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In this work, we investigate the influence of curvature on the dynamic susceptibility in FeGe nanowires, both curved and straight, hosting a skyrmionic tube texture under the action of an external bias field, using micromagnetic simulations. Our results demonstrate that both the resonance frequencies and the number of resonant peaks are highly dependent on the curvature of the system. To further understand the nature of the spin wave modes, we analyze the spatial distributions of the resonant mode amplitudes and phases, describing the differences among resonance modes observed. The ability to control the dynamic properties and frequencies of these nanostructures underscores their potential application in frequency-selective magnetic devices. [ABSTRACT FROM AUTHOR]

The influence of size on the dynamic magnetic hysteresis properties and dynamic phase transitions of Fe and F e 3 O 4 2D-circular nanodisks with varying diameters has been explored in the micromagnetic framework. This investigation is conducted under a sinusoidal dynamic magnetic field along the x-direction by solving the Landau-Lifshitz-Gilbert (LLG) equation with the OOMMF software at zero temperature. The dynamic hysteresis of nanodisks is profoundly impacted by both the frequency and amplitude of the applied external field, along with the particle size. As particle size decreases, there is an observed increase in the frequency values at which the transition to dynamic ordered frequency occurs in Fe nanodisks, whereas a decrease is noted in the transition frequency values of F e 3 O 4 nanodisks. [ABSTRACT FROM AUTHOR]

Magnetic hysteresis measurements are routinely made in the Earth and planetary sciences to identify geologically meaningful magnetic recorders, and to study variations in present and past environments. Interpreting magnetic hysteresis data in terms of domain state and paleomagnetic stability are major motivations behind undertaking these measurements, but the interpretations remain fraught with challenges and ambiguities. To shed new light on these ambiguities, we have undertaken a systematic micromagnetic study to quantify the magnetic hysteresis behavior of room‐temperature magnetite as a function of particle size (45–195 nm; equivalent spherical volume diameter) and shape (oblate, prolate and equant); our models span uniformly magnetized single domain (SD) to non‐uniformly magnetized single vortex (SV) states. Within our models the reduced magnetization associated with SV particles marks a clear boundary between SD (≥0.5) and SV (<0.5) magnetite. We further identify particle sizes and shapes with unexpectedly low coercivity and coercivity of remanence. These low coercivity regions correspond to magnetite particles that typically have multiple possible magnetic domain state configurations, which have been previously linked to a zone of unstable magnetic recorders. Of all the hysteresis parameters investigated, transient hysteresis is most sensitive to particles that exhibit such domain state multiplicity. When experimental transient hysteresis is compared to paleointensity behavior, we show that increasing transience corresponds to more curved Arai plots and less accurate paleointensity results. We therefore strongly suggest that transient behavior should be more routinely measured during rock magnetic investigations. Plain Language Summary: Characterizing the magnetic properties and behavior of natural materials in the Earth and planetary sciences is key to identifying reliable magnetic recorders and variations in the environment. One standard method for achieving this is through room‐temperature measurements of magnetic hysteresis. However, the interpretation of magnetic hysteresis data remains one of the most challenging aspects of rock magnetism. To improve our understanding of magnetic hysteresis data, we have systematically investigated how the hysteresis properties of distributions of randomly oriented magnetite change as a function of particle size and shape and how this can help us quantify the contents of natural materials and identify rocks that may give unreliable magnetic signals. We model prolate, oblate and equant magnetite particles in the size range 45–195 nm. We show that magnetic hysteresis defines a clear boundary between simple uniform magnetic structures and more complex non‐uniform magnetic structures. We also identify the sizes and morphologies of magnetic particles that are likely to have unstable remanent magnetizations. These unstable particles are associated with distinctive hysteresis behavior, suggesting that hysteresis data can be used to identify rock samples dominated by such behavior. Key Points: Magnetic hysteresis is micromagnetically modeled for oblate, prolate and equant magnetite particles (45–195 nm)Hysteresis loop shape is highly dependent on particle size/shape due to complex switching behavior of single vortex statesTransient magnetic hysteresis is a powerful tool for identifying stable remanent magnetizations but is currently infrequently reported [ABSTRACT FROM AUTHOR]

The ability of rocks to hold a reliable record of the ancient geomagnetic field depends on the structure and stability of magnetic domain‐states contained within constituent particles. In paleomagnetic studies, the Day plot is an easily constructed graph of magnetic hysteresis parameters that is frequently used to estimate the likely magnetic recording stability of samples. Often samples plot in the region of the Day plot attributed to so‐called pseudo‐single‐domain particles with little understanding of the implications for domain‐states or recording fidelity. Here we use micromagnetic models to explore the hysteresis parameters of magnetite particles with idealized prolate and oblate truncated‐octahedral geometries containing single domain (SD), single‐vortex and occasionally multi‐vortex states. We show that these domain states exhibit a well‐defined trend in the Day plot that extends from the SD region well into the multi‐domain region, all of which are likely to be stable remanence carriers. We suggest that although the interpretation of the Day plot and its variants might be subject to ambiguities, if the magnetic mineralogy is known, it can still provide some useful insights about paleomagnetic specimens' dominant domain state, average particle sizes and, consequently, their paleomagnetic stability. Plain Language Summary: Ancient magnetic field recordings from rocks, provide information about the early habitability of Earth and formation of the Solar System. Key to understanding the reliability of these magnetic recordings is knowing the particle size and shape of a rock's constituent magnetic minerals. Small magnetite particles (⪅100 nm) are magnetically uniform, but as particle size increases the magnetic structures become non‐uniform and increasingly complex. These different types of structures are termed domain states, and yield different magnetic hysteresis responses, often summarized on a so‐called "Day" diagram—a commonly used diagnostic of domain state (or particle size). The position of particles in the size 100–1,000 nm on the Day plot is poorly quantified. This is a problem, as it has been shown in the last 5 years, that this particle size range carries the most stable magnetic recordings, lasting potentially longer than the age of the universe. These particles contain vortex‐like magnetic structures. Using a numerical micromagnetic algorithm, this is the first comprehensive study to quantify the magnetic response of vortex structures on the Day plot. We show that careful use of the Day plot can provide insight into the ability of a sample to retain recordings of the ancient geomagnetic field. Key Points: Vortex domain states for magnetite predominantly have pseudo‐single‐domain Day plot characteristicsIf the mineralogy is known, the Day plot can provide an estimate on the dominance of stable remanence carriers [ABSTRACT FROM AUTHOR]

The ability to characterize three-dimensional (3D) magnetization distributions in nanoscale magnetic materials and devices is essential to fully understand their static and dynamic magnetic properties. Phase contrast techniques in the transmission electron microscope (TEM), such as electron holography and electron ptychography, can be used to record two-dimensional (2D) projections of the in-plane magnetic induction of 3D nanoscale objects. Although the 3D magnetic induction can in principle be reconstructed from one or more tilt series of such 2D projections, conventional tomographic reconstruction algorithms do not recover the 3D magnetization within a sample directly. Here, we use simulations to describe the basis of an improved model-based algorithm for the tomographic reconstruction of a 3D magnetization distribution from one or more tilt series of electron optical phase images recorded in the TEM. The algorithm allows a wide range of physical constraints, including a priori information about the sample geometry and magnetic parameters, to be specified. It also makes use of minimization of the micromagnetic energy in the loss function. We demonstrate the reconstruction of the 3D magnetization of a localized magnetic soliton — a hopfion ring — and discuss the influence of noise, choice of magnetic constants, maximum tilt angle and number of tilt axes on the result. The algorithm can in principle be adapted for other magnetic contrast imaging techniques in the TEM, as well as for other magnetic characterization techniques, such as those based on X-rays or neutrons. [ABSTRACT FROM AUTHOR]

AbstractIn the article, based on the theory of micromagnetism, a special case of the moment theory of elasticity for an anisotropic material and the Hadamard compatibility condition for deformations, a microstructural model of the behavior of the Heusler alloy with shape memory in magnetic and force fields is constructed. The dynamics of the magnetic process is described by the Landau-Lifshitz-Gilbert equation. Using the Galerkin procedure, variational equations corresponding to the differential relations of the magnetic problem are written out and the finite element method is used for numerical realization of these equations. Considering the problems of the moment theory of elasticity we come to a special case of this theory in which, under the action of force and magnetic fields, the tensor of moment stresses initiated only by the magnetic field is zero. Using this result, the general expression for the state equation is constructed that describes, within the framework of small deformations, the behavior of an anisotropic material under force and magnetic influences. The stress tensor in this equation consists of a symmetric part and a skew-symmetric part, depending only on the magnetic field. For the problem of deformable solid mechanics with such a state equation, the Lagrange variational equation was constructed, which made it possible to reduce the requirements for the smoothness of the desired solution by an order of magnitude. The general expression for the state equation has been concretized for an anisotropic material having one axis of symmetry of the fourth order and two axes of symmetry of the second order. The solution of the Hadamard compatibility equation for deformations made it possible to determine the sliding surfaces and sliding directions in a tetragonal crystal cell of a ferromagnetic material in the martensitic state, along which a shift occurs, leading to the appearance and dissipation of a twinned structure. The microstructural model, constructed using all these positions, is used to describe experimental results on the combined action of magnetic and force fields on the Heusler alloy. [ABSTRACT FROM AUTHOR]

Abstract Magnetic hysteresis measurements are routinely made in the Earth and planetary sciences to identify geologically meaningful magnetic recorders, and to study variations in present and past environments. Interpreting magnetic hysteresis data in terms of domain state and paleomagnetic stability are major motivations behind undertaking these measurements, but the interpretations remain fraught with challenges and ambiguities. To shed new light on these ambiguities, we have undertaken a systematic micromagnetic study to quantify the magnetic hysteresis behavior of room‐temperature magnetite as a function of particle size (45–195 nm; equivalent spherical volume diameter) and shape (oblate, prolate and equant); our models span uniformly magnetized single domain (SD) to non‐uniformly magnetized single vortex (SV) states. Within our models the reduced magnetization associated with SV particles marks a clear boundary between SD (≥0.5) and SV (

Abstract The ability of rocks to hold a reliable record of the ancient geomagnetic field depends on the structure and stability of magnetic domain‐states contained within constituent particles. In paleomagnetic studies, the Day plot is an easily constructed graph of magnetic hysteresis parameters that is frequently used to estimate the likely magnetic recording stability of samples. Often samples plot in the region of the Day plot attributed to so‐called pseudo‐single‐domain particles with little understanding of the implications for domain‐states or recording fidelity. Here we use micromagnetic models to explore the hysteresis parameters of magnetite particles with idealized prolate and oblate truncated‐octahedral geometries containing single domain (SD), single‐vortex and occasionally multi‐vortex states. We show that these domain states exhibit a well‐defined trend in the Day plot that extends from the SD region well into the multi‐domain region, all of which are likely to be stable remanence carriers. We suggest that although the interpretation of the Day plot and its variants might be subject to ambiguities, if the magnetic mineralogy is known, it can still provide some useful insights about paleomagnetic specimens' dominant domain state, average particle sizes and, consequently, their paleomagnetic stability.

Bloch points in magnetic materials are attractive entities in view of magnetic information transport. Here, Bloch point configuration has been investigated and experimentally determined in a magnetic trilayer (Gd12Co88/Nd17Co83/Gd24Co76) with carefully adjusted composition within the ferrimagnetic GdxCo1-x alloys in order to engineer saturation magnetization, exchange length, and interlayer couplings (ferromagnetic vs antiferromagnetic). X-ray vector magnetic tomography has allowed us to determine experimentally Bloch point polarity (related to topological charge) and Bloch point helicity γ (determined by magnetostatic energy). At the bottom layer (close to the ferromagnetic interface), Bloch points adopt a standard circulating configuration with helicity γ close to π/2. Within the top layer (with much lower saturation magnetization), Bloch points nucleate within a Neel-like exchange spring domain wall created by the antiferromagnetic coupling and adopt an uncommon hyperbolic configuration, characterized by much larger helicity angles. Our results indicate a path for Bloch point engineering in future applications adjusting material parameters and domain wall characteristics.

In this work, we investigate the influence of curvature on the dynamic susceptibility in FeGe nanowires, both curved and straight, hosting a skyrmionic tube texture under the action of an external bias field, using micromagnetic simulations. Our results demonstrate that both the resonance frequencies and the number of resonant peaks are highly dependent on the curvature of the system. To further understand the nature of the spin wave modes, we analyze the spatial distributions of the resonant mode amplitudes and phases, describing the differences among resonance modes observed. The ability to control the dynamic properties and frequencies of these nanostructures underscores their potential application in frequency-selective magnetic devices.

In the article, within the framework of the theory of micromagnetism, a magneto-elastic microstructural model of the behavior of a ferromagnetic material with the shape memory (Heusler alloy) in a magnetic field is constructed. The dynamics of the magnetic process is described by the Landau-Lifshitz-Gilbert equation. Using the Galerkin procedure, variational equations corresponding to the differential relations of the magnetic problem are written out. A martensitic structure of the type "herringbone" (a twinned variant of martensite) with magnetic domains located at an angle of 180 ° is considered. 90-degree magnetic domain walls are the boundaries of the twins. The finite element method models the formation of these walls and the distribution of the magnetization vector in them. The evolution of this magnetic structure is investigated, namely the movement and interaction of 180-degree magnetic domain walls when an external magnetic field is applied in various directions. The problems of the formation of twins from the martensite variants (the twinning problem) and their disappearance (the detwinning problem) in an external magnetic field are considered. On the plane separating the two variants of martensite, the Hadamard compatibility condition (or twinning equation) is written out. The solution of this equation is given and the processes, that this equation describes, are considered in detail. The detwinning condition for a ferroalloy with the shape memory in a magnetic field is proposed and the processes related to the reorientation (detwinning) of the martensitic variants forming a twin are discussed. The problems considered earlier without taking into account the detwinning process are supplemented by this process, the magnetization curves are constructed and the components of the strain tensor are determined. The results obtained are in good agreement with the known experimental data. [ABSTRACT FROM AUTHOR]

The paper presents three-dimensional micromagnetic simulation of the magnetization reversal process of permalloy film with additional relief elements of step-like shape made of the same material. It is shown that in the course of magnetization reversal in a constant magnetic field, the initial magnetization distribution containing a C-shaped domain wall is transformed into a magnetization reversal zone filled with vortex structures. In this case, the magnetization reversal dynamics becomes irregular. The peculiarities of the dynamics of the magnetization reversal zone (change in magnetization reversal rate, temporary or final cessation of motion) are revealed for different types of surface relief elements (strips, linear or two-dimensional arrays of rectangular parallelepipeds of different sizes). Visualization methods of vortex structures are described. [ABSTRACT FROM AUTHOR]

By combining volume sensitive high resolution Magnetic Force Microscopy with surface sensitive X-ray Photoemission Electron Microscopy, we resolved the depth profile of a weak stripe magnetic texture and its evolution upon in-plane magnetization reversal. In contrast to previous reports, we show that the conventional weak stripe texture undergoes a well-defined undulation while the magnetic field is reversed to negative after in plane positive saturation. This transformation is strongly impacting the flux closure caps domains and a staggered Néel caps texture appears. Thanks to quantitative agreement with micro-magnetic simulations, we demonstrate that the existence of both the instability and the staggered Néel caps is intrinsic in negative applied field after positive in plane saturation. This reversal mode is characterized by a checker board pattern of alternating surface magnetic charges and by a longitudinal modulation of the in-plane component of magnetization similar to the oscillatory buckling reversal mode reported in elongated soft magnetic nanostructures.

The magnetic transition processes with symmetry breaking are studied theoretically on one magnetic moment model and more complex magnetic systems. It is shown that the symmetry‐breaking processes behave much more deterministic than could be deduced from energy considerations. The two qualitatively different cases of dynamical control are identified depending on robustness against thermal fluctuations. The first case is robust against thermal fluctuations even if the magnetic transition control parameter is changed adiabatically. The second case is unstable against temperature fluctuations and the result of symmetry breaking is random. If the magnetic transition control parameter is rapidly changed, control over the final magnetic state is regained. This phenomenon is related to precession of magnetic momentum, and therefore, it is not noticeable in methods considering and minimizing energy of the magnetic systems directly. [ABSTRACT FROM AUTHOR]

By combining volume sensitive high resolution Magnetic Force Microscopy with surface sensitive X-ray Photoemission Electron Microscopy, we resolved the depth profile of a weak stripe magnetic texture and its evolution upon in-plane magnetization reversal. In contrast to previous reports, we show that the conventional weak stripe texture undergoes a well-defined undulation while the magnetic field is reversed to negative after in plane positive saturation. This transformation is strongly impacting the flux closure caps domains and a staggered Néel caps texture appears. Thanks to quantitative agreement with micro-magnetic simulations, we demonstrate that the existence of both the instability and the staggered Néel caps is intrinsic in negative applied field after positive in plane saturation. This reversal mode is characterized by a checker board pattern of alternating surface magnetic charges and by a longitudinal modulation of the in-plane component of magnetization similar to the oscillatory buckling reversal mode reported in elongated soft magnetic nanostructures. Zigzaging magnetic weak stripes have been observed in CoFeB thin films. The characteristics of this new magnetic texture and its origins are revealed thanks to MFM and XMCD-PEEM measurements combined to micromagnetic simulations. [ABSTRACT FROM AUTHOR]

Magnetic metamaterials have demonstrated promising perspectives to improve the efficiency of magnetic flux concentrators. In this work, the effects of downscaling these devices for on‐chip integration is investigated. The influence of the non‐linear magnetic response of the ferromagnetic components, their magnetic irreversibility, the formation of magnetic domains, as well as the effects of geometry and size of the devices are scrutinized. The results demonstrate that the implementation of metasurfaces at the microscale opens up new technological possibilities for enhancing the performance of magnetic field detectors and remotely charging small electric devices, thus paving the way toward new approaches in information and communication technologies. [ABSTRACT FROM AUTHOR]

This work analyzes the magnetic configurations of cylindrical nanowires with a bulk Dzyaloshinskii–Moriya interaction and easy-plane anisotropy. We show that this system allows the nucleation of a metastable toron chain even when no out-of-plane anisotropy exists in the nanowire's top and bottom surfaces, as usually required. The number of nucleated torons depends on the nanowire length and the strength of an external magnetic field applied to the system. The size of each toron depends on the fundamental magnetic interactions and can be controlled by external stimuli, allowing the use of these magnetic textures as information carriers or nano-oscillator elements. Our results evidence that the topology and structure of the torons yield a wide variety of behaviors, revealing the complex nature of these topological textures, which should present an exciting interaction dynamic, depending on the initial conditions. [ABSTRACT FROM AUTHOR]

We present a micromagnetic investigation of the spin dynamics at remanence (zero applied field) in a periodic square artificial spin ice (ASI) prepared four different microstates (i.e., with zero, two or four magnetic charges at the vertex). The ASI elements consist of permalloy elliptical dots with a fixed long axis, and a variable width and interdot separation. For each vertex configuration, we compute the equilibrium ground state at zero applied field by relaxing a previously set magnetic configuration (microstate). After the excitation of such ground state, we perform a Fourier analysis obtaining frequency spectra and space phase profiles. We discuss the behavior of the spectra in changing the system's microstate and geometry, with reference to the spin mode space profiles, magnetization configuration, and effective internal field. Our results draw a correlation between ASI macrospin orientation at vertex and a few important dynamic properties like a phase-shift in the mode profiles or the frequency gap between the edge and fundamental modes. We suggest a few specific experiments to validate of our predictions, as well as applications in the field of interferometric magnonic devices. We believe that our results can help, from the fabrication stage, in tailoring the appropriate ASI geometry for specific application purposes. [ABSTRACT FROM AUTHOR]

In this article, a microstructural model of the Heusler alloy with the shape memory effect caused by the application of an external magnetic field is constructed. The dynamics of the magnetization process are described using the Landau–Lifshitz–Gilbert equation. For the numerical implementation of the model using the finite element method, the variational equations corresponding to the differential formulation of the magnetic problem are used. Such an approach makes it possible to reduce (weaken) the requirements for the smoothness of the sought solution. The problem of magnetization of single crystals of the Ni2MnGa alloy, which has a "herringbone"-type martensitic structure (a twinned variant of martensite), is considered. In each element of the twin, the magnetic domains with walls of a certain thickness are formed. The motion and interaction of these walls and the rotation of magnetization vector in the walls and domains under the action of the external differently directed magnetic fields are studied. These processes in the Heusler alloy are also accompanied by the detwinning process. A condition for the detwinning of a ferromagnetic shape memory alloy in a magnetic field is proposed, and the effect of the reorientation (detwinning) of martensitic variants forming a twin on the magnetization of the material and the occurrence of structural (detwinning) deformation in it are taken into account. First, the processes of magnetization and structural deformation in a single grain are considered at different angles between the anisotropy axes of twinned variants and the external magnetic field. For these cases, the magnetization curves are constructed, and the deformed states are identified. The model described such experimental facts as the detwinning process and the jump in magnetization on these curves as a result of this process. It was shown that the jump occurred at a certain magnitude of the strength of the applied external magnetic field and a certain direction of its action relative to the twinning system. Then, based on the obtained results, deformed states arising due to the detwinning process were determined for various (isotropic and texture-oriented) polycrystalline samples, and magnetization curves taking into account this process were constructed for these materials. [ABSTRACT FROM AUTHOR]

The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell's equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results. [ABSTRACT FROM AUTHOR]

Paleointensities are key to understanding the formation and evolution of Earth and are determined from rocks which record magnetic fields upon cooling; however, experimental protocols for estimating paleointensities frequently fail. The primary reason is that laboratory protocols assume that rocks are dominated by uniformly magnetized, single‐domain grains, instead of much more common non‐uniformly magnetized grains. Our model for larger grains shows a multiplicity of stable domain states; with preferred states changing as a function of temperature. We show that domain state distribution depends on the thermal history of the sample—in nature and the laboratory. From numerical thermomagnetic modeling, we show that particles with non‐uniform domain states will theoretically fail standard experimental paleointensity protocols, preventing us from determining reliable ancient geomagnetic field intensities. We propose that recognizing this type of behavior, and the resulting bias, will yield more reliable paleointensity records, and a better understanding of the Earth. Plain Language Summary: Variations in geomagnetic field intensity are recorded in rocks over billion‐year time scales and constrain many theories of early Earth evolution. The primary theory of magnetic recording in rocks is based on our understanding of uniformly magnetized particles with two stable directions. This theory allows us to design experiments that recover geomagnetic paleointensities. However, it has long been acknowledged that most magnetic particles in rocks are not uniformly magnetized and, importantly, have multiple stable directions. In this work, we construct a consistent theory for thermomagnetic recording in non‐uniformly magnetized grains and demonstrate how rocks containing these grains can result in a sample from which it is impossible to retrieve the ancient field intensity. We also discuss experimental laboratory measurements that are used to detect such poor recorders. Key Points: Micromagnetic models of elongate 100 nm magnetite grains suggest several stable domain states that change as a function of temperatureThe preferred state is also a function of cooling rate and so blocking and unblocking of remanence depends on the thermal historySuch particles will likely have unequal blocking and unblocking temperatures causing Arai plot curvature and biased paleointensity estimates [ABSTRACT FROM AUTHOR]

Our study addresses the problematics of magnetic skyrmions, nanometer-size vortex-like swirling topological defects, broadly studied today for applications in classic, neuromorphic and quantum information technologies. We tackle some challenging issues of material properties versus skyrmion stability and manipulation within a multiple-scale modeling framework, involving complementary ab-initio and micromagnetic frameworks. Ab-initio calculations provide insight into the anatomy of the magnetic anisotropy, the Dzyaloshinskii–Moriya asymmetric exchange interaction (DMI) and their response to a gating electric field. Various multi-layered heterostructures were specially designed to provide electric field tunable perpendicular magnetization and sizeable DMI, which are required for skyrmion occurrence. Landau–Lifshitz–Gilbert micromagnetic calculations in nanometric disks allowed the extraction of material parameter phase diagrams in which magnetic textures were classified according to their topological charge. We identified suitable ranges of magnetic anisotropy, DMI and saturation magnetization for stabilizing skyrmionic ground states or writing/manipulating them using either a spin-transfer torque of a perpendicular current or the electric field. From analyzing the different contributions to the total magnetic free energy, we point out some critical properties influencing the skyrmions' stability. Finally, we discuss some experimental issues related to the choice of materials or the design of novel magnetic materials compatible with skyrmionic applications. [ABSTRACT FROM AUTHOR]

Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau–Lifshitz equation, Quart. Appl. Math., 76, 383–405, 2018) proposed two novel predictor-corrector methods for the Landau–Lifshitz–Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau–Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties. [ABSTRACT FROM AUTHOR]

Spin-orbit torque (SOT) driven domain wall motion has attracted significant attention as the basis for a variety of spintronic devices due to its potential use as a high speed, low power means to manipulate the magnetic state of an object. While most previous attention has focused on ultrathin films wherein the material thickness is significantly less than the magnetic exchange length, recent reports have suggested unique dynamics may be achieved in intermediate and high thickness films. We used micromagnetic modelling to explore the role of the vertically non-uniform spin textures associated with the domain wall in nanowires of varying thickness on SOT driven domain wall motion. We found large velocity asymmetries between Bloch chiralities near the current density required for reversal of the Bloch component of the magnetization and linked these asymmetries to a gradual reorientation of the domain wall structure which drives a non-negligible, chiral Néel component of the domain wall. We further explored the influence of saturation magnetization, film thickness, the Dzyaloshinskii-Moriya interaction, and in-plane fields on domain wall dynamics. These results provide a framework for the development of SOT based devices based on domain wall motion in nanowires beyond the ultrathin film limit. [ABSTRACT FROM AUTHOR]

The MATLAB-based software tool MuMag2022 is presented for the analysis of magnetic-field-dependent unpolarized small-angle neutron scattering (SANS) data of bulk ferromagnets such as elemental nanocrystalline ferromagnets, magnetic nanocomposites or magnetic steels. On the basis of the micromagnetic theory for the magnetic SANS cross section, the program analyzes unpolarized total (nuclear and magnetic) SANS data within the approach-to-saturation regime. The main features of MuMag2022 are the estimation of the exchangestiffness constant, and of the strength and spatial structure of the magnetic anisotropy field and the magnetostatic field due to longitudinal magnetization fluctuations. MuMag2022 is open source and available as a standalone executable for Windows at https://mumag.uni.lu. [ABSTRACT FROM AUTHOR]

In this article, based on the theory of micromagnetism, a microstructural model of the behavior of the Heusler alloy in a magnetic field is constructed. The dynamics of the magnetic process is described by the Landau–Lifshitz–Gilbert equation. Using the Galerkin procedure, variational equations corresponding to the differential relations of the magnetic problem are written out. For numerical simulation, we consider the problem of magnetization of a Ni 2 MnGa alloy polytwin crystals, each grain of which is a twinned variant of martensite and has pronounced anisotropic properties. First, we consider the process of magnetization of a single grain, when an external magnetic field is applied at different angles to the anisotropy axes of twinned variants, and then, based on the results obtained, we plot magnetization curves for various (isotropic and texture-oriented) polycrystalline samples. This paper does not consider the process of detwinning, which can occur in such a material during the magnetization at a sufficiently high external field strength. [ABSTRACT FROM AUTHOR]

We present a micromagnetic investigation of the spin dynamics at remanence (zero applied field) in a periodic square artificial spin ice (ASI) prepared four different microstates (i.e., with zero, two or four magnetic charges at the vertex). The ASI elements consist of permalloy elliptical dots with a fixed long axis, and a variable width and interdot separation. For each vertex configuration, we compute the equilibrium ground state at zero applied field by relaxing a previously set magnetic configuration (microstate). After the excitation of such ground state, we perform a Fourier analysis obtaining frequency spectra and space phase profiles. We discuss the behavior of the spectra in changing the system’s microstate and geometry, with reference to the spin mode space profiles, magnetization configuration, and effective internal field. Our results draw a correlation between ASI macrospin orientation at vertex and a few important dynamic properties like a phase-shift in the mode profiles or the frequency gap between the edge and fundamental modes. We suggest a few specific experiments to validate of our predictions, as well as applications in the field of interferometric magnonic devices. We believe that our results can help, from the fabrication stage, in tailoring the appropriate ASI geometry for specific application purposes.

This work analyzes the magnetic configurations of cylindrical nanowires with a bulk Dzyaloshinskii–Moriya interaction and easy-plane anisotropy. We show that this system allows the nucleation of a metastable toron chain even when no out-of-plane anisotropy exists in the nanowire’s top and bottom surfaces, as usually required. The number of nucleated torons depends on the nanowire length and the strength of an external magnetic field applied to the system. The size of each toron depends on the fundamental magnetic interactions and can be controlled by external stimuli, allowing the use of these magnetic textures as information carriers or nano-oscillator elements. Our results evidence that the topology and structure of the torons yield a wide variety of behaviors, revealing the complex nature of these topological textures, which should present an exciting interaction dynamic, depending on the initial conditions.

This paper is devoted to an optimal control problem of a coupled spin drift-diffusion Landau–Lifshitz–Gilbert system describing the interplay of magnetization and spin accumulation in magnetic-nonmagnetic multilayer structures, where the control is given by the electric current density. A variational approach is used to prove the existence of an optimal control. The first-order necessary optimality system for the optimal solution is derived in one space-dimension via Lagrange multiplier method. Numerical examples are reported to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

• Bloch points are studied in a magnetic GdCo/NdCo/GdCo' trilayer with engineered interlayer coupling and exchange lengths. • BP polarity and helicity, γ linked to topological charges and magnetostatics, obtained by X-ray vector magnetic tomography. • At bottom GdCo, BPs adopt the standard circulating configuration with helicity γ close to. π / 2. • At top GdCo, BPs adopt an uncommon hyperbolic configuration with high helicity angles within an exchange spring DW. Bloch points in magnetic materials are attractive entities in view of magnetic information transport. Here, Bloch point configuration has been investigated and experimentally determined in a magnetic trilayer (Gd 12 Co 88 /Nd 17 Co 83 /Gd 24 Co 76) with carefully adjusted composition within the ferrimagnetic Gd x Co 1-x alloys in order to engineer saturation magnetization, exchange length, and interlayer couplings (ferromagnetic vs antiferromagnetic). X-ray vector magnetic tomography has allowed us to determine experimentally Bloch point polarity (related to topological charge) and Bloch point helicity γ (determined by magnetostatic energy). At the bottom layer (close to the ferromagnetic interface), Bloch points adopt a standard circulating configuration with helicity γ close to π / 2. Within the top layer (with much lower saturation magnetization), Bloch points nucleate within a Neel-like exchange spring domain wall created by the antiferromagnetic coupling and adopt an uncommon hyperbolic configuration, characterized by much larger helicity angles. Our results indicate a path for Bloch point engineering in future applications adjusting material parameters and domain wall characteristics. [ABSTRACT FROM AUTHOR]

We solve the Landau-Lifshitz-Gilbert equation in the finite-temperature regime, where thermal fluctuations are modeled by a random magnetic field whose variance is proportional to the temperature. By rescaling the temperature proportionally to the computational cell size Δ x (T → T Δ x / a eff , where a eff is the lattice constant) [M. B. Hahn, J. Phys. Comm., 3:075009, 2019], we obtain Curie temperatures T C that are in line with the experimental values for cobalt, iron and nickel. For finite-sized objects such as nanowires (1D) and nanolayers (2D), the Curie temperature varies with the smallest size d of the system. We show that the difference between the computed finite-size T C and the bulk T C follows a power-law of the type: (ξ 0 / d) λ , where ξ 0 is the correlation length at zero temperature, and λ is a critical exponent. We obtain values of ξ 0 in the nanometer range, also in accordance with other simulations and experiments. The computed critical exponent is close to λ = 2 for all considered materials and geometries. This is the expected result for a mean-field approach, but slightly larger than the values observed experimentally. • Simulation of temperature effects using a stochastic Landau-Lifshitz-Gilbert equation. • Accurate determination of the Curie temperature for Cobalt, Nickel, and Iron. • Determination of the scaling law of the Curie temperature with the size of the system, for nanowire and nanolayer geometries. [ABSTRACT FROM AUTHOR]

The 2:14:1-type rare-earth (RE)-Fe-B permanent magnets prepared by the dual alloy method have been found to possess much superior magnetic properties to those prepared by the single alloy method, providing an appealing route to promote the utilization of high-abundance RE elements Ce and La and balance the use of the RE source. However, the relationship between magnetic interactions among different 2:14:1 main phases and superior magnetic properties is still unclear. In this study, we investigated the magnetic interactions and reversal field distribution in these magnets using first-order reversal curve (FORC) images. The FORC images showed that (Nd, Pr)27.8(La, Ce)2.7FebalM1.4B1.0 (S-9) and (Nd, Pr)19.5(La, Ce)11.0FebalM1.4B1.0 (S-36) have the characteristics of multiple main phases. The reverse magnetic fields corresponding to the soft and hard main phases, as well as the associated exchange coupling, were highly dependent on the LaCe content. The higher the LaCe content, the weaker the exchange coupling and the more asynchronous the demagnetization process. In addition, the FORC images indicated that the magnetization reversal process also varies with LaCe content, where the nucleation and propagation of reversed domains dominant in the S-9 magnet, while the domain propagation in the S-36 magnet is considerably suppressed. Additional micro-magnetic simulations also revealed that the coercivity and exchange coupling of multi-main-phase magnets decrease with increasing LaCe content, correlating well with the experimental results. These findings may not only contribute to a better understanding of the complex magnetic interactions between the soft and hard phases and how they affect macroscopic magnetic properties but also help in improving the magnetic performance of the RE-Fe-B magnets with high LaCe content. [ABSTRACT FROM AUTHOR]

The motion of a domain wall in a magnetic film with an antidot array leads to the formation of a vortex magnetization reversal zone moving along the sample. A description of the magnetization dynamics in the film is obtained by micromagnetic simulations based on the Landau–Lifshitz–Gilbert equation for various film thickness and hole arrangement and size. We show that the magnetization reversal zone either passes through the antidot array or “freezes” on it. Using a special technique for visualizing vortex structures, based on the calculation of topological charges of two types, we establish the relationship between the magnetization dynamics and the types of vortex structures. [ABSTRACT FROM AUTHOR]

In this article, a microstructural model of the Heusler alloy with the shape memory effect caused by the application of an external magnetic field is constructed. The dynamics of the magnetization process are described using the Landau–Lifshitz–Gilbert equation. For the numerical implementation of the model using the finite element method, the variational equations corresponding to the differential formulation of the magnetic problem are used. Such an approach makes it possible to reduce (weaken) the requirements for the smoothness of the sought solution. The problem of magnetization of single crystals of the Ni2MnGa alloy, which has a “herringbone”-type martensitic structure (a twinned variant of martensite), is considered. In each element of the twin, the magnetic domains with walls of a certain thickness are formed. The motion and interaction of these walls and the rotation of magnetization vector in the walls and domains under the action of the external differently directed magnetic fields are studied. These processes in the Heusler alloy are also accompanied by the detwinning process. A condition for the detwinning of a ferromagnetic shape memory alloy in a magnetic field is proposed, and the effect of the reorientation (detwinning) of martensitic variants forming a twin on the magnetization of the material and the occurrence of structural (detwinning) deformation in it are taken into account. First, the processes of magnetization and structural deformation in a single grain are considered at different angles between the anisotropy axes of twinned variants and the external magnetic field. For these cases, the magnetization curves are constructed, and the deformed states are identified. The model described such experimental facts as the detwinning process and the jump in magnetization on these curves as a result of this process. It was shown that the jump occurred at a certain magnitude of the strength of the applied external magnetic field and a certain direction of its action relative to the twinning system. Then, based on the obtained results, deformed states arising due to the detwinning process were determined for various (isotropic and texture-oriented) polycrystalline samples, and magnetization curves taking into account this process were constructed for these materials.

Our study addresses the problematics of magnetic skyrmions, nanometer-size vortex-like swirling topological defects, broadly studied today for applications in classic, neuromorphic and quantum information technologies. We tackle some challenging issues of material properties versus skyrmion stability and manipulation within a multiple-scale modeling framework, involving complementary ab-initio and micromagnetic frameworks. Ab-initio calculations provide insight into the anatomy of the magnetic anisotropy, the Dzyaloshinskii–Moriya asymmetric exchange interaction (DMI) and their response to a gating electric field. Various multi-layered heterostructures were specially designed to provide electric field tunable perpendicular magnetization and sizeable DMI, which are required for skyrmion occurrence. Landau–Lifshitz–Gilbert micromagnetic calculations in nanometric disks allowed the extraction of material parameter phase diagrams in which magnetic textures were classified according to their topological charge. We identified suitable ranges of magnetic anisotropy, DMI and saturation magnetization for stabilizing skyrmionic ground states or writing/manipulating them using either a spin-transfer torque of a perpendicular current or the electric field. From analyzing the different contributions to the total magnetic free energy, we point out some critical properties influencing the skyrmions’ stability. Finally, we discuss some experimental issues related to the choice of materials or the design of novel magnetic materials compatible with skyrmionic applications.

Femtosecond laser (FL) induced terahertz (THz) source is a new type of THz source based on injecting FL beams into ferromagnetic thin films by nonlinear effects to generate THz wave. It has a wider bandwidth compared to the traditional THz source, which provides higher flexibility and tunability in the application. In this paper, the three-temperature model and the stochastic Landau Lifshitz Gilbert equation at the atomic level are applied to simulate THz wave generation in Fe thin film induced by FL. Simulation results show that under a FL irradiance of 2 J m -2 , the maximum demagnetization of the Fe thin film reaches 8.7%. The electromagnetic waves generated completely cover the THz band (0.1-10 THz), which fully satisfied the application requirements of the THz technology, verifying the feasibility of FL inducing the Fe thin film as a THz source. However, when the Fe thin film is overheated, it will be difficult for FL to excite valuable THz waves. Therefore, additional cooling devices are needed to keep the THz source in a workable temperature state, or to use ferromagnetic materials with magnetic moments that can quickly recover to saturation., (© 2024 IOP Publishing Ltd.)

A microstructural model of behavior of ferromagnetic material (Heusler alloy) in a magnetic field is constructed within the framework of the theory of micromagnetism. The process dynamics is described by the Landau—Lifshitz—Hilbert equation. The Galerkin procedure is used to assign variational equations to differential relations. A herringbone-type martensitic structure (twinned variant of martensite) with magnetic domains arranged at an angle of 180° is considered. The twin boundaries act as 90-degree magnetic domain walls. The evolution of this magnetic structure is investigated, namely the motion and interaction of the 180-degree walls of this magnetic domain in the presence of an external magnetic field applied in different directions. The finite element method simulates the formation of these walls and the magnetization vector distribution in them. [ABSTRACT FROM AUTHOR]

Spin-orbit torque (SOT) driven domain wall motion has attracted significant attention as the basis for a variety of spintronic devices due to its potential use as a high speed, low power means to manipulate the magnetic state of an object. While most previous attention has focused on ultrathin films wherein the material thickness is significantly less than the magnetic exchange length, recent reports have suggested unique dynamics may be achieved in intermediate and high thickness films. We used micromagnetic modelling to explore the role of the vertically non-uniform spin textures associated with the domain wall in nanowires of varying thickness on SOT driven domain wall motion. We found large velocity asymmetries between Bloch chiralities near the current density required for reversal of the Bloch component of the magnetization and linked these asymmetries to a gradual reorientation of the domain wall structure which drives a non-negligible, chiral Néel component of the domain wall. We further explored the influence of saturation magnetization, film thickness, the Dzyaloshinskii-Moriya interaction, and in-plane fields on domain wall dynamics. These results provide a framework for the development of SOT based devices based on domain wall motion in nanowires beyond the ultrathin film limit.

In this work, we present an analysis of skyrmion dynamics considering Dzyaloshinskii–Moriya interactions in an STNO device with a double-disk geometry. Three regimes were observed as a function of geometric parameters and the electric current density: (i) the skyrmion is annihilating at the system’s border; (ii) the skyrmion moves in a non-circular trajectory alternating its position between the two disks, and (iii) the skyrmion only rotates inside a one-disk subsystem. For the annihilation state, we found that the transient time decays within a stretched exponential law as a function of the electric current. Our results show a 2D state diagram that can guide new experimental work in order to obtain these specific behaviors for new applications based on skyrmion dynamics.

Abstract Numerical generation of physical states is essential to all scientific research fields. The role of a numerical generator is not limited to understanding experimental results; it can also be employed to predict or investigate characteristics of uncharted systems. A variational autoencoder model is devised and applied to a magnetic system to generate energetically stable magnetic states with low local deformation. The spin structure stabilization is made possible by taking the explicit magnetic Hamiltonian into account to minimize energy in the training process. A significant advantage of the model is that the generator can create a long‐range ordered ground state of spin configuration by increasing the role of stabilization even if the ground states are not necessarily included in the training process. It is expected that the proposed Hamiltonian‐guided generative model can bring about great advances in numerical approaches used in various scientific research fields.

Numerical generation of physical states is essential to all scientific research fields. The role of a numerical generator is not limited to understanding experimental results; it can also be employed to predict or investigate characteristics of uncharted systems. A variational autoencoder model is devised and applied to a magnetic system to generate energetically stable magnetic states with low local deformation. The spin structure stabilization is made possible by taking the explicit magnetic Hamiltonian into account to minimize energy in the training process. A significant advantage of the model is that the generator can create a long‐range ordered ground state of spin configuration by increasing the role of stabilization even if the ground states are not necessarily included in the training process. It is expected that the proposed Hamiltonian‐guided generative model can bring about great advances in numerical approaches used in various scientific research fields. [ABSTRACT FROM AUTHOR]

Magnetic force microscopy (MFM) is a widespread technique for imaging magnetic structures with a resolution of some 10 nanometers. MFM can be calibrated to obtain quantitative (qMFM) spatially resolved magnetization data in units of A/m by determining the calibrated point spread function of the instrument, its instrument calibration function (ICF), from a measurement of a wellknown reference sample. Beyond quantifying the MFM data, a deconvolution of the MFM image data with the ICF also corrects the smearing caused by the finite width of the MFM tip stray field distribution. However, the quality of the calibration depends critically on the calculability of the magnetization distribution of the reference sample. Here, we discuss a Ti/Pt/Co multilayer stack that shows a stripe domain pattern as a suitable reference material. A precise control of the fabrication process, combined with a characterization of the sample micromagnetic parameters, allows reliable calculation of the sample's magnetic stray field, proven by a very good agreement between micromagnetic simulations and qMFM measurements. A calibrated qMFM measurement using the Ti/Pt/Co stack as a reference sample is shown and validated, and the application area for quantitative MFM measurements calibrated with the Ti/Pt/Co stack is discussed. [ABSTRACT FROM AUTHOR]

We have investigated the dynamical hysteresis behaviors of C o , F e , and Ni nanodisks under the effect of the time-dependent magnetic field by solving the Landau-Lifshitz-Gilbert equation with the OOMMF software. A nanodisk with a diameter of 360 n m constructed in the xy plane is exposed to a sinusoidal x-directed magnetic field with varying frequency and amplitude values. All nanodisks exhibit intricate frequency-dependent variations in the hysteresis loop area, remanent magnetization, and coercive field. Our simulation results emphasize that while the dynamical order occurs exclusively in the x direction for Fe nanodisks, dynamical order is observed in the y and z directions for Ni nanodisks within a specific frequency range. • Dynamic phase transitions of Co, Fe, and Ni are discussed in the micromagnetic framework. • Time series and hysteresis curves have been systematically investigated for each nanodisk type. • The dynamic hysteresis properties have been calculated as functions of frequency. [ABSTRACT FROM AUTHOR]