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Devices based on magnetic domain wall (DW) motion are among the most promising candidates for future spintronic data storage, logic, and neuromorphic devices. One critical issue that hinders DW motion is the presence of structural inhomogeneities such as interface intermixing, interface roughness, or crystalline texture, that cause a distribution of magnetic properties in the material. As a result, domain walls undergo local pinning leading to the so-called creep regime at low driving forces. Even beyond this creep regime, the DW motion remains affected by defects. Thus, to achieve fast and efficient DW motion, good control over the intrinsic defects in the material must be obtained.&lt;br/&gt;Ultra-thin CoFeB/MgO-based structures with perpendicular magnetic anisotropy are considered the most promising candidates for the future generation of Magnetic Random Access Memory (MRAM) devices. These films are usually annealed at high temperatures (300-400°C) to crystallize the CoFeB layer and get high Tunnelling Magnetoresistance (TMR). This treatment at high temperatures results in the presence of undesirable intrinsic defects.&lt;br/&gt;To address this issue, we have developed a very promising process solution based on post-growth He&lt;sup&gt;+&lt;/sup&gt; ion irradiation to tailor the structural properties of ultrathin magnetic films at the atomic level and improve their performances [1-4]. The utilization of light ions provides the precise control of inter-atomic displacements through the low energy transfer. The key feature of the technology is the post-growth control of structural properties and the related magnetic properties. To perform these irradiation techniques, we have used a compact ion-irradiation facility (Helium-S® from Spin-Ion Technologies), capable of ultra-fast He&lt;sup&gt;+&lt;/sup&gt; ion irradiation on 1-inch wafers with energies ranging between 5-30 keV [5].&lt;br/&gt;&lt;br/&gt;Here, we have studied the influence of two different processes of crystallization in W-CoFeB-MgO structure on domain wall dynamics. The first process is based on the combination of He&lt;sup&gt;+&lt;/sup&gt; ion irradiation at low fluences (10&lt;sup&gt;14&lt;/sup&gt; ions/cm&lt;sup&gt;2&lt;/sup&gt; to 10&lt;sup&gt;15&lt;/sup&gt; ions/cm&lt;sup&gt;2&lt;/sup&gt;) with annealing. The second process consists of pure thermal annealing. For both processes, annealing temperatures ranging from RT to 400°C have been used. Our results first demonstrate that ion irradiation can induce full crystallization of the CoFeB-MgO materials at much lower temperatures (&amp;lt;250°C) than the pure annealing process (&amp;gt;350°C). In addition, higher anisotropy values can be obtained, which is consistent with sharper interfaces. Domain wall dynamics measured by Kerr microscopy have been compared for both processes. Strong differences have been evidenced in both the creep (few nm/s) and flow regime (few m/s). In particular, the ion-irradiation-induced crystallization process results in higher domain wall velocities as shown in Figure 1 for the creep regime where structural defects have a strong influence. This is consistent with a much lower density of structural inhomogeneities in the material crystallized by ion irradiation. Finally, we will discuss the crystallization process under ion irradiation and show preliminary results of DW motion in magnetic wires where local ion irradiation can also be used to minimize edge defects.&lt;br/&gt;Our results show that He&lt;sup&gt;+&lt;/sup&gt; ion irradiation is a very versatile tool to enhance magnetic materials and devices.

**References:**

[1] J. Fassbender, D. Ravelosona, and Y. Samson, &lt;i&gt;J. of Phys. D.: Appl. Phys.&lt;/i&gt;, Vol. 37, no. 16 (2004)&lt;br/&gt;[2] T. Devolder, I. Barisic, S. Eimer et al., &lt;i&gt;J. of Appl. &lt;/i&gt;&lt;i&gt;Phys.&lt;/i&gt;, Vol. 113, no. 20, p. 203912 (2013)&lt;br/&gt;[3] L. Herrera Diez, F. García-Sanchez, J.-P. Adam et al., &lt;i&gt;Appl. Phys. Lett.&lt;/i&gt;, vol 107, p. 032401 (2015)&lt;br/&gt;[4] X. Zhao, B. Zhang, N. Vernier et al., &lt;i&gt;Appl. Phys. Lett.&lt;/i&gt;, vol. 115, p.122404 (2019)&lt;br/&gt;[5] www.spin-ion.com

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/689f25424d7c3fb6596488bc040d25c7.jpg)
Domain wall velocity as a function of the applied out-of-plane magnetic field in the creep regime for the regular annealing process (400°C, 1h) and the process based on He+ irradiation at lower annealing temperatures.



INTERMAG 2021

 Enhancing domain wall motion in W/CoFeB/MgO ultrathin films through He+-irradiation-induced crystallization

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Devices based on magnetic domain wall (DW) motion are among the most promising candidates for future spintronic data storage, logic, and neuromorphic devices. One critical issue that hinders DW motion is the presence of structural inhomogeneities such as interface intermixing, interface roughness, or crystalline texture, that cause a distribution of magnetic properties in the material. As a result, domain walls undergo local pinning leading to the so-called creep regime at low driving forces. Even beyond this creep regime, the DW motion remains affected by defects. Thus, to achieve fast and efficient DW motion, good control over the intrinsic defects in the material must be obtained. Ultra-thin CoFeB/MgO-based structures with perpendicular magnetic anisotropy are considered the most promising candidates for the future generation of Magnetic Random Access Memory (MRAM) devices. These films are usually annealed at high temperatures (300-400°C) to crystallize the CoFeB layer and get high Tunnelling Magnetoresistance (TMR). This treatment at high temperatures results in the presence of undesirable intrinsic defects. To address this issue, we have developed a very promising process solution based on post-growth He+ ion irradiation to tailor the structural properties of ultrathin magnetic films at the atomic level and improve their performances [1-4]. The utilization of light ions provides the precise control of inter-atomic displacements through the low energy transfer. The key feature of the technology is the post-growth control of structural properties and the related magnetic properties. To perform these irradiation techniques, we have used a compact ion-irradiation facility (Helium-S® from Spin-Ion Technologies), capable of ultra-fast He+ ion irradiation on 1-inch wafers with energies ranging between 5-30 keV [5]. Here, we have studied the influence of two different processes of crystallization in W-CoFeB-MgO structure on domain wall dynamics. The first process is based on the combination of He+ ion irradiation at low fluences (1014 ions/cm2 to 1015 ions/cm2) with annealing. The second process consists of pure thermal annealing. For both processes, annealing temperatures ranging from RT to 400°C have been used. Our results first demonstrate that ion irradiation can induce full crystallization of the CoFeB-MgO materials at much lower temperatures (&lt;250°C) than the pure annealing process (&gt;350°C). In addition, higher anisotropy values can be obtained, which is consistent with sharper interfaces. Domain wall dynamics measured by Kerr microscopy have been compared for both processes. Strong differences have been evidenced in both the creep (few nm/s) and flow regime (few m/s). In particular, the ion-irradiation-induced crystallization process results in higher domain wall velocities as shown in Figure 1 for the creep regime where structural defects have a strong influence. This is consistent with a much lower density of structural inhomogeneities in the material crystallized by ion irradiation. Finally, we will discuss the crystallization process under ion irradiation and show preliminary results of DW motion in magnetic wires where local ion irradiation can also be used to minimize edge defects. Our results show that He+ ion irradiation is a very versatile tool to enhance magnetic materials and devices.

**References:**

[1] J. Fassbender, D. Ravelosona, and Y. Samson, J. of Phys. D.: Appl. Phys., Vol. 37, no. 16 (2004) [2] T. Devolder, I. Barisic, S. Eimer et al., J. of Appl. Phys., Vol. 113, no. 20, p. 203912 (2013) [3] L. Herrera Diez, F. García-Sanchez, J.-P. Adam et al., Appl. Phys. Lett., vol 107, p. 032401 (2015) [4] X. Zhao, B. Zhang, N. Vernier et al., Appl. Phys. Lett., vol. 115, p.122404 (2019) [5] www.spin-ion.com

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/689f25424d7c3fb6596488bc040d25c7.jpg)
Domain wall velocity as a function of the applied out-of-plane magnetic field in the creep regime for the regular annealing process (400°C, 1h) and the process based on He+ irradiation at lower annealing temperatures.

technical paper

The rate of thermally activated transitions can generally be described by the Arrhenius law, as k=f0 exp(-ΔE/kBT). In the previous expression, ΔE is the internal energy barrier, and f0 is a prefactor that encompasses both dynamical and entropic contributions. In this work, we show how the activation entropy can play a crucial role in thermally activated magnetic transitions. To that end, we study two types of transitions involving solitons either as the stable state or the transition state, namely, skyrmion annihilation, and domain-wall mediated magnetization reversal in nanodisks. The aforementioned transitions pertain to the class of rare events, i.e., they are unlikely to arise in the course of a Langevin simulation, because the mean waiting time between events is much larger than the integration timestep. In this case, one possible approach is to use a form of reaction rate theory to calculate the energy barrier and the Arrhenius prefactor. To this end, we apply Langer’s theory [1], which is an extension of the Kramers method to many dimensions. Alternatively, one may use a path sampling method such as forward flux sampling [2] in order to sample the transition path ensemble from the system’s dynamics and extract a transition rate. In this work, we employ both methods in conjunction to study the thermal stability of magnetic skyrmions and nanodisks, for which the two methods show a good agreement. Magnetic skyrmions are particle-like, two-dimensional, non-collinear solitonic spin textures with a nontrivial topology. In ultrathin films and multilayers, small skyrmions are stabilized by an interfacial form of the Dzyaloshinskii-Moriya interaction (DMI). Skyrmion-based designs currently appear very promising for novel spintronics devices, e.g., for racetrack memories and logic gates, reservoir computing, or probabilistic computing [3]. Isolated skyrmions exist as metastable excitations of the collinear magnetic ground state and can be longed-lived. For applications, a stability of up to 10 years at room temperature is a technological requirement, and the ability to accurately predict that stability is therefore crucial. We find that skyrmions can be stabilized not only by an internal energy barrier, but also by a large activation entropy. This stems from the fact that the metastable skyrmion state possesses a large entropy compared to the transition state, because skyrmions exhibit stable internal modes of deformation [4]. When applying a destabilizing magnetic field perpendicular to the skyrmion core, the skyrmion lifetime decreases quickly with the field amplitude. This results from a shrinking of the skyrmion, which simultaneously reduces the the energy barrier, as well as the activation entropy through the suppression of the stable internal modes (Fig. 1) [5]. Next, we compute the mean waiting times between thermally activated domain-wall mediated magnetization reversals in a perpendicularly magnetized nanodisk. We use parameters that resemble that of a free CoFeB layer, as typically used in magnetoresistive random access memories [6]. By varying the perpendicular anisotropy and the interfacial DMI, we find that the Arrhenius prefactor can take extreme, seemingly non-physical values up to 1021 Hz, which is orders of magnitude beyond the typically assumed value of 109 Hz, and vary drastically as a function of material parameters. We show that the prefactor behaves like an exponential of the energy barrier, which stems from a linear variation of the activation entropy with the energy barrier (Fig. 2) [7]. This phenomenon, known as the Meyer-Neldel rule, or entropy-enthalpy compensation, has been reported across diverse fields of the natural sciences for over a century—in semiconductors, chemical reactions, biological death rates, etc [8]. In magnetism, it was only recently observed in the decay of the skyrmion lattice [9]. This result implies that the transition state, which, in the case of the magnetization reversal, is a domain wall found in the center of the disk, tend to possess a very large entropy compared to the collinear ground state. This is verified by the existence of gapless, low energy modes propagating along domain walls [10]. These findings suggest that the Arrhenius prefactor, which is typically taken as an attempt frequency within the characteristic timescale of a magnetic system's dynamics, may in some cases contain a very large entropic contribution which should not be neglected. This seems to be particularly true in magnetic transitions involving solitons.

**References:**

[1] J. S. Langer, Annals of Physics, Vol. 54, p.258 (1969) [2] R. J. Allen, P. B. Warren, and P. R. ten Wolde, Phys. Rev. Lett., Vol. 94, p.018104 (2005) [3] A. Fert et al., Nat. Nanotechnol., Vol. 3, p.152 (2013) ; Prychynenko et al., Phys. Rev. Appl., Vol. 9, p.014034 (2018) ; D. Pinna et al, Phys. Rev. Appl., Vol. 9, p.064018 (2018) [4] L. Desplat, D. Suess, J.-V. Kim et al., Phys. Rev. B, Vol. 98, p.134407 (2018) [5] L. Desplat, C. Vogler, D. Suess et al., Phys. Rev. B, Vol. 101, p.060403(R) (2020) [6] J. Sampaio, A. Khvalkovskiy, M. Kuteifan et al., Appl. Phys. Lett., Vol. 108, p.112403 (2016) [7] L. Desplat and J-.V. Kim, Phys. Rev. Lett., Vol 125, p.107201 (2020) ; L. Desplat and J-.V. Kim, Phys. Rev. Appl., Vol. 14, p.064064 (2020) [8] A. Yelon, B. Movaghar, and R. Crandall, Rep. Prog. Phys., Vol. 69, p.1145 (2006) [9] J. Wild, T. N. Meier, S. Pollath et al., Sci. Adv., Vol. 3, p.e1701704 (2017) [10] F. Garcia-Sanchez, P. Borys, R. Soucaille, et al., Phys. Rev. Lett., Vol. 114, p.247206 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3d6013214984a1f3ea7698c4d79f4b7a.jpg)
Arrhenius prefactor for skyrmion collapse as a function of the perpendicular external magnetic field in units of the exchange coupling constant, Jex. The prefactor is computed through Langer's theory, forward flux sampling (FFS), and direct Langevin simulations.

![](https://assets.underline.io/uploads/markdown_image/1/image/1c239dc0374a9439b358445c11e10d48.jpg)
Activation entropy for domain-wall mediated magnetization reversal in a nanodisk at T = 300 K as a function of the activation energy normalized by the exchange constant, A, and the disk thickness d, for variations of the DMI, D, and of the anisotropy, K. The inset shows the corresponding variation of the Arrhenius prefactor, f0.

 Entropic effects and solitons in thermally activated magnetic transitions INVITED

The topological charge or skyrmion number Q is a useful measure to characterize the topology of spin textures such as vortices and skyrmions in two-dimensional systems. When the magnetization field m(r) is projected onto the unit sphere, Q measures the number of times the magnetic moments wrap around this sphere. For vortices and merons, Q = ±1/2, while for skyrmions, Q = ±1. Higher-order half- and full-integer charges are also possible. In numerical micromagnetism, a common approach involves solving the equations of motions for m(r) that is discretized on a regular grid. The underlying assumption is that cell-to-cell variations in m are sufficiently small such that the exchange energy, proportional to the square of the magnetization gradient, remains meaningful. However, issues can arise under certain conditions, such as in the nucleation and annihilation of vortices and skyrmions, or in the stochastic dynamics under random fields, where large cell-to-cell variations in the magnetization can reduce the accuracy of the finite-difference approximations of the spatial derivatives used to estimate Q, which can result in spurious deviations from half- and full-integer values of Q. Here, we will describe a lattice-based implementation for the open source micromagnetics code MuMax3 [1] that allows an accurate determination of Q, particularly under Langevin dynamics used to model finite temperatures and in nucleation and annihilation processes [2]. The implementation is based on the approach of Berg and Lüscher [3], which involves summing over the solid angle subtended by three spins, forming elementary signed triangles, that are constructed from the finite difference cells. The implementation accounts for both periodic boundary conditions and finite-sized systems. A comparison between the finite-difference and lattice-based approaches is given in Fig. 1, where the time variation of the total topological charge is shown for a system hosting a single skyrmion under a temperature of 300 K. The blue circles represent the Q computed using finite-difference derivatives, which exhibit large fluctuations about the expected value of Q = -1 that are about as large as Q itself. Moreover, the time-averaged mean Q does not coincide with -1. The solid red line represents the result of the lattice-based approach, which remains constant near the expected value of -1. More precisely, the result is limited primarily by the single-precision of the floating-point arithmetic used, e.g., we find Q = −1.0000001, −1.0000004, −1.0000008, −1.0000002, and −0.9999996 over 20-ns intervals at 400 K. We also show how the lattice-based approach is vastly superior to the finite-difference approximation in other test cases, which involve skyrmion-antiskyrmion pair nucleation due to spin-transfer torques and skyrmion fluctuations in finite dots. While the results do not necessarily call into question the validity of published work (since the topological charge is often only used as a proxy for magnetization gradients), they do highlight the care with which noninteger values of Q should be interpreted, particularly when processes such as nucleation, annihilation, and thermal fluctuations are at play. This work was partially supported by the Agence Nationale de la Recherche under contract no. ANR-17-CE24-0025 (TOPSKY) and Fonds WetenschappelijkOnderzoek (FWO-Vlaanderen) through Project No. G098917N.

**References:**

[1] A. Vansteenkiste, J. Leliaert, M. Dvornik et al., AIP Adv 4, 107133 (2014). [2] J.-V. Kim and J. Mulkers, IOP SciNotes 1, 025211 (2020). [3] B. Berg and M. Lüscher, Nucl Phys 190, 412 (1981).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d45266e543d925ddaf2fecf323e4aa0c.jpg)
Topological charge, Q, as a function of time for a skyrmion in a 1-nm thick film with periodic boundary conditions at 300 K. Blue circles represent the calculation based on a finite-difference approximation to spatial derivatives, as provided by the package ext_topologicalcharge in MuMax3, while the solid red line represents the calculation with the lattice-based approach, which is provided by the package ext_topologicalchargelattice. The top inset illustrates snapshots of the micromagnetic state.

 On quantifying the topological charge in micromagnetics using a lattice-based approach

Magnetic skyrmions are whirling magnetic structures that can be found on certain magnetic materials [1]. They are promising candidates to become the next generation of information carriers as well as basic elements in ultradense magnetic memories, logic devices or computational systems due to their small size and high mobility. Skyrmions can be stabilized in ferromagnetic ultrathin films with the aid of interfacial Dzyaloshinskii-Moriya (iDM) interaction with a heavy-metal substrate. The same mechanism allows the formation of skyrmions in multilayers with alternate ferromagnets and heavy-metals. The experimental finding of room-temperature skyrmions has boosted the potentiality of skyrmions for applications and, in consequence, the study of their current-driving dynamics at non-zero temperatures. Skyrmionic racetracks are considered one of the most reliable systems to transport skyrmions using the spin-orbit torque produced by a spin-polarized current fed in a heavy-metal substrate. The racetrack borders create a confining potential whose intensity determines the limit speed that a skyrmion can achieve inside a track without being annihilated at the border. As temperature increases, the topological protection of skyrmions is reduced, however, in some systems it is possible to stabilize and transport them. Apart from compromising stability, temperature can turn skyrmion dynamics into stochastic. As temperature increases the dispersion of the skyrmions position becomes larger, behaving like a Brownian particle. This stochastic motion sets new conditions and restrictions on the applicability of racetracks. In particular some binary questions like: Will the skyrmion be annihilated at the edge of the racetrack?, now become probabilistic questions: Which is the probability of a skyrmion to be annihilated at the edge of the racetrack? To address this kind of questions, a deterministic approach has been developed to study the dynamics of skyrmions at non-zero temperatures [2]. By solving the Fokker-Planck equation (FPE) for a skyrmion we can directly determine the probability of finding a skyrmion for a given time and position. In [2], the dynamics of a skyrmion when approaching a pinning potential and when driven along a racetrack was studied. In this work we present a natural continuation to that study: the granularity of the ferromagnet is taken into account while the skyrmion is moving along a racetrack. When nanomagnetic systems are studied from a theoretical point of view, usually we do not consider the granularity of the ferromagnetic materials that form the system, neglecting the influence of the polycrystalline structure and the presence of defects. However, in experiments we cannot avoid them. The granularity and defects difficult the motion of skyrmions at low velocities setting an unpinning velocity, adding an additional ingredient to the transport of skyrmions: they can be pinned in some point of the track [3]. We simulate the granularity of the material setting a random array of pining potentials with random (within a range) intensities. We study how the probability of a skyrmion to reach the end of track of length without being trapped or annihilated (success probability) changes as we vary the density of pining potentials, which is proportional to the density of crystal grains, and the intensity of the pining potentials, which is proportional to the difference of the magnetic properties between grains. In Fig.1 and Fig.2 we show different snapshots for the density of probability of a skyrmion being on a given position, in a simulation where 100 pin potentials are placed randomly. In Fig.2 the average intensity of the pining potentials is higher than in Fig. 1. We can observe in Fig. 1 that the probability of success is close to 1, indicating that the skyrmion will survive to its travel along the track without getting pinned. However, in Fig.2 we observe that there is a considerable probability that the skyrmion gets pined and does not reach the end of the track. In order to build real devices, we need to ensure that the probability of success is as close to 1 as possible. We study how this success probability depends on the skyrmion speed, and granularity of the track at room temperature, finding that the granularity can even help in some cases.

**References:**

[1] Albert Fert, Vincent Cros, João Sampaio, Nat. Nanotechnology, 8, 152–156(2013) [2] Josep Castell-Queralt, Leonardo González-Gómez, Carles Navau, Phys. Rev. B 101, 140404(R) (2020) [3] Barton L. Brown, Uwe C. Täuber, and Michel Pleimling, Phys. Rev. B, 100, 024410 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c29d136d6432c1a8980e5c1679287717.jpg)
Snapshots of the time evolution of the probability density of finding a skyrmion at a certain position. The corresponding snapshot times are, from left to right, 1.5/3/4.5/7.8/13.5 ns. The track length is 1 mm. There are 100 pining sites randomly distributed on the track with moderate intensity. The success probability is close to 1 (higher than when we have no granularity), with a small probability of being annihilated at the edge of the track or being trapped.

![](https://assets.underline.io/uploads/markdown_image/1/image/01ca6d0e939f9fbbd60fcf1caf2580f6.jpg)
Snapshots of the time evolution of the probability density of finding a skyrmion at a certain position. The corresponding snapshot times are, from left to right, 0.6/1.8/3.9/9/15 ns. The track length is 1 mm. There are 100 pining sites randomly distributed on the track with high intensity. The success probability is close to 0 since the trapping probability of the skyrmion is close to 1 (the skyrmion would be most likely trapped at some pinning site).

 Theoretical Study of theTransport of Skyrmions at Room Temperature in Granular Racetracks

During the last decade, the increasing demand of energy efficiency for low power electronics oriented a considerable progress in optimizing nanomagnetic and spintronic devices [1]. In this framework, magnetization switching is a fundamental process to investigate, in order to obtain a compromise between energy efficiency and speed in magnetic recording technology. In particular, magnetic random access memories (MRAMs) have proved to represent a valid alternative solution with respect to conventional non volatile electronic memories in terms of speed, scalability and low operation power [2-3]. Concerning the latter aspect, a significantly important issue is the design of memory cells with low enough operation power (e.g. within the IoT paradigm) but capable of reliable switching within a prescribed delay time, which may vary depending on applications (from fractions of ns to seconds and beyond). This means, in terms of nanomagnet physics, that magnetization switching for low energy barriers compared to the energy of thermal fluctuations, in the presence of a small/moderate external excitation (magnetic field/spin-polarized current) becomes a central issue to investigate. This occurs because, when the external current/field pulse is applied, the energy barrier which guarantees the required data retention at zero excitation (usually &gt; 60 kBT) is lowered, and thermal fluctuations may trigger and drive the magnetization reversal process. The analysis of stochastic magnetization dynamics [4] is usually performed by using two approaches which are complementary. On one hand, one can use the so-called Langevin dynamics approach, where the magnetization switching process of a large number of nanomagnet replicas is numerically studied via the stochastic Landau-Lifshitz-Gilbert (LLG) equation and the relevant information are extracted from the ensemble statistics [5-6]. On the other hand, the second approach exploits the fact that the stochastic LLG equation is associated with a drift-diffusion equation, referred to as Fokker-Planck (FP) equation, which describes the time evolution of the magnetization transition probability density function (pdf). In this paper, we numerically solve the FP equation on the unit sphere without any symmetry assumption on the particle. To this end, we have developed two methods: the spectral-collocation (SM) [7] and the FEM [8] methods, respectively. We show that both methods accurately predict the dominant relaxation time by comparing the numerical results with the asymptotic analytical theory [5] valid for large energy barriers ΔG with respect to the thermal energy kBT. The first method consists in choosing a grid of points and expanding the solution using a set of smooth basis functions. The grid on the unit sphere is constructed by using as coordinates the component of magnetization along the z-axis and the azimuthal angle associated to the rotation around the z-axis itself. The grid points are equispaced in the azimuth angle and are distributed as Chebyshev nodes in the z variable. For the FEM, a triangulated unit sphere is considered and it is obtained recursively bisecting the edges starting from a tetrahedral polyhedron [10]. The mesh has vertices pi, i=1,...,Np, which are the nodes where the unknown pdf is located. The results were tested for different triangulations with an increasing number of points in order to verify the convergence of the numerical FEM method. The computational approaches are validated comparing the slowest thermal relaxation time constant for an ensemble of magnetic particles, and the approximate analytical expression given in [5], as shown in Fig.(1). This time constant is of interest because it determines the thermal stability of a magnetic memory. We verify the FEM code in the following conditions: Np=(8194, 65536, 131074) until reaching the desired convergence with respect to the analytical results in the limit of large energy barriers. The SM method shows a high accuracy from small to intermediate energy barriers, whereas for higher energy barriers the method requires an increasing number of grid points, on the unit sphere, limiting the possibility to compute the relaxation time to slightly above 26 kBT, after which the numerical results deviate from the analytical prediction. Conversely the method is accurate for low energy barriers and it achieves machine precision in the limit case of zero energy barrier(pure diffusion on the unit sphere). The FEM method exhibits a lower accuracy in the relaxation time computation, one can see that it covers a broader range of energy barriers up to 35 kBT. The numerical tests presented here run on a PC using MATLAB. In conclusion, the developed methods allow accurate analysis of thermal switching processes for generic particles under different operating conditions depending on the moderately large or small amplitude of the energy barrier in the presence of external excitation. It is possible to show (not reported here due to limited space) that they allow to straightforwardly determine the switching times distributions [9], instrumental to assess the write-error rates of MRAM cells [11], in the intermediate regime between thermal and ballistic switching which is of great interest for applications.

**References:**

[1] B. Dieny et al., Nature Electronics 3 (8), 446-459, (2020). [2] D. Apalkov et al., Proceedings of the IEEE, Vol. 104, No. 10, (2016). [3] S. Bhatti et al., Materials Today, Volume 20, Number 9, (2017). [4] G. Bertotti, I.D. Mayergoyz, C. Serpico, Nonlinear Dynamics in Nanosystems, Elsevier, (2009). [5] W.F. Brown, Physical Review, vol. 130, no. 5, (1963). [6] D.B. Reeves, J.B. Weaver, Applied PhysicsLetters, vol. 107, issue 22, (2019). [7] D. P. Ansalone, C. Ragusa, M. d’Aquino, C. Serpico and G. Bertotti, IEEE Transactions on Magnetics, vol. 45, (2009). [8] N.V. Peskov, Physica B: Condensed Matter, vol. 599, (2020). [9] M. d’Aquino et al., Journal of Magnetism and Magnetic Materials 475, 652-661, (2019). [10] S. Maddali S. Maddali, SphereMesh, https://github.com/siddharth-maddali/SphereMesh, (2021). [11] M. d'Aquino, S. Perna, C. Serpico, Physica B: Condensed Matter, Volume 577, (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c9a9725762d961cc4018595a8d70f838.jpg)
Fig. 1. Comparison between numerical results and analytical approximated values, for the following parameters values: T=300K, α=0.01and keff=0.3569 takes into account shape and crystalline anisotropy, Ms=8 105 A/m and no applied field. The number of unknowns are Np=131074 for the FEM and Np=21632 for SC methods.

 Numerical Solution of the Fokker-Planck Equation by Spectral Collocation and FEM Methods for Stochastic Magnetization Dynamics

Micromagnetics complements theoretical and experimental methods to support research in magnetism. The Object-Oriented MicroMagnetic Framework (OOMMF) [1] is a micromagnetic simulation tool solving non-linear time-dependent partial differential equations using the finite-difference method. It is probably the most widely used simulation tool in the micromagnetics community. It was written in C++, wrapped with Tcl, and driven through configuration files that follow the Tcl syntax. The typical computational workflow the user must follow to simulate a particular problem is to write a configuration file. After that, the user runs OOMMF by providing the configuration file to the OOMMF executable. When the OOMMF run is complete, results are saved in OOMMF-specific file formats. Finally, the user analyzes the result files. One of the specific goals of a computational micromagnetic study is parameter-space exploration. More precisely, the user repeats the simulation for different values of input parameters by changing them in the configuration file. It is often difficult to automate this, and it is challenging for the user to keep a log of all steps performed in the entire micromagnetic study. Besides, postprocessing and analysis of results is performed outside OOMMF, using techniques and scripts that are mostly developed by the user, or carried out manually. Consequently, it is hard to track, record, and convey the exact simulation procedure. Without this information, resulting publications are generally not reproducible. To address this situation, we developed a Python interface to the OOMMF executable to expose OOMMF to a general-purpose programming language supported by Jupyter. This allows us to conduct micromagnetic simulations from within the Jupyter notebook [2] to capitalize on the benefits of this environment. We developed a set of Python libraries we refer to as Ubermag, which expose the computational capabilities of OOMMF so that it can be controlled from Python. These Python libraries provide a domain-specific language to define a micromagnetic problem. A micromagnetic model, defined using the domain-specific language, is not aware of the particular simulation tool that will perform the actual micromagnetic simulation, and it is only used to describe the model. When a simulation is required, the model is translated into the OOMMF configuration file, the OOMMF executable is called, and the output files are read. By exposing the micromagnetic simulation capabilities to Python and driving the research from Jupyter Notebook, we have available all the benefits of the Jupyter research environment and all tools from Python's scientific stack are readily available. Ubermag and the Jupyter environment simplify the efforts to make micromagnetics publications reproducible. For each figure in the publication, one notebook can be provided. Using Binder [3], the community can inspect and re-run all the calculations in the cloud and make the publication reproducible.

**References:**

[1] M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interag. Rep. NISTIR 6376, Natl. Inst. Stand. Technol. Gaithersburg, MD, 1999. [2] T. Kluyver et al. Jupyter Notebooks - a publishing format for reproducible computational workflows. Positioning and Power in Academic Publishing: Players, Agents and Agendas, 87-90, 2016. [3] https://mybinder.org/

**Images:**

 Towards reproducible micromagnetic workflows using Ubermag

Machine learning (ML) entered the field of computational micromagnetics only recently [1,2,3]. The main objective of these new approaches is the automatization of solutions of parameter-dependent problems in micromagnetism such as fast respond curve estimation modeled by the Landau-Lifschitz-Gilbert (LLG) equation. Data-driven models for the prediction of solutions of time- and parameter-dependent partial differential equations require high dimensional training data-structures. ML in this case is by no means a straight-forward trivial task, it needs algorithmic and mathematical innovation. Our work [4] introduces theoretical and computational conceptions of certain kernel-based dimensionality reduction approaches for efficient machine learning of the solution trajectories of LLG via the notion of low-dimensional feature space integration, Fig.1. We will mainly concentrate on computational aspects, specifically, kernel methods will be studied with focus on numerically stable and efficient implementation of kernel ridge regression and kernel principal component analysis via low-rank treatment of certain dense operations. A second line follows neural network autoencoders as dimensional reduction for the training data with focus on smooth latent space variable description suitable for the feature space integration scheme. This will be doable due to a certain modification of contractive regularization that maintains feasibility despite the typically large computational demands coming with such approaches. We verify and compare numerically by means of a NIST standard problem, Fig.2.

**References:**

[1] A. Kovacs, J. Fischbacher, H. Oezelt, M. Gusenbauer, L. Exl, F. Bruckner, D. Suess, T. Schrefl. Learning magnetization dynamics, Journal of Magnetism and Magnetic Materials 491 (2019): 165548. doi: http://10.1016/j.jmmm.2019.165548 [2] L. Exl, N.J. Mauser, T. Schrefl, D. Suess. Learning time-stepping by nonlinear dimensionality reduction to predict magnetization dynamics, Communications in Nonlinear Science and Numerical Simulation 84 (2020): 105205. doi: 10.1016/j.cnsns.2020.105205 [3] L. Exl, J. Fischbacher, A. Kovacs, H. Oezelt, M. Gusenbauer, K. Yokota, T. Shoji, G. Hrkac, and T. Schrefl, Magnetic microstructure machine learning analysis, Journal of Physics: Materials, 2 (2019): 014001. doi: 10.1088/25157639/aaf26d [4] Exl, L., Mauser, N. J., Schaffer, S., Schrefl, T., & Suess, D. (2020). Prediction of magnetization dynamics in a reduced dimensional feature space setting utilizing a low-rank kernel method. arXiv preprint arXiv:2008.05986. https://arxiv.org/pdf/2008.05986.pdf

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7cd58bb91ad23869dce8b0b6893d6b77.jpg)
Fig.1: Time-integration scheme in low-dimensional feature space.

![](https://assets.underline.io/uploads/markdown_image/1/image/548c264f8d7c23ee60cf93515379b54d.jpg)
Fig.2: Snapshots for comparison of simulated and predicted magnetization states for varying ranks r.

 Machine learning methods for the prediction of micromagnetic magnetization dynamics

The exchange interaction between two magnetic materials may lead to a symmetry breaking manifesting itself as a unidirectional anisotropy induced in one layer (usually ferromagnetic material F) by another one less susceptible to an external magnetic field (usually antiferromagnetic material AF). The unidirectional anisotropy in the F layer leads to a hysteresis loop shifted along the magnetic field axis and the phenomenon is called exchange bias (EB). Despite its seeming simplicity, the EB has an intricate microscopic mechanism, which has been approached by many models of different complexity dealing with perfect or imperfect interface, polycrystalline structure of F and AF, and bulk spin configuration in the AF layer [1]. For real-life applications e.g. sensors, memory cells or spintronic devices, the number of significant parameters affecting coercivity and EB field of the F layer could become too large to be accounted by simplified models, so computer simulation becomes a method of choice. The increasing complexity leads to more realistic and universal models but makes it difficult to evaluate their performance and limitations, study interconnections between model’s parameters, and make a fast prediction on how the system with a given set of parameters would behave. In this study we approach this problem by applying methods of machine learning to a dataset obtained by a micromagnetic model representing a polycrystalline bilayer film consisting of F and AF layers. As AF state cannot be readily set in the continuous media approach, we adopted the model described in [2] by substituting AF with F material, for which contributions to Zeeman and magnetostatic energies were disabled. The example of the simulated hysteresis loop along with the geometry is shown in Fig. 1. Micromagnetic simulation was performed using MuMax3 software with GPU acceleration [3]. As an income several material and structural parameters have been chosen including saturation magnetization (same for F and AF), uniaxial magnetic anisotropy constants of F and AF, exchange constants in F and AF, attenuation coefficient of exchange constant between F and AF, thickness of the F layer, and average crystallites size (8 variables in total). As a result of approximately 1800 hysteresis loops simulations, for each configuration (each parameter has been varied randomly within a realistic range) two outcome parameters have been extracted: coercivity and EB field. The model has been validated with a smaller set of simulations where single parameter has been varied while fixing others at realistic values typical for a popular FeNi/FeMn system. Principal component analysis (PCA) allowed identifying readily input parameters having the strongest positive and negative correlations with the output parameters. Correlation coefficients for saturation magnetization, the thickness of the F layer, and the attenuation coefficient were significant for both coercivity and EB field, whereas the coefficient for AF anisotropy constant was among the largest only for EB field. Both PCA and T-SNE (t-distributed stochastic neighbor embedding) data visualizations suggested the possibility of building a data approximator. As EB field and coercivity are known to have complex nonlinear dependencies on some parameters, we chose the gradient boosting, which is a suitable algorithm for the current task. In this work Catboost open source library based on decision trees was used. The dataset was split into two parts for training the model and testing it using the root mean square error (RMSE) metric. The trained model could predict EB field for a test set of 500 examples rather well with RMSE = 0.04, whereas higher RMSE = 0.13 was achieved for coercivity, which could also be considered a good result. To demonstrate the applicability of the model for fast search of optimal coercivity and EB field, we employed Broyden–Fletcher–Goldfarb–Shanno and differential evolution algorithms realized in SciPy library to find sets of input parameters giving the largest values of output parameters. This approach allowed us to identify several areas of the highest coercivity and EB field values exceeding those in the training set, which were confirmed by additional micromagnetic simulations. Although the RMSE was large, the model can be further improved by including more examples of computer simulation to resolve the areas of interest. In conclusion, machine learning algorithms were successfully used for in-depth analysis of a bilayer micromagnetic model with exchange bias. The model capable of predicting values of coercivity and exchange bias using material and structural parameters as an input was successfully created and trained. The proposed approach was demonstrated to be suitable for building optimization algorithms allowing to find areas of interest in multidimensional space of income parameters giving the desired values coercivity and EB field. The demonstrated approach can be useful for the analysis of computer models as well as real systems with exchange bias. This work was supported by the Russian Science Foundation (project No 19-72-00141).

**References:**

[1] F. Hellman, A. Hoffmann, Y. Tserkovnyak et al., Rev. Mod. Phys. Vol. 89, a.n. 025006 (2017) [2] J. De Clercq, A. Vansteenkiste, M. Abes et al., J. Phys. D: Appl. Phys., Vol. 49, a.n. 435001 (2016) [3] A. Vansteenkiste, J. Leliaert, M. Dvornik et al. AIP Advances, Vol. 4, a.n. 107133 (2014)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a36a3f75ae7126384b7de6c275129056.jpg)
Figure 1. Hysteresis loop simulated using a micromagnetic model of a bilayer with exchange bias consisting of F and AF layers. In the inset a slice of the geometry with Voronoi tessellation is shown. The colors are given to differentiate the crystallites.

 Combined micromagnetic simulation and machine learning approach to analysis of polycrystalline bilayer system with exchange bias

Smartphone-based indoor localization methods draw increased interest and are frequently employed for position estimation of users inside enclosures like malls, conference halls, and crowded venues. Magnetic sensing has recently attracted considerable attention due to its pervasiveness and autonomy, introducing two main positioning approaches utilizing magnetic sensors. The first approach is identifying the magnetic field characteristics of the surrounding infrastructure as signatures, thus creating a specific magnetic fingerprint. This scheme does not require additional hardware deployment, while interferences and anomalies from indoor infrastructure and objects greatly improve magnetic field discernibility. Yet, this approach requires knowledge of magnetic maps or computationally expensive calculations for applying simultaneous localization and mapping (SLAM). Furthermore, some environments may cause strong electromagnetic interference that differs from the underline magnetic fingerprint, leading to localization errors. The second approach uses active magnetic flux transmitters as markers, which is energy consuming and requires expensive transmitter units. In our recent work, we introduced the Permanent Magnets Superstructure for Indoor Localization method (PMSIL). This approach allows simple, low-power, and robust smartphone localization in unknown-magnetic-fingerprint locations by deploying permanent magnets in pre-known locations. Practically, we embed small-volume-large-moment permanent magnets in given locations inside the building and arrange them in a single specific geometric configuration. This results in a supervised super-structure magnetic signature pattern, constituting unambiguous magnetic environments. The localization algorithm learned this unique pattern during the training stage using artificial intelligence (AI) methods and detected it in ongoing data streams during real-time localization by applying binary classification. Thus, localization is based on smartphone motion rather than on the static positioning of the magnetometer. In this study, we propose an extended version of the PMSIL method, called Multiclass-PMSIL (M-PMSIL), in which magnets are arranged in a variety of configurations in pre-known locations. These constellations generate a bank of distinct patterns that are learned by the multiclass AI-based algorithms during training and recognized on-the-fly during localization. This extension permits covering a broader area in which the user can be localized. As M-PMSIL is landmark-based, we draw comparisons to three competing state-of-the-art approaches for indoor localization with landmark-based magnetism mechanisms. We also refer to localization methods based either on spatial-temporal sequence matching or on fusion with motion. The database is recorded by a user strolling back and forth with a hand-held smartphone across a 100 m long and 3 m wide in-door corridor. Along this corridor, three neodymium magnets with different magnetic moment norms are positioned in a row, with 3 m separating between them. During the experiment, these magnets are permuted in all 6 possible combinations and every new combination is relocated along the corridor, creating 6 distinct magnetic patterns. For the sake of generalization, recordings are made in different walking paces between 0.5 to 1.5 m/s, and different crossing distances from the magnets of 0.5 m, 1m, and 1.5 m. A snippet of data recorded by the smartphone indoor can be seen in Fig. 1, including 2 of the 6 magnet constellations used in this study. The training set includes 40 minutes of randomly selected recorded data that contains 10 passes across each of 6 different magnetic patterns, while the test set contains 20 minutes of data. The 6 classes, i.e., magnetic permutations to be localized, are balanced for both disjoint sets. In this study, 6 AI-based architectures are used for localization. Their performance is compared in Fig. 2 using the receiving operating characteristic (ROC) curve, allowing analysis of the trade-off between true positive rate (TPR) and false-positive rate (FPR) in various operation points. We may observe that the long short-term memory (LSTM) network produced the best classification results, followed by the gated recurrent unit (GRU) and recurrent neural network (RNN) models. The least performant methods are the fully connected deep neural network (DNN), and support vector machine (SVM), with and without principal component analysis (PCA). To quantify the multiclass classification accuracy of the methods, we derive the maximal sum of TPR and FPR from the ROC curve. LSTM shows 95% accuracy, while GRU and RNN reach 90% and 87%, respectively. DNN shows 82%, SVM-PCA method reaches 80%, and SVM produces 75% accuracy. Our approach achieves mean localization error (MLE) lower than 1 m, supersedes competing landmark-based methods, as UnLoc and MapCraft achieved MLE between 1-2 m. The IODetector was able to obtain classification accuracy as high as 82% indoor, which falls short to the 95% obtained by the LSTM. Additional methods, rooted in spatial-temporal sequence matching or on fusion with motion, also show lower MLE and classification accuracy compared to our study.

**References:**

[1] S. He and K. Shin, ACM Comput. Surv., Vol. 50, p.1-37 (2017) [2] K. Watanabe, K., Hiroi, K., Kamiyama, et al., Journal of Information Processing., Vol. 27, p.10–24, (2019) [3] E. Weiss and R. Alimi, Low-Power and High-Sensitivity Magnetic Sensors and Systems. Artech House, (2018) [4] T. N. Sainath, O. Vinyals, A. Senior, et al., IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.4580–4584, (2015) [5] J. N. Mandrekar, J. Thorac. Oncol., vol. 5, pp. 1315–1316, (2010)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ccecf57041a3306dc63b5f676accee9b.jpg)
Fig 1: Norm value of three-axial magnetic signals recorded with a smartphone. Rectangles confine measurements of three magnets positioned in a row, in two distinct magnet constellations.

![](https://assets.underline.io/uploads/markdown_image/1/image/995c407af3fee196a6aae9cf4d4d9671.jpg)
Fig 2: Multiclass classification performance of six AI methods.

 Multiclass Permanent Magnets Superstructure for Indoor Localization using Artificial Intelligence

At the microscopic scale, magnetic systems can be described by the formalism of micromagnetics, which is used for e.g. micromagnetics of rare-earth efficient permanent magnets [1]. In a micromagnetic model, the governing equations must be discretized and represented by a number of interactions taking place between the discretized moments in the system. These interactions can be described by different magnetic fields, such as an exchange- and an anisotropy field. Of these, it is the calculation of the demagnetization field that is the most time consuming part [2]. There are in general three methods that have been applied to calculate the demagnetization field: finite difference-based fast Fourier transform methods, tensor-grid methods and finite-element methods. All methods have inherent advantages and shortcomings, e.g. the finite difference Fourier-based approaches are extremely fast but unfortunately require that the magnetic moments lie on a regular grid [3], while the finite element method allows easy meshing of complex geometries but also requires that an environment outside the object of interest is modelled. Here we present a novel open-source micromagnetism framework called MagTense that uses an analytically exact approach to calculate the full demagnetization tensor in real space for computations [4]. This ensures that the calculated magnetic field is correct, and at the same time allows for easy meshing of complex geometries. In MagTense the discretization approach used subdivides the region Ω into N "tiles", adjacent to each other. The magnetization field is then represented by a vector of 3N components, three for each tile. The demagnetization field generated by each tile is computed analytically by assuming that the tile is uniformly magnetized. The analytical expression of the demagnetization tensor is known for many different tiles, such as the rectangular cuboid (rectangular prism) [5] and the tetrahedron [6], thus allowing great flexibility to build meshes for any geometry. The total demagnetization field at any point is obtained by summing the individual contributions from all the tiles in the magnet domain Ω. Besides the calculation of the demagnetization field, the other components of the effective field must also be calculated, i.e. the exchange-, external- and anisotropy field. For the case of a Cartesian mesh of rectangular cuboids the exchange interaction field can easily be computed by applying standard finite difference schemes. However, methods are also available for the case of unstructured polyhedral meshes. Using MagTense we have calculated the solution to the μmag standard micromagnetic problems 2, 3 and 4 and find that the MagTense framework accurately predicts the solution to each of these [7]. For example for standard problem 3, we get side length at the crossover energy between the vortex and flower state of 8.477, which is in line with two of the published μmag results. We have also implemented the MagTense framework to support both CPU and GPU calculations, where in the latter the demagnetization calculations are performed on the GPU using the CUDA framework. On μmag standard problem 4, we see a decrease in total simulation time of a factor of 100 for the GPU implementation. To demonstrate the use of MagTense on a sample with multiple grains we show in Fig. 1 the magnetization evolution of a thin slice of a multi-grain material starting from a random state. The structure has been generated using a centroidal Voronoi tessellation. The hard magnetic phase in the five grains is characterized by a strong anisotropy, corresponding to the direction indicated by the thick black arrows, while the grain-boundary is soft-phase with isotropic properties. Inside the grains we observe the emergence of magnetic domains aligned to the easy axis, while at the grain-boundary we see the formation of magnetic vortexes. To simulate realistic grain samples, we have also utilized a sintering modeling framework based on kinetic Monte Carlo [8] to generate fully realistic microstructures of sintered powder compacts. We then subdivide these grain structures into magnetic tiles for simulating the magnetic properties of the complete sample. Shown in Fig. 2 (a) is an example of multi-grain sample generated using the sintering modeling framework. The colours indicate different grains, gray corresponding to the grain boundary soft-phase. The thickness of the grain boundary phase can be specified as an input parameter. Fig. 2 (b) and 2(c) show the tetrahedral and Cartesian meshes, respectively, applied to this geometry. The colours indicate the distance of a point from the nearest grain boundary, which in both cases was used to control the resolution of the mesh. In this way the mesh is finer at the boundary between grains which is where the nucleation is expected to occur during a field-reversal process.

**References:**

[1] J. Fischbacher, A. Kovacs, M. Gusenbauer, H. Oezelt, L. Exl, S. Bance, and T. Schrefl., J. Phys. D: Appl. Phys, 51(19):193002, (2018). [2] C. Abert, L. Exl, G. Selke, A. Drews, T. Schrefl, Journal of Magnetism and Magnetic Materials,326:176–185, (2013). [3] A. Vansteenkiste, Ben Van de Wiele, Journal of Magnetism and Magnetic Materials, 323(21):2585–2591, (2011). [4] R. Bjørk and K. K. Nielsen, Magtense - a micromagnetism and magnetostatic framework,doi.org/10.11581/DTU:00000071, https://www.magtense.org, (2019). [5] R. Bjørk, E. B. Poulsen, K. K. Nielsen, A. R. Insinga, “MagTense: a micromagnetic frameworkusing the analytical demagnetization tensor”, submitted, (2021). [6] A. Smith, K. K. Nielsen, D. V. Christensen, C. R. H. Bahl, R. Bjørk, J. Hattel. Journal ofApplied Physics, 107(10):103910, (2010). [7] K. K. Nielsen, A. R. Insinga, R. Bjørk, IEEE Magnetics Letters, 10:8918242, (2019). [8] V. Tikare, M. Braginsky, D. Bouvard, A. Vagnon, Comp. Mat. Sci., 48:317-25, (2010).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/21f2425d866e8b1eb830bc6b71276f78.jpg)
Spontaneous evolution of magnetic domains in a multi-grain structure. The color indicate the local direction of magnetization.

![](https://assets.underline.io/uploads/markdown_image/1/image/368893183811e0823acd0ef821e19e69.jpg)
Simulated sintered powder sample and corresponding adaptive meshes.

 Simulating sintered magnets using the full demagnetization tensor with MagTense

The numerical studies of magnetization dynamics in micromagnetic systems with spatially distributed magnetization crucially rely on the computation of magnetostatic fields generated by the ferromagnet. The solution of the magnetostatic problem can be written as convolution of an appropriate Green’s function with the magnetization vector field. Since the Green’s function is long-range, after discretization of the problem in N cells, the direct numerical computation of the convolution integral has a computational cost scaling as N2. In this work, we use, as it is often done in micromagnetics, the Fast Fourier Transform (FFT) method to reduce the computational cost scaling to N logN. We consider micromagnetic systems with thin-film geometry where the magnetization can be considered constant across the thickness of the magnet. In these situations, only the field averaged across the thickness is relevant to the magnetostatic interactions and both the magnetization and the averaged field inside the magnet depend only on two spatial variables. The usual approach to deal with this type of problem is to use FFT to compute the discretized version of the convolution integral relating the field and the magnetization [1]. The dual approach to the usual one, is based on the discretization of the integral relation between the field (averaged across the thickness) and the magnetization in the reciprocal space (k-space). The connection in the k-space is given by the Fourier transform of Green function of the magnetostatic problem which, in case of infinite thin-films, can be written in a special form taking into account the thin-film geometry [2]. The analysis in the k-space has been extensively used in connection with linear spin wave dynamics in thin films working as magnonic waveguide, where all the relation of practical use (e.g dispersion relations), are easily obtained by algebraic manipulations. When the dimensions of the thin film are finite, the reconstruction of the magnetostatic field with the infinite film Green's function, is altered by the presence of replicas of the thin film in the ordinary space due to the sampling operation in the k-space. This effect is usually smoothed by using zero-padding, which increases the distance between two interacting replicas weakening their coupling. In this work, a fast convolution algorithm in the k-space able to remove the influence of replicas in the reconstruction of the magnetostatic field in thin films of finite size, is proposed and applied to study linear spin wave dynamics. It is based on the use of the Fourier transform of truncated Green function, which has been recently used to compute volume potentials defined on free-space in a finite (truncated) domain [3]. Such method, guarantees a regularization of the Green function by cutting off the interaction in physical space beyond the domain of interest and this in turn produces spectral accuracy [5] for the Fourier transform evaluation by FFT algorithm. The accuracy of the proposed method is studied by comparing the results with analytical calculations and with those obtained by fast convolution with the infinite thin film Green's function as a function of mesh size. For example, in the figure, the computation of the demagnetizing factor along the x direction for a thin film of dimensions Lx = 300nm, Ly = 100nm and Lz = 3nm as a function of the mesh dimension (kxmax = kymax = kmax) and discretization step-size (Δkx = Δky = Δk) in the k-space is considered. With solid lines, the curves in the infinite film approximation (infin.) are indicated, while curves with symbols results from the proposed approach (trunc.). The horizontal dotted line, represents the analytical value of the demagnetizing factor evaluated with Newell’s formula [6]. Computation with Δk&lt; π/300 nm-1, for the duality property of the Fourier transform, are equivalent to use zero padding in the ordinary space. This produces a substantial improving of estimation accuracy with respect to the analytical value, in case of infinite film approximation. As a benchmark of the proposed approach, it results that the curves computed with the truncated Green function, show a weak dependence from Δk for Δk&gt;π/R and collapse into the same one for Δk&lt;π/R, where R is the radius of a disk region which defines the compact support of the truncated Green function. This, as a confirmation of the fact, that when the region that exceeds the support of the truncated Green's function is zero padded, there is no effect on the computation of the magnetostatic field. Moreover, for sufficient value of kmax the asymptotic value tends towards analytical one, since when Δk&lt;π/R the influence of replicas on the magnetostatic field computation is removed.

**References:**

[1] R.D. McMichael, M.J. Donahue, D.G. Porter, J.Eicke, Journal of Applied Physics, vol. 85, pp. 5816-5818, (1999). [2] V. Scalera et al., IEEE Transactions on Magnetics, doi: 10.1109/TMAG.2020.3014383. [3] F. Vico, L. Greengard, M. Ferrando, Journal of Computational Physics, Vol. 323, p. 191-203 (2016). [4] B.A. Kalinikos, A.N.Slavin, J. Phys. C: Solid State Phys. Vol. 19, p.7013-7033 (1986). [5] L.N. trefethen, SIAM, Philadelphia, (2000). [6] A.J. Newell, W. Williams, D.J. Dunlop, J. Geophysical Research, Vol. 98, pp. 9551-9555 (1993).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0c3a6bfd333a895ba9655d5de5798653.jpg)
Comparison of the demagnetizing factor computed analytically with Newell’s formula [6] and numerically with k-space FFT convolution with the infinite film Green's function (infin.) and with the truncated Green's function (trunc.), for different mesh dimension kmax and step-size Δk in the k-space.

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