Strong influence of magnetic field and non-uniform stress on elastic modulus and transition temperatures of twinned Ni-Fe(Co)-Ga alloy.

The magnetization value and electric resistivity of the single-crystalline sample of Ni50Fe19Co4Ga27 shape memory alloy were measured. The elastic modulus was determined by the Dynamic Mechanical Analysis (DMA). The characteristic temperatures of martensitic transformation (MT) of the alloy were estimated from the temperature dependences of magnetization, electric resistivity and elastic modulus. A significant disparity between MT temperatures resulting from DMA and those estimated from magnetic and resistivity measurements was discovered. It was argued that the discrepancy is caused by the non-uniform mechanical stressing of twinned single crystal by the DMA analyzer. Moreover, the DMA measurements revealed a significant decrease of the elastic modulus of twinned martensite under the applied magnetic field of 1.5 kOe. To explain this effect, the temperature-dependent Young's modulus of twinned crystal lattice was computed. The computations showed that the experimentally observed field-induced change of the elastic modulus is caused by the stress-assisted detwinning of the crystal lattice by the applied magnetic field.

Dipolar skyrmions and antiskyrmions of arbitrary topological charge at room temperature.

Magnetic skyrmions are localized, stable topological magnetic textures that can move and interact with each other like ordinary particles when an external stimulus is applied. The efficient control of the motion of spin textures using spin-polarized currents opened an opportunity for skyrmionic devices such as racetrack memory and neuromorphic or reservoir computing. The coexistence of skyrmions with high topological charge in the same system promises further possibilities for efficient technological applications. In this work, we directly observe dipolar skyrmions and antiskyrmions with arbitrary topological charge in Co/Ni multilayers at room temperature. We explore the dipolar-stabilized spin objects with topological charges of up to 10 and characterize their nucleation process, their energy dependence on the topological charge and the effect of the material parameters on their stability. Furthermore, our micromagnetic simulations demonstrate spin-transfer-induced motion of these spin objects, which is important for their potential device application.

magnum.np: a PyTorch based GPU enhanced finite difference micromagnetic simulation framework for high level development and inverse design.

magnum.np is a micromagnetic finite-difference library completely based on the tensor library PyTorch. The use of such a high level library leads to a highly maintainable and extensible code base which is the ideal candidate for the investigation of novel algorithms and modeling approaches. On the other hand magnum.np benefits from the device abstraction and optimizations of PyTorch enabling the efficient execution of micromagnetic simulations on a number of computational platforms including graphics processing units and potentially Tensor processing unit systems. We demonstrate a competitive performance to state-of-the-art micromagnetic codes such as mumax3 and show how our code enables the rapid implementation of new functionality. Furthermore, handling inverse problems becomes possible by using PyTorch's autograd feature.

Topology optimization of a heat-assisted magnetic recording write head to reduce transition curvature using a binary optimization algorithm utilizing the adjoint method.

In this work, the possibility to reduce transition curvature in heat-assisted magnetic recording, using a conventional write head design, by shaping the recording field to counteract the circular profile of the heat pulse is investigated. Topology optimization of the head tip is performed in order to create the desired cross-track field profile for increasing distances from the write head tip. For the topology optimization, the adjoint method is utilized to calculate the necessary gradients and a binary optimization scheme is proposed. The optimizations are performed considering linearized material parameters reducing the computational complexity and the results are compared to optimizations incorporating the full non-linear material behavior. The optimized field profiles are evaluated for their influence on the read-back process. To do so, switching probability phase diagrams are calculated and the curvature parameter, the signal to noise ratio and the channel bit density are extracted. The presented results show that while transition curvature can be reduced by shaping the cross-track profile of the write field, this alone does not consequently lead to an improvement of the read back process. Therefore, completely new head designs, considering additional parameters have to be investigated.

Domain Wall Automotion in Three-Dimensional Magnetic Helical Interconnectors.

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.

A topology optimization algorithm for magnetic structures based on a hybrid FEM-BEM method utilizing the adjoint approach.

A method to optimize the topology of hard as well as soft magnetic structures is implemented using the density approach for topology optimization. The stray field computation is performed by a hybrid finite element-boundary element method. Utilizing the adjoint approach the gradients necessary to perform the optimization can be calculated very efficiently. We derive the gradients using a "first optimize then discretize" scheme. Within this scheme, the stray field operator is self-adjoint allowing to solve the adjoint equation by the same means as the stray field calculation. The capabilities of the method are showcased by optimizing the topology of hard as well as soft magnetic thin film structures and the results are verified by comparison with an analytical solution.

Complex free-space magnetic field textures induced by three-dimensional magnetic nanostructures.

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.

Dipolar-stabilized first and second-order antiskyrmions in ferrimagnetic multilayers.

Skyrmions and antiskyrmions are topologically protected spin structures with opposite vorticities. Particularly in coexisting phases, these two types of magnetic quasi-particles may show fascinating physics and potential for spintronic devices. While skyrmions are observed in a wide range of materials, until now antiskyrmions were exclusive to materials with D2d symmetry. In this work, we show first and second-order antiskyrmions stabilized by magnetic dipole-dipole interaction in Fe/Gd-based multilayers. We modify the magnetic properties of the multilayers by Ir insertion layers. Using Lorentz transmission electron microscopy imaging, we observe coexisting antiskyrmions, Bloch skyrmions, and type-2 bubbles and determine the range of material properties and magnetic fields where the different spin objects form and dissipate. We perform micromagnetic simulations to obtain more insight into the studied system and conclude that the reduction of saturation magnetization and uniaxial magnetic anisotropy leads to the existence of this zoo of different spin objects and that they are primarily stabilized by dipolar interaction.

Non-Planar Geometrical Effects on the Magnetoelectrical Signal in a Three-Dimensional Nanomagnetic Circuit.

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 modeling, 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 noncollinear 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.

Strayfield calculation for micromagnetic simulations using true periodic boundary conditions.

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.

Intrinsic DMI-free skyrmion formation and robust dynamic behaviors in magnetic hemispherical shells.

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.

Coexistence of distinct skyrmion phases observed in hybrid ferromagnetic/ferrimagnetic multilayers.

Materials hosting magnetic skyrmions at room temperature could enable compact and energetically-efficient storage such as racetrack memories, where information is coded by the presence/absence of skyrmions forming a moving chain through the device. The skyrmion Hall effect leading to their annihilation at the racetrack edges can be suppressed, for example, by antiferromagnetically-coupled skyrmions. However, avoiding modifications of the inter-skyrmion distances remains challenging. As a solution, a chain of bits could also be encoded by two different solitons, such as a skyrmion and a chiral bobber, with the limitation that it has solely been realized in B20-type materials at low temperatures. Here, we demonstrate that a hybrid ferro/ferri/ferromagnetic multilayer system can host two distinct skyrmion phases at room temperature, namely tubular and partial skyrmions. Furthermore, the tubular skyrmion can be converted into a partial skyrmion. Such systems may serve as a platform for designing memory applications using distinct skyrmion types.

Magnetic Position System Design Method Applied to Three-Axis Joystick Motion Tracking.

This manuscript discusses the difficulties with magnetic position and orientation (MPO) system design and proposes a general method for finding optimal layouts. The formalism introduces a system quality measure through state separation and reduces the question "How to design an MPO system?" to a global optimization problem. The latter is then solved by combining differential evolution algorithms with magnet shape variation based on analytical computations of the field. The proposed formalism is then applied to study possible realizations of continuous three-axis joystick motion tracking, realized with just a single magnet and a single 3D magnetic field sensor. The computations show that this is possible when a specific design condition is fulfilled and that large state separations as high as 1mT/∘ can be achieved under realistic conditions. Finally, a comparison to state-of-the-art design methods is drawn, computation accuracy is reviewed critically, and an experimental validation is presented.

Control of the noncollinear interlayer exchange coupling.

Interlayer exchange coupling in transition metal multilayers has been intensively studied for more than three decades and is incorporated into almost all spintronic devices. With the current spacer layers, only collinear magnetic alignment can be reliably achieved; however, controlling the coupling angle has the potential to markedly expand the use of interlayer exchange coupling. Here, we show that the coupling angle between the magnetic moments of two ferromagnetic layers can be precisely controlled by inserting a specially designed magnetic metallic spacer layer between them. The coupling angle is controlled solely by the composition of the spacer layer. Moreover, the biquadratic coupling strength, responsible for noncollinear alignment, is larger than that of current materials. These properties allow for the fabrication and study of not yet realized magnetic structures that have the potential to improve existing spintronic devices.

Microscopic Origin of Magnetization Reversal in Nanoscale Exchange-Coupled Ferri/Ferromagnetic Bilayers: Implications for High Energy Density Permanent Magnets and Spintronic Devices.

Giant exchange bias shifts of several Tesla have been reported in ferrimagnetic/ferromagnetic bilayer systems, which could be highly beneficial for contemporary high energy density permanent magnets and spintronic devices. However, the lack of microscopic studies of the reversal owing to the difficulty of measuring few nanometer-wide magnetic structures in high fields precludes the assessment of the lateral size of the inhomogeneity in relation to the intended application. In this study, the magnetic reversal process of nanoscale exchange-coupled bilayer systems, consisting of a ferrimagnetic TbFeCo alloy layer and a ferromagnetic [Co/Ni/Pt] N multilayer, was investigated. In particular, minor loop measurements, probing solely on the reversal characteristics of the softer ferromagnetic layer, reveal two distinct reversal mechanisms, which depend critically on the thickness of the ferromagnetic layer. For thick layers, irreversible switching of the macroscopic minor loop is observed. The underlying microscopic origin of this reversal process was studied in detail by high-resolution magnetic force microscopy, showing that the reversal is triggered by in-plane domain walls propagating through the ferromagnetic layer. In contrast, thin ferromagnetic layers show a hysteresis-free reversal, which is nucleation-dominated due to grain-to-grain variations in magnetic anisotropy of the Co/Ni/Pt multilayer and an inhomogeneous exchange coupling with the magnetically hard TbFeCo layer, as confirmed by micromagnetic simulations.

Micromagnetic Simulations of Submicron Vortex Structures for the Detection of Superparamagnetic Labels.

We present a numerical investigation on the detection of superparamagnetic labels using a giant magnetoresistance (GMR) vortex structure. For this purpose, the Landau-Lifshitz-Gilbert equation was solved numerically applying an external z-field for the activation of the superparamagnetic label. Initially, the free layer's magnetization change due to the stray field of the label is simulated. The electric response of the GMR sensor is calculated by applying a self-consistent spin-diffusion model to the precomputed magnetization configurations. It is shown that the soft-magnetic free layer reacts on the stray field of the label by shifting the magnetic vortex orthogonally to the shift direction of the label. As a consequence, the electric potential of the GMR sensor changes significantly for label shifts parallel or antiparallel to the pinning of the fixed layer. Depending on the label size and its distance to the sensor, the GMR sensor responds, changing the electric potential from 26.6 mV to 28.3 mV.

Additive Manufactured Polymer-Bonded Isotropic NdFeB Magnets by Stereolithography and Their Comparison to Fused Filament Fabricated and Selective Laser Sintered Magnets.

Magnetic isotropic NdFeB powder with a spherical morphology is used to 3D print magnets by stereolithography (SLA). Complex magnets with small feature sizes in a superior surface quality can be printed with SLA. The magnetic properties of the 3D printed bonded magnets are investigated and compared with magnets manufactured by fused filament fabrication (FFF), and selective laser sintering (SLS). All methods use the same hard magnetic isotropic NdFeB powder material. FFF and SLA use a polymer matrix material as binder, SLS sinters the powder directly. SLA can print magnets with a remanence of 388 mT and a coercivity of 0.923 T. A complex magnetic design for speed wheel sensing applications is presented and printed with all methods.

Spin Torque Efficiency and Analytic Error Rate Estimates of Skyrmion Racetrack Memory.

In this paper, the thermal stability of skyrmion bubbles and the critical currents to move them over pinning sites were investigated. For the used pinning geometries and the used parameters, the unexpected behavior is reported that the energy barrier to overcome the pinning site is larger than the energy barrier of the annihilation of a skyrmion. The annihilation takes place at boundaries by current driven motion, as well as due to the excitation over energy barriers, in the absence of currents, without forming Bloch points. It is reported that the pinning sites, which are required to allow thermally stable bits, significantly increase the critical current densities to move the bits in skyrmion-like structures to about jcrit = 0.62 TA/m². The simulation shows that the applied spin transfer model predicts experimentally obtained critical currents to move stable skyrmions at room temperature well, which is in contrast to simulations based on spin orbit torque that predict significantly too low critical currents. By calculating the thermal stability, as well as the critical current, we can derive the spin torque efficiency η = ΔE/Ic = 0.19 kBT300/µA, which is in a similar range to the simulated spin torque efficiency of MRAM structures. Finally, it is shown that the stochastic depinning process of any racetrack-like device requires an extremely narrow depinning time distribution smaller than ~6% of the current pulse length to reach bit error rates smaller than 10-9.

Additive Manufactured and Topology Optimized Passive Shimming Elements for Permanent Magnetic Systems.

A method to create a highly homogeneous magnetic field by applying topology optimized, additively manufactured passive shimming elements is investigated. The topology optimization algorithm can calculate a suitable permanent and nonlinear soft magnetic design that fulfills the desired field properties. The permanent magnetic particles are bonded in a polyamide matrix and they are manufactured with a low-cost, end-user 3D printer. Stray field measurements and an inverse stray field simulation framework can determine printing and magnetization errors. The customized shimming elements are manufactured by a selective melting process which produces completely dense soft magnetic metal parts. The methodology is demonstrated on a simple example of two axial symmetric cylindrical magnets, which generates a high inhomogeneous magnetic field. In this case, the maximum magnetic field density is 25 mT and the the homogeneity can be increased by a factor of 35 or down to 6‰. Simulation and measurement results point out a good conformity. Additional topology optimizations of more than one shimming element layer show the opportunity to make the manufactured magnetic system even more homogeneous.

3D Printing of Polymer-Bonded Rare-Earth Magnets With a Variable Magnetic Compound Fraction for a Predefined Stray Field.

Additive manufacturing of polymer-bonded magnets is a recently developed technique, for single-unit production, and for structures that have been impossible to manufacture previously. Also, new possibilities to create a specific stray field around the magnet are triggered. The current work presents a method to 3D print polymer-bonded magnets with a variable magnetic compound fraction distribution. This means the saturation magnetization can be adjusted during the printing process to obtain a required external field of the manufactured magnets. A low-cost, end-user 3D printer with a mixing extruder is used to mix permanent magnetic filaments with pure polyamide (PA12) filaments. The magnetic filaments are compounded, extruded, and characterized for the printing process. To deduce the quality of the manufactured magnets with a variable magnetic compound fraction, an inverse stray field framework is developed. The effectiveness of the printing process and the simulation method is shown. It can also be used to manufacture magnets that produce a predefined stray field in a given region. This opens new possibilities for magnetic sensor applications. This setup and simulation framework allows the design and manufacturing of polymer-bonded permanent magnets, which are impossible to create with conventional methods.