United States

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

MMM 2022

Electronic Heat Bath Simulations for Ultra fast Spin Dynamics

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

poster

#### Welcome to the 67th Annual Conference on Magnetism and Magnetic Materials, MMM 2022!

The Conference was held from October 31 to November 4 and offers both in-person and on-demand content.

MMM 2022 is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a tutorial, and several special sessions. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

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Ferrimagnets (FiMs) persist in ultra-fast dynamics similar to antiferromagnets but are as easy to be electrically manipulated as ferromagnets, leading to a promising approach for achieving ultrafast devices with feasible tunability. Furthermore, spin-chain is of fundamental importance in condensed-matter physics, revealing unique properties such as quantum phase transition, edge effect, etc.[1] Although the dynamics of FiM have been widely described by a two sublattices macro-spin model[2], it hardly captures the sophisticated physical properties in multi-spin systems such as spin chains. Hence, we developed an atomistic spin model to investigate the spin dynamics in the FiM spin chain deployed by current-induced spin-transfer torque(STT). Our simulation results showed that in the finite FiM spin chain, first, the oscillation started from opposite-spin spatial region, defined as "oscillation core (OC)". Next, the oscillation spatial region would expand and finally only some specific spins formed stable oscillation (Fig.1). To further understand this phenomenon, we first introduce one OC into the spin chain. When excited by STT, the exchange mode can be further classified into non-flipped exchange mode (region I), critical exchange mode (region II), and flipped exchange mode (region III). In region I and III, the frequency increased linearly with different slops as the current density increased, yet in region II, the frequency slightly decreased, indicating the competition among exchange field (Hex), uniaxial anisotropy, and STT (Fig.2a). Next, when two OCs interacted, we find that smaller distance gives lager frequency, which is attributed to different Hex. Hence, proper choice of distance allows us to tune the Hex, achieving large-angle precession with THz frequency(Fig.2b). In summary, our work offers a novel understanding of FiM oscillation and proposes a strategy to construct energy-efficient spintronic oscillators with THz frequency. (This work at the National University of Singapore is supported by MOE-2019-T2-2-215 and FRC-A-8000194-01-00) 

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

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

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

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

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

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

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

Co Layer Resolved Measurement of Dzyaloshinskii

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

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

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

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

Torsion and curvature induced effects in magnetic twisted nanotubes

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

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

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

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

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

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

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

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

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

Fig1. Clock-wise vortex probability. 

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

Study of temperature influences on the vortex chirality probability in asymmetric disk array

Magnetic vortex have attracted extensive attention in the field of information storage because of their topological spin structures with chiral bistable states[1,2]. If the vortex core polarity and vortex circulation sense can be precisely controlled simultaneously in a nanodisk, which will be more beneficial to realize the multi-bit ultrahigh density storage[3,4]. In this paper, a reliable control scheme for magnetic vortex chirality is proposed by optimizing the structure of Pac-Man-like nanodisk, as shown in Fig1(a)-(b). The results show that the polarity and circulation of the vortex can be precisely controlled simultaneously by simply changing the direction of the global magnetic field, and even the chiral states of the vortex can be determined by simply detecting the stray field distribution on the surface of the nanodisk, as shown in Fig1(c). The optimized Pac-Man-like nanodisk can provide a simplified experimental method for the control and detection of magnetic vortex chirality, which will be beneficial to the realization of multi-bit magnetic storage or magnetic logic technology in the future. 
**References** 
[1]H.-S. Han, S. Lee, D.-H. Jung, et al. Appl. Phys. Lett. 117(4), 042401(2020). 
[2]K. A. Zvezdin, E. G. Ekomasov. Phys. Met. Metallogr. 123(3), 201(2022). 
[3]J. A. Fernandez-Roldan, R. P. del Real, C. Bran, et al. Phys. Rev. B. 102(2), 024421 (2020). 
[4]D. Yu, J. Kang, J. Berakdar, C. Jia, NPG Asia Mater. 12(1), 1(2020). 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/40b03adf7fd13fe6cef5e9a8eb1d7001.png) 
Fig. 1. (a) and (b) show that the four chiral states of vortex can be precisely controlled. (c) It is shown that the vortex chirality can be determined by the distribution of the stray field on the
surface of the nanodisk.

Reliable control of magnetic vortex chirality in asymmetrically optimized magnetic nanodisk

Permanent magnet machines have many advantages such as high efficiency,high power density and so on,it have been widely used in the fields of aerospace,wind power generation and electric vehicles. However,permanent magnet machines have the disadvantage that the air-gap magnetic field is difficult to be adjusted.To effectively solve this problem,many scholars have proposed a hybrid excitation machine ,in which the electrically excited magnetic potential plays an auxiliary role in adjusting the air-gap magnetic field [1].The hybrid pole permanent magnet machine was proposed[2] ,which has a simple structure and high electric excitation efficiency.It has a permanent magnet and ferromagnetic pole structure on the rotor,but it has brush structure.A new hybrid excited consequent pole brushless Doubly Fed Machine (HE-CPBDFM) is proposed in this paper,and the performance of the machine is studied. Fig.1 gives the structure of the HE-CPBDFM.There are two sets of windings on the stator of the machine,a power winding with 3 pole pairs and a control winding with 1 pole pair;the number of permanent magnets on the rotor is the same as the number of poles of the control winding,and the polarity is the same,and the mechanical angle of the ferromagnetic pole with a smaller circumference between the two permanent magnets is 60°.The direction of the magnetic field synthesized by the control winding is facing the axis of the ferromagnetic pole,and the air gap magnetic field is adjusted by changing the magnitude of the excitation of the control winding. Keeping the external 220Ω load on the power winding side unchanged,the relationship curves between the power winding output voltage and the control winding excitation current under different excitation are shown in Fig.2. From Fig.2,it can be seen that the power winding output voltage increases with the increase of excitation current,and when the excitation current increases to 4A,the voltage growth rate slows down with the increase of machine saturation. 
**References** 
[1] Ullah S, McDonald S P, Martin R, et al. “A permanent magnet assist, segmented rotor, switched reluctance drive for fault tolerant aerospace applications,”. IEEE Trans. on
Ind. Appl., vol. 55, no. 1, pp. 298-305,Jan./Feb.2018. 
[2] Q. Zhang, S. R.Huang, G. D. Xie. “Design and experimental verification of hybrid excitation machine with isolated magnetic paths,”. IEEE Trans. Energy Conversion, vol. 25, no. 4,
pp. 993-1000, Dec.2010. 
[3]Ayub M, Jawad G, Kwon B. “Consequent-pole hybrid excitation brushless wound field synchronous machine with fractional slot concentrated winding,”. IEEE Trans. Magn., DOI:
10.1109/TMAG.2018.2890509,2019. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/49f53eebc5552b4be295d867b745c562.png) 
Fig.1.Machine structure diagram 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/575bb0a645f2838d41805acb58bb6cbc.png) 
Fig.2.The relationship between the output voltage of the power winding and the excitation current of the control winding under different excitations

Design and research of the New Hybrid excited consequent pole Brushless Doubly Fed Machine

Fractals are composed of self-similar structures across different length scales, which look similar under different magnifications. They possess a non-integer dimension with a self-similarity and scale invariance, which is commonly referred to as the Hausdorff dimension or fractal dimension. Romanesque cauliflower, fern, frost, and snowflakes are popular examples of naturally observed fractals. The concept of fractals namely, Sierpinski square and triangles are also implemented in magnetic systems to study the modification of spinwave dynamics [1, 2]. It is also important to study their magnetization reversal process for their further use in spintronics. Here we have investigated the magnetization reversal process of a Sierpinski triangle which is a nonperiodic but ordered fractal structure. To understand the magnetization reversal process of Sierpinski triangles with the increase of the iteration level of the fractal structure are studied using a micromagnetic simulator namely object oriented micromagnetic framework (OOMMF) [3]. The magnetic parameters used for the simulation are taken from Ref. [1]. The hysteresis loop (M-H curve) is calculated for a structure going from simple geometric structures toward a Sierpinski triangle. TRI-0 is an equilateral triangle with the side length of 3 μm. To generate TRI-1, an equilateral triangle is removed from the center of TRI-0. To further generate, TRI-2 and TRI-3 an iteration process with a scaling factor of ½ is used. Due to the structurization, the coercivity and saturation magnetization are increased significantly with the increase of the iteration number as shown in Fig. 1(a). The simulated magnetization images during the reversal for Tri-0 and Tri-4 are shown in Fig. 1 (b) and (c) for three different magnetic fields. The interesting thing to note is that magnetization reversal occurs by the creation of a vortex. The observations have a great impact on their applications in spintronic and magnonic devices. 
**References** 
1. J. Zhou, M. Zelent, Z. Luo, V. Scagnoli, M. Krawczyk, L. J. Heyderman, S. Saha, Phys. Rev. B105, 174415 (2022) 
2. C. Swoboda, M. Martens, and G. Meier, Phys. Rev. B 91, 064416 (2015). 
3. M. Donahue and D. G. Porter, “OOMMF User’s guide, Version 1.0,” NIST Interagency Report No. 6376 _National Institute of Standard and Technology, Gaithersburg, MD, 1999_,
URL: http://math.nist.gov/oommf. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/6a9590125a8d592aabcecf804a683feb.png) 

Fig. 1 (a) Simulated hysteresis loops for different samples. Simulated magnetization images at various applied magnetic field values marked by ‘1’, ‘2’ and ‘3’ for (b) Tri-0 (c) Tri-4 [a
single triangle is taken from the whole structure]

Magnetization reversal dynamics of a triangular shaped magnetic fractals

Analysis of magnetization reversal in ferromagnetic nanodots of different shapes with dimension in the range of 100 nm in the presence of Dzyaloshinskii-Moriya Interaction (DMI) is extremely important for technological application [1]. Though a few studies have explored the effect of DMI on the magnetization reversal mechanism in nanostructures, to disentangle the influence of shape and DMI requires further systematic studies [2-4]. Here, we investigate the effect of DMI (D) and in-plane bias field (H) on magnetization reversal in perpendicularly magnetized isolated square, circular, and triangular nanodot using the micromagnetic simulation Mumax3 [5]. The side length/diameter of the nanodot is chosen as 128 nm and the thickness as 1 nm and the material parameters of Co is selected. Our results indicate that the coercive field monotonically decreases as the strength of D increases and loop becomes asymmetric in the presence of both D and H in all three shapes. Interestingly, for D = 0 and H ≠  0, nucleation of the reverse domain occurs at the central region of the nanodot (Figs. 1(a-c)), and in case of D ≠  0 and H ≠  0 chiral nucleation occurs at the far edge (Figs. 1(d-f)). Due to lateral asymmetry of the triangular nanodot, the total energy value is smaller for the triangular nanodots (for -100 mT < H < +100 mT, and -1 mJ/m2 < D < + 1 mJ/m2) thereby making it preferred for energy efficient magnetization reversal as compared to other two shapes. However, at sufficiently higher value of the magnitude of H (> 200 mT) and D (> 1 mJ/m2), due to the laterally symmetric sides of the square the energy barrier for the magnetization reversal reduces drastically (cf. Fig.1(f)). These findings indicate that the effect of D and H in lowering the energy barrier for the magnetization reversal is intricately related to the symmetry associated with the shape of the nanodots. 

**References** [1] F. Hellman, A. Hoffmann, Y. Tserkovnyak et al., Rev. Mod. Phys., Vol.89, p.025006 (2017) 
[2] S. Pizzini, J. Vogel, S. Rohart et al., Phys. Rev. Lett., Vol.113, p.047203 (2014) 
[3] D.-S. Han, N.-H. Kim, J.-S. Kim et al., Nano Lett., Vol.16, p.4438 (2016) 
[4] Syamlal S K, Hari Prasanth Perumal, and Jaivardhan Sinha., Mater. Lett., Vol.303, p.130492 (2021) 
[5] A. Vansteenkiste, J. Leliaert, M. Dvornik et al., AIP Adv., Vol.4, p.107133 (2014) 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/55a7db279d3bf5b28b8f2b50d2d2ee8a.png) 
Figure 1: Snapshot of the spin configuration for square, circular and triangular nanodot during down to up (D-U) switching at H = -100 mT and D = 0 (a-c), H = -100 mT and D = 3 mJ/m2
(d-f). (g) Variation of total energy as a function of in-plane bias field for square, circular, and triangular nanodots at different value of D is plotted during D-U switching.

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