United States

Far from thermodynamic equilibrium, nonlinear systems can be characterized by a stationary regime known as a limit cycle. In magnetic systems with a uniaxial symmetry driven with a circularly polarized field, this regime is referred to as the P-mode [1]. P-modes have been experimentally observed in the non-linear ferromagnetic resonance [2]. Stable P-modes can play a role of the equilibrium state for the elementary excitations around them. We show that these excitations can be used in cavity magnonics to explore unique features of non-Hermitian physics such as mode attraction and exceptional points.&lt;br/&gt;&lt;br/&gt;Level attraction describes a mode coalescence that can take place in open systems. It indicates development of an instability region in the energy spectrum of the coupled system bounded by exceptional points [3]. This regime has been recently observed in a number of experiments in dissipative cavity magnonics [4].&lt;br/&gt;&lt;br/&gt;Here, we present a theory of mode attraction in periodically driven systems. Our theory can be used in conjunction with numerical micromagnetics to describe non-Markovian Floquet systems [5], which allows us to determine the regions of stability and instability and associate them with mode repulsion and attraction. In magnetic systems, our approach works for quantum and semiclassical dynamics, which makes possible to use it in combination with micromagnetic simulations.&lt;br/&gt;&lt;br/&gt;We apply this approach to demonstrate the level attraction between the excitations around the P-modes and the photons inside a microwave cavity. In a system driven with a circularly polarized field, the excitations around the P-mode form two side bands around the driving frequency (see figure). We show that when the lower side band couples to the cavity mode, it forms a mode attraction spectrum. Finally we discuss possibilities for experimental observation and predicted dependences on driving field amplitude.

![](https://assets.underline.io/uploads/markdown_image/1/image/7edfde7c526bc0c7f19799421eaed99c.jpg)

Circular component of the precessing spin driven with the circularly polarized field. The central peak is the P-mode with the frequency of the driving field. Two side bands correspond to the excitation around the P-mode. Coupling to the side band with the negative frequency with respect to the driving field gives mode attraction.

2022 Joint MMM-INTERMAG

Mode attraction in Floquet systems with memory: application to cavity magnonics

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.

Far from thermodynamic equilibrium, nonlinear systems can be characterized by a stationary regime known as a limit cycle. In magnetic systems with a uniaxial symmetry driven with a circularly polarized field, this regime is referred to as the P-mode [1]. P-modes have been experimentally observed in the non-linear ferromagnetic resonance [2]. Stable P-modes can play a role of the equilibrium state for the elementary excitations around them. We show that these excitations can be used in cavity magnonics to explore unique features of non-Hermitian physics such as mode attraction and exceptional points. Level attraction describes a mode coalescence that can take place in open systems. It indicates development of an instability region in the energy spectrum of the coupled system bounded by exceptional points [3]. This regime has been recently observed in a number of experiments in dissipative cavity magnonics [4]. Here, we present a theory of mode attraction in periodically driven systems. Our theory can be used in conjunction with numerical micromagnetics to describe non-Markovian Floquet systems [5], which allows us to determine the regions of stability and instability and associate them with mode repulsion and attraction. In magnetic systems, our approach works for quantum and semiclassical dynamics, which makes possible to use it in combination with micromagnetic simulations. We apply this approach to demonstrate the level attraction between the excitations around the P-modes and the photons inside a microwave cavity. In a system driven with a circularly polarized field, the excitations around the P-mode form two side bands around the driving frequency (see figure). We show that when the lower side band couples to the cavity mode, it forms a mode attraction spectrum. Finally we discuss possibilities for experimental observation and predicted dependences on driving field amplitude.

![](https://assets.underline.io/uploads/markdown_image/1/image/7edfde7c526bc0c7f19799421eaed99c.jpg)

Circular component of the precessing spin driven with the circularly polarized field. The central peak is the P-mode with the frequency of the driving field. Two side bands correspond to the excitation around the P-mode. Coupling to the side band with the negative frequency with respect to the driving field gives mode attraction.

technical paper

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].

Magnonic Goos-Hänchen Effect Induced by One Dimensional Solitons.

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].

![](https://assets.underline.io/uploads/markdown_image/1/image/de487d7d7d62cb5d0d87e2ecb9540270.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/b68a6549d2544751c06524130eaae699.jpg)

Measuring the dispersion relations of spin wave bands using time-of-flight spectroscopy

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

Spin-wave spectroscopy methods, such as electron-spin-resonance [1], and ferromagnetic resonance [2], are vital tools for materials characterization and chemical monitoring. Together with momentum-sensitive techniques, such as Brillouin light scattering, they are also central to spintronics experiments and the design of spintronic and magnonic devices. Here, we demonstrate a new technique to spatially resolve spin-waves on the single nanometer scale with the potential for atomic resolution by transmission electron microscopy (TEM). The precessional torque of a spin excitation is coupled to a highly coherent electron beam, which reveals localized resonant spin excitations in transmission electron microscopy. This technique can be applied in-situ and in-operando on its own and together with conventional spectroscopy methods. As a model system, we present first results of magnonic networks comprising ferromagnetic nanoparticles [3,4] and observe the spatial distributions of various spin-resonance modes. Financially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 405553726 – TRR 270 and the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant No. 856538, project “3D MAGiC”)

First Results on Atomically Resolved Spin-Wave Spectroscopy by Transmission Electron Microscopy

We present focusing of spin waves with spherical lenses in YIG by creating a magnonic refractive index change via Ga+ focused ion beam (FIB) irradiation. Direct FIB irradiation in YIG induces an ion-dose-dependent change in the effective magnetization Meff [1], and we use this to locally modify the refractive index. The lenses were designed using the lensmaker’s equation where the relative refraction index is estimated from experiments as nrel=λ0/λFIB= 4.2μm/2.5μm = 1.68. In the experiments, we use an in-house sputtered 100 nm thick YIG film on GGG and obtain the largest △Meff for a dose of 1×1013 ions/cm2. For larger doses, the change in △Meff is reversed and damping increases significantly. The measured spin-wave pattern of two plano-convex spherical lenses is presented in Fig. 1. We use longitudinal time-resolved MOKE microscopy to image the spin-wave patterns in forward volume configuration (isotropic propagation). Bending of the wavefront is visible inside the lenses, as well as the shorter wavelengths. Ion irradiation in thin films to manipulate their magnetic properties has been used in many magnetic materials/compositions but is particularly challenging in insulators. Researchers in our field, to our knowledge, have not FIB irradiated YIG to locally modify the spin-wave dispersion relation, or to pattern YIG without etching. Using spin waves for analog computing (such as optical signal processing) requires precise local modification of the magnetic properties, and we recently demonstrated the potential of FIB patterning by fabricating a concave grating for spin waves realized with higher ion doses that locally destroy magnetic properties of YIG [2]. Furthermore, gradually changing refractive-index maps can be realized by irradiating a dose-gradient, giving access to create gradient-index magnonic elements like GRIN lenses [3],[4],[5].

![](https://assets.underline.io/uploads/markdown_image/1/image/522825ca15aa1d16debf523a689aec24.jpg)

Time-resolved MOKE image of a 20 (left) and a 10 μm (right) thick spin-wave lens fabricated via FIB. The irradiation pattern d-shaped lens is indicated in semitransparent red, the groundline of a coplanar antenna for coherent excitation in gray. The initially applied wavelength is 4.3 μm, and shortened to about 2.5 μm inside the lenses due to the FIB induced Meff increase.

Spin-wave optics fabricated by FIB irradiation of YIG films

Magnon has a potential to be a new information carrier because of its Joule-heating-less and long-distance propagation in ferromagnets [1]. Recently, Das et al. demonstrated that magnon in Y3Fe5O12 (YIG) is absorbed into an adjacent ferromagnet, Ni-Fe alloy (Py), and the absorption can be modulated by the magnetization direction of the Py [2]. Although this study makes progress toward realization of magnonic device, a detailed principle of magnon absorption into metals has not been sufficiently clarified yet. Here, we study magnon propagation with several middle strips (MSs), Py, SiO2, and metals on SiO2 to explore the mechanisms of magnon absorption into metals. Figure 1 shows a schematic of the device. Two Ti/Au coplanar waveguides with the center-to-center distance of 300 μm and an MS with a width of 240 μm were fabricated on a YIG substrate (5 μm in thickness). An external magnetic field (B) applied to the device was swept from 50 mT to 140 mT. AC electric current with the power of 0 dBm and frequency of 5 GHz was injected into the antenna using a vector network analyzer and magnons were excited by the induced AC magnetic field. Magnons propagating in the YIG were partially absorbed by the MS and detected by the other antenna as an induced AC magnetic field. We measured the S21 parameter and calculated |S21| (= ∫|S21|dB) to compare the amounts of the detected magnons. To clarify whether the magnon absorption is an interfacial effect between YIG and Py, we measured the samples with Py and SiO2/Py-MS. Figure 2 shows the comparison of |S21| among the bare YIG, SiO2-MS, Py-MS, and SiO2/Py-MS samples. The SiO2 and Py thicknesses are 10 nm and 16 nm. The magnons detected in the Py and SiO2/Py-MS samples were less than that in the bare YIG sample, and the decrease of Σ|S21| of the sample with the SiO2/Py-MS was not much different from that of the sample with the Py-MS. These results indicate that the absorption is not originated simply from the interfacial effect between YIG and ferromagnet such as spin exchange interaction. We will discuss a possible mechanism of magnon absorption.

![](https://assets.underline.io/uploads/markdown_image/1/image/54eb6400dcc6e7749716481aad767dc9.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/c7a5f1985ae67ac64dbe95c065718351.jpg)

Investigation of magnon absorption from Y3Fe5O12 into metals

Investigation of magnon absorption from Y3Fe5O12 into metals

The cross-integration of spin-wave and superconducting technologies is a promising method for creating novel hybrid devices for future information processing technologies to store, manipulate, or convert data in both classical and quantum regimes. Such hybrid devices can exhibit interesting properties such as new resonance spectra features induced by the interchange between superconducting microwave lines and ferromagnetic resonance dynamics of yttrium iron garnet (Y3Fe5O12, YIG) films [1]. On the other hand, hybrid magnon-polariton systems have been typically studied using bulk YIG and three-dimensional microwave photon cavities [2,3]. Limitations in YIG growth have thus far inhibited its incorporation into CMOS compatible technology such as high quality factor superconducting quantum technology. To overcome this impediment, we have used Plasma Focused Ion Beam (PFIB) technology—taking advantage of precision placement down to the micron-scale—to integrate YIG with NbN superconducting microwave devices. Example hybrid devices are shown in Fig.1. Ferromagnetic resonance has been measured at milliKelvin temperatures on PFIB-processed YIG samples using planar microwave circuits. Furthermore, we demonstrate strong coupling between superconducting resonator and YIG ferromagnetic resonance modes by maintaining reasonably low loss while reducing the system down to the micron scale. This achievement of strong coupling on-chip is a crucial step toward fabrication of functional hybrid quantum devices that advantage from spin-wave and superconducting components. For more information, please see our recent publication [4]. This work was supported by the European Research Council under the Grant Agreements 648011 and 648604, the EPSRC through grant EP/M024423/1, the Leverhulme Trust, and the University of Glasgow.

![](https://assets.underline.io/uploads/markdown_image/1/image/2c0c030f03b19b6b02146d7b7ace69cf.jpg)

Strong magnon-photon coupling with chip-integrated YIG in the zero-temperature limit

Strong magnon-photon coupling with chip-integrated YIG in the zero-temperature limit

New fascinating phenomena involving coupled spin waves and phonons are observed at the magnetoelastic resonance as acoustically-assisted spin pumping [1, 2] and magnetization swiching [3, 4]. In this context, Surface Acoustic waves (SAW), a mature technology widely used in today's sensors, filters and microwave circuitry trigger efficiently magnetization dynamics: resonant effects are obtained with appropriate intensities and directions of the external magnetic field (Bext). This tunability is an asset for magnonic applications of SAW-FMR (ferromagnetic resonance). Recently, SAW-FMR has been obtained in Fe epitaxially grown on GaAs(001), a spintronic and magnonic compatible magnetoelastic and piezoelectric heterostucture.[5] Here, we show that SAW propagation in Fe thin films is strongly affected by the SWs modes dispersion in the reciprocal space, even at very low K-values. It turns out that the Bext-dependence of the sound velocity depends a lot of the angle between Bext and the SAW vector directions. In particular, the resonant SAW-FMR is possible only in the backward configuration. Furthermore, we give a theoretical description of the observed phenomena by introducing calculated dispersion curves into a phenomenological and approximated approach derived by LLG equations. [6,7] This approach catches the physics of SAW propagation in a Fe thin films, as attested by the comparison with quasi-exact elastic calculations. We give a very good description of the SAW-SW interaction that facilitates the design of magnonic devices where SWs are generated by SAW Interdigital transducers (IDT) and not by much more energy-dissipative inductive antenna.

Effects of Spin Wave Dispersion on Surface Acoustic Wave Velocity

Interference-based magnonic devices – which exploit eigenmode transportation of ordered spins, i.e. spin waves – are potential candidates to overcome the increased power consumption and Joule heating of traditional information storage and transport devices. In this work, we present a custom-made time-resolved magneto-optical Kerr effect (TR-MOKE) microscope for visualizing locally and time-resolved the dynamic magnetization, i.e. propagating spin waves. Using femtosecond laser pulses synchronized to the phase of a microwave source, which is used to drive spin-wave dynamics, we demonstrate time-resolved 2D spin-wave images with a spatial resolution &lt; 500 nm. To investigate magnonic interference behaviors, we fabricate a diffraction grating in a 200 nm thick ferrimagnetic YIG film by argon ion-beam etching. A coplanar waveguide (CPW) located parallel to the grating, is used to coherently excite spin-waves which are imaged by TR-MOKE. Our results represent the experimental realization of a self-imaging phenomenon applied to spin-waves, well-known within the field of linear optics as the Talbot effect [1-3], where it finds several applications in optical measurement techniques. We investigate the spin-wave Talbot effect for different orientations of the external magnetic field with respect to the YIG plane and spin-wave propagation direction. Each geometry displays different interferometric behaviors, in agreement with recent theoretical predictions [2]. Furthermore, as depicted in Fig. 1, we demonstrate that the interference pattern can be tuned through careful selection of the external magnetic field strength. This work contributes to the understanding of spin-wave interference behaviors for enhancing the performance of future magnonic devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/1b2a2492a50567ffd239f789cdf02461.jpg)

Fig. 1: The top sketch indicates the sample with the ground line of the CPW (yellow stripe) and the diffraction grating (black squares). The lower images display the MOKE signal, which is indicated by the gray scale, at various external magnetic fields. It can be seen that the localization of the interference maxima can be shifted by changing the external magnetic field strength. Here, the magnetic field was applied along the Y-direction.

Mode attraction in Floquet systems with memory: application to cavity magnonics

Downloads

Next from 2022 Joint MMM-INTERMAG

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

Similar lecture

The thermodynamic properties of exchange stiffness

Mode attraction in Floquet systems with memory: application to cavity magnonics

.css-70qvj9{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;}Downloads

Next from 2022 Joint MMM-INTERMAG

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

Similar lecture

The thermodynamic properties of exchange stiffness

Downloads