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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

2022 Joint MMM-INTERMAG

Chiral Hinge Magnons in Second-Order Topological Magnon Insulators

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

Vortices are familiar phenomena in fluids, distinguishable for example in tornadoes, hurricanes, whirlpools and rings. In ferromagnets, vortices are naturally forming flux-closure states characterized by a curling of the magnetic moments around a very stable and narrow core [1]. The study of vortices is motivated by both fundamental and technological interest. Indeed, their topological properties endow vortices with rich and surprising dynamics, such as the ultrafast reversal of the vortex core [2], making them fascinating objects of study. These properties also establish them as promising candidates for applications in data storage and signal processing. Over the past decades, vortices have been extensively studied in laterally confined thin-film elements, where the magnetization is accessible with two-dimensional imaging methods. Recent developments have enabled non-destructive imaging of three-dimensional magnetic structures at the nanoscale. In particular, using hard X-ray tomography [3], it is now possible to directly image the internal structure within bulk micromagnets. For example, in GdCo2 pillars, we could image the structure surrounding micromagnetic singularities, known as Bloch points, which were predicted over 50 years ago. Recently, we have uncovered three-dimensional structures forming closed vortex loops. The cross-section of these loops consists of a vortex–antivortex pair and, based on magnetic vorticity (a quantity analogous to hydrodynamic vorticity), we identified these configurations as magnetic vortex rings [4]. Such structures have been predicted to exist as transient states in exchange ferromagnets [5]; in contrast, the observed vortex rings exist as static configurations, and we attribute their stability to the dipolar interaction. In addition, we observe stable vortex loops intersected by point singularities at which the magnetization within the vortex and antivortex cores reverses. Such structures can exist because a non-zero divergence of the magnetization is allowed in ferromagnets; they have no analogue in incompressible fluids. Incidentally, it is known since the 1970s that ‘Bloch lines’ can assume complex structures, but these could only be inferred from surface observations combined with theory. Our results shed new light onto the rich physics of bulk magnets and open possibilities for further studies of complex three-dimensional solitons, enabling the development of applications based on three-dimensional magnetic structures. Thus, the title of this talk could as well be, ‘Magnetic Vortices: Back to the Third Dimension’.

Magnetic Vortices: into the Third Dimension

Much effort has focused on achieving micron-scale spin propagation using diffusive spin waves, but their exponential decay with distance makes efficient spin transport challenging. An alternative form of magnetization dynamics – commonly referred to as spin superfluidity [1] or exchange mediated spin transport [2,3] – may enable more efficient transport owing to its topologically-protected easy-plane precession of the magnetization and associated linear decay of spin flow. Our work addresses the open question of how defects impact this easy-plane spin transport at low drive current densities. We performed micromagnetic simulations on easy-plane precessional dynamics in synthetic antiferromagnet nanostrips with and without notch defects. The material parameters were taken from our experimentally measured, low-damping FeV alloys [4]. We view the dynamics as a train of interacting, homochiral domain walls (DWs) rather than as hydrodynamic flow [2]. We observe the transition from (1) isolated, diffusively propagating DWs, which are susceptible to pinning, to (2) a fluid-like flow of a closely packed DW train that is unimpeded by pinning defects. This transition is quantified by computing the superfluid velocity ▽Φ, where Φ is the in-plane azimuthal angle of the magnetization, at a position beyond the notch defect (see Fig. 1). In Fig. 2, ▽Φ differs between the perfect and notched nanostrips as a result of pinning of the DWs at low drive current densities, Jc &lt; 5 x 1011 A/m2. As Jc increases, ▽Φ becomes identical between the two geometries, indicating that the DWs have effectively condensed to form a fluid flowing unaffected by the notch defect. Our findings are promising for future realization of superfluid-like spin transport via easy-plane precession in ferromagnetic metal devices, which are compatible with established fabrication techniques [5].

![](https://assets.underline.io/uploads/markdown_image/1/image/b29ae1464b62923fd11997a38ae0428e.jpg)

Figure 1: Micromagnetic snapshots of the perfect and notched nanostrips at Jc = 4 x 1011 A/m2, labeled with location of notch defect at the center and where ▽Φ is calculated.


![](https://assets.underline.io/uploads/markdown_image/1/image/329eb8448836511a5d16e9a2b0225809.jpg)

Figure 2: Superfluid velocity ▽Φ as a function of charge current density Jc for the perfect (black squares) and notched (red circle) nanostrips computed at a fixed position beyond the notch defects.

Diffusive and Fluid-like Motion of Homochiral Domain Walls in Easy-Plane Magnetic Strips

As three-dimensional nanomagnetism evolves, novel non-trivial magnetic textures emerge as appealing information carriers for recording and spintronics based on curved nanosystems, and particularly on Cylindrical Nanowires (NWs) [1,2]. One of the most fascinating candidates that is likely to reach the high velocities required for fast recording technologies is the Bloch Point (BP) domain wall (DW). Recently, theoretical evidence indicated that BPs in NWs could reach high velocities close to 2 km/s in the magnonic regime [3]. While the observation of the BP DW in cylindrical NWs is no longer recent [2], scarce numerical studies that combine both current and Oersted field have been published in NWs [4], despite first attempts to measure DW velocities are in progress [5]. In this work we evaluate the dynamics of the BP DW under both current directions in a Ni NW with 100 nm in diameter. We investigate two cases: i) pre-nucleated BP Wall, and ii) the BP DW originated from the transformation of a Vortex-Antivortex DW. The effects of both spin-polarized current and Oersted field are considered. We discuss in detail the role of the chirality of the BP in relation to the Oersted field, also reported previously in precursors of BPs [4]. We show that while the pre-nucleated DW with the same chirality as that of the Oersted field propagates always against the current direction, the BP originated either from the transformation of the BP with the opposite chirality or from the vortex-antivortex DW can either stop the propagation or propagate parallel to the current. We attribute this effect to the role played by the inertia of the BP in its dynamics. Finally, we provide values of the velocities achieved by the BP in the NW as a function of applied current in Fig. 1. We conclude that BPs with vanishing momentum propagate opposite to the current with velocities that may be suppressed by the Oersted field. Importantly for spintronic applications, both momentum and inertial mass play a major role in the dynamics of BPs that has not been envisaged up to know.

![](https://assets.underline.io/uploads/markdown_image/1/image/6ff528b21478a289ec02bc86a4e8eae4.jpg)

The average velocity of a head-to-head BPW driven by a spin-polarized current in a Ni nanowire, far from the ends of the nanowire, as a function of the current density.

Oersted-field- and current- induced Bloch Point domain wall dynamics in cylindrical Ni nanowires

Highly-magnetostrictive amorphous microwires show bistable behaviour, i.e. magnetization reverses by depinning/propagation of a single Domain Wall, DW, along the entire wire. This response makes these microwires an ideal system for fundamental studies on DWs dynamics [1,2]. On the other hand, the classical Matteucci effect, ME, consists of the induction of an emf voltage at the ends of the microwire under applied axial magnetic field [3], and requires a net azimuthal component of magnetization, Mφ. That emf is associated with the DW propagation and its amplitude is proportional to the DW speed, vDW, and the time derivative of Mφ, tailored by the torsion magnetoelastic anisotropy. emf has been measured in a bistable FeSiB microwire (12mm-diam, 10cm-long) under torsion (maximum of ξmax=57 πrad/m) and axial driving magnetic field, Hdr (square wave). The setup is designed to generate unidirectional motion of the DW that unpins from the same end irrespective of the Hdr direction [4], and its speed is quantified by the voltage induced in tiny pickup coils along the microwire. The clockwise/counterclockwise, CW/CCW, torsion dependences of emf and vDW are experimentally determined for parallel/antiparallel Hdr. Asymmetric behaviour is observed for both, the sense of x and the Hdr direction (Fig.1). The DW dynamics is analysed in terms of Landau-Lifschitz-Gilbert eq. considering a bell-like DW (Fig.2). The observed asymmetries are numerically interpreted to derive from the magnetochiral effect related to the DW chirality. The switching of the Hdr direction changes the Head-to-Head, H-H, or Tail-to-Tail, T-T, DW character and chirality (i.e., Mφ circulation). As a consequence of this magnetochiral effect, we proof that H-H and T-T DWs propagate at different velocities, vDW, as experimentally observed. Here we provide a first evidence of DWs magnetochiral effects in microwires that offers an opportunity for investigation of topological properties of spin-textures in microwires that has not been envisaged up to now.

![](https://assets.underline.io/uploads/markdown_image/1/image/3aa87066cea42d5386ea12ea8fba69a6.jpg)

CW, ξ+, and CCW, ξ-, torsion dependence of DW speed under Hdr = 232 A/m

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Scheme of a longitudinal slice with a propagating bell-shaped DW. Magnetization in the domains makes a small angle with the microwires axis

Magnetochiral effect of single domain wall propagation in the Matteucci effect of magnetostrictive microwire

The presence of a Dzyaloshinskii-Moriya interaction (DMI) in perpendicularly magnetised thin films stabilises Néel type domain walls of fixed chirality, as opposed to the Bloch walls that are formed in the absence of a DMI [1]. Néel walls of fixed chirality have been shown to be driven efficiently in the same direction by electric currents through spin-orbit torques [2,3]. Later, a large tuning of the DMI with an applied voltage was demonstrated [4], which can be used e.g. for electric field control of magnetic domain wall motion via the modulation of the DMI [5]. Here, we demonstrate an alternative mechanism for the control of domain wall type: using micromagnetic simulations and analytical modelling, we show that the presence of a uniaxial in-plane magnetic anisotropy can also lead to the formation of Néel walls in the absence of a DMI. It is possible to abruptly switch between Bloch and Néel walls via a small modulation of both the in-plane, but also the perpendicular magnetic anisotropy [6]. This opens up a route towards efficient electric field control of the domain wall type with tiny applied voltages, as the magnetic anisotropy can be modulated via the direct voltage controlled magnetic anisotropy (VCMA) mechanism [7], or via magnetoelastic anisotropy induced through coupling to a piezoelectric element [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/7bef5f5df8869fd862ccaab94b5a3626.jpg)

Fig. 1 In a perpendicularly magnetised thin film, domain walls (DWs) are generally of Bloch type (bottom left). The presence of a DMI (top left), stabilises Néel walls. Alternatively, and in-plane magnetic anisotropy can also form Néel walls (bottom right). An increase in the perpendicular magnetic anisotropy switches the DW type back to Bloch (top right).

Switching between Magnetic Bloch and Néel Domain Walls with Anisotropy Modulations

Spin waves are promising to develop post-CMOS and/or green technologies due to their sub-micron wavelengths in the GHz range, their frequency tuning with magnetic or electric fields and their small energy dissipation during propagation. Many spin waves based applications have already been demonstrated for data processing and other microwave devices (filters, resonators, delay lines…). To reach these objectives, magnonic crystals (MC) consisting of a periodic modulation of magnetic properties are among the most promising geometries. One key point for applications with MC is to remove the static bias field as most devices operate at zero field. So technological developments imply to find solutions to obtain well-defined spin waves modes at remanence, with the possibility to modify purposely the spin wave spectrum to achieve different functionalities. While several approaches for reconfigurable states have been proposed [1-4], a “material approach” consisting in exploiting the crystal anisotropy of magnetic materials has been poorly explored up to now. Here we show by micromagnetic simulations that materials such as Co2MnSi Heusler alloy with strong cubic anisotropy can be used to obtain very different spin waves modes at remanence in a simple and classical geometry (Fig. 1 and Fig. 2). In particular, we demonstrate the variation of remanent configuration after saturation with a weak external field H0 (&lt; 50 Oe) in different directions. This allows to decrease the amplitude, extinct or shift the frequencies of the different modes depending if they correspond to edge or volume modes. We will show the advantage of using cubic anisotropy with respect to isotropic material by comparing the amplitude variation of the different modes. Finally, we will also show that proper design of the crystal with respect to the cubic anisotropy directions can achieve a microwave response very robust against magnetic perturbation.

![](https://assets.underline.io/uploads/markdown_image/1/image/53c4c32972d9d1d95c4dd4277757c834.jpg)

Fig. 1: Remanent magnetic states for different H0 directions.The dot edge size is 300 nm and the spacing is 900 nm.


![](https://assets.underline.io/uploads/markdown_image/1/image/5d2d5ca648dfd24fea180bab3925633e.jpg)

Fig. 2: Corresponding FFT spectrum for a RF field along x. The calculated spatial profiles of the spin waves modes at particular frequencies are shown for the configuration A.

Reconfigurable dynamic states in Heusler based magnonic crystals

Optical frequency comb is a light source consisting of a series of discrete and equally spaced frequency lines. Since its inception, optical frequency comb has dramatically improved the accuracy of frequency metrology [1] and thus leads to one half of the Nobel Prize in Physics in 2005. Recently, an analogue of optical frequency comb in an acoustic system has been realized in experiment [2], which provides vast opportunities for both fundamental and applied research in phononics. On the other hand, magnons, as elementary excitations in the ordered magnets, are akin to photons and phonons as bosonic quasiparticles. However, the frequency comb of magnons has not been realized to date. The fundamental difficulty is the smallness of nonlinearity in a normal ferromagnet, which hinders the form of frequency comb structure. In this work [3], we realize a novel magnonic frequency comb induced by the nonlinear coupling between magnetic skyrmion and spin waves excited by a microwave field. Here the nonlinear interaction is significantly enhanced by the topological structure of skyrmions and thus leads to the generation of frequency comb through three-magnon process. The mode spacing of frequency combs is equal to the frequency of the skyrmion breathing mode, which can be efficiently tuned by both magnetic and electric means. The essential physics can be further extended to a wide class of magnetic solitons including antiskyrmions, domain walls, and vortices. Our results provide a novel avenue to study frequency comb physics combining the advantages of magnons and general magnetic textures and may further benefit its applications in the spin-wave calibration and metrology.

![](https://assets.underline.io/uploads/markdown_image/1/image/7f0b08234acae8f54655225ca4fd19be.jpg)

Magnonic Frequency Comb Through Nonlinear Magnon-Skyrmion Scattering

Magnonic Frequency Comb Through Nonlinear Magnon-Skyrmion Scattering

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).

![](https://assets.underline.io/uploads/markdown_image/1/image/1e52f3b7192171a9e9215e6b90add9f3.jpg)

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