Thiele Equation Research Articles

Magnetic Vortex Core String Gyrotropic Oscillations in Thick Cylindrical Dots

The nonuniform magnetic vortex gyrotropic oscillations along the cylindrical dot thickness were calculated. A generalized Thiele equation was used for describing the vortex core motion including magnetostatic and exchange forces. The magnetostatic interaction was accounted for in a local form. This allowed reducing the Thiele equation of motion to the Schrödinger differential equation and analytically determining the spin eigenmode spatial profiles and eigenfrequencies using the Liouville–Green method for the high-frequency modes. The mapping of the Schrödinger equation to the Mathieu equation was used for the low-frequency gyrotropic mode. The lowest-frequency gyrotropic mode transformed to the dot faces localized mode, increasing the dot thickness. The vortex gyrotropic modes are described for a wide range of the dot thicknesses according to the concept of the turning points in the magnetostatic potential. This approach allows treating the vortex localized modes (turning points) and nonlocalized modes within a unified picture.

Ampere–Oersted field splitting of the nonlinear spin-torque vortex oscillator dynamics

We investigate the impact of the DC current-induced Ampère–Oersted field on the dynamics of a vortex based spin-torque nano-oscillator. In this study we compare micromagnetic simulations performed using mumax^3 and our analytical model based on the Thiele equation approach. The latter is improved by adding two important corrections to the Thiele equation approach. The first is related to the magneto-static contribution and depends on the aspect ratio of the magnetic dot. The second is a full analytical description of the Ampère–Oersted field contribution. The model describes quantitatively the simulation results in the resonant regime as well as the impact of the Ampère–Oersted field. Depending on the relative orientation between the vortex in-plane curling magnetisation (chirality) and the Ampère–Oersted field a strong splitting phenomenon appears in the fundamental properties (frequency and vortex core position) of the nano-oscillator. Thus, we show that the Ampère–Oersted field should not be neglected as it has a high impact on the spin-torque vortex oscillator dynamics.

Asymmetric Motion of Magnetic Skyrmions in Ferromagnetic Nanotubes Induced by a Magnetic Field

Magnetic skyrmions, featuring topological stability and low driving current density, are believed to be a promising candidate of information carriers. One of the obstacles to application is the skyrmion Hall effect, which can lead to the annihilation of moving skyrmions at the lateral boundary of thin-film tracks. In order to resolve this issue, it was recently proposed to exploit ferromagnetic nanotubes as alternative skyrmion guides. In this work, we investigate the field-effect of current-driven skyrmion motion in nanotubes using micromagnetic simulations. It is found that, in the presence of an axial field, the skyrmion motion becomes asymmetric in tubes. This is fundamentally different from the flat strip, in which a field has little influence on the skyrmion dynamics. Based on the dissipation tensor determined by the spin texture of the skyrmions, the solution of the Thiele equation is obtained, yielding a perfect match with simulations. We argue that the asymmetry of the skyrmion dynamics originates from the curvature of the nanotube.

Dynamics and stability of skyrmions in a bent nano-beam

Magnetic skyrmions are attracting much attention due to their nontrivial topology and high mobility to electric current. Nevertheless, suppression of the skyrmion Hall effect and maintaining high velocity of skyrmions with low energy cost are two major challenges concerning skyrmion-based spintronic devices. Here we show theoretically that in a nano-beam suffering appropriate bending moment, both Bloch-type and Néel-type skyrmions move with a vanishing Hall angle under a current density smaller than that required when the bending is absent. Moreover, bending alone can be used to move skyrmions, whose velocity is solved analytically from the Thiele equation. Generally speaking, inhomogeneous elastic fields affect the stability and dynamics of skyrmions, where the local stability is dominantly determined by the local bulk stress. These findings throw new light on how to drive skyrmions in a straight line with lower energy cost, which is vital for utilizing skyrmions as information carriers.

Skyrmion Dynamics in the Presence of Deformation

The nonlinear dynamics of current-driven skyrmions are essential for designing skyrmion-based spintronic devices. However, Thiele's approach uses a rigid-body assumption that is not suitable here, as skyrmions may change shape at high speeds. This study extends Thiele's equation by including the radial and tangential force components. With this approach, one may depict the inner structure of a current-driven skyrmion and propose a way to suppress deformation with an in-plane magnetic field. This extended analytic formulation may help to describe the nonlinear dynamics of other spin textures as well, and to promote applications based on such dynamics.

Velocity increase of skyrmion motion by constructing wedge nanotracks

Spin waves (SWs) have been proved to be promising candidates for driving skyrmion motion, while the small velocity limits its further applications. Herein, we demonstrated theoretically the regulation of SW-driven skyrmion dynamics on wedge nanotracks. It is found that the edge potential barrier acts as an extra driving force and increases the skyrmion velocity by nearly 60%. By analyzing Thiele equation and velocity fitting function, we determined the optimal inclined angle and gave the quantitative expression of the increased velocity with respect to material and geometric parameters. Our results are useful for understanding the dynamic behaviors of SW-driven skyrmion and may provide guidelines for designing skyrmionic racetrack memory.

Skyrmion-Antiskyrmion Racetrack Memory in Rank-One DMI Materials

Chiral magnetic skyrmions, localized and topologically protected vortex-like magnetic textures that can be found in chiral magnets, are currently under intense study as an entity for information storage and processing. A recent study showed that so-called rank-one materials can host both skyrmions and antiskyrmions at the same energy. In such systems the Dzyaloshinskii-Moriya interaction, in general a tensorial quantity, is reduced to only one non-zero component. The presence of both skyrmions and antiskyrmions allows for the investigation of the possible interplay between them. Here, we investigate the stability and interaction of skyrmions and antiskyrmions as well as their transport properties subject to spin-orbit torque for a model system described by an atomistic spin-lattice Hamiltonian employing the simulation software Spirit. The spin-orbit torque driven spin-dynamics described by the Landau-Lifshitz-Gilbert equation is compared to the effective one of the Thiele equation. We demonstrate that, even though skyrmions and antiskyrmions can be seen as antiparticles, a rather dense arrangement of both along a memory track is possible, enabling their use as representations of the binary data bits “0” and “1” in a memory device.

Geometry and symmetry in skyrmion dynamics

The uniform motion of chiral magnetic skyrmions induced by a spin-transfer torque displays an intricate dependence on the skyrmions' topological charge and shape. We reveal surprising patterns in this dependence through simulations of the Landau-Lifshitz-Gilbert equation with Zhang-Li torque and explain them through a geometric analysis of Thiele's equation. In particular, we show that the velocity distribution of topologically non-trivial skyrmions depends on their symmetry: it is a single circle for skyrmions of high symmetry and a family of circles for low-symmetry configurations. We also show that the velocity of the topologically trivial skyrmions, previously believed to be the fastest objects, can be surpassed, for instance, by antiskyrmions. The generality of our approach suggests the validity of our results for exchange frustrated magnets, bubble materials, and others.

Diffusion of a magnetic skyrmion in two-dimensional space

Two-dimensional magnetic skyrmions are particle-like magnetic domains in magnetic thin films. The kinetic property of magnetic skyrmions at finite temperature is well described by the Thiele equation, including a stochastic field and a finite mass. This study examines the validity of the constant-mass approximation by comparing the Fourier spectrum of the Brownian motions described by the Thiele equation and the Landau–Lifshitz–Gilbert equation. Then, the expression of the diffusion flow is derived from the four-dimensional Fokker–Planck equation by considering Chandrasekhar's solution for the Thiele dynamics. Finally, a method to observe the gyro-diffusion is discussed.

Skyrmion Dynamics at Finite Temperatures: Beyond Thiele's Equation.

Magnetic textures are often treated as quasiparticles following Thiele's equation of motion. We demonstrate via spin model simulations of the current-driven and Brownian motion of ferromagnetic skyrmions that the existing theory based on Thiele's equation is insufficient to describe the dynamics of skyrmions at finite temperatures. We propose an extended equation of motion that goes beyond Thiele's equation by taking into account the coupling of the skyrmion to the magnonic heat bath leading to an additional dissipative term that is linear in temperature. Our results indicate that this so-far-neglected magnon-induced friction dominates for finite temperatures and Gilbert damping values typical for thin films and multilayers.

Random walk of antiferromagnetic skyrmions in granular films

The impressive absence of the skyrmion Hall effect of antiferromagnetic (AFM) skyrmions in clean systems has raised high expectations of AFM skyrmions as information carriers and the potential applications in data storage technology. Here, we report an undesired behavior of current-driven AFM skyrmion dynamics in granular thin films. Understanding the motion of AFM skyrmions in disordered systems is crucial because disorders are inevitable in all materials. Micromagnetic simulations show that AFM skyrmions deviate from their trajectory in the clean limit. Interestingly, the transverse (to the current direction) motion is diffusive, i.e., the mean square of transverse displacement is proportional to the traveling time. The longitudinal (along with the current direction) motion is always hindered by disorders. An effective theory based on the stochastic Thiele equation can account for the observed phenomena. At a very strong disorder above a critical value that depends on the current density, AFM skyrmions are pinned. Our findings should be important for future spintronic devices based on AFM skyrmions.

Skyrmions near defects

We study the impact of an exchange-reducing defect on a skyrmion in a thin film of finite thickness. Attraction of the skyrmion to a defect is demonstrated in a lattice model by computing the micromagnetic energy accounting for the exchange, Dzyaloshinskii–Moriya interaction, magnetic anisotropy, and dipole–dipole coupling. The spiraling dynamics of the skyrmion toward the defect is illustrated by solving numerically the full Landau–Lifshitz–Gilbert equations on a lattice and, independently, by solving the Thiele equation, with the two methods in agreement with each other. Depinning of the skyrmion by the current is investigated. We find that the skyrmion deforms when it is close to the defect. Deformation is small in the parameter space far from the phase boundary that determines stability of skyrmions. It increases dramatically near the phase boundary, leading to the transformation of the skyrmion by the defect into a snake-like magnetic domain.

Anisotropic Skyrmion Diffusion Controlled by Magnetic-Field-Induced Symmetry Breaking

Diffusion of particles has wide repercussions ranging from particle-based soft matter systems to solid state systems with particular electronic properties. Recently, in the field of magnetism, diffusion of magnetic skyrmions, topologically stabilized quasi-particles, has been demonstrated. Here we show that by applying a magnetic in-plane field and therefore breaking the symmetry of the system, the skyrmion diffusion becomes anisotropic with faster diffusion parallel to the field axis and slower diffusion perpendicular to it. We furthermore show that the absolute value of the applied field controls the absolute values of the diffusion coefficients so that one can thereby uniquely tune both the orientation of the diffusion and its strength. Based on the stochastic Thiele equation, we can explain the observed anisotropic diffusion as a result of the elliptical deformation of the skyrmions by the application of the in-plane field.

Magnetic skyrmion binning

When spin polarised electrons flow through a magnetic texture a transfer torque is generated. We examine the effect of this torque on skyrmions and skyrmion bags, skyrmionic structures of arbitrary integer topological degree, in thin ferromagnetic films. Using micromagnetic simulations and analysis from the well known Thiele equation we explore the potential for sorting or binning skyrmions of varying degrees mechanically. We investigate the applicability of the Thiele equation to problems of this nature and derive a theory of skyrmion deflection ordered by topological degree.

Dynamics of state-wise prospective reserves in the presence of non-monotone information

In the presence of monotone information, the stochastic Thiele equation describing the dynamics of state-wise prospective reserves is closely related to the classic martingale representation theorem. When the information utilized by the insurer is non-monotone, the classic martingale theory does not apply. By taking an infinitesimal approach, we derive a generalized stochastic Thiele equation that allows for information discarding. En passant, we solve some open problems for the classic case of monotone information. The results and their implication in practice are illustrated via examples where information is discarded upon and after stochastic retirement.

Current-driven skyrmion movement in a curved nanotrack

We report the results of complex studies concerning skyrmion motion in U-shaped and V-shaped nano-tracks simulated with MuMax3. The Thiele equation was used for description of skyrmion velocity components. It was shown that both size and velocity of a skyrmion vary depending on its position in the system, featuring distinct values for the straight and curved branches of the track. The degree of velocity variation can be effectively controlled by geometry of the track (including track width and its curvature radius) as well as the density of the driving current applied to the system.

Ultrafast Ratchet Dynamics of Skyrmions by Defect Engineering in Materials with Poor Conductivity Under Gigahertz Magnetic Fields

The ratchet motion of magnetic skyrmions driven by microwave magnetic fields, with the motion direction and speed tunable by field parameters, provides a promising route to drive magnetic skyrmions in materials with poor conductivity. However, as an indirect motion, skyrmion ratchet-motion speed is generally low in comparison with the direct motions driven by currents. Toward practical applications, it is necessary to ask if there are mechanisms to realize ultrafast ratchet motion of magnetic skyrmions and how such a motion can be integrated into racetrack-type skyrmion devices. In this work, we explore the effects of defects and edges on the ratchet motion of magnetic skyrmions under time-varying magnetic fields in GHz. We demonstrate that the ratchet motion of skyrmion is not only guided along the defect tracks or edges, but also with a remarkable speed up (with a factor over 10) compared with that in the bulk region. The skyrmion ratchet-motion speed reaches 100 m/s along a straight defect track and edge and ${10}^{9}$ rad/s along a circular edge under a field of approximately 50 mT, comparable to those direct motions driven by currents. Moreover, the skyrmion ratchet motion along the defect track or edge can be facilely controlled by the field and defect parameters. Analysis based on the time-averaged Thiele equation of skyrmion verifies that such a speed-up effect is due to the increased time-averaged driving force perpendicular to the skyrmion motion when it approaches the defect track or edge, analogous to that discovered in direct motions driven by currents. The generality of our conclusions has been examined for the ratchet motion of Bloch and N\'eel-type skyrmions driven by a variety of time-varying magnetic fields, and for systems with open edges or defect tracks with modified Dzyaloshinskii-Moriya or exchange interactions and anisotropy.

Controlled motion of skyrmions in a magnetic antidot lattice

Future spintronic devices based on skyrmions will require precise control of the skyrmion motion. We show that this goal can be achieved through the use of magnetic antidot arrays. With micromagnetic simulations and semi-analytical calculations based on Thiele equation, we demonstrate that an antidot array can guide the skyrmions in different directions depending on the parameters of the applied current pulse. Despite the fixed direction of the net driving current, due to the non-trivial interplay between the repulsive potential introduced by the antidots, the skyrmion Hall effect and the non-uniform current distribution, full control of skyrmion motion in a 2D lattice can be achieved. Moreover, we demonstrate that the direction of skyrmion motion can be controlled by tuning only a single parameter of the current pulse, i.e. current magnitude. For lower current magnitudes the skyrmion can be moved perpendicularly to the current direction, and can overcome the skyrmion Hall effect. For larger current magnitudes, the skyrmion Hall effect can be effectively suppressed and skyrmions can move parallel to the applied current.

Versatile Motion Control of a Magnetic Skyrmion Pair with Spin Transfer Torques

We studied the motion of a skyrmion pair under the action of independently controllable spin transfer torques. A skyrmion pair made of two coupled skyrmions with opposite skyrmion numbers moves along the direction of the current flow when the currents flowing in the two skyrmions, are identical while it moves perpendicular to the direction of current flow where the currents flow in opposite directions. Manipulating all the degrees of freedom for the strengths of both currents enables the directional motion of the skyrmion pair. We used the Thiele equation to investigate how the strengths of the currents are related with the motion of the skyrmion, and we obtained a tensor that provided the relation between the currents and the force that binds the two skyrmions, and we were able to realize a designed motion. We performed a dynamic simulation of the motion of the skyrmion pair and compared the results with those calculated using the Thiele equation.

Interaction of a Magnetic Vortex with Magnetic Anisotropy Nonuniformity

The problem of propagation of a magnetic inhomogeneity in the form of a magnetic vortex near a defect simulated by a crystallite with uniaxial anisotropy has been solved theoretically. The defect (crystallite) is implanted into a homogeneous 2D ferromagnetic matrix. Apart from the anisotropy energy, the term responsible for the existence of a centrosymmetric potential is included into the total energy. For calculations, we have used the method of collective variables (Thiele equation). We have considered the variants of bidirectional and unidirectional anisotropy of the crystallite. Analysis of the equations of motion for different directions of the anisotropy axis of the implanted defect has revealed the variety in the behavior of the vortex core as a quasiparticle. The vortex core can be trapped by the defect with equilibrium position of the vortex at rest directly on the crystallite or during its motion at a certain distance from it. It is shown that for a small damping parameter and in the case when the defect anisotropy axis lies in the plane of the magnet, the vortex moves so as if its core experiences the action of the repulsive axially symmetric potential.