Spin-wave Dispersion Relation Research Articles

Spatial control of hybridization-induced spin-wave transmission stop band

Spin-wave (SW) propagation close to the hybridization-induced transmission stop band is investigated within a trapezoid-shaped 200 nm thick yttrium iron garnet film using time-resolved magneto-optic Kerr effect microscopy and broadband spin-wave spectroscopy, supported by micromagnetic simulations. The gradual reduction of the effective field within the structure leads to local variations of the SW dispersion relation and results in a SW hybridization at a fixed position in the trapezoid where the propagation vanishes, since the SW group velocity approaches zero. By tuning external field or frequency, spatial control of the spatial stop band position and spin-wave propagation is demonstrated and utilized to gain transmission control over several microstrip lines.

Persistent magnetic coherence in magnets.

When excited, the magnetization in a magnet precesses around the field in an anticlockwise manner on a timescale governed by viscous magnetization damping, after which any information carried by the initial actuation seems to be lost. This damping appears to be a fundamental bottleneck for the use of magnets in information processing. However, here we demonstrate the recall of the magnetization-precession phase after times that exceed the damping timescale by two orders of magnitude using dedicated two-colour microwave pump-probe experiments for a Y3Fe5O12 microstructured film. Time-resolved magnetization state tomography confirms the persistent magnetic coherence by revealing a double-exponential decay of magnetization correlation. We attribute persistent magnetic coherence to a feedback effect, that is, coherent coupling of the uniform precession with long-lived excitations at the minima of the spin-wave dispersion relation. Our finding liberates magnetic systems from the strong damping in nanostructures that has limited their use in coherent information storage and processing.

Spin wave dynamics in CoFeB bilayers with wedged Au spacer

This paper presents the experimental studies of ferromagnetic layers separated by a heavy metal wedge. The studied system consists of two thin layers of CoFeB separated by a wedged Au spacer. By performing Brillouin light scattering measurements of the spin-wave dispersion relations, and dependences on the magnetic field, we extract magnetic parameters in the studied ferromagnetic layers as a function of heavy metal thickness. We concluded that for thin ferromagnetic double layers, the magnetic properties strongly depend on the thickness of the wedge gold layer spacer. In particular, the presence of perpendicular magnetic anisotropy diminishes as the thickness of the magnetic material is increased, and at the spacers thicker than 2.5 nm the dynamic coupling between propagating spin waves in both layers is negligible. These findings have potential advantages for the development of future spintronic devices.

Electrically excited magnetoelastic waves and magnetoacoustic resonance in ferromagnetic films with voltage-controlled magnetic anisotropy

Ferromagnet-dielectric heterostructures may possess interfacial magnetic anisotropy, which varies with the electric field created in the dielectric interlayer. Such a voltage-controlled magnetic anisotropy (VCMA) represents an efficient tool for the excitation of spin waves without the use of microwave magnetic fields. In ferromagnets with a significant magnetoelastic coupling between spins and strains, however, the magnetization precession in the spin wave induces elastic vibrations. Here we report micromagnetoelastic simulations of the coupled spin and acoustic dynamics generated in the thick Fe film brought into contact with an MgO nanolayer subjected to a microwave voltage. It is found that the electrically induced periodic modulation of VCMA gives rise to the excitation of both spin and elastic waves in the magnetostrictive Fe film. Remarkably, the magnetoelastic coupling makes it possible to generate propagating spin waves having frequencies well below the ferromagnetic resonance frequency. Furthermore, when the voltage frequency approaches a value ${\ensuremath{\nu}}^{*}$ at which the dispersion relations of pure spin and transverse elastic waves cross, the excitation of traveling magnetoelastic waves with a specific hybrid dispersion occurs in the Fe film. To explain the variety of observed phenomena, we propose a generalized dispersion relation for magnetoelastic waves, which accounts for a possible difference between the wave numbers of their magnetic and elastic components. The analysis of the simulation data also reveals that frequency dependencies of the amplitudes of magnetic oscillations and elastic vibrations display significant anomalies around the crossing-point frequency ${\ensuremath{\nu}}^{*}$, which originate from the electrically induced magnetoacoustic resonance.

Spin wave dispersion relation engineering by magnonic crystals with arbitrary symmetry

The use of metasurfaces to engineer the response of magnetic materials is of utmost importance in the field of magnon-spintronics. Here, we demonstrate a method to fabricate one- and two-dimensional magnonic crystals with arbitrary symmetry and use it to engineer the amplitude-frequency characteristic of magnetostatic surface spin waves excited in a magnetic material. The technique is based on the gentle microablation of the sample surface by focused femtosecond laser pulses. Tightly focused illumination allows using modest pulse energy while achieving micrometer precision. By raster scanning the incident laser spot on the sample surface, we control the shape and size of the building blocks constituting the unit cell of the crystal along with its symmetry and lattice parameter. Remarkable and controlled changes in the measured transmission characteristics reveal the strong and complex symmetry-dependent interaction of the spin waves with Bravais and non-Bravais lattices. The described single-step microfabrication method facilitates and speeds up the realization of integrated spintronics components and provides an efficient tool to explore complex magnetic dynamics in scattering lattices.

Excitation and reception of magnetostatic surface spin waves in thin conducting ferromagnetic films by coplanar microwave antennas. Part I: Theory

A fully self-consistent model for the excitation and reception of magnetostatic surface waves in thin ferromagnetic films by a set of coplanar antennas was developed and implemented numerically. The model assumes that the ferromagnetic film is highly conducting and is interfaced with non-magnetic metallic films, but is also suitable for modeling magneto-insulating films. Perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interaction can be included at both interfaces of the ferromagnetic layer. The model calculates the coupling impedances between the different strips constituting the coplanar antennas. In some situations, this leads to a frequency non-reciprocity between counter-propagating waves even in the case of no asymmetry in the spin-wave dispersion relation. Several intermediate results of the model were checked numerically and the final output of the model, given as the scattering parameters, S11, S12, and S21 of the antenna system, were in good agreement with previous experimental studies.

Spin wave dispersion relations and isofrequency curve calculations using micromagnetic simulations

Dispersion relations and isofrequency curves are of critical importance for understanding the behavior of waves, including what frequencies can be excited, how the waves will propagate, and how waves in one system will couple to another. Here, we present methods to extract the dispersion relations and isofrequency curves automatically and conveniently, each from a single micromagnetic simulation run. These methods have significant advantages in that they provide a means to obtain rapid insight into spin wave behavior in complex situations where analytic approaches are difficult or impossible. We present multiple examples to illustrate the methodology and discuss specific issues that need to be considered for the different situations.

Quantum Spin-Wave Theory for Non-Collinear Spin Structures, a Review

In this review, we trace the evolution of the quantum spin-wave theory treating non-collinear spin configurations. Non-collinear spin configurations are consequences of the frustration created by competing interactions. They include simple chiral magnets due to competing nearest-neighbor (NN) and next-NN interactions and systems with geometry frustration such as the triangular antiferromagnet and the Kagomé lattice. We review here spin-wave results of such systems and also systems with the Dzyaloshinskii–Moriya interaction. Accent is put on these non-collinear ground states which have to be calculated before applying any spin-wave theory to determine the spectrum of the elementary excitations from the ground states. We mostly show results obtained by the use of a Green’s function method. These results include the spin-wave dispersion relation and the magnetizations, layer by layer, as functions of T in 2D, 3D and thin films. Some new unpublished results are also included. Technical details and discussion on the method are shown and discussed.

Nonreciprocal spin wave propagation in bilayer magnonic waveguide

Here we present the frequency-selective spin-wave propagation in the magnonic waveguide consisting of two layers with different magnetization saturation. We show that the multimode spin-wave propagation can be performed inside the bilayer structure within two separate frequency range. At the same time this process is accompanied by strong nonreciprocity in spin-wave behavior with the relation to direction of the spin-wave propagation and bias magnetic field. Using the microwave spectroscopy we demonstrate the coexistence of two frequency range in the two samples of pure/modified yttrium iron garnet bilayers with the different orientation of layers with the respect of the substrate. Using the numerical model based on the magnetostatic dispersion relation for spin waves in the bilayer the nonreciprocal behavior of spin-waves was elucidated and its properties for the confined magnonic bilayer stripe was obtained. In particular, it was shown that the narrowing of the stripe leads to the pronounced variation of the nonreciprocity effect for the separate modes in the spin-wave spectra. The experimental data are in good agreement with the predicted by the proposed theoretical model. The proposed concept of bilayer spin-wave waveguide can underlie the fabrication of the magnonic interconnection bus with the support of multiple frequency band operational regimes.

Spin-Wave Dispersion Measurement by Variable-Gap Propagating Spin-Wave Spectroscopy

Knowledge of the spin-wave dispersion relation is a prerequisite for the explanation of many magnonic phenomena as well as for the practical design of magnonic devices. Spin-wave dispersion measurement by established optical techniques such as Brillouin light scattering or the magneto-optical Kerr effect at ultralow temperatures is often forbiddingly complicated. By contrast, microwave spectroscopy can be used at all temperatures but it usually lacks spatial and wave-number resolution. Here we develop a variable-gap-propagating-spin-wave-spectroscopy (VGPSWS) method for the deduction of the dispersion relation of spin waves in a wide frequency and wave-number range. The method is based on the phase-resolved analysis of the spin-wave transmission between two antennas with variable spacing, in conjunction with theoretical data treatment. We validate the method for in-plane magnetized $\mathrm{Co}$-$\mathrm{Fe}$-$\mathrm{B}$ and yttrium iron garnet thin films in $\mathbf{k}\ensuremath{\perp}\mathbf{B}$ and $\mathbf{k}\ensuremath{\parallel}\mathbf{B}$ geometries by deducing the full set of material and spin-wave parameters, including spin-wave dispersion, hybridization of the fundamental mode with the higher-order perpendicular standing spin-wave modes, and surface spin pinning. The compatibility of microwaves with low temperatures makes this approach attractive for cryogenic magnonics at the nanoscale.

Dispersion relation of spin wave in chiral helimagnet under stationary optical vortex radiation

We investigate a dispersion relation of a spin wave of an effectively one-dimensional chiral helimagnet CrNb3S6, which is under stationary radiation of a resonant optical vortex beam coaxial to that spiral magnetic order axis. We derive a classical magnetic structure by solving Landau–Lifshitz–Gilbert equation and calculate the collective excitation on it. We show that the optical vortex radiation makes the energy gap at k ∼ 0, and the exceptional points appear in the dispersion relation. Because such a dispersion relation implies the dynamical properties with instability, our result implies a possibility to control spin wave transport.

Channeling of spin waves in antiferromagnetic domain walls

We theoretically calculated ultrafast propagations of spin waves channeled in antiferromagnetic Bloch-type domain walls based on a phenomenological theory developed by Hals et al. [Phys. Rev. Lett. 106, 107206 (2011)]. Our analytical derivation demonstrates that the dispersion relation of such channeled spin waves exhibits an extremely high group velocity, up to \ensuremath{\sim}18.7 km/s, without any forbidden gap. Further, such electromagnetic-wave-like dispersion relation allows for dispersionless and translational transmission of trains of low-frequency, long-wavelength digital signals in the channel. Our results offer guidelines for the development of ultrafast information signal processing in nanoscale magnonic circuits composed of antiferromagnetic domain walls.

Dispersion relation of nutation surface spin waves in ferromagnets

Inertia effects in magnetization dynamics are theoretically shown to result in a different type of spin waves, i.e. nutation surface spin waves, which propagate at terahertz frequencies in in-plane magnetized ferromagnetic thin films. Considering the magnetostatic limit, i.e. neglecting exchange coupling, we calculate dispersion relation and group velocity, which we find to be slower than the velocity of conventional (precession) spin waves. In addition, we find that the nutation surface spin waves are backward spin waves. Furthermore, we show that inertia causes a decrease of the frequency of the precession spin waves, namely magnetostatic surface spin waves and backward volume magnetostatic spin waves. The magnitude of the decrease depends on the magnetic properties of the film and its geometry.

Heisenberg Exchange and Dzyaloshinskii–Moriya Interaction in Ultrathin Pt(W)/CoFeB Single and Multilayers

We present results of the analysis of Brillouin light-scattering (BLS) measurements of spin waves performed on ultrathin single and multirepeat CoFeB layers with adjacent heavy metal layers. From a detailed study of the spin-wave dispersion relation, we independently extract the Heisenberg exchange interaction (also referred to as symmetric exchange interaction), the Dzyaloshinskii–Moriya interaction (DMI, also referred to as antisymmetric exchange interaction), and the anisotropy field. We find a large DMI in CoFeB thin films adjacent to a Pt layer and nearly vanishing DMI for CoFeB films adjacent to a W layer. Furthermore, the influence of the dipolar interaction on the dispersion relation and on the evaluation of the Heisenberg exchange parameter is demonstrated. Eventually, an experimental analysis of the impact of the DMI on the spin-wave lifetime is presented.

Compact tunable YIG-based RF resonators

We report on the design, fabrication, and characterization of compact tunable yttrium iron garnet (YIG)-based RF resonators based on μm-sized spin-wave cavities. Inductive antennas with both ladder and meander configurations were used as transducers between spin waves and RF signals. The excitation of ferromagnetic resonance and standing spin waves in the YIG cavities led to sharp resonances with quality factors up to 350. The observed spectra were in excellent agreement with a model based on the spin-wave dispersion relations in YIG, showing a high magnetic field tunability of about 29 MHz/mT.

Nanoscale magnonic Fabry-P\xe9rot resonator for low-loss spin-wave manipulation

Active control of propagating spin waves on the nanoscale is essential for beyond-CMOS magnonic computing. Here, we experimentally demonstrate reconfigurable spin-wave transport in a hybrid YIG-based material structure that operates as a Fabry-Pérot nanoresonator. The magnonic resonator is formed by a local frequency downshift of the spin-wave dispersion relation in a continuous YIG film caused by dynamic dipolar coupling to a ferromagnetic metal nanostripe. Drastic downscaling of the spin-wave wavelength within the bilayer region enables programmable control of propagating spin waves on a length scale that is only a fraction of their wavelength. Depending on the stripe width, the device structure offers full nonreciprocity, tunable spin-wave filtering, and nearly zero transmission loss at allowed frequencies. Our results provide a practical route for the implementation of low-loss YIG-based magnonic devices with controllable transport properties.

Electrical spin-wave spectroscopy in nanoscale waveguides with nonuniform magnetization

Spin wave modes in magnetic waveguides with the width down to 320 nm have been studied by electrical propagating spin-wave spectroscopy and micromagnetic simulations for both longitudinal and transverse magnetic bias fields. For longitudinal bias fields, a 1.3 GHz wide spin-wave band was observed in agreement with analytical dispersion relations for uniform magnetization. However, the transverse bias field led to several distinct bands, corresponding to different quantized width modes, with both negative and positive slopes. Micromagnetic simulations showed that, in this geometry, the magnetization was nonuniform and tilted due to the strong shape anisotropy of the waveguides. Simulations of the quantized spin-wave modes in such nonuniformly magnetized waveguides resulted in spin wave dispersion relations in good agreement with the experiments.

A spin-wave driven skyrmion diode under transverse magnetic fields

A diode, a device allowing unidirectional signal transmission, lies at the heart of modern information technology as a fundamental element of logic operations. Here, we study the motion of the skyrmion driven by the spin wave (SW) in the presence of a transverse magnetic field. We show that the external magnetic field leads to a shift of SW dispersion relation and induces an asymmetric skyrmion propagation when SWs are injected from opposite sides. Based on this finding, we propose the concept of a SW-driven skyrmion diode. We numerically compute the spin-wave transport coefficients to quantify the number of reflected magnons, which explains the skyrmion velocity asymmetry. Our results offer a new insight to design skyrmion devices embracing chiral SWs.

Spin-Wave Emission from Vortex Cores under Static Magnetic Bias Fields.

We studied the influence of a static in-plane magnetic field on the alternating-field-driven emission of nanoscale spin waves from magnetic vortex cores. Time-resolved scanning transmission X-ray microscopy was used to image spin waves in disk structures of synthetic ferrimagnets and single ferromagnetic layers. For both systems, it was found that an increasing magnetic bias field continuously displaces the wave-emitting vortex core from the center of the disk toward its edge without noticeably altering the spin-wave dispersion relation. In the case of the single-layer disk, an anisotropic lateral expansion of the core occurs at higher magnetic fields, which leads to a directional rather than radial-isotropic emission and propagation of waves. Micromagnetic simulations confirm these findings and further show that focusing effects occur in such systems, depending on the shape of the core and controlled by the static magnetic bias field.

Observation of the dispersion relations for quantized coherent spin waves excited by a microwave antenna

The propagation of coherent spin waves in permalloy microstrips was investigated using a heterodyne-magneto-optical Kerr effect technique. The two-dimensional imaging reveals the mode interference with a high-order spin-wave mode along the microstrip width. The spin-wave dispersion relations across the microstrip width were deduced from fast Fourier transformations on the complex spin-wave amplitude of the one-dimensional scan, and multiple spin-wave modes were observed at the edges of the microstrips. These experimental results provide direct evidence for the quantization of spin-wave dispersion due to the finite width of the microstrips and enables us to evaluate the attenuation length for each quantized spin-wave mode.