France



The discovery of a significant interfacial Dzyaloshinskii-Moriya Interaction (iDMI) in multi-layer thin films launched a fervent research effort into its impact on the behavior of chiral magnetic objects including domain walls (DWs).[1] The field driven creep of these Dzyaloshinskii domain walls as observed by Kerr microscopy remains a popular tool for evaluating the strength of the iDMI, but results are notoriously challenging to interpret.&lt;br/&gt;&lt;br/&gt;In this talk, we focus on experimental and theoretical investigations into the asymmetric field-driven expansion of magnetic domains in thin films with iDMI, which provide a fascinating landscape to understand creep of 1D elastic manifolds with an internal order parameter.[2] We will review early models of creep based on the DW energy, which matched well with certain experimental observations – namely, that an in-plane field (H&lt;sub&gt;x&lt;/sub&gt;) coupled with a perpendicular driving field (H&lt;sub&gt;z&lt;/sub&gt;) leads to asymmetric expansion of the domain as well as minima in the plot of x-velocity vs H&lt;sub&gt;x&lt;/sub&gt;. However, other common observations were not explained by this model including that, at large values of H&lt;sub&gt;x&lt;/sub&gt;, the high energy wall orientations reached or exceeded the velocity of low energy orientations. To explain this, Pellegren at el considered the more general elastic energy of the DW, given by the stiffness [σ(θ) + σ&#39;&#39;(θ)].[3] By introducing a dispersive stiffness, which accounts for the finite lengthscale over which the wall deforms, it was indeed shown that the elastic energy of opposite extrema on the domain converge as H&lt;sub&gt;x &lt;/sub&gt;→ ∞. This more general model explains some of the features that have also been explained by an attempt frequency that depends on wall chirality (via a chiral damping parameter).[4] The possible role of chiral damping will be discussed briefly.&lt;br/&gt;&lt;br/&gt;In more recent experiments, Brock et al showed that, in some cases – especially for weak DMI, the domain adopts a highly preferred growth direction with a dendritic structure that is neither parallel or anti-parallel to H&lt;sub&gt;x&lt;/sub&gt;.[5] The results cannot be explained by a chiral Bloch component to the DW, which would violate the symmetry of the system. Instead, we identify a steady-state DW substructure that results from the z-field torque on the internal DW magnetization (see figure 1c). This breaks the energetic mirror symmetry along the x-axis and leads to orientations of the DW with vanishing elastic energy (i.e. rapidly increased creep velocity). Remarkable agreement between an augmented form of the dispersive stiffness model (now accounting for steady-state configurations) and the experimental measurements is found. The material parameters that give rise to this seemingly anomalous behavior will be discussed. Other important factors such as internal DW excitations, field-dependent pinning, and a field-dependent DW width will be considered in this presentation as well.

**References:**

[1] Thiaville, A., Rohart, S., Jué, É., Cros, V. and Fert, A., 2012. Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. &lt;i&gt;EPL (Europhysics Letters)&lt;/i&gt;, &lt;i&gt;100&lt;/i&gt;(5), p.57002.&lt;br/&gt;&lt;br/&gt;[2] Fournier, J. B. &quot;Generalized gibbs-thomson equation and surface stiffness for materials with an orientational order parameter.&quot; &lt;i&gt;Physical review letters&lt;/i&gt; 75.5 (1995): 854.&lt;br/&gt;&lt;br/&gt;[3] Pellegren, J.P., Lau, D. and Sokalski, V., 2017. Dispersive stiffness of Dzyaloshinskii domain walls. &lt;i&gt;Physical Review Letters&lt;/i&gt;, &lt;i&gt;119&lt;/i&gt;(2), p.027203.&lt;br/&gt;&lt;br/&gt;[4] Jué, E., Safeer, C.K., Drouard, M., Lopez, A., Balint, P., Buda-Prejbeanu, L., Boulle, O., Auffret, S., Schuhl, A., Manchon, A. and Miron, I.M., 2016. Chiral damping of magnetic domain walls. &lt;i&gt;Nature materials&lt;/i&gt;, &lt;i&gt;15&lt;/i&gt;(3), pp.272-277.&lt;br/&gt;&lt;br/&gt;[5] Brock, J., Medapalli, R. and Fullerton, E., 2020. Chirality-driven growth of stripe domains in Co/Ni/Pt-based multilayers. &lt;i&gt;Bulletin of the American Physical Society&lt;/i&gt;, &lt;i&gt;65&lt;/i&gt;.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d041ea67aa2fad976e5c8d2fe441cb85.jpg)
Fig 1. Schematic of a Néel bubble subject to external magnetic field. Arrows indicate the direction of internal magnetization along the circumference. Green lines indicate time-reversal mirror planes, m&#39;. a) No external field. Magnetic point symmetry: ∞m&#39;m’. b) H&lt;sub&gt;x&lt;/sub&gt; only. Magnetic point symmetry: m’. c) H&lt;sub&gt;x&lt;/sub&gt; and H&lt;sub&gt;z&lt;/sub&gt; field. Magnetic point symmetry: 1.



INTERMAG 2021

Dzyaloshinskii Domain Wall Creep INVITED

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The discovery of a significant interfacial Dzyaloshinskii-Moriya Interaction (iDMI) in multi-layer thin films launched a fervent research effort into its impact on the behavior of chiral magnetic objects including domain walls (DWs).[1] The field driven creep of these Dzyaloshinskii domain walls as observed by Kerr microscopy remains a popular tool for evaluating the strength of the iDMI, but results are notoriously challenging to interpret. In this talk, we focus on experimental and theoretical investigations into the asymmetric field-driven expansion of magnetic domains in thin films with iDMI, which provide a fascinating landscape to understand creep of 1D elastic manifolds with an internal order parameter.[2] We will review early models of creep based on the DW energy, which matched well with certain experimental observations – namely, that an in-plane field (Hx) coupled with a perpendicular driving field (Hz) leads to asymmetric expansion of the domain as well as minima in the plot of x-velocity vs Hx. However, other common observations were not explained by this model including that, at large values of Hx, the high energy wall orientations reached or exceeded the velocity of low energy orientations. To explain this, Pellegren at el considered the more general elastic energy of the DW, given by the stiffness [σ(θ) + σ''(θ)].[3] By introducing a dispersive stiffness, which accounts for the finite lengthscale over which the wall deforms, it was indeed shown that the elastic energy of opposite extrema on the domain converge as Hx → ∞. This more general model explains some of the features that have also been explained by an attempt frequency that depends on wall chirality (via a chiral damping parameter).[4] The possible role of chiral damping will be discussed briefly. In more recent experiments, Brock et al showed that, in some cases – especially for weak DMI, the domain adopts a highly preferred growth direction with a dendritic structure that is neither parallel or anti-parallel to Hx.[5] The results cannot be explained by a chiral Bloch component to the DW, which would violate the symmetry of the system. Instead, we identify a steady-state DW substructure that results from the z-field torque on the internal DW magnetization (see figure 1c). This breaks the energetic mirror symmetry along the x-axis and leads to orientations of the DW with vanishing elastic energy (i.e. rapidly increased creep velocity). Remarkable agreement between an augmented form of the dispersive stiffness model (now accounting for steady-state configurations) and the experimental measurements is found. The material parameters that give rise to this seemingly anomalous behavior will be discussed. Other important factors such as internal DW excitations, field-dependent pinning, and a field-dependent DW width will be considered in this presentation as well.

**References:**

[1] Thiaville, A., Rohart, S., Jué, É., Cros, V. and Fert, A., 2012. Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. EPL (Europhysics Letters), 100(5), p.57002. [2] Fournier, J. B. "Generalized gibbs-thomson equation and surface stiffness for materials with an orientational order parameter." Physical review letters 75.5 (1995): 854. [3] Pellegren, J.P., Lau, D. and Sokalski, V., 2017. Dispersive stiffness of Dzyaloshinskii domain walls. Physical Review Letters, 119(2), p.027203. [4] Jué, E., Safeer, C.K., Drouard, M., Lopez, A., Balint, P., Buda-Prejbeanu, L., Boulle, O., Auffret, S., Schuhl, A., Manchon, A. and Miron, I.M., 2016. Chiral damping of magnetic domain walls. Nature materials, 15(3), pp.272-277. [5] Brock, J., Medapalli, R. and Fullerton, E., 2020. Chirality-driven growth of stripe domains in Co/Ni/Pt-based multilayers. Bulletin of the American Physical Society, 65.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d041ea67aa2fad976e5c8d2fe441cb85.jpg)
Fig 1. Schematic of a Néel bubble subject to external magnetic field. Arrows indicate the direction of internal magnetization along the circumference. Green lines indicate time-reversal mirror planes, m'. a) No external field. Magnetic point symmetry: ∞m'm’. b) Hx only. Magnetic point symmetry: m’. c) Hx and Hz field. Magnetic point symmetry: 1.

technical paper

Insomnia is a common sleep disorder that affects a great percentage of the population. Magnetic stimulation technology is a new non-invasive and painless treatment for brain disorders. However, the effectiveness of magnetic stimulation technology in the treatment of insomnia has not been fully studied. In this paper, magnetic stimulation was used to stimulate acupuncture points (acupoints). 64-channel electroencephalography (EEG) signals under different states (healthy subjects' resting state, sub-healthy insomnia subjects' resting state before and after magnetic stimulation) were recorded. The brain functional networks based on EEG were constructed and analyzed to investigate the effects of magnetic stimulation on insomnia. The results showed that after magnetic stimulation, the connection and characteristic parameters of brain function network in EEG beta band of insomniacs were improved. Keywords: Magnetic stimulation; Electroencephalography (EEG); Brain functional network; Insomnia I. Introduction Insomnia is a functional disorder caused by the imbalance of brain function. Magnetic stimulation technology has the advantage of safe and painless, and it could stimulate deep tissues and nerves [1]. The synchronization of EEG signals could reflect the correlation between various brain regions [2]. II. Materials and methods Eighteen (nine insomnia and nine healthy) right-handed subjects were chosen in this experiment. According to traditional Chinese medicine, Neiguan (PC6), Shenmen (HT7) and Sanyinjiao (SP6) acupoints were selected in this experiment. Stimulation frequency was 1Hz. Intensity was 1.76 T (80% of the maximal). Each subject was stimulated for three days. 64-channel EEG signals were recorded before and after magnetic stimulation. Pearson correlation coefficient of EEG beta band was calculated. III. Results For insomnia subjects, after magnetic stimulation, the connections and the characteristic parameters of brain in EEG beta band were tending to those of healthy subjects, as shown in Fig.1 and Table 1. IV. Conclusion The results showed that after magnetic stimulation, insomnia subjects’ connections and characteristic parameters of brain functional network were improved. It suggested that magnetic stimulation of acupoints might be an effective treatment for insomnia.

**References:**

[1] Penghui S, Hua L, Siran L, Sleep Med., vol.56, pp.157-163 (2019) [2] Yunna K, Chorom B, Sunhye C, Int J Psychophysiol.,vol.124, pp.26-32 (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a6ef9c96488e69c84623f29c30b4f744.jpg)
Fig.1 Node degree of brain information maps of EEG beta band under different sates

![](https://assets.underline.io/uploads/markdown_image/1/image/f13a86538b506441b80d8949c8f8b44c.jpg)
Table 1 Characteristic parameters of brain functional networks of EEG beta band under different states

Magnetic Stimulation Reduced Electroencephalography Beta Band Excitability in Insomniacs

The first-order reversal curve (FORC) method is one of the most appealing magnetic characterization methods that has been used over decades for detailed analyses of nanoscaled magnetic systems. These detailed analyses are accompanied by numerous magnetic features in FORC heat-maps that usually cannot be fully explained1. In this direction, several theoretical studies have been done to provide a coherent framework to demonstrate those features1,2. However, due to computational limitations, the theoretical models consider mostly perfect arrangements of ideal nanomagnets, such as magnetic nanowires (MNWs), that do not match the real experiments3,4. Here, the FORC heat-maps of several MNW arrays were modeled and discussed. First, several MNW arrays with different arrangements (e.g. hexagonal, square, and random) are modeled and the effect of arrangement on the interaction fields (Hu) are explored. Next, the FORC heat-maps of these MNW arrays are calculated to visualize the importance of the arrangement on the magnetic characteristics of these systems. Then, geometrical imperfections (e.g. variations in shape and size) of MNWs were modeled and randomly assigned to investigate their effects on the FORC heat-maps. Figure 1 shows simulation results for different MNWs arrangements and geometrical imperfections that dictate the Hu among MNWs and the FORC heat-maps, respectively. It was found that the average interaction field (Huave) in hexagonal arrangements is higher than those in square arrangements regardless of the filling factor (defined as the area of MNWs to the total area of the simulation box). That is because the hexagonal arrangement has an odd number of MNWs in which the middle one is always in a frustrated state. The MNWs in random arrangements have significantly higher Hu variations (defined as the difference between the interaction field maximum (Humax) and minimum (Humin)) because the interwire distance in this arrangement is not constant, as opposed to hexagonal and square arrangements. Thus, the Hu is less uniform, and the mean-field effect theory ceased to be a reasonable approximation for simulating these randomly arranged MNWs. Moreover, modeling the geometrical imperfections as variations in coercivity (Hc) will result in presence of the well-known “Wish-bone” and “T-shape” features in the FORC heat-maps. Interestingly, we show that the presence of these features not only depends on the variation of the Hc but also depends on the relative strength of both Hc and Hu.

**References:**

1 C.-I. Dobrotă and A. Stancu, J. Appl. Phys., 2013, 113, 043928. 2 D. A. Gilbert, G. T. Zimanyi, R. K. Dumas, M. Winklhofer, A. Gomez, N. Eibagi, J. L. Vicent and K. Liu, Sci. Rep., 2014, 4, 1–5. 3 M. R. Zamani Kouhpanji, A. Ghoreyshi, P. B. Visscher and B. J. H. Stadler, Sci. Rep., 2020, 10, 15482. 4 M. R. Zamani Kouhpanji and B. J. H. Stadler, Nano Express, 2020, 1, 010017.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/18107c60ea58df5b3d2fffb6001ab6fa.jpg)
Figure 1: Subfigures (a) and (b) show the average interaction field (Hu) and the difference between the maximum and minimum Hu for various MNW arrangements. Subfigures (c) and (d) show the FORC heat-maps for the hexagonal arrangement, where the filling factor and average coercivity (Hc) are 5% and 100 Oe, respectively, for both cases. The Hc standard deviation is 10 Oe and 50 Oe for (c) and (d), respectively. As the ratio of Hu standard deviation (which is constant here because both cases have the same filling factor) to Hc standard deviation decreases from 10 to 2, the FORC heat-map of MNWs changes from “T-shape”, (c), to “Wish-bone”, (d), respectively.

Exploring effects of magnetic nanowire arrangements and imperfections on first-order reversal curve heat-maps

For more than 30 years, the properties of Co/Pt multilayers have intrigued magneticians for their high coercivity and well-developed perpendicular magnetic anisotropy (PMA) [1], with a view to applications in magneto-optical recording and perpendicular magnetic recording media and MRAM, to name but a few. The magnetic and structural properties of this system usually rely on the induction of &lt;111&gt; crystallographic texture and do vary substantially with thickness of the component layers within and number of repetitions of the Co/Pt bi-layer building block [2]. The quality of PMA typically deteriorates beyond 10-20 repetitions and mechanisms of magnetisation reversal complicate as a result. Here we describe the measurement of dynamic hysteresis loops from essentially DC to about 1 kHz in this type of multilayer using an in-house developed MOKE dynamic hysteresiograph and their interpretation within a model of viscous dynamics, accounting for both hysteretic switching and magnetisation rotation. The samples are prepared by DC magnetron sputtering in a Shamrock multi-chamber deposition tool and have the following structure SiO2/Ta(5)/Pt(2)/[Co(0.4)/Pt(0.8)]10/Co(0.4)/Pt(2), where the dimensions in brackets are given in nanometre. The measurement setup, shown on fig. 2, consists of an optical system utilizing a crossed polarizer/analyser close-to-polar geometry, with a CW He-Ne laser light source (focused down to about 30 μm) and high-speed biased (&gt; 1 MHz bandwidth) photo-diode detector. The magnetic field excitation is provided by an air-gap resonant coil, connected in series to a variable capacitance bank and a high-bandwidth linear operational power amplifier. The system is able to swing over 60 V and at least 5 A of current and deliver fields &gt; 250 mT at low frequency and 150 mT at 1 kHz. Fundamental frequency current resonance is maintained by adjusting the magnitude of the series capacitance, for each magnetic field ramp rate of interest. Two typical hysteresis loops are shown on fig. 1, demonstrating two distinct regions – one of relatively abrupt magnetisation reversal, followed by another, consisting primarily of magnetisation rotation. The behaviour of the normalized differential susceptibility is illustrated on fig. 2, for a number of frequencies in the region 5 – 810 Hz (linear ramp rates of up to 100 T/s). The same is computed using a statistical β-spline (of 3rd order) interpolation of the normalized hysteresis curves, in time domain. The peak in the differential susceptibility is becoming lower and broader as the frequency increases, while the plateau (associated with rotation) remains constant in amplitude, while shifting towards larger fields. To account for this behaviour a model is constructed, comprised of a statistical ‘normal’ hysteronic distribution and constant ‘rectangular profile’ plateau, time-convolved with a function (convolution kernel), which is a decaying exponent for positive times and zero for negative times, the time-constant of which is related to both the magnetic viscosity of the system and the magnetic field ramp rate. The resulting integral model yields excellent non-linear least squares fits to the experimental data, as demonstrated of fig. 2, and allows for the quantification of the quasi-static coercive field Hc, its distribution ΔHc, the rotational torque amplitude ΔHr and the magnetic viscosity parameter η, at different magnetic field ramp rates. The scaling of the amplitude of the model is essentially constant in the frequency range DC-1 kHz, while the extracted velues of the average coercive field, the width of the hysteronic distribution and effective viscosity parameter, all scale linearly, with factors of 25 %, 50 % and 500 %, respectively. As the rotational plateau of the normalized differential susceptibility is essentially unchanged up to field ramp rate of ~100 T/s, it can be infurred that both the magnetisation reversal and rotational dynamics (ferromagnetic resonance frequencies are expected to range betweeen 0.5 and 3 GHz, in the field region of interest) is governed by timescales much shorter than 1 ms, and the probing spot size chosen, is sufficiently small, to not impact significantly the results even for domain wall viscous propagation velocities as low as 0.1 m/s. It is hoped that the here-illustrated simple dynamic model of the dynamic hysteresis in perpendicular anisotropy thin films will also help with the parametrisation, interpretation and analysis (in broad regions of field, frequency and temperature) of experimental data on systems of great practical interest, other than Co/Pt, such as L10 FePt, FeNi and CoPd, to name just a few.

**References:**

[1] C.-J.Lin, G.L.Gorman, C.H.Lee, R.F.C.Farrow, E.E.Marinero, H.V.Do, H.Notarys and C.J.Chien “Magnetic and structural properties of Co/Pt multilayers”, Journal of Magnetism and Magnetic Materials 93 194-206(1991). [2] D. M. Donnet, V. G. Lewis, J. N. Chapman, K. O'Grady and H. W. van Kesteren “Microstructure and hysteresis in Co/Pt multilayers”, Journal of Applied Physics 26 (1993).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ed09a37c8867302bb2be444eb0563174.jpg)
(left) Illustrative dynamic hysteresis loops of a Co/Pt multilayer, measured via MOKE, at two different frequencies of the triangular profile drive field (5 Hz and 810 Hz). (right) Views of the various components of the experimental setup and, including the optical and magnetic field sub-systems and the sample mounting. Also shown is a real-time oscillogram of the field and MOKE sample signal, for sinusoidal excitation.

![](https://assets.underline.io/uploads/markdown_image/1/image/059da582912ae82a6d2610b805cecf02.jpg)
(left) Normallized differential susceptibility of the Co/Pt multilayer described in the text, at several different frequencies in the range (5 - 810 Hz). (right) A non-linear least squares fit to the data obtained at 440 Hz, using the integral model described in the text and the corresponding fittting parameters, as an inset.

A dynamic model for the hysteresis of CoPt multilayers and its use for the interpretation of MOKE hysteresis loops aquired at different field ramp rates

We use micromagnetic simulations based on the stochastic Landau-Lifshitz-Gilbert (sLLG) equation to calculate dynamic magnetic hysteresis loops relevant to magnetic nanoparticle hyperthermia. Our simulation model is inspired by a preclinical study reported by Dennis et al. [1] using iron oxide nanoparticles for cancer treatment in mice. The nanoparticles consist of nanorods nominally 6.7 nm x 20 nm x 47 nm in size. Instead of simulating atomic spins, micromagnetics suggests assuming uniformly magnetized cells with dimensions of the cubic unit cell (a0=0.839 nm), resulting in a rod composed of 10752 cells. The approximation that all spins within a cell are uniformly aligned via a relatively strong exchange interaction is valid at zero temperature for cell sizes smaller than the exchange length. At finite temperature, spin decorrelation must be taken into account, and a coarse-graining procedure is required to determine how micromagnetic parameters vary with cell size and temperature. To avoid unwanted heating of surrounding healthy tissues due to Eddy currents, restrictions on the applied field frequency and amplitude limit the field sweep rate to SR=4Hmax f &lt; 0.25 Oe/ns [2, 3]. Simulating a rod made of cells with size a0 takes months, so using fewer but larger cells is computationally attractive. Our study models the rods using larger cells (of length ab = b a0 for b &gt; 1) to be able to simulate complex nanoparticles in a reasonable time (see Fig. 1). As the cell size affects the system’s simulated properties, magnetic parameters need to be scaled to keep the model’s magnetic response, i.e., m-H hysteresis loops, invariant with the simulation cell size. Scaling also appears within the realm of phase transitions and critical phenomena, driven by thermodynamic considerations, resulting in the powerful renormalization group (RG) approach based on the correlation length. In the present study, we use the RG approach of Grinstein and Koch [4] to determine how the field H(b), anisotropy constant K(b), and exchange constant A(b) should scale with b (see Fig. 1) at finite temperature. We modify their approach to describe the scaling of the magnetization m(b) by introducing a phenomenological parameter δ that accounts for the nonlinear magnetization temperature dependence [5]. We also extend their approach to include magnetostatic interactions that were not considered in Grinstein and Koch’s work. The scaling algorithm produces nearly identical loops over several decades in the model grain volume, as shown in the top four curves of Fig. 2; for details, see Ref. [6]. Larger cell sizes also allow using a larger Δt when solving the equations of motion. Using the scaling approach decreases the rod’s simulation time from months to days. At the heart of the LLG equation is α, responsible for describing energy loss from the macroscopic motions of the local magnetization to microscopic thermal motion [7]. A larger α signifies a faster loss of energy and a shorter relaxation time required to align the magnetic moments to the field and results in a smaller hysteresis loop. Likewise, a slower SR is equivalent to a longer measurement time and consequently yields a smaller loop. Considering Sharrock's work [8], which relates the coercive field to the relaxation time and attempt frequency, we argue that, for the small α, so long as SR/α = constant, the coercive field should remain the same. In Fig. 2, we plot the m-H hysteresis loops for two sets of values for SR and α, such that SR/α =2.5, and interestingly even for large values of α, the loops overlap. Using this approach reduces the simulation time from days to hours, for details see Ref. [5]. To summarize, we show here that the RG approach of Grinstein and Koch yields a scaling of magnetostatic, exchange and anisotropy parameters that, with some modification for the rescaling of the resulting magnetization, yields finite temperature nanorod hysteresis loops that are, to approximately 10%, invariant with cell size. Scaling results hold even to the point where the nanorod is represented by a single magnetization vector that experiences only anisotropy and self-demagnetization. In addition to the computational speedup resulting from the use of fewer micromagnetic cells, the invariance of loops when SR/α is fixed provides another avenue for computational efficiency by allowing one to use a larger SR than the target value. This paves the way for efficient simulations of nanoparticles (composed of nanorods) and their response to AC fields [6], in support of experimental studies of hyperthermia.

**References:**

[1] C. L. Dennis, et al. Nanotechnology, 20(39), 395103 (2009). [2] S. Dutz, and R. Hergt, Int. J. Hyperth., 29(8), 790-800 (2013). [3] R. Hergt, and S. Dutz, J. Magn. Magn. Mater., 311(1), 187-192 (2007). [4] G. Grinstein, and R. H. Koch, Phys. Rev. Lett., 90(20), 207201 (2003). [5] R. Behbahani, M. L. Plumer and I. Saika-Voivod, J. Phys.: Condens. Matter, 32, 35LT01 6pp (2020). [6] R. Behbahani, M. L. Plumer and I. Saika-Voivod, arXiv:2010.08848 (2020). [7] T. L. Gilbert, IEEE Trans. Magn., 40(6), 3443-3449 (2004). [8] M. Sharrock, and J. McKinney, IEEE Trans. Magn., 17(6), 3020-3022 (1981).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/53d389db281adb19e35bb9b508f104a5.jpg)
Figure 1. Coarse-grained modelling of a magnetite nanorod. The smallest micromagnetic cell models eight atomic spins within a cubic unit cell of length a0 = 0.839 nm with a single magnetic moment. Ultimately, the nanorod is modelled as a single block, corresponding to b = 22. Spherical magnetic nanoparticles studied in Ref. 2 are made up of magnetite nanorods.

![](https://assets.underline.io/uploads/markdown_image/1/image/6ef098f20be94e527108ac4581410602.jpg)
Figure 2: Hysteresis loop invariance. The top four loops show invariance with cell size when magnetic parameters are scaled according to a modified RG prescription in addition to the magnetostatic interactions scaling. The last two loops show the invariance of MH loops when the ratio SR/α is constant.

Coarse-graining in micromagnetic simulations of dynamic hysteresis loops

First Order Reversal Curve (FORC) diagrams have over 30 years of history as a tool to characterise ensembles of magnetic particles reaching from applications in geophysics to detect the prevalence of carriers of paleomagnetic signals [1] to the measurement of magnetic recording media properties [2]. Using FORC diagrams as a purely quantitative tool remains an open topic of active research (see e.g. [3]), but they have proven to be useful as semi-quantitative instrument to determine traces of magnetic phases in a material mixture [4]. Based on this present understanding we introduce the idea of using FORC diagrams to detect the crystalline structure of the cobalt binder phase in tungsten carbide (WC-Co), a hard metal used in a wide range of high-tech applications (e.g. drills or cutting inserts) [5]. The goal is to interpret FORC diagrams from experimental measurements by micromagnetic simulations as a source of “big data” for a fast characterisation of tungsten carbide binders. Our proposed method goes significantly beyond state-of-the art techniques used for characterisation of WC-Co in particular, but also makes a contribution to the interpretability of FORC diagrams in general by applying machine learning to the large amount of data points available through FORCs. Here, we present our first results towards such an automated interpretation of FORC diagrams, which requires a substantial amount of training data from micromagnetic simulations. Such simulations provide the opportunity to assemble mixtures of particles with a known distribution of various properties, which should serve as labelled training data. We use energy minimization [6] to compute the magnetization curves of Co particles with sizes of 100 nm and uniaxial anisotropy. A reversal curve is computed starting at each point Hr on the major loop where the magnetization M has changed by a defined amount. Depending on the alignment of the anisotropy axes with respect to the field H different numbers and configurations of domain wall nucleation-propagation-annihilation sequences occur during a full reversal loop, leading to a large variation in the number of generated FORCs. To mitigate the necessary computational efforts to acquire the training data of large ensembles of more than 100 particles we developed a Python-based FORC analysis framework and used scikit-learn’s Random Forest Regressor [7] to predict the surface M(H,Hr) from a subset of simulations. Our first results for an ensemble of non-interacting cubes with varying anisotropy axes show that the FORC diagrams calculated from the predicted M(H,Hr) surface give a valid representation of the FORC diagram calculated directly from the fully simulated ensemble (Fig.1). This approach will enable us to speed up the generation of training data considerably, which provides the basis to understand the microstructure of WC-Co samples containing Co particles with dimensions beyond the 100 nm scale by using FORCs as a fast and non-destructive characterisation tool. Fig. 2 shows two examples of simulated FORCs and their respective FORC diagrams. In Fig. 2a the external field is aligned more parallel to the particle’s easy axis than in Fig. 2b. As FORC diagrams only show pronounced peaks for irreversible processes, they can be directly associated with the jumps present in the FORCs shown in the upper part. Analytical models to explain the peaks’ positions and heights can only be found for simplified models. Following [8], the two peaks in Fig. 2a can be modelled analytically using a curvilinear hysteresis loop, but this model does not reveal the exact reversal process in the material. Combining micromagnetic simulations with machine learning will enable us to train models using realistic assumptions about the material of interest. As a first step the presented approach will make it possible to get data for the interpretation of FORC diagrams within minutes instead of weeks by replacing time-consuming micromagnetic simulations with sufficiently trained machine learning models. The financial support by the Austrian Federal Ministry of Climate Action, Environment, Energy, Mobility, Innovation and Technology (BMK) in the KI-Carbide project (#877141) is gratefully acknowledged.

**References:**

[1] A. R. Muxworthy and D. J. Dunlop: Earth and Planet. Sci. Lett. 203, pp 369-382 (2002) [2] M. Winklhofer and G. T. Zimanyi, J. Appl. Phys. 99, 08E710 (2006) [3] D. A. Gilbert, G.T. Zimanyi, R.K. Dumas et al., Sci. Rep. 4, 4204 (2014) [4] C. Carvallo et al.: Phys. Earth Planet. Inter. 154, pp 308–322 (2006) [5] J. García et al., Int. J. Refr. Met. 80, pp 40-68 (2019) [6] L. Exl, J. Fischbacher, A. Kovacs et al., Comp. Phys. Comm. 235, pp 179-186 (2019) [7] F. Pedregosa et al, JMLR 12, pp 2825-2830 (2011) [8] C. Pike and A. Fernandez, J. Appl. Phys. 85, pp 6668-6676 (1999)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9de8549ddbc61514a201eecc9b7f0b4d.jpg)
Fig. 1 FORC diagrams computed (a) directly from simulating a full ensemble of 125 non-interacting hcp-Co particles with varying alignment of the anisotropy axis with respect to the external field and (b) by using only a subset of 20 simulations as input to a Random Forest Regressor to predict the M(H, Hr) surface

![](https://assets.underline.io/uploads/markdown_image/1/image/2c361c16a150eeb7f7d6915c3daf9b09.png)
Fig. 2 Examples of simulated FORCs and calculated FORC diagrams for a 100 nm cube of hcp Co (uniaxial anisotropy). Colorbars indicate the peak height of the FORC distribution (see also Figure 1). The external field in (a) is aligned more parallel to the easy axis than in (b). The visible peaks in the FORC diagrams (bottom) correspond to irreversible jumps of the magnetization (top).

Using a Random Forest Regressor to Predict First-Order Reversal Curves of hcp-Co Particle Ensembles

The Stoner-Wohlfarth model [1], first published in 1948, describes the magnetic behaviour of anisotropic, ferromagnetic particles, which magnetize via coherent rotation. This widely used model predicts the crossing of the ascending and descending branches of the magnetization curve when the applied field is close to the magnetic hard-axis. However, experimental observation of this hysteresis branch crossing has not been acknowledged in the literature. In fact, the crossing of hysteresis branches has been labelled as being nonphysical [2] and a problem [3] with the Stoner-Wohlfarth model. Here we show that the crossing of the ascending and descending branches of the magnetization curve does occur and is consistent with the Stoner-Wohlfarth model. Contrary to previous reports, we show that the crossing of hysteresis branches is physically realistic [4] and violates no fundamental law.

**References:**

[1] Stoner, E.C. and Wohlfarth, E.P., Phil. Trans. Roy. Soc. of London Series A, 240, no. 826, pp.599-624 (1948). [2] Vajda, F. and Della Torre, E., IEEE Trans. Mag., 28, no. 5, pp.2611-2613 (1992). [3] Tannous, C. and Gieraltowski, J., Physica B, 403, pp.3563-3570 (2008). [4] Heslop, D. and Roberts, A., Jour. Goephys. Res., 117, B03103 (2012).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/960e20c09fe918c9d9087883ca10b05a.jpg)
Figure 1: The reduced x- and y-components of the magnetization as a function of applied field for an angle of α=2o. Circles and triangles represent the x- and y-components of the measured magnetization, respectively. Solid and dashed lines represent numeric solutions to a temperature-dependant Stoner-Wohlfarth model. Ascending and descending data are shown in red and blue, respectivey. Black circles highlight the hysteresis branch crossings.

Hysteresis branch crossing and the Stoner-Wohlfarth model

I. Introduction The iron losses, permeability and magnetostriction of electrical steel sheet are sensitive to mechanical stress, and the presence of mechanical stress may bring out the deformation of the magnetostrictive hysteresis loops, the increase of core loss, and the aggravation of vibration and noise in electrical machines or transformers. The accurate prediction of magnetic characteristics under stress is significant for optimum design of high power density motor. The multi-scale domain energy models [1-2], such as the assembled domain structure model (ADSM) have been conducted to represent the magnetic hysteretic behavior in an electrical steel sheet based on the minimization principle of total magnetic energy. In comparison with some phenomenon models such as Preisach, the domain energy model interprets the magnetic domain wall movements and rotation with the magnetic field applied and builds a relevance between the microstructure and magnetic hysteresis properties. However, due to the complexity of the parameter identification, the establishment of domain model is at the cost of large computation time and its modeling accuracy to the magnetostrictive loops is also to be improved. In this paper, a multi-scale domain energy model is improved to estimate the magnetostrictive characteristics under different external stresses, in which the hysteresis energy function is introduced instead of pinning energy in order to reduce the computation time. To identify the parameters in the model, the magnetic domain wall movements in an electrical steel sheet during the magnetization process are observed by a magnetic domain observation microscope, and the magnetostrictive hysteresis loops under several external stresses are measured. The validity of proposed model is verified by comparing the measured magnetostrictive loops with computed ones. II. Experiments and Approaches A. Experiments With the presence of external magnetic field intensity H, the balance state with the total inner energy in the ferromagnetic materials keeping the minimum is disturbed, and then the wall movements and magnetic moment rotation in the magnetic domains is occurring. This magnetization process in an electrical steel sheet is observed with a polished sample of with the size of 8mm width and 8mm length, as shown in Fig.1(a), by means of a magnetic domain observation microscope, BH-782PI-SHG, Japan. Fig.1(b) represents a domain figure under a 0.5Oe field in a non-oriented silicon steel, in which the difference between grey levels indicates different magnetization direction domains. Further, the magnetic hysteresis and magnetostrictive loops with different mechanical stresses applied are measured, as shown in Fig.1(b), by a laser magnetostrictive tester from Brockhaus, Germany, in which the shape and magnitude and area of magnetostrictive loops occur to increase after applying a 4Mpa compressive stress. B. Improved Multi-Scale Domain Energy Model The magnetic domain moment is alignment with the easy directions of the crystal in a grain when the material is in the demagnetized state. According to the ADSM model each grain is assumed to be composed of six magnetic domains along the easy directions, and the unit magnetization vector m of each domain is denoted by the variables θ and Φ as a vector [sinθsinΦ, sinθcosΦ, cosθ]. The θi ,Φi and the existence probability ri ( i=1,2,…,6) of magnetization vector for each domain can be determined according to the principle of local energy minimum. The total magnetic domain energy is expressed as the summation of Zeeman energy, the crystalline anisotropic energy, the magnetostatic energy and magnetoelastic energy in the ADSM. Further in order to model the magnetic hysteresis effect, the pinning energy have been added into the total energy. In this paper, in order to estimate the behavior of magnetostrictive hysteresis behavior, a hysteresis energy density function in (1) instead of the pinning energy is employed into the total domain energy in terms of simplifying the parameter identification process. Eh=-μ0Ms/χ0(α1,imx-p+α2,i+my-p+α3,imz-p) (1) where [α1,i, α2,i, α3,i] is direction cosine of the unit magnetization vector of the i-th domain with respect to the three easy axes; (mx-p, my-p, mz-p) is three components of magnetization vector at previous moment, MS is the saturation magnetization. The hysteresis energy density function can take the effect of the magnetization history, and explain the hysteresis phenomenon as an inertia. Further, the effect of external stress on the magnetostriction in the magnetoelastic energy is described as: Eσ=-3/2λ100σ(α21,iγ21+α22,iγ23+α23,iγ23)-3λ111σ(α1,iα2,iγ1γ2+α2,iα3,iγ2γ3+α3,iα1,iγ3γ1) (2) where (γ1, γ2, γ3) is the cosine of the angle between the stress and the three easy axes; λ100 and λ111 are saturation magnetostrictive coefficients along the [100] direction and the [111] direction; σ is the stress applied on the sample; Fig. 2 shows the measured and modeled results of magnetostriction loops, and more discussion will be presented in detail in the extended paper.

**References:**

 [1] S. Ito, T. Mifune, T. Matsuo, et al. Energy-Based Magnetization and Magnetostriction Modeling of Grain-Oriented Silicon Steel under Vectorial Excitations . IEEE Trans. Magn. vol 52, no. 5, pp. 1-4, 2016. [2] D. Vanoost，S. Steentjes, J. Peuteman, et al. Magnetic hysteresis at the domain scale of a multi-scale material model for magneto-elastic behaviour. J. Magn. Magn. Mater. vol 414, no. sep., pp. 168-179, 2016. [3] T. Moses. Relevance of microstructure and texture to the accuracy and interpretation of 1 and 2 directional characterisation and testing of grain-oriented electrical steels. International Journal of Applied Electromagnetics & Mechanics, vol. 55, pp. 1-11, 2017.

**Images:**
![](https://assets.underline.io/uploads/markdown_image/1/image/fd54ab2a2bbd0678fa4e3b7e947e9359.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/7ac2dc401cdea94bc7da44b75d6694f5.jpg)

Estimation of Magnetostrictive Hysteresis Properties of Electrical Steel Sheet Under External Stress Using Multi-Scale Domain Energy Model

Magnetostriction is one of main factors causing the core vibration and noise of electromagnetic equipment, and the magnetostriction at the stator core and T-joint in transformers has the vector characteristics due to the two-dimensional (2D) rotational magnetization. Recently, the 2D vector magnetostriction models[1,2] have been researched based on the Fourier analysis of measured 2D magnetostrictive waveforms and magnetic flux density ones. But, owing to the complexity of rotational magnetization, it is more significant to describe the magnetostrictive characteristics in terms of magnetization mechanism. In this paper, with the reference to the hysteresis behavior and irreversibility in the Jiles-Atherton model, the hysteresis characteristics between the magnetostiction λ and the magnetization M are attempted to be explained by the pinning effect. After that, a 2D vector magnetostriction model is proposed and its validity is verified with experimental data from a 2D rotational magnetization tester. I. The Proposed 2D Vector Magnetostriction Model The magnetostriction λ and magnetization M are usually expressed as a square function relationship when they are static and do not consider saturation， λ=pM2 (1) A large number of experimental observations show that the change of magnetostriction λ lags behind the variation of magnetization M within one magnetization time period. For the ferromagnetic material, with the magnetic field applied, the domain wall displacement may be hindered by the pinning effect, which results from the impurity and cavity in the material, so the quantitative hysteresis behavior and irreversibility are mainly caused by overcoming the pinning. As a result, the hysteresis behavior of magnetostriction can be also explained by the pinning effect, in which the magnetostriction is mainly due to the movement of domain walls whose spontaneous magnetization is not alignment with the applied magnetic field. By adding the irreversibility of magnetostriction into (1) and taking the vector properties of 2D magnetization into account, a 2D vector magnetostriction model is proposed as λxx=pλxx[cλxxManx+(1-cλxx)Mirrλx]2/Msx2+pλyy[cλyyMany+(1-cλyy)Mirrλy]2/Msy2 λxx=qλxx[cλxxManx+(1-cλxx)Mirrλx]2/Msx2+qλyy[cλyyMany+(1-cλyy)Mirrλy]2/Msy2 (2) γxy=rγxy[cλxxManx+(1-cλxx)Mirrλx][cλyyMany+(1-cλyy)Mirrλy]/(MsxMsy) where λxx is the normal strain in x direction, λyy is the normal strain in y direction, and γxy is the shear strain; Msx and Msy are two components of saturation magnetization; Manx and Many are two components of ideal anhysteretic magnetization in the x,y direction; the parameters of cλxx, cλyy, pλxx, pλyy, qλxx, qλyy, and rrxy are to be determined from the experimental data of 2D magnetic properties; Mirrλx and Mirrλy denote the irreversible component of the magnetization and can be described as Mirrλ=Man-kλδ(dMirrλ/dHe) (3) where kλ is the pinning constant and obtained through experimental data; δ=(dHe/dt)/|dHe/dt|; He is the effective field . II. Experiments and Model Verification A 2D magnetic properties tester with a rosette strain gauge detecting the magnetostrictive strain signals in [3] is used to provide the data with identifying the parameters in (2) and (3). Fig. 1 shows the measured magnetic flux density B loci and magnetic field intensity H trajectories. Fig. 2 shows the comparison of the calculated and measured values of magnetostriction at three time moments with respect to a B trajectory with the axis ratio α=0.2, Bmax=0.9T, and the inclination angle θB=45°.

**References:**

[1] Yamagashira M，Wakabayashi D，Enokizono M．”Vector magnetic properties and 2-D magnetostriction of various electrical steel sheets under rotating flux condition,” IEEE Trans. Magn., vol. 50, no. 4, pp. 1–4, Apr. 2014, Art. no. 6100404. [2] L. Zhu, H.-S. Yoon, H.-J. Cho, D.-J. Um, and C.-S. Koh, “Finite-element analysis of magnetostriction force in power transformer based on the measurement of anisotropic magnetostriction of highly grain-oriented electrical steel sheet,” IEEE Trans. Magn., vol. 52, no. 3, pp. 1–4, Mar. 2016, Art. no. 6100304. [3] Z. Wang, Y. Zhang, D. Zhang, D. Xie, C.-S. Koh, and O. A. “Modelling of magnetostrictive property of electrical steel sheet under vectorial excitation,” IEEE Trans. Magn., vol. 55, no. 6, pp. 1–4,Jun. 2019, Art. no. 7300304.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3d6624d4a6d69ca413eb6cf7bfe3cab2.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/5fc9e227ff396f198c5a57e67c2551f3.jpg)

A 2D Vector Magnetostriction Model in an Electrical Steel Sheet Taking the Pining Hysteresis Effect into account

The treatment of hysteresis models in the numerical solution of Maxwell equations represents major issues as large computational times and significant memory space allocations are required [1][2]. Stability issues of the iterative solution are also a non-negligible source of concern. The Jiles-Atherton (J-A) [3] and the Preisach models [4] are the most used and commented hysteresis models. On the one hand, memory management in the Jiles-Atherton model is simple but convergence often goes through large temporal discretization. On the other hand, Preisach model can work with a limited time discretization but it requires huge memory space and complex memory management. In [5][6], authors proposed an interesting way to take into account hysteresis solving both the J-A and the Preisach models issues. Hysteresis is taken into account through the following hypothesis: the slope dB/dH (reciprocally dH/dB) inside and over the major hysteresis loop envelope is supposed to be dependent on only three parameters: the magnetic excitation field H, the magnetic induction field B, the sign of the time derivation of the model input (B or H). Based on this simple assumption, a new way to take into account the quasi-static scalar hysteresis for ferromagnetic materials was described. This new method also called DSHM for “Derivative Static Hysteresis Model” is an elegant alternative technique, it can be implemented with a simple memory management and reduced memory allocation demands in respect to more classical approaches. Convergence and precise simulation results can be reached with a relatively coarse temporal discretization. This hysteresis taken into account is also easily reversible (from B(H) to H(B)) and easily exportable. Fig. 1 – Illustration of the DSHM matrix. Up to now, the filling step of the DSHM matrix (Fig. 1) has always been through a set of experimental first-order reversal curves (as illustrated in [5]). It is evident that getting such experimental results can be complicated in particular situations, including electromechanical converters design or electromagnetic nondestructive testing but in this study we propose to fill the DSHM model’s matrix with a reduced number of experimental data thanks to assumptions also used to parameter the Preisach and the Jiles-Atherton models. The principles of the new method can be detailed as follows: _ The Preisach or the J-A models are implemented first. It implies to determine an adequate combination of parameters for the J-A model or an adequate triangle density for the Preisach model. Such implementation can be done with a limited number of experimental results (a single major hysteresis cycle in the extreme case). _ Reversal curves are plotted numerically using the classic hysteresis models. _ The numerical reversal curves are used to fill the DSHM matrix. The DSHM model provides correct simulation results even under unsymmetrical waveform situations. Implemented in a nonlinear dynamic finite differences numerical scheme for the resolution of the magnetic field diffusion through the cross section of a magnetic lamination [7][8], it allows considerable speed-up of the simulation process while conserving very reasonable level of accuracy. To summarize: the hysteresis law is considered in a generalized input vector space, in which an interpolation matrix is constructed with the columns and rows denoting respectively the discrete values of H and B and whose values stand for the dB/dH (reciprocally dH/dB) slope at the corresponding points. In the end of this study, the resolution of the non-linear magnetic field diffusion through a ferromagnetic lamination is chosen as a benchmark simulation. It allows to compare the performances of space discretized based numerical solvers and to confirm the improved efficiency of the proposed method versus the J-A and the Preisach ones.

**References:**

[1] P. Zhou & al., IEEE Trans. On Mag. Dev., vol. 40, Iss. 4, pp. 1803 – 1809, (2004) [2] B. Ducharne & al., J. Phys. D: Appl. Phys., Vol. 40, pp. 551-555, (2007) [3] D.C. Jiles & al., J. App. Phys. 55, pp. 2115, (1984) [4] Y. Bernard & al., vol. 25, Iss. 1-4, pp. 729-733, (2007) [5] R. Scorretti & al., proceedings of compumag 2011, 8. Mat. modelling, 11. Num. techniques, (2011) [6] T-P Do & al. & al., proceedings of compumag 2009, (2009) [7] B. Gupta & al., IEEE Trans. on. Mag, Vol. 54, Iss. 3, (2018) [8] M. A. Raulet & al., IEEE Trans. on Mag., vol. 40, n° 2, pp. 872 – 875, (2004)

**Images:**
![](https://assets.underline.io/uploads/markdown_image/1/image/8cdc4ee4e2870b0f565ca7b2609bb94f.jpg)

Optimized magnetic hysteresis management for electromagnetic space discretization simulation tool.

Ferromagnetic materials are used in a wide range of electromagnetic applications (energy converters, sensors, inductances …). Ferromagnetic materials are characterized by a strong answer under the influence of a magnetic field [1]. These properties are of huge advantages in many applications but they are also strongly non-linear and frequency dependent. In this domain, we observe a growing interest in the development of simulation tools reducing the experimental campaigns and improving the knowledge and the performances. Accurate simulation results can only be obtained by coupling precise electromagnetic equations to the exact material laws (hysteresis, saturation, frequency dependence). Under the influence of an external magnetic excitation, the local magnetic state through the ferromagnetic specimen is ruled by the combination of both the magnetic domain kinetics and the external magnetic field diffusion. Fig. 1, Illustration of the ferromagnetic hysteresis frequency dependence. The usual methods for the simulation of the magnetic behavior are all based on the separation of the magnetic contributions, where the microscopic Eddy currents due to the domain wall motions and the macroscopic ones due to the extern magnetic field variations are considered separately [2]. This separation remains artificial, since practically both losses mechanisms occurs simultaneously and interact on each other. Alternative solutions for the simulation of these phenomena have already been proposed through the resolution of an anomalous fractional magnetic field diffusion (1, 2 or 3D depending on the experimental situation) [3][4]. The fractional order constitutes an additional degree of freedom in the simulation scheme. It is identified through comparisons to experimental results. By adjusting precisely this order, very accurate local and global simulation results can be obtained on a very broad frequency bandwidth [5]. It allows to predict precisely the dynamic magnetic behavior of classic ferromagnetic components. Fractional diffusion equation is an interesting method but this is not the only way to take into account the hysteresis frequency dependence using fractional derivative operators. Lump models based on time fractional differential equations is another option [6][7]. In both methods (the fractional diffusion equation and the time fractional differential equation), a numerical resolution of the Riemann-Liouville definition is required. Recent works have demonstrated the effectiveness of using quadrature techniques to approximate the Riemann–Liouville definition for fractional derivatives in the context of nonlinear viscoelastic model [8]. In this manuscript, the quadrature methods are implemented and tested in the case of the ferromagnetic hysteresis. Theoretical results are compared to experimental ones, and conclusions and perspectives are drawn such as possible improvements.

**References:**

[1] R. M. Bozorth, Wiley-IEEE Press, Michigan University, (1993) [2] G. Bertotti, IEEE Trans. Magn., vol. 24, iss. 1, pp. 621-630, (1988) [3] B. Ducharne & al., Com. in Nonlin. Sci. and Num. Sim., vol. 92, 105450, (2020) [4] B. Ducharne & al., The Eur. Phys. J. Plus, 135:325, (2020) [5] B. Ducharne & al., under revision, AIP advances, (2021) [6] B. Zhang & al., IEEE Trans. Magn., Iss. 99, pp. 1-5, (2018) [7] B. Zhang & al., IEEE Trans. Magn., Vol. 54 iss. 3, (2017) [8] P.R. Miles & al., J. Int. Mat. Syst and Struct., (2020)

**Images:**
![](https://assets.underline.io/uploads/markdown_image/1/image/819e21a27f3363aeffe79437929b7dbb.jpg)

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