France



Research into geometrical frustration over the past two decades has revealed fundamentally new exotic behavior [1]. The magnetic interactions of the structure, coupled with the specific geometry causes frustration in the system, referred to as artificial spin ices. Several geometries exist, but for the purpose of this research, the kagome arrangement was used. In general, the geometry of these frustrated systems follows the ice rule, that being two interacting moments point in while two point out. The kagome however follows the modified ice rule of two moments pointing in while one points out or vice versa. These systems appear as 2D analogues to the 3D structure and it is the lattice configuration along with the specific frustration created that causes these exotic behaviors. Presented here are the magnetoresistances obtained for kagome magnetically frustrated systems in a single vertex and lattice format. Both geometries are compared in a magnetically connected format and a magnetically disconnected format in order to observe the exchange and dipolar energies. The structures were simulated with parameters for permalloy, using BORIS [2], and dimensions of 1µm long, 100nm wide and 10nm thick. The aim was to evaluate the magneto resistance, which was done assuming the current density aligns with the long axis and a basic resistor network.&lt;br/&gt;By comparing the simulated data for the connected and disconnected single vertex with the measurements taken previously for a lattice, a greater understanding of the effect of the disconnected versus connected structures can be achieved.&lt;br/&gt;Measurements of the kagome lattice showed that the disconnected lattice had a change in magnetoresistance that was 2~3 times greater than the connected lattice, experimentally. The simulations of the single vertex’s magnetoresistance differ by approximately an order of magnitude to those of the experiments. While the simulations are not for the full lattice, they shed some light on the experimental results obtained and the single vertex interactions that occur within the lattice. The simulation of the hybrid vertex appears to mirror that of the experimental data obtained for the hybrid lattice while the simulation of the connected vertex appears to differ somewhat from the experimental data obtained for the connected lattice. This highlights the implications of the experimental work in that the connected kagome has more complex interactions throughout the lattice while the magnetically disconnected hybrid is rather a simplified system. The subtle difference in the geometries of the connected and disconnected lattices aids in the explanation of the effects and interplay of both the spin structures and the current density.&lt;br/&gt;This work demonstrates the potential for using simulations to expose magnetic interactions in a lattice type structure and other interesting geometries, along with the enhanced magnetoresistance signals that were obtained.

**References:**

References:&lt;br/&gt;[1] Nisoli, C., Moessner, R., &amp; Schiffer, P. (2013). Colloquium: Artificial spin ice: Designing and imaging magnetic frustration. &lt;i&gt;Reviews of Modern Physics&lt;/i&gt;, &lt;i&gt;85&lt;/i&gt;(4), 1473.&lt;br/&gt;[2] Lepadatu, Serban. (2019). Boris Computational Spintronics User Manual.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bc02a8200bfe4ba73c8d18a80397bf3d.jpg)
Figure 1. SEM images of an electrically connected but magnetically disconnected (hybrid) lattice (left). Also shown is the standard honeycomb lattice for comparison.

![](https://assets.underline.io/uploads/markdown_image/1/image/65475fb3f43153712262934a633adf6d.jpg)
Figure 2: Simulated single vertex and experimental lattice for the magnetically disconnected, electronically connected (hybrid) kagome plotted against the applied magnetic field. Data shown is the positive part of a full sweep of the magnetic field i.e. from a positive applied field through to the negative of the same applied field.

INTERMAG 2021

Micromagnetic simulations of artificial spin ice lattices and vertices

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2022 Joint MMM-INTERMAG

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Research into geometrical frustration over the past two decades has revealed fundamentally new exotic behavior [1]. The magnetic interactions of the structure, coupled with the specific geometry causes frustration in the system, referred to as artificial spin ices. Several geometries exist, but for the purpose of this research, the kagome arrangement was used. In general, the geometry of these frustrated systems follows the ice rule, that being two interacting moments point in while two point out. The kagome however follows the modified ice rule of two moments pointing in while one points out or vice versa. These systems appear as 2D analogues to the 3D structure and it is the lattice configuration along with the specific frustration created that causes these exotic behaviors. Presented here are the magnetoresistances obtained for kagome magnetically frustrated systems in a single vertex and lattice format. Both geometries are compared in a magnetically connected format and a magnetically disconnected format in order to observe the exchange and dipolar energies. The structures were simulated with parameters for permalloy, using BORIS [2], and dimensions of 1µm long, 100nm wide and 10nm thick. The aim was to evaluate the magneto resistance, which was done assuming the current density aligns with the long axis and a basic resistor network. By comparing the simulated data for the connected and disconnected single vertex with the measurements taken previously for a lattice, a greater understanding of the effect of the disconnected versus connected structures can be achieved. Measurements of the kagome lattice showed that the disconnected lattice had a change in magnetoresistance that was 2~3 times greater than the connected lattice, experimentally. The simulations of the single vertex’s magnetoresistance differ by approximately an order of magnitude to those of the experiments. While the simulations are not for the full lattice, they shed some light on the experimental results obtained and the single vertex interactions that occur within the lattice. The simulation of the hybrid vertex appears to mirror that of the experimental data obtained for the hybrid lattice while the simulation of the connected vertex appears to differ somewhat from the experimental data obtained for the connected lattice. This highlights the implications of the experimental work in that the connected kagome has more complex interactions throughout the lattice while the magnetically disconnected hybrid is rather a simplified system. The subtle difference in the geometries of the connected and disconnected lattices aids in the explanation of the effects and interplay of both the spin structures and the current density. This work demonstrates the potential for using simulations to expose magnetic interactions in a lattice type structure and other interesting geometries, along with the enhanced magnetoresistance signals that were obtained.

**References:**

References: [1] Nisoli, C., Moessner, R., & Schiffer, P. (2013). Colloquium: Artificial spin ice: Designing and imaging magnetic frustration. Reviews of Modern Physics, 85(4), 1473. [2] Lepadatu, Serban. (2019). Boris Computational Spintronics User Manual.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bc02a8200bfe4ba73c8d18a80397bf3d.jpg)
Figure 1. SEM images of an electrically connected but magnetically disconnected (hybrid) lattice (left). Also shown is the standard honeycomb lattice for comparison.

![](https://assets.underline.io/uploads/markdown_image/1/image/65475fb3f43153712262934a633adf6d.jpg)
Figure 2: Simulated single vertex and experimental lattice for the magnetically disconnected, electronically connected (hybrid) kagome plotted against the applied magnetic field. Data shown is the positive part of a full sweep of the magnetic field i.e. from a positive applied field through to the negative of the same applied field.

technical paper

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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)

Ferromagnetic hysteresis model using fractional derivative resolution developed for the simulation of viscoelastic phenomena.

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.

Dzyaloshinskii Domain Wall Creep INVITED

Despite numerous recent studies focusing on the dynamics of pinned chiral magnetic textures [1-6], their interactions with weak pinning disorder is far from being understood. Basic issues such as the length-scale and the strength of interaction between domain walls (DWs) and defects in magnetic materials remain open. Our works [7] evidence a strong correlation between controlled variations of DW width and the characteristic length of pinning. The DW-disorder interaction strength is also shown to depend on DW energy and width. These findings should be also relevant for a wide variety of elastic interfaces moving in weak pinning disordered media. We explore the magnetic field-driven motion of domain walls with different chiralities in thin ferromagnetic films made of Pt/Co/Pt, Au/Co/Pt, and Pt/Co/Au. The displacement of DWs was observed by polar Kerr microscopy. An in-plane magnetic field Hx was used to control the magnetization direction in the DW, as well as its width. A pulse out-of-plane field H was used to move DW. The measured velocity corresponds to the ratio between the measured DW displacement and the pulse duration. The velocity curves obtained for Pt/Co/Au with different values of Hx are reported in Fig. 1. Their analysis with the self-consistent description of the universal creep and depinning regimes [8] was used to extract parameters (depinning field Hd and temperature Td) characterizing the interaction between domain wall and weak random pinning disorder of the films. Surprisingly, the values of Hd and Td vary (not-shown) with the in-plane magnetic field Hx, which strongly suggests that the pinning properties of DWs depend on their magnetic texture, in contrast to the usual assumption found in the literature [1-6]. To predict accurately the variation of DW energy and structure with in-plane and DMI fields, we have calculated the orientation of the magnetization M (x) for a DW plane perpendicular to Hx, from numerical micromagnetic calculations (MuMax3), using the micromagnetic parameters of the three samples (not-shown). For the DW width, we both use the geometrical Hubert and dynamic Thiele definitions. Upon varying Hx away from the magnetic field H_DMI associated to the Dzyaloshinskii-Moriya interaction (DMI), the magnetization within the DW progressively switches from Bloch to Néel configuration, and the DW width increases due to the Zeeman contribution, as expected. The DW surface energy decreases as Hx departs fromH_DMI, in qualitative agreement with the predictions of simplified models [3-5]. In order to discuss the DW pinning at microscopic scale as a function of in-plane field Hx, we use standard scaling arguments [8] based on the variation of the energy produced by the deformation of a DW. The DW energy is assumed to be controlled by the gain of Zeeman energy due to magnetization reversal, the elastic energy produced by the increase of DW length, and the pinning energy due to the fluctuations of DW energy produced by the weak disorder. From scaling arguments, we obtain the relations which link the characteristic range and force of the interaction between DW and pinning defects, to the measured depinning field Hd(Hx), temperature Td(Hx), and the predicted DW surface energy. For a comparison (see Fig. 2) with the predicted variations of the DW width, the values of the pinning range were rescaled with a constant factor, which is the only adjustable parameter. As it can be observed, there is a strong correlation between the variations of the pinning range and the DW width. This strong correlation suggests that the variations of domain wall structure, controlled by an in-plane field, modify the characteristic lengthscale of pinning, whatever the DW chirality and the interaction strength between domain wall and pinning defects.

**References:**

[1] S.-G. Je, D.-H. Kim, et al., Phys. Rev. B 88, 214401 (2013). [2] R. Lavrijsen, D. M. F. Hartmann, et al., Phys. Rev. B 91, 104414 (2015). [3] D. Lau, V. Sundar, et al., Phys. Rev. B 94, 060401 (2016). [4] J. P. Pellegren, D. Lau, and V. Sokalski, Phys. Rev. Lett. 119, 027203 (2017). [5] D. Lau, J. P. Pellegren, et al., Phys. Rev. B 98, 184410 (2018). [6] D. M. F. Hartmann, R. A. Duine, et al. , Phys. Rev. B 100, 094417 (2019). [7] P. Géhanne, S. Rohart, A. Thiaville, and V. Jeudy, Phys. Rev. Res. 2, 043134 (2020). [8] V. Jeudy, R. Diaz Pardo, et al., Phys. Rev. B 98, 054406 (2018).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/4cb670d00ac41c65fa6b4dd07aff8bed.jpg)
Domain wall velocity versus out-of-plane magnetic field measured for a Pt/Co/Au film and different values of the in-plane field. The solid and dash lines are predictions for the creep and depinning regimes. The stars correspond to the coordinates of the depinning thresholds Hd(Hx).

![](https://assets.underline.io/uploads/markdown_image/1/image/1ff28aae80963ce5024f1005ff81dcb7.jpg)
Domain wall widths (Hubert: solid line) and (Thiele: dash line), and correlation length of the pinning versus in-plane magnetic field for the three films. For each film, the correlation length was rescaled to highlight its strong correlation with the DW widths.

Interaction of chiral domain walls with pinning disorder in thin ferromagnetic films

I will present an overview of our recent results on modelling field-driven dynamics of domain walls in ferromagnetic thin films and strips with perpendicular magnetic anisotropy, focusing on the effect of internal degrees of freedom on the domain wall dynamics. In the geometries considered, the dynamics associated with the internal degrees of freedom are given by that of the magnetisation at the domain wall mid-point, and is manifested as repeated nucleation, propagation and annihilation of topological defects known as Bloch lines within the moving domain wall. Our aim is to understand the effect of such internal dynamics on the large-scale dynamics of domain walls. First, I will discuss micromagnetic simulations of magnetic field-driven domain wall dynamics in thin ferromagnetic strips with perpendicular magnetic anisotropy which demonstrate that in wide enough strips the Walker breakdown - the sudden drop of domain wall velocity as the field is increased above the Walker field - is a multistep process: It consists of several distinct velocity drops separated by short linear parts of the velocity vs field curve. These features originate from the repeated nucleation, propagation, and annihilation of an increasing number of Bloch lines within the domain wall as the driving field magnitude is increased [1]. Second, I will consider micromagnetic simulations of Barkhausen noise in disordered Pt/Co/Pt thin films due to precessional motion of domain walls. In this regime the domain walls contain Bloch lines which repeatedly nucleate, propagate, and annihilate within the domain wall during the Barkhausen jumps. In addition to bursts of domain wall propagation, the in-plane Bloch line dynamics within the domain wall exhibits crackling noise and constitutes the majority of the overall spin rotation activity [2]. Finally, I will present our very recent work on developing a reduced model of line-like domain walls in thin films with perpendicular magnetic anisotropy including a description of internal degrees of freedom, i.e., Bloch lines and their dynamics. The model, which is computationally much more efficient than full micromagnetic simulations, is applied to study the depinning transition of field-driven domain walls in disordered ferromagnetic thin films [3].

**References:**

[1] J. Hütner, T. Herranen, and L. Laurson, Phys. Rev. B 99, 174427 (2019). [2] T. Herranen and L. Laurson, Phys. Rev. Lett. 122, 117205 (2019). [3] A. Skaugen and L. Laurson, in preparation (2020).

**Images:**

On dynamics of domain walls with internal degrees of freedom

With recent advancements in fabrication techniques, in particular through FEBID technology, three-dimensional (3D) nanomagnetism has emerged as a new field of interest [1]. Over the past decades, extensive studies on patterned two-dimensional thin-film elements have demonstrated that the properties of magnetic nanostructures are significantly affected by their shape, such that certain magnetic features can be generated and modified by changing the geometry. Following this principle, the access to an additional degree of freedom in 3D nanomagnetism can introduce new effects unknown in planar structures and may open up possibilities for new devices with artificially designed properties [2]. In this context, a promising category of novel nanomagnetic nano-architectures with remarkable potential for applications are 3D arrays of interconnected magnetic nanowires. The assembly of magnetic nanowires into an interconnected array can lead to interactions and magnetic structures with features known from artificial spin ice (ASI) lattices. A particular micromagnetic situation occurs at the intersections of the wires, where the magnetization must adapt to a change of both the geometry and the magnetization direction. At the vertices, characteristic magnetic configurations and phenomena of magnetic frustration unfold. Here we show that these frustrated states can strongly impact the overall properties of a bucky-ball type geometry, which can be considered a prototype of a 3D ASI. Using finite-element micromagnetic simulations, we investigate the hysteretic behavior, the static magnetization structure, and the high-frequency properties of the magnetization in artificial buckyball-type nanostructures. The buckyball geometry [3, 4] acts as a model system to study the transition from two-dimensional to three-dimensional ASI structures. It has, on one hand, important similarities with a two-dimensional Kagomé lattice, while, on the other hand, it also incorporates distinct three-dimensional properties resulting from the spatial arrangement of the nanowires and the spherical shape of the network that they form. The geometry consists of 90 nanowires connected at 60 vertices, resulting in a closed network of hexagons and pentagons. Our model system is made of soft-magnetic Co, and its size is typically in the micron range. More specifically, we have investigated various buckyball-type geometries with individual nanowire lengths ranging between 25 nm and 250 nm. At zero field, the static magnetization structure of the magnetic bucky-balls is characterized by their configurations at the vertices. Due to the prevailing impact of shape anisotropy, each nanowire is magnetized along its axis. Based on the orientation of the magnetization in the three nanowires meeting at a vertex, the magnetic structures at the vertices can be classified into four types: two-out / one-in, two-in / one-out, three-out, and three-in. A specific magnetic charge, +1,-1,+3 and -3, can be assigned to each of these configurations, as is done in Kagomé lattices [5]. Neglecting the sign of the charges, one can distinguish between ice-rule obeying single-charge and defect-like triple-charge configurations. By means of hysteresis loop simulations, we find that triple-charged vertices can be created and removed through minor loops: If the field is reduced to zero from saturation, two triple-charge vertex configurations form at diametrically opposite sites. These triple charge defects dissolve after applying a moderate negative field, and they do not reappear if the field is subsequently removed again. Hence, we show that in these 3D ASI geometries, it is possible to inject and to remove magnetic triple-charge defect structures in a controlled way by varying the external magnetic field within a specific range. Going further in the study, we analyze the fundamental high-frequency modes of the magnetization in these geometries and their dependence on the magnetic structure. To extract the modes, the equilibrium states at zero field are excited by a picosecond field pulse, leading to a magnetic ringing over several nanoseconds. From these high-frequency dynamics, the magnetic modes are derived by using spatially-resolved windowed Fourier analysis [6]. We observe a distinct correlation between the distribution of magnetic charges at the vertices and the frequency of the modes. The modes can be distinguished as oscillations that are localized either at the vertices or within the nanowires. In particular, we find that the dynamics of the triple-charge states is very different from the dynamics at the single-charge vertices. The triple-charge states generate a strong, often dominating signal at low frequencies, which is absent in configurations without triple-charge defects. Moreover, we show that the frequency of this mode can be varied in an almost linear fashion by an external magnetic field. The combination of this feature with the particular hysteretic properties of the system opens a promising perspective to develop tailored devices with desired high-frequency properties that can be manipulated by varying the type of magnetic frustrations.

**References:**

[1] A. Fernández-Pacheco et al., Nat Commun 8, 15756 (2017) [2] P. Fischer et al., APL Materials 8, 010701 (2020) [3] L. Keller and M. Huth, Beilstein J. Nanotechnol. 9, 2581 (2018) [4] C. Donnelly et al., Phys. Rev. Lett. 114, 115501 (2015) [5] F. Montaigne et al., Sci. Rep. 4 : 5702 (2014) [6] M. Yan, R. Hertel, C. M. Schneider, Phys. Rev. B 76, 094407 (2007)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d48aacc7c01215a7aa016921fa518153.jpg)
Fig.1: Finite-element discretization of a 3D buckyball-type nanostructure. Depending on the buckyball size, we use up to 1.9 million elements.

![](https://assets.underline.io/uploads/markdown_image/1/image/7c6422b1b28117f73a2f02635cba7c87.jpg)
Fig. 2: a) Different zero-field configurations can be accessed through hysteresis loops and minor loops, in particular the triple-charge state and the single-charge state. b) Simulated Fourier spectrum of the magnetic modes of the triple-charge and single-charge state.

Micromagnetics of Frustrated States and High-frequency Modes in Artificial Buckyball Nanostructures

The manipulation of magnetic textures in ferromagnets (FM) has been the focus of intense research in recent years. More recently, this focus has been displaced towards the use of antiferromagnets (AFM), and ferrimagnets (FiM), which are more robust against external perturbation and can sustain faster magnetization dynamics. In this context, simplified models have been proved very useful to describe and clarify the phenomenology on FM [1], AFM [2], and FiM [3] materials giving in some cases also good quantitative predictions. Nevertheless, AFM and FiM are described by two sublattices strongly couple through an exchange interaction while FM are described by only one magnetic lattice. In order to recover the FM case from the FiM description, one needs not only to set the second sublattice magnetization to zero but also its corresponding energy densities. It is possible to describe all these cases in a unified framework by a simple renormalization of the energy densities to the squared of the sublattices magnetization modulus. To give an example, considering Ku1 and Ku2 the common uniaxial anisotropy parameters, we substitute them by ku1*(MS1)2 and ku2*(MS2)2 respectively. This procedure is followed for all the energy density terms and included in Petaspin micromagnetic solver [4]. Here, we focus on the statics and dynamics of domain walls (DWs) when excited by spin-orbit torque (SOT). Regarding the static states of AFM, we show that as in FM, AFM also may exhibit periodic patterns due to the interfacial Dzyaloshinskii-Moriya interaction (iDMI) [5]. In this case, the ‘effective’ inhomogeneous exchange accounting for the two intralattice inhomogeneous exchange contributions and the one interlattice inhomogeneous exchange plays the role of the exchange interaction in FM. Correspondingly, the ‘effective’ iDMI to consider is the sum of the iDMI parameter of the two sublattices. Additionally, we show that similarly to FM, iDMI promotes the nucleation of new domains at the edges of and AFM strip even for moderate current densities [6]. Differently from FM, in AFM this effect is more efficient due to the robustness of the Néel DW against SOT excitation dynamics. This effect is one of our main results since it could limit the maximum DW velocity in and AFM racetrack memory because the nucleation of new domains would change the stored information, thus limiting the maximum current density that can be applied. We have also compared full micromagnetic simulations with a collective coordinates (CC) model which also includes the DW width changes due to the SOT and the spatial tilting of the domain wall [7] for different saturation magnetization ratios. The CC model properly predicts the domain wall width at equilibrium. On the other hand, when the magnetic texture is excited by a current through the SOT we can find three scenarios: (i) when the exchange and anisotropy parameters are larger enough, the DW dynamics can be described by the CC model even if we neglect the DW tilting and width variation. When we neglect these effects we talk about the ‘rigid CC model’. (ii) The cases where the exchange is lower but the anisotropy is large enough cannot be properly described by the rigid CC model. In these cases, the effect of the tilting and the DW width reduced the stationary DW velocity and, thus, they should be taken into account to describe the dynamics. This is shown in Fig. 1 where the solid linear line corresponding to the rigid CC model cannot predict the stationary DW velocities from the full micromagnetic simulations. Nonetheless, the dashed line corresponding to the CC model follows the saturation of the velocity due to the DW tilting. (iii) For low anisotropy parameters the excitation of self-oscillation threshold can be found for moderate current densities (red line in Fig. 2). We observe that before this threshold, where the Gilbert damping is almost compensated by the antidamping term induced by the SOT, the DW velocity increases faster than the usual linear trend being able to reach much larger velocities. Neither the rigid CC model nor the CC model can describe this effect and these cases required a full micromagnetic approach. From a fundamental point of view, this super-linear trend is another of our main results. The threshold current for self-oscillations depends on the anisotropy [8] which can be tuned, opening a path to exploit the super-linear DW mobility under moderate current densities.

**References:**

[1] S. Emori, U. Bauer, S.-M. Ahn, et.al., Nat. Mater., Vol. 12, 611 (2013). [2] H. V Gomonay and V. M. Loktev, Phys. Rev. B, Vol. 81, 144427 (2010). [3] L. Sánchez-Tejerina, et. al., IEEE Magn. Lett., Vol. 11, 2505005 (2020). [4] A. Giordano, G. Finocchio, L. Torres, et. al., J. Appl. Phys. , Vol. 111, 07D112 (2012). [5] R. Tomasello, et al., Phys. Rev B, Vol. 102, 224432 (2020) [6] L. Sánchez-Tejerina, et. al., Phys. Rev. B, Vol. 101, 14433 (2020). [7] E. Martinez, S. Emori, N. Perez, et. al., J. Appl. Phys., Vol. 115, 213909 (2014). [8] V. Puliafito, R. Khymyn, M. Carpentieri, et. al., Phys. Rev. B, Vol. 99, 024405 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/161fbbefb0726562a28ef3d0cc27a4c9.jpg)
Fig. 1 Domain wall velocity near the angular momentum compensation point as a function of the current density for a case with low exchange and large anisotropy. Dots stand for full micromagnetic simulations, solid lines for the rigid domain wall model, and the dashed line for the non-rigid domain wall model.

![](https://assets.underline.io/uploads/markdown_image/1/image/bc6cd85cb0155057350b7edfb0a26663.jpg)
Fig. 2 Domain wall velocity near the angular momentum compensation point as a function of the current density for a case with low exchange and low anisotropy. Dots stand for full micromagnetic simulations, solid lines for the rigid domain wall model, and the dashed line for the non-rigid domain wall model.

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