A novel formulation for the numerical computation of magnetization modes in complex micromagnetic systems

The small oscillation modes in complex micromagnetic systems around an equilibrium are numerically evaluated in the frequency domain by using a novel formulation, which naturally preserves the main physical properties of the problem. The Landau–Lifshitz–Gilbert (LLG) equation, which describes magnetization dynamics, is linearized around a stable equilibrium configuration and the stability of micromagnetic equilibria is discussed. Special attention is paid to take into account the property of conservation of magnetization magnitude in the continuum as well as discrete model. The linear equation is recast in the frequency domain as a generalized eigenvalue problem for suitable self-adjoint operators connected to the micromagnetic effective field. This allows one to determine the normal oscillation modes and natural frequencies circumventing the difficulties arising in time-domain analysis. The generalized eigenvalue problem may be conveniently discretized by finite difference or finite element methods depending on the geometry of the magnetic system. The spectral properties of the eigenvalue problem are derived in the lossless limit. Perturbation analysis is developed in order to compute the changes in the natural frequencies and oscillation modes arising from the dissipative effects. It is shown that the discrete approximation of the eigenvalue problem obtained either by finite difference or finite element methods has a structure which preserves relevant properties of the continuum formulation. Finally, the generalized eigenvalue problem is solved for a rectangular magnetic thin-film by using the finite differences and for a linear chain of magnetic nanospheres by using the finite elements. The natural frequencies and the spatial distribution of the natural modes are numerically computed.     

Micromagnetic study of magnetic domain structure and magnetization reversal in amorphous wires with circular anisotropy

In this work we present a detailed numerical investigation on the magnetic domain formation and magnetization reversal mechanism in sub-millimeter amorphous wires with negative magnetostriction by means of micromagnetic calculations. The formation of circular magnetic domains surrounding a multidomain axially oriented central nucleus was observed for the micromagnetic model representing the amorphous wire. The magnetization reversal explained by micromagnetic computations for the M-H curve is described in terms of a combined nucleation-propagation−rotational mechanism after the saturated state. Results are interpreted in terms of the effective magnetic anisotropy.     

Influence of three-dimensional inhomogeneities of the magnetic parameters on the dynamics of vortex-like domain walls

The interaction of a vortex-like domain wall moving in an external magnetic field with a three-dimensional periodic chain of cubic volumes with high values of the saturation magnetization and magnetic anisotropy constant has been investigated theoretically. It has been found that the result of the interaction depends on the initial distance between the wall and the region of inhomogeneity of magnetic parameters at the moment of turning on the external magnetic field. The pinning of domain walls near the regions with high values of the saturation magnetization and magnetic anisotropy constant has been investigated, and the anisotropy of the corresponding depinning fields has been revealed. The method of investigation is the numerical micromagnetic simulation.     

Nonlinear oscillations of magnetization for ferromagnetic particles in the vortex state and their ordered arrays

The dynamics of magnetization oscillations with a considerable amplitude and a radial symmetry in small ferromagnetic particles in the form of a thin disk with a magnetic vortex has been investigated. The collective variables that describe radially symmetric oscillations of the magnetization dynamics for particles in the vortex state are introduced, and the dependence of the particle energy is studied as a function of these variables. The analytical expressions describing the frequency of magnetization oscillations with a radial symmetry, including nonlinear oscillations, are derived using the collective variables. It is shown that the presence of a magnetic field oriented perpendicular to the particle plane reduces the oscillation frequency and can lead to hybridization of this mode with other modes of spin oscillations, including the mode of translational oscillations of the vortex core. The soliton solutions describing the propagation of collective oscillations along the chain of magnetic particles are found.     

Two-dimensional micromagnetic simulation of domain structures in films with combined anisotropy

The transformation of the domain structure of micrometer-thick films with variations in the induced uniaxial anisotropy constant with the easy magnetization axis perpendicular to the film surface has been investigated using numerical micromagnetic simulation in the framework of a two-dimensional model of the magnetization distribution. The case where the tetra-axial crystallographic anisotropy exists in the film with uniaxial magnetic anisotropy has been considered. The transformation of the open domain structure into the structure with a magnetic flux closed inside the sample has been investigated in detail, and new types of 109-degree and 90-degree vortex-like domain walls and periodic domain structures have been obtained.     

High-frequency dynamics of Co/Fe multilayers with stripe domains

Within the framework of two-dimensional (2D) numerical micromagnetic simulations, the equilibrium magnetization configuration and the high-frequency (0.1–30 GHz) linear response of Co/Fe multilayers have been investigated in detail. Due to the perpendicular anisotropy of Co layers, a stripe domain pattern can develop through the whole multilayer, the characteristics of which depend on the magnitude of the perpendicular anisotropy, the respective thicknesses of Co and Fe layers and the number of Co/Fe bilayers in the stack. One of the most striking features associated with the layering effect is the ripening aspect of the static magnetization configuration across the multilayers which induces complicated dynamic susceptibility spectra including surface modes and volume modes strongly confined within the inner Fe layers. The effect of the cubic magnetocrystalline anisotropy of Fe layers and the influence of a nonuniform perpendicular magnetic anisotropy within the Co layers on the static and dynamic magnetic properties of Co/Fe multilayers are then analyzed quantitatively.     

Time-resolved measurement of propagating spin waves in ferromagnetic thin films

We measure the propagation of spatially localized spin waves in NiFe thin films through local inductive detection of the dynamic magnetization. A pulsed magnetic field excites a linear superposition of spin wave modes with a distribution that is predominantly driven by the spatial dependence of the in-plane excitation field. The results of numerical micromagnetic calculations exhibit excellent agreement with experiment and show that a comprehensive account of spatial nonuniformity and propagation is necessary to accurately measure the intrinsic damping rate.     

Thin ferromagnetic nanodisk in transverse magnetic field

The magnetization of a ferromagnetic nanodisk is studied using micromagnetic modeling. It is demonstrated that, under an external magnetic field applied perpendicular to the disk surface, magnetic phase transitions can occur between uniform states, between uniform and vortex states, and between vortex states with different directions of polarization. A simple variation model is proposed describing the observed magnetic states quantitatively.     

Localized spin-wave resonance modes of ferromagnetic microstrips in the field of a magnetic probe

Some results of the micromagnetic modeling of forced magnetization oscillations in planar microstrips of NiFe with easy plane anisotropy and Co/Pt with perpendicular easy axis anisotropy in the field of a magnetic spherical probe are considered. It has been shown that the probe field provokes the appearance of a hedgehog–antivortex coupling state in the NiFe strips, due to its lateral components and a skyrmion magnetization state in the Co/Pt layer. These effects destroy spatial magnetization oscillations in the microstrips and lead to the appearance of additional resonances in the spectrum of oscillations corresponding to the modes localized in the probe field.     

Micromagnetic Analysis of Edge Effects in a Thin Magnetic Film during Local Excitation of Magnetization Oscillations

Russian Physics Journal - The method of numerical micromagnetic simulation was used to study the magnetization dynamics of a thin film with uniaxial magnetic anisotropy during ferromagnetic...     

具有倾斜极化层的自旋阀结构中磁翻转以及磁振荡模式的微磁模拟

基于自由层与钉扎层均为垂直磁各向异性的自旋阀结构,采用微磁学模拟与傅里叶分析相结合的技术,研究了极化层磁矩小角度倾斜情形下自由层磁矩的进动翻转特性.通过沿样品垂直膜面方向同时施加电流与磁场,观察到自由层磁矩垂直膜面方向分量的平均值随磁场的演化翻转曲线中出现了多个凹槽.模拟研究结果表明:在一定的电流范围内,凹槽出现的位置与电流大小无关;而在固定的应用电流下,凹槽出现的位置将会受到样品厚度的影响;在凹槽区域内,非一致进动模式、自旋驻波模式、局域自旋波模式等多种磁振荡模式被激发.通过傅里叶分析,得到了各种磁振荡模式的频谱,频谱中的频率分布体现出了倍频以及间谐波的频率特性.     

Micromagnetism in a planar system with a random magnetic anisotropy and two-dimensional magnetic correlations

The hysteresis loops and the micromagnetic structure of a ferromagnetic nanolayer with a randomly oriented local easy magnetization axis and two-dimensional magnetization correlations are studied using a micromagnetic simulation. The properties and the micromagnetic structure of the nanolayer are determined by the competition between the anisotropy and exchange energies and by the dipole–dipole interaction energy. The magnetic microstructure can be described as an ensemble of stochastic magnetic domains and topological magnetization defects. Dipole–dipole interaction suppresses the formation of topological magnetization defects. The topological defects in the magnetic microstructure can cause a sharper change in the coercive force with the crystallite size than that predicted by the random magnetic anisotropy model.     

Tailoring the magnetic properties of Fe asymmetric nanodots

Asymmetric dots as a function of their geometry have been investigated using three-dimensional (3D) object oriented micromagnetic framework (OOMMF) code. The effect of shape asymmetry of the disk on coercivity and remanence is studied. Angular dependence of the remanence and coercivity is also addressed. Asymmetric dots are found to reverse their magnetization by nucleation and propagation of a vortex, when the field is applied parallel to the direction of asymmetry. However, complex reversal modes appear when the angle at which the external field is applied is varied, leading to a non-monotonic behavior of the coercivity and remanence.     

The magnetic structure of nanodisks with a crown

We investigate the magnetic properties of arrays composed of Pd/Co/Pd nanodisks with diameters of 450, 650, and 850 nm, respectively, and having crowns fabricated by means of electron-beam lithography. The experimental results show that vortex nucleation and the annihilation of critical fields decrease as nanostructure diameter increases. A feature of the investigated nanostructures is a vortex state with magnetization higher than for flat nanodisks. This is in good agreement with micromagnetic simulations of disk + ring nanosystems.     

矩形磁性纳米点动力学反磁化过程的微磁学研究

采用微磁学模拟方法研究了初始态为C形磁结构的矩形CoFe纳米点在方波脉冲场作用下的动力学反磁化过程.研究发现,随着脉冲场强的增强,磁体的反磁化模式发生了改变.当场强较弱时反磁化过程通过畴壁移动-单涡旋的形成和移动来完成;当场强较大时反磁化过程模式转变为畴壁移动-双涡旋的形成与移动;在更强的场强下反磁化过程通过畴壁的移动-多涡旋的形成与湮没来实现.由于反磁化模式随场强的变化而改变,反磁化时间随场强的增大出现振荡变化现象. 关键词： 动力学反磁化过程 反磁化时间 微磁学模拟     

Ultrafast nanomagnetic toggle switching of vortex cores

We present an ultrafast route for a controlled, toggle switching of magnetic vortex cores with ultrashort unipolar magnetic field pulses. The switching process is found to be largely insensitive to extrinsic parameters, like sample size and shape, and it is faster than any field-driven magnetization reversal process previously known from micromagnetic theory. Micromagnetic simulations demonstrate that the vortex core reversal is mediated by a rapid sequence of vortex-antivortex pair creation and annihilation subprocesses. Specific combinations of field-pulse strength and duration are required to obtain a controlled vortex core reversal. The operational range of this reversal mechanism is summarized in a switching diagram for a 200 nm Permalloy disk.     

Mechanically driven domain wall movement in magnetoelastic nanomagnets

Magnetic domain walls are fundamental objects arising in ferromagnetic materials, largely investigated both through micromagnetic simulations and experiments. While current- and field-based techniques for inducing domain wall propagation have been widely studied for fundamental understanding and application-oriented purposes, the possibility to manipulate domain walls using mechanical stress in magnetoelastic materials has only recently drawn interest. Here, a complete analytical model describing stress-induced transverse domain wall movement in ferromagnetic nanostripe with variable cross-section is presented. This approach yields a nonlinear integro-differential equation describing the magnetization field. Its numerical implementation, based on the nonlinear relaxation method, demonstrates the possibility to precisely control the position of a domain wall through mechanical action.