Asymmetric spin-wave dispersion on Fe(110): direct evidence of the Dzyaloshinskii-Moriya interaction

Magnetization States and Coupled Spin-Wave Modes in Concentric Double Nanorings

Concentric multiple nanorings have previously been fabricated and investigated mainly for their different static magnetization states. Here, we present a theoretical analysis for the magnetization dynamics in double nanorings arranged concentrically, where there is coupling across a nonmagnetic spacer due to the long-range dipole-dipole interactions. We employ a microscopic, or Hamiltonian-based, formalism to study the discrete spin waves that exist in the magnetic states where the individual rings may be in either a vortex or an onion state. Numerical results are shown for the frequencies and the spatial amplitudes (with relative phase included) of the spin-wave modes. Cases are considered in which the magnetic materials of the rings are the same (taken to be permalloy) or two different materials such as permalloy and cobalt. The dependence of these properties on the mean radial position of the spacer were studied, showing, in most cases, the existence of two distinct transition fields. The special cases, where the radial spacer width becomes very small (less than 1 nm) were analyzed to study direct interfaces between dissimilar materials and/or effects of interfacial exchange interactions such as Ruderman-Kittel-Kasuya-Yoshida coupling. These spin-wave properties may be of importance for magnetic switching devices and sensors.

Generation and Propagation of Ultrafast Terahertz Magnons in Atomically Architectured Nanomagnets

Utilizing ultrafast terahertz (THz) magnons, the quanta of collective magnetic excitations, as carriers may provide a promising alternative to overcome the problems associated with electrical losses in nanoelectronic devices and circuits. However, efficient excitation of propagating coherent THz magnons in magnonic nanowaveguides is an essential requirement for the development of such devices. Here, by growing ultrathin ferromagnetic nanostructures on a reconstructed surface, we create well-ordered periodic magnetic nanostripes. We demonstrate that such atomically architectured nanowaveguides not only provide a versatile platform for an efficient generation of THz magnons but also allow for their fast propagation. Our results reveal the complex nature of the spin dynamics within such designed nanowaveguides and pave the way for designing ultrafast magnon-based logic devices with THz operation frequencies.

Asymmetric Magnon Frequency Comb

We theoretically show the asymmetric spin wave transmission in a coupled waveguide-skyrmion structure, where the skyrmion acts as an effective nanocavity allowing the whispering gallery modes for magnons. The asymmetry originates from the chiral spin wave mode localized in the circular skyrmion wall. By inputting two-tone excitations and mixing them in the skyrmion wall, we observe a unidirectional output magnon frequency comb propagating in the waveguide with a record number of teeth (>50). This coupled waveguide-cavity structure turns out to be a universal paradigm for generating asymmetric magnon frequency combs, where the cavity can be generalized to other magnetic structures that support the whispering gallery mode of magnons. Our results advance the understanding of the nonlinear interaction between magnons and magnetic textures and open a new pathway to exploring the asymmetric spin wave transmission and to steering the magnon frequency comb.

Giant Spin-Orbit Induced Magnon Nonreciprocity in Ultrathin Ferromagnets

The propagation characteristics of fermionic and bosonic quasiparticles determine the fundamental transport properties of solids and are of great technological relevance for designing logic devices. In particular, nonreciprocity, which describes that a quasiparticle flows preferably along a certain direction of a symmetry path, is an essential requirement to realize logic architectures, e.g., switches, diodes, transistors, etc. Here we introduce a mechanism, which leads to giant nonreciprocity of ultrafast terahertz magnons in ferromagnetic films with a large spin-orbit coupling. The mechanism is based on the competition between the exchange and spin-orbit scattering. We anticipate that the effect can be used to excite nonreciprocal or even unidirectional magnons in a large class of ultrathin films and nanostructures grown on substrates with a large spin-orbit coupling.

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.

Spin Waves in Ferromagnetic Nanorings with Interfacial Dzyaloshinskii-Moriya Interactions: II. Directional Effects

A theory is presented to study the effect of interfacial Dzyaloshinskii-Moriya interactions (DMIs) on the static and dynamic magnetic properties in single-layered ferromagnetic nanorings. A microscopic (Hamiltonian-based) approach is used that also includes the antisymmetric DMI besides the competing symmetric (bilinear) exchange interactions, magnetic dipole-dipole interactions, and an applied magnetic field. Here, the axial vector of the DMI is taken to be in the plane of the nanoring (by contrast with earlier studies) and we explore cases where it is either parallel or perpendicular to the in-plane magnetic field. Significantly, with this orientation for the DMI axial vector, the inhomogeneous static magnetization is tilted to have a component perpendicular to the plane giving a surface texture. This effect is studied in both the low-field vortex and high-field onion states. There is a consequent modification to the discrete set of spin-wave modes in both states through their frequencies and spatial amplitudes. We present combined analytical and numerical results for the static properties and dynamical magnetization in ferromagnetic nanorings, including the variation with applied field.

Topological magnons driven by the Dzyaloshinskii-Moriya interaction in the centrosymmetric ferromagnet Mn5Ge3

The phase of the quantum-mechanical wave function can encode a topological structure with wide-ranging physical consequences, such as anomalous transport effects and the existence of edge states robust against perturbations. While this has been exhaustively demonstrated for electrons, properties associated with the elementary quasiparticles in magnetic materials are still underexplored. Here, we show theoretically and via inelastic neutron scattering experiments that the bulk ferromagnet Mn5Ge3 hosts gapped topological Dirac magnons. Although inversion symmetry prohibits a net Dzyaloshinskii-Moriya interaction in the unit cell, it is locally allowed and is responsible for the gap opening in the magnon spectrum. This gap is predicted and experimentally verified to close by rotating the magnetization away from the c-axis with an applied magnetic field. Hence, Mn5Ge3 realizes a gapped Dirac magnon material in three dimensions. Its tunability by chemical doping or by thin film nanostructuring defines an exciting new platform to explore and design topological magnons. More generally, our experimental route to verify and control the topological character of the magnons is applicable to bulk centrosymmetric hexagonal materials, which calls for systematic investigation.

Probing of three-dimensional spin textures in multilayers by field dependent X-ray resonant magnetic scattering

In multilayers of magnetic thin films with perpendicular anisotropy, domain walls can take on hybrid configurations in the vertical direction which minimize the domain wall energy, with Néel walls in the top or bottom layers and Bloch walls in some central layers. These types of textures are theoretically predicted, but their observation has remained challenging until recently, with only a few techniques capable of realizing a three dimensional characterization of their magnetization distribution. Here we perform a field dependent X-ray resonant magnetic scattering measurements on magnetic multilayers exploiting circular dichroism contrast to investigate such structures. Using a combination of micromagnetic and X-ray resonant magnetic scattering simulations along with our experimental results, we characterize the three-dimensional magnetic texture of domain walls, notably the thickness resolved characterization of the size and position of the Bloch part in hybrid walls. We also take a step in advancing the resonant scattering methodology by using measurements performed off the multilayer Bragg angle in order to calibrate the effective absorption of the X-rays, and permitting a quantitative evaluation of the out of plane (z) structure of our samples. Beyond hybrid domain walls, this approach can be used to characterize other periodic chiral structures such as skyrmions, antiskyrmions or even magnetic bobbers or hopfions, in both static and dynamic experiments.

Nonreciprocal collective magnetostatic wave modes in geometrically asymmetric bilayer structure with nonmagnetic spacer

Nonreciprocity, i.e. inequivalence in amplitudes and frequencies of spin waves propagating in opposite directions, is a key property underlying functionality in prospective magnonic devices. Here we demonstrate experimentally and theoretically a simple approach to induce frequency nonreciprocity in a magnetostatically coupled ferromagnetic bilayer structure with a nonmagnetic spacer by its geometrical asymmetry. Using Brillouin light scattering, we show the formation of two collective spin wave modes in Fe₈₁Ga₁₉/Cu/Fe₈₁Ga₁₉ structure with different thicknesses of ferromagnetic layers. Experimental reconstruction and theoretical modeling of the dispersions of acoustic and optical collective spin wave modes reveal that both possess nonreciprocity reaching several percent at the wavenumber of 22 × 10⁴ rad cm^-1. The analysis demonstrates that the shift of the amplitudes of counter-propagating coupled modes towards either of the layers is responsible for the nonreciprocity because of the pronounced dependence of spin wave frequency on the layers' thickness. The proposed approach enables the design of multilayered ferromagnetic structures with a given spin wave dispersion for magnonic logic gates.

Omnidirectional flat bands in chiral magnonic crystals

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.

Spanning Fermi arcs in a two-dimensional magnet

The discovery of topological states of matter has led to a revolution in materials research. When external or intrinsic parameters break symmetries, global properties of topological materials change drastically. A paramount example is the emergence of Weyl nodes under broken inversion symmetry. While a rich variety of non-trivial quantum phases could in principle also originate from broken time-reversal symmetry, realizing systems that combine magnetism with complex topological properties is remarkably elusive. Here, we demonstrate that giant open Fermi arcs are created at the surface of ultrathin hybrid magnets where the Fermi-surface topology is substantially modified by hybridization with a heavy-metal substrate. The interplay between magnetism and topology allows us to control the shape and the location of the Fermi arcs by tuning the magnetization direction. The hybridization points in the Fermi surface can be attributed to a non-trivial mixed topology and induce hot-spots in the Berry curvature, dominating spin and charge transport as well as magneto-electric coupling effects.

It has been predicted that elemental Iron, with low dimensionality, will be a topological metal hosting Weyl nodes. Here, Chen et al. grow iron on tungsten, a heavy metal with a strong spin-orbit interaction, and using momentum microscopy, show the emergence of giant open Fermi arcs which can be shaped by varying the magnetization of the iron.

Magnonic Activity of Circularly Magnetized Ferromagnetic Nanotubes Induced by Dzyalonshinskii-Moriya Interaction

Magnonic activity, a chiral effect in magnetization dynamics, was recently reported in ferromagnetic nanotubes. Being a perfect analogy to the optical activity, it refers to the continuous rotation of a standing-waves pattern formed in the circumferential direction during the wave propagation along the tube. This effect only occurs when the tube is longitudinally magnetized. Here we report that a similar phenomenon can also take place in circularly magnetized nanotubes with the presence of Dzyalonshinskii-Moriya interaction (DMI). While in the former case, the chiral-symmetry breaking is caused by the curvilinear shape of the tube, it is attributed to the intrinsic asymmetry of the DMI in the latter one. We present the results obtained in both numerical simulations and semi-analytical calculations, which are in great agreement. This work provides new aspects for the manipulation of spin waves, which may bear potential applications in the development of novel spintronic devices.

Fundamentals of Curvilinear Ferromagnetism: Statics and Dynamics of Geometrically Curved Wires and Narrow Ribbons

Low-dimensional magnetic architectures including wires and thin films are key enablers of prospective ultrafast and energy efficient memory, logic, and sensor devices relying on spin-orbitronic and magnonic concepts. Curvilinear magnetism emerged as a novel approach in material science, which allows tailoring of the fundamental anisotropic and chiral responses relying on the geometrical curvature of magnetic architectures. Much attention is dedicated to magnetic wires of Möbius, helical, or DNA-like double helical shapes, which act as prototypical objects for the exploration of the fundamentals of curvilinear magnetism. Although there is a bulk number of original publications covering fabrication, characterization, and theory of magnetic wires, there is no comprehensive review of the theoretical framework of how to describe these architectures. Here, theoretical activities on the topic of curvilinear magnetic wires and narrow nanoribbons are summarized, providing a systematic review of the emergent interactions and novel physical effects caused by the curvature. Prospective research directions of curvilinear spintronics and spin-orbitronics are discussed, the fundamental framework for curvilinear magnonics are outlined, and mechanically flexible curvilinear architectures for soft robotics are introduced.

Tuning Dzyaloshinskii-Moriya interaction via an electric field at the Co/h-BN interface

The Dzyaloshinsky-Moriya interaction (DMI) at the Co/h-BN interface can emerge and be enhanced by applying a downward electric field. The height of the Co atom relative to the h-BN layer with the electric field determines the variation of DMI. One half reduction of J₁ is beneficial to generate skyrmions. Tuning the DMI by an electric field sheds new light for research on skyrmions.

Strain-Driven Dzyaloshinskii-Moriya Interaction for Room-Temperature Magnetic Skyrmions

Dzyaloshinskii-Moriya interaction in magnets, which is usually derived from inversion symmetry breaking at interfaces or in noncentrosymmetric crystals, plays a vital role in chiral spintronics. Here we report that an emergent Dzyaloshinskii-Moriya interaction can be achieved in a centrosymmetric material, La_{0.67}Sr_{0.33}MnO_{3}, by a graded strain. This strain-driven Dzyaloshinskii-Moriya interaction not only exhibits distinctive two coexisting nonreciprocities of spin-wave propagation in one system, but also brings about a robust room-temperature magnetic skyrmion lattice as well as a spiral lattice at zero magnetic field. Our results demonstrate the feasibility of investigating chiral spintronics in a large category of centrosymmetric magnetic materials.

Nonreciprocal Transmission of Incoherent Magnons with Asymmetric Diffusion Length

Unequal transmissions of spin waves along opposite directions provide useful functions for signal processing. So far, the realization of such nonreciprocal spin waves has been mostly limited at a gigahertz frequency in the coherent regime via microwave excitation. Here we show that, in a magnetic bilayer stack with chiral coupling, tunable nonreciprocal propagation can be realized in spin Hall effect-excited incoherent magnons, whose frequencies cover the spectrum from a few gigahertz up to terahertz. The sign of nonreciprocity is controlled by the magnetic orientations of the bilayer in a nonvolatile manner. The nonreciprocity is further verified by measurements of the magnon diffusion length, which is unequal along opposite transmission directions. Our findings enrich the knowledge on magnetic relaxation and diffusive transport and can lead to the design of a passive directional signal isolation device in the diffusive regime.

Nonreciprocity of spin waves in the conical helix state

Nonreciprocity emerges in nature and in artificial objects from various physical origins, being widely utilized in contemporary technologies as exemplified by diode elements in electronics. While most of the nonreciprocal phenomena are realized by employing interfaces where the inversion symmetry is trivially lifted, nonreciprocal transport of photons, electrons, magnons, and possibly phonons also emerge in bulk crystals with broken space inversion and time reversal symmetries. Among them, directional propagation of bulk magnons (i.e., quanta of spin wave excitation) is attracting much attention nowadays for its potentially large nonreciprocity suitable for spintronic and spin-caloritronic applications. Here, we demonstrate nonreciprocal propagation of spin waves for the conical spin helix state in Cu₂OSeO₃ due to a combination of dipole and Dzyaloshinskii-Moriya interactions. The observed nonreciprocal spin dispersion smoothly connects to the hitherto known magnetochiral nonreciprocity in the field-induced collinear spin state; thus, all the spin phases show diode characteristics in this chiral insulator.

Steering magnonic dynamics and permeability at exceptional points in a parity-time symmetric waveguide

Tuning the magneto optical response and magnetic dynamics are key elements in designing magnetic metamaterials and devices. This theoretical study uncovers a highly effective way of controlling the magnetic permeability via shaping the magnonic properties of coupled magnetic waveguides separated by a nonmagnetic spacer with strong spin–orbit interaction (SOI). We demonstrate how a spacer charge current leads to enhancement of magnetic damping in one waveguide and a decrease in the other, constituting a bias-controlled magnetic parity–time (PT) symmetric system at the verge of the exceptional point where magnetic gains/losses are balanced. We find phenomena inherent to PT-symmetric systems and SOI-driven interfacial structures, including field-controlled magnon power oscillations, nonreciprocal propagation, magnon trapping and enhancement as well as an increased sensitivity to perturbations and abrupt spin reversal. The results point to a new route for designing magnonic waveguides and microstructures with enhanced magnetic response.

The ability to guide and control magnons is central to their potential in future information processing. Here, using a combination of computations and analytical approaches, the authors propose a magnonic waveguide with a unique gain and loss mechanism.

Nonreciprocal thermal transport in a multiferroic helimagnet

Breaking of spatial inversion symmetry induces unique phenomena in condensed matter. In particular, by combining this symmetry with magnetic fields or another type of time-reversal symmetry breaking, noncentrosymmetric materials can be made to exhibit nonreciprocal responses, which are responses that differ for rightward and leftward stimuli. However, the effect of spatial inversion symmetry breaking on thermal transport in uniform media remains to be elucidated. Here, we show nonreciprocal thermal transport in the multiferroic helimagnet TbMnO₃ The longitudinal thermal conductivity depends on whether the thermal current is parallel or antiparallel to the vector product of the electric polarization and magnetization. This phenomenon is thermal rectification that is controllable with external fields in a uniform crystal. This discovery may pave the way to thermal diodes with controllability and scalability.

Magnonic crystals: towards terahertz frequencies

This topical review presents an overview of the recent experimental and theoretical attempts on designing magnonic crystals for operation at different frequencies. The focus is put on the microscopic physical mechanisms involved in the formation of the magnonic band structure, allowed as well as forbidden magnon states in various systems, including ultrathin films, multilayers and artificial magnetic structures. The essential criteria for the formation of magnonic bandgaps in different frequency regimes are explained in connection with the magnon dynamics in such structures. The possibility of designing small-size magnonic crystals for operation at ultrahigh frequencies (terahertz and sub-terahertz regime) is discussed. Recently discovered magnonic crystals based on topological defects and using periodic Dzyaloshinskii-Moriya interaction, are outlined. Different types of magnonic crystals, capable of operation at different frequency regimes, are put within a rather unified picture.

Switchable giant nonreciprocal frequency shift of propagating spin waves in synthetic antiferromagnets

The nonreciprocity of propagating spin waves, i.e., the difference in amplitude and/or frequency depending on the propagation direction, is essential for the realization of spin wave-based logic circuits. However, the nonreciprocal frequency shifts demonstrated so far are not large enough for applications because they originate from interfacial effects. In addition, switching of the spin wave nonreciprocity in the electrical way remains a challenging issue. Here, we show a switchable giant nonreciprocal frequency shift of propagating spin waves in interlayer exchange-coupled synthetic antiferromagnets. The observed frequency shift is attributed to large asymmetric spin wave dispersion caused by a mutual dipolar interaction between two magnetic layers. Furthermore, we find that the sign of the frequency shift depends on relative configuration of two magnetizations, based on which we demonstrate an electrical switching of the nonreciprocity. Our findings provide a route for switchable and highly nonreciprocal spin wave-based applications.

Chiral Spin-Wave Velocities Induced by All-Garnet Interfacial Dzyaloshinskii-Moriya Interaction in Ultrathin Yttrium Iron Garnet Films

Spin waves can probe the Dzyaloshinskii-Moriya interaction (DMI), which gives rise to topological spin textures, such as skyrmions. However, the DMI has not yet been reported in yttrium iron garnet (YIG) with arguably the lowest damping for spin waves. In this work, we experimentally evidence the interfacial DMI in a 7-nm-thick YIG film by measuring the nonreciprocal spin-wave propagation in terms of frequency, amplitude, and most importantly group velocities using all electrical spin-wave spectroscopy. The velocities of propagating spin waves show chirality among three vectors, i.e., the film normal direction, applied field, and spin-wave wave vector. By measuring the asymmetric group velocities, we extract a DMI constant of 16  μJ/m^{2}, which we independently confirm by Brillouin light scattering. Thickness-dependent measurements reveal that the DMI originates from the oxide interface between the YIG and garnet substrate. The interfacial DMI discovered in the ultrathin YIG films is of key importance for functional chiral magnonics as ultralow spin-wave damping can be achieved.

Correlation between Dzyaloshinskii-Moriya interaction and orbital angular momentum at an oxide-ferromagnet interface

We report on the Dzyaloshinskii-Moriya (DMI) interaction at the interface between a ferromagnet and an oxide. We demonstrate experimentally that oxides can give rise to DMI. By comparison of systems comprised of Pt/Co90Fe10/oxide and Cu/Co90Fe10/oxide, we also show how oxidation of one interface can enhance and add to the total DMI of that generated by the Pt interface. This is due to the fact that the DMI on both interfaces promotes left-handed chirality. Finally, by use of ferromagnetic resonance spectroscopy, we show that the DMI and the spectroscopic splitting factor, which is a measure of the orbital momentum, are correlated. This indicates the importance of hybridization and charge transfer at the oxide interface for the DMI.

Experimental Realization of Atomic-Scale Magnonic Crystals

We introduce a new approach of materials design for terahertz magnonics making use of quantum confinement of terahertz magnons in layered ferromagnets. We show that in atomically designed multilayers composed of alternating atomic layers of ferromagnetic metals one can efficiently excite different magnon modes associated with the quantum confinement in the third dimension, i.e., the direction perpendicular to the layers. We demonstrate experimentally that the magnonic band structure of these systems can be tuned by changing the material combination and the number of atomic layers. We realize the idea of opening band gaps, with a size of up to several tens of millielectronvolts, between different terahertz magnon bands and thereby report on the first step toward the realization of atomic-scale magnonic crystals.

Exceptional points in classical spin dynamics

Non-conservative physical systems admit a special kind of spectral degeneracy, known as exceptional point (EP), at which eigenvalues and eigenvectors of the corresponding non-Hermitian Hamiltonian coalesce. Dynamical parametric encircling of the EP can lead to non-adiabatic evolution associated with a state flip, a sharp transition between the resonant modes. Physical consequences of the dynamical encircling of EPs in open dissipative systems have been explored in optics and photonics. Building on the recent progress in understanding the parity-time ([Formula: see text])-symmetric dynamics in spin systems, we use topological properties of EPs to implement chiral non-reciprocal transmission of a spin through the material with non-uniform magnetization, like helical magnet. We consider an exemplary system, spin-torque-driven single spin described by the time-dependent non-Hermitian Hamiltonian. We show that encircling individual EPs in a parameter space results in non-reciprocal spin dynamics and find the range of optimal protocol parameters for high-efficiency asymmetric spin filter based on this effect. Our findings offer a platform for non-reciprocal spin devices for spintronics and magnonics.

Flat Bands, Indirect Gaps, and Unconventional Spin-Wave Behavior Induced by a Periodic Dzyaloshinskii-Moriya Interaction

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.

Spin waves in disordered materials

We present an efficient methodology to study spin waves in disordered materials. The approach is based on a Heisenberg model and enables calculations of magnon properties in spin systems with disorder of an arbitrary kind and concentration of impurities. Disorder effects are taken into account within two complementary approaches. Magnons in systems with substitutional (uncorrelated) disorder can be efficiently calculated within a single-site coherent potential approximation for the Heisenberg model. From the computation point of view the method is inexpensive and directly applicable to systems like alloys and doped materials. It is shown that it performs exceedingly well across all concentrations and wave vectors. Another way is the direct numerical simulation of large supercells using a configurational average over possible samples. This approach is applicable to systems with an arbitrary kind of disorder. The effective interaction between magnetic moments entering the Heisenberg model can be obtained from first-principles using a self-consistent Green function method within the density functional theory. Thus, our method can be viewed as an ab initio approach and can be used for calculations of magnons in real materials.

Asymmetric and Symmetric Exchange in a Generalized 2D Rashba Ferromagnet

Dzyaloshinskii-Moriya interaction (DMI) is investigated in a 2D ferromagnet (FM) with spin-orbit interaction of Rashba type at finite temperatures. The FM is described in the continuum limit by an effective s-d model with arbitrary dependence of spin-orbit coupling (SOC) and kinetic energy of itinerant electrons on the absolute value of momentum. In the limit of weak SOC, we derive a general expression for the DMI constant D from a microscopic analysis of the electronic grand potential. We compare D with the exchange stiffness A and show that, to the leading order in small SOC strength α_{R}, the conventional relation D=(4mα_{R}/ℏ)A, in general, does not hold beyond the Bychkov-Rashba model. Moreover, in this model, both A and D vanish at zero temperature in the metal regime (i.e., when two spin sub-bands are partly occupied). For nonparabolic bands or nonlinear Rashba coupling, these coefficients are finite and acquire a nontrivial dependence on the chemical potential that demonstrates the possibility to control the size and chirality of magnetic textures by adjusting a gate voltage.

Effects of the Dzyaloshinsky-Moriya interaction on nonequilibrium thermodynamics in the XY chain in a transverse field

We examine the effects of the Dzyaloshinsky-Moriya (DM) interaction on the nonequilibrium thermodynamics in an anisotropic XY spin chain, which is driven out of equilibrium by a sudden quench of the control parameter of the Hamiltonian. By analytically evaluating the statistical properties of the work distribution and the irreversible entropy production, we investigate the influences of the DM interaction on the nonequilibrium thermodynamics of the system with different parameters at various temperatures. We find that depending on the anisotropy of the system and the temperature, the DM interaction may have different impacts on the nonequilibrium thermodynamics. Interestingly, the critical line induced by the DM interaction can be revealed via the properties of the nonequilibrium thermodynamics. In addition, our results suggest that the strength of the DM interaction can be detected experimentally by studying the nonequilibrium thermodynamics.

Theory of the Interfacial Dzyaloshinskii-Moriya Interaction in Rashba Antiferromagnets

In antiferromagnetic (AFM) thin films, broken inversion symmetry or coupling to adjacent heavy metals can induce Dzyaloshinskii-Moriya (DM) interactions. Knowledge of the DM parameters is essential for understanding and designing exotic spin structures, such as hedgehog Skyrmions and chiral Néel walls, which are attractive for use in novel information storage technologies. We introduce a framework for computing the DM interaction in two-dimensional Rashba antiferromagnets. Unlike in Rashba ferromagnets, the DM interaction is not suppressed even at low temperatures. The material parameters control both the strength and the sign of the interfacial DM interaction. Our results suggest a route toward controlling the DM interaction in AFM materials by means of doping and electric fields.

The interfacial nature of proximity-induced magnetism and the Dzyaloshinskii-Moriya interaction at the Pt/Co interface

The Dzyaloshinskii-Moriya interaction has been shown to stabilise Nèel domain walls in magnetic thin films, allowing high domain wall velocities driven by spin current effects. The interfacial Dzyaloshinskii-Moriya interaction (IDMI) occurs at the interface between ferromagnetic and heavy metal layers with strong spin-orbit coupling, but details of the interaction remain to be understood and the role of proximity induced magnetism (PIM) in the heavy metal is unknown. Here IDMI and PIM are reported in Pt determined as a function of Au and Ir spacer layers in Pt/Co/Au,Ir/Pt. Both interactions are found to be sensitive to sub-nanometre changes in the spacer thickness, correlating over sub-monolayer spacer thicknesses, but not for thicker spacers where IDMI continues to change even after PIM is lost.

Emergent dynamic chirality in a thermally driven artificial spin ratchet

Modern nanofabrication techniques have opened the possibility to create novel functional materials, whose properties transcend those of their constituent elements. In particular, tuning the magnetostatic interactions in geometrically frustrated arrangements of nanoelements called artificial spin ice can lead to specific collective behaviour, including emergent magnetic monopoles, charge screening and transport, as well as magnonic response. Here, we demonstrate a spin-ice-based active material in which energy is converted into unidirectional dynamics. Using X-ray photoemission electron microscopy we show that the collective rotation of the average magnetization proceeds in a unique sense during thermal relaxation. Our simulations demonstrate that this emergent chiral behaviour is driven by the topology of the magnetostatic field at the edges of the nanomagnet array, resulting in an asymmetric energy landscape. In addition, a bias field can be used to modify the sense of rotation of the average magnetization. This opens the possibility of implementing a magnetic Brownian ratchet, which may find applications in novel nanoscale devices, such as magnetic nanomotors, actuators, sensors or memory cells.

Antiskyrmions stabilized at interfaces by anisotropic Dzyaloshinskii-Moriya interactions

Chiral magnets are an emerging class of topological matter harboring localized and topologically protected vortex-like magnetic textures called skyrmions, which are currently under intense scrutiny as an entity for information storage and processing. Here, on the level of micromagnetics we rigorously show that chiral magnets can not only host skyrmions but also antiskyrmions as least energy configurations over all non-trivial homotopy classes. We derive practical criteria for their occurrence and coexistence with skyrmions that can be fulfilled by (110)-oriented interfaces depending on the electronic structure. Relating the electronic structure to an atomistic spin-lattice model by means of density functional calculations and minimizing the energy on a mesoscopic scale by applying spin-relaxation methods, we propose a double layer of Fe grown on a W(110) substrate as a practical example. We conjecture that ultra-thin magnetic films grown on semiconductor or heavy metal substrates with C 2v symmetry are prototype classes of materials hosting magnetic antiskyrmions.Skyrmions, localized defects in the magnetization, can be stabilised in materials by the Dzyaloshinskii-Moriya interaction (DMI). Hoffmann et al. predict that, when the DMI is anisotropic, antiskyrmions can be formed and coexist with skyrmions, enabling studies and exploitation of their interactions.

Theory of magnons in spin systems with Dzyaloshinskii-Moriya interaction

We study in this paper magnetic properties of a system of quantum Heisenberg spins interacting with each other via a ferromagnetic exchange interaction J and an in-plane Dzyaloshinskii-Moriya interaction D. The non-collinear ground state due to the competition between J and D is determined. We employ a self-consistent Green'function theory to calculate the spin-wave spectrum and the layer magnetizations at finite T in two and three dimensions as well as in a thin film with surface effects. Analytical details and the validity of the method are shown and discussed. Numerical solutions are shown for realistic physical interaction parameters. Discussion on possible experimental verifications is given.

Interfacial Dzyaloshinskii-Moriya Interaction in Pt/CoFeB Films: Effect of the Heavy-Metal Thickness

We report the observation of a Pt layer thickness dependence on the induced interfacial Dzyaloshinskii-Moriya interaction in ultrathin Pt(d_{Pt})/CoFeB films. Taking advantage of the large spin-orbit coupling of the heavy metal, the interfacial Dzyaloshinskii-Moriya interaction is quantified by Brillouin light scattering measurements of the frequency nonreciprocity of spin waves in the ferromagnet. The magnitude of the induced Dzyaloshinskii-Moriya coupling is found to saturate to a value of 0.45  mJ/m^{2} for Pt thicknesses larger than ∼2  nm. The experimental results are explained by analytical calculations based on the three-site indirect exchange mechanism that predicts a Dzyaloshinskii-Moriya interaction at the interface between a ferromagnetic thin layer and a heavy metal. Our findings open up a way to control and optimize chiral effects in ferromagnetic thin films through the thickness of the heavy-metal layer.

Interface-Induced Phenomena in Magnetism

This article reviews static and dynamic interfacial effects in magnetism, focusing on interfacially-driven magnetic effects and phenomena associated with spin-orbit coupling and intrinsic symmetry breaking at interfaces. It provides a historical background and literature survey, but focuses on recent progress, identifying the most exciting new scientific results and pointing to promising future research directions. It starts with an introduction and overview of how basic magnetic properties are affected by interfaces, then turns to a discussion of charge and spin transport through and near interfaces and how these can be used to control the properties of the magnetic layer. Important concepts include spin accumulation, spin currents, spin transfer torque, and spin pumping. An overview is provided to the current state of knowledge and existing review literature on interfacial effects such as exchange bias, exchange spring magnets, spin Hall effect, oxide heterostructures, and topological insulators. The article highlights recent discoveries of interface-induced magnetism and non-collinear spin textures, non-linear dynamics including spin torque transfer and magnetization reversal induced by interfaces, and interfacial effects in ultrafast magnetization processes.

Probing of the interfacial Heisenberg and Dzyaloshinskii-Moriya exchange interaction by magnon spectroscopy

This Topical Review presents an overview of the recent experimental results on the quantitative determination of the magnetic exchange parameters in ultrathin magnetic films and multilayers grown on different substrates. The experimental approaches for probing both the symmetric Heisenberg and the antisymmetric Dzyaloshinskii-Moriya exchange interaction in ultrathin magnetic films and at interfaces are discussed in detail. It is explained how the experimental spectrum of magnetic excitations can be used to quantify the strength of these interactions.

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.

Spin Wave Power Flow and Caustics in Ultrathin Ferromagnets with the Dzyaloshinskii-Moriya Interaction

The Dzyaloshinskii-Moriya interaction in ultrathin ferromagnets can result in nonreciprocal propagation of spin waves. We examine theoretically how spin wave power flow is influenced by this interaction. We show that the combination of the dipole-dipole and Dzyaloshinskii-Moriya interactions can result in unidirectional caustic beams in the Damon-Eshbach geometry. Morever, self-generated interface patterns can also be induced from a point-source excitation.

Spin-Orbit Twisted Spin Waves: Group Velocity Control

We present a theoretical and experimental study of the interplay between spin-orbit coupling (SOC), Coulomb interaction, and motion of conduction electrons in a magnetized two-dimensional electron gas. Via a transformation of the many-body Hamiltonian we introduce the concept of spin-orbit twisted spin waves, whose energy dispersions and damping rates are obtained by a simple wave-vector shift of the spin waves without SOC. These theoretical predictions are validated by Raman scattering measurements. With optical gating of the density, we vary the strength of the SOC to alter the group velocity of the spin wave. The findings presented here differ from that of spin systems subject to the Dzyaloshinskii-Moriya interaction. Our results pave the way for novel applications in spin-wave routing devices and for the realization of lenses for spin waves.

Asymmetric Hysteresis for Probing Dzyaloshinskii-Moriya Interaction

The interfacial Dzyaloshinskii-Moriya interaction (DMI) is intimately related to the prospect of superior domain-wall dynamics and the formation of magnetic skyrmions. Although some experimental efforts have been recently proposed to quantify these interactions and the underlying physics, it is still far from trivial to address the interfacial DMI. Inspired by the reported tilt of the magnetization of the side edge of a thin film structure, we here present a quasi-static, straightforward measurement tool. By using laterally asymmetric triangular-shaped microstructures, it is demonstrated that interfacial DMI combined with an in-plane magnetic field yields a unique and significant shift in magnetic hysteresis. By systematic variation of the shape of the triangular objects combined with a droplet model for domain nucleation, a robust value for the strength and sign of interfacial DMI is obtained. This method gives immediate and quantitative access to DMI, enabling a much faster exploration of new DMI systems for future nanotechnology.

Spin-Cherenkov effect in a magnetic nanostrip with interfacial Dzyaloshinskii-Moriya interaction

Spin-Cherenkov effect enables strong excitations of spin waves (SWs) with nonlinear wave dispersions. The Dzyaloshinskii-Moriya interaction (DMI) results in anisotropy and nonreciprocity of SWs propagation. In this work, we study the effect of the interfacial DMI on SW Cherenkov excitations in permalloy thin-film strips within the framework of micromagnetism. By performing micromagnetic simulations, it is shown that coherent SWs are excited when the velocity of a moving magnetic source exceeds the propagation velocity of the SWs. Moreover, the threshold velocity of the moving magnetic source with finite DMI can be reduced compared to the case of zero DMI. It thereby provides a promising route towards efficient spin wave generation and propagation, with potential applications in spintronic and magnonic devices.

Gauge invariance and geometric phase in nonequilibrium thermodynamics

We show the link between U(1) lattice gauge theories and the off-equilibrium thermodynamics of a large class of nonlinear oscillators networks. The coupling between the oscillators plays the role of a gauge field, or connection, on the network. The thermodynamical forces that drive energy flows are expressed in terms of the curvature of the connection, analogous to a geometric phase. The model, which holds both close and far from equilibrium, predicts the existence of persistent energy and particle currents circulating in closed loops through the network. The predictions are confirmed by numerical simulations. Possible extension of the theory and experimental applications to nanoscale devices are briefly discussed.

All-Electrical Measurement of Interfacial Dzyaloshinskii-Moriya Interaction Using Collective Spin-Wave Dynamics

Dzyaloshinskii-Moriya interaction (DMI), which arises from the broken inversion symmetry and spin-orbit coupling, is of prime interest as it leads to a stabilization of chiral magnetic order and provides an efficient manipulation of magnetic nanostructures. Here, we report all-electrical measurement of DMI using propagating spin wave spectroscopy based on the collective spin wave with a well-defined wave vector. We observe a substantial frequency shift of spin waves depending on the spin chirality in Pt/Co/MgO structures. After subtracting the contribution from other sources to the frequency shift, it is possible to quantify the DMI energy in Pt/Co/MgO systems. The result reveals that the DMI in Pt/Co/MgO originates from the interfaces, and the sign of DMI corresponds to the inversion asymmetry of the film structures. The electrical excitation and detection of spin waves and the influence of interfacial DMI on the collective spin-wave dynamics will pave the way to the emerging field of spin-wave logic devices.

New perspectives for Rashba spin-orbit coupling

In 1984, Bychkov and Rashba introduced a simple form of spin-orbit coupling to explain the peculiarities of electron spin resonance in two-dimensional semiconductors. Over the past 30 years, Rashba spin-orbit coupling has inspired a vast number of predictions, discoveries and innovative concepts far beyond semiconductors. The past decade has been particularly creative, with the realizations of manipulating spin orientation by moving electrons in space, controlling electron trajectories using spin as a steering wheel, and the discovery of new topological classes of materials. This progress has reinvigorated the interest of physicists and materials scientists in the development of inversion asymmetric structures, ranging from layered graphene-like materials to cold atoms. This Review discusses relevant recent and ongoing realizations of Rashba physics in condensed matter.

Thickness dependence of the interfacial Dzyaloshinskii-Moriya interaction in inversion symmetry broken systems

In magnetic multilayer systems, a large spin-orbit coupling at the interface between heavy metals and ferromagnets can lead to intriguing phenomena such as the perpendicular magnetic anisotropy, the spin Hall effect, the Rashba effect, and especially the interfacial Dzyaloshinskii-Moriya (IDM) interaction. This interfacial nature of the IDM interaction has been recently revisited because of its scientific and technological potential. Here we demonstrate an experimental technique to straightforwardly observe the IDM interaction, namely Brillouin light scattering. The non-reciprocal spin wave dispersions, systematically measured by Brillouin light scattering, allow not only the determination of the IDM energy densities beyond the regime of perpendicular magnetization but also the revelation of the inverse proportionality with the thickness of the magnetic layer, which is a clear signature of the interfacial nature. Altogether, our experimental and theoretical approaches involving double time Green's function methods open up possibilities for exploring magnetic hybrid structures for engineering the IDM interaction.

Atomistic spin dynamics and surface magnons

Atomistic spin dynamics simulations have evolved to become a powerful and versatile tool for simulating dynamic properties of magnetic materials. It has a wide range of applications, for instance switching of magnetic states in bulk and nano-magnets, dynamics of topological magnets, such as skyrmions and vortices and domain wall motion. In this review, after a brief summary of the existing investigation tools for the study of magnons, we focus on calculations of spin-wave excitations in low-dimensional magnets and the effect of relativistic and temperature effects in such structures. In general, we find a good agreement between our results and the experimental values. For material specific studies, the atomistic spin dynamics is combined with electronic structure calculations within the density functional theory from which the required parameters are calculated, such as magnetic exchange interactions, magnetocrystalline anisotropy, and Dzyaloshinskii-Moriya vectors.

Narrow Magnonic Waveguides Based on Domain Walls

The channeling of spin waves with domain walls in ultrathin ferromagnetic films is demonstrated theoretically and through micromagnetics simulations. It is shown that propagating excitations localized to the wall, which appear in the frequency gap of bulk spin wave modes, can be guided in curved geometries and propagate in close proximity to other channels. For Néel-type walls arising from a Dzyaloshinskii-Moriya interaction, the channeling is strongly nonreciprocal and group velocities can exceed 1  km/s in the long wavelength limit for certain propagation directions. The channeled modes represent an unusual analogy of whispering gallery waves that are one dimensional and nonreciprocal with this interaction. Moreover, a sufficiently strong Dzyaloshinskii-Moriya interaction can create a degeneracy of channeled and propagating modes at a critical wave vector.

Magnon-driven domain-wall motion with the Dzyaloshinskii-Moriya interaction

We study domain-wall (DW) motion induced by spin waves (magnons) in the presence of the Dzyaloshinskii-Moriya interaction (DMI). The DMI exerts a torque on the DW when spin waves pass through the DW, and this torque represents a linear momentum exchange between the spin wave and the DW. Unlike angular momentum exchange between the DW and spin waves, linear momentum exchange leads to a rotation of the DW plane rather than a linear motion. In the presence of an effective easy plane anisotropy, this DMI induced linear momentum transfer mechanism is significantly more efficient than angular momentum transfer in moving the DW.

Direct observation of the Dzyaloshinskii-Moriya interaction in a Pt/Co/Ni film

The interfacial Dzyaloshinskii-Moriya interaction in an in-plane anisotropic Pt(4  nm)/Co(1.6  nm)/Ni(1.6  nm) film has been directly observed by Brillouin spectroscopy. It is manifested as the asymmetry of the measured magnon dispersion relation, from which the Dzyaloshinskii-Moriya interaction constant has been evaluated. Linewidth measurements reveal that the lifetime of the magnons is asymmetric with respect to their counter-propagating directions. The lifetime asymmetry is dependent on the magnon frequency, being more pronounced, the higher the frequency. Analytical calculations of the magnon dispersion relation and linewidth agree well with experiments.