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Vortices are familiar phenomena in fluids, distinguishable for example in tornadoes, hurricanes, whirlpools and rings. In ferromagnets, vortices are naturally forming flux-closure states characterized by a curling of the magnetic moments around a very stable and narrow core [1]. The study of vortices is motivated by both fundamental and technological interest. Indeed, their topological properties endow vortices with rich and surprising dynamics, such as the ultrafast reversal of the vortex core [2], making them fascinating objects of study. These properties also establish them as promising candidates for applications in data storage and signal processing.&lt;br/&gt;&lt;br/&gt;Over the past decades, vortices have been extensively studied in laterally confined thin-film elements, where the magnetization is accessible with two-dimensional imaging methods. Recent developments have enabled non-destructive imaging of three-dimensional magnetic structures at the nanoscale. In particular, using hard X-ray tomography [3], it is now possible to directly image the internal structure within bulk micromagnets. For example, in GdCo&lt;sub&gt;2 &lt;/sub&gt;pillars, we could image the structure surrounding micromagnetic singularities, known as Bloch points, which were predicted over 50 years ago. Recently, we have uncovered three-dimensional structures forming closed vortex loops. The cross-section of these loops consists of a vortex–antivortex pair and, based on magnetic vorticity (a quantity analogous to hydrodynamic vorticity), we identified these configurations as magnetic vortex rings [4]. Such structures have been predicted to exist as transient states in exchange ferromagnets [5]; in contrast, the observed vortex rings exist as static configurations, and we attribute their stability to the dipolar interaction. In addition, we observe stable vortex loops intersected by point singularities at which the magnetization within the vortex and antivortex cores reverses. Such structures can exist because a non-zero divergence of the magnetization is allowed in ferromagnets; they have no analogue in incompressible fluids.&lt;br/&gt;&lt;br/&gt;Incidentally, it is known since the 1970s that ‘Bloch lines’ can assume complex structures, but these could only be inferred from surface observations combined with theory. Our results shed new light onto the rich physics of bulk magnets and open possibilities for further studies of complex three-dimensional solitons, enabling the development of applications based on three-dimensional magnetic structures. Thus, the title of this talk could as well be, ‘Magnetic Vortices: Back to the Third Dimension’.

2022 Joint MMM-INTERMAG

Magnetic Vortices: into the Third Dimension

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INTERMAG 2021

Vortices are familiar phenomena in fluids, distinguishable for example in tornadoes, hurricanes, whirlpools and rings. In ferromagnets, vortices are naturally forming flux-closure states characterized by a curling of the magnetic moments around a very stable and narrow core [1]. The study of vortices is motivated by both fundamental and technological interest. Indeed, their topological properties endow vortices with rich and surprising dynamics, such as the ultrafast reversal of the vortex core [2], making them fascinating objects of study. These properties also establish them as promising candidates for applications in data storage and signal processing. Over the past decades, vortices have been extensively studied in laterally confined thin-film elements, where the magnetization is accessible with two-dimensional imaging methods. Recent developments have enabled non-destructive imaging of three-dimensional magnetic structures at the nanoscale. In particular, using hard X-ray tomography [3], it is now possible to directly image the internal structure within bulk micromagnets. For example, in GdCo2 pillars, we could image the structure surrounding micromagnetic singularities, known as Bloch points, which were predicted over 50 years ago. Recently, we have uncovered three-dimensional structures forming closed vortex loops. The cross-section of these loops consists of a vortex–antivortex pair and, based on magnetic vorticity (a quantity analogous to hydrodynamic vorticity), we identified these configurations as magnetic vortex rings [4]. Such structures have been predicted to exist as transient states in exchange ferromagnets [5]; in contrast, the observed vortex rings exist as static configurations, and we attribute their stability to the dipolar interaction. In addition, we observe stable vortex loops intersected by point singularities at which the magnetization within the vortex and antivortex cores reverses. Such structures can exist because a non-zero divergence of the magnetization is allowed in ferromagnets; they have no analogue in incompressible fluids. Incidentally, it is known since the 1970s that ‘Bloch lines’ can assume complex structures, but these could only be inferred from surface observations combined with theory. Our results shed new light onto the rich physics of bulk magnets and open possibilities for further studies of complex three-dimensional solitons, enabling the development of applications based on three-dimensional magnetic structures. Thus, the title of this talk could as well be, ‘Magnetic Vortices: Back to the Third Dimension’.

technical paper

The 15th Joint MMM-INTERMAG Conference (2022 Joint) 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 plenary session, and an evening session, with about 1500 presentations overall. 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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Magnetic domain walls (DWs) in ferromagnetic nanostrips with perpendicular magnetic anisotropy have widely been a subject of research, not only due to its intrinsic fundamental interest but also to develop spintronic devices where information is coded in domains between walls [1,2]. To that purpose, ferrimagnetic materials (FiM) are recently attracting much attention due to their fast DW dynamics, almost zero interaction with external fields and a wide availability in nature. Controlling the current-driven DW along curved FiMs strips is essential to develop DW-based circuitry, but it faces problems such as DW distortion [3]. A proper design for the strip geometry may overcome these problems, as shape is intimately connected with DW motion via spin currents and interfacial phenomena in the structure. Studying the influence of temperature (or composition) is also key for developing real applications. Therefore, understanding the influence of shape and temperature to fully control the DW motion is nowadays a fundamental topic in spintronics research. In this work we perform a micromagnetic (μm) study on DW motion in ring-like FiM strips, characterising DW terminal velocities and inertial times, as a function of temperature, width, curvature of the strip (see insets in Fig. 1(a)) and density currents (J). Our full μm model treats spin dynamics in each sublattice separately, which yields results close to ‘realistic’ FiM. Results show that strips with low width and high curvature give high DW velocities (VDW &gt; 1000 m/s), with ultra-short response times (τ &lt;&lt;0.1 ns, see Fig.1 (b)). This is optimal for developing compact and high-speed tracks for domains since opposite DWs travel at lower but similar speeds, leading to negligible domain distortion. Our results will help to understand curvature effects to develop efficient and reliable DW-based devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/8036e1ee8c0a9ee895bd4c2baa9dc144.jpg)

Fig.1. (a) DW velocity as a function of J from sub-lattice i = 1 in a FiM strip of width w = 256 nm and different effective radius re (nm) at the angular momentum compensation temperature TA = 260 K. (b) Instantaneous DW velocities for three values of J and for a curved strip of re = 384 nm and w = 256 nm at T = TA. Relaxation times (τ) are extracted from the equation (solid curve) fitted to the data (symbols).

Current-Driven Domain Wall Motion in Curved Ferrimagnetic Strips Above and Below the Angular Momentum Compensation

Recent years have seen an increase of interest in the dynamics of ferrimagnetic materials, in particular in the motion of domain walls [1-4]. Ferrimagnetic materials represent unique systems where the ease of manipulating the spins with applied magnetic fields is combined with exchange-driven acceleration of the internal spin dynamics. Of particular interest is the temperature range around the magnetic and spin compensation points, finely balancing both magnetic moment and angular momentum of the system, and leading to a very particular character of magnetic switching by the domain wall motion. Here we present our studies of the temperature-dependent domain wall dynamics in the temperature range covering both angular momentum and magnetization compensation points in garnet film, and reaching up to the Curie temperature [5]. This is made possible by the very low coercive field of the material, not exceeding 5 mT even in the close vicinity of magnetization compensation temperature. The sample used in our experiments is a single-crystal thin film of magnetic garnet. A very unusual behavior of the domain-wall dynamics is observed, such as extraordinarily high mobility of the domain walls at very low fields. Drastic changes of both domain-wall velocity and mobility by more than 2 orders of magnitude are observed across a relatively small temperature range, related to the delicate balance of magnetization and spins of iron and rare-earth sublattices [5]. Therefore, such fine tuning of the corresponding momenta could be the key for developing ultrafast ferrimagnetic spintronic devices. The work was supported by the Russian Foundation for Basic Research (Projects No. 18-52-16006, 18-29-27020) and the Government of the Russian Federation (Project No. 075-15-2019-1874).

Domain wall dynamics across magnetic and spin compensation points in ferrimagnets

Topological spin structures, such as domain walls (DW) and skyrmions, have intriguing features and can play a key role in spintronics. In both ferromagnets (FM) and antiferromagnets (AFM) a DW, driven by applied torque, behaves like a solid particle, which can be described by collective variables: DW position and the angle of the magnetic order vector, albeit the dynamic properties are dramatically different. For an AFM the DW energy as a function of its momentum is the same as that of a standard relativistic particle (Fig. 1), because of the Lorentz invariance of the AFM dynamics [1]. For FMs the gyrotropic dynamic term couples two degrees of freedom that slows down a FM DW and results in the periodic dependence of the DW energy on momentum (Fig. 1). Thus, the AFM DW is stable up to higher speeds (~10km/s in contrast to ~100m/s for FMs). We demonstrate that the DW dynamics in ferrimagnetic (FiM) materials near the spin compensation point is entirely different from the FM and AFM cases [2]. The dispersion law for FiM DWs exhibits two crossed branches with opposite group velocities (Fig. 1), where each branch contains an endpoint [3]. Driven by a field-like torque, a FiM DW accelerates to a higher speed and momentum than its FM counterpart [4] and can reach the endpoint. Wherein it falls into a region of instability, where the internal dynamics of the DW is excited, and the collective variables approach fails. We show by micromagnetic simulations that the dynamical state of a DW in this case covers a wide frequency range, far beyond the magnon bandgap of a FiM. In this way the FiM DW driven by external field-like torque acts as an efficient source of the spin waves (SW), propagating out of the DW region. We show that by varying the applied torque and spin uncompensation of a FiM one can reach three qualitatively different regimes of the excited SWs: i) frequency-comb spectrum, shown in Fig.2, ii) chaotic behavior of the DW with the broad SW spectrum, iii) double-frequency SW excitation.

![](https://assets.underline.io/uploads/markdown_image/1/image/d58bb938c75083b271d7c47eff1fa103.jpg)


Fig.1 Dependence of the energy on momentum for a DW in FM, AFM (gray), and FiM near compensation point (blue) nano-strips.

![](https://assets.underline.io/uploads/markdown_image/1/image/60b60e08628ebcc51a436f9ca0f5c150.jpg)

Fig.2 Spectra of the dynamic state inside the DW (a) and magnons emitted outside (b).

Nonstationary domain wall dynamics in almost compensated ferrimagnets: a novel scenario

In recent years, current-induced domain wall motion (CIDWM) is attractive research field in spintronics because it’s expected to enable magnetic non-volatile memory. Our group focuses on Mn4N and its based nitrides as new candidates for CIDWM application. Especially, Mn4-xNixN nanowire recorded a DW velocity of 3,000 m/s only by spin-transfer torque [1]. We attributed this to the compensation of Mn4-xNixN around x=0.2 [2]. In spite of the excellent CIDWM, magneto-transport properties in Mn4-xNixN have scarcely been reported. In this work, we report both experimental and theoretical investigations of these properties with anisotropic magnetoresistance (AMR) measurement and calculation of PDOS with Vienna Ab initio Simulation Package (VASP). Figure 1 shows Fourier coefficients of AMR measurements in Mn4-xNixN on SrTiO3(001) substrates and energy state of d electrons under various crystal fields. C2 represents the coefficient of cos2θ component and C4 does that of cos4θ, where θ is the angle made by the current and the magnetization. Above 100K, small C2 components dominate AMR while C4 drastically became large below 100K. This represents the emergence of the tetragonal-like crystal field below 100K because it depends on the difference in PDOS between dxy and dyz, dzx states [3]. However, this trend was weakened by large Ni composition. Figure 2 shows PDOSs of d orbitals in Mn and Ni at corner site (I-site) in Mn4-xNixN (x=0, 0.5). By Ni substitution, spin polarization of d orbitals decreased at Fermi level (EF). This leads to decrease in both AMR ratio and C4 components, derived from the theory of s-d scattering [4]. In the previous study, the emergence of C4 components in Mn4N was attributable to the possible external effect such as lattice strain caused by the thermal expansion [3,4]. However, since all structures were set to be cubic in our calculation, we expect this is attributable to intrinsic change in PDOS, and dramatically affected by Ni doping. In the talk, we’re going to further investigate the modulation of Hall effect in Mn4-xNixN by Ni composiiton.

![](https://assets.underline.io/uploads/markdown_image/1/image/733d7a68846d668c0d9e1632ad6ca552.jpg)

Fig. 1 Fourier coeffcients of AMR in Mn4-xNixN and energy state of d orbitals.


![](https://assets.underline.io/uploads/markdown_image/1/image/df111606651ce41db44558dbc23cd7fa.jpg)


Fig. 2 PDOS of d orbitals of Mn at corner site in Mn4-xNixN (x=0, 0.5)

Experimental and theoretical approach to magneto-transport properties in Mn4-xNixN with high domain wall mobility

Spintronic devices are promising for neuromorphic computing due to their greater energy efficiency, high endurance compared to phase change memory and resistive random-access memory [1]. Popular spintronic memristors involve movement of a domain wall (DW) in a perpendicularly anisotropic (PMA) racetrack to achieve variable conductance to implement synaptic weights for deep neural networks (DNNs). However, energy efficient voltage controlled DW devices driven by spin orbit torque with Neel DW stabilized by Dzyaloshinskii-Moriya interaction show stochastic behavior and therefore provide limited states (see Fig.1 [2]). This is modelled rigorously with micromagnetics. To work with these DW devices and achieve high testing accuracy, the DNNs need to be trained with specific algorithms. In this study, we propose both in-situ and ex-situ training algorithms of DNN based on 2-, 3- and 5-state DW synaptic devices. For in-situ training, a mixed precision framework [3] is adopted where low precision synaptic conductances are tuned based on the quantized value of high precision gradients. With our quantized training, the learning is able to tolerate large conductance variation without any considerable accuracy loss as the synaptic state changes only few times during the entire training due to large inter-state interval. Thus, high programming noise intrinsic to DW devices is accommodated in the training which reduces the required number of programming pulses and the energy to train a DNN. The programming energy required for a DNN with 5-state DW synapse is found to be 1.25 pJ/image for MNIST digit classification with ~96.7% accuracy shown in Fig.2. Ex-situ training algorithm is also proposed where the DW devices are characterized based on statistics and then the devices characteristic is inputted to the quantized training algorithm as constraints which shows ~95% accuracy for 5-state synapses for large conductance variation. This work will pave the way to implement high accuracy intelligent future battery powered IoT edge devices using DW devices. Acknowledgement: NSF grant ECCS: #1954589

![](https://assets.underline.io/uploads/markdown_image/1/image/31364699a8a16e0f648f76850d498230.jpg)

Fig. 1: Stochastic nature of DW devices for voltage control of PMA (Ku) [2]

![](https://assets.underline.io/uploads/markdown_image/1/image/17758a182c49845d7959075dda8cb138.jpg)

Fig. 2: Testing accuracy comparison with proposed training

Neuromorphic Learning of a Deep Neural Network with Low Precision and Stochastic Domain Wall Based Memristor

Skyrmions hold great promise for next generation computation due to their current-driven control, ultra-low power consumption, non-volatile characteristics, and nanoscale footprint. Any technology candidate for in-memory computing will need to use a universal set of logic gates as its building blocks such as NAND or NOR. Despite previous efforts on all-skyrmion logic gates, the logic inverter that is essential for Boolean logic operations has not yet been addressed under realistic conditions on materials, current drive and device geometries1-4 with simple current drive and robust operation against geometric defects and Joule heating. Here, we demonstrate a domain wall-gated nanoscale skyrmion logic inverter (NOT) and NOR gates based on Co/Pt bilayer cross junctions. The inverter operation is based on a logic input (0/1: skyrmion absence/presence) applied first along the vertical channel. If there is a skyrmion logic input, this skyrmion is driven by current and converted to a domain wall (DW) to fill the vertical channel. Next, a probing skyrmion is applied5 horizontally as in Fig. 1. When there is a DW in the vertical channel, the probing pulse coming from the horizontal channel is blocked (output 0, input 1). With no skyrmion in the input, a DW does not form and the horizontal probing skyrmion reaches the gate output (output: 1). A final DC pulse along the vertical channel is applied to clear the inverter for an arbitrary input in the next clock cycle. Our proposed logic inverter has small footprint (4 × 240 × 300 nm3), room temperature operation with realistic current drive (0.4-1.2 × 1012 A m-2), robust operation under imperfect sidewall and notch geometries, and higher temperature effects. The inverter consumes about 1350 kB per operation at room temperature in addition to Joule heating. Joule heating must be suppressed by using insulating magnetic layers for minimal thermal drift, although we show that the circuit equilibrates at slightly above room temperature due to heat conduction to the substrate. We also present a non-volatile NOR gate that consists of two junctions, which is sufficient to achieve all Boolean logic functionalities.

![](https://assets.underline.io/uploads/markdown_image/1/image/ae2dc196547c4f92b71fe7b9eb85df51.jpg)

Domain Wall-Gated, Cascaded and Low-Power Skyrmion Logic Gates with Robust Operation

Domain Wall-Gated, Cascaded and Low-Power Skyrmion Logic Gates with Robust Operation

Despite their importance in a variety of technologies, the high energy cost of many forms of machine learning algorithms is a significant limitation[1]. One solution to this is to employ physical devices that can naturally mimic brain-like behaviour, known as neuromorphic computing [2]. Magnetic domain walls (DWs) are strong candidates for such devices, since they have potential for both data storage and logic operations [3] but stochastic effects limit the feasibility of creating viable devices. Here, we demonstrate how tunable stochastic pinning can be used to emulate synaptic properties for specialised neuromorphic devices. First we show measurements that demonstrate the feasibility of tuning stochastic processes by applying external stimuli. Focused magneto-optic Kerr effect measurements were used to probe the pinning of DWs at notch-shaped defects in 400 nm wide Permalloy nanowires. As shown in Fig 1, by modulating the propagation field (H) by an additional amount (dH) the pinning probability of the DW at the defect site was found to depend sigmoidally on dH and is observed on separate wires. Next, we model these wires as binary stochastic synapses in a neural network with properties based on the experimental measurements. By repeatedly sampling the network the stochastic total output can be converged to an analogue value to improve performance. It is found that for a single cycle of the network it can be trained to recognise handwritten digits with an error rate of 17%. Repeated sampling of the network decreased the error rate to 8% for 128 cycles, close to mean field performance, Fig. 2. We have also found that this performance is only slightly reduced when random individual wire properties are considered and additional neural layers mitigate stochasticity. Our work illustrates how the intrinsic stochasticity of DW devices can be harnessed to provide tunable bespoke hardware for machine learning tasks.

![](https://assets.underline.io/uploads/markdown_image/1/image/914460c7d76101d1d1aed055f61194c3.jpg)

Tuning of domain wall pinning probability in notched magnetic nanowire synapse. Inset shows a schematic of the notched nanowire and applied field direction.

![](https://assets.underline.io/uploads/markdown_image/1/image/d66b3c52afeebd3d19ca384f5af9ae3c.jpg)

Test error rate of classifying handwritten digits using a single layer stochastic network for varying repeated sampling.

Machine Learning with Stochastic Magnetic Domain Wall Synapses

Classifying or predicting complex time-dependent signals is a challenging computational task. Reservoir computing (RC) is an efficient neuromorphic computing approach that is ideally suited to such tasks, and is typically implemented in software using a recurrent neural network (RNN) with fixed synaptic weights (the reservoir) connected to a single, trainable readout layer. However, more efficient implementations of RC are possible if the software RNN reservoir is substituted with a physical system with the correct properties, such as non-linear response to input signals and inherent memory [1]. In this work, we exploit the chaotic and non-linear dynamics of a single magnetic domain wall (DW) trapped between two anti-notches in a Ni nanostrip as a hardware-based reservoir. We have modelled the structure (inset of Fig. 1) using both a simple 1D model and micromagnetic simulations with an oscillating field. As shown in Fig. 1, complex oscillatory dynamics over a range of field amplitudes are observed. A chaotic and multi-period regime occurs between 0.4 kA/m and 1.2 kA/m and a non-linear regime at higher fields. Thus, the DW oscillator can be used to transform an input magnetic field to the DW position output suitable for RC. A time-multiplexed input signal is injected into the DW oscillator by modulating the applied field magnitude, and we show how this approach allows the device to perform classification tasks as a reservoir.First, we demonstrate how the scaling to convert input data into applied fields affects the classification accuracy of the sine or square wave task, showing that the best recognition rate is obtained at the edge of a chaotic regime of oscillation (Fig.2). We also demonstrate that the DW oscillator can perform complex tasks, such as spoken digit recognition, and that coupling multiple DWs together enhances the computational capabilities of the reservoir. Our work opens new avenues for research into neuromorphic computing using nanomagnetic hardware.

![](https://assets.underline.io/uploads/markdown_image/1/image/b1c2f88ba8e32e54699a48e2fafd9388.jpg)

Bifurcation plot showing the DW position once per cycle for a given applied field using the 1D model. Inset: Schematic of the system.

![](https://assets.underline.io/uploads/markdown_image/1/image/7c99dd1e515fbc149d6687d21ac4a5b5.jpg)

Classification accuracy of the Sine-Square task over a range of field centre (H0) and field width (ΔH) values.

Neuromorphic computation with a single magnetic domain wall

Magnetic skyrmions can be transferred from it’s position to a desired location using spin torques [1] and there were proposals of skyrmion reacetrack memory and logic gate applications [2]. But, the instabilities and stochastic behavior of Skyrmions due to pinning [3], temperature and applied current, and skyrmion Hall angle (SkHA) [4] makes the realization of these applications challenging. Instead, stochastic skyrmion behaviour can be used to create new computing architectures, e.g., probabilistic and reservoir computing [5,6]. The major aspect of this computation is having a constant uncorrelated signal [5] that results from a reshuffle of the skyrmions. In this study, the stochastic property in a shuffling chamber will be investigated to create uncorrelated signals. The study will guide for the creation of a better probabilistic computing devices. We use the shuffling chamber (Fig. 1) and the current density for the geometry was generated in Mathematica and masking was used to implement it in micromagnetic simulator Mumax3 [7]. At first, the height of the paths were changed to 320, 360, 400 and 520 nm, this resulted in increase on the longevity of the final skyrmion path, and the skyrmion’s projectile motion decreased, accordingly. As the height of the rectangle increases, distance between the Skyrmion and the output borderline increases too. At first, the Skyrmion motion was observed under 0 K temperature, and without a grain. The skyrmions had a projectile and predictive motion. Then, the system was run using different temperatures, grain, and different currents to get a stochastic Skyrmion motion. In our simulations, the skyrmion under higher temperature was unstable at the output of the chamber, but by using higher anisotropy constant (Ku) at the output edge of the chamber, the instability was eliminated. The correlation between input and output paths were analyzed using Pearson correlation coefficient (PCC). We vary the temperature from 0 K to 100 K temperatures and find that the PCC changes from -0.055 to -0.04.

![](https://assets.underline.io/uploads/markdown_image/1/image/09252b3e4c415b8637e54d793b896946.jpg)

Fig. 1

![](https://assets.underline.io/uploads/markdown_image/1/image/186c33c8eb11dd907aa054cce316e78c.jpg)

Fig .2

Stochastic Property of Skyrmionic Shuffling Device

We introduce a new class of primitive building blocks for realizing quantum logic elements based on nanoscale magnetization textures called skyrmions. In a skyrmion qubit, information is stored in the quantum degree of helicity, and the logical states can be adjusted by electric and magnetic fields, offering a rich operation regime with high anharmonicity. By exploring a large parameter space, we propose two skyrmion qubit variants depending on their quantized state. We discuss appropriate microwave pulses required to generate single-qubit gates for quantum computing, and skyrmion multiqubit schemes for a scalable architecture with tailored couplings. Scalability, controllability by microwave fields, operation time scales, and readout by nonvolatile techniques converge to make the skyrmion qubit highly attractive as a logical element of a quantum processor.

![](https://assets.underline.io/uploads/markdown_image/1/image/9cf0d32a1ba0d161d137ecbbaa42774d.jpg)

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