Three-dimensional magnetic nanotextures with high-order vorticity in soft magnetic wireframes.

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, χ. We demonstrate that self-standing magnetic tetrapods (homeomorphic to a sphere; χ = + 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; χ = 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; χ < 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.

Coherent Magnons with Giant Nonreciprocity at Nanoscale Wavelengths.

Nonreciprocal 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 have so far yielded moderate nonreciprocities, 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 nonreciprocal 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 nonreciprocity 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.

Chirality coupling in topological magnetic textures with multiple magnetochiral parameters.

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.

Tailoring crosstalk between localized 1D spin-wave nanochannels using focused ion beams.

1D spin-wave conduits are envisioned as nanoscale components of magnonics-based logic and computing schemes for future generation electronics. À-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-[Formula: see text]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[Formula: see text] ion irradiation. Furthermore, after milling the microstrip along its long axis by focused ion beams, creating consecutively [Formula: see text]50 and [Formula: see text]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 [Formula: see text]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.

New Dimension in Magnetism and Superconductivity: 3D and Curvilinear Nanoarchitectures.

Traditionally, the primary field, where curvature has been at the heart of research, is 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, 2D van der Waals materials, plasmonics, magnetism, and superconductivity. Here, the state of the art is summarized and prospects for future research in curvilinear solid-state systems exhibiting such fundamental cooperative phenomena as ferromagnetism, antiferromagnetism, and superconductivity are outlined. 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 research and development oriented techniques to facilitate rapid transitions from laboratory demonstrations to industry-ready prototypes and eventual products.

Spin-Wave Dynamics and Symmetry Breaking in an Artificial Spin Ice.

Artificial spin ices are periodic arrangements of interacting nanomagnets which allow investigating emergent phenomena in the presence of geometric frustration. Recently, it has been shown that artificial spin ices can be used as building blocks for creating functional materials, such as magnonic crystals. We investigate the magnetization dynamics in a system exhibiting anisotropic magnetostatic interactions owing to locally broken structural inversion symmetry. We find a rich spin-wave spectrum and investigate its evolution in an external magnetic field. We determine the evolution of individual modes, from building blocks up to larger arrays, highlighting the role of symmetry breaking in defining the mode profiles. Moreover, we demonstrate that the mode spectra exhibit signatures of long-range interactions in the system. These results contribute to the understanding of magnetization dynamics in spin ices beyond the kagome and square ice geometries and are relevant for the realization of reconfigurable magnonic crystals based on spin ices.

Tunable Magnetic Vortex Dynamics in Ion-Implanted Permalloy Disks.

Nanoscale, low-phase-noise, tunable transmitter-receiver links are key for enabling the progress of wireless communication. We demonstrate that vortex-based spin-torque nano-oscillators, which are intrinsically low-noise devices because of their topologically protected magnetic structure, can achieve frequency tunability when submitted to local ion implantation. In the experiments presented here, the gyrotropic mode is excited with spin-polarized alternating currents and anisotropic magnetoresistance measurements yield discrete frequencies from a single device. Indeed, chromium-implanted regions of permalloy disks exhibit different saturation magnetization than neighboring, 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 for further lithographical steps, suggesting ion irradiation can be a viable and cost-effective fabrication method for densely packed networks of oscillators.

Visualizing Magnetic Structure in 3D Nanoscale Ni-Fe Gyroid Networks.

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.

Magnetization Dynamics of an Individual Single-Crystalline Fe-Filled Carbon Nanotube.

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. Currently, 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 is applied in order to investigate the evolution of the ferromagnetic resonance lines as a function of the nanowire length. The results show that the FeCNT is indeed not homogeneous along the full length but is built from 300 to 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 FeCNTs as appealing candidates for spin-wave transport in magnonic applications.

Strain Anisotropy and Magnetic Domains in Embedded Nanomagnets.

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 Fe60 Al40 thin film, where local irradiation by a focused ion beam causes the formation of disordered and strongly ferromagnetic regions of A2 Fe60 Al40 . 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.

Experimental Observation of Exchange-Driven Chiral Effects in Curvilinear Magnetism.

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.

Tunnel magnetoresistance angular and bias dependence enabling tuneable wireless communication.

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. This has an impact on devices seeking to work in the 'THz gap' due to their non-trivial TMR bias dependences.

Emission and propagation of 1D and 2D spin waves with nanoscale wavelengths in anisotropic spin textures.

Spin waves offer intriguing perspectives for computing and signal processing, because their damping can be lower than the ohmic losses in conventional complementary metal-oxide-semiconductor (CMOS) circuits. Magnetic domain walls show considerable potential as magnonic waveguides for on-chip control of the spatial extent and propagation of spin waves. However, low-loss guidance of spin waves with nanoscale wavelengths and around angled tracks remains to be shown. Here, we demonstrate spin wave control using natural anisotropic features of magnetic order in an interlayer exchange-coupled ferromagnetic bilayer. We employ scanning transmission X-ray microscopy to image the generation of spin waves and their propagation across distances exceeding multiples of the wavelength. Spin waves propagate in extended planar geometries as well as along straight or curved one-dimensional domain walls. We observe wavelengths between 1 µm and 150 nm, with excitation frequencies ranging from 250 MHz to 3 GHz. Our results show routes towards the practical implementation of magnonic waveguides in the form of domain walls in future spin wave logic and computational circuits.

Origin and Manipulation of Stable Vortex Ground States in Permalloy Nanotubes.

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.

Multiplet of Skyrmion States on a Curvilinear Defect: Reconfigurable Skyrmion Lattices.

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.

Curvature-Induced Asymmetric Spin-Wave Dispersion.

In magnonics, spin waves are conceived of as electron-charge-free information carriers. Their wave behavior has established them as the key elements to achieve low power consumption, fast operative rates, and good packaging in magnon-based computational technologies. Hence, knowing alternative ways that reveal certain properties of their undulatory motion is an important task. Here, we show using micromagnetic simulations and analytical calculations that spin-wave propagation in ferromagnetic nanotubes is fundamentally different than in thin films. The dispersion relation is asymmetric regarding the sign of the wave vector. It is a purely curvature-induced effect and its fundamental origin is identified to be the classical dipole-dipole interaction. The analytical expression of the dispersion relation has the same mathematical form as in thin films with the Dzyalonshiinsky-Moriya interaction. Therefore, this curvature-induced effect can be seen as a "dipole-induced Dzyalonshiinsky-Moriya-like" effect.

Spectral analysis of topological defects in an artificial spin-ice lattice.

Arrays of suitably patterned and arranged magnetic elements may display artificial spin-ice structures with topological defects in the magnetization, such as Dirac monopoles and Dirac strings. It is known that these defects strongly influence the quasistatic and equilibrium behavior of the spin-ice lattice. Here, we study the eigenmode dynamics of such defects in a square lattice consisting of stadiumlike thin film elements using micromagnetic simulations. We find that the topological defects display distinct signatures in the mode spectrum, providing a means to qualitatively and quantitatively analyze monopoles and strings that can be measured experimentally.

Beating the walker limit with massless domain walls in cylindrical nanowires.

We present a micromagnetic study on the current-induced domain-wall motion in cylindrical Permalloy nanowires with diameters below 50 nm. The transverse domain walls forming in such thin, round wires are found to differ significantly from those known from flat nanostrips. In particular, we show that these domain walls are zero-mass micromagnetic objects. As a consequence, they display outstanding dynamic properties, most importantly the absence of a breakdown velocity generally known as the Walker limit. Our simulation data are confirmed by an analytic model which provides a detailed physical understanding. We further predict that a particular effect of the current-induced dynamics of these domain walls could be exploited to measure the nonadiabatic spin-transfer torque coefficient.