The transition from planar to three-dimensional (3D) magnetic nanostructures represents a significant advancement in both fundamental research and practical applications, offering vast potential for next-generation technologies like ultrahigh-density storage, memory, logic, and neuromorphic computing. Despite being a relatively new field, the emergence of 3D nanomagnetism presents numerous opportunities for innovation, prompting the creation of a comprehensive roadmap by leading international researchers. This roadmap aims to facilitate collaboration and interdisciplinary dialogue to address challenges in materials science, physics, engineering, and computing. The roadmap comprises eighteen sections, roughly divided into three blocks. The first block explores the fundamentals of 3D nanomagnetism, focusing on recent trends in fabrication techniques and imaging methods crucial for understanding complex spin textures, curved surfaces, and small-scale interactions. Techniques such as two-photon lithography and focused electron beam-induced deposition enable the creation of intricate 3D architectures, while advanced imaging methods like electron holography and synchrotron x-ray tomography provide nanoscale spatial resolution for studying magnetization dynamics in three dimensions. Various 3D magnetic systems, including coupled multilayer systems, artificial spin-ice, magneto-plasmonic systems, topological spin textures, and molecular magnets are discussed. The second block introduces analytical and numerical methods for investigating 3D nanomagnetic structures and curvilinear systems, highlighting geometrically curved architectures, interconnected nanowire systems, and other complex geometries. Finite element methods are emphasized for capturing complex geometries, along with direct frequency domain solutions for addressing magnonic problems. The final block focuses on 3D magnonic crystals and networks, exploring their fundamental properties and potential applications in magnonic circuits, memory, and spintronics. Computational approaches using 3D nanomagnetic systems and complex topological textures in 3D spintronics are highlighted for their potential to enable faster and more energy-efficient computing.

The nonreciprocity created by dipolar coupling, electric currents, and Dzyaloshinskii-Moriya interactions is discussed in cases where the magnon propagation direction has a component parallel to the toroidal moment. A criterion for calculating the toroidal moments is established, addressing the issue of correct origin selection by considering compensated and uncompensated magnetization distributions. This criterion is then applied to various nonreciprocal magnetic systems, with the calculations consistent with those reported in the literature and predicting the existence of nonreciprocity in a more general manner. These results broaden the physical significance of the toroidal moment and facilitate the identification and estimation of nonreciprocity in magnonic systems. This work also clarifies the interrelations between different definitions of the toroidal moment for confined structures, where a surface term arising from surface-bound currents connects these definitions without the need for time-averaging. Comparing these definitions of the toroidal moment applied to different magnetic textures demonstrates that they are always parallel but may differ in magnitude and sign. The discrepancy in the different definitions is deemed irrelevant since its direction, rather than its magnitude, primarily predicts the existence of magnon nonreciprocity.

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 finiteelement 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.

Spin waves excited in periodically modulated magnetic nanomaterials, known as magnonic crystals, exhibit characteristic band structures. These bands can be tuned by material engineering and have been attractive for potential spin-based applications. When periodic nanomaterials with handedness are introduced, spin waves inherit the chiral feature in their behavior and manifest an exciting range of novel physics, including asymmetric and unidirectional propagation, low-frequency magnonic flat bands, and indirect band gaps. This study investigates the properties of these chiral magnonic excitations. The analysis is performed in ferromagnetic films patterned with nanowires of two different materials that produce periodically modulated perpendicular magnetic anisotropy and interfacial antisymmetric exchange (Dzyaloshinskii-Moriya interaction). The low-frequency flat modes are studied using a magnonic localization diagram that distinguishes the spatial confinement degree in zones with and without antisymmetric exchange. An analytical expression is derived for the transition region in the localization diagram that outlines the zones where magnonic confinement occurs. The findings reveal the presence of flat modes with nonreciprocal magnetization oscillation amplitudes between waves with opposite propagation directions when the spin-wave localization occurs in regions with Dzyaloshinskii-Moriya interaction. Conversely, reciprocal oscillation amplitudes are observed when modes localize in the nanowires with perpendicular anisotropy. Micromagnetic simulations demonstrate the amplitude asymmetry of the flat modes, yielding perfect agreement with the theoretical predictions. This paper provides a deeper understanding of the behavior of spin-wave modes in chiral magnonic crystals and establishes a method to control their associated magnonic bands for designing spin-wave-based nanodevices.

Magnonics is a research field that has gained an increasing interest in both the fundamental and applied sciences in recent years. This field aims to explore and functionalize collective spin excitations in magnetically ordered materials for modern information technologies, sensing applications and advanced computational schemes. Spin waves, also known as magnons, carry spin angular momenta that allow for the transmission, storage and processing of information without moving charges. In integrated circuits, magnons enable on-chip data processing at ultrahigh frequencies without the Joule heating, which currently limits clock frequencies in conventional data processors to a few GHz. Recent developments in the field indicate that functional magnonic building blocks for in-memory computation, neural networks and Ising machines are within reach. At the same time, the miniaturization of magnonic circuits advances continuously as the synergy of materials science, electrical engineering and nanotechnology allows for novel on-chip excitation and detection schemes. Such circuits can already enable magnon wavelengths of 50 nm at microwave frequencies in a 5G frequency band. Research into non-charge-based technologies is urgently needed in view of the rapid growth of machine learning and artificial intelligence applications, which consume substantial energy when implemented on conventional data processing units. In its first part, the 2024 Magnonics Roadmap provides an update on the recent developments and achievements in the field of nano-magnonics while defining its future avenues and challenges. In its second part, the Roadmap addresses the rapidly growing research endeavors on hybrid structures and magnonics-enabled quantum engineering. We anticipate that these directions will continue to attract researchers to the field and, in addition to showcasing intriguing science, will enable unprecedented functionalities that enhance the efficiency of alternative information technologies and computational schemes.

Spin waves represent the collective excitations of the magnetization field within a magnetic material, providing dispersion curves that can be manipulated by material design and external stimuli. Bulk and surface spin waves can be excited in a thin film with positive or negative group velocities and, by incorporating a symmetry-breaking mechanism, magnetochiral features arise. Here we study the band diagram of a chiral magnonic crystal consisting of a ferromagnetic film incorporating a periodic Dzyaloshinskii-Moriya coupling via interfacial contact with an array of heavy-metal nanowires. We provide experimental evidence for a strong asymmetry of the spin wave amplitude induced by the modulated interfacial Dzyaloshinskii-Moriya interaction, which generates a nonreciprocal propagation. Moreover, we observe the formation of flat spin-wave bands at low frequencies in the band diagram. Calculations reveal that depending on the perpendicular anisotropy, the spin-wave localization associated with the flat modes occurs in the zones with or without Dzyaloshinskii-Moriya interaction.

Magnetochiral properties are enriched in curved magnetic nanostructures, in which dipole-dipole and exchange couplings are their physical sources. In such systems, direct implications are evidenced in the magnetization dynamics, where a noticeable frequency shift appears between two counterpropagating spin waves. In this paper, the spin-wave asymmetry induced by the curvature is theoretically studied in thick ferromagnetic nanotubes with a vortex ground state. The spin-wave spectra are obtained using semianalytical calculations and the dynamical matrix method for thin and thick nanotubes. Under the thickness increase, radial standing spin waves are observed at low frequencies, while the nonreciprocal properties are improved. Such standing waves exhibit a nonreciprocal spin-wave dispersion, but it is not as prominent as the asymmetry of the low-frequency in-phase modes. In the limit of small wave vectors, analytical expressions are reported for the spin-wave dispersion, where the resonance frequency, the frequency shift of two counterpropagating waves, and the critical field that destabilizes the vortex state are determined. The obtained frequency shift allows us to estimate the influence of thickness and curvature on the nonreciprocity of the spin waves. These results constitute a significant advance in the fundamental understanding of the spin-wave dynamics of ferromagnetic nanotubes, predicting new phenomena and providing expressions that are easy to interpret and that allow us, therefore, to promote the study of magnetization dynamics in curved structures.

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, domainwalls, their types, and energetics aswell as possible reversalmechanisms. 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. Vazquez [1, 2].

Since the beginning of the century, the possibility of having chiral spin textures in magnetic materials has been the subject of intense scientific interest. Chiral spin textures have been observed experimentally and described theoretically, bearing potential applications associated with their topological nature. This work theoretically explores the formation of chiral magnetic order in ultrathin magnetic films, where the antisymmetric Dzyaloshinskii-Moriya interaction induces a conical helix magnetization. By minimizing the internal energy of the helix, a simple model predicts the nucleation field, the pitch vector, and the cone angle that characterize the ground-state magnetization texture. It is further demonstrated that the formation of the helical order is connected with the spin waves excited close to the instability of the field-polarized state. Namely, when an in-plane magnetic field is reduced from saturation, a second-order phase transition arises when the spin-wave frequency approaches zero at a critical point where the conical helix nucleates. Interestingly, the wave vector at which the frequency becomes zero matches the pitch vector of the conical helix texture. Thus the instability point of the magnonic excitations is associated with the spin texture, as if the softened spin-wave modes crystallize in the chiral magnetic film. A critical competition among the magnetostatic and the anisotropy field is also found which influences the orbit described by a dynamic magnetization, changing it from circular to elliptical.

Asymmetric spin-wave propagation in magnetic nanostructures has received significant attention due to the potential applications of magnon-based devices. In curved nanostructures and planar multilayers, the classical dipole-dipole interaction induces a significant frequency nonreciprocity in which two coun-terpropagating waves excited at the same frequency exhibit different wavelengths. This work proposes a cylindrical synthetic antiferromagnet as a potential three-dimensional waveguide design to generate nonreciprocal spin waves. The magnetochiral properties emerge from two mechanisms: the asymmetric interlayer dipolar coupling and the asymmetric dipolar coupling of the curved membrane. It is demon-strated that the cylindrical bilayer presents a notable spin-wave asymmetry induced by the combined action of antiparallel magnetic vortices and the curvature of the inner and outer surfaces. A substantial frequency range with waves having only a negative phase velocity is predicted, where unidirectional wave propagation is allowed. It is also found that the nonreciprocity reaches a constant value as the curvature decreases, which is an essential advantage over isolated nanotubes, where the frequency shift vanishes at a large radius. Besides, analytical expressions are proposed to predict the frequency shift in the case of coupled cylindrical shells. These results are relevant from fundamental and practical points of view since magnetochirality is a crucial ingredient in visualizing future spin-wave-based logic devices.

The magnonic band structure of two-dimensional chiral magnonic crystals is theoretically investigated. The proposed metamaterial involves a three-dimensional architecture, where a thin ferromagnetic layer is in contact with a two-dimensional periodic array of heavy-metal square islands. When these two materials are in contact, an anti-symmetric exchange coupling known as the Dzyaloshinskii-Moriya interaction (DMI) arises, which generates nonreciprocal spin waves and chiral magnetic order. The Landau-Lifshitz equation and the plane-wave method are employed to study the dynamic magnetic behavior. A systematic variation of geometric parameters, the DMI constant, and the filling fraction allows the examination of spin-wave propagation features, such as the spatial profiles of the dynamic magnetization, the isofrequency contours, and group velocities. In this study, it is found that omnidirectional flat magnonic bands are induced by a sufficiently strong Dzyaloshinskii-Moriya interaction underneath the heavy-metal islands, where the spin excitations are active. The theoretical results were substantiated by micromagnetic simulations. These findings are relevant for envisioning applications associated with spin-wave-based logic devices, where the nonreciprocity and channeling of the spin waves are of fundamental and practical scientific interest.

Magnetization-graded ferromagnetic nanostrips are proposed as potential prospects to channel spin waves. Here, a controlled reduction of the saturation magnetization enables the localization of the propagating magnetic excitations in the same way that light is controlled in an optical fiber with a varying refraction index. The theoretical approach is based on the dynamic matrix method, where the magnetic nanostrip is divided into small sub-strips. The dipolar and exchange interactions between sub-strips have been considered to reproduce the spin-wave dynamics of the magnonic fiber. The transition from one strip to an infinite thin film is presented for the Damon-Eshbach geometry, where the nature of the spin-wave modes is discussed. An in-depth analysis of the spin-wave transport as a function of the saturation magnetization profile is provided. It is predicted that it is feasible to induce a remarkable channeling of the spin waves along the zones with a reduced saturation magnetization, even when such a reduction is tiny. The results are compared with micromagnetic simulations, where a good agreement is observed between both methods. The findings have relevance for envisioned future spin-wave-based magnonic devices operating at the nanometer scale.

Magnonics is a budding research field in nanomagnetism and nanoscience that addresses the use of spin waves (magnons) to transmit, store, and process information. The rapid advancements of this field during last one decade in terms of upsurge in research papers, review articles, citations, proposals of devices as well as introduction of new sub-topics prompted us to present the first roadmap on magnonics. This is a collection of 22 sections written by leading experts in this field who review and discuss the current status besides presenting their vision of future perspectives. Today, the principal challenges in applied magnonics are the excitation of sub-100 nm wavelength magnons, their manipulation on the nanoscale and the creation of sub-micrometre devices using low-Gilbert damping magnetic materials and its interconnections to standard electronics. To this end, magnonics offers lower energy consumption, easier integrability and compatibility with CMOS structure, reprogrammability, shorter wavelength, smaller device features, anisotropic properties, negative group velocity, non-reciprocity and efficient tunability by various external stimuli to name a few. Hence, despite being a young research field, magnonics has come a long way since its early inception. This roadmap asserts a milestone for future emerging research directions in magnonics, and hopefully, it will inspire a series of exciting new articles on the same topic in the coming years.

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

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

Spin-wave-based circuits and logic devices have been considered as an alternative to current electronic devices as they approach the physical limit of miniaturization. Asymmetrical propagation of spin waves, also known as nonreciprocity, provides an additional degree of freedom to these spin-wave-based devices, increasing their flexibility. In thin films, nonreciprocity can be induced by the Dzyaloshinskii-Moriya interaction (DMI) at heavy-metal/ferromagnet bilayers, and by the dipolar coupling in multilayers. Here, we show that in an antiferromagnetically coupled multilayer with interfacial DMI, the frequency nonreciprocity induced by the DMI is enhanced when both heavy metals are the same as long as the multilayer remains in an antiparallel state. Furthermore, we show that the interplay between the dipolar and Dzyaloshinskii-Moriya interaction enhances the nonreciprocity of one oscillation mode and reduces the nonreciprocity of the other. Which mode is enhanced depends on the sign of the induced Dzyaloshinskii-Moriya interaction at the interfaces and the magnetic moments of the layers. Finally, we show that it is possible to change the frequency nonreciprocity of Pt/Co/Cu/Co/Pt and Pt/Co/Cu/Py/Pt multilayers by similar to 7 GHz when applying an in-plane magnetic field of 130 mT. This includes a change in the sign of the nonreciprocity, which could be used to control the direction of the flux of information in spin-wave devices.

In this paper, by means of analytical calculations and a simple model, we studied the effect of curvature in magnetic thin stripes. Three different domain walls were observed, depending on the slab geometry. A phase diagram that shows the geometric parameters at which each domain wall is preferred was obtained. The presented methodology can also be used to obtain the magnetization behavior in other curved structures with proper aspect ratio relations.

Frequency nonreciprocity of wave phenomena describes the situation where the wave dispersion depends on the sign of the wave vector, i.e., counterpropagating waves exhibit different wavelengths for the same frequency. Such behavior has recently been observed in heavy-metal-ferromagnetic interfaces with Dzyaloshinskii-Moriya coupling, and is also known for coupled magnetic bilayers, where the nonreciprocity is enhanced when the two layers are aligned antiparallel. Besides the conventional uses of spin waves, nonreciprocity adds further functionalities, such as its potential applications in communication technologies and logic operations. In the current paper, we thus examine the spin-wave nonreciprocity induced by dipolar interactions in a coupled bilayer consisting of two ferromagnetic layers separated by a nonmagnetic spacer. We derive an easy-to-use formula to estimate the frequency difference provided by the nonreciprocity, which allows one to choose an optimal system in order to maximize the effect. For small wave numbers, the nonreciprocity scales linearly, while for larger wave vectors the nonreciprocity behaves nonmonotonically, with a well-defined maximum. The study is carried out by means of analytical calculations that are complemented by micromagnetic simulations. Furthermore, we confirm our model by experimental investigation of the spin-wave dispersion in a prototype antiparallel-coupled bilayer system. Since the relative magnetic orientation can be controlled through a bias field, the magnon nonreciprocity can then be turned on and off, which lends an important functionality to the coupled ferromagnetic bilayers.

Periodically patterned metamaterials are known for exhibiting wave properties similar to the ones observed in electronic band structures in crystal lattices. In particular, periodic ferromagnetic materials are characterized by the presence of bands and band gaps in their spin-wave spectrum at tunable GHz frequencies. Recently, the fabrication of magnets hosting Dzyaloshinskii-Moriya interactions has been pursued with high interest since properties, such as the stabilization of chiral spin textures and nonreciprocal spin-wave propagation, emerge from this antisymmetric exchange coupling. In this context, to further engineer the magnon band structure, we propose the implementation of magnonic crystals with periodic Dzyaloshinskii-Moriya interactions, which can be obtained, for instance, via patterning of periodic arrays of heavy metal wires on top of an ultrathin magnetic film. We demonstrate through theoretical calculations and micromagnetic simulations that such systems show an unusual evolution of the standing spin waves around the gaps. We also predict the emergence of indirect gaps and flat bands, effects that depend on the strength of the Dzyaloshinskii-Moriya interaction. Such phenomena, which have been previously observed in different systems, are observed here simultaneously, opening new routes towards engineered metamaterials for spin-wave-based devices.