Azimuthal Spin Wave Modes Research Articles

Static and dynamic magnetic characteristics in concentric permalloy nanorings

Magnetic nanodisks and nanorings have been focus of attention for quite some time due to their applicability in magnetic memory devices. In this paper, static and dynamic magnetic properties of Permalloy concentric thin ring nanostructures are reported as a function of the number of rings by using micromagnetic simulations. The maximum outer radius (R) and thickness (t) of the concentric rings are fixed at 100 nm and 20 nm respectively. The distance between any two neighbouring rings is always maintained equal to the width of each ring while the number of nanorings is increased starting from one to five. In-plane magnetic hysteresis loops show that in all cases, the remanence and coercivity values are found to be almost zero while vortex state is exhibited with opposite helicity in adjacent nanorings. When the number of nanorings reach five, the narrowest hysteresis loop is observed with almost no hysteresis at all. Dynamic susceptibility spectra was computed by relaxing the system from a out of plane saturated state followed by an in-plane excitation using a small field pulse. In most of the cases, the in plane magnetic field pulse excites azimuthal spin wave modes. The resonance frequency corresponding to these excitation modes depends on the remanent configuration and the number of peaks increase with increment in the number of rings. The ferromagnetic resonance mode that is common for all the samples considered, moves towards lower frequency with increase in number of concentric rings.

Robust formation of skyrmion and skyrmionium in magnetic hemispherical shells and their dynamic switching

We explored the variations of the topological magnetic textures of vortices, skyrmions, and skyrmioniums in magnetic elements of hemispherical-shell shape with respect to surface-normal uniaxial magnetic anisotropy constant ${K}_{u}$, Dzyaloshinskii-Moriya interaction (DMI) constant ${D}_{\mathrm{int}}$, and shell diameter $2R$. For given values of $2R$, the combination of ${K}_{u}$ and ${D}_{\mathrm{int}}$ plays a crucial role in the stabilization of those different spin textures. With decreasing $2R$, the geometrical confinement of hemispherical shells more significantly affects the stabilization of skyrmions owing to curvature-induced DM-like interaction. This effect is contrastingly dependent on the sign of ${D}_{\mathrm{int}}$: skyrmion formation is more favorable for positive ${D}_{\mathrm{int}}$ values, whereas it is less favorable for negative ones. A quite promising feature is that skyrmions can be stabilized even in the absence of intrinsic DMI for $2R<25\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. We also explored characteristic dynamic properties of skyrmions excited by in-plane and out-of-plane oscillating magnetic fields. Similarly to the fundamental dynamic modes found in planar dots, in-plane gyration and azimuthal spin-wave modes as well as out-of-plane breathing modes were found, but additional higher-frequency hybrid modes also appeared due to coupling between radially quantized and azimuthal spin-wave modes. Finally, we found a switching behavior of skyrmion polarity through a transient skyrmionium state using very-low-strength AC magnetic fields. This work provides further physical insight into the static and dynamic properties of skyrmions in curved-geometry nanodots and suggests potential applications to low-power-consumption and ultra-high-density information-storage devices.

Intrinsic DMI-free skyrmion formation and robust dynamic behaviors in magnetic hemispherical shells

We performed finite-element micromagnetic simulations to examine the formation of skyrmions without intrinsic Dzyaloshinskii–Moriya interaction (DMI) in magnetic hemispherical shells. We found that curvature-induced DM-like interaction allows for further stabilization of skyrmions without the DMI in curved-geometry hemispherical shells for a specific range of uniaxial perpendicular magnetic anisotropy (PMA) constant Ku. The larger the curvature of the shell, the higher the Ku value required for the formation of the skyrmions. With well-stabilized skyrmions, we also found in-plane gyration modes and azimuthal spin-wave modes as well as an out-of-plane breathing mode, similarly to previously found modes for planar geometries. Furthermore, additional higher-frequency hybrid modes were observed due to coupling between the gyration and azimuthal modes. This work provides further physical insight into the static and dynamic properties of intrinsic DMI-free skyrmions formed in curved-geometry systems.

Non-uniform along thickness spin excitations in magnetic vortex-state nanodots

We summarize our experimental findings in the arrays of Ni80Fe20 circular nanodots with diameter 300 nm and thickness 20 nm ≤ L ≤ 100 nm, probed by broadband ferromagnetic resonance spectroscopy in the absence of external magnetic field. Spin excitation modes related to the vortex core gyrotropic dynamics were observed in the gigahertz frequency range. Micromagnetic simulations revealed that they are flexure oscillations of the vortex core string with n = 0, 1, 2 nodes along the dot thickness. It was found that for L > 70 nm the intensity of more complicated n = 1 vortex gyrotropic mode is unexpectedly higher than the one of the lowest n = 0 gyrotropic mode. This behavior was clarified on the basis of the inhomogeneous vortex mode phase profiles extracted from micromagnetic simulations and calculated analytically. Precise measurements of the dependence of resonance frequency of the vortex n = 0 mode on the dot thickness demonstrated a clear maximum around L = 70 nm, that was theoretically explained by introducing a vortex mass, which is a result of the vortex distortion due to interaction with spin waves having azimuthal indices m = ±1. Finally, several azimuthal spin-wave modes having curled structure at the dot top and bottom faces were found in the spectrum of the dots with thicknesses L ≥ 40 nm.

The internal dynamic modes of an antiskyrmion in ultrathin ferromagnetic nanodisks

We report a numerical study on the internal resonant modes of an antiskyrmion structure in thin-film ferromagnetic nanodisks by using micromagnetic simulations. As the samples are excited by an oscillating magnetic field along different directions, we are able to resolve several sets of resonant eigenmodes: gyrotropic mode and azimuthal and radial spin wave modes. At the in-plane field excitation, the gyrotropic mode at the low frequency behaves consistently with Thiele’s analytic theory, while a pair of twin azimuthal spin wave modes at higher frequencies demonstrate that their splitting is strongly relevant to the coexistent gyrotropic mode. At the out-of-plane field excitation, several breathing-like radial spin wave modes have been found. In addition, we carry out a brief comparison between these modes and their counterpart modes of an isolated skyrmion structure. We also quantitatively investigate the dependence of the eigenfrequencies of these modes on the different parameter sets of the sample size and internal interactions, which exhibit complex relationships between the internal dynamic modes of an antiskrymion and the geometry and material of the sample.

Spin dynamics of anisotropic azimuthal modes in heterogeneous magnetic nanodisks

It is well known that azimuthal spin wave modes of magnetic vortex state in permalloy nanodisks have circular symmetry. Intuitively, magnetic materials having magnetocrystalline anisotropy is not compatible with the circular symmetry of the azimuthal modes. In this article, however, we report cubic azimuthal modes in heterogeneous nanodisks consisting of a permalloy core and a Fe shell. The fourfold symmetry of azimuthal modes is due to the exchange, and magneto-static, interactions between the permalloy core and the Fe shell. In comparison to the results of circular azimuthal mode, the vortex switching occurs considerably faster under the excitation of cubic azimuthal mode. The gyration path of vortex core turns into square under the influence of induced cubic anisotropy in the permalloy region. We find out periodic oscillation of the vortex core size and the gyration speed as well. Our findings may offer a new route for spintronic applications using heterogeneous magnetic nanostructures.

Characteristic dynamic modes and domain-wall motion in magnetic nanotubes excited by resonant rotating magnetic fields

We performed micromagnetic numerical calculations to explore a cylindrical nanotube's magnetization dynamics and domain-wall (DW) motions driven by eigen-circular-rotating magnetic fields of different frequencies. We discovered the presence of two different localized DW oscillations as well as asymmetric ferromagnetic resonance precession and azimuthal spin-wave modes at the corresponding resonant frequencies of the circular-rotating fields. Associated with these intrinsic modes, there exist very contrasting DW motions of different speed and propagation direction for a given DW chirality. The direction and speed of the DW propagation were found to be controllable according to the rotation sense and frequency of noncontact circular-rotating fields. Furthermore, spin-wave emissions from the moving DW were observed at a specific field frequency along with their Doppler effect. This work furthers the fundamental understanding of soft magnetic nanotubes' intrinsic dynamic modes and spin-wave emissions and offers an efficient means of manipulating the speed and direction of their DW propagations.

Imaging spin dynamics on the nanoscale using X-Ray microscopy

The dynamics of emergent magnetic quasiparticles, such as vortices, domain walls, and bubbles are studied by scanning transmission x-ray microscopy (STXM), combining magnetic (XMCD) contrast with about 25 nm lateral resolution as well as 70 ps time resolution. Essential progress in the understanding of magnetic vortex dynamics is achieved by vortex core reversal observed by sub-GHz excitation of the vortex gyromode, either by ac magnetic fields or spin transfer torque. The basic switching scheme for this vortex core reversal is the generation of a vortex-antivortex pair. Much faster vortex core reversal is obtained by exciting azimuthal spin wave modes with (multi-GHz) rotating magnetic fields or orthogonal monopolar field pulses in x and y direction, down to 45 ps in duration. In that way unidirectional vortex core reversal to the vortex core 'down' or 'up' state only can be achieved with switching times well below 100 ps. Coupled modes of interacting vortices mimic crystal properties. The individual vortex oscillators determine the properties of the ensemble, where the gyrotropic mode represents the fundamental excitation. By self-organized state formation we investigate distinct vortex core polarization configurations and understand these eigenmodes in an extended Thiele model. Analogies with photonic crystals are drawn. Oersted fields and spin-polarized currents are used to excite the dynamics of domain walls and magnetic bubbles. From the measured phase and amplitude of the displacement of domain walls we deduce the size of the non-adiabatic spin-transfer torque. For sensing applications, the displacement of domain walls is studied and a direct correlation between domain wall velocity and spin structure is found. Finally the synchronous displacement of multiple domain walls using perpendicular field pulses is demonstrated as a possible paradigm shift for magnetic memory and logic applications.

Azimuthal-spin-wave-mode-driven vortex-core reversals

We studied, by micromagnetic numerical calculations, asymmetric vortex-core reversals driven by the m = −1 and m = +1 azimuthal spin-wave modes' excitations in soft magnetic circular nano-disks. We addressed the similarities and differences between the asymmetric core reversals in terms of the temporal evolutions of the correlated core-motion speed, locally concentrated perpendicular gyrofield, and magnetization dip near the original vortex core. The criterion for the core reversals was found to be the magnetization dip that must reach the out-of-plane magnetization component, mz = −p, with the initial polarization p, where p = +1 (−1) for the upward (downward) core magnetization. The core-motion speed and the associated perpendicular gyrofield, variable and controllable with static perpendicular field, Hz, applied perpendicularly to the disk plane, must reach their threshold values to meet the ultimate core-reversal criterion. Also, we determined the Hz strength and direction dependence of the core-switching time and threshold exciting field strength required for the core reversals, whose parameters are essential in the application aspect. This work offers deeper insights into the azimuthal spin-wave-driven core-reversal dynamics as well as an efficient means of controlling the azimuthal-modes-driven core reversals.

Azimuthal spin wave modes in an elliptical nanomagnet with single vortex configuration

In comparison with uniformly magnetized states, vortex structures demonstrate a rich frequency spectrum of spin-wave (SW) excitations. However, a detailed theoretical description of the magnetic modes is generally still a challenge due to the difficulty of analytic calculation, except for the well-defined symmetric circular states. In contrast, the method of micromagnetic simulations combined with Fourier analysis is shown to be very powerful for gaining insight into the nature of magnetic excitation modes. Vortex excitation modes have been reported to be directly influenced by the geometric symmetry of the elements and/or the nature of the initial perturbation of pulse field. In order to understand how the reduced symmetry affects the vortex SW modes, we perform the micromagnetic simulations on vortex modes excited in a submicron-sized thin ellipse. In order to excite the spin-wave modes, a short in-plane Gaussian field pulse is applied along the short axis direction. After the pulse, the off-centered vortex core moves following an elliptical trajectory around its equilibrium position. Simulations provide the time evolution of the local magnetizations (at each discretization point) and dynamics of the spatially averaged magnetization. To determine the mode frequencies, the spectrum is obtained from the average magnetization through Fourier transformation from time domain the frequency domain. By means of Fourier analysis, a variety of azimuthal SW modes can be observed in the excitation spectrum. The ellipse in single vortex state has a twofold rotational symmetry with a rotation of πup around the z-axis (out-of plane) and can be described by the C2 group. The observed azimuthal modes can be divided into two categories according to their symmetry. Two modes occur alternately with increasing azimuthal number, indicating that the magnetic excitation modes remain to keep the symmetry of the ellipse structure. Their frequencies are found to increase linearly with the azimuthal index number. An increase of the SW frequency with increasing number of nodal planes is rather well known, which results from the competition between exchange and dipolar energy terms. According to the temporal evolution of the ellipse's spatially averaged energy densities, our micromagnetic simulation shows that the average exchange energy is significantly higher than the magnetostatic energy, suggesting that the exchange interaction plays a more important role in the excitation modes. The exchange energy density is mainly focused on the core origin while the largest contribution of the magnetostatic energy is distributed near the long axis. Thus, we can conclude that the exchange interaction provides the principal contribution to the vortex energy in such small ellipses with a single vortex state, resulting in the increasing frequency versus the azimuthal number, that is observed.

Excited eigenmodes in magnetic vortex states of soft magnetic half-spheres and spherical caps

We studied the magnetization dynamics of excitation modes in special geometrical confinements of soft magnetic half-spheres and spherical caps in magnetic vortex states using finite-element micromagnetic numerical calculations. We found additional fine features of the zeroth- and first-order gyrotropic modes and asymmetric m = +1 and m = −1 azimuthal spin-wave modes, which detailed information is unobtainable from two-dimensional mesh-cell based numerical calculations. Moreover, we examined the perpendicular bias field dependence of the excited eigenmodes, which data provide for an efficient means of control over the excited modes. Such numerical calculations offer additional details or new underlying physics on dynamic features in arbitrary-shape magnetic nano-elements such as half-spheres and spherical caps in magnetic vortex states.

Unidirectional sub-100-ps magnetic vortex core reversal

The magnetic vortex structure, an important ground state configuration in micron and sub-micron sized ferromagnetic thin film platelets, is characterized by a curling in-plane magnetization and, in the center, a minuscule region with out-of-plane magnetization, the vortex core, which points either up or down. It has already been demonstrated that the vortex core polarity can be reversed with external AC magnetic fields, frequency-tuned to the (sub-GHz) gyrotropic eigenmode or to (multi-GHz) azimuthal spin wave modes, where reversal times in the sub-ns regime can be realized. This fast vortex core switching may also be of technological interest as the vortex core polarity can be regarded as one data bit. Here we experimentally demonstrate that unidirectional vortex core reversal by excitation with sub-100 ps long orthogonal monopolar magnetic pulse sequences is possible in a wide range of pulse lengths and amplitudes. The application of such short digital pulses is the favourable excitation scheme for technological applications. Measured phase diagrams of this unidirectional, spin wave mediated vortex core reversal are in good qualitative agreement with phase diagrams obtained from micromagnetic simulations. The time dependence of the reversal process, observed by time-resolved scanning transmission X-ray microscopy indicates a switching time of 100 ps and fits well with our simulations. The origin of the asymmetric response to clockwise and counter clockwise excitation which is a prerequisite for reliable unidirectional switching is discussed, based on the gyromode - spin wave coupling.

Azimuthal spin-wave modes in rolled-up permalloy microtubes: Tuneable mode frequency, mode patterns, and mode splitting

We investigated rolled-up permalloy microtubes using microwave absorption spectroscopy. The resonance frequencies of the tubes are tuneable by the external magnetic field. Up to four resonant azimuthal modes have been found and were attributed to constructive spin-wave interference in axial and perpendicular magnetization geometry. We show that the degeneracy of the mode can be lifted by applying a magnetic field in specified directions. The resonances of the system are described analytically. Thereby, an effective layer thickness and a demagnetization field are used to account for the multilayer structure and the curved surface, respectively. This modeling approach is supported by micromagnetic simulations.

Spin Wave Excitations and Winter’s Magnons in Vertically Coupled Vortex State Permalloy Dots

Circular soft magnetic dots are the main elements of many proposed novel spintronics devices, capable of fascinating spin-based electronics applications, from extremely sensitive magnetic field sensors, to current-tunable microwave vortex oscillators. Here, we investigate static and broadband dynamic magnetization responses of vertically coupled Permalloy (Py) magnetic dots in the vortex state in layered nanopillars (experiment and simulations), which were explored as a function of in-plane magnetic field and interlayer separation. Under reduction of magnetic field from saturation for the field range just above vortex-vortex ground state. We observe a metastable double vortex state for each of the dots. In this state, novel kinds of spin waves (Winter’s magnons along domain walls between vortex cores and half-edge antivortex) are excited. For dipolarly coupled circular Py(25 nm)/Cu(20 nm)/Py(25 nm) trilayer nanopilars of diameter 600 nm, a small in-plane field splits the eigenfrequencies of azimuthal spin wave modes inducing an abrupt transition between acoustic (in-phase) and optic (out-of-phase) kinds of the low-lying coupled spin wave modes. Qualitatively similar changes (although more gradual and at higher values of in-plane fields) occur in the exchange coupled Py(25 nm)/Cu(1 nm)/Py(25 nm) trilayer nanopillars. These findings are in qualitative agreement with micromagnetic dynamic simulations.

Broadband probing magnetization dynamics of the coupled vortex state permalloy layers in nanopillars

Broadband magnetization response of coupled vortex state magnetic dots in layered nanopillars was explored as a function of in-plane magnetic field and interlayer separation. For dipolarly coupled circular Py(25 nm)/Cu(20 nm)/Py(25 nm) nanopillars of 600 nm diameter, a small in-plane field splits the eigenfrequencies of azimuthal spin wave modes inducing an abrupt transition between in-phase and out-of-phase kinds of the low-lying coupled spin wave modes. The critical field for this splitting is determined by antiparallel chiralities of the vortices in the layers. Qualitatively similar (although more gradual) changes occur also in the exchange coupled Py(25 nm)/Cu(1 nm)/Py(25 nm) tri-layer nanopillars. These findings are in qualitative agreement with micromagnetic dynamic simulations.

Azimuthal Spin Wave Modes Excited in an Elliptical Nanomagnet With Vortex Pair States

The ferromagnetic thin-film elements can be prepared to form a ?diamond? remanent state that contains two vortices with opposite circulation and with the same or opposite core polarities. The vortex polarity could influence their normal modes through the magnetostatic interactions between two vortices. In this paper, micromagnetic simulations, combined with Fourier techniques, are employed to study the spin-wave (SW) modes excited in a submicron Permalloy ellipse with a vortex pair configuration. After the application of an inplane Gaussian field pulse, it is found that a series of azimuthal SW modes is excited and the high-order harmonics of SW modes are strongly dependent of the core polarities due to the dynamic magnetic interactions between the two vortices.

Interactions of Spin Waves with a Magnetic Vortex

We have investigated azimuthal spin-wave modes in magnetic vortex structures using time-resolved Kerr microscopy. Spatially resolved phase and amplitude spectra of ferromagnetic disks with diameters from 5 microm down to 500 nm reveal that the lowest order azimuthal spin-wave mode splits into a doublet as the disk size decreases. We demonstrate that the splitting is due to the coupling between spin waves and the gyrotropic motion of the vortex core.