions are topological spin textures that exhibit particle-like dynamics under current-driven forces such as spin-orbit torques (SOT). In frustrated magnets, competing exchange interactions along with the Dzyaloshinskii-Moriya interaction (DMI) can aid in stabilizing textures of different topologies, like skyrmions and antiskyrmions [1], which exhibit different motion under SOT, such as trochoidal trajectories and skyrmion-antiskyrmion pair generation, in addition to the usual rectilinear motion. Such behavior results from a departure from the frozen-core limit, where additional dynamics in the (anti)skyrmion helicity couples to the core position [2]. Here, we discuss the role of thermally driven helicity dynamics in the Brownian motion of skyrmions under weak DMI and SOT. Depending on the magnitude of the DMI, which governs the strength of the restoring force experienced by the helicity, with respect to kBT, the thermal drift of the skyrmion core under weak SOT can differ greatly from the dynamics at 0 K (Fig. 1). This behavior arises from the thermal drift and diffusion of the helicity under SOT, which is analogous to the well-known problem of Brownian motion in a tilted washboard potential, where the helicity can be assimilated to an oscillator phase in a periodic potential with a linear force term. Under weak SOT or strong DMI, we recover a ''phase-locked'' solution in which the helicity is pinned, resulting in rectlinear drift. For strong SOT or weak DMI, we find a ''free-running'' solution, where the helicity varies continuously in time and results in an amplified Brownian diffusion of the skyrmion core. In between these two regimes, thermally-activated hopping of the helicity between energy minima occurs. These results highlight the importance of helicity dynamics and may have bearing on dipole-stabilized bubbles for which drive-dependent skyrmion Hall angles [3] and low drift velocities have been reported [4].

![](https://assets.underline.io/uploads/markdown_image/1/image/4499995d079e5c36223e0c31052e4788.jpg)


Fig. 1. Comparison between SOT-driven skyrmion motion at 0 K (black line) and 50 different realisations of the Brownian motion at 10 K (colored lines), along with the average thermal drift at 10 K (purple line).

Anomalous thermal drift of skyrmions in frustrated magnets under spin-orbit torques

In order to develop energy efficient race track memory devices, it is crucial to create and controllably move chiral spin textures, namely, skyrmions in magnetic nanotrack. In recent time, intense efforts have been dedicated to understand the mechanism of domain wall pair to skyrmion conversion in different junction geometry that may fit into the memory applications [1,2]. Here, we study the mechanism of DW pair to skyrmion conversion in a typical junction geometry which consists of the combination of two narrow nanotracks (length – 800 nm, thickness – 1 nm, width varying from 20 nm to 60 nm in steps of 20 nm) and a wide track (width 200 nm). The separation between narrow channels are selected as 10 nm, 30 nm and 50 nm. The micromagnetic simulations have been performed using Ubermag to investigate the magnetization dynamics and the Zhang-Li torque evolver is implemented in Landau Lifshitz Gibert equation to drive the DWs using spin polarized current [3]. For a fixed Gilbert damping parameter (α = 0.3) and different values of non-adiabatic spin torque parameter (β = 0.2, 0.3 and 0.5) we investigate the DW to skyrmion conversion in these nanotracks. Interestingly, under the influence of spin torque, the DW pair gets converted to skyrmion only if the geometrical constraints allow the favorable instability to set in at the junction between optimally separated narrow channels and wide channel (cf. Fig. 1(a-d)) [4]. For certain width and separation between the narrow channels, the DW pair to skyrmion conversion cannot be observed for sufficiently large time scale (cf. Fig. 2(a-d)). We believe these results will be helpful in designing the skyrmion based racetrack memory where the controlled creation of skyrmion is expected to play crucial role.

![](https://assets.underline.io/uploads/markdown_image/1/image/af3301ae9ac634fc6c9411eabcfd6b01.jpg)

Fig 1: Snapshot of DW pair to skyrmions conversion from two narrow channels for α = β = 0.3, width of the narrow nanochannel = 60 nm, separation between the narrow channel = 30 nm

![](https://assets.underline.io/uploads/markdown_image/1/image/ed5b1617a9c98ac080b569cc57b93e0f.jpg)

Fig 2: DW dynamics from two channels for α = β = 0.3, width of the narrow nanochannel = 40 nm, separation between the narrow channel = 10 nm


Influence of Geometry on Domain-Wall Pair to Skyrmion Conversion in Typical Magnetic Nanochannel

For their applications in spintronics, skyrmions need to be controlled one by one, and therefore appear as excitations over a uniform ground state. This is why in most demonstrations, the application of an external field is required. In this study, we show by fine-tuning the micromagnetic parameters, through their control by the ferromagnetic layer thickness, multilayers can be optimized to host zero field skyrmion excitations. We study Pt/Co/Au based heterostructures with varying Co thickness t. Close to the spin reorientation transition (t = 1.6 nm), the domain wall energy vanishes and the ground state is a stripe phase. To favor the uniform state, t is reduced to increase the magnetic anisotropy, until samples show a 100% remanence. Below 1.45 nm, skyrmions could be obtained and stabilized during the magnetization reversal process, and remain stable at zero field. This bistable situation offers a perfect playground to study current-induced nucleation in zero field, which we study in the vicinity of a point contact. Starting from a uniform state, current pulses lead to the formation of magnetic textures. However, to avoid the formation of elongated structures, similar to stripes, Co thickness needs to be further reduced. At 1.2 nm, the magnetic anisotropy is sufficiently large to cut elongated structures into well-defined skyrmions, which could be obtained with good reproducibility. Their topological nature is proven through the observation of the skyrmion Hall effect: depending on the orientation of the initial state, skyrmions are deflected in one or the other directions. Velocities of about 50 m/s induced by SOT, using a current density of 3.7 x 1012 A/m2 in 1µm wide tracks were also observed. This opens an important perspective towards skyrmionic devices at zero field, in particular concerning stacks that are not sensitive to external fields such as ferrimagnets and antiferromagnets. Figure: Zero field MFM images of 3 µm wide magnetic nanotracks, after successive application of 3.7 x 1012 A/m2, 3 ns long current pulses starting from a saturated track. In the first serie (a-c) the sample is initially magnetized downward, in the second one (d-f), it is magnetized upward, evidencing the gyrotropic deflection.

![](https://assets.underline.io/uploads/markdown_image/1/image/15d074009eea6ac625bfde657bff2eb1.jpg)

Figure: Zero field MFM images of 3 µm wide magnetic nanotracks, after successive application of 3.7 x 1012 A/m2, 3 ns long current pulses starting from a saturated track. In the first serie (a-c) the sample is initially magnetized downward, in the second one (d-f), it is magnetized upward, evidencing the gyrotropic deflection.

Current induced skyrmion nucleation at zero field

Magnetic skyrmions present interesting and unique pseudo-particle behaviors which arises from their topological protection [1-2]. Key among these behaviors is their resonant dynamics, under microwave fields, which include both breathing and gyration modes. Due to their small size, and magnetic-only contrast of skyrmions, coupled with high frequency dynamics – in the GHz regime – it is challenging to do in-situ measurements on these excitations. This work reports the use of small angle neutron scattering (SANS) to capture the dynamics of hybrid skyrmions stabilized in Fe/Gd multilayers by means of dipolar interactions. Using perpendicular DC fields and in-plane RF fields, we explored the gyration modes of hybrid skyrmions away, below, at, and above resonance [Fig.1(a-d)]. We find the dynamic modes of hybrid skyrmions contribute to the SANS signal in two ways: first, the scattered neutrons incur additional transverse momentum as a result of scattering from a moving source, and second, the gyration skyrmions disrupt ordering of the lattice giving rise to enhancement form-factor scattering. Ferromagnetic resonance measurements confirm the RF and DC field conditions for these gyration modes. Our results offer new insights into the nanoscale dynamics of magnetic skyrmions and present a unique use of SANS to probe magnetization dynamics.

![](https://assets.underline.io/uploads/markdown_image/1/image/a7eefda0d3ae85f8c787ee1cd94ec130.jpg)

Fig.1 2D SANS pattern of magnetic skyrmions under different magnetic fields and excitation frequencies: (a) Static state (H=0, RF=0), (b) below-resonance state (H=150 mT, RF=1.2 GHz), (c) at-resonance state (H=150 mT, RF=1.3 GHz) and (d) above-resonance state (H=150 mT, RF=1.4 GHz). All 2D images are plotted with the same intensity scale.

In-situ GHz Dynamics of Skyrmions Probed with SANS

Skyrmions have clear advantages for future spintronic devices over domain walls in terms of information carrying and storage, however there are still many open questions left and inherent challenges in the transition to applications. Here we study the effect of an additional interfacial spin-transfer torque (ISTT) [1], as well as the well-established spin-orbit torque (SOT), on skyrmion collections [2] – group of skyrmions dense enough that they are not isolated from one another – in magnetic multilayers, by comparing modelling using Boris [3], with experimental results. We show that skyrmion collective motion has additional effects not reproduced when compared to modelled isolated skyrmions [1]. In particular, as the skyrmion collection density increases the effect of local disorder is reduced, leading to reduced threshold current densities, and a sharper dependence of skyrmion Hall Angle (SkHA) on velocity in the depinning regime, in agreement with experimental results. When including both the ISTT and SOT, the small SkHA observed in experiment are in good agreement, also explaining the observed diameter-independent SkHA [4], as seen in Figure 1. Contrary to the model which considers the SOT as the only driving torque, and which predicts a monotonically increasing SkHA with decreasing diameter, inclusion of ISTT results in SkHA closely aligned to experimental results, particularly when the effect of local disorder and skyrmion-skyrmion interactions are taken into account. Our results shed further light on collective current-driven skyrmion motion in magnetic multilayers and highlight the significance of the ISTT in understanding current-driven skyrmion motion in magnetic multilayers.

![](https://assets.underline.io/uploads/markdown_image/1/image/219d9cf02b4f240924efc702e30f317a.jpg)

Figure 1. Collective SkHA dependence on skyrmion diameter for (a) N = 50 Sk/µm2 and (b) N = 100 Sk/µm2. (c) Experimentally obtained SkHA from skyrmion collections with a range of diameters with an average current density of 2×1011 A/m2. The overlaid dashed and solid lines show the ideal isolated skyrmion case – independent of current density – for the SOT and ISTT+SOT, respectively. The solid blue line in panel (b) represents a Savitzky-Golay fit to the simulated skyrmion motion, for a current density of Jc = 2×1011 A/m2.

Collective Skyrmion Motion Under the Influence of an Additional Interfacial Spin-Transfer Torque

Co/Gd based synthetic ferrimagnets have received considerable attention owing to the coexistence of both pronounced spin-orbitronic effects and all-optical switching (AOS), allowing for novel, hybrid devices [1,2]. Previous studies [3,4] have shown that spin-orbit torque (SOT) driven domain wall motion in these 3d-4f ferrimagnets system is most efficient at angular momentum compensation. Following this notion, in this study, we explore the stack engineering of Co/Gd based synthetic ferrimagnets to achieve high SOT driven domain wall motion at room temperature[5]. Here, we experimentally demonstrate that magnetization compensation can be achieved and tuned by varying layer thicknesses in Co/Gd(x)/Co/Gd quadlayer systems (see inset of Fig. 1), which shows AOS in the full range of Gd thickness[5]. Fig. 1 (a) shows the coercivity and polar magneto-optical Kerr effect (MOKE)-signal in such a quadlayer system where the middle Gd layer is wedged. The sign change of the MOKE signal and divergence of the coercivity indicate a compensated system. The CIDWM velocity as a function of Gd thickness of this stack has been investigated, after structuring a series of domain wall conduits from another wedge sample grown in the same batch. In Fig. 1 (b) it can be observed that for low to moderate current densities the optimum in the velocity can be found at compensation. As the current density is increased the optimum in the velocity is found to shift towards larger Gd thicknesses, likely due to the difference in temperature dependence of the magnetization in Co and Gd leading to a shift in the compensation thickness due to joule heating during the applied current pulses. Our study shows a significant improvement of the room temperature domain wall velocity in synthetic ferrimagnetic systems through stack engineering. Further studies to explore the role of the weak exchange link provided by the proximity induced Gd magnetization profile are in progress.

![](https://assets.underline.io/uploads/markdown_image/1/image/abbc4d28d2f8fc22cfb8a86256cef998.jpg)

Fig. 1: Coercivity, polar MOKE signal (a) and current-driven domain wall velocity (b) as a function of Gd thickness in a Pt(4)/Co(0.6)/Gd(x)/Co(0.7)/Gd(1.5) multilayer system (thicknesses in nm in parenthesis), and x indicates Gd layer thickness.

![](https://assets.underline.io/uploads/markdown_image/1/image/439e7460be2e15697bd7be423caa409c.jpg)

Fig. 1 (a) The XRD pattern and (b) the FC magnetic moment as a function of temperature at 100 Oe, of LNMO films. The sample numbers represent the annealing time in minutes.

Fast Current-Induced Domain Wall Motion in Compensated Co/Gd Quadlayers

In recent years, there was an increased interest in using antiferromagnetic (AFM) materials for THz frequency sources. The operation of such AFM structures is based on the AFM resonance, which can have THz frequencies, exceeding the frequency ferromagnetic (FM) resonances by order of magnitude [1,2]. Recent experiments showed that it is possible to generate such signal in AFM FeRh/Pt bi-layers by optically exciting them with a sub-picosecond laser pulse [3]. The signal in the AFM state can be originated from a partial phase transition induced by the laser heat from the AF to the FM phase [4]. Such transformation occurs on a sub picosecond time scale [5-7]. Using atomistic simulations, it is possible to reproduce such transition by including both the bilinear and biquadratic exchanges in the Hamiltonian [7,8]. However, to model larger spatial structures, a coarse-grained model for the dynamic of the magnetization is desirable. Here, we present a micromagnetic formulation to model the magnetization dynamics and phase transition in FeRh at low and high temperatures, including the transition near the Curie temperature. The formulation is based on a mean field approximation (MFA), where each macrospin is modeled by two coupled sublattices, including the atomistic biquadratic interactions described as a micromagnetic inter-sublattice molecular field with a cubic dependence of the magnetization. To validate the model, we compare our simulations of the phase transitions in the static and dynamic case with the existing literature. The results show an agreement with the experiments and atomistic modeling, displaying the characteristic hysteretic behavior for the magnetization (Fig. 1), and a similar dynamic response to an external thermal pulse (Fig.2).

![](https://assets.underline.io/uploads/markdown_image/1/image/514f2aed032312bb7acd410db166547c.jpg)

Equilibrium magnetization as a function of temperature.

![](https://assets.underline.io/uploads/markdown_image/1/image/25e2660f89da1d9cb11ba63fe22f9af2.jpg)

Dynamic response of the magnetization to an optical pulse (shaded area) for different values of the atomistic damping coefficient α

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