Numerical Micromagnetic Simulations Research Articles

Realization of Hadamard gate with twisted magnon modes in synthetic antiferromagnets

Manipulating the polarization of spin waves highlights the potential of antiferromagnetic magnonics in encoding and handling magnon information with high fidelity. Here, we propose a flexible approach to mutually convert polarization states (i.e., Hadamard gate) by incorporating a topological degree of freedom, intrinsic orbital angular momentum (OAM), into twisted spin wave modes within synthetic antiferromagnetic nanodisks. The polarization states of spin waves and the implementation of magnonic logic operations can be electrically read out through combined spin pumping and inverse spin Hall effect, as demonstrated by numerical micromagnetic simulations for CoFeB-based synthetic antiferromagnets. Our findings present an exciting possibility of parallel magnonic computing utilizing topologically protected and magnetic damping-resistance OAM of twisted magnons.

The iron garnet stripe domain structure “refraction” effect at the electrode location

The new type of magnetoelectric effect observed at the domain walls in iron garnet films is considered. This effect manifests itself as a kind of refraction of the stripe domain structure at the charged stripe electrode. The numerical micromagnetic simulation taking into account inhomogeneity of electric field from the stripe electrode as well as magnetic anisotropy of the (210) iron garnet films reproduces the main features of the shape of the domain walls in the vicinity of the electrode edges.

Micromagnetic study of inertial spin waves in ferromagnetic nanodots

Here we report the possibility to excite ultra-short spin waves in ferromagnetic thin-films by using time-harmonic electromagnetic fields with terahertz frequency. Such ultra-fast excitation requires to include inertial effects in the description of magnetization dynamics. In this respect, we consider the inertial Landau-Lifshitz-Gilbert (iLLG) equation and develop analytical theory for exchange-dominated inertial spin waves. The theory predicts a finite limit for inertial spin wave propagation velocity, as well as spin wave spatial decay and lifetime as function of material parameters. Then, guided by the theory, we perform numerical micromagnetic simulations that demonstrate the excitation of ultra-short inertial spin waves (20 nm long) propagating at finite speed in a confined magnetic nanodot. The results are in agreement with the theory and provide the order of magnitude of quantities observable in realistic ultra-fast dynamics experiments.

Explaining Browns paradox in NdFeB magnets from micromagnetic simulations

We present the results of a systematic investigation into how the coercivity and maximum energy product of NdFeB permanent magnets are affected by magnetic and geometric microscopic properties. The results are based on numerical micromagnetic simulations carried out with the open source numerical framework MagTense. We considered artificially generated realistic microstructures for which we can control the number of crystal grains as well as the thickness of the intergrain region. Based on variations of the exchange constant, easy axis orientation, grain boundary width and intergrain material properties, the results indicate that while all of these can contribute to a reduction of coercivity, the easy axis orientation has the largest influence. For this, if the easy axis orientation is distributed within a cone with an opening angle of 15°, that is enough to reduce the coercivity by 1 T. Regarding the maximum energy product, the width of the grain boundary layer as well as the easy axis orientation were seen to have the largest influence, with the exchange constant only very weakly influencing the maximum energy product. Our analysis thus methodically clarifies the different factors contributing to the reduction of the value of the coercivity and maximum energy product when compared to the theoretical limit, a discrepancy known as Brown’s paradox.

Classical analog of qubit logic based on a magnon Bose–Einstein condensate

Advances in quantum computing and telecommunications stimulate the search for classical systems allowing partial implementation of a similar functionality under less stringent environmental conditions. Here, we present a classical version of several quantum bit (qubit) functionalities using a two-component magnon Bose–Einstein condensate (BEC) formed at opposite wavevectors in a room-temperature yttrium-iron-garnet ferrimagnetic film. Employing micromagnetic numerical simulations, we show the use of wavelength-selective parametric pumping to controllably initialize and manipulate the two-component BEC. Next, by modeling the interaction of this BEC with a pulse- and radio-frequency-driven dynamic magnonic crystal we translate the concept of Rabi-oscillations into the wavevector domain and demonstrate how to manipulate the magnon-BEC system regarding the polar and azimuthal angles in the Bloch sphere representation. We hope that our study provides a significant stimulus on the boundary between qubit functionality and classical systems of interacting BECs, which use a subset of qubit-based algorithms.

Electric readout of Bloch sphere spanned by twisted magnon modes

We present a magnonic type of Bloch sphere based on twisted spin-wave (magnon) eigenmodes with opposite intrinsic orbital angular momentum, which is topology-protected and damping-resistant. Taking advantage of the release of the chiral degeneracy of magnons by dynamic dipolar interactions and/or interfacial Dzyaloshinskii–Moriya interactions in ferromagnetic nanodisks, we show how these magnonic “qubit” states can be precisely launched and electrically detected through combined spin pumping and inverse spin Hall effect. The experimental feasibility is verified using full-edged numerical micromagnetic simulations for FeB nanodisks. Our investigations demonstrate the potential of twisted spin waves for magnonic information encoding in a flexible and realizable approach.

Emergent magnonic singularities in anti parity-time symmetric synthetic antiferromagnets

We study the impact of the transition across the anti-parity-time (anti-) symmetry breaking point on the low-energy spin excitations in a synthetic antiferromagnet (SyAFM) that consists of two magnetic layers coupled antiferromagnetically via the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. When varying the interlayer (RKKY) coupling strength a degeneracy point (exceptional point) is reached beyond which the system enters the anti- symmetry-broken phase. This phase is marked by the emergence of a finite net magnetization and low-lying excitations beyond the Goldstone modes in the anti- symmetry-preserved phase. For systems hosting an interfacial Dzyaloshinskii–Moriya interaction, we find a bound state in the continuum with a maximal coherent superposition of SyAFM excitations without any radiation. Analytical results for linearized models are corroborated with full numerical micromagnetic simulations endorsing the robustness and generalities of the theory predictions.

Phase shifter based on voltage-controlled magnetic domain walls

A controllable phase shifter is an essential part of spin-wave (SW) logic devices. Magnetic domain walls (DWs) as magnonic waveguides have been used to study SW propagation in ultrathin ferromagnetic films. In this study, we present a channel for SW propagation that relies on magnetic DWs as natural waveguides and realize phase manipulation by voltage-controlled magnetic anisotropy (VCMA) at a lower excitation frequency with micromagnetic simulations. The system model is a double-layer magnetic film structure, containing two oppositely magnetized domains and an upper layer (Co20Fe60B20) coupled with the lower layer ([Co/Pd]) through exchange interaction. The pinning structure can effectively maintain the state of the waveguide during later operation without an additional magnetic field. By numerical micromagnetic simulation, we can clearly see that the exchange-dominated isotropic magnetostatic forward volume like SWs are confined in the Bloch-type DW. The phase accumulation by a propagating SW depends on the accumulated phase wave vector (k) and the distance traveled. Therefore, SWs accumulate more phase through DW waveguides covered by electrodes with different lengths. The phase of the SW is shifted after the SW passes through the voltage region, compared with the waveguide where a voltage is unapplied. As a result, the established phase shifter shows a phase difference of 2π by applying a voltage (E = 1 V/nm), and the length of the applied electrode is about 644 nm.

Ferromagnetic Resonance in Permalloy Metasurfaces

Permalloy films with one-dimensional profile modulation of submicron periodicity are fabricated based on commercially available DVD-R discs and studied using ferromagnetic resonance method and micromagnetic numerical simulations. The main resonance position shows in-plane angular dependence which is strongly reminiscent of that in ferromagnetic films with uniaxial magnetic anisotropy. The main signal and additional low-field lines are attributed to multiple standing spin-wave resonances defined by the grating period. The results may present interest in magnetic metamaterials and magnonics applications.

Chiral magnetic domain walls under transverse fields: A semi-analytical model

An analytical model for the domain wall structure in ultrathin films with perpendicular easy axis and interfacial Dzyaloshinskii-Moriya interaction, submitted to an arbitrary in-plane magnetic field, is presented. Its solution is simplified to the numerical minimization of an analytic function of just one variable. The model predictions are compared to numerical micromagnetic simulations, using parameters of existing samples, revealing a very good agreement. Remaining differences are analyzed, and partly corrected. Differences with the predictions of the simplest model, usually found in the literature, in which only the domain wall moment’s in-plane orientation can vary, are exemplified. The model allows accurate computations, as a function of in-plane field module and orientation, of the domain wall tension and width, quantities controlling the creep motion of domain walls in such films.

Domain periodicity in an easy-plane antiferromagnet with Dzyaloshinskii-Moriya interaction

Antiferromagnetic spintronics is a promising emerging paradigm to develop high-performance computing and communications devices. From a theoretical point of view, it is important to implement simulation tools that can support a data-driven development of materials having specific properties for particular applications. Here, we present a study focusing on antiferromagnetic materials having an easy-plane anisotropy and interfacial Dzyaloshinskii-Moriya interaction (IDMI). An analytical theory is developed and benchmarked against full numerical micromagnetic simulations, describing the main properties of the ground state in antiferromagnets and how it is possible to estimate the IDMI from experimental measurements. The effect of the IDMI on the electrical switching dynamics of the antiferromagnetic element is also analyzed. Our theoretical results can be used for the design of multi-terminal heavy metal/antiferromagnet memory devices.

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

The method of numerical micromagnetic simulation was used to study the magnetization dynamics of a thin film with uniaxial magnetic anisotropy during ferromagnetic resonance (FMR) excitation in it on the local sites by a linearly polarized high-frequency magnetic field emitted from the 1 mm opening in the metallic screen of the transmission line. It was established that non-uniformities of demagnetizing fields emerging near the film edges lead not only to the change in the FMR field, but also to the change in the value and direction of the uniaxial anisotropy field. The distribution of non-uniformities of the magnetic anisotropy characteristics over the area of 60 nm thick permalloy film that was measured on the scanning FMR spectrometer agrees well with the micromagnetic simulation results. Demagnetizing fields were proved to be the main cause for the edge effects in magnetic films.

Bloch oscillations of backward volume magnetostatic spin waves

We have used numerical micromagnetic simulations to propose a feasible candidate system in which Bloch oscillations of spin waves could be observed experimentally. Our simulations demonstrate these phenomena for backward volume magnetostatic spin waves (BVMSWs) in a film of yttrium-iron-garnet in a spatially varying bias magnetic field comprising a sinusoidal and gradient contributions. Despite the complex character of the BVMSW dispersion relation, the spin-wave packets are distinctly confined by the field gradient, while showing only minor broadening over the simulation time.

Wannier-Stark ladder spectrum of Bloch oscillations of magneto-dipole spin waves in graded 1D magnonic crystals

We have used the method of Wannier functions to calculate the frequencies and profiles of spin waves localized in one-dimensional magnonic crystals due to a gradient in the bias magnetic field. This localization of spin waves is analogous to the phenomenon of Bloch oscillations of quantum-mechanical electrons in crystals in a uniform electric field. As a convenient yet realistic model, we consider backward volume magnetostatic spin waves in a film of yttrium-iron garnet in a bias magnetic field comprising spatially uniform, cosine and gradient contributions. The spin-wave spectrum is shown to have the characteristic form of a Wannier–Stark ladder. The analytical results are verified using those obtained using numerical micromagnetic simulations. The physics of spin-wave Bloch oscillations combines the topics of magnonic crystals and graded magnonic index — the two cornerstones of modern magnonics.

Bias-free reconfigurable magnonic phase shifter based on a spin-current controlled ferromagnetic resonator

Controllable phase modulation plays a pivotal role in the research of magnonic logic gates. Here we propose a reconfigurable spin-current controlled magnonic phase shifter based on a ferromagnetic resonator. The proposed phase shifter requires no magnetic bias field during operation. The device is directly configured over the waveguide while keeping the original structure of the waveguide unaffected. Numerical micromagnetic simulations show that the phase shifter could yield either a -phase or no shift depending on the magnetization configuration of the resonator, which can be controlled by a current pulse. Moreover, the phase-shifting operation could be affected by spin current. At different input current density, the device could be either used as a dynamic controlled phase shifter or a spin-wave valve. Finally, a XNOR magnonic logic gate is demonstrated using the proposed phase shifter. Our work can be a beneficial step to enhance the functionality and compatibility of the magnonic logic circuits.

Microwave magnetization dynamics in ferromagnetic spherical nanoshells

Ferromagnetic particles of nano- and micrometer-size range are in high demand owing to their promising technological applications in magnetic memory storage, biomedical treatment, and microwave devices. New absorbing materials are required to possess the properties of low density and strong absorption in a broad frequency band. The dynamic susceptibility spectra of nanosized spherical shells supporting onion and vortex magnetization configurations are studied as a function of particle radius $40--125\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ and shell thickness $10--115\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ by means of numerical micromagnetic simulation. For the saturated shell, the frequencies of higher-order spin-wave modes are found to rapidly approach the subterahertz frequency regime ($200--300\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$) when the geometry is transformed from the single-domain sphere to the spherical shell. The frequency of each mode is proportional to ${R}_{2}^{\ensuremath{-}2}$ when the sphere radius outer ${R}_{2}$ is below 30 nm in reasonable agreement with the exchange approximation. Thickness-dependent vortex core gyrotropic modes were studied in the absence of an external bias field in the frequency range $0.1--30\phantom{\rule{0.16em}{0ex}}\mathrm{GHz}$. The size dependence of the vortex spin excitations is complicated by mixing of the modes at different particle sizes. For shells with large thickness and outer radii above ${R}_{2}\ensuremath{\ge}100\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$ the amplitudes of both $n=0$ and $n=1$ vortex flexural modes are less intense than even higher-frequency resonances. However, the $n=0$ mode becomes the dominant mode in the dynamic susceptibility when either the radius or thickness of the particle becomes sufficiently small.

Influence of spin-orbit and spin-Hall effects on the spin-Seebeck current beyond linear response: A Fokker-Planck approach

We study the spin transport theoretically in heterostructures consisting of a ferromagnetic metallic thin film sandwiched between heavy-metal and oxide layers. The spin current in the heavy metal layer is generated via the spin Hall effect, while the oxide layer induces at the interface with the ferromagnetic layer a spin-orbital coupling of the Rashba type. Impact of the spin Hall effect and Rashba spin-orbit coupling on the spin Seebeck current is explored with a particular emphasis on nonlinear effects. Technically, we employ the Fokker-Planck approach and contrast the analytical expressions with full numerical micromagnetic simulations. We show that when an external magnetic field is aligned parallel (antiparallel) to the Rashba field, the spin-orbit coupling enhances (reduces) the spin pumping current. In turn, the spin Hall effect and the Dzyaloshinskii-Moriya interaction are shown to increase the spin pumping current.

Bullets and droplets: Two-dimensional spin-wave solitons in modern magnonics (Review Article)

In this review, we consider theoretical and experimental results related to the properties of two-dimensional spin-wave (SW) solitons, so-called SW bullets and SW droplets. Such nonlinear self-localized SW modes possess very interesting physical properties, and could have practical applications in modern and future magnonics and spintronics. The experimental and theoretical results presented in this review have undeniably proven the existence of SW bullets in magnetic films and confined magnetic nanostructures (magnetic nanocontacts), and have elucidated the essential distinctions between the properties of the one-dimensional nonlinear SW solitons, two-dimensional nonlinear SW bullets and linear spin wave packets, as well as the possibility of self-generation, parametrical excitation, and phase-conjugation of the SW bullets, similar to the case of the conventional linear spin waves. Also, in this review we presented experimental results demonstrating the nucleation, dynamics, and annihilation of two-dimensional strongly nonlinear SW “droplets” in spin-torque-driven magnetic nanocontacts. The properties of these exotic nonlinear objects are analyzed using recently developed theoretical models and illustrated by micromagnetic numerical simulations.

Magnetic Skyrmions in Thickness-Modulated Films

The possibility of the formation of lattices of magnetic skyrmions in thickness-modulated films with perpendicular magnetic anisotropy has been studied experimentally. The samples constitute a regular rectangular lattice (300 nm in period) of cylindrical bulges (150 nm in diameter) formed in a Co/Pt multilayer film by electron lithography. The spatial symmetry of a magnetic force microscopy signal indicates the formation of magnetic skyrmions in the process of remagnetization of the system. Formed skyrmions remain stable in the absence of an external magnetic field. The experimental data are in good agreement with the numerical micromagnetic simulation of the system.

Determination of the out-of-plane anisotropy contributions (first and second anisotropy terms) in amorphous Nd-Co thin films by micromagnetic numerical simulations

Amorphous Nd-Co thin films exhibit stripe shaped periodic magnetic domains with local out-of-plane magnetization components due to their perpendicular magnetic anisotropy. This anisotropy has been quantified in a fairly simple way by reproducing the experimental magnetization curves by means of micromagnetic numerical simulations. The simulations show that the first (K1) and second (K2) anisotropy constants must be used to properly describe the variation of the stripe domains with the in plane applied magnetic field. A strong temperature dependence of both K1 and K2 has been obtained between 10 K and room temperature. This anisotropy behavior is characteristic of two magnetically coupled 3d-4f sublattices with competing anisotropies.