Thiele Equation Research Articles

Inertia, diffusion, and dynamics of a driven skyrmion

Skyrmions recently discovered in chiral magnets are a promising candidate for magnetic storage devices because of their topological stability, small size ($\sim 3-100$nm), and ultra-low threshold current density ($\sim 10^{6}$A/m$^2$) to drive their motion. However, the time-dependent dynamics has hitherto been largely unexplored. Here we show, by combining the numerical solution of the Landau-Lifshitz-Gilbert equation and the analysis of a generalized Thiele's equation, that inertial effects are almost completely absent in skyrmion dynamics driven by a time-dependent current. In contrast, the response to time-dependent magnetic forces and thermal fluctuations depends strongly on frequency and is described by a large effective mass and a (anti-) damping depending on the acceleration of the skyrmion. Thermal diffusion is strongly suppressed by the cyclotron motion and is proportional to the Gilbert damping coefficient $\alpha$. This indicates that the skyrmion position is stable, and its motion responds to the time-dependent current without delay or retardation even if it is fast. These findings demonstrate the advantages of skyrmions as information carriers.

Magnetic Vortices Induced by a Monopole Tip

A ferromagnetic monolayer with an easy-plane anisotropy scanned by a magnetic tip that is moved with constant velocity ν is studied using atomistic computer simulations. The spin dynamics are treated using the Landau-Lifshitz-Gilbert equation. To study the influence of the tip's field, it is modeled by a monopole field instead of a dipole field, which is a common near-field approximation of a scanning probe microscopy cantilever. The magnetic structures induced by the moving tip are analyzed with respect to the strength of the coupling as well as the scanning velocity, and the energy dissipation is calculated. The results agree with calculations in a continuum model using Thiele's equation, as well as with earlier results obtained from simulations using a dipolar tip. The quantitative influence of the field is illustrated using energetic arguments.

A strategy for the design of skyrmion racetrack memories

Magnetic storage based on racetrack memory is very promising for the design of ultra-dense, low-cost and low-power storage technology. Information can be coded in a magnetic region between two domain walls or, as predicted recently, in topological magnetic objects known as skyrmions. Here, we show the technological advantages and limitations of using Bloch and Néel skyrmions manipulated by spin current generated within the ferromagnet or via the spin-Hall effect arising from a non-magnetic heavy metal underlayer. We found that the Néel skyrmion moved by the spin-Hall effect is a very promising strategy for technological implementation of the next generation of skyrmion racetrack memories (zero field, high thermal stability, and ultra-dense storage). We employed micromagnetics reinforced with an analytical formulation of skyrmion dynamics that we developed from the Thiele equation. We identified that the excitation, at high currents, of a breathing mode of the skyrmion limits the maximal velocity of the memory.

Coupled oscillations of vortex cores confined in a ferromagnetic elliptical disk

By solving the Thiele equation with simultaneous application of a radio-frequency (rf) magnetic field $({\mathbit{h}}_{\mathrm{rf}})$ and an rf spin current $({\mathbit{j}}_{\mathrm{sp}})$, the dynamic susceptibility of exchange-coupled vortices in response to ${\mathbit{h}}_{\mathrm{rf}}$ and ${\mathbit{j}}_{\mathrm{sp}}$ was obtained. It was found that the four eigenmodes expected for two vortices trapped in a magnetic elliptical disk were coupled to different components of ${\mathbit{h}}_{\mathrm{rf}}$ and ${\mathbit{j}}_{\mathrm{sp}}$. As a consequence, orthogonal ${\mathbit{h}}_{\mathrm{rf}}$ and ${\mathbit{j}}_{\mathrm{sp}}$ (which are simultaneously generated by the application of an rf current to an elliptical disk) can excite two modes with different eigenfrequencies. This result suggests that a fieldlike nonadiabatic torque caused by an rf spin current can be spectroscopically distinguished from the one caused by the rf magnetic field.

Nonlinear magnetic vortex dynamics in a circular nanodot excited by spin-polarized current.

We investigate analytically and numerically nonlinear vortex spin torque oscillator dynamics in a circular magnetic nanodot induced by a spin-polarized current perpendicular to the dot plane. We use a generalized nonlinear Thiele equation including spin-torque term by Slonczewski for describing the nanosize vortex core transient and steady orbit motions and analyze nonlinear contributions to all forces in this equation. Blue shift of the nano-oscillator frequency increasing the current is explained by a combination of the exchange, magnetostatic, and Zeeman energy contributions to the frequency nonlinear coefficient. Applicability and limitations of the standard nonlinear nano-oscillator model are discussed.

Exchange interaction between vortex and antivortex

The dynamic properties of interacting vortex—antivortex pairs in thin film are studied by analytical calculations. Analytical expressions for the magnetization vector distribution of vortex—antivortex pairs and the trivortex states are given. The magnetic states of the vortices are treated as having rigid structures, i.e., the vortex maintains its spin distribution when moving. The trajectories of the vortex cores are calculated by the Thiele's equation. It is found that the vortex—antivortex pair rotates around each other when they have opposite polarities, however, vortex and antivortex cores move along straight lines when they have the same polarity. The frequency of the rotation decreases with increasing the distance between the two cores of vortex—antivortex pair, and it has a lower value when a third vortex is introduced.

Thermally assisted current-driven skyrmion motion

We study the behavior of skyrmions in thin films under the action of stochastic torques arising from thermal fluctuations. We find that the Brownian motion of skyrmions is described by a stochastic Thiele's equation and its corresponding Fokker-Planck equation. The resulting Fokker-Planck equation is recognized as the one for a high-friction Brownian particle which has been studied extensively in different physical contexts. It is shown that thermal fluctuations favor the skyrmion motion allowing a finite mobility even in presence of pinning traps. We calculate explicitly the mobility tensor of skyrmions in linear response to an electric current finding that it increases with temperature.

Interaction between magnetic vortex cores in a pair of nonidentical nanodisks

The coupling of two nonidentical magnetic nanodisks, i.e., with different vortex gyrotropic frequencies, is studied. From the analytical approach, the interactions between the nanodisks along x and y directions (the coupling integrals) were obtained as a function of distance. From the numerical solution of Thiele's equation, we derived the eigenfrequencies of the vortex cores as a function of distance. The motion of the two vortex cores and, consequently, the time dependence of the total magnetization M(t) were derived both using Thiele's equation and by micromagnetic simulation. From M(t), a recently developed method, the magnetic vortex echoes, analogous to the Nuclear Magnetic Resonance spin echoes, was used to compute the distance dependence of the magnetic coupling strength. The results of the two approaches differ by approximately 10%; using one single term, a dependence with distance found is broadly in agreement with studies employing other techniques.

Control of skyrmion magnetic bubble gyration

The skyrmion magnetic bubble in a ferromagnetic disk exhibits hypocycloidal gyrations contrary to the vortex gyration, showing a simple circular trajectory. To describe the hypocycloidal bubble gyration, a mass term is needed in Thiele's equation. In this study, we analytically derived both mass and spring constant term, which are crucial parameters for describing the bubble gyration. Values obtained by these analytic expressions were consistent with those obtained by simulations. We could find the dependences of these two terms on several external parameters, including the bubble radius. Especially using the radius's dependence, we could obtain regular polygonlike trajectories such as a square and a triangle confirmed by the numerical simulations. Based on this effective method to control the bubble gyration, the regular polygonlike trajectories of this skyrmion magnetic bubble make it possible to study the bubble gyration without time-resolved experiments.

Energy dissipation of moved magnetic vortices

A two-dimensional easy-plane ferromagnetic substrate, interacting with a dipolar tip which is magnetised perpendicularly with respect to the easy plane is studied numerically by solving the Landau-Lifshitz Gilbert equation. The dipolar tip stabilises a vortex structure which is dragged through the system and dissipates energy. An analytical expression for the friction force in the limit based on the Thiele equation is presented. The limitations of this result which predicts a diverging friction force in the thermodynamic limit, are demonstrated by a study of the size dependence of the friction force. While for small system sizes the dissipation depends logarithmically on the system size, it saturates at a specific velocity-dependent value. This size can be regarded as an effective vortex size and it is shown how this effective vortex size agrees with the infinite extension of a vortex in the thermodynamic limit. A magnetic friction number is defined which represents a general criterion for the validity of the Thiele equation and quantifies the degree of nonlinearity in the response of a driven spin configuration.

Wave modes of collective vortex gyration in dipolar-coupled-dot-array magnonic crystals

Lattice vibration modes are collective excitations in periodic arrays of atoms or molecules. These modes determine novel transport properties in solid crystals. Analogously, in periodical arrangements of magnetic vortex-state disks, collective vortex motions have been predicted. Here, we experimentally observe wave modes of collective vortex gyration in one-dimensional (1D) periodic arrays of magnetic disks using time-resolved scanning transmission x-ray microscopy. The observed modes are interpreted based on micromagnetic simulation and numerical calculation of coupled Thiele equations. Dispersion of the modes is found to be strongly affected by both vortex polarization and chirality ordering, as revealed by the explicit analytical form of 1D infinite arrays. A thorough understanding thereof is fundamental both for lattice vibrations and vortex dynamics, which we demonstrate for 1D magnonic crystals. Such magnetic disk arrays with vortex-state ordering, referred to as magnetic metastructure, offer potential implementation into information processing devices.

Gyrotropic frequencies of pinned vortices in magnetic nanodisks

Recent experimental data have indicated that magnetic vortices can be pinned to an impurity at the surface of a nanodisk. Unpinned vortices exhibit a gyrotropic precession frequency that is approximately proportional to the aspect ratio of the disk, $L/R$, where $L$ and $R$ are the disk thickness and radius, respectively. More recent experimental data have indicated that the gyrotropic frequency of pinned vortices has an approximate $R/L$ dependence for the thin disk, and the vortex is pinned at only one surface of the disk. Pinning of the vortex to one surface of the disk results in a significant contribution to the exchange energy, which is used along with the magnetostatic energy in the Thiele equation to calculate the frequency. This energy results in the $R/L$ dependence as seen in experimental thin disk measurements.

Phase diagram for magnetic vortex core switching studied by ferromagnetic absorption spectroscopy and time-resolved transmission x-ray microscopy

We investigate the switching criteria of magnetic vortices in micron-sized Permalloy squares. The vortices are excited by high frequency magnetic fields. Continuous core reversal is demonstrated for a wide range of frequencies and amplitudes of excitation by ferromagnetic absorption spectroscopy and for selected frequencies and amplitudes with time-resolved scanning x-ray microscopy. The boundary of this switching regime is derived from the Thiele equation when a critical velocity of ${v}_{\mathrm{crit}}\ensuremath{\approx}250\phantom{\rule{0.222222em}{0ex}}{\mathrm{ms}}^{\ensuremath{-}1}$ is considered.

Phase locking of vortex-based spin-torque nanocontact oscillators by antivortices

Magnetic vortices formed at a nanocontact undergo gyrotropic oscillations when driven by a spin-torque providing potential applications as microwave nano-oscillators; however, to increase the power output it is necessary to use an array of synchronized oscillators. Here a theory is developed for two nanocontact oscillators that are interacting through an intermediate antivortex. The confining potential for the two vortices formed at each nanocontact as well as the antivortex is the Oersted-Ampere field about each nanocontact. Solution of the Thiele equation for this system indicates that the nanocontact vortex oscillators will be phase locked over a wide range of nanocontact currents.

A mechanism to pin skyrmions in chiral magnets

We propose a mechanism to pin skyrmions in chiral magnetic thin films by introducing local maxima of magnetic exchange strength as pinning centers. The local maxima can be realized by engineering the local density of itinerant electrons. The stationary properties and the dynamical pinning and depinning processes of an isolated skyrmion around a pinning center are studied. We carry out numerical simulations of the Landau–Lifshitz–Gilbert (LLG) equation and find a way to control the position of an isolated skyrmion in a pinning center lattice using electric current pulses. The results are verified by a Thiele equation analysis. We also find that the critical current to depin a skyrmion, which is estimated to have order of magnitude 107–108 A m−2, has linear dependence on the pinning strength.

ПІДХІД НА ОСНОВІ АКТУАРНИХ МАТЕМАТИЧНИХ МОДЕЛЕЙ ДО ЗАДАЧ СТРАХОВОЇ МЕДИЦИНИ

In this work there is developed an approach based on actuarial mathematical models for the problems of health insurance. Models are presented in terms of Kolmogorov and Thiele equations.

Oscillation frequency of magnetic vortex induced by spin-polarized current in a confined nanocontact structure

We studied the oscillation frequency associated with gyrotropic motion of magnetic vortex in a Permalloy nanodot driven by an out-of-plane spin-polarized current injected through a nanocontact by micromagnetic simulations and analytical calculations. The analytical results were calculated by the Thiele's equation, where both the forces corresponding to the spin-transfer-torque and the Oersted field accompanying the current were taken into account. Variation curves of oscillation frequency with vortex core position, nanocontact radius, and current density were given. Good agreement with analytical calculations and micromagnetic simulations was found.

Magnetic Vortex Core Oscillations in Multi Point Contact Spin Valve Stacks

We study gyration frequencies of magnetic vortices in double point contact spin valve stacks by combining Thiele's equation and micromagnetics. We apply numerical integration and error control techniques to find stable, in-plane orbits in the rigid vortex model as a function of applied current densities and point contact geometry. The results show that the velocity of the vortex depends strongly on the restoring potential. The angular velocity (i.e., frequency) increases as the radius of one of the point contacts is decreased, due to a narrower current density profile. In addition we observe that multiple oscillation trajectories are possible depending on the current distribution between the point contacts.

Thermal vortex dynamics in thin circular ferromagnetic nanodisks

The dynamics of gyrotropic vortex motion in a thin circular nanodisk of soft ferromagnetic material is considered. The demagnetization field is calculated using two-dimensional Green's functions for the thin film problem and fast Fourier transforms. At zero temperature, the dynamics of the Landau-Lifshitz-Gilbert equation is simulated using fourth order Runge-Kutta integration. Pure vortex initial conditions at a desired position are obtained with a Lagrange multipliers constraint. These methods give accurate estimates of the vortex restoring force constant $k_F$ and gyrotropic frequency, showing that the vortex core motion is described by the Thiele equation to very high precision. At finite temperature, the second order Heun algorithm is applied to the Langevin dynamical equation with thermal noise and damping. A spontaneous gyrotropic motion takes place without the application of an external magnetic field, driven only by thermal fluctuations. The statistics of the vortex radial position and rotational velocity are described with Boltzmann distributions determined by $k_F$ and by a vortex gyrotropic mass $m_G=G^2/k_F$, respectively, where $G$ is the vortex gyrovector.

Nonlinear magnetic vortex gyration

Nonlinear motion of magnetic vortices in soft magnetic squares is investigated by micromagnetic simulations, analytical calculations, and broadband ferromagnetic resonance measurements. Increasing amplitudes of alternating unidirectional spin currents and magnetic fields cause the shapes of the trajectories to approach the shape of the sample and shift the resonance frequency of vortex gyration. We show that it is the thickness of the magnetic element that determines whether the resonance frequency shifts to the red or to the blue. A nonlinear equation of motion based on the Thiele equation reproduces the simulated and measured results.