France



The early development of photonic devices [1, 2] is leading the way for their magnonic analogs [3] such as spin-wave (SW) generators [4], couplers [5, 6], and phase shifters [7]. Such basic functional components are necessary for the development of miniaturized and energy-effcient magnetic circuits for processing high-frequency microwave signals. Our study goes along further development of SW devices, to demonstrate circulation [8] and other functionalities using ferromagnetic conduits coupled by resonators in analogy to photonic channel add-drop filters [1].&lt;br/&gt;&lt;br/&gt;We investigate the propagation behavior of SW in a cobalt (Co) bi-layer structure separated by a non-magnetic material and coupled with a permalloy (Py) stripe placed between two layers. The layers have a thickness of 5 nm, the separation between the stripe and the layers is of 10 nm and the stripe has a thickness of 50 nm. All elements are saturated in the same in-plane direction, perpendicular to the propagation of SWs. We investigate numerically the SW dynamics governed by the Landau-Lifshitz-Gilbert equation, solved numerically using the software COMSOL Multiphysics. We study the SWs dynamics with two different approaches. In the first approach, we investigate the SW dispersion relation of the system in the frequency domain. The lattice constant of the system is set at 500 nm. To analyze the origin of different type of interactions and possible functionalization we calculate the dispersion relations for two layers, the stripe only, three layers, and finally for the two layers coupled with the Py stripe. The dispersion relations allow to determine propagation properties, the coupling between the modes and their hybridization. We found that the coupling between the elements is strong at some frequencies at which suitable conditions for control of propagating SWs are formed.&lt;br/&gt;&lt;br/&gt;As a second approach, we used the time-domain simulations to demonstrate functionality of the proposed system. We excite the system with an antenna generating SWs with a single frequency. We found interesting propagation properties for a 19-GHz spin wave. The excitation of a wave in the top layer generates an edge mode in the Py resonator (Fig. 1a) and the SW is fully transmitted to the bottom layer propagating in the left. The excitation in the bottom layer (Fig. 1b) is not transferring to the top layer even if the Py resonator is weakly excited. Due to inversion symmetry with respect to the horizontal mid-plane, the analogous properties are for SW excited on the right side of the resonator. Thus, we demonstrated the functionality of a circulator in the system which is suitable for further miniaturization through decreasing the size of a resonant element. The proposed system composed of two thin-film SW conduits coupled by a ferromagnetic resonator can operate with different functionalities in dependence on the frequency. The next investigations, at other frequencies, positioned at proximity of other Py resonances, offer other functionalities, like magnetostatic screening from the direct coupling between the waveguides. The operation can be further tuned by changing the magnetization characteristics, magnetic field, and by exploring the third dimension.&lt;br/&gt;&lt;br/&gt;The research leading to these results has received funding from the Polish National Science Centre, project No. UMO-2018/30/Q/ST3/00416.

**References:**

[1] S. Fan et al. In: Opt. Exp. 3 (Aug. 1998), pp. 4-11.&lt;br/&gt;[2] S. Fan et al. In: Phys. Rev. Lett. 80 (Feb. 1998), pp. 960-963.&lt;br/&gt;[3] A. Barman and A. Haldar. In: Solid State Phys. 65 (May 2014), pp. 1-108.&lt;br/&gt;[4] P. Gruszecki et al. In: Solid State Phys. 6 (Mar. 2016), p. 22367.&lt;br/&gt;[5] A. Sadovnikov et al. In: Appl. Phys. Lett. 107 (Nov. 2015), p. 202405.&lt;br/&gt;[6] Q. Wang et al. In: Nature Electronics 3 (Dec. 2020), pp. 1-10.&lt;br/&gt;[7] Z. Zhang et al. In: J. Phys. D 53.10 (Dec. 2019), p. 105002.&lt;br/&gt;[8] K. Szulc et al. In: Phys. Rev. Appl. 14 (Sept. 2020), p. 12.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8ea59d302777763a68e7427139755587.jpg)
Fig.1



INTERMAG 2021

 Spin-wave Circulation in a Ferromagnetic Resonator with Two Adjacent Layers

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The early development of photonic devices [1, 2] is leading the way for their magnonic analogs [3] such as spin-wave (SW) generators [4], couplers [5, 6], and phase shifters [7]. Such basic functional components are necessary for the development of miniaturized and energy-effcient magnetic circuits for processing high-frequency microwave signals. Our study goes along further development of SW devices, to demonstrate circulation [8] and other functionalities using ferromagnetic conduits coupled by resonators in analogy to photonic channel add-drop filters [1]. We investigate the propagation behavior of SW in a cobalt (Co) bi-layer structure separated by a non-magnetic material and coupled with a permalloy (Py) stripe placed between two layers. The layers have a thickness of 5 nm, the separation between the stripe and the layers is of 10 nm and the stripe has a thickness of 50 nm. All elements are saturated in the same in-plane direction, perpendicular to the propagation of SWs. We investigate numerically the SW dynamics governed by the Landau-Lifshitz-Gilbert equation, solved numerically using the software COMSOL Multiphysics. We study the SWs dynamics with two different approaches. In the first approach, we investigate the SW dispersion relation of the system in the frequency domain. The lattice constant of the system is set at 500 nm. To analyze the origin of different type of interactions and possible functionalization we calculate the dispersion relations for two layers, the stripe only, three layers, and finally for the two layers coupled with the Py stripe. The dispersion relations allow to determine propagation properties, the coupling between the modes and their hybridization. We found that the coupling between the elements is strong at some frequencies at which suitable conditions for control of propagating SWs are formed. As a second approach, we used the time-domain simulations to demonstrate functionality of the proposed system. We excite the system with an antenna generating SWs with a single frequency. We found interesting propagation properties for a 19-GHz spin wave. The excitation of a wave in the top layer generates an edge mode in the Py resonator (Fig. 1a) and the SW is fully transmitted to the bottom layer propagating in the left. The excitation in the bottom layer (Fig. 1b) is not transferring to the top layer even if the Py resonator is weakly excited. Due to inversion symmetry with respect to the horizontal mid-plane, the analogous properties are for SW excited on the right side of the resonator. Thus, we demonstrated the functionality of a circulator in the system which is suitable for further miniaturization through decreasing the size of a resonant element. The proposed system composed of two thin-film SW conduits coupled by a ferromagnetic resonator can operate with different functionalities in dependence on the frequency. The next investigations, at other frequencies, positioned at proximity of other Py resonances, offer other functionalities, like magnetostatic screening from the direct coupling between the waveguides. The operation can be further tuned by changing the magnetization characteristics, magnetic field, and by exploring the third dimension. The research leading to these results has received funding from the Polish National Science Centre, project No. UMO-2018/30/Q/ST3/00416.

**References:**

[1] S. Fan et al. In: Opt. Exp. 3 (Aug. 1998), pp. 4-11. [2] S. Fan et al. In: Phys. Rev. Lett. 80 (Feb. 1998), pp. 960-963. [3] A. Barman and A. Haldar. In: Solid State Phys. 65 (May 2014), pp. 1-108. [4] P. Gruszecki et al. In: Solid State Phys. 6 (Mar. 2016), p. 22367. [5] A. Sadovnikov et al. In: Appl. Phys. Lett. 107 (Nov. 2015), p. 202405. [6] Q. Wang et al. In: Nature Electronics 3 (Dec. 2020), pp. 1-10. [7] Z. Zhang et al. In: J. Phys. D 53.10 (Dec. 2019), p. 105002. [8] K. Szulc et al. In: Phys. Rev. Appl. 14 (Sept. 2020), p. 12.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8ea59d302777763a68e7427139755587.jpg)
Fig.1

technical paper

Spin wave (SW) based computing i.e. magnonics relies on propagating SW as information carriers. Being quasiparticles, magnons (the SW quanta) have a fine lifetime characterizing the exponential decay of the magnons population. Consequently, up to now, all magnonic devices operate within a short time-window and on short propagation distances. Finding a SW amplification paradigm is hence a prerequisite for the development of magnonics as a credible CMOS alternative. Two schemes are currently envisioned for SW amplification: Parametric pumping1 and spin-orbit-torque amplification (SOT)2. Here we will discuss SOT amplification in BiYIG/Pt bilayer3 (Fig.1). Using micro Brillouin light scattering spectroscopy (µ-BLS) on a 500 nm wide Pt/BiYIG waveguides, we take advantage of the strong BLS signal of BiYIG (nearly two orders of magnitude the one of YIG3) to extensively study the influence of SOT on coherent propagating magnon modes. By passing current in the Pt layer, the spin accumulation resulting from the spin-Hall effect induces a positive torque that compensates the Gilbert damping of BiYIG when the magnetic field orientation is perpendicular to the waveguide axis. At subcritical values, the current has a linear effect that lowers the effective damping, thus significantly increasing the decay length of the propagating magnons, similar to what was observed previously in a YIG/Pt waveguide4. However, by increasing the excitation frequency of these propagating magnons for a constant external field, we observe a gradual but strong decrease of the modulation efficiency of their lifetime in our BiYIG/Pt waveguide. Once the current in the Pt layer reaches a threshold value, the vanishing effective damping allows for the onset of the auto-oscillation regime5 where the thermal magnons get incoherently amplified and the magnetization auto-oscillates in the main uniform mode. However, coherent propagating magnons strongly interact with the auto-oscillations6, preventing the observation of their propagation at continuous supercritical current. To circumvent these non-linear interactions, we study the auto-oscillation transient regime for the first time by performing time-resolved µ-BLS measurements. When the SOT is switched on, the uniform auto-oscillation grows exponentially, in agreement with the Slavin & Tiberkevitch non-linear auto-oscillator model7 in the linear regime limit. The onset of non-linearities only occurs when the auto-oscillation reaches its stable orbital, after few hundreds of nanoseconds typically. Using a rf and DC pulse scheme, we propagate rf-excited magnons before the onset of non-linearities at supercritical currents and observe the amplification of propagating magnons (Fig. 2-a). Similarly to what was observed in the continuous regime at subcritical currents, the modulation efficiency of the propagating magnons lifetime strongly decreases with frequency (Fig. 2-b). This experiment sheds new light on the influence of SOT on propagating magnons and pave the way toward a magnonic spin-wave amplifier using DC current.

**References:**

1 T. Brächer, P. Pirro, and B. Hillebrands, Phys. Rep. 699, 1 (2017). 2 V.E. Demidov, S. Urazhdin, A. Anane, V. Cros, and S.O. Demokritov, J. Appl. Phys. 127, 170901 (2020). 3 L. Soumah, N. Beaulieu, J. Ben Youssef, and A. Anane, Nat. Commun. 9, 3355 (2018). 4 M. Evelt, V.E. Demidov, and A. Anane, Appl. Phys. Lett. 108, 172406 (2016). 5 M. Collet, V. Cros, A. Anane, G. de Loubens, and O. Klein, Nat. Commun. 7, 10377 (2016). 6 B. Divinskiy, S. Urazhdin, S.O. Demokritov, and V.E. Demidov, Nat. Commun. 10, 5211 (2019). 7 A. Slavin and V. Tiberkevich, IEEE Trans. Magn. 45, 1875 (2009).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7c591b0f58bbf93bc6d6866a53fdf516.jpg)
Fig. 1 : Schematic representation of the 0.5µm-wide BiYIG/Pt waveguide. The DC current is applied in the Pt along the waveguide axis and a 300nm-wide strip rf-antenna is used to excite propagating magnons. The µ-BLS laser spot is focused from the back of the sample.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8aea6e40370d823a1c720f70ddf0448.jpg)
Fig. 2 : Spin-wave intensity as a function of the distance to the antenna. The experimental curves (solid lines) are linearly fitted (dotted lines) in log scale for 4 frequencies (only 2 shown here) 4.925 GHz (a), 4.95 GHz, 4.975 GHz and 5.0 GHz (b) and 4 DC voltages corresponding to one subcritical current (1.7V) and three supercritical currents (2.2, 2.3 and 2.4V). The FMR is constant during the pulse with value 4.885 ± 0.01 GHz. 
The positive slope at supercritical currents at 4.925 GHz indicates an amplification of the propagating spin-waves (negative effective damping) due to SOT. The damping compensation efficiency gradually decreases as the frequency is increased, and the spin-waves are no longer amplified at 5.0 GHz for the applied voltages.

 Insight on the effect of SOT on propagating magnons: Spin-wave amplification and frequency dependent efficiency

Artificial neural networks (NN) running on traditional computing hardware tend to be relatively slow and power-hungry. One possible solution to this issue is to create hardware with neurons that spike in a manner similar to biological neurons. For example, Intel [1] and IBM [2] have recently fabricated neuromorphic chips based on spiking NN. Still, neurons in these chips are fabricated with multiple silicon transistors, which limits power efficiency of such an approach. The development of neuromorphic chips would benefit from a nanoscale, single element synthetic neuron that behaves in a manner that closely resembles a biological neuron. Such a synthetic neuron can be fabricated with nanometer dimensions from an antiferromagnetic (AFM) material mounted on platinum [3]. These “AFM neurons” are spin Hall oscillators driven by a sub-threshold spin current that can produce ultra-short (~ 5 ps) spikes in response to a weak external stimulus. In traditional spiking NN, synapses that connect artificial neurons are formed by complex active circuits. A much simpler approach would be to use passive memristors as artificial synapses [4]. However, memristors create symmetric coupling between neurons, which allows a spike produced by one neuron to travel to all neighboring neurons, both forwards and backwards. Unfortunately, this bi-directional signal propagation makes use of standard machine learning algorithms impossible. Therefore, it would be interesting to design NN in which unidirectional spike propagation is possible even with symmetrically coupled neurons. Here, we demonstrate that the unidirectional signal propagation is realizable in symmetrically-coupled NNs based on AFM neurons. This regime is possible due to an effective inertia that originates from exchange coupling between the two magnetic sublattices [3] and creates a relatively long refractory period - an interval of time after a neuron fires when it is unable to fire again. We show, that it is possible to adjust synapse weights such that each neuron is still in its refractory period when it receives a backward propagating signal from its neighbors. Then, a simple chain of neurons receiving an input at one end carry the signal to the opposite end without any reflections. However, for such a simple chain of AFM neurons, an input can occur at either end, and thus spikes would be able to travel in both directions. It would be beneficial to create an ‘isolator’ neuromorphic circuit, which would allow the propagation of spikes in only one direction. Figure 1 depicts the schematic of a possible ‘isolator’ circuit composed of AFM neurons. This figure uses circles to represent AFM neurons. The isolator is composed of neurons 4, 5, and 6. In the figure, double arrows represent symmetrical synapses with arrow width proportional to the coupling strength. This isolator allows spikes to be carried “forwards” from neuron 1 to 9, but stops signals carried “backwards” from neuron 9 to 1, as illustrated by the results of numerical simulations shown in Fig. 2. The isolator functions as follows. When the spike is initiated in neuron 1 (Fig. 2(a)), it travels along the chain to neuron 4. Then, the spike generated by neuron 4 travels to both neuron 5 and neuron 6. Neuron 5 also generates a spike, which is carried to neuron 6. The synaptic weights connecting to neuron 6 are small enough that a single input is not sufficient to induce a spike in the AFM neuron. However, when both spikes combine at neuron 6, they provide enough stimulus for neuron 6 to fire. The spike can then continue to propagate from neuron 6 to 9. When traveling backwards (Fig. 2(b)), a spike begins at neuron 9 and travels unidirectionally to neuron 6. Neuron 6 will generate a spike, but the synaptic weights from neuron 6 to 4/5 are insufficient to induce an additional spike. This, in effect, prevents the backward travel of spikes through the isolator. Using the proposed isolator NN, it is possible to design a number of other neuromorphic circuits supporting unidirectional information propagation, for example, neuromorphic combiners and unidirectional memory rings. The creation of such circuits with symmetric synapses shows hope for AFM neurons to be used to create more realistic spiking NN. The delay caused by the exchange inertia in the AFM neurons plays a key role in any neuromorphic circuit constructed with these neurons. These physically realistic neurons will increase the possibility for more functional spiking neural network hardware.

**References:**

[1] M. Davies et al., Loihi: A Neuromorphic Manycore Processor with On-Chip Learning, in IEEE Micro, vol. 38, no. 1, pp. 82-99, January/February 2018, doi: 10.1109/MM.2018.112130359. [2] M. V. DeBole et al., TrueNorth: Accelerating From Zero to 64 Million Neurons in 10 Years, in Computer, vol. 52, no. 5, pp. 20-29, May 2019, doi: 10.1109/MC.2019.2903009. [3] R. Khymyn, I. Lisenkov, J. Voorheis, O. Sulymenko, O. Prokopenko, V. Tiberkevich, J. Akerman, and A. Slavin, Ultra-fast artificial neuron: generation of picosecond-duration spikes in a current-driven antiferromagnetic auto-oscillator, Scientific Reports 8, 15727 (2018). [4] Sung Hyun Jo, Ting Chang, Idongesit Ebong, Bhavitavya B. Bhadviya, Pinaki Mazumder, and Wei Lu Nanoscale Memristor Device as Synapse in Neuromorphic System, Nano Lett. 2010, 10, 4, 1297–1301

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/eb70901938b9e7ce0e61457aefc96cde.jpg)
Fig 1: Chain of spiking neurons, configured as an isolator circuit. Gray circles represent neurons, and black arrows represent coupling between neurons. Thick black arrows represent symmetric synaptic weight tuned such that a spike cannot flow backwards. Thin black double arrows represent symmetric synaptic weight weak coupling.

![](https://assets.underline.io/uploads/markdown_image/1/image/7460602315b4c8ef8afb70fa1a01a721.jpg)
 
Fig 2: Results of simulation of AFM neurons configured as an isolator circuit. (a) Initial spike is in neuron 1. The spike travels through the chain, to neuron 9. (b) The initial spike is at neuron 9. The spike travels to neuron 6, then is unable to continue.

 Antiferromagnetic Artificial Neural Networks with Symmetric Coupling

The mutual synchronization of spin-torque nano-oscillators (STNOs) arranged in a large array improves the individual properties of each STNO, increasing the output-power and reducing the linewidth of the emitted signal. These improvements are required to implement STNOs in wireless communication schemes [1] and furthermore open applications for different oscillator based neuromorphic computing schemes [2]. Many theoretical and experimental studies considered single STNO devices [3], while experimentally the synchronization of a large number of STNOs remains a challenging task [4]. This is because the synchronized state and the phase-locking depend on the geometry, device resistance, input signals, coupling mechanism, among others [5]. As a first step to guide experiments on the synchronization behavior of STNO arrays, we have undertaken a theoretical study, combining numerical and analytical approaches based on the spin-wave formalism [3]. The STNO devices considered here are of circular cross-section (100nm in diameter) with a perpendicular polarizer and an in-plane magnetized free layer under a strong out-of-plane field. This field is larger than the saturation field. The free layer magnetization is orientated out-of-plane leading to an out-of-plane precession mode that is advantageous to induce strong dipolar coupling between STNOs, which is the coupling mechanism considered here. We derived analytically the coupled equations of amplitudes and phases where the latter are similar to the Kuramoto-phase equation, but limited to a finite number of devices. Solving for the stationary states, this model provides analytical expressions for the locking-range, the stationary amplitudes, and the frequency of oscillations of the synchronized state. These solutions depend on the array geometry, number N of STNOs in the array, separation between STNOs, distribution of the DC current, as well as the different magnetic material parameters, and the external magnetic field. In this work, we investigate the solution of these equations for two specific array geometries: a straight line and a ring structure (N=3,6 and 10), shown in figure 1b. Our results show that the locking-range depends on the separation among the STNOs (because this scales the coupling strength), but it is independent of the angle among them. Furthermore, we show that the locking-range is larger when the STNOs are placed in a ring structure (or triangle for N=3) as compared to a straight-line structure. Finally, the most interesting result is that for the ring structure, we identify two dynamic modes that represent a standing wave for the STNO phases along the ring structure (observed for N&gt;5). These two dynamics modes are shown in Fig. 2 and are called here (a) the in-phase mode, with identical phases for all STNOs, and (b) the out-of-phase mode, where the phase difference between two adjacent STNOs is 2π/N. We study how these states and their existence range are modified upon variation of the current density through the different STNOs (identical and non-identical currents), the STNO separation, and the number of STNOs in the ring. The analytical results are in good agreement to the numerical solutions of the Landau-Lifshitz-Gilbert (LLG) equation, as long as the separation among the STNOs is above a certain critical value. For smaller separations, the dipolar interactions become too strong and non-linear effects would require to go beyond the first-order approximations used to solve the amplitude and phase equations. The results reported here represent a first step to validate the phase equations and to define their limitations, and can be used to address now the Kuramoto equations to find solutions for large scale structures (N very large). We acknowledge financial support in Chile from FONDECYT 1200867, and Financiamiento Basal para Centros Científicos y Tecnológicos de Excelencia AFB 180001. D. M.-A. acknowledges Postdoctorado FONDECYT 2018, 3180416. M. A. Castro acknowledges Conicyt-PCHA/Doctorado Nacional/2017-21171016. The project was supported in part by ERC Advanced Grant MAGICAL No. 669204

**References:**

[1] T. Chen, et al., Proceedings of the IEEE., 104, 1919-1945(2016). [2] J. Grollier, et al., Nat. Electron. 3, 360 (2020). [3] A. Slavin and V., Tiberkevich, IEEE Trans. Magn., 45,1875(2009). [4] S. Tsunegi, et al., Scientific Reports., 8, 13475(2018). [5] R. Lebrun, et al., Nat. Commun. 8, 15825 (2017).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/67e3f596fc435dd303beec991462b132.jpg)
Fig.1: (a) Schematics of an STNO device and (b) two configurations for a coupled STNO array.

![](https://assets.underline.io/uploads/markdown_image/1/image/fdc26dc3ca23f437e028887120de5d0d.jpg)
Fig. 2: Schematics of the two dynamic modes obtained for the ring structure. The snapshots show the instantaneous distribution of the x- and y- components of the magnetization in three consecutive time steps t1 < t2 < t3, for (a) the in-phase and (b) the out-of-phase modes.

 Mutual synchronization of an array of spin-torque oscillators with perpendicular polarizer

It has been shown that a rapid cooling of Yttrium-Iron-Garnet (YIG)/Pt nanostructures preheated by a DC electric current pulse sent through the Pt layer leads to an overpopulation of a magnon gas and to the subsequent Bose-Einstein condensation (BEC) of magnons [1]. Hereby, the application of a sufficiently-long DC heating pulse was used to generate a high population of magnons being in equilibrium with the phononic system. The subsequent rapid decrease in the temperature resulted in the break of the equilibrium, in the over-population of magnons over the whole magnon spectrum, and, in the following, in a redistribution of magnons from higher to lower energies. Finally, if the elevated temperature was high enough, and the cooling process fast enough, the chemical potential reached the minimal energy of the system and the BEC was triggered [1]. In these experiments, a Pt or Al layer was used to heat the YIG nanostructure. For a Pt heater, in addition to the Ohmic heating effects, one expects the generation of a spin-polarized current transverse to the YIG/Pt interface due to the spin Hall effect (SHE) [2]. The spin current, consequently, is known to act on the magnetization dynamics in the YIG via spin-transfer torque (STT). Whereas in our previous studies the induced disequilibrium was found to be mainly given by the temperature increase and subsequent rapid cooling, we observe a significant contribution of the spin Hall effect (SHE) in a new set of samples. The larger effect of the SHE-STT was reached by a modification of the structures (see Fig. 1) with respect to those in the original study [1], i.e. by a decrease in the YIG thickness, and via using a sputtered Pt layer instead of an epitaxially grown Pt layer studies to enhance the spin Hall angle [3]. It was found that SHE-induced STT enhances the BEC formation at the bottom of the magnon spectrum – in the minimum frequency of the fundamental spin-wave mode, and excites a nonlinear magnon mode – the spin-wave bullet, located below the linear magnon spectrum [4]. We investigated the dynamics of these magnon states employing Brillouin light scattering (BLS) spectroscopy. A sketch of the experimental setup and SEM images of the investigated structure is depicted in Fig. 1. The application of a 50-ns-long DC-pulse to a 7-nm- thin Pt layer on top of a 34-nm-thin and 2-μm-wide YIG strip results in the SHE-STT. At sufficiently large amplitudes of the DC-pulse applied, the bullet mode is formed during the DC pulse at a frequency around 1.8 GHz below the frequency of the fundamental mode, see Fig. 2b. The pronounced magnon density peak in the left panel in Fig. 2c (solid orange line), measured at the time just before the current pulse is switched off, clearly demonstrates the bullet mode formation. The SHE-STT nature of the mode is confirmed by the variation of the magnetisation orientation with respect to the current direction. The consequent switching of the DC pulse off results in the rapid cooling-induced BEC of magnons at the frequency around 6 GHz - see Fig. 2c. The results of the control experiment with no SHE-STT contribution under periodically changing polarity of the applied current (Fig. 2d) are in agreement with the original findings presented in Ref. [1]. The interplay between the magnon condensate and the SHE-STT bullet mode, after the current pulse was gone, attracts special attention. Initially, in the rapid cooling phase of the process, thermally-induced excess magnons populate both the bullet mode and the condensate, see Fig. 2a. Further, both modes decay slower than expected from their lifteime indicating that they are still pumped from the gas. Finally, about 30 ns after the DC pulse is turned off, a qualitatively new process begins, resulting in the redistribution of magnons from the bullet mode to the condensate. This phenomenon manifests itself in an elevation of the condensate density accompanied by an increased decay of the bullet mode, and apparently occurs when the chemical potential of the magnon gas [1] decreases below the bullet mode frequency. The results suggest a path towards the application of macroscopic quantum states in spintronic devices.

**References:**

[1] M. Schneider, T. Brächer, D. Breitbach, V. Lauer, P. Pirro, D. A. Bozhko, H. Y. Musiienko- Shmarova, B. Heinz, Q. Wang, T. Meyer, F. Heussner, S. Keller, E. T. Papaioannou, B. Lägel, T. Löber, C. Dubs, A. N. Slavin, V. S. Tiberkevich, A. A. Serga, B. Hillebrands, and A. V. Chumak, Bose-Einstein condensation of quasiparticles by rapid cooling, Nature Nanotechnology 15, 457 (2020). [2] L. J. Cornelissen, K. J. H. Peters, G. E. W. Bauer, R. A. Duine, and B. J. van Wees, Magnon spin transport driven by the magnon chemical potential in a magnetic insulator, Phys. Rev. B 94, 014412 (2016). [3] E. Sagasta, Y. Omori, M. Isasa, M. Gradhand, L. E. Hueso, Y. Niimi, Y. Otani, and F. Casanova, Tuning the spin hall effect of pt from the moderately dirty to the superclean regime, Phys. Rev. B 94, 060412 (2016). [4] A. Slavin and V. Tiberkevich, Spin wave mode excited by spin-polarized current in a magnetic nanocontact is a standing self-localized wave bullet, Phys. Rev. Lett. 95, 237201 (2005).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/60e5026f5c65e8d34354a4da266411ef.jpg)
Colored SEM images of the investigated structures and sketch of the experimental setup. The structures consists of a 2-μm-wide and 34-nm-thick YIG strip. A 3-μm-long Platinum-heater (7 nm) on top is contacted by Ti/Au leads. The magnon intensity below the Pt-covered region is measured using time-resolved BLS spectroscopy. The shape of the Ti/Au contact pads attached to the Pt injector results in a DC current parallel to the short axis of the strip. Qualitatively the same results were obtained for another set of samples with a DC current flowing parallel to the short axis of the strip.

![](https://assets.underline.io/uploads/markdown_image/1/image/534ab70ea9e9adb9ae113bed737dcfc7.jpg)
(a) Integrated BLS intensity from (b) as a function of the time within the two indicated frequency ranges. (b) BLS intensity color-coded as a function of time and the BLS frequency. During the pulse (0 ns < t < 50 ns and U = 2.0 V) the formation of a bullet mode is visible. The inset shows a sketch of the structure. (c) BLS intensity as a function of the BLS frequency at three different time steps, indicated by the dashed lines in (b). The black dashed lines depict the thermal BLS spectrum without a DC pulse present. (d) Integrated BLS intensity color coded as a function of time and the BLS frequency. In contrast to (b), the current polarity is alternating during the pulse, effectively switching off the contribution of the SHE-STT while kepping the heating constant.

 The interplay between the rapid cooling-induced BEC and SHE-STT-driven bullet mode

Spin waves can probe the Dzyaloshinskii-Moriya interaction (DMI), which gives rise to topological spin textures, such as skyrmions. However, the DMI has not yet been reported in yttrium iron garnet (YIG) with arguably the lowest damping for spin waves. In this work, we experimentally evidence the interfacial DMI in a 7-nm-thick YIG film by measuring the nonreciprocal spin-wave propagation in terms of frequency, amplitude, and most importantly group velocities using all electrical spin-wave spectroscopy. The velocities of propagating spin waves show chirality among three vectors, i.e., the film normal direction, applied field, and spin-wave wave vector. By measuring the asymmetric group velocities, we extract a DMI constant of 16 μJ/m2, which we independently confirm by Brillouin light scattering. Thickness-dependent measurements reveal that the DMI originates from the oxide interface between the YIG and garnet substrate. The interfacial DMI discovered in the ultrathin YIG films is of key importance for functional chiral magnonics as ultralow spin-wave damping can be achieved.

**References:**

[1]. H. C. Wang, J. L. Chen, T. Liu, J. Y. Zhang, K. Baumgaertl, C. Y. Guo, Y. H. Li, C. P. Liu, P. Che, S. Tu, S. Liu, P. Gao, X. F. Han, D. P. Yu, M. Z. Wu, D. Grundler, and H. M. Yu*, Chiral Spin-Wave Velocities Induced by All-Garnet Interfacial Dzyaloshinskii-Moriya Interaction in Ultrathin Yttrium Iron Garnet Films, Phys. Rev. Lett. 124, 027203 (2020).

**Images:**

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

Quantum magnonics is a rapidly growing field offering myriad of potential applications[1,2] particularly in the realm of coherent information processing. Magnons which are the collective excitations of spins offer exotic functionalities such as the ability to couple to various dynamic systems. Realisation of strong coupling , owing to large spin densities, which is tunable into the ultrastrong coupling regime makes them suitable for coherent information exchange at much faster speeds. Apart from this the easily accessible gigahertz frequency range, simple fabrication and miniaturization provides an ideal platform for incorporation in various microwave electronic circuits. Recently[3,4,5], Synthetic Antiferromagnets(SyAF) have attracted prodigious interest due to the controllable exchange coupling in the GHz regime making them a test bed for various magnonic applications[6]. Moreover, the substantially large mode volume overlapping gives them an advantage over light-matter interactions in attaining higher a coupling strength. A SyAF consists of two ferromagnetic thin films (FM) with an antiferromagnetic exchange coupling separated by a non-magnetic spacer (NM). When the two FMs are symmetric, a mode crossing occurs between symmetry protected in-phase and out- of -phase resonance precessions via in-plane external magnetic field tuning. This is reminiscent of degeneracy between even and odd parity magnons. This degeneracy and subsequent mode crossing can be lifted by breaking the symmetry either extrinsically or intrinsically. Extrinsic symmetry breaking (SB) can be exerted by tilting the external magnetic field out-of-plane which shifts the rotational symmetry axis from the SyAF plane, and the system is no longer symmetric under two-fold rotation. This leads to hybridization of the two modes and an anticrossing gap is generated. Additionally, a coupling gap can be induced by dynamic dipolar interaction[3]. The strength of the coupling between two magnon modes is determined by the size of the gap. Maximizing the coupling efficiency is thus easily attainable by the controllability of the coupling gap by external magnetic field orientation. Apart from an extrinsic control of coupling strength the symmetry of the system can be broken intrinsically by using either two different FMs or the same FMs with different thickness. This would lead to a coupling gap even for the external applied fields applied in-plane, owing to a change of magnetization under two-fold rotation. In this work we demonstrate the control of magnon-magnon coupling in SyAFs via intrinsic and extrinsic SB. The schematic of the two modes in SyAF system is shown in Fig 1(a). We first consider a symmetrical SyAF structure CoFeB/Ru/CoFeB as in Fig 2(a). The apparent mode crossing results from a degeneracy of two non-interacting modes which is protected by a two-fold rotational symmetry. By tilting the external field (see Fig 2(b)) we break this symmetry and an anticrossing gap appears. This demonstrates the extrinsic SB. When two FMs with different thickness are considered an inherent SB exists in the system which opens a gap even for an in-plane field as in Fig 2(c). Further, breaking the symmetry as in 2(d) by tilting the field leads to an increase of the gap and thus the coupling strength. This effect can be further enhanced when using two FMs of different materials, exhibiting a large coupling gap for an in-plane field as in Fig 2(e). Here, the coupling strength far exceeds the losses, described by the frequency half-width at half maximum of the individual modes. The absolute coupling strength reaches 30% of the mode frequencies, indicating the onset of the ultrastrong coupling regime. By calculating the coupling gap as a function of external field tilt angle we can putatively claim that both intrinsic and extrinsic SB can realise strong magnon-magnon coupling in SyAFs. Further, the strong coupling leads to an entanglement between even and odd parity modes, making it attractive for quantum information applications. We generalize a theoretical model in the macro spin limit by solving the two coupled LLG(Landau Lifshitz Gilbert) equations to describe the microwave absorption spectroscopy for the SB induced coupling in SyAFs. This study should shed light on the potential of SyAFs for studying and controlling magnon-magnon coherent coupling at room temperature. Moreover large tunable coupling strengths in this system will provide a novel platform for increased control of quantum systems.

**References:**

[1] G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Proc. Natl. Acad. Sci. U.S.A. 112, 3866 (2015). [2] M. Ruggenthaler, N. Tancogne-Dejean, J. Flick, H. Appel, and A. Rubio, Nature Reviews Chemistry, 2(3), pp.1-16 (2018). [3] Y. Shiota, T. Taniguchi, M. Ishibashi, T. Moriyama, and Teruo Ono, Phys. Rev. Lett. 125, 017203 (2020). [4] A. Sud, C. W. Zollitsch, A. Kamimaki, T. Dion, S. Khan, S. Iihama, S. Mizukami, and H. Kurebayashi, Phys. Rev. B 102, 100403(R) (2020). [5] M. Ishibashi, Y. Shiota, T. Li, S. Funada, T. Moriyama, and T. Ono, Sci. Adv. 6, eaaz6931 (2020). [6] A. Guedes, M. J. Mendes, P. P. Freitas, and J. L. Martins, Journal of applied physics 99,08B703 (2006).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fda936d0116f14aa194f61f7d6f95187.jpg)
Fig 1. (a) Schematic of resonance modes in SyAFs. Two moments (m1 and m2) are coupled antiferromagnetically and canted at equilibrium. Under microwave irradiation, they precess in-phase (acoustic mode) and out-of-phase (optical mode) at different angular frequencies ωac and ωop respectively .We define θB as in the figure. (b) Schematics of the magnon-magnon coupling phenomena. When the exchange field (Bex) is small or two moments are within the film plane, the coupling strength (g) is zero and we can valve g by tuning Bex and θB.

![](https://assets.underline.io/uploads/markdown_image/1/image/7eaef5b95c81b3bc4450cc2312714f60.jpg)
Fig 2. Microwave transmission as a function of frequency.(a) For CoFeB(5)/Ru(0.5)/CoFeB(5) at θB = 90°, the two modes show mode crossing (b) At θB = 25° a gap appears. (c) In NiFe(5)/Ru(0.4)/NiFe(3) intrinsic SB occurs and the gap is seen for θB = 90° which increases as in (d) for θB = 30°. (e) CoFeB(3)/Ru(0.5)/NiFe(3) showing large gap at (e) θB = 90° and (f) θB = 20°. (g-h) Coupling strength g/2π as a function of θB which increases as the tilt angle is increased.* The numbers in brackets are the thickness of the FM layer in nm.

 Modulation of magnon-magnon coupling by inherent symmetry breaking in synthetic antiferromagnets

Kagome artificial spin ice (KASI) is a network of Ising type nanobars on a kagome lattice [1,2]. Great progress has been made in understanding defects including disordered states in KASI via quasistatic imaging techniques [1,3]. However, the magnetodynamic study of disordered states has been confined to global magnetodynamic (e.g. broadband spin wave spectroscopy) and micromagnetic simulations [4]. Furthermore, the simulation study on KASI has shown the presence of novel magnetodynamic microstates that offer excellent insights into the KASI’s disordered regime [4]. From the fundamental physics side, dynamically controlled microstates may offer a way to create Dirac strings via microwave-assisted switching interior to the KASI lattice in a controlled manner and study the disordered regime systematically. From the magnonic application perspective, experimental studies of microstates in KASIs are key towards their usage as a new type of microwave filter [5] and reprogrammable magnonic crystal [6]. We investigate spin dynamics of a KASI consisting of Ni81Fe19 nanomagnets arranged on an interconnected kagome lattice using broadband spin wave spectroscopy (Fig. a,b), magnetic force microscopy (MFM) (Fig. c), and micro-focus Brillouin light scattering (BLS) microscopy (Fig. e-f). Micro-focus BLS performed on magnetically disordered states exhibit a series of magnon resonances that depend on topological defect configurations that we image by magnetic force microscopy. Nanomagnets on a Dirac string and between a monopole-antimonopole pair show pronounced modifications in magnon frequencies. Our work is key for the creation and annihilation of Dirac strings via microwave-assisted switching and reprogrammable magnonics based on ASIs.

**References:**

1) S. Ladak, D. Read, G. Perkins, L. Cohen, and W. Branford, Nature Physics 6, 359 (2010). 2) R. Wang, C. Nisoli, R. Freitas, J. Li, W. McConville, B. Cooley, M. Lund, N. Samarth, C. Leighton, V. Crespi, et al., Nature 439, 303 (2006). 3) E. Mengotti, L. J. Heyderman, A. F. Rodriguez, F. Nolting, R. V. Hugli, and H.-B. Braun, Nat. Phys. 7, 68 (2011). 4) V. Bhat, F. Heimbach, I. Stasinopoulos, and D. Grundler, Physical Review B 93, 140401 (2016). 5) X. Zhou, G.-L. Chua, N. Singh, and A. O. Adeyeye, Advanced Functional Materials 26, 1437 (2016). 6) M. Krawczyk and D. Grundler, J. Phys.: Condens. Matter 26, 123202 (2014).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a84d0d93b7a280959772b4805fbd9680.jpg)
Grayscale AESWS spectra map of KASI for a field protocol of (a) -90 mT → +90 mT and (b) -90 mT → 45 mT → 0 mT. (c) Magnetic force microscopy images of KASI after a field protocol of −100 mT → +44 mT → 0 mT. Stray fields detected by MFM are colored in blue and red consistent with charges Q = +q and Q = −q, respectively. (d) Sketches of charge configurations (blue and red spheres) and magnetization vectors M (arrows) for configurations C1 to C5. (e) BLS intensities measured at the central position on horizontal nanobars belonging to the configurations displayed above the spectra at µ0H = 25 mT. (f) Spatially resolved BLS intensity maps at 25 mT for fixed frequencies in configurations C1 to C5 displayed above.

 Direct Observation of Magnon Modes In Kagome Artificial Spin Ice with Topological Defects

In this work, we report our micromagnetic simulation results on the correlation between spin wave modes and emergent microstates in square artificial spin ice (S-ASI), consisting of elliptical and stadium shape magnetic nanoislands. S-ASI1 is a lithographically patterned engineered dipolar coupled nanostructure array, consisting of orthogonal square sublattices of highly shape anisotropic nanoislands. These islands behave as a macrospin due to high shape anisotropy where each macrospin, according to the dumbbell model, can be considered as a magnetic dipole of charge separated by the length of the nanoisland. Recently, there is a growing interest in the low-energy collective spin excitation of magnetic nanostructures of vastly different shapes resulting in the development of the field of magnonics. These excitations or spin waves are the magnetic analogs of electromagnetic waves, phonons, and plasmons which can be treated as a perturbation in terms of phase-coherent precession of magnetization vector that propagates in coupled magnetic media. The field of magnonics offers new functionalities in modern technological devices e.g., magnonics logic, phase shifters, microwave antenna, directional coupler, SW-based multiplexer, grating, and neuromorphic computing, etc. S-ASI exhibit geometrical frustration and therefore, it shows ground state degeneracy. The microstates in S-ASI enable the possibility of tunability of associated spin-wave modes and alter the spin-wave band gap without structural deformation. This key feature of S-ASI makes it a potential candidate for reconfigurable magnonics2. The interesting dynamic behavior of ASI suggests that it can be perceived as magnonic crystals with potential applications in logic devices3, data storage4, microwave filter5, spin logic gate3, etc. Hence, detailed investigation6, 7 of spin wave dynamics and evolution of modes in ASI systems involving highly shape anisotropic nanoislands of different shapes is very important. The question of the correlation between underlying magnetic states of dipolar coupled nanoislands of shapes such as ellipse and stadium, etc., and the spin wave modes is far from properly understood. In this work, we have performed a detailed study of the dynamical behavior of emergent microstates in S-ASI, using micromagnetic simulations. For these studies, we have considered S-ASI to be consisting of four square ring-type structures with elliptical and stadium shaped nanoislands of permalloy with dimensions 300 nm × 100 nm × 25 nm and lattice constant 150 nm (edge-to-edge distance of nanoislands). The magnetization reversal behavior of stadium-shaped magnetic nanoislands in S-ASI geometry involves the formation of vortices, antivortices, etc. whereas for the case of elliptical nanoislands the reversal involves sharp switching of spins. In order to investigate the dependence of the behavior of spin wave dynamics on local magnetic behavior in the S-ASI of two different shapes, we have investigated the spin wave modes and emergent microstates in both the shapes as a function of a number of square-type rings (i.e., single ring, double ring, triple ring, and quadruple ring e.g., S-ASI) of nanoislands. For S-ASI involving elliptical nanoislands, our studies reveal that emergent microstates at remanence with net local magnetic charges at the junction of the nanoislands for head-to-head or tail-to-tail configuration enhances spin wave generation in the proximity islands up-to ∼95% as compared to islands in the proximity of zero local charges ( in head-to-tail or tail-to-head configuration). Thus, we observe that the power profiles in remanence map the local charges present in the S-ASI or ring-type structures. We observe the presence of ubiquitous switching identifier mode at ~ 18 GHz (saturation) in nanoislands with an easy axis along the external bias field. This mode shows marked power variation in the switched nanoislands with respect to un-switched nanoislands. Our simulation results show that observed spin wave spectra and calculated mode profiles remain consistent as a function of rings. Our studies of S-ASI involving stadium shaped nanoislands further show that the spin wave spectra are significantly different than for the elliptical case (see Fig. 1). In the case of stadium-shaped nanoislands, the demagnetization field is inhomogeneous which creates variation in the local field within the nanoislands whereas, in the case of elliptical nanoisland, demagnetization field variation is observed near the edges. However, for a similar magnetization configuration, we have observed several new spin-wave modes with complex mode profiles in the case of stadium shape nanoislands (Fig. 2). Thus, our study gives new insight in understanding the origin of spin wave mode behavior in terms of the evolved microstates and demonstrate spin wave modes as a tool to investigate magnetization reversal and local magnetic charges in artificial spin ice system.

**References:**

(1) S. T. Bramwell et al., Science 294, 1495–1501 (2001). (2) E. Iacocca et al., Phys. Rev. B 93, 134420 (2016). (3) H. Arava et al., Nanotech. 29, 265205 (2018). (4) Y.-L. Wang et al., Science 352, 962–966 (2016). (5) I. Huynen et al., IEEE Microw. Guided W. 9, 401–403 (1999). (6) S. Gliga et al., Phys. Rev. Lett. 110, 117205 (2013). (7) E. Iacocca et al., Phys. Rev. Appl. 13, 044047 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8fa292a767f18a77cc35d89a92aba095.jpg)
The power profile of the spin wave mode excited at ∼ 6.5 GHz in S-ASI with (left) elliptical and (right) stadium-shape nanoislands at 200mT external field.

![](https://assets.underline.io/uploads/markdown_image/1/image/16c49e9a6920a28ab2b3c0ed74f242a4.jpg)
(left) Spin wave spectra of S-ASI consisting of stadium shape and elliptical shape nanoislands saturated along the x-axis at 200mT. (right) magnetization configuration of S-ASI with the stadium and elliptical shape nanoislands.

 Dependence of spin wave modes on the geometry of nanomagnets in square artificial spin ice vertices

Spin waves, and their quanta magnons, are of great interest as potential data carriers in future low-energy computing devices [1]. The phase of a spin wave provides an additional degree of freedom, while the scalability of structures and wavelengths down to the nanometer regime [2] are further advantages. Recently, a set of magnonic data processing units was demonstrated. However, the development of each of them requires specialized investigations and, usually, one device design is suitable for one function only. Moreover, an integrated all-magnonic circuit, which is suitable for the cascading of multiple magnonic units, has not yet been demonstrated. In the first half of my talk, I will present briefly the experimental realization of a nanoscale magnonic directional coupler, which consists of two separated single-mode waveguides with a width of 350 nm as shown in Fig. 1a [3]. A U-shaped antenna is used to excite spin waves and space-resolved Brillouin Light Scattering (BLS) spectroscopy is exploited for detection. It is shown that the data is coded into the spin-wave amplitude is guided towards one of its two outputs depending on the signal frequency, magnitude, and on the magnetic field. Using micromagnetic simulations, we also propose an integrated magnonic half-adder that consists of two directional couplers and we investigate its functionality for information processing within the magnon domain. In the second part of the talk, I will present the method of inverse-design magnonics, in which any functionality can be specified first, and a feedback-based computational algorithm is used to obtain the device design [4]. Our proof-of-concept prototype is based on a rectangular ferromagnetic area which can be patterned using square shaped voids as shown in Fig. 1b. To demonstrate the universality of this approach, we explore linear, nonlinear and nonreciprocal magnonic functionalities and use the same algorithm to create a magnonic (de-)multiplexer, a nonlinear switch and a circulator. Thus, inverse-design magnonics can be used to develop highly efficient rf applications as well as Boolean and neuromorphic computing building blocks.

**References:**

A. V. Chumak, V. I. Vasyuchka, A. A. Serga, & B. Hillebrands, Nat. Phys. 11, 453 (2015). Q. Wang, B. Heinz, R. Verba, et al. Phys. Rev. Lett. 122, 247202 (2019). Q. Wang, M. Kewenig, M. Schneider, et al. Nat. Electron. 3, 765-774 (2020). Q. Wang, P. Pirro, A. V. Chumak, arXiv:2012.04544.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/12c3053bde7dbdc6eb5b8bbe6696770d.jpg)
Fig. 1 The evolution from the conventional design (a) to inverse-design (b) magnonic demultiplexers.

 Nanoscale magnonic devices: From conventional to inverse-design magnonics INVITED

Spin waves (SWs) are promising quanta for transmission and processing of information for future communication devices. Excitation and detection of beyond 20 GHz magnons with sub-100 nm wavelength are of key importance [1]. We report on the magnonic grating coupler (MGC) effect [2] for which prepared specific nanopillars. Depending on the type of MGC lattice and nanopillar diameters, magnons are emitted resonantly on the chip in a multitude of directions with different efficiency. We demonstrate the interference effect of short-wave magnons independent of propagation directions. We engineered grating couplers for broadband magnon emission by preparing 100-nm-thick permalloy nanopillars in a periodic square lattice and an aperiodic (Penrose P3) quasicrystalline arrangement [3] on insulating ferrimagnetic yttrium iron garnet (YIG) films. Characteristic lengths amounted 480 nm. Several sub-micron-wide coplanar waveguides (CPWs) with separations of 12 micrometers were integrated on top of the MGC samples to excite/detect SWs by broadband spectroscopy. The MGC sample based on the quasicrystalline arrangement allowed for SW emission omnidirectionally with a broad range of wave vectors. SWs of up to 25.3 GHz were detected in the square lattice-based MGC sample. These parameters were achieved in very small magnetic fields of only a few mT in contrast to previous studies on grating couplers [2]. SWs simultaneously excited by two emitters interfered constructively/destructively at a separate detector antenna depending on the phase difference of the two phase-coherent SWs. Thereby a binary 1/0 output operation could be realized with 70 nm-wavelength magnons over distances of more than 10 micrometers. The applied field direction and specific microwave frequency defined the SW emission direction for the interference experiment. Our study and findings on sub-100 nm magnons are important for future magnonic devices targeting at information processing via wave-logic on the nanoscale. The work was supported by SNSF grant 163016.

**References:**

[1] A.V. Chumak, V. I. Vasyuchka, A. A. Serga et al., Nature Physics, Vol.11, p. 453 (2015) [2] H. Yu, G. Duerr, R. Huber et al., Nat. Commun., Vol. 4, p. 2702 (2013) [3] S. Watanabe, V.S. Bhat, K. Baumgaertl et al., Adv. Funct. Mater., Vol. 30, p. 2001388 (2020)

**Images:**

Spin-wave Circulation in a Ferromagnetic Resonator with Two Adjacent Layers

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Spin-wave Circulation in a Ferromagnetic Resonator with Two Adjacent Layers

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