Propagating Spin Waves Research Articles

Interaction of propagating spin waves with extended skyrmions

Active control of propagating short-wavelength spin waves in perpendicularly magnetized materials is promising for designing nanoscale magnonic devices. One method of manipulating spin waves on the nanoscale is through their interaction with magnetic textures, an example of which is the magnetic skyrmion—a particle-like topological object stabilized in thin film heterostructures by the Dzyaloshinskii–Moriya interaction (DMI) and perpendicular magnetic anisotropy. In this paper, the interaction between spin waves and skyrmions is studied using micromagnetic simulations. The magnetic parameters chosen are similar to those found experimentally, leading to a skyrmion with an extended core of reversed magnetization. The effect of a propagating spin wave on the skyrmion is to cause the emission of a secondary spin wave by the skyrmion. At low frequencies, where the incoming spin wave wavelength is much larger than the skyrmion, this leads to a nearly circular re-emitted spin wave. The pattern of emission becomes increasingly complex at higher frequencies as the wavelength becomes similar to the skyrmion size due to the complex excitation of the extended core. The emitted spin wave profile can be controlled by altering the size of the skyrmion through the magnitude of the DMI, providing a method of tuning the system.

Nonreciprocal Spin-Wave Propagation in a Ferromagnet With Stepwise Interfacial Dzyaloshinskii–Moriya Interaction

In this article, boundary conditions for Landau–Lifshitz equations at the interface between two ferromagnets with different interfacial Dzyaloshinskii–Moriya interactions (iDMIs) are derived. We calculated and verified the analytical expression for the energy flux density continuity for the spin-wave (SW) propagation through the ferromagnet with stepwise iDMI considering the boundary conditions mentioned. Analytical expressions for reflection, transmission coefficients, and the nonreciprocity factor are calculated in the case of SW propagation through a ferromagnet with the stepwise iDMI. Two different types of nonreciprocity effects for SW propagation through a ferromagnet with the stepwise interface Dzyaloshinskii–Moriya interaction are revealed for opposite orientations of Dzyaloshinskii–Moriya vector. The material parameters of a ferromagnet and the stepwise iDMIs are found for which the extremely high nonreciprocity factor (>10) is expected according to the results of calculations. The results of this article deepen the knowledge about the SW propagation control in magnonic devices.

Spin-wave directional coupler based on the Bloch-type magnetic domain walls

Domain walls (DWs) as waveguides of spin waves (SWs) have been used to study SWs propagation properties in ferromagnetic thin films. In this work, we present a spin-wave (SW) directional coupler based on two Bloch-type DWs for the first time with micromagnetic simulations. The distance between the DWs is 40 nm, and the magnetization orientation is is parallel at the central positions of the two magnetic DWs. Under the dipolar interaction, the lowest-width SW mode is split into symmetric and antisymmetric modes, and the SW coupling length L is characterized to be 469nm.

Propagation of Spin Waves in Intersecting Yttrium Iron Garnet Nanowaveguides

We study experimentally the propagation of spin waves in waveguide structures consisting of two submicrometer-width yttrium iron garnet waveguides intersecting at a right angle. We show that, despite the significant spatial variations of the internal static magnet field and the in-plane anisotropy of the dispersion characteristics, the incident spin wave can efficiently pass through the microscopic cross-shaped structure and be transmitted into all its arms. This process depends strongly on the frequency of the wave and the orientation of the static magnetic field. By varying these parameters, one can achieve a controllable uniform or preferential transmission of the wave into different arms of the cross. Our results create the basis for the implementation of nanoscale magnonic networks to be used for the realization of complex non-Boolean data-processing schemes, including neuromorphic computing.

Tuning of spin-wave transmission and mode conversion in microscopic YIG waveguides with magnonic crystals

We report experimental results on spin-wave propagation, transmission gap tuning, and mode conversion in straight, curved, and Y-shaped yttrium iron garnet waveguides with magnonic crystals made of submicrometer-wide airgrooves. We observe forbidden frequency gaps with sizes up to 200 MHz in straight waveguides and narrowing of the gaps in curved and Y-shaped waveguides. The spin-wave transmission signal is strongly suppressed inside the gaps and remains high at allowed frequencies for all waveguide types. Using super-Nyquist sampling magneto-optical Kerr effect microscopy, we image symmetric and asymmetric spin-wave interference patterns, the self-focusing of propagating spin waves, and interconversions between width modes with different quantization numbers.

Hybrid magnonic-oscillator system

We propose a hybrid magnonic-oscillator system based on the combination of a spin transfer auto-oscillator and a magnonic waveguide to open new perspectives for spin-wave based circuits. The system is composed of a spin transfer oscillator based on a vortex state which is dipolarly coupled to a nanoscale spin-wave waveguide with longitudinal magnetization. In its auto-oscillating regime, the oscillator emits coherent spin waves with tunable and controllable frequencies, directions, and amplitudes into the waveguide. We demonstrate the principle of this method using micromagnetic simulations and show that reconfiguration of the system is possible by changing the chirality and polarity of the magnetic vortex. Spin waves are emitted into the waveguide with high non-reciprocity and the preferred direction depends on the core polarity of the vortex. In contrast, different vortex chiralities lead to different amplitudes of the emitted waves. Our findings open up a novel way to design an agile spintronic device for the coherent and tunable generation of propagating spin waves.

Finite-element dynamic-matrix approach for propagating spin waves: Extension to mono- and multi-layers of arbitrary spacing and thickness

In our recent work [Körber et al., AIP Adv. 11, 095006 (2021)], we presented an efficient numerical method to compute dispersions and mode profiles of spin waves in waveguides with translationally invariant equilibrium magnetization. A finite-element method (FEM) allowed to model two-dimensional waveguide cross sections of arbitrary shape but only finite size. Here, we extend our FEM propagating-wave dynamic-matrix approach from finite waveguides to the important cases of infinitely extended mono- and multi-layers of arbitrary spacing and thickness. To obtain the mode profiles and frequencies, the linearized equation of the motion of magnetization is solved as an eigenvalue problem on a one-dimensional line-trace mesh, defined along the normal direction of the layers. Being an important contribution to multi-layer systems, we introduce interlayer exchange into our FEM approach. With the calculation of dipolar fields being the main focus, we also extend the previously presented plane-wave Fredkin–Koehler method to calculate the dipolar potential of spin waves in infinite layers. The major benefit of this method is that it avoids the discretization of any non-magnetic material such as non-magnetic spacers in multilayers. Therefore, the computational effort becomes independent of the spacer thicknesses. Furthermore, it keeps the resulting eigenvalue problem sparse, which, therefore, inherits a comparably low arithmetic complexity. As a validation of our method (implemented into the open-source finite-element micromagnetic package TETRAX), we present results for various systems and compare them with theoretical predictions and with established finite-difference methods. We believe this method offers an efficient and versatile tool to calculate spin-wave dispersions in layered magnetic systems.

Nonlinear Bloch dynamics and spin-wave generation in a Bose-Hubbard ladder subject to effective magnetic field.

The two-leg magnetic ladder is the simplest and ideal model to reflect the coupling effects of lattice and magnetic field. It is of great significance to study some novel phases, topological characteristics, and chiral characteristics in condensed matter physics. In particular, the left-right leg degree of freedom can be regarded as a pseudospin, and the two-leg magnetic ladder also provides an ideal platform for the study of spin dynamics. Here the ground state, Bloch oscillations (BOs), and spin dynamics of the interacting two-leg magnetic ladder subject to an external linear force are studied by using variational approach and numerical simulation. In the absence of the external linear force, the critical condition of transition between the zero-momentum state and plane-wave state is obtained analytically, and the physical mechanism of the ground-state transition is revealed. When the external linear force presents, the occurrence of BOs excites the spin dynamics, and we reveal the chiral BOs and the accompanied spin dynamics of the system in different ground states. In particular, we further study the influence of periodically modulated linear force on BOs and spin dynamics. The frequencies of the linear force corresponding to the resonances and pseudoresonances are obtained analytically, which result in rich nonlinear dynamics. In resonances, stable and strong BOs (with larger amplitude) are observed. In pseudoresonances, because the pseudoresonance frequencies are related to the initial momentum and phase of the wave packet, a dispersion effect takes place and strong diffusion of wave packet occurs. When the frequency is nonresonant, drift and weak dispersion of wave packet occur simultaneously with the wave-packet oscillation. In all cases, the wave-packet dynamics is accompanied with periodic but anharmonic pseudospin oscillation. The BOs and spin dynamics are effectively controlled by periodically modulating the linear force.

Omnidirectional flat bands in chiral magnonic crystals

The magnonic band structure of two-dimensional chiral magnonic crystals is theoretically investigated. The proposed metamaterial involves a three-dimensional architecture, where a thin ferromagnetic layer is in contact with a two-dimensional periodic array of heavy-metal square islands. When these two materials are in contact, an anti-symmetric exchange coupling known as the Dzyaloshinskii–Moriya interaction (DMI) arises, which generates nonreciprocal spin waves and chiral magnetic order. The Landau–Lifshitz equation and the plane-wave method are employed to study the dynamic magnetic behavior. A systematic variation of geometric parameters, the DMI constant, and the filling fraction allows the examination of spin-wave propagation features, such as the spatial profiles of the dynamic magnetization, the isofrequency contours, and group velocities. In this study, it is found that omnidirectional flat magnonic bands are induced by a sufficiently strong Dzyaloshinskii–Moriya interaction underneath the heavy-metal islands, where the spin excitations are active. The theoretical results were substantiated by micromagnetic simulations. These findings are relevant for envisioning applications associated with spin-wave-based logic devices, where the nonreciprocity and channeling of the spin waves are of fundamental and practical scientific interest.

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

We report on propagating spin-wave spectroscopy measurements carried out on coplanar nano-antenna devices made from a Si/SiO2/Ru(5 nm)/Co(20)/Pt(5 nm) film. The measurements were analyzed in detail by employing newly developed theoretical modeling and de-embedding procedures. The magnetic parameters of the film were determined by complementary Brillouin light scattering and ferromagnetic resonance measurements. The propagating spin wave signals could be accounted for quantitatively for the range of externally applied magnetic fields investigated in this study: 130–1500 Oe.

Magnon Straintronics in the 2D van der Waals Ferromagnet CrSBr from First-Principles.

The recent isolation of two-dimensional (2D) magnets offers tantalizing opportunities for spintronics and magnonics at the limit of miniaturization. One of the key advantages of atomically thin materials is their outstanding deformation capacity, which provides an exciting avenue to control their properties by strain engineering. Herein, we investigate the magnetic properties, magnon dispersion, and spin dynamics of the air-stable 2D magnetic semiconductor CrSBr (TC = 146 K) under mechanical strain using first-principles calculations. Our results provide a deep microscopic analysis of the competing interactions that stabilize the long-range ferromagnetic order in the monolayer. We showcase that the magnon dynamics of CrSBr can be modified selectively along the two main crystallographic directions as a function of applied strain, probing the potential of this quasi-1D electronic system for magnon straintronics applications. Moreover, we predict a strain-driven enhancement of TC by ∼30%, allowing the propagation of spin waves at higher temperatures.

Film-penetrating transducers applicable to on-chip reservoir computing with spin waves

We have proposed a spin-wave transducer structure named film-penetrating transducers (FPTs). FPTs penetrate an on-chip magnetic film for a spin-wave transmission medium and allow flexible spatial arrangements of many exciters/detectors due to their zero-dimensional feature. We constructed four device models with different spatial arrangements of FPT/conventional exciters using a 10-nm-thick ferrimagnetic garnet film with a central FPT detector. We performed numerical experiments that combine electromagnetics with micromagnetics including thermal noise at 300 K. We evaluated important device features of FPTs, such as the signal-to-noise ratios (SNRs), input/output signal transmission efficiencies, and nonlinear phenomena of spin waves. We applied in-phase sinusoidal input currents with various amplitudes and frequencies and altered the damping strengths near the film boundaries. We obtained sufficient SNRs for the practical use of FPTs and revealed that FPTs have both higher transmission efficiencies and nonlinear strengths than conventional antennas, as the input frequency approaches the ferromagnetic resonance frequency of the film. Moreover, we observed and analyzed various nonlinear phenomena of spin waves, including beats in the time-domain waveform, components of integer harmonic frequencies, wide-range scatterings of inter-harmonic frequencies, and frequency doubling in spin precession. These characteristics probably originate from various device effects: FPTs effectively excite dipolar spin waves with large-angle precession, propagating spin waves reflect from the film boundaries, and spin waves dynamically and nonlinearly interfere with each other. This study demonstrated that FPTs have promising features for both their applications to reservoir computing and the studies on the physics of nonlinear and space-varying spin waves.

Spin-wave frequency combs

We experimentally demonstrate the generation of spin-wave frequency combs based on the nonlinear interaction of propagating spin waves in a microstructured waveguide. By means of time- and space-resolved Brillouin light scattering spectroscopy, we show that the simultaneous excitation of spin waves with different frequencies leads to a cascade of four-magnon scattering events, which ultimately results in well-defined frequency combs. Their spectral weight can be tuned by the choice of amplitude and frequency of the input signals. Furthermore, we introduce a model for stimulated four-magnon scattering, which describes the formation of spin-wave frequency combs in the frequency and time domain.

Voltage-induced terahertz magnon excitation associated with antiferromagnetic domain wall precession

Exploring a low-power method for exciting terahertz (THz) signals is of great interest in developing devices with ultrafast signal processing. We numerically investigated the THz magnon excitation from a moving antiferromagnetic (AFM) domain wall (DW) driven by a voltage-induced magnetic anisotropy energy gradient (dEa/dx). This magnon excitation originates from the DW precession induced by dEa/dx. Unlike the AFM DW precession triggered by DW acceleration as predicted in previous investigation, the dEa/dx-induced DW precession is possible when the DW acceleration is negligible. When the frequency of DW precession is greater than the frequency gap of spin-wave propagation in the AFM medium, THz magnons are resonantly excited. Because dEa/dx can be generated under a moderate DC voltage, our work provides a potential approach for developing THz spintronic devices with a low power and dissipation.

Dynamic coupling and spin-wave dispersions in a magnetic hybrid system made of an artificial spin-ice structure and an extended NiFe underlayer

We present a combined experimental and numerical study of the spin-wave dispersion in a NiFe artificial spin-ice (ASI) system consisting of an array of stadium-shaped nanoislands deposited on the top of a continuous NiFe film with non-magnetic spacer layers of varying thickness. The spin-wave dispersion, measured by wavevector resolved Brillouin light scattering spectroscopy in the Damon–Eshbach configuration, consists of a rich number of modes, with either stationary or propagating character. We find that the lowest frequency mode displays a bandwidth of ∼0.5 GHz, which is independent of the presence of the film underneath. On the contrary, the Brillouin light scattering intensity of some of the detected modes strongly depends on the presence of the extended thin-film underlayer. Micromagnetic simulations unveil the details of the dynamic coupling between the ASI lattice and film underlayer. Interestingly, the ASI lattice facilitates dynamics of the film either specific wavelengths or intensity modulation peculiar to the modes of the ASI elements imprinted in the film. Our results demonstrate that propagating spin waves can be modulated at the nanometer length scale by harnessing the dynamic mode coupling in the vertical, i.e., the out-of-plane direction of suitably designed magnonic structures.

Exceptional-Point Phase Transition in Coupled Magnonic Waveguides

Spontaneous symmetry breaking is a source of the phase transition in a variety of physical systems including such as non-Hermitian. It has recently been demonstrated that so-called exceptional points, which are singularities in non-Hermitian systems, separate the areas with symmetrical and asymmetrical eigenstates. Such separation is evidence of phase transitions in non-Hermitian systems taking place at exceptional points. In this paper, we experimentally demonstrate that exceptional-point phase transition in a system of two coupled magnonic waveguides coupled by dipole-dipole interaction occurs. Using infrared laser irradiation to control the spin-wave losses in one of the waveguides covered by a semiconductor $\mathrm{GaAs}$ layer, we observe a change in the wave numbers of the propagating spin waves and a change in the distribution of spin-wave intensity between the waveguides. Retrieving the experimental data, we find the spin-wave propagation parameters and, based on these and the use of coupled mode theory, we argue that in the system of magnonic waveguides the exceptional-point phase transition is observed.

Spin-wave nonreciprocity and formation of lateral standing spin waves in CoFeB/Ta/NiFe meander-shaped films

Studying the spin-wave (SW) propagation in 3D periodic structures opens new possibilities for joining functional units placed on the different layers of the magnonic circuitry. In the path toward 3D magnonics, the main challenge is the fabrication of large-scale 3D magnetic structures with nanometric precision control of geometry and material composition. In this work, we study the dependence on the Ta spacer thickness of the magnonic band structure, measured by Brillouin light scattering spectroscopy, of CoFeB/Ta/NiFe meander-shaped bilayers fabricated on pre-patterned Si substrate with thickness steps of 50 nm. Both propagating and stationary SW modes are observed. While the frequency of the dispersive mode slightly depends on the Ta spacer thickness, the frequency position of the three stationary modes in the lowest frequency range of the spectra significantly increases by increasing the Ta thickness. Micromagnetic calculations indicate that each of the three stationary modes is composed of a doublet of modes whose frequency separation, within each doublet, increases by increasing the mode frequency. The origin of this frequency separation is ascribed to the dynamic dipolar coupling between the magnetic layers that generate a significant frequency nonreciprocity of counterpropagating SWs. For these reasons, the investigated structures offer potential application as the nonreciprocal versatile interconnections performing the frequency selective regimes of signal propagation in magnonic circuits.

Hybridized propagating spin waves in a CoFeB/IrMn bilayer

In this work, we report the propagating spin waves hybridized between first-order and quasiuniform modes in a $\mathrm{Co}_{20}\mathrm{Fe}_{60}\mathrm{B}_{20}$ thin film capped by $\mathrm{Ir}_{25}\mathrm{Mn}_{75}$. The anticrossing gaps are observed at room temperature both in the reflection and transmission spectra, where the coupling strength can be tuned by varying the value of the in-plane wave vector at which the dispersion curves cross. The key mechanism behind this feature is theoretically ascribed to the dipole-dipole interaction by a model which accounts for many features of our experimental results. The strong coupling with a cooperativity up to 2.0 is achieved with taking the dissipation rates of two coupled branches into account. A reference CoFeB sample without IrMn reveals that the interfacial pinning effect also plays a role in the magnon-magnon hybridization. Tunable hybridization of magnons may foster the field of magnonics in information processing.

Nonreciprocal propagation of spin waves in a bilayer magnonic waveguide based on yttrium-iron garnet films

Objectives. Nonreciprocal spin wave effects can manifest themselves in metalized films of ferrite garnets. By studying the dynamics of spin waves in micro- and nano-scale magnetic films, the possibility of using multilayer dielectric films of yttrium iron garnet (YIG) to ensure the manifestation of the nonreciprocity effect is demonstrated. This approach offers advantages compared to the use of a layered YIG/metal structure due to significantly lower spin-wave losses in the two-layer YIG film consisting of layers with different values of magnetization. Such films can be used in logical elements to create controllable Mach-Zehnder interferometers based on magnonic principles. The purpose of this work is to reconcile the concept of nonreciprocal spin-wave propagation of a signal with the simultaneous manifestation of the effects arising from the propagation of spin waves in microwave guides formed by finite-width YIG films.Methods. We used an experimental microwave spectroscopy method based on a vector network analyzer along with a finite difference method to perform a numerical simulation of the dispersion characteristics of spin waves in two-layer magnonic microwave guides. An analytical model was also used to obtain a dispersion equation based on the magnetostatic approximation.Results. Based on measurements of the amplitude and phase responses, the possible coexistence of two frequency ranges for the propagation of a spin-wave signal in a two-layer magnon microwave guide based on a YIG film formed by two layers with different values of saturation magnetization was demonstrated. Regimes of nonreciprocal propagation of a spin-wave signal were revealed. A numerical model was using to study the formation mechanisms of spin wave modes in the spectrum of a two-layer structure formed due to the finite dimensions of the microwave guide. An analytical model was used to evaluate the transformation of the mode spectrum. The experimental data are in good agreement with the results of the proposed numerical and analytical models.Conclusions. The possibility of frequency-selective propagation of spin waves in a magnon microwaveguide consisting of two layers with different saturation magnetization values is demonstrated. Multimode propagation of spin waves can occur inside a two-layer structure in two frequency ranges. At the same time, this process is accompanied by a strong nonreciprocity of spin-wave signal propagation, which manifests itself in a change in the amplitude and phase responses when the direction of the external magnetic field is reversed. The proposed two-layer spin-wave waveguide concept can be used in the manufacture of magnon interconnects and magnon interferometers with the support of multiband regimes of operation.

Imaging magnonic frequency multiplication in nanostructured antidot lattices

Frequency multiplication is an essential part of electronics and optics which led to numerous indispensable applications. In this paper, we utilize a combination of scanning transmission x-ray microscopy and micromagnetic simulations to directly image magnonic frequency multiplication by means of dynamic real-space magnetization measurements. We experimentally demonstrate frequency multiplication up to the seventh order, which enables the generation of nanoscale spin waves at $6\phantom{\rule{0.16em}{0ex}}\mathrm{G}\mathrm{Hz}$ with excitation frequencies of less than $1\phantom{\rule{0.16em}{0ex}}\mathrm{G}\mathrm{Hz}$. Good agreement between the experiment and micromagnetic simulations allows us to build a micromagnetic model capable of predicting conversion efficiencies and multiplexing capabilities of the system. Furthermore, simulations reveal that more than two rows of antidots do not increase the conversion efficiency substantially. By enabling magnonic multiplexing with low input frequencies while not exceeding the size of a few microns, the device will lead to numerous applications, further advancing the capabilities of magnonic data transmission.