France



The recent discovery of anti-skyrmions[1] and elliptical block skyrmions[2] in lamellae formed from single crystals of inverse tetragonal ternary Heusler compounds makes these compounds extremely interesting.  Related binary compounds host chiral domain walls whose current induced motion in films that are only 1 unit cell thick has recently been observed.  These films were prepared using a novel method of chemical templating[3].  We show that strain can dramatically affect the current induced domain wall motion in such films, changing both its magnitude and, remarkably, even its direction.  Anti-skyrmions are extremely interesting objects that have high stability[4] and whose size can be tuned by more than an order of magnitude [5] by varying the thickness of the host lamella. These properties derive from the symmetry of the bulk Dzyaloshinskii-Moriya vector exchange interaction that stabilizes them that itself is derived from their underlying crystal structure. We have found similar properties in other materials with related symmetries[6]. The current induced motion of anti-skyrmions has not yet been realized but theoretical models suggest that the skyrmion Hall angle will vary strongly with current direction and, for certain crystal directions, will be zero.  Whilst we have found no evidence, to date, for current induced motion of such objects from volume spin-transfer torques, models show that spin-orbit torques should give rise to their motion.  For this purpose, ultra-thin 2D layers are needed that can be coupled to heavy metal layers in which spin currents can be generated.  In this talk we discuss progress in preparing high quality, epitaxial, ultra-thin films of ternary inverse tetragonal Heusler compounds that are only 1 to 2 unit cells thick and the exploration of non collinear spin textures in these films as well as their current induced motion. Finally, we discuss the first observation of fractional anti-skyrmions and Block skyrmions in nano-stripes formed from inverse tetragonal ternary Heusler compounds.

**References:**

[1] Nayak, A.K., et al., &lt;i&gt;Magnetic antiskyrmions above room temperature in tetragonal Heusler materials.&lt;/i&gt; Nature, 2017. 548(7669): p. 561-566.&lt;br/&gt;[2] Jena, J., et al., &lt;i&gt;Elliptical Bloch skyrmion chiral twins in an antiskyrmion system.&lt;/i&gt; Nat. Commun., 2020. 11: p. 1115.&lt;br/&gt;[3] Filippou, P.C., et al., &lt;i&gt;Chiral domain wall motion in unit-cell thick perpendicularly magnetized Heusler films prepared by chemical templating.&lt;/i&gt; Nat. Commun., 2018. 9: p. 4653.&lt;br/&gt;[4] Saha, R., et al., &lt;i&gt;Intrinsic stability of magnetic anti-skyrmions in the tetragonal inverse Heusler compound Mn&lt;sub&gt;1.4&lt;/sub&gt;Pt&lt;sub&gt;0.9&lt;/sub&gt;Pd&lt;sub&gt;0.1&lt;/sub&gt;Sn &lt;/i&gt;Nat. Commun., 2019. 10: p. 5305.&lt;br/&gt;[5] Ma, T., et al., &lt;i&gt;Tunable Magnetic Antiskyrmion Size and Helical Period from Nanometers to Micrometers in a D&lt;sub&gt;2d&lt;/sub&gt;Heusler Compound.&lt;/i&gt; Adv. Mater., 2020: p. 2002043.&lt;br/&gt;[6] Srivastava, A.K., et al., &lt;i&gt;Observation of Robust Néel Skyrmions in Metallic PtMnGa.&lt;/i&gt; Adv. Mater., 2020. 32: p. 1904327.

**Images:**





INTERMAG 2021

 Ultra-thin Heusler films and lamellae that host non-collinear spin textures

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The recent discovery of anti-skyrmions[1] and elliptical block skyrmions[2] in lamellae formed from single crystals of inverse tetragonal ternary Heusler compounds makes these compounds extremely interesting. Related binary compounds host chiral domain walls whose current induced motion in films that are only 1 unit cell thick has recently been observed. These films were prepared using a novel method of chemical templating[3]. We show that strain can dramatically affect the current induced domain wall motion in such films, changing both its magnitude and, remarkably, even its direction. Anti-skyrmions are extremely interesting objects that have high stability[4] and whose size can be tuned by more than an order of magnitude [5] by varying the thickness of the host lamella. These properties derive from the symmetry of the bulk Dzyaloshinskii-Moriya vector exchange interaction that stabilizes them that itself is derived from their underlying crystal structure. We have found similar properties in other materials with related symmetries[6]. The current induced motion of anti-skyrmions has not yet been realized but theoretical models suggest that the skyrmion Hall angle will vary strongly with current direction and, for certain crystal directions, will be zero. Whilst we have found no evidence, to date, for current induced motion of such objects from volume spin-transfer torques, models show that spin-orbit torques should give rise to their motion. For this purpose, ultra-thin 2D layers are needed that can be coupled to heavy metal layers in which spin currents can be generated. In this talk we discuss progress in preparing high quality, epitaxial, ultra-thin films of ternary inverse tetragonal Heusler compounds that are only 1 to 2 unit cells thick and the exploration of non collinear spin textures in these films as well as their current induced motion. Finally, we discuss the first observation of fractional anti-skyrmions and Block skyrmions in nano-stripes formed from inverse tetragonal ternary Heusler compounds.

**References:**

[1] Nayak, A.K., et al., Magnetic antiskyrmions above room temperature in tetragonal Heusler materials. Nature, 2017. 548(7669): p. 561-566. [2] Jena, J., et al., Elliptical Bloch skyrmion chiral twins in an antiskyrmion system. Nat. Commun., 2020. 11: p. 1115. [3] Filippou, P.C., et al., Chiral domain wall motion in unit-cell thick perpendicularly magnetized Heusler films prepared by chemical templating. Nat. Commun., 2018. 9: p. 4653. [4] Saha, R., et al., Intrinsic stability of magnetic anti-skyrmions in the tetragonal inverse Heusler compound Mn1.4Pt0.9Pd0.1Sn Nat. Commun., 2019. 10: p. 5305. [5] Ma, T., et al., Tunable Magnetic Antiskyrmion Size and Helical Period from Nanometers to Micrometers in a D2dHeusler Compound. Adv. Mater., 2020: p. 2002043. [6] Srivastava, A.K., et al., Observation of Robust Néel Skyrmions in Metallic PtMnGa. Adv. Mater., 2020. 32: p. 1904327.

**Images:**

technical paper

Nickel thin films (100 nm) deposited on 128o Y-cut: LiNbO3 exhibit a well-defined magnetic anisotropy upon annealing at 325o C [1]. Simulating the magnetization of these films using a temperature dependent Stoner-Wohlfarth model indicates that the films have a very narrow angular distribution of anisotropy axes [2]. Here we report a direct measurement of the angular distribution of anisotropy axes by the angular remanence technique [3]. The angular range over which the easy axis remanence drop significantly is a measure of the angular distribution of the anisotropy axes. For annealed nickel thin films on 128o Y-cut: LiNbO3 the FWHM of the remanence as a function of angle (α) is approximately 0.44o .

**References:**

[1] S.A. Mathews, N.S. Bingham, R.J. Suess, K.M. Charipar, R.C.Y. Auyeung, H. Kim, and N.A. Charipar, IEEE Trans. Mag., Vol. 55, Issue 2, (2019) [2] S.A. Mathews, A. Ehrlich, and N. A. Charipar, “Hysteresis branch crossing and the Stoner-Wohlfarth model”, Scientific Reports, 10:15141 (2020). [3] H.J. Richter and H. Hibst, Jour. Mag. Mag. Mat., 95, 118-122 (1991).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a8bbb46dcce86d5d88cbc7706792ebd7.jpg)
Figure 1: Absolute value of easy axis remanence as a function of α (angle between the hard axis and the saturating field). Open squares represent measured data. The solid line represents the model, assuming an ensemble of Stoner-Wohlfarth particles with two normal (Gaussian) distributions of anisotropy orientation.

 Angular Remanence and Anisotropy Orientation Distribution in Nickel Films on LiNbO3

In this work, we report transverse susceptibility (TS) measurements performed on nickel thin films, deposited on 128o Y-cut LiNb03 substrates and annealed at low temperature. The films exhibit a strong uniaxial anisotropy [1] and hysteresis branch crossing (HBC) [2]. The results are compared to numeric solutions to the Stoner-Wohlfarth (SW) model [3] using a finite temperature approximation [4] and a distributed anisotropy. The experimentally observed TS-curves, measured with applied field close to the hard axis, are shown to be in excellent agreement with the SW-model, confirming that the magnetization in these samples occurs via coherent rotation. As predicted by Aharoni [5], the TS curves show three peaks or cusps: two reversible peaks associated with anisotropy field, and one irreversible peak associated with the discontinuity in the transverse magnetization.

**References:**

[1] S.A. Mathews, N.S. Bingham, R.J. Suess, K.M. Charipar, R.C.Y. Auyeung, H. Kim, and N.A. Charipar, IEEE Trans. Mag., Vol. 55, Issue 2, (2019) [2] S.A. Mathews, A.C. Ehrlich, and N.A. Charipar, Scientific Reports, 10:15141 (2020) [3] Stoner, E.C. and Wohlfarth, E.P., Phil. Trans. Roy. Soc. of London Series A, 240, no. 826, pp.599-624 (1948). [4] L. Lanci and D. Kent, J. Geophys. Res., Vol. 108, No. B3, 2142 (2003). [5] A. Aharoni, E. M. Frei, S. Shtrikman, and D. Treves, Bull. Res. Counc. Isr., Sect. F 6A, 215 (1957).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3ddf7ccaf7051008650d6c58dc9c0178.jpg)
Figure 1: Measured (open squares) and simulated (solid lines) transverse susceptibility of a nickel thin film on LiNbO3. The field is applied at an angle α=4o from the hard axis. The ascending and descending curves are shown in red and blue, respectively.

 Transverse susceptibility of nickel thin films with uniaxial anisotropy

In this paper, an improved interpolating element-free Galerkin method (IIEFGM) with nonsingular weight function is introduced to calculate the magnetic field distribution of electrical machine. Since the shape function satisfied the property of Kronecker-delta function, the IIEFGM can impose the boundary conditions directly. Meanwhile, the IIEFGM adopts nonsingular weight function, which means the weight function of the conventional element-free Galerkin method (EFGM) can be used. To demonstrate the effectiveness of IIEFGM, the magnetic field distribution of a permanent magnet coreless disc machine is analyzed. The selection of weight function and the nodes distribution are discussed, the comparison with Finite element method (FEM) shows the IIEFGM has high computational accuracy. I. Introduction Meshless methods are the important supplement to the finite element method in solving boundary problems, and element-free Galerkin method is the most widely used method. The mainly difference between the EFGM and the FEM is the construction method of the shape function, however, since the shape function of EFGM does not satisfy the property of Kronecker-delta function, the imposition of boundary conditions required additional methods, such as penalty method. To solve this problem, reference [1] proposed an interpolating element-free Galerkin method (IEFGM), however, the method adopt singular weight, which makes the computation difficult. Reference [2] proposed IIEFGM with nonsingular weight function, which has the advantages of both the FEM and the IEFGM. In this paper,a permanent magnet coreless disc machine (PMCDM) is taken as an example [3] the IIEFGM is introduced to calculate the magnetic field distribution of the PMCDM, and the comparison with the FEM validated the effectiveness of the IIEFGM. II The foundation of the IIEFGM In conventional EFGM, the basis function is defined as: p1=1, p2=x, p3=y (m=3), and the trial function is uh(x)=pT(x)a(x). The coefficients a(x) can be obtained by minimizing the weighted least-square function, then the shape function N(x) can be derived. To make the shape function satisfied the property of the Kronecker-delta function, the IIEFGM modified the basis function. Then the boundary conditions can be imposed directly, and the negative exponential function is selected as the weight function. III Magnetic field calculation of the PMCDM Fig.1 shows the magnetic field distribution of the PMCDM by using IIEFGM, the problem domain is divided into the radial part and the circular part. Fig.2 depicts the main flux density on the airgap center line. It can be seen that the calculated results obtained by IIEFGM are in excellent agreement with the results with the FEM. IV Conclusion This paper introduced an improved interpolating element-free Galerkin method. The Galerkin weak form and the discretized equation are derived, the flow of the algorithm is optimized, and a weight function with higher accuracy is given. Based on IIEFGM, the magnetic field distribution of a PMCDM is analyzed, numerical results verified the effectiveness of the IIEFGM. The study results provide references for the application of IIEFGM.

**References:**

[1] Ren H , Cheng Y ,"The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems," Engineering Analysis with Boundary Elements, Vol. 36(5), p.873-880, 2012. [2]Wang J F , Sun F X , Cheng Y M, "An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems," Chinese Physics B, Vol. 21, No.9, 2012. [3]Gu C L , Wu W , Shao K R, "Magnetic field analysis and optimal design of DC permanent magnet coreless disc machine," IEEE Transactions on Magnetics, Vol. 30(5), p. 3668-3671, 1994.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/16d3efc5f003de67c4a8d3fbbb91d924.jpg)
Fig.1 Magnetic field distribution. (a) circular direction. (b) radial direction

![](https://assets.underline.io/uploads/markdown_image/1/image/4fb8054f5ba97bdbc11fc680df6f9a1b.jpg)
Fig.2 Main flux density on airgap center line. (a) circular direction. (b) radial direction

 Application of an improved interpolating element-free Galerkin method in magnetic field calculation

Introduction Hysteresis of magnetic materials is a complicated physical phenomenon. The appropriate mathematical expression which can describe the hysteresis mechanism is desired in the design of electromagnetic devices, such as transformers and reactors. It could enable researchers to model the hysteresis accurately, and hence improve the performance and efficiency of the devices. This paper modifies the Bouc-Wen model to model the hysteresis of magnetic materials, and accords the parameters of the model with physical meaning. At present, the magnetic hysteresis models are classified as mathematical models and physical models. The Preisach model, as a mathematical model, can describe the hysteresis phenomenon, but it cannot describe the magnetization process of the materials[1]. The domain structure model, which has a clear physical meaning and takes into account of energies related to the growth of the domain, can describe the hysteresis well, but it only studies in microscopic range, and is complicated to be employed in engineering applications[2]. To model the magnetic properties efficiently and accurately, a practical model which can correlate model parameters describing magnetization process with the hysteresis phonemonen responsible to the applied magnetic field is needed. The model proposed in this paper has the feature to simulate the magnetization process in general by adjusting the parameters according to the materials characteristic in applied magnetic fields. Theorem Ferromagnetism reveals that the internal mechanism of hysteresis is very complicated. The magnetization process mainly includes motion of the domain wall and rotation of the spontaneous magnetization of domain [3]. The process of wall motion is very similar to the kinetic process of viscous friction, such as nonlinear hysteretic mechanics Bouc-Wen model, so that the kinetics model is exploited to describe the domain wall motion. However, it cannot characterize the rotation of the domain magnetization[4]. The model is modified by introducing a term which can characterize the rotation process. The improved model includes the four resistances encountered when the domain wall moves and the domain magnetization rotates during the magnetization process: i) the resistance caused by the increasing energy of the domain wall, which is linearly related to the displacement of the domain wall; ii) the resistance caused by the local eddy current, which is proportional to the moving velocity of the domain wall; iii) the resistance caused by the rotation of the domain magnetization; and iv) the hysteretic restoring force caused by material defects, which is proportional to the hysteretic displacement. The hysteretic displacement which describes the inelastic behavior of domain wall motion is calculated by the differential equation of the Bouc-Wen model. The improved model can be used to calculate the magnetic hysteresis under high and low magnetic field strengths. The parameters of the model representing the damping coefficients of each resistance are identified by the particle swarm algorithm. It is found that the damping coefficient is a function of the strength of the applied field under quasi-static magnetization, and the shape parameters of the model are constants. Since eddy current will cause the reactive force during the motion of domain wall and rotation of domain magnetization under dynamic excitation, the model must be modified to take the account of dynamic excitation into damping coefficients. A function of damping coefficients related to applied field strength and frequency is discussed in detail in the full paper. Experiment In order to simulate magnetization process using the improved model, the oriented silicon steel B27R090 with relatively simple domain structure is used. The Single Sheet Test is used as the experimental platform to measure the magnetic properties of the sample under quasi-static and dynamic magnetizations. Fig. 1 and Fig.2 show the measured hysteresis loops and the predicted hysteresis loops under sinusoidal excitation at 5 Hz and 50 Hz. The predicted hysteresis loops calculated by the improved Bouc-Wen model are in good agreement with the measured loops when the flux density is between 0.3 T and 1.4 T. The calculated total losses using the improved Bouc-Wen model is also in good agreement with the the measurement results. The full paper will discuss the results in detail.

**References:**

[1] Mayergoyz I D, IEEE Transactions on Magnetics., Vol. 22, p. 603-608(1986) [2] Sudo M, Mifune T and Matsuo T, IEEE Transactions on Magnetics., Vol. 49, p. 1829-1832 (2013) [3] Bertotti G, IEEE Transactions on Magnetics., Vol. 24, p. 621-630 (2002) [4] Ikhouane F, José Rodellar, Nonlinear Dynamics., Vol. 42, p. 63-78 (2005)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d8bb2736bdfccbddfea7878208a89264.jpg)
Fig.1 Measured (solid lines) and calculated (dashed lines) hysteresis loops under sinusoidal excitation with amplitudes of flux density of 0.3 T, 0.7 T, and 1.4 T at frequencies of 5 Hz

![](https://assets.underline.io/uploads/markdown_image/1/image/11057d74d6a094ebc1633ff25b2967f4.jpg)
Fig.2 Measured (solid lines) and calculated (dashed lines) hysteresis loops under sinusoidal excitation with amplitudes of flux density of 0.3 T, 0.7 T, and 1.4 T at frequencies of 50 Hz

 An Improved Hysteresis Model Based on Bouc-Wen Model Under Quasi-Static and Dynamic Magnetizations

—A circular polarized vortex beam generating based on leaky-wave antenna with substrate integrated waveguide(SIW) is proposed in this paper. Comparing with phased array antenna, the leaky-wave antenna is working without complicated feeding network and phase shifter. Then the unit is designed to realize circular polarization. Finally The loop leaky-wave antenna with SIW is designed to generate vortex beam and works at 14.5 GHz. The simulated gain reaches 11.4 dBi with generating -1 mode orbital angular momentum(OAM) of vortex beam. Vortex electromagnetic wave are electromagnetic waves with orbital angular momentum Wave. The most important characteristic of orbital angular momentum is its orthogonality in different modes. The degree of freedom of electromagnetic wave includes frequency and polarization. It is amazing that the orthogonality of vortex electromagnetic wave provides extra freedom for electromagnetic wave. Therefore, vortex electromagnetic wave has the potential to be used in communication and imaging systems[1]-[3]. In this paper, A circular polarized vortex beam generating based on leaky-wave antenna with substrate integrated waveguide(SIW) is proposed. Because the leaky-wave antenna is working without complicated feeding network and phase shifter. And the unit is designed to realize circular polarization. Then the radiation pattern of loop leaky-wave antenna is derived from the leaky-wave unit. Finally The loop leaky-wave antenna with SIW is proved to radiate -1 mode vortex electromagnetic wave with circular polarization in the simulation of CST Studio. The simulation model is showed as Fig. 1 a), the reason the loop leaky-wave antenna is three-quarter circle is that CST Studio limits the wave port should be put at the coordinate axis. And from [4], we know a part of circle to radiating is enough to radiate vortex beam. The microstrip line is designed for impedance matching to transmit microwave to the loop leaky-wave antenna. The S11 and S21 is showed in Fig. 1 b. It is showed the S11 is lower than -10dB and S21 is also low, because the most energy is radiated to open space. Fig. 2 a) is the phase distribution in far field at 14.5GHz. As we can see, the phase distribution is -1 mode vortex beam form. The radius of loop antenna determines the phase difference between adjacent unit, and in this paper, the radius is 52mm to make the OAM mode is -1. Fig. 2 b) shows the amplitude distribution in far field. The distribution is not uniform, which is caused by deletion of loop leaky-wave antenna. If the loop leaky-wave antenna is complete circle, the distribution will be improved. A circular polarized vortex beam generating based on leaky-wave antenna with substrate integrated waveguide (SIW) is proposed in this paper. Comparing with phased array antenna, the leaky-wave antenna is working without complicated feeding network and phase shifter. We proof linear polarized unit can radiate circular polarized beam but the beam is not clean, so we design unit with circular polarized radiation. Finally, the loop leaky-wave antenna with SIW is designed to transmit vortex beam and works at 14.5 GHz. The simulated gain reaches 11.4 dBi with generating -1 mode vortex beam.

**References:**

[1] Yan, Y. , Xie, G. , Lavery, M. P. J. "High-capacity millimetre-wave communications with orbital angular momentum multiplexing." Nature Communications vol. 5, pp. 4876, 2014. [2] Shilie, Zheng, Xiaonan, et al. "Orbital angular momentum mode-demultiplexing scheme with partial angular receiving aperture." Optics express. vol. 23, no. 9, pp.12251-12257, 2015. [3] Bai, Q. , A. Tennant , and B. Allen . "Experimental circular phased array for generating OAM radio beams." Electronics Letters 50.20:1414-1415, 2014. [4] Xiong, Xiaowen , et al. "Direct Generation of OAM Mode-Group and Its Application in LoS-MIMO System." IEEE Communications Letters PP.99(2020):1-1.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/399fb16ca0487d903477aca5d9a245b8.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/e87535880351c24dc4b35b3f36a74626.jpg)

 Circular Polarized Vortex Beam Generating based on Leaky-wave Antenna with Substrate Integrated Waveguide

It is known that the assessment of local magnetization in objects exhibiting hysteresis is crucial to the accurate design and/or performance estimation of a wide range of electromagnetic devices. Among the widely used approaches to carry out the afore-mentioned assessment tasks, the integral equations approach has been widely adopted (See, for instance [1]). In the past, it was demonstrated that an elementary hysteresis operator may be constructed using a two-node Hopfield Neural Network (HNN) having step activation functions and positive inter-node feedback [2]. The approach was further extended in [3] to construct a 2D vector hysteresis operator using a tri-node Hopfield Neural Network (HNN) having orientation-dependent inter-node feedback values. Utilization of this extension was previously partially utilized in 2D field computations by coupling the proposed approach with time consuming iterative integral equation methodologies [4]. The purpose of this paper is to offer a novel 2D field computation approach in media exhibiting hysteresis that solely utilizes HNNs. The approach is based on the novel incorporation of domain-domain interaction in the overall HNN energy formulation. Consequently, 2D field computation may be accomplished by employing typical HNN automated energy minimization algorithms. Consider a HNN comprised of N clusters of tri-node HNNs. The total applied field Ht at any node u in HNN #i having a triangular median given by (xi, yi) may be given by: Htiu=(H+∑j=1,N & j≒i hji)●eiu+∑v=1,3 & v≒u fivkcopl│eiu●eiv│, (1) hij=-▽(∑s=1,3 (fjsejs●Rjp)/(4π│Rjp│3))│(xi,yi) , Rjp=(x-xj)ex+(y-yj)ey, (2) where H is the external applied field, eab is the unit vector denoting the orientation of node #b in HNN #a, kcopl is the internal tri-node coupling factor using which the switching value of the elementary operator may be tuned, while f represent the HNN step activation function given by: f(Ht)={+1 if Ht&gt;0, -1 if Ht&lt;0, Unchanged if Ht=0}. (3) It should be mentioned here that, in HNN terminology, all coupling factors between any two nodes in the overall HNN may be deduced from (1) and (2). The proposed implementation yields node outputs corresponding to, an automatically achieved, minimized value of the HNN energy E (in line with the magnetostatic energy) given by: E=-∑i=1,N(∑u=1,3[(H+∑j=1,N & j≒i hji)●fiueiu +1/2 ∑v=1,3 & v≒u fiufiv│eiu●eiv│]). (4) Local magnetization values of any tri-node HNN, denoted by HNN #i, representing a triangular subdivision may be deduced from: Mi(H)=∑u=1,3 fiueiu. (5) Hence, total output of the overall HNN representing the magnetiation of the magnetized body under consideration may thus be expressed using superposition in the form: M(H)=∑i=1,NMi(H). (6) In order to demonstrate the proposed methodology, consider the case of a pair of tri-node HNN (i.e., N=2) as shown in Fig. 1.a. Sample results for this particular case are given in Figs. 1 and 2 for the two tri-node HNNs representing triangular subdivisions whose vertices have the coordinates (1.0,0.5), (-0.5,0.5), (-0.5,-0.5) and (1.0,0.5), (1.0,-0.5), (-0.5,-0.5), respectively. In Figs. 1.b and 1.c results corresponding to x-axis cyclic and rotational applied inputs are demonstrated when the two triangles join to form a rectangular shape. Variations of the x-axis cyclic input-output curves resulting from the change in the vertical spacing Dy and/or horizontal spacing Dx between both triangular subdivisions are further demonstrated in Fig. 2. Results clearly highlight the impact of the incorporation of domain-domain interaction in the introduced field computation approach. It should be mentioned that using such an approach HNN modules, which are widely available in a number of mathematical simulation tools, may be utilized in 2D field computation in media exhibiting hysteresis. More details of the approach for arbitrary 2D objects, demonstration of local magnetization vector plots, simulations and comparisons with finite element analysis tools will be presented in the full paper.

**References:**

[1] A.A. Adly, I.D. Mayergoyz and R.D. Gomez, IEEE Transactions on Magnetics, Vol. 29, p. 2380 (1993). [2] A.A. Adly and S.K. Abd-El-Hafiz, Journal of Magnetism and Magnetic Materials, Vol. 263, No.3, p. 301 (2003). [3] A.A. Adly and S.K. Abd-El-Hafiz, Proceedings of ECTICON, Paper No. 1067, p. 1 (2016). [4] A.A. Adly and S.K. Abd-El-Hafiz, Journal of Magnetism and Magnetic Materials, Vol. 434, p. 151 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e3a633f16b17803e9942186d7f4db975.jpg)
Fig. 1. (a) Two identical tri-node HNNs having different orientations, (b) Superposition of the HNN outputs for Dx=Dy=0 corresponding to a cyclic input along the x-direction, and (c) Superposition of the HNN outputs for Dx=Dy=0 corresponding to a rotational input.

![](https://assets.underline.io/uploads/markdown_image/1/image/fe05ca9c7dace532cec2ca4a8f6d47c4.jpg)
Fig. 2. Superposition of the HNN outputs corresponding to a cyclic input along the x-direction for; (a) Dx=Dy=infinity (i.e., no interaction), (b) Dx=1.0, Dy=0, and (c) Dx=0.0, Dy=-1.0.

 Field Computation in Media Exhibiting Hysteresis Using Hopfield Neural Networks

In this paper, to accurately analysis the magnetic properties of high temperature superconductor, the elemental operator model is proposed. The calculated and measured results of silver-sheathed Bi2223 superconducting tape samples under external magnetic field are compared to show the effectiveness and accuracy of the proposed method. I. Introduction The hysteresis phenomenon of high temperature superconductor (HTS) is caused by the pinning of magnetic flux due to various defects, as illustrated in Fig.1, such as voids, normal inclusions, grain boundaries, and compositional variations. The consequence of flux-pinning is that the flux remains constant in the interior of the HTS until a new flux is applied. This phenomenon constitutes a memory in the HTS, which can be well described by Bean’s critical-state model. Based on the mechanisms of superconductor, the HTS magnetic characteristic can be described via an infinite set of antimagnetic operators and magnetic operators. The property of antimagnetic operator and magnetic operator can be obtained by the antimagnetic curves and elementary hysteresis loops of different switching values of magnetic field, respectively[1,2]. In this paper, in the framework of the elemental operator and wind-rose method, as mentioned in [3], the underlying principle, experimental verification, and the error analysis of the elemental operator have been introduced. II. Elemental Operator Method In the proposed model, based on the mechanisms of HTS, it is assumed that lots of interacting magnetic elemental operators and antimagnetic operators. The magnetic elemental operators, which possesses its own biaxial anisotropy and anisotropy coefficient are enclosed in the HTS material. The interacting elemental operator which possesses its own biaxial anisotropy and anisotropy coefficient. Then, the magnetization direction of each biaxial elemental operator is determined from the various energy contributions of the material anisotropy and applied field. Similar to the determination method of Stoner-Wohlfarth (S-W) particle, the stable orientation of the magnetization in the HTS structure can be obtained by minimizing the energy of the elemental operator. With this wind–rose curve, graphical interpretation can be given for the magnetization process of this biaxial elemental operator. In this Hx-Hy plane, four regions can be separated by the wind–rose curve, and different region corresponding to different number of the energy minima exists. The energy barrier of elemental operator under different magnetic field can decide the position of energy minima, which means the stable orientation of the magnetization, Besides, the determination of the actual energy minimum is related to the magnetization history. Based on the energy barrier diagrams of the elemental operator, the number of energy minimum and the stable orientation of the magnetization can be obtained. When the magnetic field applies inside the wind–rose curve, with the increase of magnetic field, four, three and two energy minima may exist corresponding to different magnetic field magnitudes. When the magnetic field applies outside the wind–rose curve, the magnetization only has one stable orientation as there is only one energy minimum. For the bulk magnetic hysteresis, a collection of biaxial elemental operators can be considered with different easy axis orientations and weights associated to each operator. To simulate the magnetic properties of different HTS materials by this model, the different distribution functions and parameters are needed. Each HTS has an optimal distribution function and parameters, which deliver the best results. With different parameters of the distribution function, this model can be employed to predict the magnetic characteristic of different HTS materials. III. Experimental verification The experiment was carried out on a Bi-2223 multi-filamentary tape with an external magnetic field parallel to a wide surface of the specimen and perpendicular to the long axis of filaments, and the magnetization versus the applied magnetic field, M−H, over the HTS sample at the temperature of 77.3 K was recorded, as shown in Fig.2. The magnetization curves measured and simulated by the presented method, respectively. The comparison illustrated that the elemental operator model that we have proposed agrees accurately with the experimental results for both the limiting hysteresis loop and major loops.

**References:**

[1] N. N. Duan, W. J. Xu, S. H. Wang, et al., Journal of Applied Physics, vol. 117, no. 17, p. 17A718( 2015). [2] N. N. Duan, W. J. Xu, S. H. Wang, et al., IEEE Transactions on Magnetics, vol. 51, no. 3, p. 7300904(2015). [3] W. J. Xu, N. N. Duan, S. H. Wang, et al., IEEE Transactions on Industrial Electronics, vol. 64, no.3, pp. 2459-2467(2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/41dabfe68862d3b320947bd99a3b67c8.jpg)
Different state of HTS under external magnetic field

![](https://assets.underline.io/uploads/markdown_image/1/image/4ebaada7745baf31ece68e99909b43c4.jpg)
Hysteresis loops of Bi2223 HT3-PA sample at 77K

 Magnetic Characteristic Analysis of High Temperature Superconductors by the Elemental Operator Model

1. Introduction Iron loss calculation is important for efficiency prediction of electric machines. To establish a general iron loss model which is applicable to a wide range of magnetization and frequency, the physical phenomena such as micro-eddy currents, pinning effect, etc. during the dynamic domain wall motion should be one option [1]. The authors have proposed a dynamic iron loss modeling method considering the pinning effect macroscopically and the calculation accuracy is confirmed to be improved [2]. However, in the electromagnetic field analysis, the static hysteresis phenomenon was not considered. In [3], a method considering the static hysteresis phenomenon in the electromagnetic field analysis by introducing the reactive magnetic field Hhys is proposed. In this paper, we applied this method and proposed a method to calculate Hhys simply by using the Fourier series expansion. The iron losses and the dynamic hysteresis loops under different frequencies are obtained and compared with the measured ones. 2. Analysis model and method The analysis model is one non-oriented silicon steel (JIS C 2550-1986: 50A470, thickness: 0.5mm). Only half region is analyzed due to symmetry. Sinusoidal flux are imposed by using the Dirichlet boundary conditions. The nonlinear eddy current analysis with A-Φ (A: magnetic vector potential and Φ: electrical scalar potential) method is carried out to obtain the flux distribution and eddy current distribution. The static hysteresis phenomenon is considered by using the reactive magnetic field Hhys expressed by a Fourier series. The B-H hysteresis loops due to the static hysteresis phenomenon and eddy currents are calculated using the obtained flux distribution. B of each time step is calculated by averaging all the flux density in the steel region and H is calculated by averaging all the flux density in the coating region then divided by the permeability in the vacuum. 3. Results and Discussions The obtained iron losses are shown in Fig. 1 and compared with the measured ones. It shows that the iron loss calculation accuracy is much improved by considering the static hysteresis phenomenon. The obtained dynamic hysteresis loops under different frequencies are shown in Fig. 2 and compared with the measured ones. It shows that by considering the static hysteresis phenomenon in the electromagnetic field analysis, the accuracy of dynamic hysteresis loops are much improved. More investigation will be given in the full paper.

**References:**

[1] 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. [2] W. Guan, H. Qian, F. Fang, Y. Luo, J. Yang, B. Chen, Y. Gao, and K. Muramatsu, “Numerical modeling of excess loss in SiFe sheet considering pinning effect,” IEEE Trans. on Applied Superconductivity, vol. 26, no. 7, Art. no. 0608804, 2016. [3] K. Yamazaki and Y. Sakamoto, “Electromagnetic field analysis considering reaction field caused by eddy currents and hysteresis phenomenon in laminated cores,” IEEE Trans. on Magn., vol. 54, no. 3, Art. no. 1300204, 2018.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8fc2099b05a6fb55c4adc9d809686c93.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/689cdb0f0ae1f69e2fd9431c96c48d80.jpg)

 Iron loss calculation of non-oriented silicon steel considering hysteresis using Fourier series expansion

Magnetic films with perpendicular magnetic anisotropy (PMA) are an important subject of research, as they are used in nonvolatile memory devices and sensors. Moreover, they are also interesting as ferromagnetic systems with nontrivial configurations of magnetization. This is due to the fact that in PMA materials the influence of magnetocrystalline anisotropy and magnetostatic self-interaction are opposite to each other. Being projected onto the film plane, the emerging magnetization distribution can look like a stripe or bubble domain state, or as a labyrinth domain structure [1]. Along with this, the magnetization near the film boundaries differs from the magnetization inside the film, and the domain walls (DWs) have a vortex structure [2]. As shown in [3, 4], this system may contain a large number of vertical and horizontal Bloch lines (VBLs and HBLs), as well as Bloch points (BPs). In this work, we numerically investigate the process of magnetization reversal in a Co(001) PMA film containing an antidot array [5] (a system of columnar defects). In our case, these defects are square holes. We show that these holes can lead to appearance of stable HBLs located on the DWs and pinned to adjacent holes. The formation of HBLs and VBLs is accompanied by the transfer of the topological charge. We numerically calculated the distribution of magnetization in a 180° DW located between two holes. For this, we have minimized the total magnetic energy in a cuboid with dimensions X = 150 nm, Y = 200 nm, Z = 800 nm, using the mumax3 [6] solver (Y is the film thickness). Periodic boundary conditions were imposed along the z axis. To visualize HBLs and VBLs, we numerically calculated the density of the gyrotropic vector [3]. The topological charges of BPs were calculated by integrating the gyrotropic vector over closed surfaces. The following magnetization distributions were found: (a) two VBLs without BPs, (b) the HBL connecting holes and containing a BP, (c) the HBL connecting holes and not containing a BP. The domain structure in a cuboid with dimensions X = 1024 nm, Y = 1024 nm and Z = 200 nm, containing an antidot array was also simulated (Z is the film thickness). The period of the rectangular array was chosen to be 100 nm or 200 nm. Hysteresis loops were calculated by minimizing the total energy including the Zeeman energy. The magnetic field step was chosen to be 250 Oe. The initial condition for each step was chosen as the result of the previous energy minimization. The simulation results are shown in Figs. 1, 2. The static domain structure in Fig. 1 was obtained for the period of the array 100 nm. We can see that the complex configuration of magnetization within the antidot array entails a nontrivial distribution of HBLs and BPs. In Fig. 2 the hysteresis loops are shown for the case of continuous film and for antidot arrays with periods 100 nm and 200 nm (the magnetic field is perpendicular to the film surface).

**References:**

[1] L. Fallarino, O. Hovorka and A. Berger. Phys. Rev. B, Vol. 94, 064408 (2016). [2] M. Kisielewski, A. Maziewski, V. Zablotskii. JMMM Vol. 316, p. 277 (2007). [3] E. Zh. Baykenov, I. M. Izmozherov, V. V. Zverev. Bull. Russ. Acad. Sci.: Phys. Vol. 83, p. 809 (2019). [4] I. M. Izmozherov, E. Zh. Baykenov, V. V. Zverev. Phys. Met. Metallogr. Vol. 122, No. 3 (2021) (in print). [5] T. Schneider, M. Langer, J. Alekhina, et al. Sci. Rep. Vol. 7, 41157 (2017). [6] A. Vansteenkiste, J. Leliaert, M. Dvornik, et al. AIP Advances. Vol. 4, 107133 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7ebecceb44c66a9aacc2926578898de6.jpg)
Fig.1. Domain structure in an antidot Co film. The HBLs are marked.

![](https://assets.underline.io/uploads/markdown_image/1/image/58c058a74cb35089c9aba5435b426f7d.jpg)
Fig.2. Hysteresis loops for the case of continuous film and for antidot arrays with periods 100 nm and 200 nm.

 3D Simulations of Domain Wall Structure and Hysteresis in Cobalt Films with Columnar Defects

There is electromagnetic interference in the DC power supply system of metro, which will produce electrochemical corrosion to the adjacent metal structure of rail. Buried metal pipeline is one of the main objects of electrochemical corrosion. Due to the characteristics of buried pipeline laying and the randomness of electromagnetic leakage, it is difficult to directly and effectively analyze the corrosion law of buried pipeline under electromagnetic interference. In addition, due to the lack of corrosion detection, it is easy to cause metro operation accidents. 1.Introduction In recent years, many scholars have indirectly evaluated the electromagnetic interference degree and electrochemical corrosion of buried metal by the pipe-to-soil voltage [1-2]. Due to the characteristics of buried pipeline laying and the randomness of electromagnetic distribution, it is difficult to directly and effectively analyze the corrosion law. Some scholars have tried to investigate the electromagnetic interference of buried pipelines from the perspective of electrochemical corrosion mechanism [3-4]. However, the existing research is mainly focused on the static corrosion problem, which is lack of considering the operation of metro vehicles, so there is no direct basis for quantitative analysis of the corrosion area of buried pipelines. In this paper, the model and verification platform for dynamic electromagnetic interference are established, and the impact of metro vehicle operation state on corrosion area is analyzed quantitatively, as well as the cumulative effect of electrochemical corrosion under weak electromagnetic interference. 2.Simulation experiment This research set up an experimental platform for simulating stray current corrosion under the motion of subway vehicles as shown in Fig.1. Based on the corrosion experiment and simulation of buried pipeline, the dynamic corrosion process and law of buried pipeline metal are studied under the action of Variable electromagnetic interference. The graphite electrode (positive) indicates a moving metro vehicle, and its movement is realized by a slider block controlled by a stepper motor. The movement process of the slider block simulates the actual movement of the subway vehicle. The programmable DC power supply provides dynamic current that leaks during the movement of the graphite electrode (positive). The leakage current of the positive graphite electrode forms an electrochemical corrosion environment in the simulated soil solution, and finally flows back through two graphite electrodes (negative), and the buried pipeline specimen was placed in a simulated soil solution. The above process truly simulates the double side power supply system of metro. The finite element simulation model was used to simulate the entire process of buried pipeline corrosion under the dynamic electromagnetic interference. Finally, the experimental results and simulation results were compared and analyzed. 3.Results When the graphite electrode (positive electrode) moves to the 40th second, the corrosion current density distribution of buried pipeline is shown in Figure 2 (a). When the corrosion current value is negative, it indicates the current density of the outgoing pipeline, and when the corrosion current value is positive, it indicates the current density of the incoming pipeline. Therefore, the corresponding anode area and cathode area are divided on the buried pipeline. The metal of buried pipeline in anode area is corroded by electronic loss caused by corrosion current. The pipeline metal in anode area is corroded by the electronic loss caused by corrosion current. The closer the anode area of the buried pipeline is, the greater the corrosion current density of the outgoing pipeline is. Obviously, the metal surface of both ends of the buried pipeline is seriously corroded. With the frequent start and stop of metro vehicles, the anode area of buried pipeline is constantly changing. 4.Conclusions The results show that under the dynamic electromagnetic interference, the metal of buried pipeline is bipolar, and the anode area (corrosion area) changes with the running state of metro vehicles. Therefore, in the case of no excavation detection of buried pipelines, the corrosion area caused by electromagnetic interference can be quantitatively identified by analyzing the operation of metro vehicles. And the cumulative effect of stray current cannot be neglect under the condition of uniform speed of metro vehicles.

**References:**

1.Zakowski, W. Sokólski, Corrosion Science., Vol.41, p.2213(1999). 2.S. R.Allahkaram, M. Isakhani-Zakaria, Journal of Natural Gas Science and Engineering.,Vol.26, p.453(2015). 3.K. Tang, Materials Today Communications., Vol.20, p.100564(2019). 4.S. Qian, Y. F. Cheng, Construction and Building Materials.,Vol.148, p.675(2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/90129ab39a06aa2fb7896aeea1e13316.jpg)
Fig.1 Corrosion experiment device for buried pipelines under stray current

![](https://assets.underline.io/uploads/markdown_image/1/image/a8c5a08f26844c87bed15fc0555feda5.jpg)
Fig.2 Comparative analysis of stray current corrosion of buried pipelines under electromagnetic interference

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