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.

Non-reciprocal wave propagation arises in systems with broken time-reversal symmetry and is key to the functionality of devices, such as isolators or circulators, in microwave, photonic and acoustic applications. In magnetic systems, collective wave excitations known as magnon quasiparticles so far yielded moderate non-reciprocities, mainly observed by means of incoherent thermal magnon spectra, while their occurrence as coherent spin waves (magnon ensembles with identical phase) is yet to be demonstrated. Here, we report the direct observation of strongly non-reciprocal propagating coherent spin waves in a patterned element of a ferromagnetic bilayer stack with antiparallel magnetic orientations. We use time-resolved scanning transmission x-ray microscopy (TR-STXM) to directly image the layer-collective dynamics of spin waves with wavelengths ranging from 5 µm down to 100 nm emergent at frequencies between 500 MHz and 5 GHz. The experimentally observed non-reciprocity factor of these counter-propagating waves is greater than 10 with respect to both group velocities and specific wavelengths. Our experimental findings are supported by the results from an analytic theory and their peculiarities are further discussed in terms of caustic spin-wave focusing.

Additive nanotechnology enable curvilinear and three-dimensional (3D) magnetic architectures with tunable topology and functionalities surpassing their planar counterparts. Here, we experimentally reveal that 3D soft magnetic wireframe structures resemble compact manifolds and accommodate magnetic textures of high order vorticity determined by the Euler characteristic, $\chi$. We demonstrate that self-standing magnetic tetrapods (homeomorphic to a sphere; $\chi=+2$) support six surface topological solitons, namely four vortices and two antivortices, with a total vorticity of +2 equal to its Euler characteristic. Alternatively, wireframe structures with one loop (homeomorphic to a torus; $\chi = 0$) possess equal number of vortices and antivortices, which is relevant for spin-wave splitters and 3D magnonics. Subsequent introduction of $N$ holes into the wireframe geometry (homeomorphic to an $N$-torus; $\chi < 0$) enables the accommodation of a virtually unlimited number of antivortices, which suggests their usefulness for non-conventional (e.g. reservoir) computation. Furthermore, complex stray-field topologies around these objects are of interest for superconducting electronics, particle trapping and biomedical applications.

We present a study of the piezostrain-tunable gyrotropic dynamics in Co40Fe40B20 vortex microstructures fabricated on a 0.7Pb[Mg1/3Nb2/3]O3-0.3PbTiO3 single-crystal substrate. Using field-modulated-spin-rectification measurements, we demonstrate large frequency tunability (up to 45%) in individual microdisks accessed locally with low surface voltages, and magnetoresistive readout. With increased voltage applied to the substrate, we observe a gradual decrease of the vortex-core gyrotropic frequency associated with the contribution of the strain-induced magnetoelastic energy. The frequency tunability strongly depends on the disk size, with increased frequency downshift for disks with larger diameter. Micromagnetic simulations suggest that the observed size effects originate from the joint action of the strain-induced magnetoelastic and demagnetizing energies in large magnetic disks. These results enable a selective energy-efficient tuning of the vortex gyrotropic frequency in individual vortex-based oscillators with all-electrical operation.

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.

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.

We present results of direct maskless magnetic patterning of ferromagnetic nanostructures using a cobalt focused ion beam (FIB) system. The liquid metal ion source of the FIB was made of a Co36Nd64 alloy. A Wien mass filter allows for selecting the ion species. Using the FIB, we implanted narrow tracks of Co ions into a nominal 5000×1000×50 nm3 permalloy strip. We observed the Co-induced changes of the magnetic properties by measuring the sample with microresonator ferromagnetic resonance before and after the implantation. Regions as small as 50 nm can be implanted up to concentrations of at.-10 % near the surface. This allows for easy magnetic modification of edge-localized spin waves with a lateral resolution otherwise hard to reach. The direct-write maskless FIB process is quick and convenient for optical measurement techniques, as it does not involve the virtually impossible removal of ion-hardened resist masks one would face when using lithography with broad-beam ion implantation

Chiral effects originate from the lack of inversion symmetry within the lattice unit cell or sample’s shape. Being mapped onto magnetic ordering, chirality enables topologically non-trivial textures with a given handedness. Here, we demonstrate the existence of a static 3D texture characterized by two magnetochiral parameters being magnetic helicity of the vortex and geometrical chirality of the core string itself in geometrically curved asymmetric permalloy cap with a size of 80 nm and a vortex ground state. We experimentally validate the nonlocal chiral symmetry breaking effect in this object, which leads to the geometric deformation of the vortex string into a helix with curvature 3 μm−1 and torsion 11 μm−1. The geometric chirality of the vortex string is determined by the magnetic helicity of the vortex texture, constituting coupling of two chiral parameters within the same texture. Beyond the vortex state, we anticipate that complex curvilinear objects hosting 3D magnetic textures like curved skyrmion tubes and hopfions can be characterized by multiple coupled magnetochiral parameters, that influence their statics and field- or current-driven dynamics for spin-orbitronics and magnonics.

We present an experimental and numerical study of three-magnon splitting in a micrometer-sized magnetic disk with the vortex state strongly deformed by static in-plane magnetic fields. Excited with a large enough power at frequency fRF, the primary radial magnon modes of a cylindrical magnetic vortex can decay into secondary azimuthal modes via spontaneous three-magnon splitting. This nonlinear process exhibits selection rules leading to well-defined and distinct frequencies fRF/2±Δf of the secondary modes. Here, we demonstrate that three-magnon splitting in vortices can be significantly modified by deforming the magnetic vortex with in-plane magnetic fields, leading to a much richer three-magnon response. We find that, with increasing field, an additional class of secondary modes is excited which are localized to the highly-flexed regions adjacent to the displaced vortex core. While these modes satisfy the same selection rules of three-magnon splitting, they exhibit a much lower three-magnon threshold power compared to regular secondary modes of a centered vortex. The applied static magnetic fields are small (≃ 10 mT), providing an effective parameter to control the nonlinear spectral response of confined vortices. Our work expands the understanding of nonlinear magnon dynamics in vortices and advertises these for potential neuromorphic applications based on magnons.

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.

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.

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.

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.

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.

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.

The influence of the geometrical curvature of chiral magnetic films on the static and dynamic properties of hosted skyrmions are studied theoretically. We predict the effects of the curvature-induced drift of skyrmions under the action of the curvature gradients without any external stimuli. The strength of the curvature-induced driving force essentially depends on the skyrmion type, N\'eel or Bloch, while the trajectory of motion is determined by the type of magnetic ordering: ferro- or antiferromagnetic. During the motion along the surface, skyrmions experience deformations which depend on the its type. In the small-curvature limit, using the collective-variable approach we show, that the driving force acting on a N{\'e}el skyrmion is linear with respect to the gradient of the mean curvature. The driving acting on a Bloch skyrmion is much smaller: it is proportional to the product of the mean curvature and its gradient. In contrast to the fast N{\'e}el skyrmions, the dynamics of the slow Bloch skyrmions is essentially affected by the skyrmion profile deformation. For the sake of simplicity we restrict ourselves to the case of zero Gaussian curvature and consider cylindrical surfaces of general type. Equations of motion for ferromagnetic and antiferromagnetic skyrmions in curved magnetic films are obtained in terms of collective variables. All analytical predictions are confirmed by numerical simulations.

Traditionally, the primary field, where curvature has been at the heart of research, was the theory of general relativity. In recent studies, however, the
impact of curvilinear geometry enters various disciplines, ranging from solid-state physics over soft-matter physics, chemistry and biology to mathematics,
giving rise to a plethora of emerging domains such as curvilinear nematics, curvilinear studies of cell biology, curvilinear semiconductors,
superfluidity, optics, two-dimensional van der Waals materials, plasmonics, magnetism and superconductivity. Here, we summarize the state of the art
and outline prospects for future research in curvilinear solid-state systems exhibiting such fundamental cooperative phenomena as ferromagnetism,
antiferromagnetism and superconductivity. Highlighting the recent developments and current challenges in theory, fabrication and characterization of
curvilinear micro- and nanostructures, special attention is paid to perspective research directions entailing new physics and to their strong application
potential. Overall, the perspective is aimed at crossing the boundaries between the magnetism and superconductivity communities and drawing attention
to the conceptual aspects of how extension of structures into the third dimension and curvilinear geometry can modify existing and aid launching
novel functionalities. In addition, the perspective should stimulate the development and dissemination of R&D-oriented techniques to facilitate rapid
transitions from laboratory demonstrations to industry-ready prototypes and eventual products.

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

Under certain conditions, spin waves can be channeled into a broad angular spectrum of wave vectors, where the direction
of the group velocity becomes independent of those wave vectors. Such highly focused waves are called caustic waves,
whose properties can be manipulated by anisotropies or chiral interactions, like the Dzyaloshinskii-Moriya interaction. In this
paper, we theoretically study the focusing features of the spin waves induced by the dipole-dipole interaction in synthetic
antiferromagnets. For stacked systems, the dipolar interaction causes a noticeable frequency nonreciprocity when the
magnetizations in both films are antiparallelly aligned, and then the focusing properties of the spin waves are enhanced. The
role of thicknesses and magnetic graduation along the film's normal are systematically analyzed. We found that the degree
of focalization of the spin waves can be manipulated by increasing the layers' thickness. Also, we show that the low- and
high-frequency modes exhibit different focalization properties; the low-frequency mode manifests a similar behavior to the
heavy-metal/ferromagnet systems with interfacial Dzyaloshinskii-Moriya interaction, while the high-frequency one tends the
generate almost reciprocal interference patterns along one axis. In the case of magnetization-graded synthetic
antiferromagnets, we demonstrate that the graduation slightly influences the low-frequency mode, while the focusing and
nonreciprocal dynamic properties of the high-frequency ones are notoriously altered. The theoretical calculations are
compared with micromagnetic simulations, where a good agreement is found between both methods. Our results
demonstrate that a synthetic antiferromagnetic system allows for controlling the propagation of spin waves, assisting in the
transfer of angular momentum and energy.

We experimentally demonstrate the formation of room-temperature skyrmions with radii of about 25 nm in easy-plane anisotropy multilayers with an interfacial Dzyaloshinskii-Moriya interaction (DMI). We detect the formation of individual magnetic skyrmions by magnetic force microscopy and find that the skyrmions are stable in out-of-plane fields up to about 200 mT. We determine the interlayer exchange coupling as well as the strength of the interfacial DMI. Additionally, we investigate the dynamic microwave spin excitations by broadband
magnetic resonance spectroscopy. From the uniform Kittel mode we determine the magnetic anisotropy and lowdamping α < 0.04. We also find clear magnetic resonance signatures in the nonuniform (skyrmion) state. Our findings demonstrate that skyrmions in easy-plane multilayers are promising for spin-dynamical applications.

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.

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.

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.

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.

Magnetization switching using in-plane charge current recently has been widely investigated in heavy metal/ferromagnet bilayers with the switching mechanism usually attributed to the action of the spin-orbit coupling. Here we study in-plane current induced magnetization switching in model epitaxial bilayers that consist of Au(001) and Fe(001) grown on MgO(001). We use the planar Hall effect combined with magnetooptical Kerr effect (MOKE) microscopy to investigate magnetic properties of the bilayers and current-induced switching. We show that a current density beyond 1.4×10^7 A/cm can be employed for reproducible electrical switching of the magnetization between multiple stable states that correspond to different arrangements of magnetic domains with magnetization direction along one of the in-plane easy magnetization axes of the Fe(001) film. Lower current densities result in stable intermediate transversal resistances which are interpreted based on MOKE-microscopy investigations as resulting from the current-induced magnetic domain structure that is formed in the area of the Hall cross. We find that the physical mechanism of the current-induced magnetization switching of the Au/Fe/MgO(001) system at room temperature can be fully explained by the Oersted field, which is generated by the charge current flowing mostly through the Au layer.

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 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.

Artificial spin ices are periodic arrangements of interacting nanomagnets that have been successfully used to investigate emergent phenomena in the presence of geometric frustration. Recently, it has become clear that artificial spin ices equally have the potential to be used as building blocks for creating functional materials, such as magnonic crystals and ratchets, in addition to supporting a large number of programmable magnetic states. In this context, we investigate the magnetization dynamics in a system exhibiting asymmetric magnetostatic interactions owing to locally broken structural symmetry. We find that this gives rise to a rich spectrum that can be tuned through an external field. We also determine the evolution of the observed excitation modes, starting with building blocks and evolving into larger arrays, highlighting the role of symmetry breaking in defining the mode spectrum of the system. Concurrently, the increasing complexity of the spectrum leads to the existence of a large number of modes over a narrow range of frequencies. These results contribute to the understanding of magnetization dynamics in spin ice systems beyond the kagome and square ice geometries with a view towards the realization of reconfigurable magnonic crystals based on spin ices.

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 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.

Long-range spin transport in magnetic systems can be achieved by means of exchange-mediated spin textures with robust topological winding - a phenomenon referred to as spin superfluidity. Its experimental signatures have been discussed in antiferromagnets which are nearly free of dipolar interaction. In ferromagnets, which present with non-negligible dipole fields, however, realization of such spin transport has remained a challenge. Using micromagnetic simulations, we investigate exchange-mediated spin transport in extended thin ferromagnetic films. We uncover a two-fluidstate, in which the long-range spin transport by spin textures co-exists with and is stabilized by spin waves, as well as a soliton-screened spin transport regime at high spin injection biases. Both states are associated with distinct spin texture reconstructions near the spin injection region and sustain spin transport over large distances.

Here, we present an analytical and numerical model describing the magnetization dynamics in MgO-based spin-torque nano-oscillators with an in-plane magnetized polarizer and an out-of-plane free layer. We introduce the spin-transfer torque asymmetry by considering the cosine angular dependence of the magnetoresistance between the two magnetic layers in the stack. For the analytical solution, dynamics are determined by assuming a circular precession trajectory around the direction perpendicular to the plane, as set by the effective field, and calculating the energy integral over a single precession period. In a more realistic approach, we include the bias dependence of the tunnel magnetoresistance, which is assumed empirically to be a piecewise linear function of the applied voltage. The dynamical states are found by solving the stability condition for the Jacobian matrix for out-of-plane static states. We find that the bias dependence of the tunnel magnetoresistance, which is an inseparable effect in every tunnel junction, exhibits drastic impact on the spin-torque nano-oscillator phase diagram, mainly by increasing the critical current for dynamics and quenching the oscillations at high currents. The results are in good agreement with our experimental data published elsewhere.

Spin waves have the potential to be used as a next-generation platform for data transfer and processing as they can reach wavelengths in the nanometer range and frequencies in the terahertz range. To realize a spin-wave device, it is essential to be able to manipulate the amplitude as well as the phase of spin waves. Several theoretical and recent experimental works have also shown that the spin-wave phase can be manipulated by the transmission through a domain wall (DW). Here, we study propagation of spin waves through a DW by means of micro-focused Brillouin light scattering microscopy (μBLS). The 2D spin-wave intensity maps reveal that spin-wave transmission through a Néel DW is influenced by a topologically enforced circular Bloch line in the DW center and that the propagation regime depends on the spin-wave frequency. In the first regime, two spin-wave beams propagating around the circular Bloch line are formed, whereas in the second regime, spin waves propagate in a single central beam through the circular Bloch line. Phase-resolved μBLS measurements reveal a phase shift upon transmission through the domain wall for both regimes. Micromagnetic modeling of the transmitted spin waves unveils a distortion of their phase fronts, which needs to be taken into account when interpreting the measurements and designing potential devices. Moreover, we show that, by means of micromagnetic simulations, an external magnetic field can be used to move the circular Bloch line within the DW and to manipulate spin-wave propagation.
The authors thank R. Schäfer and O. Fruchart for the discussions on the DW classification.
This research was supported by the CEITEC Nano+ project (No. CZ.02.1.01/0.0/0.0/16013/0001728) and Austrian Science Fund (FWF) project I1937. M. Staňo acknowledges support by the ESF under the project CZ.02.2.69/0.0/0.0/19_074/0016239. CzechNanoLab project LM2018110 funded by MEYS CR is gratefully acknowledged for the financial support of the measurement and sample fabrication at the CEITEC Nano Research Infrastructure.

Spectrum of spin eigenmodes localized on a ferromagnetic skyrmion pinned by a geometrical defect (bump) of magnetic films is studied theoretically. By means of direct numerical solution of the corresponding eigenvalue problem and finite element micromagnetic simulations we demonstrate, that the curvature can induce localized modes with higher azimuthal and radial quantum numbers, which are absent for planar skyrmions (for the same parameters). The eigenfrequencies of all modes, except the breathing and gyromodes decreases with increasing curvature. Due to the translational symmetry break, the zero translational mode of the skyrmion gains a finite frequency and forms the gyromode, which describes the uniform rotation of skyrmions around the equilibrium position. In order to treat the gyromotion analytically we developed a Thiele-like collective variable approach. We show that Neel skyrmions in curvilinear films experience a driving force originating from the gradient of the mean curvature. The gyrofrequency of the pinned skyrmion is proportional to the second derivative of the mean curvature at the point of equilibrium.

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.

Arrays of interacting 2D nanomagnets display unprecedented electromagnetic properties via collective effects, demonstrated in artificial spin ices and magnonic crystals. Progress toward 3D magnetic metamaterials is hampered by two challenges: fabricating 3D structures near intrinsic magnetic length scales (sub-100 nm) and visualizing their magnetic configurations. Here, we fabricate and measure nanoscale magnetic gyroids, periodic chiral networks comprising nanowire-like struts forming three-connected vertices. Via block copolymer templating, we produce Ni75Fe25 single-gyroid and double-gyroid (an inversion pair of single-gyroids) nanostructures with a 42 nm unit cell and 11 nm diameter struts, comparable to the exchange length in Ni–Fe. We visualize their magnetization distributions via off-axis electron holography with nanometer spatial resolution and interpret the patterns using finite-element micromagnetic simulations. Our results suggest an intricate, frustrated remanent state which is ferromagnetic but without a unique equilibrium configuration, opening new possibilities for collective phenomena in magnetism, including 3D magnonic crystals and unconventional computing.

Nanoscale, low-phase noise, tunable transmitter-receiver links are key for enabling the progress of wireless communi-cation. We demonstrate that vortex-based spin-torque nano-oscillators, which are intrinsically low-noise devices due to their topologically-protected magnetic structure, can achieve frequency tunability when submitted to local ion im-plantation. In the experiments presented here, the gyrotropic mode is excited with spin-polarized alternating currents and anisotropic magnetoresistance measurements yield discreet frequencies from a single device. Indeed, chromium-implanted regions of permalloy disks exhibit different saturation magnetisation than the surrounding, non-irradiated areas, and thus different resonance frequency, corresponding to the specific area where the core is gyrating. Our study proves that such devices can be fabricated without the need of further lithographical steps, suggesting ion irradiation can be a viable and cost-effective fabrication method for densely-packed networks of oscillators.

The concept of curvature and chirality in space and time are foundational for the understanding of the organic life and formation of matter in the Universe. Chiral interactions but also curvature effects are tacitly accepted to be local. A prototypical condensed matter example is a local spin-orbit- or curvature-induced Rashba or Dzyaloshinskii-Moriya interactions. Here, we introduce a chiral effect, which is essentially non-local and resembles itself even in static spin textures living in curvilinear magnetic nanoshells. Its physical origin is the non-local magnetostatic interaction. To identify this interaction, we put forth a self-consistent micromagnetic framework of curvilinear magnetism. Understanding of the non-local physics of curved magnetic shells requires a curvature-induced geometrical charge, which couples the magnetic sub-system with the curvilinear geometry. The chiral interaction brings about a non-local chiral symmetry breaking effect: it introduces handedness in an intrinsically achiral material and enables the design of magnetolectric and ferrotoroidic responses.

In the last decade, two revolutionary concepts in nanomagnetism emerged from research for storage technologies and advanced information processing. The first suggests the use of magnetic domain walls in ferromagnetic nanowires to permanently store information in domain-wall racetrack memories. The second proposes a hardware realization of neuromorphic computing in nanomagnets using nonlinear magnetic oscillations in the gigahertz range. Both ideas originate from the transfer of angular momentum from conduction electrons to localized spins in ferromagnets, either to push data encoded in domain walls along nanowires or to sustain magnetic oscillations in artificial neurones. Even though both concepts share a common ground, they live on very different timescales which rendered them incompatible so far. Here, we bridge both ideas by demonstrating the excitation of magnetic auto-oscillations inside nanoscale domain walls using pure spin currents. This Letter will shed light on the current characteristic and spatial distribution of the excited auto-oscillations.

The main origin of the chiral symmetry breaking and, thus, for the magnetochiral effects in magnetic materials is associated with an antisymmetric exchange interaction, the intrinsic Dzyaloshinskii-Moriya interaction (DMI). Recently, numerous inspiring theoretical works predict that the bending of a thin film to a curved surface is often sufficient to induce similar chiral effects. However, these originate from the exchange or magnetostatic interactions and can stabilize noncollinear magnetic structures or influence spin-wave propagation. Here, we demonstrate that curvature-induced chiral effects are experimentally observable rather than theoretical abstraction and are present even in conventional soft ferromagnetic materials. We show that, by measuring the depinning field of domain walls in the simplest possible curve, a flat parabolic stripe, the effective exchange-driven DMI interaction constant can be quantified. Remarkably, its value can be as high as the interfacial DMI constant for thin films and can be tuned by the parabola’s curvature.

Nanoscale modifications of strain and magnetic anisotropy can open pathways to engineering magnetic domains for device applications. A periodic magnetic domain structure can be stabilized in sub-200 nm wide linear as well as curved magnets, embedded within a flat non-ferromagnetic thin film. The nanomagnets are produced within a non-ferromagnetic B2-ordered Fe60Al40 thin film, where local irradiation by a focused ion beam causes the formation of disordered and strongly ferromagnetic regions of A2 Fe60Al40. An anisotropic lattice relaxation is observed, such that the in-plane lattice parameter is larger when measured parallel to the magnet short-axis as compared to its length. This in-plane structural anisotropy manifests a magnetic anisotropy contribution, generating an easy-axis parallel to the short axis. The competing effect of the strain and shape anisotropies stabilizes a periodic domain pattern, in linear as well as spiral nanomagnets, providing a versatile and geometrically controllable path to engineering the strain and thereby the magnetic anisotropy at the nanoscale.

The magnetization state of 1D magnetic nanoparticle (NP) chains plays a key role in a wide range of applications ranging from diagnosis and therapy in medicine to actuators, sensors, and quantum recording media. The interplay between the exact particle orientation and the magnetic anisotropy is in turn crucial for controlling the overall magnetization state with high precision. Here, a 3D description of the magnetic structure of one-NP-wide chains is reported. Here, two complementary experimental techniques are combined, magnetic force microscopy (MFM) and electronic holography (EH) which are sensitive to out-of-plane and in-plane magnetization components, respectively. The approach to micromagnetic simulations is extended, which provides results in good agreement with MFM and EH. The findings are at variance with the known results on unidirectional NP assemblies, and show that magnetization is rarely strictly collinear to the chain axis. The magnetic structure of one-NP-wide chains can be interpreted as head-to-head magnetic domain structures with off-axis magnetization components, which is very sensitive to morphological defects in the chain structure such as minute size variation of NPs, tiny misalignment of NPs, and/or crystal orientation with respect to easy magnetization axis.

The magnetization dynamics of individual Fe-filled multiwall carbon-nanotubes (FeCNT), grown by chemical vapor deposition, are investigated by microresonator ferromagnetic resonance (FMR) and Brillouin light scattering (BLS) microscopy and corroborated by micromagnetic simulations. Up to now, only static magnetometry measurements are available. They suggest that the FeCNTs consist of a single-crystalline Fe nanowire throughout the length. The number and structure of the FMR lines and the abrupt decay of the spin-wave transport seen in BLS indicate, however that the Fe filling is not a single straight piece along the length. Therefore a stepwise cutting procedure was applied in order to investigate the evolution of the ferromagnetic resonance lines as a function of the nanowire length. Our results show that the FeCNT is indeed not homogeneous along the full length but is built from 300-400 nm long single-crystalline segments. These segments consist of magnetically high quality Fe nanowires with almost the bulk values of Fe and with a similar small damping in relation to thin films, promoting the FeCNTs as appealing candidates for spin-wave transport in magnonic applications.

Here, we demonstrate numerically that shape anisotropy in MgO-based spin-torque nano-oscillators consisting of an out-of-plane magnetized free layer and an in-plane polarizer is necessary to stabilize out-of-plane magnetization precession without the need of external magnetic fields. As the in-plane anisotropy is increased, a gradual tilting of the magnetization towards the in-plane easy direction is introduced, favouring zero-field dynamics over static in-plane states. Above a critical value, zero-field dynamics are no longer observed. The optimum ratio of in-plane shape to out-of-plane uniaxial anisotropy, for which large angle out-of-plane zero-field dynamics occur within the widest current range, is reported.

We present the generation of whispering gallery magnons with unprecedented high wave vectors via nonlinear 3-magnon scattering in a μm-sized magnetic vortex state disc. These modes exhibit a strong localisation at the perimeter of the disc and practically zero amplitude in an extended area around the vortex core. They originate from the splitting of the fundamental radial magnon modes, which can be resonantly excited in a vortex texture by an out-of-plane microwave field. We shed light on the basics of this non-linear scattering mechanism from experimental and theoretical point of view. Using Brillouin light scattering (BLS) microscopy, we investigated the frequency and power dependence of the 3-magnon splitting. The spatially resolved mode
profiles give evidence for the localisation at the boundaries of the disc and allow for a direct determination of the modes wavenumber.

Curvature effects in magnetism offer appealing possibilities to obtain new magnetic textures at the nanoscale due to the interplay between exchange and magnetostatic interactions. Experimentally and theoretically, curvature driven changes of static magnetic properties in parabolic nanostripes have been addressed here. The shape of a parabolic stripe is tuned to cover broad range of widths and curvatures allowing to construct a phase diagram of magnetic equilibrium states. For this, joint experimental, i.e., soft X-ray imaging, and theoretical studies are carried out. Analytical calculations in the framework, when non-local magnetostatic effects are neglected, coincide with the experimental and simulation results in a broad range of parameters. The results give confidence in the applicability of the existing theoretical framework for further analytical considerations of equilibrium magnetization states of curvilinear nanomagnets.

Spin-transfer torques (STTs) can be exploited in order to manipulate the magnetic moments of nanomagnets, thus allowing for new consumer-oriented devices to be designed. Of particular interest here are tuneable radio-frequency (RF) oscillators for wireless communication. Currently, the structure that maximizes the output power is an Fe/MgO/Fe-type magnetic tunnel junction (MTJ) with a fixed layer magnetized in the plane of the layers and a free layer magnetized perpendicular to the plane. This structure allows for most of the tunnel magnetoresistance (TMR) to be converted into output power. Here, we experimentally and theoretically demonstrate that the main mechanism sustaining steady-state precession in such structures is the angular dependence of the magnetoresistance. The TMR of such devices is known to exhibit a broken-linear dependence versus the applied bias. Our results show that the TMR bias dependence effectively quenches spin-transfer-driven precession and introduces a non-monotonic frequency dependence at high applied currents. Thus we expect the bias dependence of the TMR to have an even more dramatic effect in MTJs with Mn-Ga-based free layers, which could be used to design wireless oscillators extending towards the ‘THz gap’, but have been experimentally shown to exhibit a non-trivial TMR bias dependence.

Spin waves offer intriguing novel perspectives for computing and signal processing, since their damping can be lower than the Ohmic losses in conventional CMOS circuits. For controlling the spatial extent and propagation of spin waves on the actual chip, magnetic domain walls show considerable potential as magnonic waveguides. However, low-loss guidance of spin waves, in particular around angled tracks, remains to be shown. Here we experimentally demonstrate that such advanced control of propagating spin waves can be obtained using natural features of magnetic order in an interlayer exchange-coupled, anisotropic ferromagnetic bilayer. Using Scanning Transmission X-Ray Microscopy, we image the generation of spin wave and their subsequent propagation across distances exceeding multiple times the wavelength, in extended planar geometries as well as along one-dimensional domain walls, which can be straight and curved. These results show routes towards the practical implementation of magnonic waveguides employing domain walls in future spin wave logic and computational circuits.

The curvature of a magnetic membrane was presented as a means of inducing nonreciprocities in the spin-wave (SW) dispersion relation [see Otalora et al. Phys. Rev. Lett. 117, 227203 (2017) and Otalora et al. Phys. Rev. B 95, 184415 (2017)], thereby expanding the toolbox for controlling SWs. In this paper, we further complement this toolbox by analytically showing that the membrane curvature is also manifested in the absorption of SWs, leading to a difference in the frequency linewidth (or lifetime) of counterpropagating magnons. Herein, we studied the nanotubular case, predicting changes of approximately greater than 10% and up to 20% in the frequency linewidth of counterpropagating SWs for a wide range of nanotube radii ranging from 30 nm to 260 nm and with a thickness of 10 nm. These percentages are comparable to those that can be extracted from experiments on heavy metal/magnetic metal sandwiches, wherein linewidth asymmetry results from an interfacial Dzyaloshinskii-Moriya interaction (DMI). We also show that the interplay between the frequency linewidth and group velocity leads to asymmetries in the SW decay length, presenting changes between 10% and 22% for counterpropagating SWs in the frequency range of 2-10 GHz. For the case of the SW dispersion relation, the predicted effects are identified as the classical dipole-dipole interaction, and the analytical expression of the frequency linewidth has the same mathematical form as in thin films with the DMI. Furthermore, we present limiting cases of a tubular geometry with negligible curvature such that our analytical model converges to the case of a planar thin film known from the literature. Our findings represent a step forward toward the realization of three-dimensional curvilinear magnonic devices.

We present a detailed study on the static magnetic properties of individual permalloy nanotubes (NTs) with hexagonal cross-sections. Anisotropic magnetoresistance (AMR) measurements and scanning transmission X-ray microscopy (STXM) are used to investigate their magnetic ground states and its stability. We find that the magnetization in zero applied magnetic field is in a very stable vortex state. Its origin is attributed to a strong growth-induced anisotropy with easy axis perpendicular to the long axis of the tubes. AMR measurements of individual NTs in combination with micromagnetic simulations allow the determination of the magnitude of the growth-induced anisotropy for different types of NT coatings. We show that the strength of the anisotropy can be controlled by introducing a buffer layer underneath the magnetic layer. The magnetic ground states depend on the external magnetic field history and are directly imaged using STXM. Stable vortex domains can be introduced by external magnetic fields and can be erased by radio-frequency magnetic fields applied at the center of the tubes via a strip line antenna.

Typically, the chiral magnetic Skyrmion is a single-state excitation. Here we propose a system, where multiplet of Skyrmion states appears and one of these states can be the ground one. We show that the presence of a localized curvilinear defect drastically changes the magnetic properties of a thin perpendicularly magnetized ferromagnetic film. For a large enough defect amplitude a discrete set of equilibrium magnetization states appears forming a ladder of energy levels. Each equilibrium state has either a zero or a unit topological charge; i.e., topologically trivial and Skyrmion multiplets generally appear. Transitions between the levels with the same topological charge are allowed and can be utilized to encode and switch a bit of information. There is a wide range of geometrical and material parameters, where the Skyrmion level has the lowest energy. Thus, periodically arranged curvilinear defects can result in a Skyrmion lattice as the ground state.