Search

This archive contains the raw data as well as the Tetrax (www.tetrax.software) Jupyter notebooks to produce the data that has been analyzed and used for the manuscript: Curvature-induced parity loss and hybridization of magnons: Exploring the connection of flat and tubular magnetic shells, Physical Review B 110, 134428 (2024), published on 17 October, 2024.

This paper delves into the connection between flat and curvilinear magnetization dynamics. For this, we numerically study the evolution of the magnon spectrum of rectangular waveguides upon rolling its cross section up to a full tube. Magnon spectra are calculated over a wide range of magnetization states using a finite-element dynamic-matrix method, which allows us to trace the evolution of the magnon frequencies and several critical magnetic fields with increasing curvature. By analyzing the parity of the higher-order magnon modes, we find a curvature-induced mode heterosymmetry that originates from a chiral contribution to the exchange interaction and is related to the Berry phase of magnons in closed loops. Importantly, this curvature-induced parity loss has profound consequences for the linear coupling between different propagating magnons, allowing for hybridization between initially orthogonal modes. In this context, we demonstrate the integral role of edge modes in forming the magnon spectrum in full tubes. Our findings provide theoretical insights into curvilinear magnetization dynamics and are relevant for interpreting and designing experiments in the field.

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)

We present a systematic micromagnetic study of standing spin-wave modes in infinitely long Permalloy strips with rectangular cross-section. Using a finite-element dynamic-matrix method, we first calculate the eigenfrequencies and the corresponding eigenvectors (mode profiles), as a function of the in-plane magnetic field applied across the strip. The ferromagnetic resonance spectra is computed from the mode profiles, assuming a homogeneous radio-frequency excitation, equivalently to an experimental ferromagnetic resonance measurement. The investigation of the field-dependent mode profiles enables for the classification of the observed resonances, here focusing mostly on the \textit{true edge mode} localized at the vicinity of strip edges. Furthermore, we study the mode localization in pairs of 50-nm-thick Permalloy strips as a function of the strip width and their lateral separation. For closely spaced strips, the spatial profile of the quasi-uniform mode is substantially modified due to a significant hybridization with the edge-localized standing spin-wave modes of the neighbouring strip. We show that a wide-range-tunability of the localized edge-mode resonances can be achieved with a precise control of the magnetostatic coupling between the strips. Extreme sensitivity of the edge mode frequency on the bias field demonstrates a potential of the edge resonances for field sensing. Furthermore, for narrow strips ($\approx$100~nm in width), due to the reduced number of the allowed confined modes, a field-controllable switching between the resonances localized either in the strip center or at the edges of the strips can be achieved.

Perpendicular magnetic anisotropy thin film systems are well known for their periodic magnetic stripe domain structures. In this study, we focus on investigating the behavior of [Co(3.0 nm)/Pt(0.6 nm)]X multilayers within the transitional regime from preferred in-plane to out-of-plane magnetization orientation. Particularly, we examine the sample with X = 11 repetitions, which exhibits a remanent state characterized by a significant presence of both out-of-plane (OOP) and in-plane (IP) magnetization components, here referred to as the “tilted” stripe domain state. Vector vibrating sample magnetometry and magnetic force microscopy are used to investigate this specific sample and its unusual out-of-plane reversal behavior. Through experimental data analysis and micromagnetic simulations of the tilted magnetization system, we identify a single point of irreversibility during an out-of-plane external magnetic field sweep. This behavior is qualitatively similar to the reversal of a Stoner-Wohlfarth particle or of an IP magnetized disk with remanent vortex structure, since both show distinct points of irreversibility as well. Such a collective response to an external field is typically not observed in conventional OOP or IP systems, where the reversal process often involves independent nucleation, propagation, and annihilation of individual domains. Finally, we show that our findings are not at all restricted to Co/Pt multilayers, but are a quite general feature of transitional in-plane to out-of-plane magnetization systems.

We present a systematic micromagnetic study of standing spin-wave modes in infinitely long Permalloy strips with rectangular cross-section. Using a finite-element dynamic-matrix method, we first calculate the eigenfrequencies and the corresponding eigenvectors (mode profiles), as a function of the in-plane magnetic field applied across the strip. The ferromagnetic resonance spectra is computed from the mode profiles, assuming a homogeneous radio-frequency excitation, equivalently to an experimental ferromagnetic resonance measurement. The investigation of the field-dependent mode profiles enables for the classification of the observed resonances, here focusing mostly on the \textit{true edge mode} localized at the vicinity of strip edges. Furthermore, we study the mode localization in pairs of 50-nm-thick Permalloy strips as a function of the strip width and their lateral separation. For closely spaced strips, the spatial profile of the quasi-uniform mode is substantially modified due to a significant hybridization with the edge-localized standing spin-wave modes of the neighbouring strip. We show that a wide-range-tunability of the localized edge-mode resonances can be achieved with a precise control of the magnetostatic coupling between the strips. Extreme sensitivity of the edge mode frequency on the bias field demonstrates a potential of the edge resonances for field sensing. Furthermore, for narrow strips ($\approx$100~nm in width), due to the reduced number of the allowed confined modes, a field-controllable switching between the resonances localized either in the strip center or at the edges of the strips can be achieved.

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.

Perpendicular anisotropy thin film systems are well known for their periodic magnetic stripe domain structures. In this study, we focus on investigating the behavior of [Co(3.0 nm)/Pt(0.6 nm)]\textsubscript{$X$} multilayers within the transitional regime from preferred in-plane (IP) to out-of-plane (OOP) magnetization orientation, particularly, we examine the sample with $X=11$ repetitions, which exhibits a remanent state characterized by a significant presence of both OOP and IP magnetization components, here referred to as the "tilted" stripe domain state*. Using vibrating sample magnetometry, magnetic force microscopy and micromagnetic simulations we investigate this specific sample and find an unusual OOP field reversal behavior via a remanent parallel stripe domain state and a single point of irreversibility. While the reversal via distinct points of irreversibility is qualitatively similar to that of a nano-sized Stoner Wohlfarth particle or a vortex reversal in a micron-sized IP magnetized disk, our system is macroscopic. Finally, we show that this characteristic behavior is a rather general feature of transitional IP to OOP systems. \newline *[L. Fallarino et al., Phys. Rev. B 99, 024431 (2019)]

Periodic magnetic stripe domain patterns are a prominent feature of perpendicular anisotropy thin film systems. Here, we focus on the behavior of [Co(3.0 nm)/Pt(0.6 nm)]\textsubscript{$X$} multilayers within the transitional regime from preferred in-plane (IP), $X=6$, to out-of-plane (OOP), $X=22$, magnetization orientation, particularly, we examine a sample with $X=11$ repetitions, which exhibits a remanent state characterized by a significant presence of both OOP and IP magnetization components, here referred to as the "tilted" stripe domain state*. We investigate this specific sample with vibrating sample magnetometry, magnetic force microscopy and micromagnetic simulations, and find an unusual OOP field reversal behavior via a remanent parallel stripe domain state and a single point of irreversibility. Finally, we show that this characteristic reversal behavior is a rather general feature of transitional IP to OOP systems by comparing the Co/Pt multilayers with c-axis single Co thin films and Fe/Gd multilayers. \newline *[L. Fallarino et al., Phys. Rev. B 99, 024431 (2019)]

We present a systematic micromagnetic study of standing spin-wave modes in infinitely long Permalloy strips with rectangular cross-section. Using a finite-element dynamic-matrix method, we first calculate the eigenfrequencies and the corresponding eigenvectors (mode profiles), as a function of the in-plane magnetic field applied across the strip. The ferromagnetic resonance spectra is computed from the mode profiles, assuming a homogeneous radio-frequency excitation, equivalently to an experimental ferromagnetic resonance measurement. The investigation of the field-dependent mode profiles enables for the classification of the observed resonances, here focusing mostly on the \textit{true edge mode} localized at the vicinity of strip edges. Furthermore, we study the mode localization in pairs of 50-nm-thick Permalloy strips as a function of the strip width and their lateral separation. For closely spaced strips, the spatial profile of the quasi-uniform mode is substantially modified due to a significant hybridization with the edge-localized standing spin-wave modes of the neighbouring strip. We show that a wide-range-tunability of the localized edge-mode resonances can be achieved with a precise control of the magnetostatic coupling between the strips. Extreme sensitivity of the edge mode frequency on the bias field demonstrates a potential of the edge resonances for field sensing. Furthermore, for narrow strips ($\approx$100~nm in width), due to the reduced number of the allowed confined modes, a field-controllable switching between the resonances localized either in the strip center or at the edges of the strips can be achieved.

We demonstrate excitation of the gyrotropic mode in a magnetostrictive vortex by time-varying strain. The vortex dynamics is driven by a time-varying voltage applied to the piezoelectric substrate and detected electrically by spin rectification at subthreshold values of rf current. When the frequency of the time-varying strain matches the gyrotropic frequency at given in-plane magnetic field, the strain-induced in-plane magnetic anisotropy leads to a resonant excitation of the gyration dynamics in a magnetic vortex. We show that nonlinear gyrotropic dynamics can be excited already for moderate amplitudes of the time-varying strain.

We demonstrate excitation of the gyrotropic mode in a magnetostrictive vortex by time-varying strain. The vortex dynamics is driven by a time-varying voltage applied to the piezoelectric substrate and detected electrically by spin rectification at subthreshold values of rf current. When the frequency of the time-varying strain matches the gyrotropic frequency at given in-plane magnetic field, the strain-induced in-plane magnetic anisotropy leads to a resonant excitation of the gyration dynamics in a magnetic vortex. We show that nonlinear gyrotropic dynamics can be excited already for moderate amplitudes of the time-varying strain.

Dzyaloshinskii-Moriya interaction, also known as an antisymmetric exchange interaction, is the main source of chiral symmetry breaking effects in micromagnetic systems [1]. The later manifests itself in magnetic materials and layer stacks with structural space inversion symmetry breaking, where it leads to the formation of non-trivial chiral and topological spin textures (e.g. skyrmions, bubbles, homochiral spirals and domain walls). Such textures potentially could be utilized for prospective spintronic devices as a bit carrier. Still, tailoring of magnetochirality is only done by the selection of materials and adjustment of their composition in layer stacks. Alternatively, we demonstrate that space inversion symmetry breaking of the magnetic order parameter appears in geometrically curved systems [2]. In curvilinear ferromagnets, curvature governs the appearance of geometry-induced chiral and anisotropic responses, which introduce a new toolbox to create artificial chiral nanostructures from achiral magnetic materials suitable for the stabilization of non-trivial chiral textures [2,3]. Moreover, curvilinear geometry also leads to the appearance of non-local chiral effects, that arise from the asymmetry of the top and bottom surfaces and existence of both in- and out-of-plane magnetization components of different parity with respect to the reflection procedure [4]. Recently, we demonstrate the existence of non-local chiral effects in geometrically curved asymmetric permalloy cap with the vortex texture [5]. We find that the equilibrium vortex core obtain both bend and curling deformation, that are dependent on the geometric symmetries and magnetic parameters. References [1] A. Fert, N. Reyren and V. Cros, Nature Reviews Materials 2, 17031 (2017). [2] D. Makarov, O. M. Volkov, A. Kákay, O. V. Pylypovskyi, B. Budinská and O. V. Dobrovolskiy, Adv. Mater. 34, 2101758 (2021). [3] O. M. Volkov, A. Kákay, F. Kronast, I. Mönch, M.-A. Mawass, J. Fassbender and D. Makarov, Ph

Symmetry effects are fundamental in condensed matter physics as they define not only interactions but also resulting responses for the intrinsic order parameter depending on its transformation properties with respect to the operations of space and time reversal. Magnetic materials or layer stacks with structural space inversion symmetry breaking obtained much research attention due to the appearance of chiral Dzyaloshinskii-Moriya interaction (DMI) [1,2]. The latter manifests itself in the formation of non-trivial chiral and topological spin textures (e.g. skyrmions, bubbles, homochiral spirals and domain walls), that are envisioned to be utilized for prospective spintronic devices. At present, tailoring magnetochirality is done by the selection of materials and adjustment of their composition. Alternatively, space inversion symmetry breaking of the magnetic order parameter appears in geometrically curved systems [3]. In curvilinear ferromagnets, curvature governs the appearance of geometry-induced chiral and anisotropic responses, which introduce a new toolbox to create artificial chiral nanostructures from achiral magnetic materials suitable for the stabilization of non-trivial chiral textures [4,5]. Recently, much attention was dedicated to the exchange interaction, which enables curvature-induced extrinsic DMI as was proposed theoretically and validated experimentally for the case of conventional achiral magnetic materials [6]. Here, we demonstrate the existence of non-local chiral effects in geometrically curved asymmetric permalloy cap with the vortex texture. Using the full-scale simulation of the asymmetric nanodots we study how the vortex texture is changing with respect to the introduced sample asymmetry. We find that the equilibrium vortex core obtain both bend and curling deformation, that are dependent on the geometric symmetries and magnetic parameters. We relate the observed changes in the vortex string to the non-local chiral effects, that arise from

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)

We present results for direct maskless magnetic patterning of ferromagnetic nanostructures using a special liquid metal alloy ion source for focused ion beam (FIB) systems. We used a Co36Nd64 alloy as the FIB source [1]. A Wien mass filter allows for quick switching between the ion species in the alloy without changing the FIB source. A 5000×1000×50 nm3 permalloy strip served as the sample. Using the FIB we implanted a 300-nm-wide track with Co ions (see Fig.1). We observed the Co-induced changes by measuring the sample with microresonator ferromagnetic resonance before and after the implantation. Structures as small as 30 nm can be implanted up to a concentration of 10 % near the surface. Such lateral resolution is hard to reach for other lithographic methods. This allows for easy magnetic modification of edge-localized spin waves. In another set of samples, we implanted Dy ions to locally increase the damping in a stripe pattern of ~120-nm-wide strips with 400 nm periodicity on a total area of 1×1 mm². Thus, the Gilbert damping parameter can be easily increased by one order of magnitude with a lateral resolution of about 100 nm. In contrast to electron beam lithography in combination with broad-beam ion implantation, the maskless FIB process does not require the cumbersome and difficult removal of the ion-hardened resist if optical measurements like BLS or TR-MOKE are needed.

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.

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.

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.

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

Symmetry effects are key building blocks of condensed matter physics as they define not only interactions but also resulting re- sponses for the intrinsic order parameter. Namely, in magnetism geometric curvature governs the appearance of chiral and anisotropic responses [1], that introduce a new toolbox to create artificial chi- ral nanostructures from achiral magnetic materials [2,3]. Here, we demonstrate both theoretically and experimentally the existence of non-local chiral effects in geometrically curved asymmetric permalloy caps with the vortex texture. We find that the equilibrium vortex core obtain bend and curling deformation, that are dependent on the geometric symmetries and magnetic texture parameters. [1] D. D. Sheka et al., Comm. Phys. 3, 128 (2020). [2] O. M. Volkov et al., Phys. Rev. Lett, 123, 077201 (2019). [3] D. Makarov et al., Adv. Mater. 34, 2101758 (2022).

Dzyaloshinskii-Moriya interaction, also known as an antisymmetric exchange interaction, is the main source of chiral symmetry breaking effects in micromagnetic systems [1]. The later manifests itself in magnetic materials and layer stacks with structural space inversion symmetry breaking, where it leads to the formation of non-trivial chiral and topological spin textures (e.g. skyrmions, bubbles, homochiral spirals and domain walls). Such textures potentially could be utilized for prospective spintronic devices as a bit carrier. Still, tailoring of magnetochirality is only done by the selection of materials and adjustment of their composition in layer stacks. Alternatively, we demonstrate that space inversion symmetry breaking of the magnetic order parameter appears in geometrically curved systems [2]. In curvilinear ferromagnets, curvature governs the appearance of geometry-induced chiral and anisotropic responses, which introduce a new toolbox to create artificial chiral nanostructures from achiral magnetic materials suitable for the stabilization of non-trivial chiral textures [2,3]. Moreover, curvilinear geometry also leads to the appearance of non-local chiral effects, that arise from the asymmetry of the top and bottom surfaces and existence of both in- and out-of-plane magnetization components of different parity with respect to the reflection procedure [4]. Recently, we demonstrate the existence of non-local chiral effects in geometrically curved asymmetric permalloy cap with the vortex texture [5]. We find that the equilibrium vortex core obtain both bend and curling deformation, that are dependent on the geometric symmetries and magnetic parameters. References [1] A. Fert, N. Reyren and V. Cros, Nature Reviews Materials 2, 17031 (2017). [2] D. Makarov, O. M. Volkov, A. Kákay, O. V. Pylypovskyi, B. Budinská and O. V. Dobrovolskiy, Adv. Mater. 34, 2101758 (2021). [3] O. M. Volkov, A. Kákay, F. Kronast, I. Mönch, M.-A. Mawass, J. Fassbender and D. Makarov, Ph

Symmetry effects are fundamental in condensed matter physics as they define not only interactions but also resulting responses for the intrinsic order parameter depending on its transformation properties with respect to the operations of space and time reversal. Magnetic materials or layer stacks with structural space inversion symmetry breaking obtained much research attention due to the appearance of chiral Dzyaloshinskii-Moriya interaction (DMI) [1,2]. The latter manifests itself in the formation of non-trivial chiral and topological spin textures (e.g. skyrmions, bubbles, homochiral spirals and domain walls), that are envisioned to be utilized for prospective spintronic devices. At present, tailoring magnetochirality is done by the selection of materials and adjustment of their composition. Alternatively, space inversion symmetry breaking of the magnetic order parameter appears in geometrically curved systems [3]. In curvilinear ferromagnets, curvature governs the appearance of geometry-induced chiral and anisotropic responses, which introduce a new toolbox to create artificial chiral nanostructures from achiral magnetic materials suitable for the stabilization of non-trivial chiral textures [4,5]. Recently, much attention was dedicated to the exchange interaction, which enables curvature-induced extrinsic DMI as was proposed theoretically and validated experimentally for the case of conventional achiral magnetic materials [6]. Here, we demonstrate the existence of non-local chiral effects in geometrically curved asymmetric permalloy cap with the vortex texture. Using the full-scale simulation of the asymmetric nanodots we study how the vortex texture is changing with respect to the introduced sample asymmetry. We find that the equilibrium vortex core obtain both bend and curling deformation, that are dependent on the geometric symmetries and magnetic parameters. We relate the observed changes in the vortex string to the non-local chiral effects, that arise from

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

The archive contains the experimental (MR and EDFMR) and simulated (mumax3) data related to the publication "Piezostrain as a Local Handle to Control Gyrotropic Dynamics of Magnetic Vortices" (Physical Review Applied 20(2023), 024080).

We demonstrate excitation of the gyrotropic mode in a magnetostrictive vortex by time-varying strain. The vortex dynamics is driven by a time-varying voltage applied to the piezoelectric substrate and detected electrically by spin rectification at subthreshold values of rf current. When the frequency of the time-varying strain matches the gyrotropic frequency at given in-plane magnetic field, the strain-induced in-plane magnetic anisotropy leads to a resonant excitation of the gyration dynamics in a magnetic vortex. We show that nonlinear gyrotropic dynamics can be excited already for moderate amplitudes of the time-varying strain.

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.

The interplay between geometry and topology of the order parameter is crucial properties in soft and condensed matter physics, including cell membranes [1], nematic crystals [2,3], superfluids [4], semiconductors [5], ferromagnets [6] and superconductors [7]. Until recently, in the case of magentism, the influence of the geometry on the magnetization vector fields was addressed primarily by the design of the sample boundaries, aiming to tailor anisotropy of the samples. With the development of novel fabrication techniques allowing to realize complex 3D architectures, not only boundary effects, but also local curvatures can be addressed rigorously for the case of ferromagnets and antiferromagnets. It is shown that curvature governs the appearance of geometry-induced chiral and anisotropic responses [6-8]. Here we provide experimental confirmations of the existence of local and non-local curvature-induced chiral interactions of the exchange and magnetostatic origin in conventional soft ferromagnetic materials. Namely, we will present the experimental validation of the appearance of exchange-driven Dzyaloshinskii-Moriya interaction interaction (DMI, local effect) for the case of conventional achiral yet geometrically curved magnetic materials [9,10]. This curvature induced DMI is predicted to stabilize skyrmions [11] and skyrmionium states [12]. Furthermore, we will address the impact of nonlocal magnetostatic interaction on the properties of curvilinear ferromagnets, which enables the stabilization of topological magnetic textures [13,14], realization of high-speed magnetic racetracks [15] and curvature-induced asymmetric spin-wave dispersions in nanotubes [16]. Furthermore, symmetry analysis demonstrates the possibility to generate a fundamentally new chiral symmetry breaking effect, which is essentially nonlocal [13]. Thus, geometric curvature of thin films and nanowires is envisioned as a toolbox to create artificial chiral nanostructures from achiral magnetic mate

Perpendicular anisotropy thin film systems are well known for their highly periodic magnetic stripe domains. Here we study [Co(3.0 nm)/Pt(0.6 nm)]X multilayers in the regime of transitional in-plane to out-of-plane anisotropy. For this we vary the number of repeats X in order to tune the remanent state from purely in-plane (IP) via tilted stripe domains (tilted), i.e. with significant out-of plane as well as in-plane magnetization component, to fully out-of-plane stripe domains (OOP). Vibrating Sample Magnetometry and Magnetic Force Microscopy are used to investigate three characteristic samples with X = 6,11 and 22, which represent the three above mentioned remanent states, respectively. In contrast to fully in-plane or fully out-of-plane systems experimental data and corresponding micromagnetic simulation of the tilted magnetization regime (X=11) reveals fully reversible field regions as well as distinct points of irreversibility during an external field sweep. This collective reversal behavior seems at first sight somewhat counter intuitive for a macroscopic system and has qualitative similarities with microscopic systems, such as the Stoner Wohlfarth particle and the vortex reversal in an in-plane magnetized disk, which both show as well distinct points of irreversibility.

Conventional magnetic nanoscale devices are based on planar thin films and straight racetracks hosting magnetic topological solitons. Recent progress in fabrication and characterization methods allows to realise and study of complex-shaped planar and three-dimensional (3D) architectures. In the planar case, boundaries of nanodots lead to the formation of inhomogeneous textures, such as vortices and antivortices. In 3D, the magnetostatic interaction favours a spatially inhomogeneous shape anisotropy, which acts as easy-axis anisotropy along wires or hard axis of anisotropy perpendicular to the film surface. These interactions track the sample geometry and enable curvature-induced symmetry-breaking effects, such as topology-induced magnetization patterning and emergent anisotropic and chiral responses of the Dzyaloshinskii-Moriya interaction (DMI) type [1,2]. Curvature-induced magnetic responses can be classified as being local or nonlocal. In ferromagnets, local effects stem from the exchange interaction and DMI. The curvature-induced DMI originates from exchange: it is linear in curvatures and has the symmetry of the interfacial DMI. Its strength can be comparable with typical values of the intrinsic DMI. This is experimentally confirmed by the stabilization of chiral domain walls (CDW) on the apex of a Permalloy parabola-shaped stripe [3]. The strength of the CDW depinning field gives an estimation for the curvature-induced DMI constant and can be tuned by the geometry. In contrast to curvature itself, also curvature gradients offer a possibility to pin CDW, which was studied with an example of a circular indentation with a conic cross-section profile. This geometry supports circular CDWs described by the forced skyrmion equation, where the effective force acts as the stabilizing factor for large-radius skyrmion and skyrmionium states [4]. The magnetostatic interaction is a source of novel curvature-induced chiral effects, which are essentially nonlocal, in contras

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.

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.

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.

Following the recent developments in materials science and sample fabrication magnetic nanowires and nanotubes became an intensively studied research field in magnetism. However, it should be mentioned that the driving force behind can be attributed to the theoretical, both analytical and numerical predictions of novel magnetic textures and interesting features, such as chiral domain wall motion, the Spin-Cherenkov effect or the curvature-induced magnetochiral effects in general. In this chapter, the static properties of tubular nanomagnets will be reviewed, including magnetic configurations, domain walls, their types, and energetics as well as possible reversal mechanisms. The dynamical properties section is divided into two parts. The first part will guide you through the domain wall motion related to magnetochiral effects. The second part will discuss the general aspects of spin-wave propagation. Aspects, being static or dynamic, related to magnetochiral effects or curvature and topology will be addressed mostly. For those interested in a summary of experimental methods to fabricate tubular samples, an overview of all possible techniques one can use to characterize or measure magnetic tubes, or in a guide through all the analytical and numerical formalism developed to investigate the static and dynamic properties of magnetic nanotubes, we kindly ask to read these recently published excellent books by M. Vázquez [1, 2].

Magnetic nanostructures needed for magnonics and spintronics are usually processed by conventional lithography techniques in combination with lift-off or broad-beam ion etching. However, it has been shown [1] that the quality and shape of the structures’ edges play an important role for the magnetization dynamics as structures become smaller and smaller. Furthermore, regarding optical measurement techniques, hard-to-remove resist masks that become hardened by ion etching are problematic. Direct-writing focused ion beams (FIB) do not have these issues. In addition, using non-standard ion species opens various paths for local magnetic patterning, i.e., influencing the magnetic properties locally. I will present results for maskless magnetic patterning of ferromagnetic nanostructures using He and Ne ions as well as a few liquid metal alloy ion sources (LMAIS) for FIB systems. He/Ne FIBs are well established and commercially available. Irradiation of (paramagnetic) FeAl films by Ne ions creates local ferromagnetic nanostructures caused by disorder that are embedded in a paramagnetic matrix [2]. The precise Ne FIB also enables us to trim the edges of magnetic nanostructures enhancing their magnetic fidelity and creating certain localized magnon states at the edges of the samples. Using specifically developed LMAIS, like e.g., Co36Nd64, CoDy, or CuDy [3,4] in combination with a Wien mass filter offers further new paths for magnetic patterning. I will present results on the modification of Ni80Fe20 (permalloy) strip samples. Using the CoNd LMAIS a narrow track of Co ions was implanted. The induced magnetic changes were measured with microresonator ferromagnetic resonance (FMR) before and after the implantation. Structures as small as 30 nm can be implanted up to a concentration of 10 % near the surface. Such lateral resolution is hard to reach for other lithographic methods. Using Dy ions one can locally increase the Gilbert damping parameter of the magnetization dynamics

Conventional magnetic nanoscale devices are based on planar thin films and straight racetracks hosting magnetic topological solitons. Recent progress in fabrication and characterization methods allows to realise and study of complex-shaped planar and three-dimensional (3D) architectures. In the planar case, boundaries of nanodots lead to the formation of inhomogeneous textures, such as vortices and antivortices. In 3D, the magnetostatic interaction favours a spatially inhomogeneous shape anisotropy, which acts as easy-axis anisotropy along wires or hard axis of anisotropy perpendicular to the film surface. These interactions track the sample geometry and enable curvature-induced symmetry-breaking effects, such as topology-induced magnetization patterning and emergent anisotropic and chiral responses of the Dzyaloshinskii-Moriya interaction (DMI) type [1,2]. Curvature-induced magnetic responses can be classified as being local or nonlocal. In ferromagnets, local effects stem from the exchange interaction and DMI. The curvature-induced DMI originates from exchange: it is linear in curvatures and has the symmetry of the interfacial DMI. Its strength can be comparable with typical values of the intrinsic DMI. This is experimentally confirmed by the stabilization of chiral domain walls (CDW) on the apex of a Permalloy parabola-shaped stripe [3]. The strength of the CDW depinning field gives an estimation for the curvature-induced DMI constant and can be tuned by the geometry. In contrast to curvature itself, also curvature gradients offer a possibility to pin CDW, which was studied with an example of a circular indentation with a conic cross-section profile. This geometry supports circular CDWs described by the forced skyrmion equation, where the effective force acts as the stabilizing factor for large-radius skyrmion and skyrmionium states [4]. The magnetostatic interaction is a source of novel curvature-induced chiral effects, which are essentially nonlocal, in contras

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.

The interplay between geometry and topology of the order parameter is crucial properties in soft and condensed matter physics, including cell membranes [1], nematic crystals [2,3], superfluids [4], semiconductors [5], ferromagnets [6] and superconductors [7]. Until recently, in the case of magentism, the influence of the geometry on the magnetization vector fields was addressed primarily by the design of the sample boundaries, aiming to tailor anisotropy of the samples. With the development of novel fabrication techniques allowing to realize complex 3D architectures, not only boundary effects, but also local curvatures can be addressed rigorously for the case of ferromagnets and antiferromagnets. It is shown that curvature governs the appearance of geometry-induced chiral and anisotropic responses [6-8]. Here we provide experimental confirmations of the existence of local and non-local curvature-induced chiral interactions of the exchange and magnetostatic origin in conventional soft ferromagnetic materials. Namely, we will present the experimental validation of the appearance of exchange-driven Dzyaloshinskii-Moriya interaction interaction (DMI, local effect) for the case of conventional achiral yet geometrically curved magnetic materials [9,10]. This curvature induced DMI is predicted to stabilize skyrmions [11] and skyrmionium states [12]. Furthermore, we will address the impact of nonlocal magnetostatic interaction on the properties of curvilinear ferromagnets, which enables the stabilization of topological magnetic textures [13,14], realization of high-speed magnetic racetracks [15] and curvature-induced asymmetric spin-wave dispersions in nanotubes [16]. Furthermore, symmetry analysis demonstrates the possibility to generate a fundamentally new chiral symmetry breaking effect, which is essentially nonlocal [13]. Thus, geometric curvature of thin films and nanowires is envisioned as a toolbox to create artificial chiral nanostructures from achiral magnetic mate

Perpendicular anisotropy thin film systems are well known for their highly periodic magnetic stripe domains. Here we study [Co(3.0 nm)/Pt(0.6 nm)]X multilayers in the regime of transitional in-plane to out-of-plane anisotropy. For this we vary the number of repeats X in order to tune the remanent state from purely in-plane (IP) via tilted stripe domains (tilted), i.e. with significant out-of plane as well as in-plane magnetization component, to fully out-of-plane stripe domains (OOP). Vibrating Sample Magnetometry and Magnetic Force Microscopy are used to investigate three characteristic samples with X = 6,11 and 22, which represent the three above mentioned remanent states, respectively. In contrast to fully in-plane or fully out-of-plane systems experimental data and corresponding micromagnetic simulation of the tilted magnetization regime (X=11) reveals fully reversible field regions as well as distinct points of irreversibility during an external field sweep. This collective reversal behavior seems at first sight somewhat counter intuitive for a macroscopic system and has qualitative similarities with microscopic systems, such as the Stoner Wohlfarth particle and the vortex reversal in an in-plane magnetized disk, which both show as well distinct points of irreversibility.