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

Interplay of geometrical and spin chiralities in 3D twisted magnetic ribbons

Chirality is a ubiquitous and fundamental asymmetry in nature1,2. Recently, the interaction of chiral objects with spin currents has attracted enormous attention from both scientific and technological perspectives3-5. Of particular interest is the current-driven motion of chiral topological excitations such as chiral magnetic domain walls in chiral three-dimensional magnetic structures that could allow for high-density memory-storage devices. Here we use state-of-the-art multiphoton lithography6,7 to create three-dimensional chiral magnetic ribbons and perform current-induced motion of chiral domain walls. The ribbons are designed to have a clockwise or anticlockwise chiral twist with a variable magnitude. We find that domain walls can either pass through the ribbon or are impeded, depending on their chirality and configuration and the geometrical chiral twist of the ribbon. The interplay between the magnetic exchange energy and the geometrical twist generates a torsional field that favours chiral Bloch-type walls rather than the Néel-type wall favoured by the intrinsic magnetic properties of the magnetic ribbon itself. Furthermore, the interplay of spin chirality and chiral twist results in a non-reciprocal domain wall motion, namely, a domain wall filter or diode8-10. Our findings show how the interplay between geometrical and spin chiralities can lead to new functionalities that could allow for innovative chiral spintronics.

Magnetic Solitons in Hierarchical 3D Magnetic Nanoarchitectures of Nanoflower Shape

Curvilinear magnetism emerged as a new route to tailor properties of magnetic solitons by the choice of geometry and topology of a magnetic architecture. Here, we develop an anodized aluminum oxide template-based approach to realize hierarchical 3D magnetic nanoarchitectures of nanoflower shape. The technique provides defect-free regular arrays of magnetic nanoflowers of tunable shape with a period of 400 nm over cm2 areas. We combined advanced magnetic imaging methods with micromagnetic simulations to study complex magnetic states in nanoflowers originating due to magnetostatics-driven symmetry break in curvilinear nanomembranes. An interaction between surface and volume magnetostatic charges in 3D curved nanoflowers leads to the stabilization of asymmetric and shifted vortices as well as states with two Bloch lines. Ordered large area arrays of complex-shaped magnetic nanoarchitectures developed in this work are relevant for prospective research on 3D magnonics and spintronics.

Realization of skyrmion shift register

The big data explosion demands novel data storage technology. Among many different approaches, solitonic racetrack memory devices hold great promise for accommodating nonvolatile and low-power functionalities. As representative topological solitons, magnetic skyrmions are envisioned as potential information carriers for efficient information processing. While their advantages as memory and logic elements have been vastly exploited from theoretical perspectives, the corresponding experimental efforts are rather limited. These challenges, which are key to versatile skyrmionic devices, will be studied in this work. Through patterning concaved surface topography with designed arrays of indentations on standard Si/SiO2 substrates, we demonstrate that the resultant non-flat energy landscape could lead to the formation of hexagonal and square skyrmion lattices in Ta/CoFeB/MgO multilayers. Based on these films, one-dimensional racetrack devices are subsequently fabricated, in which a long-distance deterministic shifting of skyrmions between neighboring indentations is achieved at room temperature. Through separating the word line and the bit line, a prototype shift register device, which can sequentially generate and precisely shift complex skyrmionic data strings, is presented. The deterministic writing and long-distance shifting of skyrmionic bits can find potential applications in transformative skyrmionic memory, logic as well as the in-memory computing devices.

Spatially reconfigurable antiferromagnetic states in topologically rich free-standing nanomembranes

Antiferromagnets hosting real-space topological textures are promising platforms to model fundamental ultrafast phenomena and explore spintronics. However, they have only been epitaxially fabricated on specific symmetry-matched substrates, thereby preserving their intrinsic magneto-crystalline order. This curtails their integration with dissimilar supports, restricting the scope of fundamental and applied investigations. Here we circumvent this limitation by designing detachable crystalline antiferromagnetic nanomembranes of α-Fe2O3. First, we show-via transmission-based antiferromagnetic vector mapping-that flat nanomembranes host a spin-reorientation transition and rich topological phenomenology. Second, we exploit their extreme flexibility to demonstrate the reconfiguration of antiferromagnetic states across three-dimensional membrane folds resulting from flexure-induced strains. Finally, we combine these developments using a controlled manipulator to realize the strain-driven non-thermal generation of topological textures at room temperature. The integration of such free-standing antiferromagnetic layers with flat/curved nanostructures could enable spin texture designs via magnetoelastic/geometric effects in the quasi-static and dynamical regimes, opening new explorations into curvilinear antiferromagnetism and unconventional computing.

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.

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.

Unveiling the three-dimensional magnetic texture of skyrmion tubes

Magnetic skyrmions are stable topological solitons with complex non-coplanar spin structures. Their nanoscopic size and the low electric currents required to control their motion has opened a new field of research, skyrmionics, that aims for the usage of skyrmions as information carriers. Further advances in skyrmionics call for a thorough understanding of their three-dimensional (3D) spin texture, skyrmion-skyrmion interactions and the coupling to surfaces and interfaces, which crucially affect skyrmion stability and mobility. Here, we quantitatively reconstruct the 3D magnetic texture of Bloch skyrmions with sub-10-nanometre resolution using holographic vector-field electron tomography. The reconstructed textures reveal local deviations from a homogeneous Bloch character within the skyrmion tubes, details of the collapse of the skyrmion texture at surfaces and a correlated modulation of the skyrmion tubes in FeGe along their tube axes. Additionally, we confirm the fundamental principles of skyrmion formation through an evaluation of the 3D magnetic energy density across these magnetic solitons.

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.

Intrinsic DMI-free skyrmion formation and robust dynamic behaviors in magnetic hemispherical shells

We performed finite-element micromagnetic simulations to examine the formation of skyrmions without intrinsic Dzyaloshinskii–Moriya interaction (DMI) in magnetic hemispherical shells. We found that curvature-induced DM-like interaction allows for further stabilization of skyrmions without the DMI in curved-geometry hemispherical shells for a specific range of uniaxial perpendicular magnetic anisotropy (PMA) constant K_u. The larger the curvature of the shell, the higher the K_u value required for the formation of the skyrmions. With well-stabilized skyrmions, we also found in-plane gyration modes and azimuthal spin-wave modes as well as an out-of-plane breathing mode, similarly to previously found modes for planar geometries. Furthermore, additional higher-frequency hybrid modes were observed due to coupling between the gyration and azimuthal modes. This work provides further physical insight into the static and dynamic properties of intrinsic DMI-free skyrmions formed in curved-geometry systems.

Nucleation and stability of skyrmions in three-dimensional chiral nanostructures

We studied the magnetization evolution in three-dimensional chiral nanostructures, including nanotubes and circularly curved thin films, by micromagnetic simulations. We found that in a nanotube skyrmions can be formed by broken of the helical stripes on the left and right sides of the nanotube, and the formation of skyrmions doesn't correspond to any abrupt change of topological number. Skyrmions can exist in a large range of magnetic field, and the thinner nanotube has a larger field range for skyrmion existence. The configuration of a skyrmion in nanotubes is different from the one in thin film. From the outer to the inner circular layer, the size of the skyrmion becomes larger, and the deformation becomes more obvious. In circularly curved magnetic films with fixed arc length, there are three kinds of hysteresis processes are found. For the curved films with a large radius, the magnetization evolution behavior is similar to the case in two-dimensional thin films. For the curved films with a small radius, the skyrmions are created by broken of the helical stripes on the left and right sides of the curved film. For the curved film with a medium radius, no skyrmion is formed in the hysteresis process.

Writing 3D Nanomagnets Using Focused Electron Beams

Focused electron beam induced deposition (FEBID) is a direct-write nanofabrication technique able to pattern three-dimensional magnetic nanostructures at resolutions comparable to the characteristic magnetic length scales. FEBID is thus a powerful tool for 3D nanomagnetism which enables unique fundamental studies involving complex 3D geometries, as well as nano-prototyping and specialized applications compatible with low throughputs. In this focused review, we discuss recent developments of this technique for applications in 3D nanomagnetism, namely the substantial progress on FEBID computational methods, and new routes followed to tune the magnetic properties of ferromagnetic FEBID materials. We also review a selection of recent works involving FEBID 3D nanostructures in areas such as scanning probe microscopy sensing, magnetic frustration phenomena, curvilinear magnetism, magnonics and fluxonics, offering a wide perspective of the important role FEBID is likely to have in the coming years in the study of new phenomena involving 3D magnetic nanostructures.

Ultrafast domain wall propagation due to the interfacial Dzyaloshinskii-Moriya interaction

It is shown that the interplay between curvature and interfacial Dzyalonshinsky-Moriya interaction (DMI) is a pathway to ultrafast domain wall (DW) dynamics in ferromagnetic nanotubes. In this work, we theoretically study the effect that interfacial DMI has on the average velocity of a vortex DW in thin ferromagnetic nanotubes grown around a core composed of heavy atoms. Our main result shows that by delaying the Walker breakdown instability, the DW average velocity is of the order of 10³ m s^-1, which is greater than usual values for these systems. The remarkable velocities achieved through this configuration could greatly benefit the development of spintronic devices.

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.