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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.&lt;br/&gt;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.&lt;br/&gt;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 Nd&lt;sub&gt;2&lt;/sub&gt;Fe&lt;sub&gt;14&lt;/sub&gt;B particle. The difference between the finite element solution and the neural network estimate was 0.9 percent. We computed the lowest eigenvalue of Brown&#39;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 Sm&lt;sub&gt;2&lt;/sub&gt;Fe&lt;sub&gt;17&lt;/sub&gt;N&lt;sub&gt;3&lt;/sub&gt; as function of defect size was 0.4 percent as compared to the analytic solution [3].&lt;br/&gt;Because of the ability to solve multiple problems simulateneously, physics informed neural networks have a great potential for optimization and tailoring the magnet&#39;s internal structure to achieve a specific demagnetiation curve.&lt;br/&gt;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.

2022 Joint MMM-INTERMAG

Physics informed neural networks for computational magnetism

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INTERMAG 2021

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.

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.

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Far from thermodynamic equilibrium, nonlinear systems can be characterized by a stationary regime known as a limit cycle. In magnetic systems with a uniaxial symmetry driven with a circularly polarized field, this regime is referred to as the P-mode [1]. P-modes have been experimentally observed in the non-linear ferromagnetic resonance [2]. Stable P-modes can play a role of the equilibrium state for the elementary excitations around them. We show that these excitations can be used in cavity magnonics to explore unique features of non-Hermitian physics such as mode attraction and exceptional points. Level attraction describes a mode coalescence that can take place in open systems. It indicates development of an instability region in the energy spectrum of the coupled system bounded by exceptional points [3]. This regime has been recently observed in a number of experiments in dissipative cavity magnonics [4]. Here, we present a theory of mode attraction in periodically driven systems. Our theory can be used in conjunction with numerical micromagnetics to describe non-Markovian Floquet systems [5], which allows us to determine the regions of stability and instability and associate them with mode repulsion and attraction. In magnetic systems, our approach works for quantum and semiclassical dynamics, which makes possible to use it in combination with micromagnetic simulations. We apply this approach to demonstrate the level attraction between the excitations around the P-modes and the photons inside a microwave cavity. In a system driven with a circularly polarized field, the excitations around the P-mode form two side bands around the driving frequency (see figure). We show that when the lower side band couples to the cavity mode, it forms a mode attraction spectrum. Finally we discuss possibilities for experimental observation and predicted dependences on driving field amplitude.

![](https://assets.underline.io/uploads/markdown_image/1/image/7edfde7c526bc0c7f19799421eaed99c.jpg)

Circular component of the precessing spin driven with the circularly polarized field. The central peak is the P-mode with the frequency of the driving field. Two side bands correspond to the excitation around the P-mode. Coupling to the side band with the negative frequency with respect to the driving field gives mode attraction.

Mode attraction in Floquet systems with memory: application to cavity magnonics

Extending nanostructures into the third dimension has become a major research avenue in modern magnetism and spintronics because of geometry-, curvature-, and topology-induced phenomena. Here, we introduce nanovolcanoes as on-demand engineered nano-architectures for 3D magnon spintronics. 40 nm-thick Co-Fe nanovolcanoes with diameters down to 300 nm were fabricated by the focused electron beam-induced deposition (FEBID) technique (Fig. 1) and studied by perpendicular spin-wave resonance (PSWR) spectroscopy. The spin-wave eigenfrequencies of the nanovolcanoes are notably different from the eigenfrequencies of the nanodisks and nanorings they are built from, because of the strongly non-uniform internal magnetic field. The experimental findings were elucidated with the aid of micromagnetic simulations, which indicate that the rings encircling the volcano craters lead to an effective confinement of the lower-frequency eigenmodes under the volcano crater, while the highest-frequency eigenmodes are confined in the ring area (Fig. 2). Accordingly, extension of 2D nanodisks into 3D nanovolcanoes allows to obtain same response in low frequency part of the spectrum in elements with 30% smaller footprints [1]. The nanovolcanoes can be viewed as multi-mode resonators with potential applications in telecom industry. Also, we demonstrated the engineering of the magnetic properties of CoFe-based nanodisks fabricated by FEBID. The material composition was tuned in situ via the e-beam waiting time in the deposition process and their post-growth irradiation with Ga ions. The saturation magnetization Ms of the nanodisks was deduced from PSWR measurements. The achieved variation of Ms in the broad range from 720 to 1430 emu/cm3 continuously bridges the gap between the Ms values of widely used magnonic materials such as Permalloy and CoFeB [2].

![](https://assets.underline.io/uploads/markdown_image/1/image/ca3695146d0c42bbf0d0b907eb2d1eb1.jpg)

Fig. 1. Atomic force microscopy image of the nanovolcano with the outer diameter of 300 nm and the crater diameter of 200 nm. Inset: cross-sectional line scan.

![](https://assets.underline.io/uploads/markdown_image/1/image/355d13301bed6b9caa1885c2c3af1f2e.jpg)

Fig. 2. Calculated spatial dependences of the spin-wave eigenfunctions for (a) 40 nm-thick nanovolcano NV300/200, (b) 20 nm-thick nanodisk ND300, and (c) 20 nm-thick nanoring NR300/200 at the out-of-plane bias magnetic field of 1.2 T.

Spin-wave spectra of 3D nanovolcanoes fabricated by focused electron beam-induced deposition

The appearance of spontaneous coherence is a fundamental feature of a Bose–Einstein condensate (BEC) and an essential requirement for possible applications of the condensates for classical and quantum computing. Using a magnon BEC in a magnetic crystal, such computing can be performed even at room temperature. Using specially shaped bulk samples of yttrium iron garnet crystal, we developed a novel efficient method of BEC investigation by direct detection of microwave radiation emitted by parametrically driven magnon gas (Fig.1a-b). By using this method, we studied the process of coherence formation in a magnon BEC and demonstrated that no coherent BEC state could be found during the action of a strong magnetic pumping field. At the same time, after the pumping is terminated, the overpopulated magnon gas evolves toward BEC and reaches full coherence, with the width of the magnon radiation spectrum decreasing by more than two orders of magnitude (Fig. 1c). The residual bandwidth is mainly determined by the lifetime of magnons, as expected for a fully coherent BEC consisting of a single magnon state. This direct demonstration of the magnon BEC coherence [1] brings the implementation of room temperature BEC-based computing closer. Moreover, the magnon BEC coupling with dynamic stray fields outside the sample is enabled by a proper choice of the sample shape giving direct spectroscopic access to the BEC. Such an approach can function as a convenient tool for integrating magnetic quantum systems into electrical environments.

![](https://assets.underline.io/uploads/markdown_image/1/image/08092bb8bbc4772d1216c4df47190393.jpg)

(a) Experimental setup for microwave detection of magnon BEC
dynamics.
(b) Bulk BEC mode and one of the edge magnon modes in a cuboid YIG sample. The monotonic blue line schematically shows the profile of the static magnetic field H within the YIG sample. Letters denote three field values: A – deeply inside the sample, B – at the point where the bulk BEC mode becomes evanescent, C – at the sample edge.
(c) Magnon dispersion curves in the middle of the sample (at point A) and near the edge (at point B). The green, blue, and red signal intensity lines show the microwave power spectra registered during 1 µs interval before the end of pumping as well as 2 µs and 4 µs after the pump pulse is turned off.

Evolution of Room-Temperature Magnon Gas toward Coherent Bose–Einstein Condensate

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)

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