Landau-Lifshitz-Gilbert Equation Research Articles

A micromagnetic-mechanically coupled phase-field model for fracture and fatigue of magnetostrictive alloys

Magnetostrictive alloys are usually brittle materials with micromagnetic structures. Their structural reliability and durability depend on the complex micromagnetic-mechanical coupling at smaller length scales encompassing the evolution of micromagnetic structures. Herein we propose a micromagnetic-mechanically coupled phase-field model for fracture and fatigue behavior of magnetostrictive alloys with evolution of the micromagnetic structure. The thermodynamically-consistent model is derived from microforce theory, laws of thermodynamics, and Coleman–Noll analysis. The evolution of crack phase-field and magnetization-vector order parameters that are fully coupled is governed by history field dependent Allen–Cahn and Landau–Lifshitz–Gilbert equations, respectively. The model is extended to fatigue by introducing a degradation prefactor for the fracture energy as a function of positive elastic energy. One-dimensional analyses are then presented to anatomize the crack driving forces in terms of fully coupled micromagnetic-mechanical and pure mechanical driving force. We demonstrate the model capabilities by finite-element numerical studies on the micromagnetic domain evolution during the crack propagation and the influence of external magnetic field for type-I, type-II, and three-point bending fracture, as well as for the fracture of a single-edge notched specimen with an elliptical inclusion. The simulation result shows that depending on how micromagnetic domains are switched under micromagnetic-mechanical coupling, the magnetic field can enhance or decrease the critical load. In the presence of inclusion with larger fracture toughness, a crack is found to nucleate in the tri-junction of multi-domain micromagnetic structure owing to the high elastic strain around the tri-junction point. It is further found that a suitable magnetic field promoting magnetization-vector rotation around the crack tip could remarkably improve the fracturing load and fatigue life. The results demonstrate the model promising for the study of micromagnetic-mechanically coupled fracture and fatigue in magnetostrictive alloys.

Finite element analysis of pulse sharpening effect of gyromagnetic nonlinear transmission line based on Landau–Lifshitz–Gilbert equation

Ferrite-loaded gyromagnetic nonlinear transmission line (GNLTL) provides a possible option to compress an input pulse to a narrower width for its remarkable sharpening effect. However, it is difficult to accurately predict the output of the GNLTL due to the complex interaction between the magnetic moment of ferrite and the bias magnetic field. In this paper, a finite element model of the GNLTL is established based on the Landau–Lifshitz–Gilbert equation to investigate the performance of the GNLTL. To validate this model, a prototype is used for experimental comparison. The result demonstrates good agreement between experiment and simulation. This paper further explores the influence of the bias magnetic field and the length of the GNLTL on the output pulse. Moreover, a method to sharpen the falling edge is proposed based on the reflection and superposition of the GNLTL output. Simulation and experimental results show its effectiveness and feasibility.

Entanglement between Micro-Magnetism, electromagnetism and the Tensor Magnetic Phase Theory (TMPT) – Symmetry, conservation and invariance laws analysis at low frequency

Ferromagnetic materials show magnetic structures with domains and walls. Thanks to decades of research regarding the origin and behaviour of magnetic domains, we now possess a general foundation which has been verified experimentally in single crystals and powders. The governing equations at the microscopic scale were built in the 1960 s when Brown published calculations of the magnetic moments distribution inside domain walls. This micromagnetic theory uses the so called LLG ‘Landau-Lifshitz-Gilbert’ equation and can include the damping effects. The LLG equation requires a coupling to the field derived from energy contributions: exchange, anisotropy, magnetostriction, stray-field … and the anti-eddy field. At the macroscopic scale, such behaviours are lumped in a homogenized magnetization law for the electromagnetic field equations inside larger polycrystals. Therefore, the inhomogeneous magnetic material nature is always ignored. The Tensor Magnetic Phase Theory (TMPT) describes the magnetic structure and the magnetization thanks to a tensor variable at a mesoscopic scale. The material structuring is then explained thanks to an energy balance which will be discussed.This paper presents the results of investigations on entangled relationships between the micromagnetic theory, the electromagnetic theory and what is called the Tensor Magnetic Phase Theory (TMPT), which statistically describes the magnetic structures of soft ferromagnetic materials. It examines a connection between the TMPT and the LLG by deriving the main energy terms. Then, the TMPT must stay compatible and coupled to the Maxwell equations at low frequencies with volume and surface connections. Additionally, this paper investigates the way to derive the domains structuring and magnetization laws through the Lagrange principle and the corresponding conservation laws with invariants linked to the Nœther theorem. Finally, the TMPT must be discussed while checking its coherence and formulation when changing the reference frame.

Solutions to the Landau–Lifshitz–Gilbert equation in the frequency space: Discretization schemes for the dynamic-matrix approach

The dynamic matrix method addresses the Landau–Lifshitz–Gilbert (LLG) equation in the frequency domain by transforming it into an eigenproblem. Subsequent numerical solutions are derived from the eigenvalues and eigenvectors of the dynamic matrix. In this work we explore discretization methods needed to obtain a matrix representation of the dynamic operator, a fundamental counterpart of the dynamic matrix. Our approach opens a new set of linear algebra tools for the dynamic matrix method and expose the approximations and limitations intrinsic to it. Moreover, our discretization algorithms can be applied to various discretization schemes, extending beyond micromagnetism problems. We present some application examples, including a technique to obtain the dynamic matrix directly from the magnetic free energy function of an ensemble of macrospins, and an algorithmic method to calculate numerical micromagnetic kernels, including plane wave kernels. We also show how to exploit symmetries and reduce the numerical size of micromagnetic dynamic-matrix problems by a change of basis. This procedure significantly reduces the size of the dynamic matrix by several orders of magnitude while maintaining high numerical precision. Additionally, we calculate analytical approximations for the dispersion relations in magnonic crystals. This work contributes to the understanding of the current magnetization dynamics methods, and could help the development and formulations of novel analytical and numerical methods for solving the LLG equation within the frequency domain.

Magnetic tunnel junction random number generators applied to dynamically tuned probability trees driven by spin orbit torque

Perpendicular magnetic tunnel junction (pMTJ)-based true-random number generators (RNGs) can consume orders of magnitude less energy per bit than CMOS pseudo-RNGs. Here, we numerically investigate with a macrospin Landau–Lifshitz-Gilbert equation solver the use of pMTJs driven by spin–orbit torque to directly sample numbers from arbitrary probability distributions with the help of a tunable probability tree. The tree operates by dynamically biasing sequences of pMTJ relaxation events, called ‘coinflips’, via an additional applied spin-transfer-torque current. Specifically, using a single, ideal pMTJ device we successfully draw integer samples on the interval [0, 255] from an exponential distribution based on p-value distribution analysis. In order to investigate device-to-device variations, the thermal stability of the pMTJs are varied based on manufactured device data. It is found that while repeatedly using a varied device inhibits ability to recover the probability distribution, the device variations average out when considering the entire set of devices as a ‘bucket’ to agnostically draw random numbers from. Further, it is noted that the device variations most significantly impact the highest level of the probability tree, with diminishing errors at lower levels. The devices are then used to draw both uniformly and exponentially distributed numbers for the Monte Carlo computation of a problem from particle transport, showing excellent data fit with the analytical solution. Finally, the devices are benchmarked against CMOS and memristor RNGs, showing faster bit generation and significantly lower energy use.

Temporal High-Order Accurate Numerical Scheme for the Landau–Lifshitz–Gilbert Equation

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.

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

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

Magneto-mechanical properties of sintered NdFeB under impact load: Impact-induced demagnetization experiment, constitutive model and micromagnetic simulation

With the widespread applications of sintered NdFeB, the impact-prone scenarios are often encountered. Such impacts will cause magnetic disorder inside the magnet and further demagnetization. At present, research on impact-induced demagnetization of sintered NdFeB mostly focuses on experimental investigations, lacking clear theoretical expressions and visual reproduction of the microscopic demagnetization process. To end this, this paper uses the impact-induced demagnetization experiment, theoretical expression, and micromagnetic simulation to reveal and quantify the demagnetization behavior of NdFeB. An impact-induced demagnetization experimental platform capable of producing corresponding load amplitudes is first established. The experimental results demonstrate the existence of reversible and irreversible demagnetization of NdFeB under impact. A magneto-mechanical constitutive model for sintered NdFeB considering stress history is further developed using the principle of thermodynamic, pinning effect and proximity effect. The residual magnetization intensity between the numerical simulation and the experimental results have good agreement. Finally, the demagnetization behavior of the magnet is visually displayed through the micromagnetic simulation. Employing the LLG equation and kinematic equations for elastic waves, a set of magnetization dynamics equations is established to analyze impact-induced demagnetization characteristics in sintered NdFeB. These results will provide the reference for the application of NdFeB in impact environment.

Antiferromagnetic and nutation resonance frequencies of antiferromagnets at an arbitrary strength of the applied dc field

Nutation and precession resonances in an antiferromagnet subjected to a dc magnetic field are investigated by employing coupled linearized inertial Landau–Lifshitz–Gilbert equations describing the dynamics of magnetizations of antiferromagnet sublattices with uniaxial magnetocrystalline anisotropy. Analytical expressions for the eigenfrequencies of such an antiferromagnet are obtained for the longitudinal and transverse directions of the external dc field and for different ranges of its strength. The effect of inertia on the values of the resonant frequencies is shown for all possible states of the antiferromagnet in both the longitudinal and transverse directions of the external field. The estimated resonant frequencies are compared with those obtained from the numerical solution of the system of undamped inertial Landau–Lifshitz–Gilbert equations for closed trajectories of sublattice magnetizations. The good agreement of both independent estimations is demonstrated.

Nonlinear response of magnetization obeying inertial dynamics to superimposed dc and ac fields and dynamic magnetic hysteresis

The nonlinear ac stationary responses of magnetization and the dynamic magnetic hysteresis of ferromagnetic particles with uniaxial anisotropy acted on by both external dc and ac magnetic fields are evaluated by employing the inertial Landau–Lifshitz–Gilbert equation. Analytical expressions are obtained for the components of the second- and third-order nonlinear susceptibility tensors using the perturbation method. The dynamic nonlinear susceptibility strongly depends on the dc field strength. The effect of inertia on the area and slope of the dynamic magnetic hysteresis loop is demonstrated. In addition to nutation resonance, the main nonlinear effects previously observed in the ferromagnetic resonance region (frequency doubling and appearance of weak subharmonic resonance peaks) are predicted in the nutation resonance frequency region.

Magnetic dynamics of strained non-collinear antiferromagnet

In this work, we theoretically study the switching and oscillation dynamics in strained non-collinear antiferromagnet (AFM) Mn3X (X = Sn, Ge, etc.). Using the perturbation theory, we identify three separable dynamic modes—one uniform and two optical modes, for which we analytically derive the oscillation frequencies and effective damping. We also establish a compact, vector equation for describing the dynamics of the uniform mode, which is in analogy to the conventional Landau–Lifshitz–Gilbert (LLG) equation for ferromagnet but captures the unique features of the cluster octuple moment. Extending our model to include spatial inhomogeneity, we are able to describe the excitations of dissipative spin wave and spin superfluidity state in the non-collinear AFM. Furthermore, we carry out numerical simulations based on coupled LLG equations to verify the analytical results, where good agreements are reached. Our treatment with the perturbative approach provides a systematic tool for studying the dynamics of non-collinear AFM and is generalizable to other magnetic systems in which the Hamiltonian can be expressed in a hierarchy of energy scales.

Midpoint geometric integrators for inertial magnetization dynamics

We consider the numerical solution of the inertial version of Landau-Lifshitz-Gilbert equation (iLLG), which describes high-frequency nutation on top of magnetization precession due to angular momentum relaxation. The iLLG equation defines a higher-order nonlinear dynamical system with very different nature compared to the classical LLG equation, requiring twice as many degrees of freedom for space-time discretization. It exhibits essential conservation properties, namely magnetization amplitude preservation, magnetization projection conservation, and a balance equation for generalized free energy, leading to a Lyapunov structure (i.e. the free energy is a decreasing function of time) when the external magnetic field is constant in time. We propose two second-order numerical schemes for integrating the iLLG dynamics over time, both based on implicit midpoint rule. The first scheme unconditionally preserves all the conservation properties, making it the preferred choice for simulating inertial magnetization dynamics. However, it implies doubling the number of unknowns, necessitating significant changes in numerical micromagnetic codes and increasing computational costs especially for spatially inhomogeneous dynamics simulations. To address this issue, we present a second time-stepping method that retains the same computational cost as the implicit midpoint rule for classical LLG dynamics while unconditionally preserving magnetization amplitude and projection. Special quasi-Newton techniques are developed for solving the nonlinear system of equations required at each time step due to the implicit nature of both time-steppings. The numerical schemes are validated on analytical solution for macrospin terahertz frequency response and the effectiveness of the second scheme is demonstrated with full micromagnetic simulation of inertial spin waves propagation in a magnetic thin-film.

Strain‐controlled dynamics of transverse domain walls in hybrid piezoelectric–magnetostrictive heterostructures: An asymptotic approach

AbstractThe present work deals with an analytical study of the statical and dynamical behavior of transverse magnetic domain walls in hybrid bilayer piezoelectric/magnetostrictive heterostructures under the combined influence of spin‐polarized electric currents, transverse, and axial applied magnetic fields. The magnetostrictive material is considered to be isotropic and linearly elastic. We perform the investigation within the framework of the one‐dimensional Landau–Lifshitz–Gilbert equation. By employing the regular perturbation asymptotic expansion approach, we derive the explicit analytical expression of the most significant dynamic features, namely, traveling wave profile, DW velocity, and displacement under the influence of small and moderate transverse magnetic fields. Finally, we illustrate numerically the analytical results and delineate the domain wall motion for metallic and semiconductor ferromagnets.

Bifurcation to complex dynamics in largely modulated voltage-controlled parametric oscillator

An experimental demonstration of a parametric oscillation of a magnetization in a ferromagnet was performed recently by applying a microwave voltage, indicating the potential to be applied in a switching method in non-volatile memories. In the previous works, the modulation of a perpendicular magnetic anisotropy field produced by the microwave voltage was small compared with an external magnetic field pointing in an in-plane direction. A recent trend is, however, opposite, where an efficiency of the voltage controlled magnetic anisotropy (VCMA) effect is increased significantly by material research and thus, the modulated magnetic anisotropy field can be larger than the external magnetic field. Here, we solved the Landau–Lifshitz–Gilbert equation numerically and investigated the magnetization dynamics driven under a wide range of the microwave VCMA effect. We evaluated bifurcation diagrams, which summarize local maxima of the magnetization dynamics. For low modulation amplitudes, the local maximum is a single point because the dynamics is the periodic parametric oscillation. The bifurcation diagrams show distributions of the local maxima when the microwave magnetic anisotropy field becomes larger than the external magnetic field. The appearance of this broadened distribution indicates complex dynamics such as chaotic and transient-chaotic behaviors, which were confirmed from an analysis of temporal dynamics.

Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ2

This paper focuses on establishing the existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya (DM) interaction and V flow in two‐dimensional domains, including the torus and Euclidean space. The primary objective is to establish global existence results by investigating the conditions under which a smooth solution exists for all time, provided that the norm of the initial magnetization gradient is sufficiently small. Rigorous mathematical proofs for these global existence results are provided by combining analytical techniques and energy estimates. These findings enhance our understanding of solution behavior and regularity in the Landau–Lifshitz–Gilbert equation with DM interaction, shedding light on its dynamics in different spatial domains.

Microscopic theory of current-induced skyrmion transport and its application in disordered spin textures

Introduction: Magnetic skyrmions hold great promise for realizing compact and stable memory devices that can be manipulated at very low energy costs via electronic current densities.Methods: In this work, we extend a recently introduced method to describe classical skyrmion textures coupled to dynamical itinerant electrons. In this scheme, the electron dynamics is described via nonequilibrium Green’s function (NEGF) within the generalized Kadanoff–Baym ansatz, and the classical spins are treated via the Landau–Lifshitz–Gilbert equation. Here, the framework is extended to open systems by the introduction of a non-interacting approximation to the collision integral of NEGFs. This, in turn, allows us to perform computations of the real-time response of skyrmions to electronic currents in large quantum systems coupled to electronic reservoirs, which exhibit linear scaling in the number of time steps. We use this approach to investigate how electronic spin currents and dilute spin disorder affect skyrmion transport and the skyrmion Hall drift.Results: Our results show that the skyrmion dynamics is sensitive to a specific form of the spin disorder, such that different disorder configurations lead to qualitatively different skyrmion trajectories for the same applied bias.Discussion: This sensitivity arises from the local spin dynamics around the magnetic impurities, a feature that is expected not to be well-captured by phenomenological or spin-only descriptions. At the same time, our findings illustrate the potential of engineering microscopic impurity patterns to steer skyrmion trajectories.

Wideband loss separation model of nanocrystalline materials considering microscopic magnetization mechanism

The core loss directly determines the performance of high-frequency transformers. This paper presents an enhanced method for calculating core losses by integrating the LLG equation and Maxwell's diffusion equation. By the method, the losses caused by the rotation of the magnetic moment can be obtained. The loss attributed to domain wall displacement is derived from the total loss by subtracting losses due to magnetic moment rotation, further categorized into the hysteresis loss and the residual loss. The observation of the dynamic variation of nanocrystalline magnetic domains with applied excitation verifies the feasibility of the improved method. Finally, the causes of residual loss generation are also analyzed the model accuracy is verified by comparing the measured and modelled values and the traditional loss separation model.

Research on the simulation accuracy of static hysteresis loops of electrical steels using an improved simplified LLG equation

For tackling the problem of large errors in simulating the static inner symmetrical minor hysteresis loops through the Simplified Landau-Lifshitz-Gilbert (S-LLG) equation, a modification by improving the grain structure arrangement and introducing additional hysteresis energy function is proposed. With the proposed model, only a unified set of parameters identified by four static hysteresis loops is needed to predict hysteresis loops under arbitrarily different magnetic flux density levels. To verify the global simulation capability of the proposed model, the simulation results of hysteresis loops are compared with the measured data of a grain-oriented (GO) steel. The obtained results show that the model has good simulation ability for hysteresis loops under different magnetic flux densities for both sinusoidal and harmonic excitation conditions.

Thermal effect on microwave pulse-driven magnetization switching of Stoner particle

We investigate the cosine-chirped microwave pulse (cosine CMP)-driven magnetization switching of a nanoparticle or stoner particle at a finite temperature in the framework of the stochastic Landau–Lifshitz-Gilbert equation. Numerical results reveal that the ultrafast and efficient magnetization switching is robust even at room temperature, and hence we estimate the maximal temperature at which the magnetization switching is still valid. The maximal temperature increases with the enlargement (by increasing cross-sectional area) of the nanoparticle/stoner particle volume to a certain value, and afterward, the maximal temperature decreases with the further increment of the nanoparticle size. Initially, the shape anisotropy (approximated by the easy-plane) coefficient does not become dominant although the stoner particle volume increases, which plays a role in increasing thermal stability (maximal temperature), and later the shape anisotropy field becomes dominant, which opposes the uniaxial anisotropy, i.e., reduces the energy barrier, which reduces the maximal temperature. For smaller volumes, the parameters of cosine CMP show a decreasing trend with temperature. The initial frequency requirement significantly decreases with shape anisotropy. Therefore, these findings might be useful to realize cosine CMP-driven fast and energy-efficient magnetization switching in device applications.

Regulation of static and dynamic magnetic properties of amorphous FeCoZr composition gradient films by Zr doping

A series of amorphous FeCoZr composition gradient monolayer films with varying Zr contents was prepared on the Si(100) substrate using RF magnetron sputtering. The effects of the Zr sputtering power PZr on the static and dynamic magnetic properties of FeCoZr films were systematically investigated. The results demonstrate that the introduction of the Zr element as a composition gradient into FeCo films not only improves the soft magnetic properties of the films but also enhances their in-plane uniaxial magnetic anisotropy. In particular, the doping of Zr elements leads to the destruction of FeCo lattice, inducing a transformation of the films from polycrystalline to amorphous state, resulting in a significant decrease in coercivity (Hc reduced by 82%) and surface roughness (Ra reduced by 78%). In addition, as PZr increases from 30 to 70 W, the anisotropy fields Hk of the films increase from 128 to 340 Oe, and the resonance frequency fr increases from 4.24 to 5.23 GHz. By fitting the permeability spectrum using the LLG equation, it is found that FeCoZr composition gradient films exhibit a lower damping coefficient α of around 0.011–0.014, indicating the reduction of energy loss during magnetization dynamics. These findings highlight the potential applications of FeCoZr composition gradient films in the field of high-frequency microwaves.