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The presence of a Dzyaloshinskii-Moriya interaction (DMI) in perpendicularly magnetised thin films stabilises Néel type domain walls of fixed chirality, as opposed to the Bloch walls that are formed in the absence of a DMI [1]. Néel walls of fixed chirality have been shown to be driven efficiently in the same direction by electric currents through spin-orbit torques [2,3]. Later, a large tuning of the DMI with an applied voltage was demonstrated [4], which can be used e.g. for electric field control of magnetic domain wall motion via the modulation of the DMI [5].&lt;br/&gt;&lt;br/&gt;Here, we demonstrate an alternative mechanism for the control of domain wall type: using micromagnetic simulations and analytical modelling, we show that the presence of a uniaxial in-plane magnetic anisotropy can also lead to the formation of Néel walls in the absence of a DMI. It is possible to abruptly switch between Bloch and Néel walls via a small modulation of both the in-plane, but also the perpendicular magnetic anisotropy [6]. This opens up a route towards efficient electric field control of the domain wall type with tiny applied voltages, as the magnetic anisotropy can be modulated via the direct voltage controlled magnetic anisotropy (VCMA) mechanism [7], or via magnetoelastic anisotropy induced through coupling to a piezoelectric element [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/7bef5f5df8869fd862ccaab94b5a3626.jpg)

Fig. 1 In a perpendicularly magnetised thin film, domain walls (DWs) are generally of Bloch type (bottom left). The presence of a DMI (top left), stabilises Néel walls. Alternatively, and in-plane magnetic anisotropy can also form Néel walls (bottom right). An increase in the perpendicular magnetic anisotropy switches the DW type back to Bloch (top right).

2022 Joint MMM-INTERMAG

Switching between Magnetic Bloch and Néel Domain Walls with Anisotropy Modulations

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The presence of a Dzyaloshinskii-Moriya interaction (DMI) in perpendicularly magnetised thin films stabilises Néel type domain walls of fixed chirality, as opposed to the Bloch walls that are formed in the absence of a DMI [1]. Néel walls of fixed chirality have been shown to be driven efficiently in the same direction by electric currents through spin-orbit torques [2,3]. Later, a large tuning of the DMI with an applied voltage was demonstrated [4], which can be used e.g. for electric field control of magnetic domain wall motion via the modulation of the DMI [5]. Here, we demonstrate an alternative mechanism for the control of domain wall type: using micromagnetic simulations and analytical modelling, we show that the presence of a uniaxial in-plane magnetic anisotropy can also lead to the formation of Néel walls in the absence of a DMI. It is possible to abruptly switch between Bloch and Néel walls via a small modulation of both the in-plane, but also the perpendicular magnetic anisotropy [6]. This opens up a route towards efficient electric field control of the domain wall type with tiny applied voltages, as the magnetic anisotropy can be modulated via the direct voltage controlled magnetic anisotropy (VCMA) mechanism [7], or via magnetoelastic anisotropy induced through coupling to a piezoelectric element [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/7bef5f5df8869fd862ccaab94b5a3626.jpg)

Fig. 1 In a perpendicularly magnetised thin film, domain walls (DWs) are generally of Bloch type (bottom left). The presence of a DMI (top left), stabilises Néel walls. Alternatively, and in-plane magnetic anisotropy can also form Néel walls (bottom right). An increase in the perpendicular magnetic anisotropy switches the DW type back to Bloch (top right).

technical paper

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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)

Skyrmion Qubits: A New Class of Quantum Logic Elements Based on Nanoscale Magnetization

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’.

Magnetic Vortices: into the Third Dimension

Much effort has focused on achieving micron-scale spin propagation using diffusive spin waves, but their exponential decay with distance makes efficient spin transport challenging. An alternative form of magnetization dynamics – commonly referred to as spin superfluidity [1] or exchange mediated spin transport [2,3] – may enable more efficient transport owing to its topologically-protected easy-plane precession of the magnetization and associated linear decay of spin flow. Our work addresses the open question of how defects impact this easy-plane spin transport at low drive current densities. We performed micromagnetic simulations on easy-plane precessional dynamics in synthetic antiferromagnet nanostrips with and without notch defects. The material parameters were taken from our experimentally measured, low-damping FeV alloys [4]. We view the dynamics as a train of interacting, homochiral domain walls (DWs) rather than as hydrodynamic flow [2]. We observe the transition from (1) isolated, diffusively propagating DWs, which are susceptible to pinning, to (2) a fluid-like flow of a closely packed DW train that is unimpeded by pinning defects. This transition is quantified by computing the superfluid velocity ▽Φ, where Φ is the in-plane azimuthal angle of the magnetization, at a position beyond the notch defect (see Fig. 1). In Fig. 2, ▽Φ differs between the perfect and notched nanostrips as a result of pinning of the DWs at low drive current densities, Jc &lt; 5 x 1011 A/m2. As Jc increases, ▽Φ becomes identical between the two geometries, indicating that the DWs have effectively condensed to form a fluid flowing unaffected by the notch defect. Our findings are promising for future realization of superfluid-like spin transport via easy-plane precession in ferromagnetic metal devices, which are compatible with established fabrication techniques [5].

![](https://assets.underline.io/uploads/markdown_image/1/image/b29ae1464b62923fd11997a38ae0428e.jpg)

Figure 1: Micromagnetic snapshots of the perfect and notched nanostrips at Jc = 4 x 1011 A/m2, labeled with location of notch defect at the center and where ▽Φ is calculated.


![](https://assets.underline.io/uploads/markdown_image/1/image/329eb8448836511a5d16e9a2b0225809.jpg)

Figure 2: Superfluid velocity ▽Φ as a function of charge current density Jc for the perfect (black squares) and notched (red circle) nanostrips computed at a fixed position beyond the notch defects.

Diffusive and Fluid-like Motion of Homochiral Domain Walls in Easy-Plane Magnetic Strips

As three-dimensional nanomagnetism evolves, novel non-trivial magnetic textures emerge as appealing information carriers for recording and spintronics based on curved nanosystems, and particularly on Cylindrical Nanowires (NWs) [1,2]. One of the most fascinating candidates that is likely to reach the high velocities required for fast recording technologies is the Bloch Point (BP) domain wall (DW). Recently, theoretical evidence indicated that BPs in NWs could reach high velocities close to 2 km/s in the magnonic regime [3]. While the observation of the BP DW in cylindrical NWs is no longer recent [2], scarce numerical studies that combine both current and Oersted field have been published in NWs [4], despite first attempts to measure DW velocities are in progress [5]. In this work we evaluate the dynamics of the BP DW under both current directions in a Ni NW with 100 nm in diameter. We investigate two cases: i) pre-nucleated BP Wall, and ii) the BP DW originated from the transformation of a Vortex-Antivortex DW. The effects of both spin-polarized current and Oersted field are considered. We discuss in detail the role of the chirality of the BP in relation to the Oersted field, also reported previously in precursors of BPs [4]. We show that while the pre-nucleated DW with the same chirality as that of the Oersted field propagates always against the current direction, the BP originated either from the transformation of the BP with the opposite chirality or from the vortex-antivortex DW can either stop the propagation or propagate parallel to the current. We attribute this effect to the role played by the inertia of the BP in its dynamics. Finally, we provide values of the velocities achieved by the BP in the NW as a function of applied current in Fig. 1. We conclude that BPs with vanishing momentum propagate opposite to the current with velocities that may be suppressed by the Oersted field. Importantly for spintronic applications, both momentum and inertial mass play a major role in the dynamics of BPs that has not been envisaged up to know.

![](https://assets.underline.io/uploads/markdown_image/1/image/6ff528b21478a289ec02bc86a4e8eae4.jpg)

The average velocity of a head-to-head BPW driven by a spin-polarized current in a Ni nanowire, far from the ends of the nanowire, as a function of the current density.

Oersted-field- and current- induced Bloch Point domain wall dynamics in cylindrical Ni nanowires

Highly-magnetostrictive amorphous microwires show bistable behaviour, i.e. magnetization reverses by depinning/propagation of a single Domain Wall, DW, along the entire wire. This response makes these microwires an ideal system for fundamental studies on DWs dynamics [1,2]. On the other hand, the classical Matteucci effect, ME, consists of the induction of an emf voltage at the ends of the microwire under applied axial magnetic field [3], and requires a net azimuthal component of magnetization, Mφ. That emf is associated with the DW propagation and its amplitude is proportional to the DW speed, vDW, and the time derivative of Mφ, tailored by the torsion magnetoelastic anisotropy. emf has been measured in a bistable FeSiB microwire (12mm-diam, 10cm-long) under torsion (maximum of ξmax=57 πrad/m) and axial driving magnetic field, Hdr (square wave). The setup is designed to generate unidirectional motion of the DW that unpins from the same end irrespective of the Hdr direction [4], and its speed is quantified by the voltage induced in tiny pickup coils along the microwire. The clockwise/counterclockwise, CW/CCW, torsion dependences of emf and vDW are experimentally determined for parallel/antiparallel Hdr. Asymmetric behaviour is observed for both, the sense of x and the Hdr direction (Fig.1). The DW dynamics is analysed in terms of Landau-Lifschitz-Gilbert eq. considering a bell-like DW (Fig.2). The observed asymmetries are numerically interpreted to derive from the magnetochiral effect related to the DW chirality. The switching of the Hdr direction changes the Head-to-Head, H-H, or Tail-to-Tail, T-T, DW character and chirality (i.e., Mφ circulation). As a consequence of this magnetochiral effect, we proof that H-H and T-T DWs propagate at different velocities, vDW, as experimentally observed. Here we provide a first evidence of DWs magnetochiral effects in microwires that offers an opportunity for investigation of topological properties of spin-textures in microwires that has not been envisaged up to now.

![](https://assets.underline.io/uploads/markdown_image/1/image/3aa87066cea42d5386ea12ea8fba69a6.jpg)

CW, ξ+, and CCW, ξ-, torsion dependence of DW speed under Hdr = 232 A/m

![](https://assets.underline.io/uploads/markdown_image/1/image/83f8fc99b2ba0f6840b6466bd0d04e32.jpg)

Scheme of a longitudinal slice with a propagating bell-shaped DW. Magnetization in the domains makes a small angle with the microwires axis

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