Magnetic charge propagation upon a 3D artificial spin-ice

Field-Induced Antiferromagnetic Correlations in a Nanopatterned Van der Waals Ferromagnet: A Potential Artificial Spin Ice

Nano-patterned magnetic materials have opened new venues for the investigation of strongly correlated phenomena including artificial spin-ice systems, geometric frustration, and magnetic monopoles, for technologically important applications such as reconfigurable ferromagnetism. With the advent of atomically thin 2D van der Waals (vdW) magnets, a pertinent question is whether such compounds could make their way into this realm where interactions can be tailored so that unconventional states of matter can be assessed. Here, it is shown that square islands of CrGeTe3 vdW ferromagnets distributed in a grid manifest antiferromagnetic correlations, essential to enable frustration resulting in an artificial spin-ice. By using a combination of SQUID-on-tip microscopy, focused ion beam lithography, and atomistic spin dynamic simulations, it is shown that a square array of CGT island as small as 150 × 150 × 60 nm3 have tunable dipole-dipole interactions, which can be precisely controlled by their lateral spacing. There is a crossover between non-interacting islands and significant inter-island anticorrelation depending on how they are spatially distributed allowing the creation of complex magnetic patterns not observable at the isolated flakes. These findings suggest that the cross-talk between the nano-patterned magnets can be explored in the generation of even more complex spin configurations where exotic interactions may be manipulated in an unprecedented way.

Direct visualization of domain wall pinning in sub-100 nm 3D magnetic nanowires with cross-sectional curvature

The study of 3D magnetic nanostructures has uncovered rich phenomena including the stabilization of topological spin textures using nanoscale curvature, controlled spin-wave emission, and novel ground states enabled by collective frustrated interactions. From a technological perspective, 3D nanostructures offer routes to ultrahigh density data storage, massive interconnectivity within neuromorphic devices, as well as mechanical induction of stem cell differentiation. However, the fabrication of 3D nanomagnetic systems with feature sizes down to 10 nm poses a significant challenge. Here we present a means of fabricating sub-100 nm 3D ferromagnetic nanowires, with both cross-sectional and longitudinal curvature, using two-photon lithography at a wavelength of 405 nm, combined with conventional deposition. Nanostructures with lateral features as low as 70 nm can be rapidly and reproducibly fabricated. A range of novel domain walls, with anti-vortex textures and hybrid vortex/anti-vortex textures are enabled by the cross-sectional curvature of the system, as demonstrated by micromagnetic simulations. Magnetic force microscopy experiments in an externally applied magnetic field are used to image the injection and pinning of domain walls in the 3D magnetic nanowire. At specific field values, domain walls are observed to hop from trap to trap, providing a direct means to probe the local energy landscape.

Tailoring magnon modes by extending square, kagome, and trigonal spin ice lattices vertically via interlayer coupling of trilayer nanomagnets

In this work high-frequency magnetization dynamics and statics of artificial spin-ice lattices with different geometric nanostructure array configurations are studied where the individual nanostructures are composed of ferromagnetic/non-magnetic/ferromagnetic trilayers with different non-magnetic thicknesses. These thickness variations enable additional control over the magnetic interactions within the spin-ice lattice that directly impacts the resulting magnetization dynamics and the associated magnonic modes. Specifically the geometric arrangements studied are square, kagome and trigonal spin ice configurations, where the individual lithographically patterned nanomagnets (NMs) are trilayers, made up of two magnetic layers ofNi81Fe19of 30 nm and 70 nm thickness respectively, separated by a non-magnetic copper layer of either 2 nm or 40 nm. We show that coupling via the magnetostatic interactions between the ferromagnetic layers of the NMs within square, kagome and trigonal spin-ice lattices offers fine-control over magnetization states and magnetic resonant modes. In particular, the kagome and trigonal lattices allow tuning of an additional mode and the spacing between multiple resonance modes, increasing functionality beyond square lattices. These results demonstrate the ability to move beyond quasi-2D single magnetic layer nanomagnetics via control of the vertical interlayer interactions in spin ice arrays. This additional control enables multi-mode magnonic programmability of the resonance spectra, which has potential for magnetic metamaterials for microwave or information processing applications.

Ultrastrong magnon-magnon coupling and chiral spin-texture control in a dipolar 3D multilayered artificial spin-vortex ice

Strongly-interacting nanomagnetic arrays are ideal systems for exploring reconfigurable magnonics. They provide huge microstate spaces and integrated solutions for storage and neuromorphic computing alongside GHz functionality. These systems may be broadly assessed by their range of reliably accessible states and the strength of magnon coupling phenomena and nonlinearities. Increasingly, nanomagnetic systems are expanding into three-dimensional architectures. This has enhanced the range of available magnetic microstates and functional behaviours, but engineering control over 3D states and dynamics remains challenging. Here, we introduce a 3D magnonic metamaterial composed from multilayered artificial spin ice nanoarrays. Comprising two magnetic layers separated by a non-magnetic spacer, each nanoisland may assume four macrospin or vortex states per magnetic layer. This creates a system with a rich 16N microstate space and intense static and dynamic dipolar magnetic coupling. The system exhibits a broad range of emergent phenomena driven by the strong inter-layer dipolar interaction, including ultrastrong magnon-magnon coupling with normalised coupling rates ofΔ f ν = 0.57 , GHz mode shifts in zero applied field and chirality-control of magnetic vortex microstates with corresponding magnonic spectra.

Collective Spin-Wave Dynamics in Gyroid Ferromagnetic Nanostructures

Expanding upon the burgeoning discipline of magnonics, this research elucidates the intricate dynamics of spin waves (SWs) within three-dimensional nanoenvironments. It marks a shift from traditionally used planar systems to exploration of magnetization configurations and the resulting dynamics within 3D nanostructures. This study deploys micromagnetic simulations alongside ferromagnetic resonance measurements to scrutinize magnetic gyroids, periodic chiral configurations composed of chiral triple junctions with a period in nanoscale. Our findings uncover distinctive attributes intrinsic to the gyroid network, most notably the localization of collective SW excitations and the sensitivity of the gyroid's ferromagnetic response to the orientation of the static magnetic field, a correlation closely tied to the crystallographic alignment of the structure. Furthermore, we show that for the ferromagnetic resonance, multidomain gyroid films can be treated as a magnonic material with effective magnetization scaled by its filling factor. The implications of our research carry the potential for practical uses such as an effective, metamaterial-like substitute for ferromagnetic parts and lay the groundwork for radio frequency filters. The growing areas of 3D magnonics and spintronics present exciting opportunities to investigate and utilize gyroid nanostructures for signal processing purposes.

Realization and Control of Bulk and Surface Modes in 3D Nanomagnonic Networks by Additive Manufacturing of Ferromagnets

The high-density integration in information technology fuels the research on functional 3D nanodevices. Particularly ferromagnets promise multifunctional 3D devices for nonvolatile data storage, high-speed data processing, and non-charge-based logic operations via spintronics and magnonics concepts. However, 3D nanofabrication of ferromagnets is extremely challenging. In this work, an additive manufacturing methodology is reported, and unprecedented 3D ferromagnetic nanonetworks with a woodpile-structure unit cell are fabricated. The collective spin dynamics (magnons) at frequencies up to 25 GHz are investigated by Brillouin Light Scattering (BLS) microscopy and micromagnetic simulations. A clear discrepancy of about 10 GHz is found between the bulk and surface modes, which are engineered by different unit cell sizes in the Ni-based nanonetworks. The angle- and spatially-dependent modes demonstrate opportunities for multi-frequency signal processing in 3D circuits via magnons. The developed synthesis route will allow one to create 3D magnonic crystals with chiral unit cells, which are a prerequisite toward surface modes with topologically protected properties.

3D Magnonic Conduits by Direct Write Nanofabrication

Magnonics is a rapidly developing domain of nanomagnetism, with application potential in information processing systems. Realisation of this potential and miniaturisation of magnonic circuits requires their extension into the third dimension. However, so far, magnonic conduits are largely limited to thin films and 2D structures. Here, we introduce 3D magnonic nanoconduits fabricated by the direct write technique of focused-electron-beam induced deposition (FEBID). We use Brillouin light scattering (BLS) spectroscopy to demonstrate significant qualitative differences in spatially resolved spin-wave resonances of 2D and 3D nanostructures, which originates from the geometrically induced non-uniformity of the internal magnetic field. This work demonstrates the capability of FEBID as an additive manufacturing technique to produce magnetic 3D nanoarchitectures and presents the first report of BLS spectroscopy characterisation of FEBID conduits.

Nonlinear multi-magnon scattering in artificial spin ice

Magnons, the quantum-mechanical fundamental excitations of magnetic solids, are bosons whose number does not need to be conserved in scattering processes. Microwave-induced parametric magnon processes, often called Suhl instabilities, have been believed to occur in magnetic thin films only, where quasi-continuous magnon bands exist. Here, we reveal the existence of such nonlinear magnon-magnon scattering processes and their coherence in ensembles of magnetic nanostructures known as artificial spin ice. We find that these systems exhibit effective scattering processes akin to those observed in continuous magnetic thin films. We utilize a combined microwave and microfocused Brillouin light scattering measurement approach to investigate the evolution of their modes. Scattering events occur between resonance frequencies that are determined by each nanomagnet's mode volume and profile. Comparison with numerical simulations reveals that frequency doubling is enabled by exciting a subset of nanomagnets that, in turn, act as nanosized antennas, an effect that is akin to scattering in continuous films. Moreover, our results suggest that tunable directional scattering is possible in these structures.

Nucleation and Stability of Toron Chains in Non-Centrosymmetric Magnetic Nanowires

This work analyzes the magnetic configurations of cylindrical nanowires with a bulk Dzyaloshinskii-Moriya interaction and easy-plane anisotropy. We show that this system allows the nucleation of a metastable toron chain even when no out-of-plane anisotropy exists in the nanowire's top and bottom surfaces, as usually required. The number of nucleated torons depends on the nanowire length and the strength of an external magnetic field applied to the system. The size of each toron depends on the fundamental magnetic interactions and can be controlled by external stimuli, allowing the use of these magnetic textures as information carriers or nano-oscillator elements. Our results evidence that the topology and structure of the torons yield a wide variety of behaviors, revealing the complex nature of these topological textures, which should present an exciting interaction dynamic, depending on the initial conditions.

Magnetostatic interaction between Bloch point nanospheres

Three-dimensional topological textures have become a topic of intense interest in recent years. This work uses analytical and numerical calculations to determine the magnetostatic field produced by a Bloch point (BP) singularity confined in a magnetic nanosphere. It is observed that BPs hosted in a nanosphere generate magnetic fields with quadrupolar nature. This finding is interesting because it shows the possibility of obtaining quadrupole magnetic fields with just one magnetic particle, unlike other propositions considering arrays of magnetic elements to generate this kind of field. The obtained magnetostatic field allows us to determine the interaction between two BPs as a function of the relative orientation of their polarities and the distance between them. It is shown that depending on the rotation of one BP related to the other, the magnetostatic interaction varies in strength and character, being attractive or repulsive. The obtained results reveal that the BP interaction has a complex behavior beyond topological charge-mediated interaction.

3D Nanolithography by Means of Lipid Ink Spreading Inhibition

While patterning 2D metallic nanostructures are well established through different techniques, 3D printing still constitutes a major bottleneck on the way to device miniaturization. In this work a fluid phase phospholipid ink is used as a building block for structuring with dip-pen nanolithography. Following a bioinspired approach that relies on ink-spreading inhibition, two processes are presented to build 2D and 3D metallic structures. Serum albumin, a widely used protein with an innate capability to bind to lipids, is the key in both processes. Covering the sample surface with it prior to lipid writing, anchors lipids on the substrate, which ultimately allows the creation of highly stable 3D lipid-based scaffolds to build metallic structures.

Curvature-induced stabilization and field-driven dynamics of magnetic hopfions in toroidal nanorings

Three dimensional magnetic textures are a cornerstone in magnetism research. In this work, we analyze the stabilization and dynamic response of a magnetic hopfion hosted in a toroidal nanoring with intrinsic Dzyaloshinskii-Moriya interaction simulating FeGe. Our results evidence that unlike their planar counterparts, where perpendicular magnetic anisotropies are necessary to stabilize hopfions, the shape anisotropy originated on the torus symmetry naturally yields the nucleation of these topological textures. We also analyze the magnetization dynamical response by applying a magnetic field pulse to differentiate among several magnetic patterns. Finally, to understand the nature of spin wave modes, we analyze the spatial distributions of the resonant mode amplitudes and phases and describe the differences among bulk and surface modes. Importantly, hopfions lying in toroidal nanorings present a non-circularly symmetric poloidal resonant mode, which is not observed in other systems hosting hopfions.

3D Interconnected Magnetic Nanowire Networks as Potential Integrated Multistate Memristors

Interconnected magnetic nanowire (NW) networks offer a promising platform for three-dimensional (3D) information storage and integrated neuromorphic computing. Here we report discrete propagation of magnetic states in interconnected Co nanowire networks driven by magnetic field and current, manifested in distinct magnetoresistance (MR) features. In these networks, when only a few interconnected NWs were measured, multiple MR kinks and local minima were observed, including a significant minimum at a positive field during the descending field sweep. Micromagnetic simulations showed that this unusual feature was due to domain wall (DW) pinning at the NW intersections, which was confirmed by off-axis electron holography imaging. In a complex network with many intersections, sequential switching of nanowire sections separated by interconnects was observed, along with stochastic characteristics. The pinning/depinning of the DWs can be further controlled by the driving current density. These results illustrate the promise of such interconnected networks as integrated multistate memristors.

Magnetic Force Microscopy on Nanofibers—Limits and Possible Approaches for Randomly Oriented Nanofiber Mats

Magnetic force microscopy (MFM) belongs to the methods that enable spatially resolved magnetization measurements on common thin-film samples or magnetic nanostructures. The lateral resolution can be much higher than in Kerr microscopy, another spatially resolved magnetization imaging technique, but since MFM commonly necessitates positioning a cantilever tip typically within a few nanometers from the surface, it is often more complicated than other techniques. Here, we investigate the progresses in MFM on magnetic nanofibers that can be found in the literature during the last years. While MFM measurements on magnetic nanodots or thin-film samples can often be found in the scientific literature, reports on magnetic force microscopy on single nanofibers or chaotic nanofiber mats are scarce. The aim of this review is to show which MFM investigations can be conducted on magnetic nanofibers, where the recent borders are, and which ideas can be transferred from MFM on other rough surfaces towards nanofiber mats.

On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such a technique can also be used to obtain an upper bound, we find that in all finite degree examples we studied, another upper bound based on Schrijver inequality is tighter.

Observation of Coherent Spin Waves in a Three-Dimensional Artificial Spin Ice Structure

Harnessing high-frequency spin dynamics in three-dimensional (3D) nanostructures may lead to paradigm-shifting, next-generation devices including high density spintronics and neuromorphic systems. Despite remarkable progress in fabrication, the measurement and interpretation of spin dynamics in complex 3D structures remain exceptionally challenging. Here, we take a first step and measure coherent spin waves within a 3D artificial spin ice (ASI) structure using Brillouin light scattering. The 3D-ASI was fabricated by using a combination of two-photon lithography and thermal evaporation. Two spin-wave modes were observed in the experiment whose frequencies showed nearly monotonic variation with the applied field strength. Numerical simulations qualitatively reproduced the observed modes. The simulated mode profiles revealed the collective nature of the modes extending throughout the complex network of nanowires while showing spatial quantization with varying mode quantization numbers. The study shows a well-defined means to explore high-frequency spin dynamics in complex 3D spintronic and magnonic structures.

Magnetic charge propagation upon a 3D artificial spin-ice

Magnetic charge propagation in spin-ice materials has yielded a paradigm-shift in science, allowing the symmetry between electricity and magnetism to be studied. Recent work is now suggesting the spin-ice surface may be important in mediating the ordering and associated phase space in such materials. Here, we detail a 3D artificial spin-ice, which captures the exact geometry of bulk systems, allowing magnetic charge dynamics to be directly visualized upon the surface. Using magnetic force microscopy, we observe vastly different magnetic charge dynamics along two principal directions. For a field applied along the surface termination, local energetics force magnetic charges to nucleate over a larger characteristic distance, reducing their magnetic Coulomb interaction and producing uncorrelated monopoles. In contrast, applying a field transverse to the surface termination yields highly correlated monopole-antimonopole pairs. Detailed simulations suggest it is the difference in effective chemical potential as well as the energy landscape experienced during dynamics that yields the striking differences in monopole transport.