United States

We develop a generic all-inductive procedure to measure the dispersion relation of spin waves in a magnetic stripe. Our method works even if several spin wave branches coexist in the investigated frequency interval, provided that the branches possess sufficiently different group velocities. We first measure the microwave scattering matrix of a network composed of distant antennas inductively coupled to the spin wave bath of a magnetic stripe. The antenna-to-antenna coupling that is independent from the applied magnetic field are suppressed by an appropriate calibration to get the complex spin wave transmission coefficient versus frequency. A comparison with the expected contribution of when a single spin wave mode exists argues for the existence of several spin wave branches in the magnetic conduit (Fig. 1). After a mathematical transformation to the time-domain to get the transmission impulse response, the different spin wave branches are viewed as wavepackets that reach successively the receiving antenna after different travel times (Fig. 2) determined by their group velocities. In analogy with time-of flight spectroscopy, the wavepackets are then separated by time-gating. The time-gated responses are used to recalculate the contribution of each spin wave branch to the frequency domain scattering matrix. The dispersion relation of each branch stems from the absolute phase of the time-gated transmission parameter. The spin wave wavevector can be determined unambiguously if the results for several propagation distances are combined, so as to get the dispersion relations of each band, and in the end the full band structure of the spin waves [1].

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Measuring the dispersion relations of spin wave bands using time-of-flight spectroscopy

2022 Joint MMM-INTERMAG

Measuring the dispersion relations of spin wave bands using time-of-flight spectroscopy

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

technical paper

Optical frequency combs are well established and powerful tools for synchronization, stabilization and frequency conversion in both fundamental science and technological applications. In this work, we present experimental observations on the generation of spin-wave frequency combs. Nonlinear four-magnon scattering is known to cause parasitic losses for a driven spin-wave mode with respect to coherence and amplitude. By means of Brillouin light scattering microscopy, we demonstrate that adding a second microwave frequency causes the scattering to populate well-defined states instead of redistributing energy to the entire spin-wave band. Here, the amplitudes and frequencies of both microwave sources can be utilised in order to tune this behaviour. When reducing the sizes of the magnetic conduits, lateral confinement causes a discretisation of the eigenmode spectrum. Therefore, the formation of frequency combs can be observed even when driving the conduits into the nonlinear regime by just a single pumping frequency. The tunability of multimode spin-wave spectra, as demonstrated in our experiments, offers promising technological applications that allow to bridge the frequency gap between CMOS technology and spin waves.

Spin-wave frequency combs

Strong coupling between magnons and different quantum particles was studied extensively for quantum electrodynamics in the past few years. In this system, magnons in magnetic materials with high spin density are utilized where the “coupling strength” is collectively enhanced by square root of number of spins (N) to overcome the weaker coupling strength between individual spins and the microwave field. However, the strongly magnon-magnon [1] couplings in a confined nanomagnet remains to be revealed which is important for on chief intregration of hybride system. Here, we report the interaction between different magnon modes [2] in a magnetic dot array. The intermodal coupling in the strong coupling regime was approached with a maximum coupling strength of 0.745 GHz and cooperativity of 2.5. Furthermore, it is found that the coupling strength is highly dependent on the orientation of bias field. Micromagnetic simulation reveals that the compitetion between external bias field and inter dot dipolar interaction create a movement of domain in between the dot which results the anticrossing. The tunable coupling strength with the strength and orientation of bias field opens the opportunity of extremely controlled hybrid magnonic devices. This findings could greatly enrich the still evolving field of quantum magnonics.

![](https://assets.underline.io/uploads/markdown_image/1/image/1000a626fbb67c2a7ac444f1916e06d6.jpg)

Figure 1: (a) Bias field (Hext) dependent spin wave (SW) absorption spectra of NiFe nanodot lattice are shown at φ = 45°. (b) Distribution of internal at 400 Oe bias field strength along the 45 degree orientation (φ = 45°). The color map for the surface plots and internal field profile with the schematic of the external applied field (Hext) are given at the bottom right corner of the figure.

Strong Magnon-Magnon Coupling in Two-Dimensional Diamond Shaped Ferromagnetic Nanodots Array

Development of structuring technologies for thin-film magnetic micro- and nanostructures make it possible rapid growing number and variety of studies of mechanisms and methods of excitation and control of spin wave (SW) in magnetic materials [1]. Using of the effects of dipole coupling of spin waves in lateral stripes leads to a periodic transfer of power between the stripes, while the spin-wave coupling can be used to implement the modes of spatial-frequency selection SW [2]. In the regime of multimode propagation of a spin-wave signal in a system of laterally coupled magnetic microwave guides, it turns out to be impossible to completely ensure the transfer of the propagating SW energy from one magnetic microstrip to another. In the same time over the last few years, have become a subject of striking interest structures with periodic changes in magnetic or geometric parameters due to the great potential of the application for signal and information processing devices. Such structures are named magnonic crystals (MC). One of the options for using geometric modulation is the formation of a one-dimensional array of antidots in the magnonic structure of laterally connected magnetic stripes [3]. The disadvantage of MCs with geometric modulation in the form of a one-dimensional array of antidots is the fixed parameters after fabrication of the structure, which implies the difficulty in adjusting the SW propagation. Combining the idea of a magnonic crystal and laterally coupled magnetic waveguides avoids the disadvantages of both types of structures. In this work, we investigated the properties of spin-wave transport in a reconfigurable system of magnetic microwave guides connected laterally by a commutation connection in the form of a one-dimensional antidote array. As a material for microwave guides, we used yttrium iron garnet [Y3Fe5O12 (111)] with a saturation magnetization Ms = 1750 G due to the low damping of the spin wave. The modes of simultaneous spatial and frequency selection of spin waves by the Brillouin spectroscopy and by the method of micromagnetic simulation are shown. This work was supported by the Russian Science Foundation (Grant 20-79-10191).

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Spin wave dynamics in magnonic coupler with antidote interconnection

When interacting spins in condensed matter order ferromagnetically, their ground-state wave function is topologically trivial. Nonetheless, in two dimensions, ferromagnets can support spin excitations with nontrivial topology, an exotic state known as topological magnon insulator (TMI). In Ref. [1], we theoretically unveil and numerically confirm a ferromagnetic state in three dimensions dubbed second-order TMI, whose hallmarks are excitations at its hinges, where facets intersect. Since ferromagnetism naturally comes with broken time-reversal symmetry, the hinge magnons are chiral, rendering backscattering impossible. Hence, as shown in the figure, they trace out three-dimensional paths about the sample unimpeded by defects and are topologically protected by the spectral gap. They are remarkably robust against disorder and highly tunable by atomic-level engineering of the sample termination. We predict that a van der Waals heterostructure built from chromium trihalide and transition metal dichalcogenide monolayers exhibits second-order magnon topology. Our findings empower magnonics, the harnessing of spin waves as information carriers, with the tools of higher-order topology, a promising route to combine low-energy information transfer free of Joule heating with three-dimensional vertical integration.

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Chiral Hinge Magnons in Second-Order Topological Magnon Insulators

Chiral Hinge Magnons in Second-Order Topological Magnon Insulators

Two-particle interference, i.e., the Hong-Ou-Mandel (HOM) effect [1], is used in many quantum technologies to detect and generate entanglement [2]. HOM has no classical analog and is expressly quantum in nature, making it ideal to study fundamental quantum properties. HOM was first observed with photons in a 50:50 beam splitter (BS) [1] and has been studied in other quantum objects using various interference systems [2], but the effect has not yet been studied in quantum magnonics [3, 4]. Here, we propose a model system for two-magnon interference. The system is an elliptical pillar of 2 antiferromagnetically-coupled ferromagnetic (FM) layers (see Fig. 1 inset) with resonant frequencies ω1 and ω2. With no bias magnetic field (B=0) ω1=ω2 and the FM layers are strongly coupled, but they are virtually independent when B≠0. This means that a time-dependent magnetic field B(t) can dynamically tune the system off-resonance (ω1≠ω2) to excite and detect magnons individually in each FM layer, and on-resonance (ω1=ω2) to allow for interaction of the magnons in two FM layers. The magnonic BS operation is achieved when the system is on-resonance for some time τ such that the probability for single magnon scattering between layers is ½. Our numerical simulations in Fig. 2 show the evolution of the system initially prepared in a two-magnon unentangled state │11〉. After the BS pulse, a maximally entangled N00N state (│20〉+│02〉)/√2 is formed. The relative phase between │20〉 and │20〉 components of the N00N state can be controlled by shaping the profile of the magnetic field B(t). The magnonic BS is a reversible unitary operation so it can also detect different N00N states. In summary, we found that a dynamically tuned magnon resonance can be used to form a magnonic BS that can generate and detect entanglement by two-magnon interference.

![](https://assets.underline.io/uploads/markdown_image/1/image/841d937f4d8f5aa207d43ec90ab1cf52.jpg)

Fig. 1. Dependence of the resonant frequencies on magnetic field B. Solid lines: collective frequencies. Dashed lines and symbols: independent frequencies ω1,2. Gray highlight: strongly coupled region. Inset: elliptical FM/NM/FM pillar geometry.


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Fig. 2. Evolution of unentangled state │11〉 into entangled superposition of states (│20〉+│02〉)/√2 under a magnonic beam splitter operation (gray highlight).

Generation of Entanglement by Two-Magnon Interference

Artificial spin ices are magnetic metamaterials comprising geometrically tiled arrays of interacting nanomagnets. There has been significant recent interest1,2 in these systems as reconfigurable magnonic crystals due to their vast microstate landscape. Here we measure the spin-wave spectra of square artificial spin systems in zero field and isolate spectral effects arising from the microstate and subtle array imperfections. Crucially remanence FMR allows the study of subtle spectral signatures of the local dipolar field without being dwarfed by the applied field3. We examine this in both width-modified, where the microstate may be controlled via global field4, and normal square ASI comprising identical bars. Width modification allows well-defined transitions between vertex states by switching only selected bars. These controlled states allow the identification of precise frequency shifts and peak amplitude changes associated with specific microstates, verified via a combination of MOKE magnetometry and micromagnetic simulations. Having established the fitting procedure in the ordered states of the width-modified arrays, we apply it to the more challenging problem of disordered microstates in square ASI comprising identical bars. Here reversal is defect-mediated with avalanche-type 1D chain reversal, where a combination of all vertex types appear in intermediate states. We measure remanence FMR and show excellent agreement with micromagnetic simulation and local dipolar field from the pure microstates. We illustrate the great potential of spectral-fingerprinting remanence spin-wave spectroscopy via distinct spectral readout of three microstates with identical (zero) magnetisation. These states are indistinguishable from any magnetometry based technique, but easily discernible via remanence FMR.

![](https://assets.underline.io/uploads/markdown_image/1/image/3f06240cdf5040700626995955083270.jpg)

(a) SEM of width-modified square, (b) MOKE magnetometry hysteresis loop showing remanence measurement procedure, (c) Remanence FMR spectral heatmap of width-modified square in monopole orientation showing frequency shift due to local dipolar landscape, (d) Remanence FMR spectral heatmap of normal square sample through reversal, (e) Remanence FMR spectra of as-grown state with MFM of microstate.

Spectral-fingerprinting: Microstate readout via remanence ferromagnetic resonance in artificial spin systems

Strongly-interacting artificial spin systems are moving beyond mimicking naturally-occurring materials to find roles as versatile functional platforms, from reconfigurable magnonics to designer magnetic metamaterials. Typically artificial spin systems comprise nanomagnets with a single magnetisation texture: collinear macrospins or chiral vortices. By tuning nanoarray dimensions we achieve macrospin/vortex bistability and demonstrate a four-state metamaterial spin-system ‘Artificial Spin-Vortex Ice’ (ASVI). ASVI is capable of adopting Ising-like macrospins with strong ice-like vertex interactions, in addition to weakly-coupled vortices with low stray dipolar-field. The enhanced bi-texture microstate space gives rise to emergent physical memory phenomena, with ratchet-like vortex training and history-dependent nonlinear training dynamics. We observe vortex-domain formation alongside MFM tip vortex-writing. Tip-written vortices dramatically alter local reversal and memory dynamics. Vortices and macrospins exhibit starkly-differing spin-wave spectra with analogue-style mode-amplitude control via vortex training and mode-frequency shifts of df = 3.8 GHz. We leverage spin-wave ‘spectral fingerprinting’ for rapid, scaleable readout of vortex and macrospin populations over complex training-protocols with applicability for functional magnonics and physical memory.

![](https://assets.underline.io/uploads/markdown_image/1/image/6fe74ca92f4f20fc19dc1d93439c81eb.jpg)

a-d) FMR heatmaps of ASVI taken during 0-30 training loop field-sequence. x-shaped vortex modes emerge with increadsing intensity as training progresses.
e) Corresponding MFM images showing vortex nucleation and vortex domain growth.

![](https://assets.underline.io/uploads/markdown_image/1/image/8fb28a1a5f1c817f24cd18ee2b3e7681.jpg)

FMR spectra taken at constant -1.2 mT bias field after successive training loops. Macrospin magnon mode-amplitude decreases through training as vortex magnon mode-amplitude increases.

Reconfigurable Training, Vortex Writing and Spin-Wave Fingerprinting in an Artificial Spin-Vortex Ice

In recent years, artificial spin ice (ASI) systems have seen burgeoning interest due to their intriguing physics and potential applications in reprogrammable memory, logic and reconfigurable magnonics [1-2]. As the field progresses, direct in-depth comparisons of distinct artificial spin systems are crucial to advancing the field. While studies have investigated the effects of different lattice geometries, little comparison exists between systems comprising continuously connected nanostructures, where spin-waves (SWs) propagate via dipole-exchange interaction, and systems with nanobars disconnected at vertices where SW propagation occurs via stray dipolar-field. Here, we study the magnonic response of two kagome spin ices, a continuously connected system (c-ASI) and a disconnected system (d-ASI) with vertex gaps via Brillouin light scattering (BLS) (see Fig. 1(a) for schematic) and micromagnetic simulation [3]. Array of 25-nm-thick Ni80Fe20 (Py) kagome ASI were fabricated using electron-beam lithography and lift-off process. For both samples, the nanobars have length (l) and width (w) of 300 nm and 80 nm, respectively and d-ASI has a 50 nm vertex gap (Figs. 1(b,c)). Bias magnetic field (H) dependent SW spectra (Figs. 1(d,g)) measured by BLS reveal that both c-ASI and d-ASI exhibit frequency minimum, occurring at negative bias magnetic fields, when ramped down from a positive saturation field. A sharp jump in the mode frequencies was observed beyond the minima, linked to bar reversal via the switching field observed from the magneto-optical Kerr effect loops and the corresponding mixed magnetization state of nanobars aligned parallel and anti-parallel to the magnetic bias field. Furthermore, we elucidate striking differences in spatial mode-localization and mode quantization between dipole-exchange-mediated c-ASI and dipolar-coupled d-ASI (Figs. 1(e,f) and 1(h,i)). These observations are pertinent for the fundamental understanding of artificial spin systems and broader design and engineering of reconfigurable functional magnonic crystals.

![](https://assets.underline.io/uploads/markdown_image/1/image/760d0a123abfca6bca5645ed091efa1f.jpg)

Figure 1(a) Schematic of the BLS measurement geometry. (b,c) SEM images. (d-i) SW mode frequencies as a function of H together with the phase profiles for M1 and M3.

Comparison of Spin-Wave Modes in Connected and Disconnected Artificial Spin Ice Nanostructures using Brillouin Light Scattering

One of the prospects of magnetic nanotubes in magnetization dynamics is using them as waveguides for spin waves. Recently, we have shown that the spin-wave propagation in thin-shell nanotubes with a vortex-state equilibrium magnetization exhibits strongly nonreciprocal spin-wave transport along the nanotube [1,2]. In particular, spin waves with the same wavelength but opposite propagation direction along the tube inherit different oscillation frequencies. This dispersion asymmetry is of dipolar origin and a consequence of the geometrical magnetic pseudo charges induced by the surface curvature of the nanotube. So far, predictions on the spin-wave dispersion in magnetic nanotubes were only available for tubes with a thin shell (with thicknesses comparable to the exchange length of the material) for which the spatial profile of the spin-wave modes can be assumed to be approximately homogenous in radial direction. With the help of our newly-developed finite-element dynamic-matrix approach for spin waves propagating in waveguides with arbitrary cross section [3], we are now exploring the transition of the spin-wave dispersion and mode profiles as the thickness of the nanotube is increased. To this end, we investigate the emergence of spin-wave modes with a standing-wave character along the nanotube thickness. These modes are shown to be strongly hybridized as long as they share the same azimuthal symmetry. Moreover, they correspond to the radial modes in vortex-state disks or to the perpendicular-standing spin waves in magnetic thin films. We present that the dispersion changes drastically as the nanotube-shell thickness is increased. In particular, the dispersion asymmetry increases to the point where there is an almost completely unidirectional propagation similar to the spin-wave transport reported for exchange-coupled bilayers [4].

Spin-wave dispersion in magnetic nanotubes: Beyond the thin-shell approximation

The magnon spectral problem is solved in terms of the spectrum of a diagonalizable operator for a generic class of magnetic states that includes several types of domain walls and the chiral solitons of monoaxial helimagnets. Focusing on the isolated solitons of monoaxial helimagnets, it is shown that the spin waves scattered (reflected and transmitted) by the soliton suffer a lateral displacement analogous to the Goos-Hänchen effect of optics. The displacement is a fraction of the wavelength, but can be greatly enhanced by using an array of well separated solitons. Contrarily to the Goos-Hänchen effect recently studied in some magnetic systems, which takes place at interfaces between different magnetic systems, the effect predicted here takes place at the soliton position, what it is interesting from the point of view of applications since solitons can be created at different places and moved across the material. This kind of Goos-Hänchen displacement is a consequence of the dependence of the phase shift on the transverse component of the wave vector, which does not appear in other known one dimensional systems, as quantum mechanical particles moving in a one dimensional potential. However, it is not particular of the solitons of monoaxial helimagnets, but it is generic for a class of magnetic states, including domain walls in systems with interfacial Dzyaloshinskii-Moriya interaction. The conditions under which the soliton induced Goos-Hänchen effect takes place are discussed [1].

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