United States

Electrically driven magnetization switch utilizing voltage-controlled magnetic anisotropy (VCMA) have attracted much attention in the new generation memory, especially
for spin-orbit torque (SOT)-based magnetic random access memory (MRAM) [1,2]. For the traditional SOT switching, an additional magnetic field is needed to break the inversion
symmetry, which makes the circuit difficult to implement [3]. Combining the SOT with the VCMA effect can realize the certain switching in SOT-MRAM with lower critical switching
current [4]. With the development of the SOT-MRAM, device model is urgently needed for the device characterization. However, the published models are facing limitations as the size shrinks down
to nanometer. Also, the effect of the elevated temperature caused by repeated operation on the device is non-negligible [5]. Therefore, an effective model is needed to represent the
switching dynamic of the device concerning the influence of the nanoscale and the thermal effect.
In the paper, a compact model of three-terminal VCMA assisted SOT switching is established. The influence of the VCMA effect on the field-free SOT switching is considered by
numerically solving the LLG equations. Furthermore, a 3D model of the VCMA+SOT device is established by finite element method to demonstrate the thermoelectric behavior.
Considering the influence of the thermal effect generated in the working state on the switching characteristics, the electrothermal behavior is integrated into the compact model to show the
influence of the temperature on the switching behavior, highlighting the importance of the thermal effect for the realistic modelling of SOT switching. Additionally, considering the
influence of thermal effect, the switching error rate of the device under different working voltage conditions is evaluated. Finally, a novel voltage pulse scheme is proposed, which can
effectively shorten the inversion time and improve the reliability of the device. The established model provides strategies and guidelines for next-generation memory design and
application.&lt;br&gt;
**References** &lt;br&gt;[1] W. Kang, Y. Ran and Y. G. Zhang et al., IEEE Trans. Nanotechnol., Vol. 16, p. 387 (2017).&lt;br&gt;
[2] S. Sharmin, A. Jaiswal, and K. Roy, IEEE Trans. Electron Devices, vol. 63, p. 3493 (2016).&lt;br&gt;
[3] A. Manchon, I. M. Miron and T. Jungwirth et al., Rev. Mod. Phys., vol. 91, p. 035004 (2019).&lt;br&gt;
[4] S. Shreya and B. K. Kaushik, IEEE Trans. Electron Devices, vol. 67, p. 90 (2019).&lt;br&gt;
[5] J. J. Kan, C. Park and C. Ching et al., IEEE Trans. Electron Devices, vol. 64, p. 3639 (2017).&lt;br&gt;

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/3ae5dbed55d72a570c2440ec178e1e30.png)&lt;br&gt;
Fig.1 Schematics of the three-terminal device allowing for field-free SOT switching with an illustration of the influences of VCMA and thermal effects.

MMM 2022

Field free switching model of Spin

Electrically driven magnetization switch utilizing voltage-controlled magnetic anisotropy (VCMA) have attracted much attention in the new generation memory, especially
for spin-orbit torque (SOT)-based magnetic random access memory (MRAM) [1,2]. For the traditional SOT switching, an additional magnetic field is needed to break the inversion
symmetry, which makes the circuit difficult to implement [3]. Combining the SOT with the VCMA effect can realize the certain switching in SOT-MRAM with lower critical switching
current [4]. With the development of the SOT-MRAM, device model is urgently needed for the device characterization. However, the published models are facing limitations as the size shrinks down
to nanometer. Also, the effect of the elevated temperature caused by repeated operation on the device is non-negligible [5]. Therefore, an effective model is needed to represent the
switching dynamic of the device concerning the influence of the nanoscale and the thermal effect.
In the paper, a compact model of three-terminal VCMA assisted SOT switching is established. The influence of the VCMA effect on the field-free SOT switching is considered by
numerically solving the LLG equations. Furthermore, a 3D model of the VCMA+SOT device is established by finite element method to demonstrate the thermoelectric behavior.
Considering the influence of the thermal effect generated in the working state on the switching characteristics, the electrothermal behavior is integrated into the compact model to show the
influence of the temperature on the switching behavior, highlighting the importance of the thermal effect for the realistic modelling of SOT switching. Additionally, considering the
influence of thermal effect, the switching error rate of the device under different working voltage conditions is evaluated. Finally, a novel voltage pulse scheme is proposed, which can
effectively shorten the inversion time and improve the reliability of the device. The established model provides strategies and guidelines for next-generation memory design and
application. 
**References** [1] W. Kang, Y. Ran and Y. G. Zhang et al., IEEE Trans. Nanotechnol., Vol. 16, p. 387 (2017). 
[2] S. Sharmin, A. Jaiswal, and K. Roy, IEEE Trans. Electron Devices, vol. 63, p. 3493 (2016). 
[3] A. Manchon, I. M. Miron and T. Jungwirth et al., Rev. Mod. Phys., vol. 91, p. 035004 (2019). 
[4] S. Shreya and B. K. Kaushik, IEEE Trans. Electron Devices, vol. 67, p. 90 (2019). 
[5] J. J. Kan, C. Park and C. Ching et al., IEEE Trans. Electron Devices, vol. 64, p. 3639 (2017). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/3ae5dbed55d72a570c2440ec178e1e30.png) 
Fig.1 Schematics of the three-terminal device allowing for field-free SOT switching with an illustration of the influences of VCMA and thermal effects.

poster

#### Welcome to the 67th Annual Conference on Magnetism and Magnetic Materials, MMM 2022!

The Conference was held from October 31 to November 4 and offers both in-person and on-demand content.

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Spin torque nano oscillator(STNO) and spin Hall nano oscillator(SHNO) are the promising devices to generate spin wave with thermal stability and low power
consumption.
[1,2] But the spin wave has in nature the low power density for device applications. Thus, various spin wave excitation methods have been studied, such as synchronization of
auto-oscillation via multiple SHNO array
[4-7] and utilizing high spin Hall angle in a heavy metal.
[8] Here, we designed a permalloy (Py) ferromagnetic layer in a rectangular bar type for
spin wave excitation by auto oscillation and investigated the effects of a transition metal (Co) alloying on the excitation efficiency. The magnetic fluctuation in bar type layers of Py1-xCox
(x = 0, 0.11, 0.21, 0.29, and 0.40), fabricated by Py-Co cosputtering, is estimated from micro-Brillouin light scattering (μ-BLS) spectroscopy data, as shown in Fig. 1. And the threshold
current for auto oscillation in the bar layers is estimated from the current dependence of integral intensity of the spectra. It is found that Py-Co cosputtering reduces the threshold current for
auto oscillation, e.g., 27.3% for Py1-xCox
(x = 0.21), compared with that of Py layer. It is conjectured that the cosputtering enhances the spin Hall transparency while it causes almost no
change in the effective Gilbert damping constant in the layer. Details will be discussed in the poster presentation and conference proceeding paper. 
**References** 
[1] J. Wunderlich, B. Kaestner, and T. Jungwirth, Phys Rev Lett 94, 047204 (2005). 
[2] J. Torrejon et al., Nature 547, 428 (2017). 
[3] A. Slavin and V. Tiberkevich, IEEE Transactions on Magnetics 45, 1875 (2009). 
[4] Y. Ou, D. C. Ralph, and R. A. Buhrman, Phys Rev Lett 120, 097203 (2018). 
[5] Y. Yin et al., Physical Review B 92 (2015). 
[6] M. A. W. Schoen, D. Thonig, and J. M. Shaw, Nature Physics 12, 839 (2016). 
[7] Y. Fu et al., IEEE Transactions on Magnetics 45, 4004 (2009). 
[8] H. Mazraati et al., Applied Physics Letters 109 (2016). 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/b078253a80d9187c1b90409a52a46000.png) 
Figure 1. Brillouin light scattering (BLS) intensity spectra of Py1-xCox
(x = 0.21) (Py = permalloy) nanowires with various applied DC currents of 3.5, 4.0, 4.5, 5.0 and 5.5 mA

Effects of Permalloy Cobalt Cosputtering in a Nanowire Structure on Magnetic Fluctuation and Auto

Non-reciprocal microwave components such as circulators, isolators, and phase shifters are indispensable tools in both today’s communication systems and future quantum
computers [1-2]. However, these components rely almost entirely on field-based bulky ferrimagnet, which tend to be relatively large, off-chip, and costly to assemble. In this context,
different fields of research are investigating solutions to miniaturize these non-reciprocal devices compatible with integrated circuit technology. Among them, the field of magnonic plays a
key role in this search benefiting from a wide range of non-reciprocal properties in the propagation of spin waves [3]. Recently, unidirectional transmission of spin waves was achieved by
taking advantage of the chiral coupling between the uniform resonance of Co nanowires and exchange spin waves in a thin YIG film [4, 5]. Here, we further miniaturized this method and
demonstrate the possibility of shaping non-reciprocal spin wave beams in a continuous thin YIG film. We performed spin wave spectroscopy on a series of devices made of arrays of Co
bars of dimensions: L=10µm in length, w=200 nm in width, 40nm in thickness, and laterally spaced by a=400nm (see Fig. 1). The transmission spectra done with the external field applied
along the length of the Co bars show perfect non-reciprocity for several peaks located around the resonance frequency of the Co bars, e.g. 8.4 to 10.4 GHz range as shown in Fig. 2. These
peaks correspond to integer values of the lateral spacing between Co bars, e.g. kn=nπ/a ranging up to 80rad/µm. Additional non-reciprocal peaks occurring at lower frequencies ranging
between 2 and 6 GHz correspond to the satellite peaks of the antenna directly coupled to the YIG film. Surprisingly, some of the peaks display a reversed non-reciprocity. Finally, we make
use of this multitude of transmission peaks to characterize the k-dependence of the relaxation time and the group velocity of dipole-exchange spin waves. 
**References** 87.png)[1] W. Palmer, et al., IEEE Micr. Magazine 20, 36 (2019) 
[2] M. Devoret, et al., Superconducting circuits for quantum information,Science 339, 1169 (2013) 
[3] J. Chen, et al., J. Phys. D. Appl. Phys. 55, 123001 (2022) 
[4] H. Wang, et al., Nano Res. 14, 2133–2138 (2021) 
[5] J. Chen, et al., Phys. Rev. B 100, 104427 (2019) 
 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/94e4cd227f20528c8226450f05b99d7c.png) 

SEM image of Co nanowires with length L=10μm, width w=200nm, and period a=400nm lithographed on a 50nm YIG film with thickness, and located under 2 µm wavelength Au coplanar waveguides (CPW)

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/0d51030b4d6ea35e8b9bbc51296dd09e.png) 
Transmission spectrum ΔL21
(blue) and ΔL12
(red) at 24mT applied field.

High Wave Vector Non reciprocal spin Wave Beams

Understanding magnons in confined geometry that relates to experimental devices is rather challenging, since device heterostructures, e.g. Spin Seebeck devices, are a
combination of both ferromagnetic (FM) and antiferromagnetic (AFM) layers [1]. The size of the layers can range from hundreds to several atomic layers, hence discontinuities on a short
length scale where a change of magnetic properties is abrupt, in presence of magnetic anisotropies, bring an extra challenge in determining the magnon dynamics, dispersion as well as
prorogation in such structures.
In this work, we extend the method, used in [2] where the spin-scattering function was calculated for 1D finite chains, to 3D systems. In particular, we show how to modify this approach to
calculate the dynamics of magnon modes in finite systems using second quantisation of the Heisenberg Hamiltonian for thin film and interfaces, as well as hetero-structure between FM and
AFM layers, taking into account surface and interface anisotropies and exchange parameters. The effect of such changes in the parameters of the system leads to the appearance of confined
states at the surfaces and interfaces of the heterostructures, Fig. 1. We study the effect that such confined states have on the spin-scattering function, which can be regarded as a proxy for
neutron-scattering. 
**References** 
[1] Lin, W., et. al. Physical Review Letters, Vol.116, p.186601 (2016) 
[2] Bearisto, B., Phys. Rev. B, Vol.104, p.134415 (2021) 
[3] Pajda, M., Phys. Rev. B, Vol.64, p.174402 (2001) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/6f4fdee497bbb1a8a4599d57036da149.png) 
Spin scattering function for Fe BCC, exchange parameters are taken from [3] and experimental points (solid black dots) were reproduced from the same paper. We can see a clear resonance matching the bulk dispersion, as well as a line that corresponds to localized surface states.

Theory of confined magnons in 3D heterostructures

Magnetic modelling for ultrafast simulations at the atomistic and micromagnetic scale make common use of Langevin Dynamics to model the effect of thermal fluctuations on the procession of the magnetic moment. Such a model is simple to add to any finite-temperature simulation: a Gaussian distribution scaled against a collection of material dependent parameters are included in the effective field term in the LLG spin Hamiltonian, and at pico-second timescales is uncorrelated in space or time [1]. In general, however, the thermal fluctuations in a material are correlated through collective occupation of electron, phonon, and magnon modes in the system. Analytical solutions for these relationships under conditions of non-equilibrium require simple materials or drastic approximations, conditions ill-suited to ultrafast ferri and antiferromagnetic materials. We present an improved thermostat for ultrafast atomistic scale magnetic simulations using a computationally determined conduction band environment based on semi-classical electron-electron and electron-phonon scattering events [2][3]. Currently, our environment simulates charge and heat transport in metals on ultra-fast time scales during severe non-equilibrium caused from applied electric fields or laser-heating pulses. The electron-electron and electron-phonon relationship is parametrized into the relaxation time approximation using constants derived from experimental data [3][4]. Our environment successfully reproduces the popular two-temperature model (TTM) used to simulate laser-heating of experimental samples (fig. 1), and currently reproduces the phenomena of effective Joule heating from applied external fields and super-diffusive electron transport. The current work lays the foundation for the inclusion of electron spin into the conduction band environment, offering the ability to include correlated spin effects not possible in Langevin Dynamics. Figure 1. Simulated electron and lattice temperatures (solid) following laser exposure compared to the standard TTM (dash). The varying colours and dynamics are a result of varying the electron-electron scattering constant.

Electronic Heat Bath Simulations for Ultra fast Spin Dynamics

Ferrimagnets (FiMs) persist in ultra-fast dynamics similar to antiferromagnets but are as easy to be electrically manipulated as ferromagnets, leading to a promising approach for achieving ultrafast devices with feasible tunability. Furthermore, spin-chain is of fundamental importance in condensed-matter physics, revealing unique properties such as quantum phase transition, edge effect, etc.[1] Although the dynamics of FiM have been widely described by a two sublattices macro-spin model[2], it hardly captures the sophisticated physical properties in multi-spin systems such as spin chains. Hence, we developed an atomistic spin model to investigate the spin dynamics in the FiM spin chain deployed by current-induced spin-transfer torque(STT). Our simulation results showed that in the finite FiM spin chain, first, the oscillation started from opposite-spin spatial region, defined as "oscillation core (OC)". Next, the oscillation spatial region would expand and finally only some specific spins formed stable oscillation (Fig.1). To further understand this phenomenon, we first introduce one OC into the spin chain. When excited by STT, the exchange mode can be further classified into non-flipped exchange mode (region I), critical exchange mode (region II), and flipped exchange mode (region III). In region I and III, the frequency increased linearly with different slops as the current density increased, yet in region II, the frequency slightly decreased, indicating the competition among exchange field (Hex), uniaxial anisotropy, and STT (Fig.2a). Next, when two OCs interacted, we find that smaller distance gives lager frequency, which is attributed to different Hex. Hence, proper choice of distance allows us to tune the Hex, achieving large-angle precession with THz frequency(Fig.2b). In summary, our work offers a novel understanding of FiM oscillation and proposes a strategy to construct energy-efficient spintronic oscillators with THz frequency. (This work at the National University of Singapore is supported by MOE-2019-T2-2-215 and FRC-A-8000194-01-00) 

**References** 
[1] H. Brune, Science., Vol. 312, p.1005-1006 (2006)
[2] M. Guo, H. Zhang and R. Cheng, Phys. Rev. B., Vol. 105, p.064410 (2022) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/6a6c6abaafc634f3233c0686bd548bad.png) 
Fig.1 Schematic of FiM spin chain oscillation behavior evolved with time. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/9635db2888915fc5843feae8047b4e3c.png) 
Fig.2 Numerical results of the oscillation in the FiM spin chain with (a)one OC and (b)two OCs.

Theoretical Investigation of Tunable Ferrimagnetic Spin Chain Oscillation Based on Atomistic Spin Model Simulation

Dzyaloshinskii-Moriya interaction (DMI) that occurs in structural inversion asymmetric systems stabilizes chiral domain-walls (DWs) which is a key issue to achieve high performance spintronic applications such as memory and data storage devices with high speed and high durability [1-4]. It is therefore important to analyze the strength of DMI and thus, there have been numerous efforts devoted to quantifying the DMI both theoretically and experimentally [4-6]. Unfortunately, the inconsistency between theory and experiment inevitably occurs because the theory predicts the strength of DMI based on single interface, however, the experiments must be implemented based on at least, double interface because of Ex-Situ nature [2-6]. Here, we first, measure the strength of DMI at Pt/Co single interface in In-Situ nature. To measure the strength of DMI at single interface, we set up In-Situ Magneto-Optical-Kerr-Effect (MOKE) microscopy with UHV magnetron sputtering chamber. Then, we quantified the strength of DMI with respect to Co layer thickness. Fig.1 clearly shows the plot of HDMI with respect to tCo based on Je's method [5]. 
**References** [1] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320 (5873), 190-194. (2008). 
[2] I. E. Dzialoshinskii, Sov. Phys. JETP 5, 1259 (1957). 
[3] T. Moriya, Phys. Rev. 120, 91 (1960). 
[4] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152 (2013). 
[5] S.-G. Je, K.-J. Lee, and S.-B. Choe, Phys. Rev. B 88, 214401 (2013). 
[6] J. Cho, B. Koopmans, and C.-Y. You, Nat. Commun. 6, 7635 (2015). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/ca39e5c56db70732287c6e9d8334f74e.png) 

Fig. 1 The plot of HDMI with respect to tCo.

Co Layer Resolved Measurement of Dzyaloshinskii

Recently, researchers developed 3d magnetization models for magnetic shells with curvature and torsion [1]. For example, in a thin nanotube, the curvature induced a magnetostatic anisotropy [2]. Also, in the case of the nanocylinder, the curvature induces dipole-dipole interaction [3]. However, in these tubular systems, torsion has not been explored yet. Here we propose to study a nanocylinder with torsion that we call the twisted nanotube. In Figure 1, we present the system. It has an internal radius r, an external radius R, infinite length L, an elliptical transversal area, curvature κ and torsion τ. Figure 1. The magnetization vector in terms of the base vectors (e1 e2, e3). The parametrization of a 2D tubular surface γ, embedded in a 3D space defines these vectors. For the twisted nanotube, we compute the dipolar interaction by implementing the model for the energy of the nanocylinder [3]. Furthermore, the torsion is introduced by defining the reference frame for the magnetization, as is suggested in a recent approach developed for magnetic shells [1]. Hence, the magnetization base vectors are derived in terms of the surface curvilinear parametrization, as presented in Figure 1. With this methodology, we show that in the twisted nanotube, there are volumetric charges because of the torsion of the system. In equation (1), we present the expression for the volumetric charges, where the term Γd is the contribution of the torsion. Also, the developed model shows that the torsion does not induce surface charges. With this work, we expect to contribute to future developments of spintronic technology, for example, supporting a theoretical model for a magnonic transducer, like the one developed for the nanocylinder [4]. ▽ ● M = M sinΘ( sinp/ρ - ∂zΘ + Γd) (1) 

**References** [1] Gaididei, Yuri, Volodymyr P. Kravchuk, and Denis D. Sheka, "Curvature effects in thin magnetic shells," Physical review letters vol. 112, no. 25, pp. 257203, 2014. 
[2] Landeros, Pedro, and Álvaro S. Núñez, "Domain wall motion on magnetic nanotubes," Journal of Applied Physics vol. 108, no. 3, pp. 033917, 2010. 
[3] Otálora, J. A., et al, "Breaking of chiral symmetry in vortex domain wall propagation in ferromagnetic nanotubes," Journal of magnetism and magnetic materials vol. 341, pp. 86-92, 2013. 
[4] Salazar-Cardona, Monica M., et al, "Nonreciprocity of spin waves in magnetic nanotubes with helical equilibrium magnetization," Applied Physics Letters vol. 118, no. 26, pp. 262411,
2021. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/3463b668f320a0f6440973f22c8b35e6.png) 

Figure 1. The magnetization vector in terms of the base vectors (e1, e2, e3). The parametrization of a 2D tubular surface γ, embedded in a 3D space defines these vectors.

Torsion and curvature induced effects in magnetic twisted nanotubes

I.Introduction Understanding of excess loss in electrical steel sheets, which is the difference between the measured iron loss and the sum of static hysteresis loss and classical eddy current loss [1], is difficult as it is related to non-repeatable dynamic domain wall motion inside the steel [2], whereas it is normally agreed upon that the anomalous eddy current loss, which is due to the micro eddy current generated by domain wall motion, is one main source of the excess loss for grain-oriented steel sheets [3]. One famous model coping with the anomalous eddy current loss is the Pry and Bean domain model [4] and we have established a numerical domain model based on it [5]. However, this model cannot model the skin effect of the flux density and it cannot represent the nonlinear BH curve of the steel sheets. In this paper, a simple numerical method using multiple boundary conditions is proposed to calculate the micro eddy currents and the classical eddy currents simultaneously while considering the nonlinear B-H constitutive relation of steel sheets. II.Analysis model and Conditions The domain pattern of grain-oriented steel sheet is considered as a simple periodic one as shown in Fig. 1. Half of one periodic region is chosen as the analyzed model. Micro eddy currents are generated around the domain walls when the domain wall moves. In the proposed analysis, the overall alternating magnetization to induce the classical eddy currents is imposed by using one Dirichlet boundary condition. And the local magnetization change due to domain wall motion is imposed by using another Dirichlet boundary condition to induce the micro eddy currents. III.Analysis Results The calculated flux and eddy current distributions are shown in Fig. 2. The skin effect of flux density can be observed in the flux distribution. To show the micro eddy currents clearly, the eddy current calculated by using the proposed method is subtracted with the classical one, and the obtained micro eddy currents are shown in Fig. 2 (b)(ii). We can see that micro eddy current around the domain walls are generated. 
**References** 
[1] G. Bertotti, “Hysteresis in Magnetism: For physicists, Materials scientists, and Engineers”, Academic Press, San Diego, (1998). 
[2] A. J. Moses, “Energy efficient electrical steels: magnetic performance prediction and optimization”, Scripta Materialia, vol. 67 no. 6, pp. 560–565, 2012. 
[3] J. E. L. Bishop, “Enhanced eddy current loss due to domain displacement”, Journal of Magnetism and Magnetic Materials, vol. 49, no. 3, pp. 241-249, 1985. 
[4] R. H. Pry and C. P. Bean, “Calculation of the energy loss in magnetic sheet materials using a domain model,” Journal of Applied Physics, vol. 29, no. 3, pp. 532-533, 1958. 
[5] Y. Gao, Y. Matsuo, and K. Muramatsu, “Investigation on simple numeric modeling of anomalous eddy current loss in steel plate using modified conductivity,” IEEE Trans. on Magn.,
vol. 48, no. 2, pp, 635-638, 2012. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/8990bf10daf7abadfeb3009c1091e1a0.png) 
Fig. 1. Analysis model (D: sheet thickness). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/820e339a1d5935989673694745a9e3f3.png) 
Fig. 2. Flux and eddy current distributions.

Investigation on simple numerical modeling of anomalous eddy current loss in steel sheet using multiple boundary conditions

The hysteresis modeling of ferromagnetic materials in electrical equipment is one of the basic theoretical studies in the field of electrical engineering. In this paper, a three-dimensional vector hysteresis model is proposed based on Stacked Auto-Encoder (SAE) model and hysteresis operator space theory. Multiple 3D vector Play hysteresis operators are used to construct an hysteresis operator space to create the high-dimensional operator data. By the operator space, the nonlinear hysteresis relationship in the hysteresis data is transformed into a nonlinear mapping between the operator data and the output of the model. And the SAE model is used to characterize this nonlinear mapping. In the training of the SAE model, training set is composed of operator data and magnetic field strength data. The structure of the vector hysteresis model is determined by taking the output of the operator space as the input of the SAE model. The simulation results show that the model can effectively describe the nonlinear characteristics and anisotropic of the Soft Magnetic Composite under 3D vector excitation. In addition, this model has certain generalization ability. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/bf555ed47ded496a9734c2097a428114.png) 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/f5327d19a685f1dea2094457d0546462.png)

The Study of the Three dimensional Vector Hysteresis Model based on the SAE model and Play operator

We using Mumax3 to simulate a 5×5 asymmetric disk array. By changing the temperature, we explored the variety of energy during the evolution of the vortex and the probability of the vortex chirality. We also controled disc spacing to adjustment the magnitude of stray field . The material we used amorphous permalloy, simulation parameters were saturation magnetization 8.6x105 A/M, exchange length 13x10-12 J/m, damping constant 0.01, cell size 5x5x5 nm3, and temperatures changed from 0K to 300K. Geometric parameters were disk diameter 1000 nm, thickness 40 nm, asymmetric ratio 0.3, and disc spacing 200, 500 and 800 nm. In the previous research, vortex chirality seems to be related to a rise in total energy. When increase temperature will also increase the total energy in our simulation. We analyze the total energy, demagnetization energy and exchange energy separately then we finded the exchange energy increases significantly with temperature. It can be speculated that changes in temperature make the magnetic moments less parallel to each other, resulting in a significant increase in exchange energy and increase in the probability of CW vortices. 
**References** 
Arne Vansteenkiste, Jonathan Leliaert, Mykola Dvornik, et al. AIP Advances, 4, 107133(2014) 
Mi-Young Im, Ki-Suk Lee, Andreas Vogel, et al. NATURE COMMUNICATIONS, 5:5620(2014) 
Dustin A. Gilbert, Brian B. Maranville, Andrew L. Balk, et al. NATURE COMMUNICATIONS 6 8462 (2015) 
Mi-Young Im, Peter Fischer, Hee-Sung Han, et al. NPG Asia Materials 9 (2017) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/23ee7e6067cfba89d83c2c80460bfab0.png) 

Fig1. Clock-wise vortex probability. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/85be8c41c545d2fbcf95d1d138aac056.png) 
Fig2. MOKE image of disk array.

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