Finite Element Micromagnetic Simulations Research Articles

Parallel-in-time integration of the Landau–Lifshitz–Gilbert equation with the parallel full approximation scheme in space and time

Speeding up computationally expensive problems, such as numerical simulations of large micromagnetic systems, requires efficient use of parallel computing infrastructures. While parallelism across space is commonly exploited in micromagnetics, this strategy performs poorly once a minimum number of degrees of freedom per core is reached. We use magnum.pi, a finite-element micromagnetic simulation software, to investigate the Parallel Full Approximation Scheme in Space and Time (PFASST) as a space- and time-parallel solver for the Landau–Lifshitz–Gilbert equation (LLG). Numerical experiments show that PFASST enables efficient parallel-in-time integration of the LLG, significantly improving the speedup gained from using a given number of cores as well as allowing the code to scale beyond spatial limits.

On the magnetization of an antiferromagnetic film with uniaxial magnetocrystalline anisotropy

In this work finite element micromagnetic simulations were performed in order to study the magnetic behavior of an antiferromagnetic film with uniaxial magnetocrystalline anisotropy under the influence of external magnetic fields. The stability of the antiferromagnetic phase and metamagnetic phases arising from it, as well as the features and the evolution of the magnetic microstructure are studied in three different combinations of the easy axis and applied field directions. The investigation was carried out in a wide range of geometrical and intrinsic film’s parameters in order to derive analytical relations regarding the critical external fields.

Magnetization reversals in core–shell sphere clusters: finite-element micromagnetic simulation and machine learning analysis

Recently developed permanent magnets, featuring specially engineered microstructures of inhomogeneous magnetic phases, are being considered as cost-effective alternatives to homogeneous single-main-phase hard magnets composed of Nd2Fe14B, without compromising performance. In this study, we conducted a comprehensive examination of a core–shell sphere cluster model of Ce-substituted inhomogeneous Nd2-δCeδFe14B phases versus homogeneous magnetic phases, utilizing finite-element micromagnetic simulation and machine learning methods. This involved a meticulous, sphere-by-sphere analysis of individual demagnetization curves calculated from the cluster model. The grain-by-grain analyses unveiled that these individual demagnetization curves can elucidate the overall magnetization reversal in terms of the nucleation and coercive fields for each sphere. Furthermore, it was observed that Nd-rich spheres exhibited much broader ranges of nucleation and coercive field distributions, while Nd-lean spheres showed relatively narrower ranges. To identify the key parameter responsible for the notable differences in the nucleation fields, we constructed a machine learning regression model. The model utilized numerous hyperparameter sets, optimized through the very fast simulated annealing algorithm, to ensure reliable training. Using the kernel SHapley Additive eXplanation (SHAP) technique, we inferred that stray fields among the 11 parameters were closely related to coercivity. We further substantiated the machine learning models’ inference by establishing an analytical model based on the eigenvalue problem in classical micromagnetic theory. Our grain-by-grain interpretation can guide the optimal design of granular hard magnets from Nd2Fe14B and other abundant rare earth transition elements, focusing on extraordinary performance through the careful adjustment of microstructures and elemental compositions.

Micromagnetic Simulation of Increased Coercivity of (Sm, Zr)(Co, Fe, Cu)z Permanent Magnets

The finite element micromagnetic simulation is used to study the role of complex composition of 2:17R-cell boundaries in the realization of magnetization reversal processes of (Sm, Zr)(Co, Cu, Fe)z alloys intended for high-energy permanent magnets. A modified sandwich model is considered for the combinations of 2:7R/1:5H phase and 5:19R/1:5H phase layers as the 2:17R-cell boundaries in the alloy structure. The results of the simulation represented in the form of coercive force vs. total width of cell boundary showed the possibility of reaching the increased coercivity at the expense of 180°-domain wall pinning at the additional barriers within cell boundaries. The phase and structural states of the as-cast Sm1-xZrx(Co0.702Cu0.088Fe0.210)z alloy sample with x = 0.13 and z = 6.4 are studied, and the presence of the above phases in the vicinity of the 1:5H phase was demonstrated.

Real-space observation of standing spin-wave modes in a magnetic disk

In-plane standing spin-wave modes in a minute magnetic disk are directly observed by using time-resolved magneto-optical microscopy synchronized with microwaves. The time-resolved microscopy allowed us to obtain snapshots of standing spin-wave modes in a magnetic disk, which show a hourglass-like standing spin wave pattern. We found that the characteristic pattern is caused by spatially nonuniform magnetization and a strong microwave excitation in terms of finite element calculation and micromagnetic simulations. The technique we developed in this work allows us to access magnetization dynamics in microstructured magnets under strong microwave pumping.

Bending and Collapse: Magnetic Recording Fidelity of Magnetofossils From Micromagnetic Simulation

AbstractMagnetosome chains produced by magnetotactic bacteria are important paleoenvironmental and paleomagnetic recorders. It has been shown that magnetic properties of magnetosome chains are closely related to their morphology and chain structures; however, the in situ structures of magnetosome chains in sediments (magnetofossils) are not known. Magnetosome chains are subject to various deformations after cell dissolution and are therefore unlikely to be fully intact, obscuring their original magnetic signals. Here, we use finite element micromagnetic simulations to quantify changes in magnetic signals in response to chain deformation, in particular, as a function of variable degrees of bending and collapse. Our results indicate that bending/collapse leads to a significant coercivity reduction and domain state transition of the chain. Therefore, hysteresis parameters can be used to assess the degree of chain bending/collapse in magnetofossil‐rich sediments. Calculations of the contributions of chain bending/collapse to the post‐depositional remanent magnetization (pDRM) of magnetofossils indicate that pDRM remains both faithful to the pre‐bending/collapse natural remanent magnetization, and that the remanence of some structurally deformed magnetofossil assemblages remains thermally stable over billion‐year timescales, suggesting that even strongly deformed magnetosome chains in ancient geological materials retain faithful paleomagnetic records and thus have potentials for tracing ancient geomagnetic field variations and microbial activities on early Earth.

Switchable magnetic frustration in buckyball nanoarchitectures

Recent progress in nanofabrication has led to the emergence of three-dimensional magnetic nanostructures as a vibrant field of research. This includes the study of three-dimensional arrays of interconnected magnetic nanowires with tunable artificial spin-ice properties. Prominent examples of such structures are magnetic buckyball nanoarchitectures, which consist of ferromagnetic nanowires connected at vertex positions corresponding to those of a C60 molecule. These structures can be regarded as prototypes for the study of the transition from two- to three-dimensional spin-ice lattices. In spite of their significance for three-dimensional nanomagnetism, little is known about the micromagnetic properties of buckyball nanostructures. By means of finite-element micromagnetic simulations, we investigate the magnetization structures and the hysteretic properties of several sub-micron-sized magnetic buckyballs. Similar to ordinary artificial spin-ice lattices, the array can be magnetized in a variety of zero-field states with vertices exhibiting different degrees of magnetic frustration. Remarkably, and unlike planar geometries, magnetically frustrated states can be reversibly created and dissolved by applying an external magnetic field. This easiness to insert and remove defect-like magnetic charges, made possible by the angle-selectivity of the field-induced switching of individual nanowires, demonstrates a potentially significant advantage of three-dimensional nanomagnetism compared to planar geometries. The control provided by the ability to switch between ice-rule obeying and magnetically frustrated structures could be an important feature of future applications, including magnonic devices exploiting differences in the fundamental frequencies of these configurations.

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.

Effect of curvature on the eigenstates of magnetic skyrmions

Spectrum of spin eigenmodes localized on a ferromagnetic skyrmion pinned by a geometrical defect (bump) of magnetic films is studied theoretically. By means of direct numerical solution of the corresponding eigenvalue problem and finite element micromagnetic simulations we demonstrate, that the curvature can induce localized modes with higher azimuthal and radial quantum numbers, which are absent for planar skyrmions (for the same parameters). The eigenfrequencies of all modes, except the breathing and gyromodes decreases with increasing curvature. Due to the translational symmetry break, the zero translational mode of the skyrmion gains a finite frequency and forms the gyromode, which describes the uniform rotation of skyrmions around the equilibrium position. In order to treat the gyromotion analytically we developed a Thiele-like collective variable approach. We show that N\'{e}el skyrmions in curvilinear films experience a driving force originating from the gradient of the mean curvature. The gyrofrequency of the pinned skyrmion is proportional to the second derivative of the mean curvature at the point of equilibrium.

Visualizing Magnetic Structure in 3D Nanoscale Ni-Fe Gyroid Networks.

Arrays of interacting 2D nanomagnets display unprecedented electromagnetic properties via collective effects, demonstrated in artificial spin ices and magnonic crystals. Progress toward 3D magnetic metamaterials is hampered by two challenges: fabricating 3D structures near intrinsic magnetic length scales (sub-100 nm) and visualizing their magnetic configurations. Here, we fabricate and measure nanoscale magnetic gyroids, periodic chiral networks comprising nanowire-like struts forming three-connected vertices. Via block copolymer templating, we produce Ni75Fe25 single-gyroid and double-gyroid (an inversion pair of single-gyroids) nanostructures with a 42 nm unit cell and 11 nm diameter struts, comparable to the exchange length in Ni-Fe. We visualize their magnetization distributions via off-axis electron holography with nanometer spatial resolution and interpret the patterns using finite-element micromagnetic simulations. Our results suggest an intricate, frustrated remanent state which is ferromagnetic but without a unique equilibrium configuration, opening new possibilities for collective phenomena in magnetism, including 3D magnonic crystals and unconventional computing.

Mind the gap: Towards a biogenic magnetite palaeoenvironmental proxy through an extensive finite-element micromagnetic simulation

Magnetotactic bacteria (MTB) preserved in sediments (magnetofossils) are highly sensitive to environmental and climatic forcing and can therefore be used as an easy-to-measure palaeo-environment proxy. It is known that magnetofossils occur with various morphologies – each with its own particular magnetic signature. We present the first systematic extensive micromagnetic study relating magnetosome chain morphology to their hysteresis parameters. We modelled more than 450 different intact magnetosome chains in the finite-element micromagnetic modelling software MERRILL, relating the morphology (magnetosome size, elongation, intra-chain spacing, and chain length) to the magnetic parameters of coercivity, coercivity of remanence, and saturation magnetization, and analyzing domain-states and switching modes. It is shown that magnetic properties are far dominated by the intra-chain spacing. However, analysis of a large set of published TEM images of different MTB strains shows a strong correlation between intra-chain spacing and grain size. This correlation can in principle be used to infer MTB morphology purely from magnetic measurements, providing a framework to use biogenic magnetite as a sensitive palaeo-environment proxy. Additionally, it is found that MTB carefully control intra-chain spacing during magnetosome synthesis to maximize saturation remanence as well as magnetic stability.

Stable and manipulable Bloch point

The prediction of magnetic skyrmions being used to change the way we store and process data has led to materials with Dzyaloshinskii-Moriya interaction coming into the focus of intensive research. So far, studies have looked mostly at magnetic systems composed of materials with single chirality. In a search for potential future spintronic devices, combination of materials with different chirality into a single system may represent an important new avenue for research. Using finite element micromagnetic simulations, we study an FeGe disk with two layers of different chirality. We show that for particular thicknesses of layers, a stable Bloch point emerges at the interface between two layers. In addition, we demonstrate that the system undergoes hysteretic behaviour and that two different types of Bloch point exist. These ‘head-to-head’ and ‘tail-to-tail’ Bloch point configurations can, with the application of an external magnetic field, be switched between. Finally, by investigating the time evolution of the magnetisation field, we reveal the creation mechanism of the Bloch point. Our results introduce a stable and manipulable Bloch point to the collection of particle-like state candidates for the development of future spintronic devices.

Effect of misalignments of individual grains’ easy axis on magnetization-reversal process in granular NdFeB magnets: A finite-element micromagnetic simulation study

We studied, using finite-element micromagnetic simulations, magnetization-reversal processes of exchange-decoupled granular NdFeB magnets for a cubic model system that consists of 27 polyhedral grains of different crystallographic orientations along with non-magnetic grain boundaries. In the simulations, we varied the degree of easy-axis alignment of individual grains with different uniaxial orientations. For perfect uniaxial alignment, i.e., α = 1, the normalized remanence magnetization and the normalized coercivity along with the maximum energy product (BH)max were estimated to be 1, 0.64, and 64.71 MGOe, respectively, and were decreased to 0.59, 0.42, and 20.56 MGOe, representing reductions of 41, 34, and 68%, respectively, for a randomly oriented alignment of the grains’ easy axis, α = 0.6. From a comparison of the magnetization-reversal processes for different easy-axis alignments, i.e., α = 1, 0.98, 0.94, 0.87, 0.82, 0.79, and 0.60, it was found that all of the magnetization reversals consisted of three different sub-processes: the nucleation of a reversed domain within a grain and the expansion of nucleated domains to complete the reversal in that grain via domain-wall motion; magnetization-reversal propagation to the next grain via successive nucleation in other neighboring grains; further rotation of the magnetizations of all of the grains for cases of larger misalignment of grains’ easy axis. According to these reversal sub-processes, as the degree of easy-axis alignment, α, decreased, the nucleation of reversed domains occurred at smaller field strengths and the apparent grain-by-grain reversal propagation became slower. The lower the value of α, the lower the energy required for the nucleation of reversed domains and the more predominant the pinning of the apparent grain-by-grain reversal propagation. These reversal processes led to hard-magnet parameters such as normalized remanence magnetization and coercivity and the maximum energy product (BH)max. The values of the parameters for the perfect uniaxial alignment of grains became smaller with increasing misalignment of the grain’s easy axis. The results provide both a better understanding of micro-scale magnetization-reversal processes and guidelines for optimal design of hard-magnet microstructures for higher (BH)max values.

Automated meshing of electron backscatter diffraction data and application to finite element micromagnetics

This paper gives a procedure for automatically generating finite element meshes with an adaptive mesh size from Electron Backscatter Diffraction (EBSD) data. After describing the procedure in detail, including preliminary and image processing steps, an example application is given. The method was used to carry out finite element (FE) micromagnetic simulations based on real microstructures in the hard magnetic material, MnAl. A fast micromagnetic solver was used to compute hysteresis properties from the finite element mesh generated automatically from EBSD data. The visualization of the magnetization evolution showed that the reversal is governed by domain wall pinning at twin boundaries. The calculated coercive fields are very sensitive to changes of the Gilbert damping constant, even for low field rates.

Modeling magnetic-field-induced domain wall propagation in modulated-diameter cylindrical nanowires

Domain wall propagation in modulated-diameter cylindrical nanowires is a key phenomenon to be studied with a view to designing three-dimensional magnetic memory devices. This paper presents a theoretical study of transverse domain wall behavior under the influence of a magnetic field within a cylindrical nanowire with diameter modulations. In particular, domain wall pinning close to the diameter modulation was quantified, both numerically, using finite element micromagnetic simulations, and analytically. Qualitative analytical model for gently sloping modulations resulted in a simple scaling law which may be useful to guide nanowire design when analyzing experiments. It shows that the domain wall depinning field value is proportional to the modulation slope.

Spin-wave duplexer studied by finite-element micromagnetic simulation

We conceptually designed a robust nano-scale waveguide structure suitable for potential use as a spin-wave duplexer that allows signal propagation only of selected narrow-band frequencies and duplex transmission in a three-port device comprising a receiver, a transmitter, and their common antenna. The waveguide structure combines three different arms and a circular ring, both made of nanostrip waveguides and a single magnetic material for reliably controllable propagations of spin waves. We attribute the observed duplex transmission of spin waves of narrow pass bands to scattering of spin waves by edge solitons placed at contact areas between the arms and the circular ring. This work proposes the first concept of nano-scale magnonic duplexers operating beyond GHz-frequency ranges.

Microstructure of a Dy-free Nd-Fe-B sintered magnet with 2 T coercivity

We investigated the microstructure of the Dy-free high coercivity sintered magnet that was developed by optimizing the chemical composition of Nd-Fe-B sintered magnet with 0.1 at.%Ga doping in order to understand the reasons for the high coercivity of μ0Hc = 2.0 T and the improved squareness of 0.95 compared to a recently developed 0.5 at. %Ga doped magnet. The as-sintered 0.1Ga sample shows a relatively high coercivity of 1.5 T owing to the finer grain size of ∼3.3 μm and the higher Al concentration up to 1.3 at.% compared to post-sinter annealed 0.5Ga magnet that has a grain size of ∼5.5 μm and Al concentration of 0.9 at. %. The post-sinter annealing leads to a substantial coercivity increase from 1.5 to 2.0 T due to the formation of a thick and crystalline intergranular grain boundary (GB) phase containing 30-60 at.% of Nd. The trace Ga addition of 0.1 at.% improved the wettability of the liquids and facilitated the formation of thick GB phase during the post-sinter annealing. The good squareness is mainly attributed to the ferromagnetic nature of the intergranular GB phase as well as the strong crystal alignment to the easy axis. Finite element micromagnetic simulations of the demagnetization processes of the models incorporating experimentally determined GB thickness and chemistry well explain the simultaneous achievement of the high coercivity and good squareness observed in this magnet in contrast to the very poor squareness observed in the post-sinter annealed 0.5Ga sample.

Periodic chiral magnetic domains in single-crystal nickel nanowires

We report on experimental and computational investigations of the domain structure of ~0.2 x 0.2 x 8 {\mu}m single-crystal Ni nanowires (NWs). The Ni NWs were grown by a thermal chemical vapor deposition technique that results in highly-oriented single-crystal structures on amorphous SiOx coated Si substrates. Magnetoresistance measurements of the Ni NWs suggest the average magnetization points largely off the NW long axis at zero field. X-ray photoemission electron microscopy images show a well-defined periodic magnetization pattern along the surface of the nanowires with a period of {\lambda} = 250 nm. Finite element micromagnetic simulations reveal that an oscillatory magnetization configuration with a period closely matching experimental observation ({\lambda} = 240 nm) is obtainable at remanence. This magnetization configuration involves a periodic array of alternating chirality vortex domains distributed along the length of the NW. Vortex formation is attributable to the cubic anisotropy of the single crystal Ni NW system and its reduced structural dimensions. The periodic alternating chirality vortex state is a topologically protected metastable state, analogous to an array of 360{\deg} domain walls in a thin strip. Simulations show that other remanent states are also possible, depending on the field history. Effects of material properties and strain on the vortex pattern are investigated. It is shown that at reduced cubic anisotropy vortices are no longer stable, while negative uniaxial anisotropy and magnetoelastic effects in the presence of compressive biaxial strain contribute to vortex formation.

Vortex-chirality-dependent standing spin-wave modes in soft magnetic nanotubes

Spin-wave (SW) modes excited in cylindrical nanotubes of finite length were investigated using finite-element micromagnetic simulations. From the simulation results along with the relevant analytical interpretation, we found unique dynamic modes representative of a variety of standing SW modes. Those modes are controllable not only according to the geometric confinements of given nanotubes but also by the relative configuration of the vortex-chirality at both ends of the nanotubes. The asymmetric (symmetric) spin-wave dispersion originates from nonreciprocal (reciprocal) spin-wave propagations from the parallel (antiparallel) configuration of vortex chiralities at both ends of the nanotubes. Using a simple analytical model, we estimated the quantized dispersions of the excited modes that agree with the simulation results. This work facilitates further understanding of discrete standing SW modes in three-dimensional curvilinear nano-elements, such as cylindrical nanotubes, and opens up a broader and deeper perspective on chirality-dependent SW modes.