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We study the full 3d static micromagnetic equations via a physics-informed neural network (PINN) ansatz for the continuous magnetization configuration. PINNs are inherently mesh-free and unsupervised learning models. In our approach we can learn to minimize the total Gibbs free energy with additional conditional parameters, such as the exchange length, by a single low-parametric neural network model. In contrast, traditional numerical methods would require the computation and storage of a large number of solutions to interpolate the continuous spectrum of quasi-optimal magnetization configurations. We also consider the important and computationally expensive stray field problem via PINNs, where we use a basically linear learning ansatz, called extreme learning machines (ELM) within a splitting method for the scalar potential. This reduces the stray field training to a linear least squares problem with precomputable solution operator. We validate the stray field method by means of numerical example comparisons from literature and illustrate the full micromagnetic approach via the NIST $\mu$MAG standard problem $\#3$., Comment: 17 pages, 10 figures

We present a simple atomistic model for the description of noncollinear coupling in magnetic multilayers with hybrid spacer layers made of Ru alloyed to ferromagnetic atoms such as Fe. In contrast to previous analytical and micromagnetic models that explain the noncollinear coupling by means of lateral fluctuations in the coupling constant, the presented model accounts for atom-atom coupling in all three spatial dimensions within the spacer layer. The new model is able to accurately predict the dependence of the macroscopic bilinear and biquadratic coupling constants on the spacer-layer composition and thickness, showing much better quantitative agreement than lateral-fluctuation models. Moreover, it predicts noncollinear coupling even for infinitely stiff ferromagnetic layers which goes beyond the predictions of previous models.

Expanding nanomagnetism and spintronics into three dimensions (3D) offers great opportunities for both fundamental and technological studies. However, probing the influence of complex 3D geometries on magnetoelectrical phenomena poses important experimental and theoretical challenges. In this work, we investigate the magnetoelectrical signals of a ferromagnetic 3D nanodevice integrated into a microelectronic circuit using direct-write nanofabrication. Due to the 3D vectorial nature of both electrical current and magnetization, a complex superposition of several magnetoelectrical effects takes place. By performing electrical measurements under the application of 3D magnetic fields, in combination with macrospin simulations and finite element modelling, we disentangle the superimposed effects, finding how a 3D geometry leads to unusual angular dependences of well-known magnetotransport effects such as the anomalous Hall effect. Crucially, our analysis also reveals a strong role of the non-collinear demagnetizing fields intrinsic to 3D nanostructures, which results in an angular dependent magnon magnetoresistance contributing strongly to the total magnetoelectrical signal. These findings are key to the understanding of 3D spintronic systems and underpin further fundamental and device-based studies., Leverhulme Trust Isaac Newton Trust L’Oréal-UNESCO U.K. and Ireland Fellowship For Women In Science EPSRC Winton Program for Physics of Sustainability China Scholarship Council European Union’s Horizon 2020 research and innovation program Spanish AEI

We performed finite-element micromagnetic simulations to examine the formation of skyrmions without intrinsic Dzyaloshinskii–Moriya interaction (DMI) in magnetic hemispherical shells. We found that curvature-induced DM-like interaction allows for further stabilization of skyrmions without the DMI in curved-geometry hemispherical shells for a specific range of uniaxial perpendicular magnetic anisotropy (PMA) constant Ku. The larger the curvature of the shell, the higher the Ku value required for the formation of the skyrmions. With well-stabilized skyrmions, we also found in-plane gyration modes and azimuthal spin-wave modes as well as an out-of-plane breathing mode, similarly to previously found modes for planar geometries. Furthermore, additional higher-frequency hybrid modes were observed due to coupling between the gyration and azimuthal modes. This work provides further physical insight into the static and dynamic properties of intrinsic DMI-free skyrmions formed in curved-geometry systems.

Spintronic devices would greatly benefit from a noncollinear alignment between magnetizations of adjacent ferromagnetic layers for maximum performance and reliability. We demonstrate that such an alignment can be created and controlled by coupling two ferromagnetic layers across a magnetic spacer layer consisting of a nonmagnetic material, Ru, alloyed with a ferromagnetic element, Co. Changing the composition and thickness of the spacer layer enables the control of the relative angle between the magnetizations of the ferromagnetic layers between [Formula: see text] and [Formula: see text]. The onset of noncollinear alignment between the ferromagnetic layers coincides with the advent of magnetic order in the spacer layer. This study maps the concentration and thickness ranges of RuCo spacer layers that give rise to noncollinearity between ferromagnetic Co layers. The experimental results are successfully reproduced by simulating our structures with an atomistic model. This model assumes that Co atoms in the RuCo spacer layer have magnetic moments and that neighboring Co atoms are ferromagnetically coupled, while Co atoms separated by one or more Ru atoms are antiferromagnetically coupled.

We present a simple simulation model for analysing magnetic and frictional losses of magnetic nano-particles in viscous fluids subject to alternating magnetic fields. Assuming a particle size below the single-domain limit, we use a macrospin approach and solve the Landau-Lifshitz-Gilbert equation coupled to the mechanical torque equation. Despite its simplicity the presented model exhibits surprisingly rich physics and enables a detailed analysis of the different loss processes depending on field parameters and initial arrangement of the particle and the field. Depending on those parameters regions of different steady states emerge: a region with dominating N\'eel relaxation and high magnetic losses and another region region with high frictional losses at low fields or low frequencies. The energy increases continuously even across regime boundaries up to frequencies above the Brownian relaxation limit. At those higher frequencies the steady state can also depend on the initial orientation of the particle in the external field. The general behavior and special cases and their specific absorption rates are compared and discussed., Comment: 12 pages, 9 figures

The fundamental limits currently faced by traditional computing devices necessitate the exploration of ways to store, compute, and transmit information going beyond the current CMOS-based technologies. Here, we propose a three-dimensional (3D) magnetic interconnector that exploits geometry-driven automotion of domain walls (DWs), for the transfer of magnetic information between functional magnetic planes. By combining state-of-the-art 3D nanoprinting and standard physical vapor deposition, we prototype 3D helical DW conduits. We observe the automotion of DWs by imaging their magnetic state under different field sequences using X-ray microscopy, observing a robust unidirectional motion of DWs from the bottom to the top of the spirals. From experiments and micromagnetic simulations, we determine that the large thickness gradients present in the structure are the main mechanism for 3D DW automotion. We obtain direct evidence of how this tailorable magnetic energy gradient is imprinted in the devices, and how it competes with pinning effects that are due to local changes in the energy landscape. Our work also predicts how this effect could lead to high DW velocities, reaching the Walker limit during automotion. This work demonstrates a possible mechanism for efficient transfer of magnetic information in three dimensions., This work was supported by the EPSRC Cambridge NanoDTC EP/L015978/1, the Winton Program for the Physics of Sustainability, the project CALIPSOplus (under Grant Agreement No. 730872 from the EU Framework Programme for Research and Innovation HORIZON 2020), and by the European Community (under the Horizon 2020 Program, Contract No. 101001290, 3DNANOMAG). L.S. acknowledges support from St. Johns College of the University of Cambridge. C.D. was supported by the Leverhulme Trust (No. ECF-2018-016), the Isaac Newton Trust (No. 18-08), and the L’Oréal-UNESCO UK and Ireland Fellowship For Women In Science. A.H.-R. acknowledges support from Spanish AEI, under Project Reference No. PID2019-104604RB/AEI/10.13039/501100011033. The authors acknowledge the University of Vienna research platform MMM Mathematics–Magnetism–Materials, the FWF (Project No. I 4917), and Aragon Government through the Project Q-MAD.

We propose a novel micromagnetic standard problem calculating the coercive field for unpinning a domain wall at the interface of a multiphase magnet. This problem is sensitive to discontinuities in material parameters for the exchange interaction, the uniaxial anisotropy, and the spontaneous magnetization. We derive an explicit treatment of jump conditions at material interfaces for the exchange interaction in the finite-difference discretization. The micromagnetic simulation results are compared with analytical solutions and show good agreement. The proposed standard problem is well-suited to test the implementation of both finite-difference and finite-element simulation codes.

Contemporary spintronics devices, notably GMR and TMR-sensors with high linear ranges, stand to benefit from large bias fields in excess of 1T that have been observed in exchange-coupled double layers using rare earth-transition metal ferrimagnets as pinning layers. However, before exploiting the strong coupling the reversal process must be understood at the scales of the smallest devices, where it would be effective. Little is known about these processes to-date, owing to the technical difficulty of imaging magnetic structures smaller than 20nm in fields of several Tesla. Here we image the nanoscale magnetic structures of a double layer comprising a polycrystalline Co/Pt-multilayer (Co/Pt) which is strongly exchange-coupled to an amorphous and magnetically-hard ferrimagnetic TbFe alloy layer. High resolution magnetic force microscopy provides snapshots of the magnetic structures along various applied fields between 0 and 7T as the magnetization reversal takes place. In contrast to a magnetization reversa...

The lowest achievable blocking temperature limits magnetic ordering in highly frustrated thermally active artificial kagome spin ice. By exploiting the interfacial Dzyaloshinskii-Moriya interaction, we can lower the blocking temperature of individual nanomagnets without strongly affecting their magnetic moments, thus leaving the critical transition temperatures unchanged. Using this approach, we demonstrate that a seven-ring kagome structure consisting of 30 nanomagnets can be thermally annealed into its ground state. Furthermore, the spin-ice correlations extracted from extended kagome lattices are found to exhibit the quantitative signatures of long-range charge-order, thereby giving experimental evidence for the theoretically predicted continuous transition to a charge-ordered state.

Partial differential equations and variational problems can be solved with physics informed neural networks (PINNs). The unknown field is approximated with neural networks. Minimizing the residuals of the static Maxwell equation at collocation points or the magnetostatic energy, the weights of the neural network are adjusted so that the neural network solution approximates the magnetic vector potential. This way, the magnetic flux density for a given magnetization distribution can be estimated. With the magnetization as an additional unknown, inverse magnetostatic problems can be solved. Augmenting the magnetostatic energy with additional energy terms, micromagnetic problems can be solved. We demonstrate the use of physics informed neural networks for solving magnetostatic problems, computing the magnetization for inverse problems, and calculating the demagnetization curves for two-dimensional geometries., 22 pages, 9 figures

The Landau-Lifshitz equation governing magnetization dynamics is written in terms of the amplitudes of normal modes associated with the micromagnetic system's appropriate ground state. This results in a system of nonlinear ordinary differential equations, the right-hand side of which can be expressed as the sum of a linear term and nonlinear terms with increasing order of nonlinearity (quadratic, cubic, etc.). The application of the method to nanostructured magnetic systems demonstrates that the accurate description of magnetization dynamics requires a limited number of normal modes, which results in a considerable improvement in computational speed. The proposed method can be used to obtain a reduced-order dynamical description of magnetic nanostructures which allows to adjust the accuracy between low-dimensional models, such as macrospin, and fully spatially discretized micromagnetic models. This new paradigm for micromagnetic simulations is tested for three problems relevant to the areas of spintronics and magnonics: directional spin-wave coupling in magnonic waveguides, high power ferromagnetic resonance in a magnetic nanodot, and injection-locking in spin-torque nano-oscillators. The case studies considered the validity of the proposed approach as a new systematic method to obtain an intermediate order dynamical model based on normal modes for the analysis of magnetic nanosystems. The time-consuming calculation of the normal modes has to be done only one time for the system. These modes can be used to optimize and predict the system response for all possible time-varying external magnetic fields. This is of utmost importance for various applications where a fast and accurate system response is required, such as in electronic circuit simulation software where an accurate response of the magnetic device has to be predicted.

Density functional theory calculations have been carried out for the hard magnetic compound Y C o 5 and its related phases. The calculations provide a straightforward but accurate method, as reflected in good agreement obtained with experimental results in terms of saturation magnetization and magnetocrystalline anisotropy energy (MAE). For the calculations we used the projector augmented wave method with GGA-PBE approximation and included spin-orbit coupling and ( U , J ) potentials, which are calculated to convergence. It is shown that these properties are influenced by the crystal structure and atomic distances, hence the crystal structures have been optimized and relaxed. The influence of changes in symmetry, distances and lattice parameters are studied, as well as the effect of substituting one C o - atom by F e or C u as examples for a magnetic and a non-magnetic impurity. The comparison between Y C o 5 and hcp - C o shows the importance of atomic environments and lattice symmetry for a large MAE. The calculations predict that F e and C u both reduce MAE, but F e increases magnetization. Despite lacking 4 f - orbitals as compared to S m C o 5 , Y C o 5 exhibits a large anisotropy, originating from the contribution of the C o - atoms. We have also shortly studied the influence of 4 f - electrons for rare earths R = P r and S m on MAE for comparison. Y C o 4 F e , Y C o 4 C u , S m C o 4 F e and S m C o 4 C u are calculated to have MAE values of 2.1 M J / m 3 , 2.7 M J / m 3 , 19.2 M J / m 3 and 8 . 2 M J / m 3 at 0 K respectively.

Laser powder bed fusion is a well-established additive manufacturing method that can be used for the production of net-shaped Nd Fe B magnets. However, low coercivity has been one of the drawbacks in the laser powder bed fusion processed Nd Fe B magnets. In this work, we have demonstrated that the grain boundary diffusion process using low-melting Nd Cu, Nd Al Ni Cu, and Nd Tb Cu alloys to the selective laser sintered NdFeB magnets can results in a substantial enhancement of coercivity from 0.65 T to 1.5 T. Detailed microstructure investigations clarified that the formation of Nd-rich grain boundary phase, the introduction of Tb-rich shell at the surface of Nd2Fe14B grains, and maintaining the grain size in nano-scale are responsible for the large coercivity enhancement.

A new type of spin-torque-oscillator (STO), all-in-plane STO, is introduced for microwave assisted magnetic recording (MAMR) that consist of a spin-injection-layer (SIL) and a field-generating-layer (FGL) with an effective in-plane easy axis due to the shape anisotropy separated with a metallic spacer. In this device, electrons are injected from SIL to FGL while the magnetization of the SIL and FGL is saturated to the out-of-plane by the external magnetic field of ∼1.0 T. Micromagnetic simulations showed that the magnetization direction of SIL can be switched to the opposite direction to that of the applied external magnetic field by the use of spin-transfer-torque. This results in a larger spin accumulation in FGL and its oscillation with a large cone angle at a low bias current density. We designed SIL to reduce the critical current density, Jc, required for the magnetization switching of SIL. Materials with a smaller saturation magnetization in SIL reduce Jc. Smaller spin polarization of SIL leads to a larger spin accumulation in SIL with an opposite direction to the magnetization, resulting in a reduction of Jc. This enables magnetization switching of SIL in small Jc followed by oscillation of FGL with frequency above 20 GHz with a large out-of-plane oscillation cone angle of 45–50°. The validity of this finding was studied experimentally by developing STO with two SIL materials, Co2Fe(Al0.5Si0.5) Heusler alloy and Fe67Co33, the former has the B2 crystal structure with a large spin polarization and the latter has the A2 crystal structure with a smaller spin polarization. The magnetization configuration of SIL and FGL in STO with a diameter of ∼60 nm is investigated experimentally based on the field dependent resistance change and the oscillation behavior is discussed based on the micromagnetic simulations.

We present methods for calculating the strayfield in finite element and finite difference micromagnetic simulations using true periodic boundary conditions. In contrast to pseudo periodic boundary conditions, which are widely used in micromagnetic codes, the presented methods eliminate the shape anisotropy originating from the outer boundary. This is a crucial feature when studying the influence of the microstructure on the performance of composite materials, which is demonstrated by hysteresis calculations of soft magnetic structures that are operated in a closed magnetic loop configuration. The applied differential formulation is perfectly suited for the application of true periodic boundary conditions. The finite difference equations can be solved by a highly efficient Fast Fourier Transform method.

The design of complex, competing effects in magnetic systems-be it via the introduction of nonlinear interactions1-4, or the patterning of three-dimensional geometries5,6-is an emerging route to achieve new functionalities. In particular, through the design of three-dimensional geometries and curvature, intrastructure properties such as anisotropy and chirality, both geometry-induced and intrinsic, can be directly controlled, leading to a host of new physics and functionalities, such as three-dimensional chiral spin states7, ultrafast chiral domain wall dynamics8-10 and spin textures with new spin topologies7,11. Here, we advance beyond the control of intrastructure properties in three dimensions and tailor the magnetostatic coupling of neighbouring magnetic structures, an interstructure property that allows us to generate complex textures in the magnetic stray field. For this, we harness direct write nanofabrication techniques, creating intertwined nanomagnetic cobalt double helices, where curvature, torsion, chirality and magnetic coupling are jointly exploited. By reconstructing the three-dimensional vectorial magnetic state of the double helices with soft-X-ray magnetic laminography12,13, we identify the presence of a regular array of highly coupled locked domain wall pairs in neighbouring helices. Micromagnetic simulations reveal that the magnetization configuration leads to the formation of an array of complex textures in the magnetic induction, consisting of vortices in the magnetization and antivortices in free space, which together form an effective B field cross-tie wall14. The design and creation of complex three-dimensional magnetic field nanotextures opens new possibilities for smart materials15, unconventional computing2,16, particle trapping17,18 and magnetic imaging19., EPSRC Early Career Fellowship EP/M008517/1 Winton Program for the Physics of Sustainability Leverhulme Trust (ECF-2018-016) Isaac Newton Trust (18-08) L���Or��al-UNESCO UK and Ireland Fellowship For Women In Science 2019 European Union���s Horizon 2020 research and innovation program under Marie Sk��odowska-Curie grant ref. H2020-MSCA-IF-2016-746958 Spanish AEI under project reference PID2019���104604RB/AEI/10.13039/501100011033 German Ministerium f��r Bildung und Forschung (BMBF) through contracts 05K16WED and 05K19WE2 European Union���s Horizon 2020 research and innovation program under the Marie Sk��odowska-Curie grant agreement no. 701647 FWF project I 4917.

3D nano-architectures present a new paradigm in modern condensed matter physics with numerous applications in photonics, biomedicine, and spintronics. They are promising for the realisation of 3D magnetic nano-networks for ultra-fast and low-energy data storage. Frustration in these systems can lead to magnetic charges or magnetic monopoles, which can function as mobile, binary information carriers. However, Dirac strings in 2D artificial spin ices bind magnetic charges, while 3D dipolar counterparts require cryogenic temperatures for their stability. Here, we present a micromagnetic study of a highly-frustrated 3D artificial spin ice harboring tension-free Dirac strings with unbound magnetic charges at room temperature. We use micromagnetic simulations to demonstrate that the mobility threshold for magnetic charges is by $\SI{2}{eV}$ lower than their unbinding energy. By applying global magnetic fields, we steer magnetic charges in a given direction omitting unintended switchings. The introduced system paves a way towards 3D magnetic networks for data transport and storage, 7 pages, 4 figures and 1 table

A self-consistent micromagnetic model is proposed for simulating interactions between a superparamagnetic label particle and a low-noise GMR vortex sensor, focusing on most common operations such as label detection and saturation. For this purpose, we evaluate the combined action of vortex stray field and the applied external field. By solving the Landau–Lifshitz-Gilbert equation we find that a superparamagnetic label with a diameter of 60 nm can be successfully detected by a magnetic vortex structure assisted with an external magnetic field of 100 mT. Subsequently, the self-consistent spin diffusion model is applied to calculate electric response of a standard low-noise GMR vortex sensor to the presence of magnetically-saturated label. It is shown that the stray field of the activated label produces a detectable potential difference in the GMR sensor of 27.1 to 27.8 mV, the magnitude of which depends on location of the label with respect to the sensor surface.