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The integration of heterogeneous modular units for building large-scale quantum networks requires engineering mechanisms that allow a suitable transduction of quantum information. Magnon-based transducers are especially attractive due to their wide range of interactions and rich nonlinear dynamics, but most of the work to date has focused on linear magnon transduction in the traditional system composed of yttrium iron garnet and diamond, two materials with difficult integrability into wafer-scale quantum circuits. In this work, we present a different approach by utilizing wafer-compatible materials to engineer a hybrid transducer that exploits magnon nonlinearities in a magnetic microdisc to address quantum spin defects in silicon carbide. The resulting interaction scheme points to the unique transduction behavior that can be obtained when complementing quantum systems with nonlinear magnonics.

Over the past few decades, extensive research has been conducted on magnetic vortices due to their fundamental physical properties and potential applications as magnetic storage devices or resonators. Information can be encoded in the polarity or gyrotropic motion of the vortex core. Moreover, magnetic vortices offer a versatile spectrum of radial and azimuthal magnon modes, which exhibit interesting linear and nonlinear dynamics. One notable example is three-magnon splitting, where one mode can spontaneously split into two secondary magnon modes when excited above a threshold power. Three-magnon splitting follows specific selection rules, with the split modes having distinct frequencies and mode numbers to fulfill energy and angular momentum conservation [1]. Magnetic vortices offer the potential to stimulate these processes below their intrinsic threshold powers [2, 3], making them promising candidates for novel computing approaches such as reservoir computing. In this study, we demonstrate that the application of in-plane magnetic fields in the order of a few mT can efficiently modify three-magnon splitting [4]. Using micromagntic simulations and Brillouin-light-scattering microscopy, we show that the deformation of the vortex results in additional secondary butterfly modes that follow the same selection rules as the regular modes but exhibit different localization and much lower three-magnon splitting threshold powers. [1] K. Schultheiss et al. PRL 122, 097202 (2019) [2] L. Körber et al. PRL 125, 207203 (2020) [3] L. Körber et al. arXiv 2211.02328 (2022) [4] L. Körber et al. APL 122, 092401 (2023)

Magnons are elementary excitations in magnetic materials and undergo nonlinear multimode scattering processes at large input powers. In experiments and simulations, we show that the interaction between magnon modes of a confined magnetic vortex can be harnessed for pattern recognition. We study the magnetic response to signals comprising sine wave pulses with frequencies corresponding to radial mode excitations. Three-magnon scattering results in the excitation of different azimuthal modes, whose amplitudes depend strongly on the input sequences. We show that recognition rates above 95\% can be attained for four-symbol sequences using the scattered modes, with strong performance maintained with the presence of amplitude noise in the inputs.

We use a pure spin current originating from the spin Hall effect to generate a spin-orbit torque strongly reducing the effective damping in an adjacent ferromagnet. Because of additional microwave excitation, large spin-wave amplitudes are achieved exceeding the threshold for four-magnon scattering, thus resulting in additional spin-wave signals at discrete frequencies. Two or more modes are generated below and above the directly pumped mode with equal frequency spacing. It is shown how this nonlinear process can be controlled in magnonic waveguides by the applied dc current and the microwave pumping power. The sudden onset of the nonlinear effect after exceeding the thresholds can be interpreted as a spiking phenomenon, which makes the effect potentially interesting for neuromorphic computing applications. Moreover, we investigated this effect under microwave frequency and external field variation. The appearance of the additional modes was investigated in the time domain, revealing a time delay between the directly excited and the simultaneously generated nonlinear modes. Furthermore, spatially resolved measurements show different spatial decay lengths of the directly pumped mode and nonlinear modes.

Soft magnetic dots in the form of thin rings have unique topological properties. They can be in a vortex state with no vortex core. Here, we study the magnon modes of such systems both analytically and numerically. In an external magnetic field, magnetic rings are characterized by easy-cone magnetization and shows a giant splitting of doublets for modes with the opposite value of the azimuthal mode quantum number. The effect of the splitting can be refereed as a magnon analog of the topology-induced Aharonov-Bohm effect. For this we develop an analytical theory to describe the non-monotonic dependence of the mode frequencies on the azimuthal mode number, influenced by the balance between the local exchange and non-local dipole interactions.

Magnons are elementary excitations in magnetic materials and undergo nonlinear multimode scattering processes at large input powers. In experiments and simulations, we show that the interaction between magnon modes of a confined magnetic vortex can be harnessed for pattern recognition. We study the magnetic response to signals comprising sine wave pulses with frequencies corresponding to radial mode excitations. Three-magnon scattering results in the excitation of different azimuthal modes, whose amplitudes depend strongly on the input sequences. We show that recognition rates above 95\% can be attained for four-symbol sequences using the scattered modes, with strong performance maintained with the presence of amplitude noise in the inputs.

1D spin-wave conduits are envisioned as nanoscale components of magnonics-based logic and computing schemes for future generation electronics. A-la-carte methods of versatile control of the local magnetization dynamics in such nanochannels are highly desired for efficient steering of the spin waves in magnonic devices. Here, we present a study of localized dynamical modes in 1-$\mu$m-wide Permalloy conduits probed by microresonator ferromagnetic resonance technique. We clearly observe the lowest-energy edge mode in the microstrip after its edges were finely trimmed by means of focused Ne+ ion irradiation. Furthermore, after milling the microstrip along its long axis by focused ion beams, creating consecutively ~50 and ~100 nm gaps, additional resonances emerge and are attributed to modes localized at the inner edges of the separated strips. To visualize the mode distribution, spatially resolved Brillouin light scattering microscopy was used showing an excellent agreement with the ferromagnetic resonance data and confirming the mode localization at the outer/inner edges of the strips depending on the magnitude of the applied magnetic field. Micromagnetic simulations confirm that the lowest-energy modes are localized within $\sim$15-nm-wide regions at the edges of the strips and their frequencies can be tuned in a wide range (up to 5 GHz) by changing the magnetostatic coupling (i.e. spatial separation) between the microstrips.

Spin-based technologies can operate at terahertz frequencies but require manipulation techniques that work at ultrafast timescales to become practical. For instance, devices based on spin waves, also known as magnons, require efficient generation of high-energy exchange spin waves at nanometre wavelengths. To achieve this, a substantial coupling is needed between the magnon modes and an electro-magnetic stimulus such as a coherent terahertz field pulse. However, it has been difficult to excite non-uniform spin waves efficiently using terahertz light because of the large momentum mismatch between the submillimetre-wave radiation and the nanometre-sized spin waves. Here we improve the light–matter interaction by engineering thin films to exploit relativistic spin–orbit torques that are confined to the interfaces of heavy metal/ferromagnet heterostructures. We are able to excite spin-wave modes with frequencies of up to 0.6 THz and wavelengths as short as 6 nm using broadband terahertz radiation. Numerical simulations demonstrate that the coupling of terahertz light to exchange-dominated magnons originates solely from interfacial spin–orbit torques. Our results are of general applicability to other magnetic multilayered structures, and offer the prospect of nanoscale control of high-frequency signals.

This data publication contains the numerical and experimental data for our paper "Modification of three-magnon splitting in a flexed magnetic vortex" submitted to Applied Physics Letters. The data contains mumax3 recipes, field- and power-dependent frequency spectra and spatial mode profiles of spin waves in a ferromagnetic disk in the vortex state. All files are sorted according to the figures in which they appear in the paper.

This data publication contains the data for our publication "Pattern recognition in reciprocal space with a magnon-scattering reservoir" published in Nature Communications. The dataset is structured in folders corresponding to the different figures in the paper. Folder Fig2 and Fig2 contain the experimental data measured with Brillouin-light-scattering microscopy. The files contain the data integrated for the measurement positions described in the methods section in a csv format. Forlder Fig4 contains the evaluated numerical data presented in the corresponding figure. The raw data generated with micromagnetic simulations is too large for this dataset and is available upon request by the authors.

This dataset contains the numerical and experimental data (both raw and evaluated), labbooks associated with the measurements for our paper published in Physical Review Applied.

1D spin-wave conduits are envisioned as nanoscale components of magnonics-based logic and computing schemes for future generation electronics. A-la-carte methods of versatile control of the local magnetization dynamics in such nanochannels are highly desired for efficient steering of the spin waves in magnonic devices. Here, we present a study of localized dynamical modes in 1-$\mu$m-wide Permalloy conduits probed by microresonator ferromagnetic resonance technique. We clearly observe the lowest-energy edge mode in the microstrip after its edges were finely trimmed by means of focused Ne+ ion irradiation. Furthermore, after milling the microstrip along its long axis by focused ion beams, creating consecutively ~50 and ~100 nm gaps, additional resonances emerge and are attributed to modes localized at the inner edges of the separated strips. To visualize the mode distribution, spatially resolved Brillouin light scattering microscopy was used showing an excellent agreement with the ferromagnetic resonance data and confirming the mode localization at the outer/inner edges of the strips depending on the magnitude of the applied magnetic field. Micromagnetic simulations confirm that the lowest-energy modes are localized within $\sim$15-nm-wide regions at the edges of the strips and their frequencies can be tuned in a wide range (up to 5 GHz) by changing the magnetostatic coupling (i.e. spatial separation) between the microstrips.

1D spin-wave conduits are envisioned as nanoscale components of magnonics-based logic and computing schemes for future generation electronics. A-la-carte methods of versatile control of the local magnetization dynamics in such nanochannels are highly desired for efficient steering of the spin waves in magnonic devices. Here, we present a study of localized dynamical modes in 1-$\mu$m-wide Permalloy conduits probed by microresonator ferromagnetic resonance technique. We clearly observe the lowest-energy edge mode in the microstrip after its edges were finely trimmed by means of focused Ne+ ion irradiation. Furthermore, after milling the microstrip along its long axis by focused ion beams, creating consecutively ~50 and ~100 nm gaps, additional resonances emerge and are attributed to modes localized at the inner edges of the separated strips. To visualize the mode distribution, spatially resolved Brillouin light scattering microscopy was used showing an excellent agreement with the ferromagnetic resonance data and confirming the mode localization at the outer/inner edges of the strips depending on the magnitude of the applied magnetic field. Micromagnetic simulations confirm that the lowest-energy modes are localized within $\sim$15-nm-wide regions at the edges of the strips and their frequencies can be tuned in a wide range (up to 5 GHz) by changing the magnetostatic coupling (i.e. spatial separation) between the microstrips.

In our recent work [L. Körber, AIP Advances 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 multilayers of arbitrary spacing and thickness. To obtain the mode profiles and frequencies, the linearized equation of 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 in multilayer 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 like non-magnetic spacers in multilayers. Therefore, the computational effort becomes independent on 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 \textsc{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.

We experimentally demonstrate the generation of spin-wave frequency combs based on the non- linear 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 excita- tion 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. Frequency

We investigate how geometry influences spin dynamics in polygonal magnetic nanotubes. We find that lowering the rotational symmetry of nanotubes, by decreasing the number of planar facets, splits an increasing number spin-wave modes, which are doubly degenerate in cylindrical tubes. This symmetry-governed splitting is distinct form the topological split recently observed in cylindrical nanotubes. Doublet modes, where the azimuthal period is half-integer or integer multiple of the number of facets, split to singlet pairs with lateral standing-wave profiles of opposing mirror-plane symmetries. Moreover, the polygonal geometry facilitates the hybridization of modes with different azimuthal periods but the same symmetry, manifested in avoided level crossings. These phenomena, unimaginable in cylindrical geometry, provide new tools to control spin dynamics on the nanoscale. Our concepts can be generalized to nano-objects of versatile geometries and order parameters, offering new routes to understand and engineer dynamic responses in mesoscale physics.

Surface curvature of magnetic systems can lead to many static and dynamic effects which are not present in flat systems of the same material. These emergent magnetochiral effects can lead to frequency nonreciprocity of spin waves, which has been shown to be a bulk effect of dipolar origin and is related to a curvature-induced symmetry breaking in the magnetic volume charges. So far, such effects have been investigated theoretically mostly for thin shells, where the spatial profiles of the spin waves can be assumed to be homogeneous along the thickness. Here, using a finite-element dynamic-matrix approach, we investigate the transition of the spin-wave spectrum from thin to thick curvilinear shells, at the example of magnetic nanotubes in the vortex state. With increasing thickness, we observe the appearance of higher-order radial modes which are strongly hybridized and resemble the perpendicular-standing-waves (PSSWs) in flat films. Along with an increasing dispersion asymmetry, we uncover the curvature-induced non-reciprocity of the mode profiles. This is explained in a very simple picture general for thick curvilinear shells, considering the inhomogeneity of the emergent geometric volume charges along the thickness of the shell. Such curvature-induced mode-profile asymmetry also leads to non-reciprocal hybridization which can facilitate unidirectional spin-wave propagation. With that, we also show how curvature allows for nonlinear three-wave splitting of a higher-order radial mode into secondary modes which can also propagate unidirectionally. We believe that our study provides a significant contribution to the understanding of the spin-wave dynamics in curvilinear magnetic systems, but also advertises these for novel magnonic applications.

Nonvolatile and energy-efficient spin-based technologies call for new prospects to realize computation and communication devices that are able to operate at terahertz (THz) frequencies. In particular, the coupling of electro-magnetic radiation to a spin system is a general requirement for future communication units and sensors. Here we propose a layered metallic system, based on a ferromagnetic film sandwiched in between two heavy metals that allows a highly effective coupling of millimeter wavelength THz light to nanometer-wavelength magnon modes. Using single-cycle broadband THz radiation we are able to excite spin-wave modes with a frequency of up to 0.6 THz and a wavelength as short as 6 nm. Our experimental and theoretical studies demonstrate that the coupling originates solely from interfacial spin-orbit torques. These results are of general applicability to magnetic multilayered structures, and offer the perspective of coherent THz excitation of exchange-dominated nanoscopic magnon modes.

Data were obtained by means of Brillouin light scattering microscopy, micro magnetic simulations in MuMax3 and analytic calculations.

This data set contains the numerical raw data for "Mode splitting of spin waves in magnetic nanotubes with discrete symmetries" published in Physical Review B. The data has been obtained using our in-house developed finite-element dynamic-matrix approach for propagating spin waves [see AIP Advances 11, 095006 (2021) for details].  The folder high-res-only-k0/ contains the lateral mode profiles and frequencies of polygonal nanotubes with different number of corners at k = 0. It also contains a python script used to calculate the microwave absorption from the mode profiles. The folders poly_*/ contain only the dispersion without lateral mode profiles for the different polygonal tubes as well as a round nanotube All folders contain the geometry and mesh files as well as sparam and eparam yaml files containg the material parameters and experimental parameters, respectively. The equilibrium states are saved as m_eq.h5 hdf5 files.

This dataset contains the numerically calculated data for our publication "Finite-element dynamic-matrix approach for propagating spin waves: Extension to mono- and multilayers of arbitrary spacing and thickness" published in AIP Advances. The data is structured as folders associated with each Figure showing any data. For cases in which different methods are compared, the data is separated into additional subfolders associated with each method (same naming scheme as in manuscript).

This dataset contains the numerical data for our publication "Curvilinear spin-wave dynamics beyond the thin-shell approximation: Magnetic nanotubes as a case study" published in Physical Review B. The data consists of dispersion, magnetization ground states and mode profiles of spin waves in vortex-state magnetic nanotubes of different thicknesses, and has been calculated with the TetraX micromagnetic modeling package. All calculations are described within each subfolder by a jupyter notebook.

This repository holds the first release of TetraX. Documentation: codebase.helmholtz.cloud/micromagnetic-modeling/tetrax TetraX is a package for finite-element-method (FEM) micromagnetic modeling of magnetization statics and dynamics with the aim to provide user friendly and flexible workflows. Apart from energy minimizers and an LLG solver, it provides implementations of several FEM dynamic-matrix approaches to numerically calculate the normal modes and associated frequencies for magnetic specimen of different geometries such as confined samples, infinitely long waveguides, or infinitely extended multilayers. Next to support for ferromagnets, TetraX will also provide the first full micromagnetic package for antiferromagnets. TetraX is a package for finite-element-method (FEM) micromagnetic modeling of magnetization statics and dynamics with the aim to provide user friendly and flexible workflows. Apart from energy minimizers and an LLG solver, it provides implementations of several FEM dynamic-matrix approaches to numerically calculate the normal modes and associated frequencies for magnetic specimen of different geometries such as confined samples, infinitely long waveguides, or infinitely extended multilayers. Next to support for ferromagnets, TetraX will also provide the first full micromagnetic package for antiferromagnets.

1D spin-wave conduits are envisioned as nanoscale components of magnonics-based logic and computing schemes for future generation electronics. A-la-carte methods of versatile control of the local magnetization dynamics in such nanochannels are highly desired for efficient steering of the spin waves in magnonic devices. Here, we present a study of localized dynamical modes in 1-$\mu$m-wide Permalloy conduits probed by microresonator ferromagnetic resonance technique. We clearly observe the lowest-energy edge mode in the microstrip after its edges were finely trimmed by means of focused Ne+ ion irradiation. Furthermore, after milling the microstrip along its long axis by focused ion beams, creating consecutively ~50 and ~100 nm gaps, additional resonances emerge and are attributed to modes localized at the inner edges of the separated strips. To visualize the mode distribution, spatially resolved Brillouin light scattering microscopy was used showing an excellent agreement with the ferromagnetic resonance data and confirming the mode localization at the outer/inner edges of the strips depending on the magnitude of the applied magnetic field. Micromagnetic simulations confirm that the lowest-energy modes are localized within $\sim$15-nm-wide regions at the edges of the strips and their frequencies can be tuned in a wide range (up to 5 GHz) by changing the magnetostatic coupling (i.e. spatial separation) between the microstrips.

We experimentally demonstrate the generation of spin-wave frequency combs based on the non- linear 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 excita- tion 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. Frequency

We experimentally demonstrate the generation of spin-wave frequency combs based on the non- linear 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 excita- tion 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. Frequency

In this paper, using micromagnetic simulations, we investigate the stress-induced frequency tunability of double-vortex nano-oscillators comprising magnetostrictive and non-magnetostrictive ferromagnetic layers separated vertically by a non-magnetic spacer. We show that the relative orientations of the vortex core polarities p1 and p2 have a strong impact on the eigen-frequencies of the dynamic modes. When the two vortices with antiparallel polarities have different eigen-frequencies and the magnetostatic coupling between them is sufficiently strong, the stress-induced magnetoelastic anisotropy can lead to the single-frequency resonant gyration mode of the two vortex cores. Additionally, for the case of parallel polarities, we demonstrate that for sufficiently strong magnetostatic coupling, the magnetoelastic anisotropy leads to the coupled vortex gyration in the chaotic regime and to the lateral separation of the vortex core trajectories. These findings offer a path for achieving a fine control over gyration frequencies and trajectories in vortex-based oscillators via adjustable elastic stress, which can be easily generated and tuned electrically, mechanically or optically.

We present a numerical approach to obtain the Ferromagnetic Resonance (FMR) spectra of micrometer- and nano-sized magnetic elements by micromagnetic simulations. Mimicking common experimental conditions, a static magnetic field is applied and a linearly polarized oscillating magnetic field is used to excite magnetization dynamics. A continuous single-frequency excitation is utilized, which permits to study the steady-state dynamics in space- and time-domain. This gives direct access to resonance fields, line widths and relative amplitudes as observed in the experiments, which is not easily accessible in pulsed schemes and allows for a one-to-one identification between simulation and experiment. Similar to numerical approaches using pulsed excitations the phases, ellipticity and spatial mode profiles of the spin-wave excitations may also be accessed. Using large excitation powers we then showcase that one can additionally study nonlinear responses by this method such as the nonlinear shift of the resonance fields and the fold-over of the absorption lines. Since the dynamic susceptibility is directly determined from standard outputs of common micromagnetic codes, the presented method is robust, efficient and easy-to-use, adding to its practical importance.

We present a numerical approach to obtain the Ferromagnetic Resonance (FMR) spectra of micrometer- and nano-sized magnetic elements by micromagnetic simulations. Mimicking common experimental conditions, a static magnetic field is applied and a linearly polarized oscillating magnetic field is used to excite magnetization dynamics. A continuous single-frequency excitation is utilized, which permits to study the steady-state dynamics in space- and time-domain. This gives direct access to resonance fields, line widths and relative amplitudes as observed in the experiments, which is not easily accessible in pulsed schemes and allows for a one-to-one identification between simulation and experiment. Similar to numerical approaches using pulsed excitations the phases, ellipticity and spatial mode profiles of the spin-wave excitations may also be accessed. Using large excitation powers we then showcase that one can additionally study nonlinear responses by this method such as the nonlinear shift of the resonance fields and the fold-over of the absorption lines. Since the dynamic susceptibility is directly determined from standard outputs of common micromagnetic codes, the presented method is robust, efficient and easy-to-use, adding to its practical importance.

Analytic and numerical studies on curved magnetic nano-objects predict numerous exciting effects that can be referred to as magneto-chiral effects, which do not originate from intrinsic Dzyaloshinskii–Moriya interaction or interface-induced anisotropies. In constrast, these chiral effects stem from isotropic exchange or dipole-dipole interaction, present in all magnetic materials, which acquire asymmetric contributions in case of curved geometry of the specimen. As a result, for example, the spin-wave dispersion in round magnetic nanotubes becomes asymmetric, namely spin waves of the same frequency propagating in opposite directions along the nanotube exhibit different wavelenghts. Here, using time-resolved scanning transmission X-ray microscopy experiments, standard micromagntic simulations and a dynamic-matrix approach, we show that the spin-wave spectrum undergoes additional drastic changes when transitioning from a continuous to a discrete rotational symmetry, i.e. from round to hexagonal nanotubes, which are much easier to fabricate. The polygonal shape introduces localization of the modes both to the sharp, highly curved corners and flat edges. Moreover, due to the discrete rotational symmetry, the degenerate nature of the modes with azimuthal wave vectors known from round tubes is partly lifted, resulting in singlet and duplet modes. For comparison with our experiments, we calculate the microwave absorption from the numerically obtained mode profiles which shows that a dedicated antenna design is paramount for magnonic applications in 3D nano-structures. To our knowledge these are the first experiments directly showing real space spin-wave propagation in 3D nano objects.

Spin waves (SWs) in magnetic nanotubes have shown interesting nonreciprocal properties in their dispersion relation, group velocity, frequency linewidth, and attenuation lengths. The reported chiral effects are similar to those induced by the Dzyaloshinskii–Moriya interaction but originating from the dipole–dipole interaction. Here, we show that the isotropic-exchange interaction can also induce chiral effects in the SW transport; the so-called Berry phase of SWs. We demonstrate that with the application of magnetic fields, the nonreciprocity of the different SW modes can be tuned between the fully dipolar governed and the fully exchange governed cases, as they are directly related to the underlying equilibrium state. In the helical state, due to the combined action of the two effects, every single sign combination of the azimuthal and axial wave vectors leads to different dispersions, allowing for a very sophisticated tuning of the SW transport. A disentangle- ment of the dipole–dipole and exchange contributions so far was not reported for the SW transport in nanotubes. Furthermore, we propose a device based on coplanar waveguides that would allow to selectively measure the exchange or dipole induced SW nonreciprocities. In the context of magnonic applications, our results might encourage further developments in the emerging field of 3D magnonic devices using curved magnetic membranes.

Spin waves (SWs) in magnetic nanotubes have shown interesting nonreciprocal properties in their dispersion relation, group velocity, frequency linewidth, and attenuation lengths. The reported chiral effects are similar to those induced by the Dzyaloshinskii–Moriya interaction but originating from the dipole–dipole interaction. Here, we show that the isotropic-exchange interaction can also induce chiral effects in the SW transport; the so-called Berry phase of SWs. We demonstrate that with the application of magnetic fields, the nonreciprocity of the different SW modes can be tuned between the fully dipolar governed and the fully exchange governed cases, as they are directly related to the underlying equilibrium state. In the helical state, due to the combined action of the two effects, every single sign combination of the azimuthal and axial wave vectors leads to different dispersions, allowing for a very sophisticated tuning of the SW transport. A disentangle- ment of the dipole–dipole and exchange contributions so far was not reported for the SW transport in nanotubes. Furthermore, we propose a device based on coplanar waveguides that would allow to selectively measure the exchange or dipole induced SW nonreciprocities. In the context of magnonic applications, our results might encourage further developments in the emerging field of 3D magnonic devices using curved magnetic membranes.

Modern problems in magnetization dynamics require more and more the numerical determination of the spin-wave spectra and -dispersion in magnetic systems where analytic theories are not yet available. Micromagnetic simulations can be used to compute the spatial mode profiles and oscillation frequencies of spin-waves in magnetic system with almost arbitrary geometry and different magnetic interactions. Although numerical approaches are very versatile, they often do not give the same insight and physical understanding as analytical theories. For example, it is not always possible to decide whether a certain feature (such as dispersion asymmetry, for example) is governed by one magnetic interaction or the other. Moreover, since numerical approaches typically yield the normal modes of the system, it is not always feasible to disentangle hybridized modes. In this manuscript, we build a bridge between numerics and analytics by presenting a methodology to calculate the individual contributions to general spin-wave dispersions in a fully numerical manner. We discuss the general form of any spin-wave dispersion in terms of the effective (stiffness) fields produced by the modes. Based on a special type of micromagnetic simulation, the numerical dynamic-matrix approach, we show how to calculate each stiffness field in the respective dispersion law, separately for each magnetic interaction. In particular, it becomes possible to disentangle contributions of different magnetic interactions to the dispersion asymmetry in systems where non-reciprocity is present. At the same time, dipolar-hybridized modes can be easily disentangled. Since this methodology is independent of the geometry or the involved magnetic interactions at hand, we believe it is attractive for experimental and theoretical studies of magnetic systems where there are no analytics available yet, but also to aid the development of new analytical theories.

We present a numerical approach to efficiently calculate spin-wave dispersions and spatial mode profiles in magnetic waveguides of arbitrarily shaped cross section with any non-collinear equilibrium magnetization which is translationally invariant along the waveguide. Our method is based on the propagating-wave dynamic-matrix approach by Henry et al. and extends it to arbitrary cross sections using a finite-element method. We solve the linearized equation of motion of the magnetization only in a single waveguide cross section which drastically reduces computational effort compared to common three-dimensional micromagnetic simulations. In order to numerically obtain the dipolar potential of individual spin-wave modes, we present a plane-wave version of the hybrid finite-element/boundary-element method by Frekdin and Koehler which, for the first time, we extend to a modified version of the Poisson equation. Our method is applied to several important examples of magnonic waveguides including systems with surface curvature, such as magnetic nanotubes, where the curvature leads to an asymmetric spin-wave dispersion. In all cases, the validity of our approach is confirmed by other methods. Our method is of particular interest for the study of curvature-induced or magnetochiral effects on spin-wave transport but also serves as an efficient tool to investigate standard magnonic problems.

We present an experimental study of time refraction of spin waves propagating in microscopic waveguides under the influence of time-varying magnetic fields. Using space- and time-resolved Brillouin light scattering microscopy, we demonstrate that the broken translational symmetry along the time coordinate can be used to in- or decrease the energy of spin waves during their propagation. This allows for a broadband and controllable shift of the spin-wave frequency. Using an integrated design of spin-wave waveguide and microscopic current line for the generation of strong, nanosecond-long, magnetic field pulses, a conversion efficiency up to 39% of the carrier spin-wave frequency is achieved, significantly larger compared to photonic systems. Given the strength of the magnetic field pulses and its strong impact on the spin-wave dispersion relation, the effect of time refraction can be quantified on a length scale comparable to the spin-wave wavelength. Furthermore, we utilize time refraction to excite spin-wave bursts with pulse durations in the nanosecond range and a frequency shift depending on the pulse polarity.

We use vector Hamiltonian formalism (VHF) to study theoretically three-magnon parametric interaction (or three-wave splitting) in a magnetic disk existing in a magnetic vortex ground state. The three-wave splitting in a disk is found to obey two selection rules: (i) conservation of the total azimuthal number of the resultant spin-wave modes, and (ii) inequality for the radial numbers of interacting modes, if the mode directly excited by the driving field is radially symmetric (i.e. if the azimuthal number of the directly excited mode is m=0). The selection rule (ii), however, is relaxed in the "small" magnetic disks, due to the influence of the vortex core. We also found, that the efficiency of the three-wave interaction of the directly excited mode strongly depends on the azimuthal and radial mode numbers of the resultant modes, that becomes determinative in the case when several splitting channels (several pairs of resultant modes) simultaneously approximately satisfy the resonance condition for the splitting. The good agreement of the VHF analytic calculations with the experiment and micromagnetic simulations proves the capability of the VHF formalism to predict the actual splitting channels and the magnitudes of the driving field thresholds for the three-wave splitting.

We present an experimental study of time refraction of spin waves propagating in microscopic waveguides under the influence of time-varying magnetic fields. Using space- and time-resolved Brillouin light scattering microscopy, we demonstrate that the broken translational symmetry along the time coordinate can be used to in- or decrease the energy of spin waves during their propagation. This allows for a broadband and controllable shift of the spin-wave frequency. Using an integrated design of spin-wave waveguide and microscopic current line for the generation of strong, nanosecond-long, magnetic field pulses, a conversion efficiency up to 39% of the carrier spin-wave frequency is achieved, significantly larger compared to photonic systems. Given the strength of the magnetic field pulses and its strong impact on the spin-wave dispersion relation, the effect of time refraction can be quantified on a length scale comparable to the spin-wave wavelength. Furthermore, we utilize time refraction to excite spin-wave bursts with pulse durations in the nanosecond range and a frequency shift depending on the pulse polarity.

We use vector Hamiltonian formalism (VHF) to study theoretically three-magnon parametric interaction (or three-wave splitting) in a magnetic disk existing in a magnetic vortex ground state. The three-wave splitting in a disk is found to obey two selection rules: (i) conservation of the total azimuthal number of the resultant spin-wave modes, and (ii) inequality for the radial numbers of interacting modes, if the mode directly excited by the driving field is radially symmetric (i.e. if the azimuthal number of the directly excited mode is m=0). The selection rule (ii), however, is relaxed in the "small" magnetic disks, due to the influence of the vortex core. We also found, that the efficiency of the three-wave interaction of the directly excited mode strongly depends on the azimuthal and radial mode numbers of the resultant modes, that becomes determinative in the case when several splitting channels (several pairs of resultant modes) simultaneously approximately satisfy the resonance condition for the splitting. The good agreement of the VHF analytic calculations with the experiment and micromagnetic simulations proves the capability of the VHF formalism to predict the actual splitting channels and the magnitudes of the driving field thresholds for the three-wave splitting.

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This repository contains the data used to determine the numerical threshold fields for three-magnon scattering in a vortex-state disk used in our paper "Theory of three-magnon interaction in a vortex-state magnetic nanodot" published in Physical Review B.  For different excitation frequencies, we provide: mumax3 simulation file and table containing the time-dependent magnetic energy for simulations where the microwave-field power is decreased over time mumax3 file and resulting power spectrum for a continuous-wave excitation at a given microwave-field power above threshold spatial mode profiles (magnitude/amplitude) to identify the modes taking part in the three-magnon splitting channel

This repository contains data generated to showcase our developed micromagnetic method described in our paper "Finite-element dynamic-matrix approach for spin-wave dispersions in magnonic waveguides with arbitrary cross section". For each example in Section V, we provide .clc and .spc files containing important material and experimental parameters .geo files containing the sample geometry (gmsh files) .csv files containing the dispersions calculated using our eigensolver    only for example A: linescans of the mode profiles along the width of the waveguide only for example B: .dat file of the obtained mumax dispersion

We experimentally demonstrate the generation of spin-wave frequency combs based on the non- linear 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 excita- tion 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. Frequency

Theoretical and numerical studies on curved magnetic nano-objects predict numerous exciting effects that can be referred to as magneto-chiral effects, which do not originate from the intrinsic Dzyaloshinskii-Moriya interaction or surface-induced anisotropies. The origin of these chiral effects is the isotropic exchange or the dipole-dipole interaction present in all magnetic materials but renormalized by the curvature. Here, we demonstrate experimentally that curvature induced effects originating from the dipole-dipole interaction are directly observable by measuring spin-wave propagation in magnetic nanotubes with hexagonal cross section using time resolved scanning transmission X-ray microscopy. We show that the dispersion relation is asymmetric upon reversal of the wave vector when the propagation direction is perpendicular to the static magnetization. Therefore counter-propagating spin waves of the same frequency exhibit different wavelenghts. Hexagonal nanotubes have a complex dispersion, resulting from spin-wave modes localised to the flat facets or to the extremely curved regions between the facets. The dispersion relations obtained experimentally and from micromagnetic simulations are in good agreement. %The asymmetric spin-wave transport is present for all modes, promoting hexagonal nanotubes for magnonic applications. These results show that spin-wave transport is possible in 3D, and that the dipole-dipole induced magneto-chiral effects are significant.

One of the fascinating qualities of spin waves (or magnons), which are the elementary excitations in magnetically ordered substances, is their nonlinear behavior at moder- ate excitation powers. This makes spin waves not only an attractive model system to study general nonlinear systems, but it also provides a way to utilize nonlinear dynamics in possible technical applications. In a ferromagnetic nano disk which is magnetized in the vortex state, the spin-wave modes meet strict boundary conditions and therefore inherit a discrete spectrum. When driven with a large enough excitation field, they can decay into other spin-wave modes within well-defined channels due to a nonlinear process called three-magnon scattering [1]. The aim of this project is to explore this phenomenon within nonlinear spin-wave theory and by means of micromagnetic simulations. For this purpose, first the linear dynamics are mapped out and the possible scattering channels are predicted. The stability of these channels with respect to static external fields is studied. Within this context, exotic spin-wave modes which arise in a broken cylindrical symmetry are found. Moreover, a model to predict the temporal evolution of the spin-wave modes is developed within the classical Hamiltonian formalism for nonlinear spin-wave dynamics. Together with micromagnetic simulations, this model is applied in order to study the power-dependence of three-magnon scattering as well as to uncover a phenomenon called stimulated three-magnon scattering, which may allow for an integration of this nonlinear pro- cess in magnonics circuits. The results are compared with Brillouin light-scattering experiments which were conducted prior to this thesis or were motivated by it. Financial support of within DFG programme SCHU 2922/1-1 and KA 5069/1-1 is acknowledged.

Theoretical and numerical studies on curved magnetic nano-objects predict numerous exciting effects that can be referred to as magneto-chiral effects, which do not originate from the intrinsic Dzyaloshinskii-Moriya interaction or surface-induced anisotropies. The origin of these chiral effects is the isotropic exchange or the dipole-dipole interaction present in all magnetic materials but renormalized by the curvature. Here, we demonstrate experimentally that curvature induced effects originating from the dipole-dipole interaction are directly observable by measuring spin-wave propagation in magnetic nanotubes with hexagonal cross section using time resolved scanning transmission X-ray microscopy. We show that the dispersion relation is asymmetric upon reversal of the wave vector when the propagation direction is perpendicular to the static magnetization. Therefore counter-propagating spin waves of the same frequency exhibit different wavelenghts. Hexagonal nanotubes have a complex dispersion, resulting from spin-wave modes localised to the flat facets or to the extremely curved regions between the facets. The dispersion relations obtained experimentally and from micromagnetic simulations are in good agreement. %The asymmetric spin-wave transport is present for all modes, promoting hexagonal nanotubes for magnonic applications. These results show that spin-wave transport is possible in 3D, and that the dipole-dipole induced magneto-chiral effects are significant.

We present a combined numerical, theoretical and experimental study on stimulated three-magnon splitting in a magnetic disk in the vortex equilibrium state. Our micromagnetic simulations and Brillouin-light-scattering results confirm that three-magnon splitting can be triggered even below threshold by exciting one of the secondary modes by magnons propagating in a waveguide next to the disk. The experiments show that stimulation is possible over an extended range of excitation powers and a wide range of frequencies around the eigenfrequencies of the secondary modes. Rate-equation calculations predict an instantaneous response to stimulation and the possibility to prematurely trigger three-magnon splitting even above threshold in a sustainable manner. These predictions are confirmed experimentally using time-resolved Brillouin-light-scattering measurements and are in a good qualitative agreement with the theoretical results. We believe that the controllable mechanism of stimulated three-magnon splitting could provide a possibility to utilize magnon-based nonlinear networks as hardware for reservoir or neuromorphic computing.

We present a combined numerical, theoretical and experimental study on stimulated three-magnon splitting in a magnetic disk in the vortex equilibrium state. Our micromagnetic simulations and Brillouin-light-scattering results confirm that three-magnon splitting can be triggered even below threshold by exciting one of the secondary modes by magnons propagating in a waveguide next to the disk. The experiments show that stimulation is possible over an extended range of excitation powers and a wide range of frequencies around the eigenfrequencies of the secondary modes. Rate-equation calculations predict an instantaneous response to stimulation and the possibility to prematurely trigger three-magnon splitting even above threshold in a sustainable manner. These predictions are confirmed experimentally using time-resolved Brillouin-light-scattering measurements and are in a good qualitative agreement with the theoretical results. We believe that the controllable mechanism of stimulated three-magnon splitting could provide a possibility to utilize magnon-based nonlinear networks as hardware for reservoir or neuromorphic computing.

We report on the impact of nonlinear four-magnon scattering on magnon transport in microstructured waveguides with low magnetic damping. Using microfocused Brillouin light scattering, we analyze magnon propagation lengths in a broad range of excitation powers and observe a decrease of the attenuation length at high powers, which is consistent with the onset of nonlinear four-magnon scattering. Hence, when measuring magnon propagation lengths and deriving damping parameters from those results, one needs to be careful to stay in the linear regime. Otherwise, the intrinsic nonlinearity of magnetization dynamics may lead to a misinterpretation of magnon propagation lengths and, thus, to wrong values of the magnetic damping of the system.

We report on the impact of nonlinear four-magnon scattering on magnon transport in microstructured waveguides with low magnetic damping. Using microfocused Brillouin light scattering, we analyze magnon propagation lengths in a broad range of excitation powers and observe a decrease of the attenuation length at high powers, which is consistent with the onset of nonlinear four-magnon scattering. Hence, when measuring magnon propagation lengths and deriving damping parameters from those results, one needs to be careful to stay in the linear regime. Otherwise, the intrinsic nonlinearity of magnetization dynamics may lead to a misinterpretation of magnon propagation lengths and, thus, to wrong values of the magnetic damping of the system.

One of the fascinating qualities of spin waves (or magnons), which are the elementary excitations in magnetically ordered substances, is their nonlinear behavior at moder- ate excitation powers. This makes spin waves not only an attractive model system to study general nonlinear systems, but it also provides a way to utilize nonlinear dynamics in possible technical applications. In a ferromagnetic nano disk which is magnetized in the vortex state, the spin-wave modes meet strict boundary conditions and therefore inherit a discrete spectrum. When driven with a large enough excitation field, they can decay into other spin-wave modes within well-defined channels due to a nonlinear process called three-magnon scattering [1]. The aim of this project is to explore this phenomenon within nonlinear spin-wave theory and by means of micromagnetic simulations. For this purpose, first the linear dynamics are mapped out and the possible scattering channels are predicted. The stability of these channels with respect to static external fields is studied. Within this context, exotic spin-wave modes which arise in a broken cylindrical symmetry are found. Moreover, a model to predict the temporal evolution of the spin-wave modes is developed within the classical Hamiltonian formalism for nonlinear spin-wave dynamics. Together with micromagnetic simulations, this model is applied in order to study the power-dependence of three-magnon scattering as well as to uncover a phenomenon called stimulated three-magnon scattering, which may allow for an integration of this nonlinear pro- cess in magnonics circuits. The results are compared with Brillouin light-scattering experiments which were conducted prior to this thesis or were motivated by it. Financial support of within DFG programme SCHU 2922/1-1 and KA 5069/1-1 is acknowledged.