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The effective temperature dependence of micromagnetic parameters such as anisotropy (K), exchange stiffness (A) and the Dzyaloshinskii–Moriya interaction (DMI) are usually written as a power law of the magnetisation. For anisotropy this is well founded, based on Callen-Callen theory&lt;sup&gt;1&lt;/sup&gt;. For exchange stiffness and DMI the power laws are based on the assumption that they behave the same as anisotropy and from classical numerical models. Recently there has been a lot of interest in the temperature dependence of these properties because their value affects spin textures like skyrmions.&lt;br/&gt;&lt;br/&gt;Inferring the DMI from experimental measurements often requires first deducing the exchange stiffness by fitting Bloch&#39;s law to measurements of magnetisation at finite temperature. The problem is that Bloch&#39;s law contains the zero temperature value of exchange stiffness, but is used to fit a finite temperature range over which the value of the exchange stiffness should change significantly. Another issue is that Bloch&#39;s law is derived using quantum mechanics with magnons which obey Bose statistics – numerical models typically use classical statistics for the thermal occupation of spin waves. Classical models cannot reproduce Bloch&#39;s T&lt;sup&gt;3/2&lt;/sup&gt; law for ferromagnets or the T&lt;sup&gt;2&lt;/sup&gt; dependence of sublattice magnetisation in antiferromagnets&lt;sup&gt;2&lt;/sup&gt;. So, the classical thermodynamics of exchange is also a poor approximation for real materials in which the quantum distributions of magnons plays a large role.&lt;br/&gt;&lt;br/&gt;Here we discuss a quantum implementation of atomistic spin dynamics&lt;sup&gt;2&lt;/sup&gt;, compare with classical results and present the temperature scaling of the micromagnetic exchange stiffness for both ferromagnets and antiferromagnets calculated from magnon spectra. We show antiferromagnets have similar scaling behaviour to ferromagnets and that the results using quantum statistics doesn’t follow a power law.&lt;br/&gt;&lt;br/&gt;Figure 1 presents the thermal behaviour of the exchange stiffness as a function of reduced sublattice magnetisation of the prototypical antiferromagnet NiO used in antiferromagnetic spintronics&lt;sup&gt;3&lt;/sup&gt;. At low temperature (high magnetisation) the exchange stiffness is almost temperature independent when using quantum statistics in stark contrast to a power law.

![](https://assets.underline.io/uploads/markdown_image/1/image/7982d7754373f6c2812464262fbe6f6c.jpg)

2022 Joint MMM-INTERMAG

The thermodynamic properties of exchange stiffness

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.

The effective temperature dependence of micromagnetic parameters such as anisotropy (K), exchange stiffness (A) and the Dzyaloshinskii–Moriya interaction (DMI) are usually written as a power law of the magnetisation. For anisotropy this is well founded, based on Callen-Callen theory1. For exchange stiffness and DMI the power laws are based on the assumption that they behave the same as anisotropy and from classical numerical models. Recently there has been a lot of interest in the temperature dependence of these properties because their value affects spin textures like skyrmions. Inferring the DMI from experimental measurements often requires first deducing the exchange stiffness by fitting Bloch's law to measurements of magnetisation at finite temperature. The problem is that Bloch's law contains the zero temperature value of exchange stiffness, but is used to fit a finite temperature range over which the value of the exchange stiffness should change significantly. Another issue is that Bloch's law is derived using quantum mechanics with magnons which obey Bose statistics – numerical models typically use classical statistics for the thermal occupation of spin waves. Classical models cannot reproduce Bloch's T3/2 law for ferromagnets or the T2 dependence of sublattice magnetisation in antiferromagnets2. So, the classical thermodynamics of exchange is also a poor approximation for real materials in which the quantum distributions of magnons plays a large role. Here we discuss a quantum implementation of atomistic spin dynamics2, compare with classical results and present the temperature scaling of the micromagnetic exchange stiffness for both ferromagnets and antiferromagnets calculated from magnon spectra. We show antiferromagnets have similar scaling behaviour to ferromagnets and that the results using quantum statistics doesn’t follow a power law. Figure 1 presents the thermal behaviour of the exchange stiffness as a function of reduced sublattice magnetisation of the prototypical antiferromagnet NiO used in antiferromagnetic spintronics3. At low temperature (high magnetisation) the exchange stiffness is almost temperature independent when using quantum statistics in stark contrast to a power law.

![](https://assets.underline.io/uploads/markdown_image/1/image/7982d7754373f6c2812464262fbe6f6c.jpg)

technical paper

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.

Experimental Observation of the Spin-Wave Talbot Effect

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.

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

Extending nanostructures into the third dimension has become a major research avenue in modern magnetism and spintronics because of geometry-, curvature-, and topology-induced phenomena. Here, we introduce nanovolcanoes as on-demand engineered nano-architectures for 3D magnon spintronics. 40 nm-thick Co-Fe nanovolcanoes with diameters down to 300 nm were fabricated by the focused electron beam-induced deposition (FEBID) technique (Fig. 1) and studied by perpendicular spin-wave resonance (PSWR) spectroscopy. The spin-wave eigenfrequencies of the nanovolcanoes are notably different from the eigenfrequencies of the nanodisks and nanorings they are built from, because of the strongly non-uniform internal magnetic field. The experimental findings were elucidated with the aid of micromagnetic simulations, which indicate that the rings encircling the volcano craters lead to an effective confinement of the lower-frequency eigenmodes under the volcano crater, while the highest-frequency eigenmodes are confined in the ring area (Fig. 2). Accordingly, extension of 2D nanodisks into 3D nanovolcanoes allows to obtain same response in low frequency part of the spectrum in elements with 30% smaller footprints [1]. The nanovolcanoes can be viewed as multi-mode resonators with potential applications in telecom industry. Also, we demonstrated the engineering of the magnetic properties of CoFe-based nanodisks fabricated by FEBID. The material composition was tuned in situ via the e-beam waiting time in the deposition process and their post-growth irradiation with Ga ions. The saturation magnetization Ms of the nanodisks was deduced from PSWR measurements. The achieved variation of Ms in the broad range from 720 to 1430 emu/cm3 continuously bridges the gap between the Ms values of widely used magnonic materials such as Permalloy and CoFeB [2].

![](https://assets.underline.io/uploads/markdown_image/1/image/ca3695146d0c42bbf0d0b907eb2d1eb1.jpg)

Fig. 1. Atomic force microscopy image of the nanovolcano with the outer diameter of 300 nm and the crater diameter of 200 nm. Inset: cross-sectional line scan.

![](https://assets.underline.io/uploads/markdown_image/1/image/355d13301bed6b9caa1885c2c3af1f2e.jpg)

Fig. 2. Calculated spatial dependences of the spin-wave eigenfunctions for (a) 40 nm-thick nanovolcano NV300/200, (b) 20 nm-thick nanodisk ND300, and (c) 20 nm-thick nanoring NR300/200 at the out-of-plane bias magnetic field of 1.2 T.

Spin-wave spectra of 3D nanovolcanoes fabricated by focused electron beam-induced deposition

The appearance of spontaneous coherence is a fundamental feature of a Bose–Einstein condensate (BEC) and an essential requirement for possible applications of the condensates for classical and quantum computing. Using a magnon BEC in a magnetic crystal, such computing can be performed even at room temperature. Using specially shaped bulk samples of yttrium iron garnet crystal, we developed a novel efficient method of BEC investigation by direct detection of microwave radiation emitted by parametrically driven magnon gas (Fig.1a-b). By using this method, we studied the process of coherence formation in a magnon BEC and demonstrated that no coherent BEC state could be found during the action of a strong magnetic pumping field. At the same time, after the pumping is terminated, the overpopulated magnon gas evolves toward BEC and reaches full coherence, with the width of the magnon radiation spectrum decreasing by more than two orders of magnitude (Fig. 1c). The residual bandwidth is mainly determined by the lifetime of magnons, as expected for a fully coherent BEC consisting of a single magnon state. This direct demonstration of the magnon BEC coherence [1] brings the implementation of room temperature BEC-based computing closer. Moreover, the magnon BEC coupling with dynamic stray fields outside the sample is enabled by a proper choice of the sample shape giving direct spectroscopic access to the BEC. Such an approach can function as a convenient tool for integrating magnetic quantum systems into electrical environments.

![](https://assets.underline.io/uploads/markdown_image/1/image/08092bb8bbc4772d1216c4df47190393.jpg)

(a) Experimental setup for microwave detection of magnon BEC
dynamics.
(b) Bulk BEC mode and one of the edge magnon modes in a cuboid YIG sample. The monotonic blue line schematically shows the profile of the static magnetic field H within the YIG sample. Letters denote three field values: A – deeply inside the sample, B – at the point where the bulk BEC mode becomes evanescent, C – at the sample edge.
(c) Magnon dispersion curves in the middle of the sample (at point A) and near the edge (at point B). The green, blue, and red signal intensity lines show the microwave power spectra registered during 1 µs interval before the end of pumping as well as 2 µs and 4 µs after the pump pulse is turned off.

Evolution of Room-Temperature Magnon Gas toward Coherent Bose–Einstein Condensate

Interaction between quasiparticles of a different nature, such as magnons and phonons, leads to mixing their properties and forming hybrid quasiparticles—hybrid magnetoelastic bosons, also called magnon polarons—in the areas of intersection of individual spectral branches. In garnet ferrite films, such hybridization results in several new phenomena as the resonant increase in the efficiency of the spin Seebeck effect [1,2] and the spontaneous bottleneck accumulation [3] of the magnetoelastic quasiparticles. Similar to the Bose-Einstein magnon condensation (BEC) [4], the latter phenomenon occurs at the bottom of the magnon spectrum in yttrium iron garnet films exposed to microwaves. However, unlike the BEC, which is a consequence of the equilibrium Bose statistics, the bottleneck accumulation is determined by changing interparticle interactions. Studying the transport properties of accumulated quasiparticles [5], we found that such accumulation occurs in two frequency-distant groups: quasiphonons and quasimagnons. These quasiparticles propagate in the film plane as spatially localized beams with distinctly different directions and velocities. The developed theoretical model describes the double accumulation effect. Parametrically pumped magnons scatter through a step-by-step thermalization process towards the lower energy states. At the magnon-phonon hybridization region, the efficiency of nonlinear scattering decreases, and the downward quasiparticle flux slows down. Owing to the bottleneck effect, quasimagnons form a density peak, in which intensity saturates due to their scattering to the lowest branch of the magnon spectrum. The remaining quasimagnons, scattering to phonon states below the magnon spectrum, transform into quasiphonons with vanishing scattering amplitude. These quasiphonons form the second accumulated group. By considering the 2D magnon-phonon spectrum, we determined the wavevectors and frequencies of each group and concluded about the caustic nature of the quasimagnon beams. Due to nonzero group velocity, the accumulated quasiparticles can be used for data transfer in magnon spintronics. Funded by the ERC Advanced Grant 694709 SuperMagnonics and by the DFG within TRR173 – 268565370 (project B04).

Double Accumulation and Anisotropic Transport of Magneto-Elastic Bosons in Yttrium Iron Garnet Films

Cu(1,3-bdc) is a topological magnon insulator where the Cu ions are arranged in a 2D Kagome lattice1,2. Here, we study the magnetization dynamics of Cu(1,3-bdc) using vibrating sample magnetometry (VSM, Fig.1a) and ferromagnetic resonance (FMR, Fig.1b,c) spectroscopy. We find that the saturation magnetization MS values when an external field H is applied out-of-plane (OOP) is less than MS values when H is applied in-plane (IP) at all tested temperatures. We find similar inequivalence in the g-factor, extracted from the fitted FMR dispersion curves (Fig.2a,b) depending on the direction of H. The g-factor tensor (Fig.2c) points towards orbital momentum contribution to the magnetic moment. As shown in Fig.2d, the temperature dependence of ΔMS=MS, IP-MS, OOP, and Δg=gIP- gOOP are closely correlated at T&lt;6K but they diverge at T&gt;6K. Since the g-factor tensor is most likely mediated by orbital magnetism, the deviation of Δg from ΔMS is unexpected. We propose that this could be a result of thermally-induced non-vanishing spin chirality contributing to the electronic orbital magnetism3. Following the approach in Ref.[3],our analysis shows that the symmetry of the system allows contributions to the orbital magnetization originating from spin excitations for the OOP field, but not the IP field. The prediction of our theoretical assessment is an increase of the g-factor with temperature for OOP field, while it is expected to decay for IP field. The general trend and the change in sign of the g-factor anisotropy observed experimentally at higher temperatures is thus consistent with this theory, which suggests that the spin excitations can renormalize the fundamental quantum mechanical constant conventionally associated with the atomic orbital magnetism. Our findings bring to the forefront a so far not considered mechanism for orbital dynamics in topological magnonic systems.

![](https://assets.underline.io/uploads/markdown_image/1/image/eb9082c0d94a561e7eac0b66d95d264f.jpg)

Fig.1:a VSM M vs. H curves at different temperatures. b and c FMR profiles at different microwave frequencies for IP and OOP fields, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/1d106de32a84f8d6ee6bc552c78421e3.jpg)

Fig.2: a and b FMR dispersion curves showing Kittel fits for IP and OOP fields, respectively. c Temperature dependence of the gyromagnetic ratio γ and g-factor. d Temperature dependence of Δγ and ΔMS.

Spin Chirality-Mediated Orbital Angular Momentum in Quasi-Two-Dimensional Topological Magnon Insulator Cu(1,3-bdc)

The investigation of Spin waves (SW) and their quanta, magnons, in thin magnetic films, has emerged as an attractive field of research due to their very short wavelength reaching down to nanometers at GHz frequencies. SWs also have a rich dispersion relation, that depends on their propagation direction with respect to a magnetic field. Hence, SW -based devices are promising candidates for microwave information processing, and eventually for overcoming the limitations encountered with CMOS devices[1]. Yet, a key challenge is to achieve long propagation distances of the spin waves[2]. Recent studies using spin-orbit torques[3] where a Platinum stripe placed on top of a 20 nm thick YIG waveguide (Pt/YIG) showed that SOT could generate a full compensation of the damping, leading to auto-oscillations of the magnetization above a critical injected current density. When excited with a microwave field, spin waves propagation length was increased by a factor of 10 in the bilayer. However, amplification of propagating spin-waves was not observed yet, due to the onset of nonlinear dissipative processes above the critical current. Recently, interests turned towards materials having perpendicular magnetic anisotropy in which one can also achieve a full damping compensation along with SW emission. Here, we propose a new pathway towards SOT-based SW amplification. Using micromagnetic simulations (Mumax3), we demonstrate that pulsed SOT does induce a lossless SWs propagation when the pulse duration is shorter than 200 ns. Remarkably, amplification is achieved when injecting currents exceeding the auto-oscillation threshold (Fig.1). A key condition is however to minimize non-linear coupling. In our case, we achieved this condition in a 20 nm thick Bi-YIG film structure with a vanishing effective magnetization i.e. the out-of-plane magnetic anisotropy compensates the demagnetization field[4].

![](https://assets.underline.io/uploads/markdown_image/1/image/a7679bfc41668549787a60c203c14403.jpg)

Spin Wave Amplification using Spin-Orbit-Torques, A micromagnetic approach

The thermodynamic properties of exchange stiffness

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