Numerical Micromagnetic Simulations Research Articles

Voltage-Controlled Reconfigurable Spin-Wave Nanochannels and Logic Devices

Though confinement of spin waves (SWs) through nanochannels (NCs) is essential for the development of magnonic logic hardware, voltage-controlled reconfigurable SW nanochannels and logic devices have remained elusive. Here numerical micromagnetic simulations show that NCs of arbitrary shape and size, down to a few tens of nanometers, can be easily configured to transmit SWs with variable wave vectors as needed, even in the absence of a magnetic field. Operation of reconfigurable logic devices using the NCs is also demonstrated. This study is expected to have a large impact on the development of all-voltage-controlled low-power nanoscale magnonic devices.

Ferromagnetic Resonance Assisted Reconfiguration of Multi-Input Logic Gate Based on Edge Precession Mode

The ferromagnetic resonance (FMR) modes of a multi-input logic gate consisting of Supermalloy-based multiferroic nanomagnets is investigated by micromagnetic simulations and fast-Fourier-transform tools. We propose a clocking scheme to reconfigure the multi-input logic gate by edge-mode resonant excitations based on our numerical micromagnetic simulations. We can reconfigure the input of multi-input logic gate from “000” to “100” by adjusting only the frequency of the resonant excitation, which makes it universal for circuit layout. The miniaturization of the elements is limited. The FMR modes will not work at small dimensions due to the uniform edge modes. By applying a strain along the major axis, we realize the functionality of a multiferroic majority gate. These results may increase programming reliability and provide a guide to multiferroic logic devices applications.

Small-angle neutron scattering modeling of spin disorder in nanoparticles

Magnetic small-angle neutron scattering (SANS) is a powerful technique for investigating magnetic nanoparticle assemblies in nonmagnetic matrices. For such microstructures, the standard theory of magnetic SANS assumes uniformly magnetized nanoparticles (macrospin model). However, there exist many experimental and theoretical studies which suggest that this assumption is violated: deviations from ellipsoidal particle shape, crystalline defects, or the interplay between various magnetic interactions (exchange, magnetic anisotropy, magnetostatics, external field) may lead to nonuniform spin structures. Therefore, a theoretical framework of magnetic SANS of nanoparticles needs to be developed. Here, we report numerical micromagnetic simulations of the static spin structure and related unpolarized magnetic SANS of a single cobalt nanorod. While in the saturated state the magnetic SANS cross section is (as expected) determined by the particle form factor, significant deviations appear for nonsaturated states; specifically, at remanence, domain-wall and vortex states emerge which result in a magnetic SANS signal that is composed of all three magnetization Fourier components, giving rise to a complex angular anisotropy on a two-dimensional detector. The strength of the micromagnetic simulation methodology is the possibility to decompose the cross section into the individual Fourier components, which allows one to draw important conclusions regarding the fundamentals of magnetic SANS.

Linear-chain assemblies of iron oxide nanoparticles

We synthesized iron oxide nanoparticles using a simple hydrothermal approach and found several types of segments of their linear-chain self-assemblies as observed by field emission scanning electron microscopy. X-ray diffraction and transmission electron microscopy measurements confirm a well-defined single-phase FCC structure. Vibrating sample magnetometry measurements exhibit a ferromagnetic behavior. Micromagnetic numerical simulations show magnetic vortex states in the nanosphere model. Also, calculations of binding energies for different numbers of particles in the linear-chain assemblies explain a possible mechanism responsible for the self-assemblies of segments of the linear chains of nanoparticles. This work offers a step towards linear-chain self-assemblies of iron oxide nanoparticles and the effect of magnetic vortex states in individual nanoparticles on their binding energy.

Nanocompositional Electron Microscopic Analysis and Role of Grain Boundary Phase of Isotropically Oriented Nd-Fe-B Magnets

Nanoanalytical TEM characterization in combination with finite element micromagnetic modelling clarifies the impact of the grain misalignment and grain boundary nanocomposition on the coercive field and gives guidelines how to improve coercivity in Nd-Fe-B based magnets. The nanoprobe electron energy loss spectroscopy measurements obtained an asymmetric composition profile of the Fe-content across the grain boundary phase in isotropically oriented melt-spun magnets and showed an enrichment of iron up to 60 at% in the Nd-containing grain boundaries close to Nd2Fe14B grain surfaces parallel to the c-axis and a reduced iron content up to 35% close to grain surfaces perpendicular to the c-axis. The numerical micromagnetic simulations on isotropically oriented magnets using realistic model structures from the TEM results reveal a complex magnetization reversal starting at the grain boundary phase and show that the coercive field increases compared to directly coupled grains with no grain boundary phase independently of the grain boundary thickness. This behaviour is contrary to the one in aligned anisotropic magnets, where the coercive field decreases compared to directly coupled grains with an increasing grain boundary thickness, if Js value is > 0.2 T, and the magnetization reversal and expansion of reversed magnetic domains primarily start as Bloch domain wall at grain boundaries at the prismatic planes parallel to the c-axis and secondly as Néel domain wall at the basal planes perpendicular to the c-axis. In summary our study shows an increase of coercive field in isotropically oriented Nd-Fe-B magnets for GB layer thickness > 5 nm and an average Js value of the GB layer < 0.8 T compared to the magnet with perfectly aligned grains.

A Combined TEM/STEM and Micromagnetic Study of the Anisotropic Nature of Grain Boundaries and Coercivity in Nd-Fe-B Magnets

The nanoanalytical high resolution TEM/STEM investigation of the intergranular grain boundary phase of anisotropic sintered and rapidly quenched heavy rare earth-free Nd-Fe-B magnet materials revealed a difference in composition for grain boundaries parallel (large Fe-content) and perpendicular (low Fe content) to the alignment direction. This behaviour vanishes in magnets with a high degree of misorientation. The numerical finite element micromagnetic simulations are based on the anisotropic compositional behaviour of GBs and show a decrease of the coercive field with an increasing thickness of the grain boundary layer. The magnetization reversal and expansion of reversed magnetic domains primarily start as Bloch domain wall at grain boundaries parallel to thec-axis and secondly as Néel domain wall perpendicular to thec-axis into the adjacent hard magnetic grains. The increasing misalignment of grains leads to the loss of the anisotropic compositional behaviour and therefore to an averaged value of the grain boundary composition. In this case the simulations show an increase of the coercive field compared to the anisotropic magnet. The calculated coercive field values of the investigated magnet samples are in the order ofμ0HcJ=1.8 T–2.1 Tfor a mean grain boundary thickness of 4 nm, which agrees perfectly with the experimental data.

Current-Induced Dynamics of Multiple Skyrmions With Domain-Wall Pair and Skyrmion-Based Majority Gate Design

As an intriguing, ultrasmall, particle-like magnetic texture, skyrmions have attracted a lot of research interest for next-generation, ultradense, low-power magnetic memory/logic designs. Previous studies have demonstrated a single skyrmion domain-wall pair collision in a specially designed magnetic racetrack junction. We investigate the dynamics of multiple skyrmions with a domain-wall pair in a magnetic racetrack. The numerical micromagnetic simulation results indicate that the domain-wall pair could be pinned or depinned by a rectangular notch pinning site depending on both the number of skyrmions in the racetrack and the magnitude of driving current density. Such an emergent dynamical property could be used to implement a threshold-tunable step function in which the inputs are skyrmions and the threshold may be tuned by the driving current density. Threshold-tunable step functions are widely used in logic and neural network applications. We also present a three-input skyrmion-based majority logic gate design to demonstrate the potential application of such dynamic interactions of multiple skyrmions and domain-wall pairs.

Ultrafast annular-magnetic-field-driven vortex-core reversals

By micromagnetic numerical simulations, we investigate the dynamics of vortex-core reversal in a soft magnetic nanodisk under the excitation of annular, perpendicular, resonant magnetic fields. The non-fundamental radial modes of the nanodisk are characterized into alternating radial-phase-regions for which two adjacent regions across a node point are in antiphase. We show that radial spin-waves excited by fields applied in the in-phase regions are in phase, and therefore generate strong magnetization oscillations resulting from constructive spin-wave interference. Such annular magnetic fields can substantially speed up the vortex-core reversal and lower the threshold field amplitude in comparison with the global field. Our work provides an efficient mechanism for spin-wave excitation and ultrafast vortex-core switching.

Analytical trajectories of skyrmions in confined geometries: Skyrmionic racetracks and nano-oscillators

Magnetic skyrmions are candidates for a new generation of information carriers due to their nanometric size and topologically protected stability. The study of the dynamics of such skyrmions in racetrack geometries and, in general, in confined geometries becomes essential to achieve these goals. Here we present analytical expressions for the trajectories of skyrmions in confined geometries (including long tracks and squared samples) when driven by polarized electric currents. The force exerted by the borders is derived as a function of the material parameters (anisotropy, Dzyaloshinskii-Moriya, and exchange constants) and incorporated in the equation of motion of the skyrmions. We find the general expressions for the trajectories in several possible scenarios, and we also find simple expressions for the stationary position, the velocity, as well as for the threshold velocity of the driving electrons that could overcome the edge repulsion. In fully confined geometries, the geometric edges can produce a naturally oscillating movement, allowing for dynamical resonances if the driving current is an ac current with adequate frequency. Numerical micromagnetic simulations considering typical experimental parameters are used to corroborate the theory.

Asymmetric pinning of vortex domain walls in magnetic films in regions with lowered saturation magnetization

The asymmetric pinning of vortex domain walls in a region with a lowered saturation magnetization has been investigated in ferromagnetic films with the in-plane anisotropy. Numerical micromagnetic simulation within the two-dimensional model of the magnetization distribution has been used. The wall structure and the depinning fields in different directions have been investigated depending on the dimensions of the region with lowered saturation magnetization and the film thickness. The physical factors that determine the obtained regularities have been established.

Negative dissipation gradients in hysteretic materials.

Measuring energy dissipation on the nanoscale is of great interest not only for nanomechanics but also to understand important energy transformation and loss mechanisms that determine the efficiency of energy of data storage devices. Fully understanding the magnetic dynamics and dissipation processes in nanomagnets is of major relevance for a number of basic and applied issues from magnetic recording to spin-based sensor devices to biomedical magnetic-based hyperthermia treatments. Here we present experimental evidence for a counter-intuitive monotonical reduction of energy dissipation as the interaction between two nanomagnets is enhanced. This behavior, which takes place when spins are parallel, can be understood in terms of hysteresis phenomena involved in the reorientation of these spins. The measured magnetic losses of about a few femtowatts are in agreement with quasi-static micromagnetic numerical simulations.

Asymmetrical pinning of vortex domain walls in ferromagnetic films in areas with increased saturation magnetization

The investigation is devoted to the pinning of vortex domain walls existing in ferromagnetic films with planar anisotropy on areas with increased saturation magnetization. The method of the investigation was the numerical micromagnetic simulation in the framework of a two-dimensional model of the magnetization distribution. The anisotropy was found of the domain wall depinning field magnitude for different directions of the external field. The depinning field is investigated as a function of sizes and location of the area with the increased saturation magnetization, the difference of magnetizations of this area and the main part of the film, and film thickness. The physical causes of derived laws have been determined.

Magnonic beam splitter: The building block of parallel magnonic circuitry

We demonstrate a magnonic beam splitter that works by inter-converting magnetostatic surface and backward-volume spin waves propagating in orthogonal sections of a T-shaped yttrium iron garnet structure. The inter-conversion is enabled by the overlap of the surface and volume spin wave bands. This overlap results from the demagnetising field induced along the transversely magnetised section(-s) of the structure and the quantization of the transverse wave number of the propagating spin waves (which are therefore better described as waveguide modes). In agreement with numerical micromagnetic simulations, our Brillouin light scattering imaging experiments reveal that, depending on the frequency, the incident fundamental waveguide magnonic modes may also be converted into higher order waveguide modes. The magnonic beam splitter demonstrated here is an important step towards the development of parallel logic circuitry of magnonics.

Magnetization processes in core/shell exchange-spring structures

The magnetization reversal processes in cylindrical and spherical soft core/hard shell exchange-spring structures are investigated via the analytical nucleation theory and are verified with numerical micromagnetic simulations. At small core sizes, the nucleation of magnetic reversal proceeds via the modified bulging mode, where the transverse component of the magnetization is only semi-coherent in direction and the nucleation field contains a contribution from self-demagnetization. For large core sizes, the modified curling mode, where the magnetization configuration is vortex-like, is favored at nucleation. The preference for the modified curling mode is beneficial in that the flux-closure allows cylindrical and spherical core/shell exchange-spring elements to be densely packed into bulk permanent magnets without affecting the nucleation field, thereby offering the potential for high energy product.

Perpendicular-bias-field control of coupled vortex oscillations in nanodot networks

We studied, by combined micromagnetic numerical simulations and analytical derivations, coupled-vortex dynamics in one-dimensional (1D) arrays composed of two or more dipolar-coupled-vortex-state disks under perpendicular bias fields. We derived analytical expressions that could provide physical insights into the observed dynamic behaviors. The effects of perpendicular bias fields on the interaction strengths between the coupled-vortex oscillators and their characteristic band structures were examined and explained in terms of field strength and direction. Those effects showed that the normal modes and dispersion relations of collective vortex gyration, and the signal-transfer rate, can be tailored according to the derived explicit forms. The band width and gap of 1D coupled-vortex oscillator magnonic crystals, for example, are essential to the control of gyration-signal transfer in vortex-state dot networks. All of the analytical calculation results showed quantitatively good agreement with the micromagnetic simulation results, indicating that the perpendicular-bias-field dependence of coupled-vortex gyrations can be expressed simply as a function of the dynamic parameters under the zero field as well as the field strength and direction. This work provides not only a fundamental understanding of the effects of perpendicular bias fields on coupled-vortex oscillators but also an efficient practical means of dynamically manipulating collective vortex gyrations.

Stability of Magnetic Nanowires Against Spin-Polarized Current

Stability of ground magnetization state of a thin magnetic nanowire against longitudinal spin-polarized current is studied theoretically with dipole-dipole interaction taken into account. The critical current, minimum current at which the instability of the ground state develops, is determined. Dependence of the critical current on size and form of the transversal wire cross-section is clarified. Theoretical predictions are confirmed by numerical micromagnetic simulations.

Propulsion of a domain wall in an antiferromagnet by magnons

We analyze the dynamics of a domain wall in an easy-axis antiferromagnet driven by circularly polarized magnons. Magnons pass through a stationary domain wall without reflection and thus exert no force on it. However, they reverse their spin upon transmission, thereby transferring two quanta of angular momentum to the domain wall and causing it to precess. A precessing domain wall partially reflects magnons back to the source. The reflection of spin waves creates a previously identified reactive force. We point out a second mechanism of propulsion, which we term redshift: magnons passing through a precessing domain wall lower their frequency by twice the angular velocity of the domain wall; the concomitant reduction of magnons' linear momentum indicates momentum transfer to the domain wall. We solve the equations of motion for spin waves in the background of a uniformly precessing domain wall with the aid of supersymmetric quantum mechanics and compute the net force and torque applied by magnons to the domain wall. Redshift is the dominant mechanism of propulsion at low spin-wave intensities; reflection dominates at higher intensities. We derive a set of coupled algebraic equations to determine the linear velocity and angular frequency of the domain wall in a steady state. The theory agrees well with numerical micromagnetic simulations.

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.

Hysteresis properties of the two-dimensional structure of domain walls in magnetically triaxial films with a (110)-oriented surface plane

Structural transformations of a two-vortex asymmetric domain wall in a magnetically triaxial film with Goss-oriented (110) surface and cubic anisotropy have been studied by numerical micromagnetic simulation. It is established that transformations of the domain wall structure exhibit significant hysteresis that is accompanied by the related hysteresis of magnetization reversal in a field applied perpendicular to the easy magnetization axis.

Thin Magnetic Films with Artificial Texture on Substrate: Microwave Properties

Using the scanning spectrometer of ferromagnetic resonance (FMR) the experimental dependences of the resonance field and FMR line width of thin permalloy magnetic films, which were deposited in vacuum on the substrate with an artificial texture, were obtained. The texture was produced by putting parallel grooves using a diamond cutter on glass substrates with period from 5 to 100 μm. It was found that the presence of the texture led to a considerable increase of the resonance field and FMR line width, when the external field was directed orthogonal to the grooves. On the base of a numerical micromagnetic simulation the explanation of the nature of observable in thin magnetic films effects was given.