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We use soft x-ray vector-ptychographic tomography to determine the three-dimensional magnetization field in superparamagnetic nanoparticles self-assembled at the liquid-liquid interface and reveal the magnetic order induced by layered structure. The spins in individual nanoparticles become more aligned with increasing number of layers, resulting in a larger net magnetization. Our experimental results show a magnetic short-range order in the monolayer due to the proliferation of thermally induced magnetic vortices and a magnetic long-range order in the bilayer and trilayer, stemming from the strengthened dipolar interactions that effectively suppress thermal fluctuations. We also observe a screening effect of magnetic vortices and the attractive interaction between the magnetic vortices with opposite topological charges. Our work demonstrates the crucial role of layered structure in shaping the magnetization of nanoparticle assemblies, providing new opportunities to modulate these properties through strategic layer engineering.

It is shown micromagnetic and atomistic spin dynamics simulations can use multiple GPUs in order to reduce computation time, but also to allow for a larger simulation size than is possible on a single GPU. Whilst interactions which depend on neighbouring spins, such as exchange interactions, may be implemented efficiently by transferring data between GPUs using halo regions, or alternatively using direct memory accesses, implementing the long-range demagnetizing interaction is the main difficulty in achieving good performance scaling, where the data transfer rate between GPUs is a significant bottleneck. A multi-GPU convolution algorithm is developed here, which relies on single-GPU FFTs executed in parallel. It is shown that even for micromagnetic simulations where the demagnetizing interaction computation time dominates, good performance scaling may be achieved, with speedup factors up to 1.8, 2.5, and 3.1, for 2, 3, and 4 GPUs respectively. The code developed here can be used for any number of GPUs in parallel, with performance scaling strongly dependent on inter-GPU data transfer rate and connection topology. This is further improved in micromagnetic simulations which include a spin transport solver, obtaining speedup factors up to 1.96, 2.8, and 3.7, for 2, 3, and 4 GPUs respectively. The best case scenario is obtained for atomistic spin dynamics simulations, where the demagnetizing interaction is implemented with spin-averaged cells. Using a single workstation with 4 GPUs, it is shown atomistic spin dynamics simulations with up to 1 billion spins, and atomistic Monte Carlo simulations with up to 2 billion spins are possible, with a near-ideal performance scaling.

It is predicted magnetic skyrmions can be controllably moved on surfaces using a focused laser beam. Here an absorbed power of the order 1 mW, focused to a spot-size of the order 10 $\mu$m, results in a local temperature increase of around 50 K, and a local perpendicular strain of the order 10$^{-3}$ due to the thermo-elastic effect. For positive magneto-elastic coupling this generates a strong attractive force on skyrmions due to the magneto-elastic effect. The resultant motion is dependent on forces due to i) gradients in the local strain-induced magnetic anisotropy, ii) gradients in the effective anisotropy due to local temperature gradients, and magnetic parameters temperature dependences, and iii) Magnus effect acting on objects with non-zero topological number. Using dynamical magneto-thermo-elastic modelling, it is predicted skyrmions can be moved with significant velocities (up to 80 m/s shown), both for ferromagnetic and antiferromagnetic skyrmions, even in the presence of surface roughness. This mechanism of controllably moving single skyrmions in any direction, as well as addressing multiple skyrmions in a lattice, offers a new approach to constructing and studying skyrmionic devices with all-optical control.

Magnon eigenmodes in easy-plane antiferromagnetic insulators are linearly polarized and are not expected to carry any net spin angular momentum. Motivated by recent nonlocal spin transport experiments in the easy-plane phase of hematite, we perform a series of micromagnetic simulations in a nonlocal geometry at finite temperatures. We show that by tuning an external magnetic field, we can control the magnon eigenmodes and the polarization of the spin transport signal in these systems. We argue that a coherent beating oscillation between two orthogonal linearly polarized magnon eigenmodes is the mechanism responsible for finite spin transport in easy-plane antiferromagnetic insulators. The sign of the detected spin signal is also naturally explained by the proposed coherent beating mechanism. Our finding opens a path for on-demand control of the spin signal in a large class of easy-plane antiferromagnetic insulators.

Brain-inspired neuromorphic computing is a promising path towards next generation analogue computers that are fundamentally different compared to the conventional von Neumann architecture. One model for neuromorphic computing that can mimic the human brain behavior are spiking neural networks (SNNs), of which one of the most successful is the leaky integrate-and-fire (LIF) model. Since conventional complementary metal-oxide-semiconductor (CMOS) devices are not meant for modelling neural networks and are energy inefficient in network applications, recently the focus shifted towards spintronic-based neural networks. In this work, using the advantage of antiferromagnetic insulators, we propose a non-volatile magnonic neuron that could be the building block of a LIF spiking neuronal network. In our proposal, an antiferromagnetic domain wall in the presence of a magnetic anisotropy gradient mimics a biological neuron with leaky, integrating, and firing properties. This single neuron is controlled by polarized antiferromagnetic magnons, activated by either a magnetic field pulse or a spin transfer torque mechanism, and has properties similar to biological neurons, namely latency, refraction, bursting and inhibition. We argue that this proposed single neuron, based on antiferromagnetic domain walls, is faster and has more functionalities compared to previously proposed neurons based on ferromagnetic systems.

A three-dimensional self-consistent spin transport model is developed, which includes both tunnelling transport, as well as metallic transport. Using the spin accumulation computed either side of a tunnel barrier, spin torques are obtained, and it is shown the model reproduces both damping-like and field-like spin-transfer torques, with the expected sinusoidal angular dependence, and inverse ferromagnetic layer thickness dependence. An explicit solution to the drift-diffusion model is derived, which allows analysing the effect of both the reference and free layer thickness on the spin-transfer torque polarization and field-like coefficient. In particular, when the layers are thin, additional spin-dependent scattering contributions due to incomplete absorption of transverse spin components reduce both the damping-like and field-like spin torques. It is shown the model developed here can be used to compute the signal-to-noise ratio in realistic magnetic read-heads, where spin torque-induced fluctuations and instabilities limit the maximum operating voltage. The effect of metallic pinhole defects in the insulator layer is also analysed, which results in a mixture of metallic and tunnelling transport, and highly non-uniform charge and spin currents, requiring the full spin transport model to compute the resulting magneto-resistance and spin torques. Increasing the area covered by pinholes results in a rapid degradation of the magneto-resistance, following an inverse dependence. Moreover, the spin torque angular dependence becomes skewed, and the spin-transfer torque polarization decreases. The same results are obtained when considering tunnel junctions with a single pinhole defect, but decreasing cross-sectional area, showing that even a single pinhole defect can significantly degrade the performance of tunnel junctions and magnetic read-heads below the 40 nm node.

The performance of numerical micromagnetic models is limited by the demagnetizing field computation, which typically accounts for the majority of the computation time. For magnetization dynamics simulations explicit evaluation methods are in common use. Higher order methods call for evaluation of all effective field terms, including the demagnetizing field, at all sub-steps. Here a general method of speeding up such explicit evaluation methods is discussed, by skipping the demagnetizing field computation at sub-steps, and instead approximating it using polynomial extrapolation based on stored previous exact computations. This approach is tested for a large number of explicit evaluation methods, both adaptive and fixed time-step, ranging from 2nd order up to 5th order. The polynomial approximation order should be matched to the evaluation method order. In this case we show higher order methods with polynomial extrapolation are more accurate than lower order methods with full evaluation of the demagnetizing field. Moreover, for higher order methods we show it is possible to achieve a factor of 2 or more computation speedup with no decrease in solution accuracy.

Here we study the effect of an additional interfacial spin-transfer torque, as well as the well established spin-orbit torque and bulk spin-transfer torque, on skyrmion collections - group of skyrmions dense enough that they are not isolated from on another - in ultrathin heavy metal / ferromagnetic multilayers, by comparing modelling with experimental results. Using a skyrmion collection with a range of skyrmion diameters and landscape disorder, we study the dependence of the skyrmion Hall angle on diameter and velocity, as well as the velocity as a function of diameter. We show the experimental results are in good agreement with modelling when including the interfacial spin-transfer torque, and cannot be reproduced by using the spin-orbit torque alone. We also show that for skyrmion collections the velocity is approximately independent of diameter, in marked contrast to the motion of isolated skyrmions, as the group of skyrmions move together at an average group velocity. Moreover, the calculated skyrmion velocities are comparable to those obtained in experiments when the interfacial spin-transfer torque in included, whilst modelling using the spin-orbit torque alone shows large discrepancies with the experimental data. Our results thus show the significance of the interfacial spin-transfer torque in ultrathin magnetic multilayers, which is of similar strength to the spin-orbit torque, and both significantly larger than the bulk spin-transfer torque. Due to the good agreement with experiments, we conclude that the interfacial spin-transfer torque should be included in numerical modelling for correct reproduction of experimental results., Comment: 25pages, 5 figures

An efficient method for computing thermodynamic equilibrium states at the micromagnetic length scale is introduced, using the Markov chain Monte Carlo method. Trial moves include not only rotations of vectors, but also a change in their magnetization length. The method is parameterized using the longitudinal susceptibility, reproduces the same Maxwell-Boltzmann distribution as the stochastic Landau-Lifshitz-Bloch equation, and is applicable both below and above the Curie temperature. The algorithm is fully parallel, can be executed on graphical processing units, and efficiently includes the long range dipolar interaction. This method is generally useful for computing finite-temperature relaxation states both for uniform and non-uniform temperature profiles, and can be considered as complementary to zero-temperature micromagnetic energy minimization solvers, with comparable computation time. Compared to the dynamic approach it is shown the micromagnetic Monte Carlo method is up to almost 20 times faster. Moreover, unlike quasi-zero temperature approaches which do not take into account the magnetization length distribution and stochasticity, the method is better suited for structures with unbroken symmetry around the applied field axis, granular films, and at higher temperatures and fields. In particular, applications to finite-temperature hysteresis loop modelling, chiral magnetic thin films, granular magnetic media, and artificial spin ices are discussed.

Monte Carlo algorithms are frequently used in atomistic simulations, including for computation of magnetic parameter temperature dependences in multiscale simulations. Even though parallelization strategies for Monte Carlo simulations of lattice spin models are known, its application to computation of magnetic parameter temperature dependences is lacking in the literature. Here we show how, not only the unconstrained algorithm, but also the constrained atomistic Monte Carlo algorithm, can be parallelized. Compared to the serial algorithms, the parallel Monte Carlo algorithms are typically over 200 times faster, allowing computations in systems with over 10 million atomistic spins on a single GPU with relative ease. Implementation and testing of the algorithms was carried out in large-scale systems, where finite-size effects are reduced, by accurately computing temperature dependences of magnetization, uniaxial and cubic anisotropies, exchange stiffness, and susceptibilities. In particular for the exchange stiffness the Bloch domain wall method was used with a large crosssectional area, which allows accurate computation of the domain wall width up to the Curie temperature. The exchange stiffness for a simple cubic lattice closely follows an mk scaling at low temperatures, with k < 2 dependent on the anisotropy strength. However, close to the Curie temperature the scaling exponent tends to k = 2. Furthermore, the implemented algorithms are applied to the computation of magnetization temperature dependence in granular thin films with over 15 million spins, as a function of average grain size and film thickness. We show the average Curie temperature in such systems may be obtained from a weighted Bloch series fit, which is useful for analysis of experimental results in granular thin films.

This work discusses the design and testing of a new computational spintronics research software. Boris is a comprehensive multi-physics open-source software, combining micromagnetics modelling capabilities with drift-diffusion spin transport modelling and heat flow solver in multi-material structures. A multi-mesh paradigm is employed, allowing modelling of complex multi-layered structures with independent discretization and arbitrary relative positioning between different computational meshes. Implemented micromagnetics models include not only ferromagnetic materials modelling, but also two-sublattice models, allowing simulations of antiferromagnetic and ferrimagnetic materials, fully integrated in the multi-mesh and multi-material design approach. High computational performance is an important design consideration in Boris, and all computational routines can be executed on GPUs, in addition to CPUs. In particular a modified 3D convolution algorithm is used to compute the demagnetizing field on the GPU, termed pipelined convolution, and benchmark comparisons with existing GPU-accelerated software Mumax3 have shown performance improvements up to twice faster.

It is known that ultrafast laser pulses can be used to deterministically switch magnetisation and create skyrmions, however the deterministic creation of a single N\'eel skyrmion after ultrafast demagnetisation remains an open question. Here we show domain wall skyrmions also emerge in systems with broken inversion symmetry after exposure to an ultrafast laser pulse, carrying an integer topological charge. Whilst domain wall skyrmions do not appear in the relaxed state due to quick thermal decay following an Arrhenius law, they play a key role in controlling the final skyrmion population through annihilations with skyrmions of opposite topological charge, with the resultant skyrmion states following a Poisson distribution. Using single-shot linearly polarised laser pulses, as well as a train of circularly polarised laser pulses, we show that when a high degree of disorder is created, the possibility of nucleating a single N\'eel skyrmion is accompanied by the possibility of nucleating a skyrmion with domain wall skyrmion pair, which results in a self-annihilation collapse.

As research into magnetic thin films and spintronics devices is moving from single to multiple magnetic layers, there is a need for micromagnetics modelling tools specifically designed to efficiently handle magnetic multilayers. Here we show an exact method of computing demagnetising fields in magnetic multilayers, which is able to handle layers with arbitrary spacing, arbitrary thicknesses, and arbitrary relative positioning between them without impacting on the computational performance, or sacrificing numerical accuracy. The multilayered convolution method is a generalisation of the well-known convolution method used to compute demagnetising fields in a single magnetic body. In typical use cases, such as multilayered stacks used to study skyrmions, we show the multilayered convolution method can be up to 8 times faster, implemented both for central processors and graphics processors, compared to the simple convolution method.

It is well known that skyrmions can be driven using spin-orbit torques due to the spin-Hall effect. Here we show an additional contribution in multilayered stacks arises from vertical spin currents due to inter-layer diffusion of a spin accumulation generated at a skyrmion. This additional interfacial spin torque is similar in form to the in-plane spin transfer torque, but is significantly enhanced in ultra-thin films and acts in the opposite direction to the electron flow. The combination of this diffusive spin torque and the spin-orbit torque results in skyrmion motion which helps to explain the observation of small skyrmion Hall angles even with moderate magnetisation damping values. Further, the effect of material imperfections on threshold currents and skyrmion Hall angle is also investigated. Topographical surface roughness, as small as a single monolayer variation, is shown to be an important contributing factor in ultra-thin films, resulting in good agreement with experimental observations.

A heat-assisted multiferroic solid-state memory design is proposed and analysed, based on a PbNbZrSnTiO3 antiferroelectric substrate and Ni81Fe19 magnetic free layer. Information is stored as magnetisation direction in the free layer of a magnetic tunnel junction element. The bit writing process is contactless and relies on triggering thermally activated magnetisation switching of the free layer towards a strain-induced anisotropy easy axis. A stress is generated using the antiferroelectric substrate by voltage-induced antiferroelectric to ferroelectric phase change, and this is transmitted to the magnetic free layer by strain-mediated coupling. The thermally activated strain-induced magnetisation switching is analysed here using a three-dimensional, temperature-dependent magnetisation dynamics model, based on simultaneous evaluation of the stochastic Landau Lifshitz Bloch equation and heat flow equation, together with stochastic thermal fields and magnetoelastic contributions. The magnetisation switching probability is calculated as a function of stress magnitude and maximum heat pulse temperature. An operating region is identified, where magnetisation switching always occurs, with stress values ranging from 80 to 180 MPa, and maximum temperatures normalised to the Curie temperature ranging from 0.65 to 0.99.

A three-dimensional spin current solver based on a generalised spin drift-diffusion description, including the spin Hall effect, is integrated with a magnetisation dynamics solver. The resulting model is shown to simultaneously reproduce the spin-orbit torques generated using the spin Hall effect, spin pumping torques generated by magnetisation dynamics in multilayers, as well as the spin transfer torques acting on magnetisation regions with spatial gradients, whilst field-like and spin-like torques are reproduced in a spin valve geometry. Two approaches to modelling interfaces are analysed, one based on the spin mixing conductance and the other based on continuity of spin currents where the spin dephasing length governs the absorption of transverse spin components. In both cases analytical formulas are derived for the spin-orbit torques in a heavy metal / ferromagnet bilayer geometry, showing in general both field-like and damping-like torques are generated. The limitations of the analytical approach are discussed, showing that even in a simple bilayer geometry, due to the non-uniformity of the spin currents, a full three-dimensional treatment is required. Finally the model is applied to the quantitative analysis of the spin Hall angle in Pt by reproducing published experimental data on the ferromagnetic resonance linewidth in the bilayer geometry.

Pulsed domain wall movement is studied here in Ni80Fe20 nanowires on SiO2, using a fully integrated electrostatic, thermoelectric, and micromagnetics solver based on the Landau-Lifshitz-Bloch equation, including Joule heating, anisotropic magneto-resistance, and Oersted field contributions. During the applied pulse the anisotropic magneto-resistance of the domain wall generates a dynamic heat gradient which increases the current-driven velocity by up to 15%. Using a temperature-dependent conductivity significant differences are found between the constant voltage-pulsed and constant current-pulsed domain wall movement: constant voltage pulses are shown to be more efficient at displacing domain walls whilst minimizing the increase in temperature, with the total domain wall displacement achieved over a fixed pulse duration having a maximum with respect to the driving pulse strength., Comment: Corrected from previous 25 Jun 2016 version, due to error found in a computational routine

Domain walls in ferromagnetic nanowires are potential building-blocks of future technologies such as racetrack memories, in which data encoded in the domain walls are transported using spin-polarised currents. However, the development of energy-efficient devices has been hampered by the high current densities needed to initiate domain wall motion. We show here that a remarkable reduction in the critical current density can be achieved for in-plane magnetised coupled domain walls in CoFe/Ru/CoFe synthetic ferrimagnet tracks. The antiferromagnetic exchange coupling between the layers leads to simple N\'{e}el wall structures, imaged using photoemission electron and Lorentz transmission electron microscopy, with a width of only $\sim 100$~nm. The measured critical current density to set these walls in motion, detected using magnetotransport measurements, is $1.0 \times 10^{11}$~Am$^{-2}$, almost an order of magnitude lower than in a ferromagnetically coupled control sample. Theoretical modelling indicates that this is due to nonadiabatic driving of anisotropically coupled walls, a mechanism that can be used to design efficient domain-wall devices., Comment: Supplmentary Information as ancillary file

An effective field model is introduced here within the micromagnetics formulation, to study roughness in magnetic structures, by considering sub-exchange length roughness levels as a perturbation on a smooth structure. This allows the roughness contribution to be separated, which is found to give rise to an effective configurational anisotropy for both edge and surface roughness, and accurately model its effects with fine control over the roughness depth without the explicit need to refine the computational cellsize to accommodate the roughness profile. The model is validated by comparisons with directly roughened structures for a series of magnetization switching and domain wall mobility simulations and found to be in excellent agreement for roughness levels up to the exchange length. The model is further applied to vortex domain wall mobility simulations with surface roughness, which is shown to significantly modify domain wall movement and result in dynamic pinning and stochastic creep effects.