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Micromagnetics requires calculation of the demagnetization field, anisotropy field, applied field and exchange field. With the MagTense framework, demagnetization is calculated analytically [1] and both anisotropy and applied field are local, leaving only the exchange field in the form of a second order partial derivative. Standard micromagnetics are limited to finite element methods with large, extraneous simulation volumes or finite difference methods with homogeneous simulation grids, as the demagnetization is calculated using a fast Fourier transform [2], but with MagTense, no such limitation exists. However, here an optimal method of calculating second order partial derivatives on arbitrary meshes in micromagnetics must be determined, continuing previous research [3].&lt;br/&gt;&lt;br/&gt;We present solutions to the mumag standard micromagnetic problems (mumag) 3[4] and 4[5] using a direct second order partial derivate technique on four different meshes: Prismal, tetrahedral, grained prismal and grained tetrahedral. The grained meshes consist of voronoi generated and Lloyd iterated grain regions with lower resolution towards the center and higher towards the edges, along with high resolution intergrain regions.&lt;br/&gt;&lt;br/&gt;In fig. 1a) is shown an example of a grained prismal mesh with 9 grains and an intergrain region in gray. Of course, in the mumag standard problems considered, all regions have the same material properties, but it is easy to envision scenarios where they do not.&lt;br/&gt;&lt;br/&gt;In fig. 1b) is shown the exchange crossover length of mumag 3 as a function of resolution for three different meshes compared to published solutions. All mesh types converge to the correct value, with the tetrahedral meshes converging faster.&lt;br/&gt;&lt;br/&gt;In fig. 2 is shown the error compared to published solutions of mumag 4 as a function of resolution using regular-, 4 grained- and 9 grained prismal meshes. The unstructured meshes converge to the correct result in a way similar to a regular grid.&lt;br/&gt;&lt;br/&gt;In both cases it is apparent that while for these problems grained meshes offer few advantages, they are nonetheless correctly implemented and open up a slew of possibilities for accurately simulating complex geometries with far fewer elements in irregular meshes.

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Fig. 1

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2022 Joint MMM-INTERMAG

Solving the Standard Micromagnetic Problems using Unstructured Meshes with MagTense

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INTERMAG 2021

Micromagnetics requires calculation of the demagnetization field, anisotropy field, applied field and exchange field. With the MagTense framework, demagnetization is calculated analytically [1] and both anisotropy and applied field are local, leaving only the exchange field in the form of a second order partial derivative. Standard micromagnetics are limited to finite element methods with large, extraneous simulation volumes or finite difference methods with homogeneous simulation grids, as the demagnetization is calculated using a fast Fourier transform [2], but with MagTense, no such limitation exists. However, here an optimal method of calculating second order partial derivatives on arbitrary meshes in micromagnetics must be determined, continuing previous research [3]. We present solutions to the mumag standard micromagnetic problems (mumag) 3[4] and 4[5] using a direct second order partial derivate technique on four different meshes: Prismal, tetrahedral, grained prismal and grained tetrahedral. The grained meshes consist of voronoi generated and Lloyd iterated grain regions with lower resolution towards the center and higher towards the edges, along with high resolution intergrain regions. In fig. 1a) is shown an example of a grained prismal mesh with 9 grains and an intergrain region in gray. Of course, in the mumag standard problems considered, all regions have the same material properties, but it is easy to envision scenarios where they do not. In fig. 1b) is shown the exchange crossover length of mumag 3 as a function of resolution for three different meshes compared to published solutions. All mesh types converge to the correct value, with the tetrahedral meshes converging faster. In fig. 2 is shown the error compared to published solutions of mumag 4 as a function of resolution using regular-, 4 grained- and 9 grained prismal meshes. The unstructured meshes converge to the correct result in a way similar to a regular grid. In both cases it is apparent that while for these problems grained meshes offer few advantages, they are nonetheless correctly implemented and open up a slew of possibilities for accurately simulating complex geometries with far fewer elements in irregular meshes.

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Fig. 1

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Fig. 2

technical paper

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Interaction between quasiparticles of a different nature, such as magnons and phonons, leads to mixing their properties and forming hybrid quasiparticles—hybrid magnetoelastic bosons, also called magnon polarons—in the areas of intersection of individual spectral branches. In garnet ferrite films, such hybridization results in several new phenomena as the resonant increase in the efficiency of the spin Seebeck effect [1,2] and the spontaneous bottleneck accumulation [3] of the magnetoelastic quasiparticles. Similar to the Bose-Einstein magnon condensation (BEC) [4], the latter phenomenon occurs at the bottom of the magnon spectrum in yttrium iron garnet films exposed to microwaves. However, unlike the BEC, which is a consequence of the equilibrium Bose statistics, the bottleneck accumulation is determined by changing interparticle interactions. Studying the transport properties of accumulated quasiparticles [5], we found that such accumulation occurs in two frequency-distant groups: quasiphonons and quasimagnons. These quasiparticles propagate in the film plane as spatially localized beams with distinctly different directions and velocities. The developed theoretical model describes the double accumulation effect. Parametrically pumped magnons scatter through a step-by-step thermalization process towards the lower energy states. At the magnon-phonon hybridization region, the efficiency of nonlinear scattering decreases, and the downward quasiparticle flux slows down. Owing to the bottleneck effect, quasimagnons form a density peak, in which intensity saturates due to their scattering to the lowest branch of the magnon spectrum. The remaining quasimagnons, scattering to phonon states below the magnon spectrum, transform into quasiphonons with vanishing scattering amplitude. These quasiphonons form the second accumulated group. By considering the 2D magnon-phonon spectrum, we determined the wavevectors and frequencies of each group and concluded about the caustic nature of the quasimagnon beams. Due to nonzero group velocity, the accumulated quasiparticles can be used for data transfer in magnon spintronics. Funded by the ERC Advanced Grant 694709 SuperMagnonics and by the DFG within TRR173 – 268565370 (project B04).

Double Accumulation and Anisotropic Transport of Magneto-Elastic Bosons in Yttrium Iron Garnet Films

Cu(1,3-bdc) is a topological magnon insulator where the Cu ions are arranged in a 2D Kagome lattice1,2. Here, we study the magnetization dynamics of Cu(1,3-bdc) using vibrating sample magnetometry (VSM, Fig.1a) and ferromagnetic resonance (FMR, Fig.1b,c) spectroscopy. We find that the saturation magnetization MS values when an external field H is applied out-of-plane (OOP) is less than MS values when H is applied in-plane (IP) at all tested temperatures. We find similar inequivalence in the g-factor, extracted from the fitted FMR dispersion curves (Fig.2a,b) depending on the direction of H. The g-factor tensor (Fig.2c) points towards orbital momentum contribution to the magnetic moment. As shown in Fig.2d, the temperature dependence of ΔMS=MS, IP-MS, OOP, and Δg=gIP- gOOP are closely correlated at T&lt;6K but they diverge at T&gt;6K. Since the g-factor tensor is most likely mediated by orbital magnetism, the deviation of Δg from ΔMS is unexpected. We propose that this could be a result of thermally-induced non-vanishing spin chirality contributing to the electronic orbital magnetism3. Following the approach in Ref.[3],our analysis shows that the symmetry of the system allows contributions to the orbital magnetization originating from spin excitations for the OOP field, but not the IP field. The prediction of our theoretical assessment is an increase of the g-factor with temperature for OOP field, while it is expected to decay for IP field. The general trend and the change in sign of the g-factor anisotropy observed experimentally at higher temperatures is thus consistent with this theory, which suggests that the spin excitations can renormalize the fundamental quantum mechanical constant conventionally associated with the atomic orbital magnetism. Our findings bring to the forefront a so far not considered mechanism for orbital dynamics in topological magnonic systems.

![](https://assets.underline.io/uploads/markdown_image/1/image/eb9082c0d94a561e7eac0b66d95d264f.jpg)

Fig.1:a VSM M vs. H curves at different temperatures. b and c FMR profiles at different microwave frequencies for IP and OOP fields, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/1d106de32a84f8d6ee6bc552c78421e3.jpg)

Fig.2: a and b FMR dispersion curves showing Kittel fits for IP and OOP fields, respectively. c Temperature dependence of the gyromagnetic ratio γ and g-factor. d Temperature dependence of Δγ and ΔMS.

Spin Chirality-Mediated Orbital Angular Momentum in Quasi-Two-Dimensional Topological Magnon Insulator Cu(1,3-bdc)

The investigation of Spin waves (SW) and their quanta, magnons, in thin magnetic films, has emerged as an attractive field of research due to their very short wavelength reaching down to nanometers at GHz frequencies. SWs also have a rich dispersion relation, that depends on their propagation direction with respect to a magnetic field. Hence, SW -based devices are promising candidates for microwave information processing, and eventually for overcoming the limitations encountered with CMOS devices[1]. Yet, a key challenge is to achieve long propagation distances of the spin waves[2]. Recent studies using spin-orbit torques[3] where a Platinum stripe placed on top of a 20 nm thick YIG waveguide (Pt/YIG) showed that SOT could generate a full compensation of the damping, leading to auto-oscillations of the magnetization above a critical injected current density. When excited with a microwave field, spin waves propagation length was increased by a factor of 10 in the bilayer. However, amplification of propagating spin-waves was not observed yet, due to the onset of nonlinear dissipative processes above the critical current. Recently, interests turned towards materials having perpendicular magnetic anisotropy in which one can also achieve a full damping compensation along with SW emission. Here, we propose a new pathway towards SOT-based SW amplification. Using micromagnetic simulations (Mumax3), we demonstrate that pulsed SOT does induce a lossless SWs propagation when the pulse duration is shorter than 200 ns. Remarkably, amplification is achieved when injecting currents exceeding the auto-oscillation threshold (Fig.1). A key condition is however to minimize non-linear coupling. In our case, we achieved this condition in a 20 nm thick Bi-YIG film structure with a vanishing effective magnetization i.e. the out-of-plane magnetic anisotropy compensates the demagnetization field[4].

![](https://assets.underline.io/uploads/markdown_image/1/image/a7679bfc41668549787a60c203c14403.jpg)

Spin Wave Amplification using Spin-Orbit-Torques, A micromagnetic approach

Spin Wave Amplification using Spin-Orbit-Torques, A Micromagnetic Approach

The effective temperature dependence of micromagnetic parameters such as anisotropy (K), exchange stiffness (A) and the Dzyaloshinskii–Moriya interaction (DMI) are usually written as a power law of the magnetisation. For anisotropy this is well founded, based on Callen-Callen theory1. For exchange stiffness and DMI the power laws are based on the assumption that they behave the same as anisotropy and from classical numerical models. Recently there has been a lot of interest in the temperature dependence of these properties because their value affects spin textures like skyrmions. Inferring the DMI from experimental measurements often requires first deducing the exchange stiffness by fitting Bloch's law to measurements of magnetisation at finite temperature. The problem is that Bloch's law contains the zero temperature value of exchange stiffness, but is used to fit a finite temperature range over which the value of the exchange stiffness should change significantly. Another issue is that Bloch's law is derived using quantum mechanics with magnons which obey Bose statistics – numerical models typically use classical statistics for the thermal occupation of spin waves. Classical models cannot reproduce Bloch's T3/2 law for ferromagnets or the T2 dependence of sublattice magnetisation in antiferromagnets2. So, the classical thermodynamics of exchange is also a poor approximation for real materials in which the quantum distributions of magnons plays a large role. Here we discuss a quantum implementation of atomistic spin dynamics2, compare with classical results and present the temperature scaling of the micromagnetic exchange stiffness for both ferromagnets and antiferromagnets calculated from magnon spectra. We show antiferromagnets have similar scaling behaviour to ferromagnets and that the results using quantum statistics doesn’t follow a power law. Figure 1 presents the thermal behaviour of the exchange stiffness as a function of reduced sublattice magnetisation of the prototypical antiferromagnet NiO used in antiferromagnetic spintronics3. At low temperature (high magnetisation) the exchange stiffness is almost temperature independent when using quantum statistics in stark contrast to a power law.

![](https://assets.underline.io/uploads/markdown_image/1/image/7982d7754373f6c2812464262fbe6f6c.jpg)

The thermodynamic properties of exchange stiffness

In the last decade, the low-moment, highly spin-polarized ferrimagnets of the Mn2RuGa (MRG) family have been investigated as of their potential for application in high-frequency resonance devices [1] relying on either MTJs and CPP transport [2] or Hall bar type structures [3], utilizing STT and/or SOT. Very recently, ultra-fast switching of the magnetisation in a single pulse, all-optical toggle scheme has been shown [4]. The deeper understanding, control and optimisation of any of these approaches, require that an effective model is built, which is computationally much cheaper than, but can be calibrated based on DFT. Here we build such an effective model, starting from a description of generalized Heisenberg Hamiltonian, on the perfectly ordered structure, with two dissimilar magnetic sub-lattices, within the approximation of frozen magnons and phonons. The interactions that are taken into account are the following: isotropic exchange on both sublattices and mutual; two-fold and four-fold anisotropy, with orientation which is different for the different local-symmetry sites; Zeeman coupling to the external field; elastic coupling coefficients for both the magnetic and non-magnetic sites; distance dependencies for the exchange beyond first nearest neighbours and RKKY-type coupling. The main features of the structure and the resulting phonon, magnon and combined excitations are shown on the corresponding panels of Fig. 1. Interestingly, the system exhibits double crossing of the in and out-of-phase magnon branches (for finite k-vectors), potentially allowing for the fast relaxation of the optically excited magnons and softening of the phonon branches at high momenta. The distance dependent exchange allows for the combined elastic-magnetic excitation of the system and the potential exploitation of phonon-driven ultra-fast magnetisation control schemes.

![](https://assets.underline.io/uploads/markdown_image/1/image/9d4d12070ab5b2f6ce8a1677bd5d3f53.jpg)

(a) The structure of MRG, with the different magnetic and non-magnetic sites marked. (b) The in-phase (acoustic) and out-of-phase (optical) branches of the primary magnon dispersions along the <001> axis. (c) The main four acoustic phonon modes within the first Brillouin zone. (d) The energy of the combined frozen magnon and phonon excitation for arbitrary momenta.

An Effective Hamiltonian Model for Combined Magnon-Phonon Excitations in Ferrimagnets of the Mn2RuGa Family

Synthetic antiferromagnets (S-AFMs) have recently been shown to be ideal systems to search for Exceptional Points (EPs) [1]. In a S-AFM, EPs are the point at which the optical and acoustic magnons coalesce and are indistinguishable from one another. EPs can be reached by varying parameters, such as the exchange coupling between magnetic layers. Once an EP is reached, only a single magnon branch exists in the magnon energy spectrum, and this makes EPs an attractive prospect for controlling magnons in magnetic microstructures. Recently, we have computationally calculated and visualized EPs in S-AFMs using micromagnetic simulations [2]. We determined that dipolar interactions play a pivotal role in determining if and where EPs occur in the parameter space used to describe the S-AFM. Our results are both qualitatively and quantitatively different compared to macrospin models [3], where dipolar interactions are not considered. We are now investigating how the macrospin limit can be recovered in micromagnetic simulations if spacer layers are built into the simulation, or if the shape of the micromagnetic object is changed. By recovering the macrospin limit within micromagnetic simulations, we will demonstrate new geometric parameters that can be used to control EPs and the magnon energy spectrum within S-AFMs. Work at Oakland University was supported by U.S. National Science Foundation under Award No. ECCS-1941426.

![](https://assets.underline.io/uploads/markdown_image/1/image/824acc503fd69f0ff6f80f24f5badeb6.jpg)

FIG. 1. (a) Illustration of two micromagnetic layers in antiferromagnetic alignment. A nonmagnetic spacer layer is represented by the grey section. (b) Magnon spectra shown when simulating the antiferromagnetic state with ferromagnetic interlayer exchange stiffness. No clear presence of an EP exists in the antiferromagnetic state before it becomes unstable due to the FM exchange stiffness.

Adjusting dipolar interactions to control exceptional points in synthetic antiferromagnets

The wave reversal mode is a magnetization reversal mechanism that appears in ferromagnetic nanotubes of certain geometric parameters when an external magnetic field is applied perpendicular to their axes. The distinctive feature of this mode is that leads to well-defined S-shaped hysteresis curves [1]. In order to gain insight into the stability of this latter effect, we have performed micromagnetic simulations for permalloy and nickel nanotubes obtaining a non-monotonic behavior for coercivity as well as for remanence as a function of nanotube diameter for both materials. Motivated on these latter intriguing results, we found that measuring the area that encloses the hysteresis curve is a novel and simple strategy to identify the appearance of the wave reversal mode [2]. An additional contribution of this work is the proposal of a new magnetic phase diagram that allows determining the stability of this reversal mechanism as a function of the geometric and magnetic parameters of the tubes [2]. The authors acknowledge the access to Mendieta cluster (CCAD-UNC) and financial support from SECYT-UNC, MINCyT 2019, Fondecyt 1200302, Basal Project AFB180001 and Programa Escala Docente AUGM.

![](https://assets.underline.io/uploads/markdown_image/1/image/9dbc44ada0f75d7113ec3e993652dfd0.jpg)

Wave reversal mode: A new magnetization reversal mechanism in magnetic nanotubes

With the rise of deep learning, physics informed neural networks [1] became an alternative to finite element simulation in many fields of materials and device design. Physics informed neural networks bring several advantages as compared to traditional numerical methods for solving partial differential equations like finite differences or finite elements: 1. There is no need for mesh generation. 2. Inverse problems may be solved effectively. 3. A whole family of problems may be solved with a single neural network. We demonstrate how physics informed neural networks can be applied in magnetostatics and micromagnetics. The approach is like the Ritz method for computing the magnetic field or the magnetization. However, instead of finite element basis functions [2], we use dense neural networks to approximate the unknowns. The underlying physics is incorporated through the loss function which is minimized during training of the neural network. We first applied physics informed neural network to solve a classical inverse problem in magnetism. We searched for the magnetization in a hollow cylinder that produces a uniform field in the hole and has zero field outside. The mean relative error between the neural network estimate and the analytic solution for the magnetization was 0.27 percent. We computed the demagnetization curve of a Nd2Fe14B particle. The difference between the finite element solution and the neural network estimate was 0.9 percent. We computed the lowest eigenvalue of Brown's linearized micromagnetic euqation to get the nucleation field for a soft magnetic defect in a hard magnetic matrix. The mean realtive error for the nucleation field of an Fe inclusion in Sm2Fe17N3 as function of defect size was 0.4 percent as compared to the analytic solution [3]. Because of the ability to solve multiple problems simulateneously, physics informed neural networks have a great potential for optimization and tailoring the magnet's internal structure to achieve a specific demagnetiation curve. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged.

Physics informed neural networks for computational magnetism

The motion of electrons and therefore the topological Hall effect (THE) are governed by the flux density (B-field), as contrasted to the applied magnetic field (H-field). Maxwell's equations predict that the relevant perpendicular flux density (B-field) in laterally homogeneous thin films is µoH, independent of the magnetization M. However, transport measurements and nonvanishing THEs at H = 0 indicate that the motion of the electrons depends on M. This paradox is solved by noting that electrons are point-like objects moving in exchange-and-correlation holes. Their magnetostatic interaction with the crystalline environment is realized through Lorentz cavities [1, 2] where B = µo(H + DL M) with DL ~ 1/3, intermediate between B = µoH and B = µo(H + M). This approximation includes inhomogeneities, because the local magnetization vector M(r) is not necessarily parallel to the volume-averaged magnetization &lt;M&gt;. The separation of the THE from the magnetization contribution to the anomalous Hall-effect is therefore not straightforward and the use of &lt;M&gt; can lead to systematic errors [3-5]. For example, in nanogranular structures (Fig. 1), changes in &lt;M&gt; may be dominated by loosely coupled regions (inside the dashed circles) where conductivity, current density &lt;j&gt;, and Hall-effect contribution are disproportionally small. Another field consideration is that the THE reflects electrons adiabatically exchange-interacting with atomic magnetic moments. The corresponding Berry phase changes the motion of the electrons. The change is formally equivalent to the effect of a fictitious magnetic field, but this field does not obey Maxwell's equations in a micromagnetic sense. This work is supported by NSF-EQUATE (OIA-2044049), NSF-DMREF (No. 1729288), the NU Collaborative Initiative, and NCMN.

![](https://assets.underline.io/uploads/markdown_image/1/image/7dfda3153223eb4af3edd7e68fc7833e.jpg)

Fig. 1. Effect of structural inhomogeneities on the THE: (a) current density in a nanoparticulate thin film, (b) Bruggeman description of the current, (c) spin-structure in the interaction-domain limit, and (d) spin structure in the non-cooperative limit. The yellow region in (c) fully contributes to the magnetization but only partially to the Hall effect.

Berry-Phase Interpretation of Thin-Film Micromagnetism

An efficient method for computing thermodynamic equilibrium states at the micromagnetic length scale is introduced [1], based on the Markov chain Monte Carlo method. Trial moves include not only rotations of spins, but also a change in their magnetization length. The method is parameterized using the longitudinal susceptibility, reproduces the same Maxwell-Boltzmann distribution as the stochastic Landau-Lifshitz-Bloch equation, and is applicable both below and above the Curie temperature. The algorithm has been implemented in Boris [2], is fully parallel [3], can be executed on graphical processing units, and efficiently includes the long range dipolar interaction. Applications to finite-temperature hysteresis loop modelling, chiral magnetic thin films, granular magnetic media, and artificial spin ices are discussed. A typical application is exemplified in relation to Figure 1, where finite-temperature hysteresis loops in a 2 nm thin Co film with interfacial Dzyaloshinsky-Moriya interaction are computed. Simulation of such hysteresis loops in chiral films is problematic with quasi-zero temperature methods such as steepest descent energy minimizer, since the nucleation and annihilation of skyrmions and labyrinth domain structure is principally a thermally activated process. Furthermore, the method is also extended to two-sublattice systems, allowing modelling of ferrimagnetic and antiferromagnetic materials at the micromagnetic length scale. Using a multi-layered computation, which includes both ferromagnetic and antiferromagnetic layers, exchange bias may be simulated in bilayers, as well as more complex multi-layered structures including synthetic antiferromagnetic stacks. Moreover we show the method may also be applied to study of thermal and athermal training effects in exchange bias.

![](https://assets.underline.io/uploads/markdown_image/1/image/c87224dc80ce91d19d61166abaca5e0c.jpg)

Figure 1 - Hysteresis loops in a 2 nm thin Co film on Pt, using a uniform 300 K temperature (thin dashed line), and a Gaussian temperature profile varying from 300 K at extremities to 400 K at the centre (thick dashed line). The insets show the perpendicular magnetization component on the increasing field sweep, with blue denoting magnetization into the plane, and red out of the plane.

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Double Accumulation and Anisotropic Transport of Magneto-Elastic Bosons in Yttrium Iron Garnet Films

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Simulating sintered magnets using the full demagnetization tensor with MagTense

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