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.

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.

Scaling dependence and tailoring of the pinning field in FePt-based exchange coupled composite media.

We studied exchange coupled composite (ECC) media with an out-of-plane easy axis consisting of hard magnetic L1(0) chemically ordered FePtCu alloy films and magnetically softer [Co/Pt](N) multilayer stacks. By tailoring the structural properties of the ternary FePtCu alloy and [Co/Pt](N) multilayers, we can tune the magnetic parameters of the composite in a wide range. This allowed us to address experimentally one of the most crucial properties determining the performance of ECC media, namely the pinning field of the magnetic domain wall present at the interface between the hard and soft layers. We demonstrate that the pinning field is proportional to the difference of the magnetic anisotropy constants of the hard and soft layers, which confirms the theoretical predictions. Furthermore, we show that the pinning field can be efficiently decreased after an additional annealing step. Transmission electron microscopy investigations indicated that the origin of the observed effect is due to a heat-induced phase transformation of iron oxide present at the interface between the hard and soft layers. This study reveals that tailoring the properties of the hard/soft interface is another efficient tuning knob for optimization of the performance of ECC media.

Spin-polarized transport in ferromagnetic multilayers: An unconditionally convergent FEM integrator.

We propose and analyze a decoupled time-marching scheme for the coupling of the Landau-Lifshitz-Gilbert equation with a quasilinear diffusion equation for the spin accumulation. This model describes the interplay of magnetization and electron spin accumulation in magnetic and nonmagnetic multilayer structures. Despite the strong nonlinearity of the overall PDE system, the proposed integrator requires only the solution of two linear systems per time-step. Unconditional convergence of the integrator towards weak solutions is proved.

Steerable current-driven emission of spin waves in magnetic vortex pairs.

The efficient excitation of spin waves is a key challenge in the realization of magnonic devices. We demonstrate current-driven generation of spin waves in antiferromagnetically coupled magnetic vortices. We use time-resolved x-ray microscopy to directly image the emission of spin waves upon the application of alternating currents flowing directly through the magnetic stack. Micromagnetic simulations allow us to identify the current-driven Oersted field as the main origin of excitation, in contrast to spin-transfer torques. In our case, these internal Oersted fields have an orders of magnitude higher spin-wave excitation efficiency than commonly used stripline antennas. For magnetostrictive materials, we furthermore demonstrate that the direction of magnon propagation can be steered by increasing the excitation amplitude, which modifies the underlying magnetization profile through an additional anisotropy. The demonstrated methods allow for the efficient and tunable excitation of spin waves, marking a substantial advance concerning the design of magnonic devices.

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.

Magnetic anisotropy and GGG substrate stray field in YIG films down to millikelvin temperatures.

Quantum magnonics investigates the quantum-mechanical properties of magnons, such as quantum coherence or entanglement for solid-state quantum information technologies at the nanoscale. The most promising material for quantum magnonics is the ferrimagnetic yttrium iron garnet (YIG), which hosts magnons with the longest lifetimes. YIG films of the highest quality are grown on a paramagnetic gadolinium gallium garnet (GGG) substrate. The literature has reported that ferromagnetic resonance (FMR) frequencies of YIG/GGG decrease at temperatures below 50 K despite the increase in YIG magnetization. We investigated a 97 nm-thick YIG film grown on 500 µm-thick GGG substrate through a series of experiments conducted at temperatures as low as 30 mK, and using both analytical and numerical methods. Our findings suggest that the primary factor contributing to the FMR frequency shift is the stray magnetic field created by the partially magnetized GGG substrate. This stray field is antiparallel to the applied external field and is highly inhomogeneous, reaching up to 40 mT in the center of the sample. At temperatures below 500 mK, the GGG field exhibits a saturation that cannot be described by the standard Brillouin function for a paramagnet. Including the calculated GGG field in the analysis of the FMR frequency versus temperature dependence allowed the determination of the cubic and uniaxial anisotropies. We find that the total crystallographic anisotropy increases more than three times with the decrease in temperature down to 2 K. Our findings enable accurate predictions of the YIG/GGG magnetic systems behavior at low and ultralow millikelvin temperatures, crucial for developing quantum magnonic devices.

Combining micromagnetism and magnetostatic Maxwell equations for multiscale magnetic simulations.

Magnetostatic Maxwell equations and the Landau-Lifshitz-Gilbert (LLG) equation are combined to a multiscale method, which allows to extend the problem size of traditional micromagnetic simulations. By means of magnetostatic Maxwell equations macroscopic regions can be handled in an averaged and stationary sense, whereas the LLG allows to accurately describe domain formation as well as magnetization dynamics in some microscopic subregions. The two regions are coupled by means of their strayfield and the combined system is solved by an optimized time integration scheme.

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.

Removal of earth's magnetic field effect on magnetoelastic resonance sensors by an antisymmetric bias field.

Magnetoelastic sensors are used in a wide field of wireless sensing applications. The sensing element is a low-cost magnetostrictive ribbon whose resonant frequency depends on the measured quantity. The accuracy of magnetoelastic sensors is limited by the fact that the resonant frequency is also affected by the earth's magnetic field. In this paper we present a technique to minimize this effect by applying an antisymmetric magnetic bias field to the ribbon. The ribbon's response to external perturbation fields was measured and compared to a conventional sensor design. Our results show that the influence of the earth's magnetic field could be reduced by 77%.

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.

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.

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.

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.

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.

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.

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.