Landau – Lifshitz – Gilbert Research Articles

Electromagnetic breathing dromion-like structures in an anisotropic ferromagnetic medium

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.

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.

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.

Analytical solutions to the precession relaxation of magnetization with uniaxial anisotropy

Based on the Landau–Lifshitz–Gilbert (LLG) equation, the precession relaxation of magnetization is studied when the external field H is parallel to the uniaxial anisotropic field H k. The evolution of three-component magnetization is solved analytically under the condition of H = nH k (n = 3, 1 and 0). It is found that with an increase of H or a decrease of the initial polar angle of magnetization, the relaxation time decreases and the angular frequency of magnetization increases. For comparison, the analytical solution for H k = 0 is also given. When the magnetization becomes stable, the angular frequency is proportional to the total effective field acting on the magnetization. The analytical solutions are not only conducive to the understanding of the precession relaxation of magnetization, but also can be used as a standard model to test the numerical calculation of LLG equation.

Magnetic properties of twisted bilayer magnetic thin films with interlayer dipolar interaction

In this paper, a model of bilayer film with single-ion anisotropy in z-direction and ferromagnetic exchange interaction in the layer and dipolar interaction between layers is established. The influence of dipolar interaction intensity on the magnetic properties of the system at different interlayer twisted angles θ was studied by using Landau Lifshitz Gilbert (LLG) equation. At zero temperature it is found that the system will be in the z-direction ferromagnetic state when the interlayer dipolar interaction intensity is small, but the state will become the z-direction magnetic domain structure when the interlayer dipolar interaction intensity is greater than a certain value. When θ = 0°, the Curie temperature and coercivity of the system decrease with the increase of dipolar interaction intensity, while the resonant frequency increases with the increase of dipolar interaction intensity. When θ ≠ 0°, if the interlayer dipolar interaction intensity is small, the change of θ has little effect on the magnetic properties, but the Curie temperature, coercivity and resonance frequency are slightly lower than that at θ = 0° when the interlayer dipolar interaction is big. When θ is larger than a certain angle, the change of θ has almost no effect on coercivity and resonance frequency, which is similar to the change of the total energy of the system.

Antiferromagnetic bimeron dynamics controlled by magnetic defects

We investigate dynamics of an isolated bimeron in the antiferromagnetic system, in the presence of magnetic defects. The bimeron is driven by torques due to a spin-polarized current, which is incorporated into the Landau–Lifshitz–Gilbert (LLG) equation. We numerically solve the LLG equation, and find a breathing dynamics of the bimeron as well as its transverse motion owing to the defect potential. We also analyze the potential landscapes due to the interaction between the bimeron and a defect, and give a qualitative picture of the numerically obtained dynamics.

Fourier-spectral method for the Landau–Lifshitz–Gilbert equation in micromagnetism

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.

Studying the rate-dependent specific absorption rate in magnetic hyperthermia through multiscale simulations

In this work, the issue of whether the dynamic magnetic properties of monodispersed magnetic colloids, modeled using micromagnetic simulations, can be extrapolated to analyze magnetic particle hyperthermia data, i.e., specific absorption rate (SAR) values acquired at high frequencies of excitation fields, is addressed. Micromagnetic finite difference simulations were performed using the Object Oriented Micromagnetic Framework (OOMMF) software package in order to obtain the dynamic hysteresis loops under a 24 kA/m alternating magnetic field amplitude and for various frequencies (50–765 kHz). In OOMMF, the finite difference method was used to find the solution of the nonlinear Landau–Lifshitz Gilbert (LLG) equation, which describes the nanoparticles’ magnetization motion when applying an effective magnetic field. To create a system of randomly oriented magnetite nanoparticles having a certain volume fraction (0.02%) that coincides with the experimentally utilized concentration of 1 mg/ml, we start with a perfect simple cubic lattice with a large lattice spacing so that the particle–particle distance is large enough to neglect dipolar interactions (non-interacting nanoparticles). The system under study is a set of 40-nm magnetite nanoparticles with a lognormal size distribution. The simulations were performed assuming quasistatic conditions, an approach that is reasonable for ferromagnetic-like behavior. It is worth noting that the code considers not only the uniaxial anisotropy Ku but also the cubic magnetocrystalline one Kc as well. Kc is usually neglected in literature because the uniaxial contribution dominates, but this is not the case for magnetite since Ku = 9 kJ/m3 and Kc = −11 kJ/m3. Moreover, such an inclusion seems quite reasonable since the magnetocrystalline anisotropy is always present yet with a relative contribution. The SAR values at each frequency were determined after calculating hysteresis losses via the area of the simulated hysteresis loops. Interestingly, SAR values at low frequencies followed an exponential increase trend with a frequency indicating a deviation from the linear behavior usually reported in the literature. To validate our approach, we employed a coupled electromagnetic-thermal model based on COMSOL Multiphysics simulations that provides an accurate estimation of the magnetic field and temperature distribution within the ferrofluid. The time-dependent temperature curves are obtained after 30 min of magnetic particle hyperthermia treatment for the same alternating magnetic field amplitude used in OOMMF simulations (30 mT) and for two representative frequency values. One in the low (300 kHz) and one in the high (765 kHz) frequency regimes. The numerical curves were compared to the corresponding experimental ones and found to be in good agreement. Our findings provide new insight into the validity of dynamic micromagnetic simulation to analyze the frequency behavior of SAR within the framework of LLG and indicate that anisotropy selection plays a key role in the reliability of simulations.

A frequency-domain micromagnetic simulation module based on COMSOL Multiphysics

Micromagnetic simulation is a numerical method to solve the Landau–Lifshitz–Gilbert (LLG) equation for magnetic dynamics. Most of the mainstream micromagnetic simulation packages, including the object oriented micromagnetic framework and MuMax3, perform simulation in the time domain. Here, utilizing the frequency domain simulation capability of COMSOL Multiphysics, we developed a COMSOL-based micromagnetic simulation module that solves the LLG equation in the frequency domain, which runs much faster and more accurate than time-domain simulations. Frequency-domain simulation is ideal for finding spin wave eigenmodes and the corresponding dispersions. We verify the validity of the module using three examples in the absence of dipolar field, and the inclusion of the dipolar field can be incorporated by combining this module with the alternating current/direct current module within COMSOL.

Magnetic field detection using spin-torque nano-oscillator combined with magnetic flux concentrator

Spin-torque nano-oscillators (STNO) are studied in terms of the Landau–Lifshitz–Gilbert (LLG) equation. The effect on the limit of detectivity of an STNO concerning externally applied magnetic fields is studied with micromagnetic models by placing adjacent magnetic flux concentrators (MFCs) at different distances from the nanopillar to analyze the effect on the induced auto-oscillations and magnetization dynamics. Perpendicular STNO structures allow for different detectivities with respect to externally applied magnetic fields depending on the distance from the MFCs to the nanopillar. The optimal design of an STNO combined with MFCs is proposed to improve the limit of detectivity, where the STNO consists of two out-of-plane (OP) ferromagnetic (FM) layers separated by a MgO insulating nonmagnetic (NM) thin film, and the MFCs positioned in the vicinity of the STNO are made of permalloy. The time evolution of the free-layer magnetization is governed by the Landau–Lifshitz–Gilbert (LLG) equation. The auto-oscillations induced within the free-layer averaged magnetization are provoked by externally applied magnetic fields. In addition, the DC current-driven auto-oscillations in the STNO structure are studied as a function of the externally applied magnetic field strength, with and without MFCs. The suppression of the DC current-driven auto-oscillations is observed due to the damping effect generated by the MFCs positioned at varying distances with respect to the STNO. By placing MFCs adjacent to the STNO, the lowest detectable magnetic field strength is enhanced from 10 (μT) to 10 (nT). Therefore, it is concluded that MFCs improve the sensitivity of STNO to externally applied magnetic fields thanks to the damped magnetization dynamics. The results presented in this work could inspire the optimal design of STNO and MFC-based ultra-low magnetic field sensors based on nanoscale oscillators and spintronic diodes.

Skin effect suppression for millimeter-wave frequencies through manipulation of permeability

A new device strategy for suppressing unwanted skin effect in coplanar transmission lines at frequencies beyond 20 GHz is presented. Utilization of the anti-resonance point as a source of imaginary permeability to suppress the skin effect is proposed. This idea is backed up by Kramers-Kronig and published experiments. The principle is independent of Landau–Lifshitz-Gilbert magnetic precession damping. The present work contributes both a greater understanding of skin effect manipulation and a method for obtaining zero skin effect using known laminate materials.

Low-Energy Shared-Current Write Schemes for Voltage-Controlled Spin-Orbit-Torque Memory

Voltage-controlled (VC) spin-orbit-torque (SOT) magnetic random access memory (MRAM) is being considered as the next-generation magnetic memory with potential to achieve superior speed, power, and write error rates over existing MRAM technologies. By placing multiple VC devices on a single SOT bus, VC-SOT MRAM can also enable compact structures, in which multiple devices can be addressed individually yet programmed via a shared current. In this work, we propose two implementations of shared-current write: the horizontal shared current write (HSCW), which reduces the average SOT current per bit by the number of bits on the SOT bus, and the vertical shared current write (VSCW), which can further leverage data dependency for increased performance. We simulate the efficiency of the HSCW and VSCW using a Landau–Lifshitz–Gilbert (LLG)-based VC-SOT model and a 28-nm CMOS technology and show that HSCW and VSCW can achieve an energy saving of 74% and 40%–87%, respectively, in a 32-bit setting. Analysis of data patterns in common workloads finds that 40% of data share more than 85% common bits, for which VSCW can leverage for further improved performance.

Ab-initio study and atomistic spin model simulations of Cr2O3 thin films

Magnetoelectric random access memory (MERAM) is the next generation technology of data storage with high-density and low power. Magnetoelectric material can be controlled by voltage such as an electric field to transform spin domain directions for storing data. One of the most promising materials that possess such properties near room temperature is Cr2O3. Although, the bulk has perfect properties for MERAM, it is necessary to fabricate thin films for applicable spintronic devices, i.e., reducing the required external control voltage; however, the film magnetic and physical properties might be different from bulk. In this work, we propose that the film properties might be induced by surface effects and are confirmed with Ab-initio calculation and atomistic spin simulation. The main simulation processes are the density functional theory (DFT) calculating, mapped with the classical Heisenberg model for exchange interaction constants up to the fifth nearest neighbour (J 1-J 5). The atomistic spin model was then used to approximate the Néel temperature (T N), comparing two methods i.e., Monte Carlo simulation and the Landau–Lifshitz–Gilbert (LLG) equation. The Néel temperatures of the film were found to be slightly lower than the bulk, which is in good agreement with previous experiments. Furthermore, the spin dynamic model gives more correct result comparing to the Monte Carlo method.

Influence of Permeability Dispersion on Radiation of BAW Antenna: Modeling of Multiphysics Dynamic Coupling

Coupling of multiphysics including electrodynamics, elastodynamics, magnetodynamics, and electromagnetic (EM) radiation is involved in the operating process of bulk acoustic wave (BAW) antenna. There is still no good way to design the structure and evaluate the performance of the antenna in theory. Therefore, the multiphysics dynamic coupling model of the antenna is constructed by uniting Newton’s equation for acoustic wave, Maxwell’s equation for EM wave, and Landau–Lifshitz–Gilbert (LLG) equation for spin wave with constitutive relations of multiferroic materials to discuss the dispersion law of permeability and magnetomechanical coupling factor for the magnetostrictive layer. The importance of consistency between ferromagnetic resonance (FMR) frequency and BAW resonant frequency is demonstrated. The analysis shows that both increasing the anisotropic field or the saturation magnetization and decreasing the FMR linewidth are beneficial to improving the internal energy conversion efficiency and the average radiated power for BAW antenna. In addition, a micromagnetic module is introduced into finite element analysis (FEA) software to simulate the precession of unit magnetic moment at different frequencies. Thus, the analytical model is verified from the perspective of power absorption.

Spin-diffusion model for micromagnetics in the limit of long times

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.

First-principles calculation of the parameters used by atomistic magnetic simulations

While the ground state of magnetic materials is in general well described on the basis of spin density functional theory (SDFT), the theoretical description of finite-temperature and non-equilibrium properties require an extension beyond the standard SDFT. Time-dependent SDFT (TD-SDFT), which give for example access to dynamical properties are computationally very demanding and can currently be hardly applied to complex solids. Here we focus on the alternative approach based on the combination of a parameterized phenomenological spin Hamiltonian and SDFT-based electronic structure calculations, giving access to the dynamical and finite-temperature properties for example via spin-dynamics simulations using the Landau–Lifshitz–Gilbert (LLG) equation or Monte Carlo simulations. We present an overview on the various methods to calculate the parameters of the various phenomenological Hamiltonians with an emphasis on the KKR Green function method as one of the most flexible band structure methods giving access to practically all relevant parameters. Concerning these, it is crucial to account for the spin–orbit coupling (SOC) by performing relativistic SDFT-based calculations as it plays a key role for magnetic anisotropy and chiral exchange interactions represented by the DMI parameters in the spin Hamiltonian. This concerns also the Gilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral multispin interaction parameters of the extended Heisenberg Hamiltonian, as well as spin–lattice interaction parameters describing the interplay of spin and lattice dynamics processes, for which an efficient computational scheme has been developed recently by the present authors.

Compact Model of Domain Wall MTJ Driven by Spin Orbit Torque and Dzyaloshinskii–Moriya Interaction

Current-induced domain wall motion (CIDWM) shows promising prospects with low power, high density, high speed, and so on. Recent studies have demonstrated that magnetic tunnel junction (MTJ) based on CIDWM has great potential in mimicking a non-volatile artificial neuron and synapse. By combining the effect of the interfacial Dzyaloshinskii–Moriya interaction (DMI) to stabilize the Néel-type domain wall (DW) and the high-speed low-power advantages of spin orbit torque (SOT), the novel MTJ device is experimentally demonstrated with low threshold current and high propagation speed, which shows potential applications in the field of the artificial neuron and synapse. In this article, a compact model for CIDWM-MTJ based on SOT and DMI is presented. The model integrates the SOT mechanism for magnetization reversal and DW nucleation, CIDWM behaviors, and tunnel resistance theory of MTJ nanopillar. The micromagnetic simulation and the temporal evolution of the DW position are implemented by solving the Landau–Lifshitz–Gilbert (LLG) equation and the 1-D model. Based on the developed model, a hybrid CIDWM-MTJ/CMOS circuit is simulated for verification. The presented model combines the CIDWM dynamics and the MTJ magnetic dynamics, showing potential application for the simulation of all-spin artificial neural network (ANN).

Comparative study of magnetic properties of Mn[formula omitted] magnetic clusters in GaN using classical and quantum mechanical approach

Currently, simulations of many-body quantum systems are known to be computationally too demanding to be solved on classical computers. The main problem is that the computation time and memory necessary for performing the calculations usually grow exponentially with the number of particles N. An efficient approach to simulate many-body quantum systems is the use of classical approximation. Such approach can reduce the overall computational complexity, but it usually encounters difficulties in proper reproduction of temperature-dependent properties of a investigated system due to neglect of spin quantization. Therefore, it is timely and important to assess the validity of the classical approximation. To this end, in this work, we compare the results of numerical calculations of small Mn3+ paramagnetic clusters in GaN, where the Mn spins are treated classically with those where they are treated quantum-mechanically (crystal field model). In the first case, we solve the Landau–Lifshitz–Gilbert (LLG) equation that describes the precessional dynamics of spins represented by classical vectors. On the other hand, in the crystal field model, the state of Mn3+ ion (d4 configuration with S=2, L=2) is characterized by the set of orbital and spin quantum numbers |mS,mL>. Particular attention is paid to use numerical parameters that ensure the same single ion magnetic anisotropy in both classical and quantum approximation. Finally, a detailed comparative study of magnetization M(H,T) as a function of the magnetic field H, temperature T, number of ions in a given cluster N and the strength of super-exchange interaction J, obtained from both approaches will be presented.