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Let Mn be an embedded closed submanifold of ℝk+1 or a smooth bounded domain in ℝn, where n ≥ 3. We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time, provided that the initial map u0 is in a suitable nontrivial homotopy class with energy small enough. [ABSTRACT FROM AUTHOR]

We present an introductory review of concepts behind micromagnetic simulations, in which magnetic moments representing collections of atomic spins within a material evolve according to the Landau-Lifshitz-Gilbert equation, a generalized torque equation. This evolution is determined by a variety of interactions, including those arising from external fields, magnetostatic and exchange effects, and magnetic anisotropy. Anisotropy is a key ingredient in the Stoner-Wohlfarth model, which provides a quantitative basis for understanding magnetic hysteresis. In turn, hysteresis loops provide a basis for comparing simulations and experiments, and are important, for example, in quantifying the heating response of a sample to an oscillating external magnetic field. Micromagnetic simulations bear conceptual similarity to molecular dynamics (MD) simulations, but whereas in MD classical potentials are used to naturally model interactions between atoms and/or molecules, the choice of modelling length scale in micromagnetics is less obvious. If effective interactions are determined for, say, two crystallographic unit cells of a material, how interaction parameters should scale with micromagnetic simulation cell size, particularly at finite temperature, is still an area of research. Finally, we discuss the coupling of magnetic and mechanical degrees of freedom in simulating atomic and nanoparticle systems. This review is based, in part, on our own experience in modelling hysteretic heating of magnetite nanoparticles.

We investigated the temperature dependence of agglomeration systems consisting of interacting single-domain magnetic nanoparticles (MNPs). In our study, we utilized electron spin resonance (ESR) experiments to measure how temperature-induced change in agglomerate state affects the peak-to-peak width of ESR spectra (Hpp) and validated these observations through simulations. Micromagnetic simulations were conducted by combining the classical magnetic dipolar Metropolis Monte Carlo method with the semi-classical macro-spin Landau-Lifshitz-Gilbert finite element method, and the results were compared with experimental findings. Our research elucidates the behavior of thermally-induced disruption of MNP agglomerates. We discovered that the temperature at which MNPs are completely dispersed might correspond to one at which the temperature dependence of Hpp disappears.

The article presents the results of a qualitative study of the model of a modern magnetic memory cell, in which the spin Hall effect is used for recording. Cells of square cross-section with longitudinal anisotropy of the active layer are considered. Based on the Landau-Lifshitz-Gilbert vector equation, a mathematical model for controlling the process of writing zero and one into a cell is constructed. In the approximation of a uniform distribution of magnetization, a system of equations is derived that describes the dynamics of magnetization under the action of a magnetic field and spin current. The parameters of the qualitatively equivalent dynamics of the model are determined. It has been established that at zero currents and fields in both cases there are two main stable equilibrium positions. These equilibria, depending on the mutual orientation of the magnetization vector of the active and reference layers, correspond to zero and one, written in the cell. The transition from one cell state to another is described by solving a system of differential equations. A bifurcation diagram of a dynamical system in the variables "field–current" is constructed. It is shown that with a given configuration of the memory element, external influences transfer the magnetization to an intermediate state in the plane of the free layer, which, when the current and field are turned off, leads to writing zero or one to the memory cell. The critical switching current is estimated as a function of the applied external magnetic field. [ABSTRACT FROM AUTHOR]

We investigated the temperature dependence of agglomeration systems consisting of interacting single-domain magnetic nanoparticles (MNPs). In our study, we utilized electron spin resonance (ESR) experiments to measure how temperature-induced change in agglomerate state affects the peak-to-peak width of ESR spectra (H pp) and validated these observations through simulations. Micromagnetic simulations were conducted by combining the classical magnetic dipolar Metropolis Monte Carlo method with the semi-classical macro-spin Landau-Lifshitz-Gilbert finite element method, and the results were compared with experimental findings. Our research elucidates the behavior of thermally-induced disruption of MNP agglomerates. We discovered that the temperature at which MNPs are completely dispersed might correspond to one at which the temperature dependence of H pp disappears. [ABSTRACT FROM AUTHOR]

The rotation of a magnetic vortex in a ferromagnet with a local smooth change in the exchange interaction is considered. The frequency of this precession is found for all distances of the vortex from defect center. The rotation velocity of the vortex nonmonotonically depends on the distance to the center, reaching a maximum at a distance of the order of the characteristic dimension of the defect. The decrease in velocity at large distances agrees with the data known in the literature. Accounting of attenuation leads to the vortex escape the defect or falling onto it along a logarithmic spiral. [ABSTRACT FROM AUTHOR]

The global solution of the 2-dimensional Landau-Lifshitz-Gilbert (LLG) equation on the sphere S2 is studied. By the Hasimoto transformation, an equivalent complex-valued equation is deduced under cylindrical symmetric coordinates. Then the global H2 well-posedness of the Cauchy problem for this complex system with minimal regularity assumptions on the initial data is proved, and the well-posedness of the LLG equation is presented. [ABSTRACT FROM AUTHOR]

In this paper, a family of temporal high-order accurate numerical schemes for the Landau–Lifshitz–Gilbert (LLG) equation is proposed. The proposed schemes are developed utilizing the Gauss–Legendre quadrature method, enabling them to achieve arbitrary high-order time discretization. Furthermore, the geometrical properties of the LLG equation, such as the preservation of constant magnetization magnitude and the Lyapunov structure, are investigated based on the proposed discrete schemes. It is demonstrated that the magnetization magnitude remains constant with an error of (2p+3) order in time when utilizing a (2p+2)th-order discrete scheme. Additionally, the preservation of the Lyapunov structure is achieved with a second-order error in the temporal step size. Numerical experiments and simulations effectively verify the performance of our proposed algorithm and validate our theoretical analysis.

In this paper, we first prove the local-in-time existence of the evolutionary model for magnetoelasticity with finite initial energy by employing the nonlinear iterative approach to deal with the constraint on values of the magnetization | M (t , x) | = 1 in the Landau-Lifshitz-Gilbert (LLG) equation. We reformulate the evolutionary model near the constant equilibrium for magnetoelasticity with vanishing external magnetic field, so that a further dissipative term will be sought from the elastic stress. We thereby justify the global well-posedness to the evolutionary model for magnetoelasticity with zero external magnetic field under small size of initial data. [ABSTRACT FROM AUTHOR]

Magnetization dynamics in magnetic materials is modeled by the Landau-Lifshitz-Gilbert (LLG) equation, which is a nonlinear system of partial differential equations. In the LLG equation, the length of magnetization is conserved and the system energy is dissipative. Implicit and semi-implicit schemes have often been used in micromagnetics simulations due to their unconditional numerical stability. In more details, implicit schemes preserve the properties of the LLG equation, but solve a nonlinear system of equations per time step. In contrast, semi-implicit schemes only solve a linear system of equations, while additional operations are needed to preserve the length of magnetization. It still remains unclear which one shall be used if both implicit and semi-implicit schemes are available. In this work, using the implicit Crank-Nicolson (ICN) scheme as a benchmark, we propose to make this implicit scheme semi-implicit. Stability and convergence analysis, and numerical performance in terms of accuracy and efficiency are systematically studied. Based on these results, we conclude that a semi-implicit scheme is superior to its implicit analog both theoretically and numerically, and we recommend the semi-implicit scheme in micromagnetics simulations if both methods are available. [ABSTRACT FROM AUTHOR]

We demonstrate the manifestations of nonlinear features in magnetic dynamics and I–V characteristics of a φ0 Josephson junction in the ferromagnetic resonance region. We show that at small values of the system parameters damping, spin–orbit interaction, and Josephson-to-magnetic energy ratio, the magnetic dynamics is reduced to the dynamics of a scalar Duffing oscillator driven by the Josephson oscillations. The role of the increasing superconducting current in the resonance region is clarified. Shifting of the ferromagnetic resonant frequency and the reversal of its damping dependence due to nonlinearity are demonstrated by the full Landau–Lifshitz–Gilbert–Josephson system of equations and in its different approximations. Finally, we demonstrate the negative differential resistance in the I–V characteristics and its correlation with the fold-over effect.

The superparamagnetic behavior of magnetic nanoparticles plays a key role in the scientific and/or technological use of these particles. Blocking temperature is a remarkable quantity that allows one to determine the regime, superparamagnetic or blocked, in which the nanoparticles can be found. Even though it is a well-known fact that the blocking temperature is strongly dependent on the magnetostatic interaction, the overall effect of the dipolar interaction remains unclear. In a system of single-domain magnetic nanoparticles, placed along a linear chain, we address the effect of the dipole-dipole interaction on the blocking temperature of the system. We add disorder into the system by allowing random sizes of the particles and random directions of their uniaxial anisotropy axis. We perform numerical simulations of the stochastic Landau-Lifshitz-Gilbert equation, taking into account an explicit dependence of the saturation magnetization and uniaxial anisotropy energy density on temperature, relevant for systems where the blocking temperature is high. From ZFC magnetization measurements for different strengths of the dipolar coupling, we show that the blocking temperature is a non-monotonic function of the mean distance between particles. The blocking temperature reaches its maximum value when particles touch each other, then decreases as the mean distance between particles increases, and attains a minimum value, starting to increase again up to a constant value where the dipolar interaction becomes negligible. [ABSTRACT FROM AUTHOR]

The Landau–Lifshitz–Gilbert (LLG) equation models the temporal evolution of magnetization in continuum ferromagnets. The LLG equation has a nonconvex constraint and is highly nonlinear. In this paper, we will use the Fourier-spectral method for approximating the solution of the LLG equation with the nonconvex constraint. We consider the penalty problem and show the stability and convergence of the approximate penalty problem, and then we show the convergence of the penalty problem to a (weak) solution of the LLG equation. Computational experiments and comparison with other numerical methods are presented to demonstrate the effectiveness of the proposed method.

In this paper, we consider spin-diffusion Landau–Lifshitz–Gilbert equations (SDLLG), which consist of the time-dependent Landau–Lifshitz–Gilbert (LLG) equation coupled with a time-dependent diffusion equation for the electron spin accumulation. The model takes into account the diffusion process of the spin accumulation in the magnetization dynamics of ferromagnetic multilayers. We prove that in the limit of long times, the system reduces to simpler equations in which the LLG equation is coupled to a nonlinear and nonlocal steady-state equation, referred to as SLLG. As a by-product, the existence of global weak solutions to the SLLG equation is obtained. Moreover, we prove weak-strong uniqueness of solutions of SLLG, i.e., all weak solutions coincide with the (unique) strong solution as long as the latter exists in time. The results provide a solid mathematical ground to the qualitative behavior originally predicted by Zhang , Levy , and Fert in [44] in ferromagnetic multilayers. [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]

We discuss a mass-lumped midpoint scheme for the numerical approximation of the Landau–Lifshitz–Gilbert equation, which models the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic field contributions, our setting covers the non-standard Dzyaloshinskii–Moriya interaction, which is the essential ingredient for the enucleation and stabilization of magnetic skyrmions. Our analysis also includes the inexact solution of the arising nonlinear systems, for which we discuss both a constraint-preserving fixed-point solver from the literature and a novel approach based on the Newton method. We numerically compare the two linearization techniques and show that the Newton solver leads to a considerably lower number of nonlinear iterations. Moreover, in a numerical study on magnetic skyrmions, we demonstrate that, for magnetization dynamics that are very sensitive to energy perturbations, the midpoint scheme, due to its conservation properties, is superior to the dissipative tangent plane schemes from the literature. [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]

Iron rhodium (FeRh) undergoes a first-order anti-ferromagnetic to ferromagnetic phase transition above its Curie temperature. By measuring the anomalous Nernst effect (ANE) in (110)-oriented FeRh films on Al 2 O 3 substrates, the ANE thermopower over a temperature range of 100-350 K is observed, with similar magnetic transport behaviors observed for in-plane magnetization (IM) and out-of-plane magnetization (PM) configurations. The temperature-dependent magnetization-magnetic field strength (M-H) curves revealed that the ANE voltage is proportional to the magnetization of the material, but additional features magnetic textures not shown in the M-H curves remained intractable. In particular, a sign reversal occurred for the ANE thermopower signal near zero field in the mixed-magnetic-phase films at low temperatures, which is attributed to the diamagnetic properties of the Al 2 O 3 substrate. Finite element method simulations associated with the Heisenberg spin model and Landau-Lifshitz-Gilbert equation strongly supported the abnormal heat transport behavior from the Al 2 O 3 substrate during the experimentally observed magnetic phase transition for the IM and PM configurations. The results demonstrate that FeRh films on an Al 2 O 3 substrate exhibit unusual behavior compared to other ferromagnetic materials, indicating their potential for use in novel applications associated with practical spintronics device design, neuromorphic computing, and magnetic memory., (© 2024 Wiley‐VCH GmbH.)

Butterfly magnetoresistance (BMR) and antisymmetric magnetoresistance (ASMR) are about a butterfly-cross curve and a curve with one peak and one valley when a magnetic field is swept up and down along a fixed direction. Other than the parallelogram-shaped magnetoresistance-curve (MR-curve) often observed in magnetic memory devices, BMR and ASMR are two ubiquitous types of MR-curves observed in diversified magnetic systems, including van der Waals materials, strongly correlated systems, and traditional magnets. Here, we reveal the general principles and the picture behind the BMR and the ASMR that do not depend on the detailed mechanisms of magnetoresistance: 1) The systems exhibit hysteresis loops, common for most magnetic materials with coercivities. 2) The magnetoresistance of the magnetic structures in a large positive magnetic field and in a large negative magnetic field is approximately the same. With the generalized Ohm’s law in magnetic materials, these principles explain why most BMR appears in the longitudinal resistance measurements and is very rare in the Hall resistance measurements. Simple toy models, in which the Landau-Lifshitz-Gilbert equation governs magnetization, are used to demonstrate the principles and explain the appearance and disappearance of BMR in various experiments. Our finding provides a simple picture to understand magnetoresistance-related experiments.

Motivated by recent models of current driven magnetization dynamics, we examine the coupling of the Landau–Lifshitz–Gilbert equation and classical electron transport governed by the Vlasov–Maxwell system. The interaction is based on space-time gyro-coupling in the form of emergent electromagnetic fields of quantized helicity that add up to the conventional Maxwell fields. We construct global weak solutions of the coupled system in the framework of frustrated magnets with competing first- and second-order gradient interactions known to host topological solitons such as magnetic skyrmions and hopfions. [ABSTRACT FROM AUTHOR]

We study the homogenization of Landau–Lifshitz–Gilbert equation in a ϵ-periodic composite material formed by two constituents, separated by an imperfect interface Γ ϵ , on which we prescribe the continuity of the conormal derivatives and a jump of the solution proportional to the conormal derivative, by means of a coefficient of order ϵ γ . We use the periodic unfolding method together with extension operators for handling the nonlinearities to identify the limit problem when tuning up the parameter γ in R . [ABSTRACT FROM AUTHOR]

The dynamics of magnetization is important in spintronics, where the coupling between phonon and magnon attracts much attention. In this work, we study the angular dependence of the coupling between longitudinal-wave phonon and magnon. We investigated the magnetization dynamics using the time-resolved magneto-optical Kerr effect, which allows measuring spin-wave resonances and the magnetic echo signal. The frequency, mode number, and amplitude of the spin-wave resonance change with the out-of-plane angle of the external magnetic field. The amplitude of the magnetic echo signal caused by the strain pulse also changes with the angle. We calculate these angular dependences based on the Landau–Lifshitz–Gilbert equation and find that the angles of the external field and magnetic moment are important factors for the phonon–magnon coupling when phonon propagates in the thickness direction under the out-of-plane magnetic field. [ABSTRACT FROM AUTHOR]

The influence of Gilbert damping on the propagation of electromagnetic waves (EMWs) in an anisotropic ferromagnetic medium is investigated theoretically. The interaction of the magnetic field component of the electromagnetic wave with the magnetization of a ferromagnetic medium has been studied by solving the associated Maxwell's equations coupled with the Landau–Lifshitz–Gilbert (LLG) equation. When small perturbations are made on the magnetization of the ferromagnetic medium and magnetic field along the direction of propagation of electromagnetic wave by using the reductive perturbation method, the associated nonlinear dynamics is governed by a time-dependent damped derivative nonlinear Schrödinger (TDDNLS) equation. The Lagrangian density function is constructed by using the variational method to solve the TDDNLS equation to understand the dynamics of the system under consideration. The propagation of EMW in a ferromagnetic medium with inherent Gilbert damping admits very interesting nonlinear dynamical structures. These structures include Gilbert damping-managing symmetrically breathing solitons, localized erupting electromagnetic breathing dromion-like modes of excitations, breathing dromion-like soliton, decaying dromion-like modes and an unexpected creation-annihilation mode of excitations in the form of growing-decaying dromion-like modes. • The influence of Gilbert damping on the propagation of EMWs in an anisotropic ferromagnetic medium is investigated theoretically. • The nonlinear spin dynamics of ferromagnetic system associated with Maxwell's equations is governed by a TDDNLS equation. • The Lagrangian density function is constructed by using the variational method to solve the TDDNLS equation. • The propagation of EMW in a ferromagnetic medium with Gilbert damping admits very interesting nonlinear dynamical structures. [ABSTRACT FROM AUTHOR]

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]

The transition from classical physics to quantum mechanics has been mysterious. Here, we mathematically derive the space-independent von Neumann equation for electron spin from the classical Bloch equation. Subsequently, the space-independent Schrödinger–Pauli equation is derived in both the quantum mechanical and recently developed co-quantum dynamic frameworks. [ABSTRACT FROM AUTHOR]

In this paper, we study the thin-film limit of the micromagnetic energy functional in the presence of bulk Dzyaloshinskii–Moriya interaction (DMI). Our analysis includes both a stationary Γ -convergence result for the micromagnetic energy, as well as the identification of the asymptotic behavior of the associated Landau–Lifshitz–Gilbert equation. In particular, we prove that, in the limiting model, part of the DMI term behaves like the projection of the magnetic moment onto the normal to the film, contributing this way to an increase in the shape anisotropy arising from the magnetostatic self-energy. Finally, we discuss a convergent finite element approach for the approximation of the time-dependent case and use it to numerically compare the original three-dimensional (3D) model with the 2D thin-film limit. [ABSTRACT FROM AUTHOR]

We investigate the effect of spin torque on the switching dynamics of magnetic solitons in a weak ferromagnetic nanowire under the influence of an electromagnetic wave (EMW). The magnetization dynamics of the current-driven ferromagnetic nanowire and the EMW propagation is governed by the celebrated Landau-Lifshitz-Gilbert (LLG) vector equation and the Maxwell's equations, respectively. We recast the set of LLG and Maxwell equations onto the extended derivative nonlinear Schr o ¨ dinger (EDNLS) equation. We employ the nonlinear perturbation analysis along the lines of Kodama and Ablowitz and analyze the interplay of the Dzyaloshinskii-Moriya interaction (DMI) along with the spin transfer torque on the magnetization reversal dynamics by solving the associated evolution equations for the soliton parameters. We also demonstrate the spin-polarized current triggers an ultrafast switching of EM solitons in the ferromagnetic nanowire in the range of 0.58 - 0.12 n s , and the Gilbert damping supports the EM soliton switching to sustain indefinitely. We invoke the Jacobi elliptic function method to explore the propagation of breather-like solitonic localized modes along the ferromagnetic nanowire. [ABSTRACT FROM AUTHOR]

Landau-Lifshitz-Gilbert (LLG) spin-dynamics calculations based on the extended Heisenberg Hamiltonian is an important tool in computational materials science involving magnetic materials. LLG simulations allow to bridge the gap from expensive quantum mechanical calculations with small unit cells to large supercells where the collective behavior of millions of spins can be studied. In this work we present the AiiDA-Spirit plugin that connects the spin-dynamics code Spirit to the AiiDA framework. AiiDA provides a Python interface that facilitates performing high-throughput calculations while automatically augmenting the calculations with metadata describing the data provenance between calculations in a directed acyclic graph. The AiiDA-Spirit interface thus provides an easy way for high-throughput spin-dynamics calculations. The interface to the AiiDA infrastructure furthermore has the advantage that input parameters for the extended Heisenberg model can be extracted from high-throughput first-principles calculations including a proper treatment of the data provenance that ensures reproducibility of the calculation results in accordance to the FAIR principles. We describe the layout of the AiiDA-Spirit plugin and demonstrate its capabilities using selected examples for LLG spin-dynamics and Monte Carlo calculations. Furthermore, the integration with first-principles calculations through AiiDA is demonstrated at the example of γ–Fe, where the complex spin-spiral ground state is investigated.

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]

For the Landau-Lifshitz-Gilbert (LLG) equation of micromagnetics we study linearly implicit backward difference formula (BDF) time discretizations up to order 5 combined with higher-order non-conforming finite element space discretizations, which are based on the weak formulation due to Alouges but use approximate tangent spaces that are defined by L2-averaged instead of nodal orthogonality constraints. We prove stability and optimal-order error bounds in the situation of a sufficiently regular solution. For the BDF methods of orders 3 to 5, this requires that the damping parameter in the LLG equations be above a positive threshold; this condition is not needed for the A-stable methods of orders 1 and 2, for which furthermore a discrete energy inequality irrespective of solution regularity is proved. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau–Lifshitz–Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S 2 , at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles. [ABSTRACT FROM AUTHOR]

We address the question of global in time existence of solutions to a magnetoviscoelastic system with general initial data. We show that the notion of dissipative solutions allows to prove such an existence in two and three dimensions. This extends an earlier result for the viscoelastic subsystem to the setting which includes the magnetization vector and its evolution in terms of a Landau-Lifshitz-Gilbert equation. [ABSTRACT FROM AUTHOR]

Abstract We consider the multidimensional dimensional inhomogeneous Landau–Lifshitz–Gilbert (ILLG) equation and its degenerate case, the Schrödinger map equation. We investigate the special solutions (under large initial values) and their energy property of the ILLG and Schrödinger map equations. Until now, we had not seen a paper presenting an explicit dynamic solution of the multidimensional ILLG. Using the stereographic method, an equivalent equation of ILLG is obtained. Based on this equivalent system, we obtain some exact solutions of the ILLG equation and present some implicit solutions of the Schrödinger map equation. Based on these solutions, by a careful estimation we give the decay rate of energy density.

In this paper, we aim at a reduced 2d-model describing the observable states of the magnetization in curved thin films. Under some technical assumptions on the geometry of the thin-film, it is well-known that the demagnetizing field behaves like the projection of the magnetization on the normal to the thin film. We remove these assumptions and show that the result holds for a broader class of surfaces; in particular, for compact surfaces. We treat both the stationary case, governed by the micromagnetic energy functional, and the time-dependent case driven by the Landau–Lifshitz–Gilbert equation. [ABSTRACT FROM AUTHOR]

We study the dynamics of a domain wall (DW) in the magnetic core of thin amorphous glass-coated bistable microwires with a circular cross section containing longitudinal inhomogeneities. We use a systematic analytic approach to the problem of finding particular solutions of the continuous Heisenberg model for which we use Landau—Lifshitz—Gilbert equations. We establish a relation between the structure of a material including defects and the DW mobility that explains some experimental data. For a given defect distribution in the longitudinal direction, we study the influence of defects on DW propagation in bistable glass-coated microwires. We obtain new key formulas for the DW velocity and acceleration based on taking the average defect distribution into account. [ABSTRACT FROM AUTHOR]

We propose an unconditionally convergent linear finite element scheme for the stochastic Landau–Lifshitz–Gilbert (LLG) equation with multi-dimensional noise. By using the Doss–Sussmann technique, we first transform the stochastic LLG equation into a partial differential equation that depends on the solution of the auxiliary equation for the diffusion part. The resulting equation has solutions absolutely continuous with respect to time. We then propose a convergent θ -linear scheme for the numerical solution of the reformulated equation. As a consequence, we are able to show the existence of weak martingale solutions to the stochastic LLG equation. [ABSTRACT FROM AUTHOR]

• The antiferromagnetic exchange interaction between the spins in the core can greatly reduce the coercivity of the system, and the coercivity decreases with the increase of the interface exchange interaction. • When the anisotropy constant of the antiferromagnetic core increases gradually, the coercivity of the system varies nonlinearly. • When the transverse field is added, the increase of the transverse field has a weakening effect on the coercivity of systems, but the hysteresis loop of the system is distorted when the critical value is exceeded. The magnetization switching behavior of magnetic nanotube rings with core–shell structure is investigated by solving the Landau-Lifshitz-Gilbert (LLG) equation, focusing on the effects of the magnetic structure, interfacial exchange interactions, anisotropy constants, and transverse field on the coercivity of the system. The reduction of the coercivity of the magnetic nanorings can increase the magnetization switching rate and reduces the energy consumption when the magnetic nanorings are used as magnetic storage units. In the study, it is found that the antiferromagnetic exchange interaction between the spins in the core can greatly reduce the coercivity of the system, and the coercivity decreases with the increase of the interface exchange interaction. When the anisotropy constant of the antiferromagnetic core gradually increases, the coercivity of the system varies nonlinearly and a point of minima occurs. Finally, the effect of the transverse magnetic field on the hysteresis behavior of the system is also discussed, and it is found that the coercivity of the system decreases gradually with the increase of the transverse magnetic field, but the hysteresis loop of the system is distorted when the critical value is exceeded. [ABSTRACT FROM AUTHOR]

A theory of voltage-induced control of magnetic domain walls propagating along the major axis of a magnetostrictive nanostrip, tightly coupled with a ceramic piezoelectric, is developed in the framework of the Landau–Lifshitz–Gilbert equation. It is assumed that the strains undergone by the piezoelectric actuator, subject to an electric field generated by a dc bias voltage applied through a couple of lateral electrodes, are fully transferred to the magnetostrictive layer. Taking into account these piezo-induced strains and considering a magnetostrictive linear elastic material belonging to the cubic crystal class, the magnetoelastic field is analytically determined. Therefore, by using the classical traveling-wave formalism, the explicit expressions of the most important features characterizing the two dynamical regimes of domain-wall propagation have been deduced, and their dependence on the electric field strength has been highlighted. Moreover, some strategies to optimize such a voltage-induced control, based on the choice of the ceramic piezoelectric material and the orientation of dielectric poling and electric field with respect to the reference axes, have been proposed.

The implicit mid-point rule is a Runge–Kutta numerical integrator for the solution of initial value problems, which possesses important properties that are relevant in micromagnetic simulations based on the Landau–Lifshitz–Gilbert equation, because it conserves the magnetization length and accurately reproduces the energy balance (i.e. preserves the geometric properties of the solution). We present an adaptive step size version of the integrator by introducing a suitable local truncation error estimator in the context of a predictor-corrector scheme. We demonstrate on a number of relevant examples that the selected step sizes in the adaptive algorithm are comparable to the widely used adaptive second-order integrators, such as the backward differentiation formula (BDF2) and the trapezoidal rule. The proposed algorithm is suitable for a wider class of non-linear problems, which are linearised by Newton's method and require the preservation of geometric properties in the numerical solution. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of the Landau–Lifshitz–Gilbert equation, which describes the dynamics of the magnetization in ferromagnetic materials. In addition to the classical micromagnetic contributions, the energy comprises the Dzyaloshinskii–Moriya interaction, which is the most important ingredient for the enucleation and the stabilization of chiral magnetic skyrmions. We propose and analyze three tangent plane integrators, for which we prove (unconditional) convergence of the finite element solutions towards a weak solution of the problem. The analysis is constructive and also establishes existence of weak solutions. Numerical experiments demonstrate the applicability of the methods for the simulation of practically relevant problem sizes. [ABSTRACT FROM AUTHOR]

Highlights • Magnetophoretic force on the cluster of nanoparticles is obtained numerically. • For a given particle number, the force decreases with concentration. • Simulation results are interpreted analytically. • New analytical expression for the magnetophoretic mobility is proposed. Abstract Magnetophoretic force acting on a rigid spherical cluster of single-domain nanoparticles in a constant-gradient weak magnetic field is investigated numerically using the Langevin dynamics simulation method. Nanoparticles are randomly and uniformly distributed within the cluster volume. They interact with each other via long-range dipole-dipole interactions. Simulations reveal that if the total amount of particles in the cluster is kept constant, the force decreases with increasing nanoparticle concentration due to the demagnetizing field arising inside the cluster. Numerically obtained force values with great accuracy can be described by the modified mean-field theory, which was previously successfully used for the description of various dipolar media. Within this theory, a new expression is derived, which relates the magnetophoretic mobility of the cluster with the concentration of nanoparticles and their dipolar coupling parameter. The expression shows that if the number of particles in the cluster is fixed, the mobility is a nonmonotonic function of the concentration. The optimal concentration values that maximize the mobility for a given amount of magnetic phase and a given dipolar coupling parameter are determined. [ABSTRACT FROM AUTHOR]