Abstract
The field of magnonics, which aims at using spin waves as carriers in dataprocessing devices, has attracted increasing interest in recent years. We present and study micromagnetically a nonlinear nanoscale magnonic ring resonator device for enabling implementations of magnonic logic gates and neuromorphic magnonic circuits. In the linear regime, this device efficiently suppresses spinwave transmission using the phenomenon of critical resonant coupling, thus exhibiting the behavior of a notch filter. By increasing the spinwave input power, the resonance frequency is shifted, leading to transmission curves, depending on the frequency, reminiscent of the activation functions of neurons, or showing the characteristics of a power limiter. An analytical theory is developed to describe the transmission curve of magnonic ring resonators in the linear and nonlinear regimes, and is validated by a comprehensive micromagnetic study. The proposed magnonic ring resonator provides a multifunctional nonlinear building block for unconventional magnonic circuits.
Introduction
Spin waves are collective excitations of spin systems in magnetic materials, which can be considered as a potential data carrier in future lowenergy dataprocessing systems^{1,2,3,4}. This is due to their small wavelengths, down to a few nanometers^{5,6}, high frequencies up to a few terahertz^{7}, ultralow losses^{8,9}, and abundance of associated nonlinear phenomena^{10,11,12}. These features make spin waves highly attractive for wavebased and neuromorphic computing concepts. Several important milestones were achieved in the realization of magnonic dataprocessing units, including logic gates^{13,14,15}, majority gates^{16,17,18}, a magnon transistor^{19}, a phase shifter^{20}, building blocks for unconventional computing^{21,22,23}, auxiliary units for integrated circuits^{12}, magnonic directional couplers^{24,25,26,27}, and an integrated magnonic halfadder^{25}.
Here we propose a nanoscale nonlinear magnonic ring resonator. It is magnetic counterpart of the photonic ring resonator, which is considered as a universal unit and widely used in integrated photonic circuits^{28}, photonic quantum computing^{29}, and photonic neuromorphic computing^{30}. The concept of the magnonic ring resonator (see Fig. 1a) is similar to that of the photonic ring resonator^{31}, except that spin waves, instead of light, are used to carry information. A magnonic ring resonator of submillimeter size has been studied in the linear regime using micromagnetic simulations as reported in ref. ^{32}. Although such rather macroscopic ring resonators demonstrate certain interesting features due to multimode coupling and external field sensitivity, their functionality and size are hardly compatible with the current state of the complementary metaloxidesemiconductor technology. Moreover, the presence of multiple modes such as width modes with different coupling strengths results in less effective energy transfer and makes it impossible for the device to operate in the socalled “critical coupling” condition. Here we study the singlemode nanoscale magnonic ring resonator using the critical coupling phenomenon and demonstrate its functionality analytically and by simulation, including linear and nonlinear operation regimes, as well as their anticipated applications. Despite the fact that this is a simulation, the recent progress in the realization of singlemode magnonic nanoconduits proves that the sizes chosen here can be realized based on the current nanofabricating technology^{33}.
Results and discussion
Theory and micromagnetic simulations of the linear regime
The basic configuration of the magnonic ring resonator consists of a ring of mean radius R and width w, and a straight waveguide of same width w, as shown in Fig. 1a. The static magnetization distribution of the magnonic ring resonator obtained from micromagnetic simulation is shown in Fig. 1b (details of the micromagnetic simulation method are described in “Methods”). For a sufficiently narrow ring, here w = 100 nm, the static magnetization is in the vortex state with the magnetization lying along the ring. Such a vortex state is the ground state in the presence of zero external fields (i.e., it corresponds to the global energy minimum) and it can be easily achieved in experiments^{34,35}, for instance, by controlling the variation of the external field (see Supplementary Figs. 1 and 2). The static magnetization of a straight waveguide is uniform and is along the waveguide, which is caused by a strong shape anisotropy and by the small crosssection. We consider YttriumIronGarnet (YIG) as the material of both the waveguide and the ring: it is chosen for its low damping, allowing for longrange spinwave propagation^{36}. The used material parameters of YIG are described in “Methods.”
For the theoretical description of the power transmission in the ring resonator, we adopt a method typically used in optics and microwave electronics^{31,37}. Let us denote the complex amplitudes of input (output) spin waves in the waveguide and ring by a_{1,2} (b_{1,2}), as shown in Fig. 1a. To define a reference plane, we use the position of the minimum distance between the ring and waveguide (see the dashed line in Fig. 1a), so that all a_{i} and b_{i} are the values, which would be at the point x = 0 if continuously extrapolated in the absence of coupling. If the coupling between the ring and waveguide is lossless, which is the case for pure dipolar coupling, it is described by the unitary scattering matrix^{31}:
The parameter κ is the coupling coefficient between the straight waveguide and the ring, which shows the fraction of the spinwave amplitude coupled from the waveguide into the ring structure and vice versa. The parameter τ is the transmission coefficient across the coupling region, which shows the fraction of the spinwave amplitude passed through the coupling region. The transmission coefficient τ is different from the transmittance T, which demonstrates the final transmitted power through all the structures and accounts for the interference in the ring. Naturally, ĸ^{2} + τ^{2} = 1, which reflects the lossless nature of the coupling. The calculation of the coefficients κ and τ for the dipolar coupled waveguides and ring is presented in “Methods.”
The complex amplitudes a_{2} and b_{2} are connected by the circulation condition: a_{2} = b_{2}βe^{iθ}. Here, β = exp(−2πRΓ/v_{gr}) is the loss coefficient that describes which part of the spinwave amplitude remains in the ring after one circulation, with Γ and v_{gr} being the spinwave damping rate and group velocity in the ring, respectively (for details, see “Methods”). The parameter θ = 2πRk is the roundtrip phase accumulation, where k is the wavenumber of the spin wave in the ring. The wavenumber is determined by the input spinwave frequency, which is given by the dispersion relation ω_{k} in the ring and is normally different from the wavenumber in the straight waveguide.
Solving Eq. (1) together with the circulation condition, one finds the transmission T through the ring resonator structure:
using the phase ψ = Arg[τ]. In our case of a straight waveguide and ring of the same crosssection, the phase is ψ = 0. At the resonance frequencies, at which θ = 2πn (n = 0, 1, 2, 3, … is the number of the resonance modes in the ring), the transmission T is equal to T = (β − τ)^{2}/(1 − βτ)^{2}. The output signal vanishes completely if the transmission coefficient is equal to the loss coefficient, i.e., τ = β, which is the socalled case of “critical coupling.” The maximum transmission T in the critical case, which is reached when θ = (2n + 1)π, is equal to T = 4β^{2}/(1 + β^{2})^{2}, and increases with β. Therefore, it is desirable to work in the range of β ≈ τ → 1 to achieve a large output power, i.e., to have small losses in the ring and weak coupling between the ring and the waveguide. However, a small loss and a small coupling increase the operational time of the resonator (time to reach the dynamic equilibrium)^{38}. Thus, optimal values of the transmission coefficient and the loss coefficient are in the range β ≈ τ ~ 0.6–0.9, which is the result of the tradeoff between a large transmission and a short delay time for the ring resonator.
The coefficients κ, θ, and β depend on the spinwave frequency. The coupling coefficient κ significantly depends on the gap between waveguide and ring. In our example simulations, the minimum gap, which is the closest distance between the waveguide and the ring, is fixed to δ = 20 nm for all simulations. The ring radius determines the separation between the ring resonance frequencies. To set the loss coefficient to a value close to the optimum of β ≈ τ, in our simulations we increased the Gilbert damping in the ring structure to α_{G} = 2 × 10^{−3}. In an experiment, an increase of the YIG damping can be realized, for instance, by placing a normal metal on top of the ring to use the phenomenon of spin pumping^{39}. Please note that the parameters R, δ, and α_{G} are selected for the working frequency range from 2.6 to 2.8 GHz. These parameters can also be modified to obtain other frequency ranges that fulfill the critical coupling condition.
As an approximation of the ring dispersion relation, in principle, one can use the dispersion relation of a straight waveguide^{40}. For our case, it results in only a slight discrepancy of 80 MHz and the discrepancy becomes more negligible for R » w and kR » 1. In all the following calculations, we use a more accurate theory of the dispersion in the ring, as outlined in “Methods.” The spinwave damping rate and group velocity, which determine the loss coefficient, are calculated from the dispersion relation.
The frequency dependencies of the transmission and the loss coefficient τ and β, together with the roundtrip phase θ are shown in Fig. 2. In the chosen frequency range, the condition τ ~ β holds and the critical coupling condition is exactly satisfied at a frequency around 2.55 GHz (not shown in Fig. 2). The frequency dependence of the transmission coefficient is more pronounced because of a significant wavenumber dependence of the dynamic dipolar fields, generated by the spin waves propagating in the waveguide and the ring^{24}.
The theoretical transmission curve of the whole ring resonator calculated according to Eq. (2), as well as the results of micromagnetic simulations are shown in Fig. 3a. In the simulation, the transmission T is defined by calculating the ratio of the spinwave intensities in the straight waveguide behind and in front of the ring. Two resonance frequencies of the magnonic ring resonator are observed in this frequency range, which correspond to the 16th and 17th resonant mode. At these frequencies, the output signal is vanishing due to the destructive interference in the outgoing waveguide between the transmitted spin wave a_{1}τ and the coupledback spin wave ia_{2}κ, which acquires the roundtrip phase of θ = 2πn plus two π/2 phase shifts in the coupler (see Eq. (1)), being, in total, in antiphase to the first wave. At the resonance frequency, all the spinwave power is concentrated in the ring (see Fig. 3b). In contrast, at the frequency of 2.724 GHz, which corresponds to θ = 2π × 16.5, the constructive interference conditions are satisfied and a large part of input power is transmitted, whereas only a small amount is circulating in the ring (Fig. 3b). A strong frequency dependence of the output power is important, as it allows one to realize notch filters with a magnonic ring and it enhances the sensitivity of the system to a nonlinear frequency shift.
In addition, Fig. 3 shows the transmission curves for the rings having different radii. As expected, the resonance frequencies change and the resonance curves are shifted, while preserving their shape. The resonance frequencies, calculated analytically, are 20 MHz higher than those found from the micromagnetic simulations. Also, in the simulations, the critical coupling condition is satisfied in the range 2.65–2.68 GHz, as it is evident from the vanishing output at the resonance frequency, whereas theory indicates the critical coupling at a slightly different frequency of 2.55 GHz. These two weak discrepancies are mainly due to a slightly nonuniform width profile of the spin waves in the ring and the waveguide, and to a weak dipolar field generated by the straight waveguide, which slightly modifies the dispersion relation in the ring. Both effects are not taken into account in the theory. Furthermore, there is a certain difference in the maximum transmission energy between the simulations and the theoretical calculations, which is attributed to the propagation losses in the straight waveguide and the coupling area.
Reconfigurable magnonic ring resonator
In the previous section, the magnetization in the ring is oriented counterclockwise and the magnetization in the coupling region between ring and waveguide is aligned parallel. However, in the absence of an external field, the ring can exist in two stable magnetic configurations—clockwise and counterclockwise—as shown in the insets of Fig. 4. These two states lead to antiparallel and parallel magnetization configurations in the coupling region. The switching between these two states can be realized by controlling the tracks of the variation of the external field before reaching the remanent state (for details see Supplementary Figs. 1 and 2). The coupling strength, described by the coupling coefficient κ, depends strongly on the static magnetization configuration and affects significantly the transmission coefficient τ^{24}. The coupling strength is stronger for the antiparallel magnetization configuration and this results in the breaking of the critical coupling condition, i.e., τ ≠ β and, consequently, the transmission T at the resonance frequencies increases. Figure 4 shows the transmission curve for parallel and antiparallel alignments in which the transmissions at the resonance frequency of 2.662 GHz are 0.48% and 30.8%, respectively. The contrast between the two states can be further increased by optimizing the parameters of the ring resonator. For a future onchip magnonic device, this switching can be realized by a local Oersted field created by direct current passing through a conducting wire, which is placed on top of the ring structure. This example shows that the symmetry break caused by the direction of the magnetization allows creating magnonic functionalities that are not available in the same form in, e.g., photonics.
Nonlinearity of magnonic rings
Magnonic systems are known to involve a variety of nonlinear effects, which open a way for the development of various nonlinear powerdependent devices. In general, all the parameters, which define the operation of the magnonic ring resonator, namely θ, β, and τ, are power dependent. However, it can be shown that the main impact of nonlinearities is caused by the nonlinear phase accumulation θ = θ(b), where b is the spinwave amplitude, whereas the nonlinearities of the loss coefficient (due to the group velocity shift) and the coupling strength lead only to a small (second order) correction and, therefore, can be neglected in almost all experimentally achievable cases.
An increase of the spinwave power results in a nonlinear shift of the spinwave resonance frequency, \(\omega _{k}\left( b \right) = \omega _{k}^{\left( {lin} \right)} + W_{k}\left b \right^2\), where W_{k} is the nonlinear shift coefficient. Consequently, a wave of a constant frequency possesses a powerdependent wavenumber k ≈ k_{0} − (W_{k}/v_{gr})b^{2}, which directly affects the phase accumulation during the spinwave propagation. The integration over the ring yields the roundtrip phase θ(b_{2}) = 2πR(k_{0} − Kb_{2}^{2}), where \(K=W_{k}(1e^{4 \pi \Gamma R/v_{gr}})/4 \pi \Gamma R \approx W_{k} \beta/v_{gr}\) is the averaged coefficient of the nonlinear shift of the spinwave wavenumber. Then, Eq. (1) together with the circulation condition yields the following relation:
which implicitly determines the amplitude in the ring. The transmission T is given by the same Eq. (2), in which one should use the nonlinear phase accumulation θ(b_{2}) with the amplitude of b_{2}, found from Eq. (3).
The simulated transmission curves of the magnonic ring resonator for different excitation fields b_{e} are shown in Fig. 5 (bottom panel). A pronounced shift of the resonance frequency, at which transmission is minimum, is observed. A similar powerdependent transmission curve shift was observed in optical ring resonators^{28,41,42}. To plot the theoretical curves, we use a nonlinear frequency shift value of W = −2π × 2.6 GHz, which is calculated for a straight waveguide^{25,43}. As one can see, this approximation is reasonable and gives a similar shift of the transmission minimum position compared to the simulated results.
For a largeenough input spinwave power, the transmission curve becomes bistable (see the magenta line in Fig. 5). The appearance of bistability is clear from Eq. (3), which, by expanding the denominator near the resonance frequency, has the same structure as the nonlinear ferromagnetic resonance curve, demonstrating the foldover effect^{44,45}. In the bistability range, the exact shape of the transmission curve depends on the experimental (simulation) conditions. The solid curves are obtained when all the simulations start from the same ground state. To access the dashed curve in the simulations, we gradually decrease the excitation frequency starting outside of the bistability range with a constant spinwave amplitude. The small discrepancy between theory and simulation is mainly attributed to the nonlinear shift coefficient, which is extracted from a straight waveguide and not from the ring structure and the previously mentioned nonuniform spinwave profile in the ring structure.
Nonlinear ring resonator for neuromorphic applications
In neuromorphic systems, the socalled “activation function” plays an important role. It describes how an incoming stimulus (here, the incoming spinwave amplitude a_{1}) is transformed into the output signal (the outgoing spinwave amplitude b_{1}) via a strongly nonlinear function. Thus, if the ring resonator should serve as a building block for neuromorphic spinwave computing in a larger network composed of many resonators and interconnecting spinwave waveguides and combiners, the activation function is characterized by the transmission T of the ring resonator, which should depend strongly on the input amplitude. For instance, in socalled spiking neurosynaptic networks, the “firing” of an artificial neuron, i.e., the emission of an output signal, takes place only if the input signal overcomes a certain threshold. This essentially means that the transmission T should be significantly large only if a certain incoming spinwave input amplitude is overcome. Wavebased neuromorphic computing using such kind of function has been recently demonstrated for a full neuronal network in optics^{30}.
To characterize the activation function of the magnonic ring resonator, the spinwave excitation frequency is fixed to f = 2.662 GHz, which coincides with the 16th resonance in the linear regime, as depicted in Fig. 5 by the vertical dashed line. The output spinwave intensity nonlinearly depends on the input power, because this frequency does not correspond anymore to a resonance in the nonlinear regime. Figure 6a shows the relative transmission power for f = 2.662 GHz as a function of the excitation field amplitude b_{e} and the dynamic outofplane component of magnetization m_{z} (top axis). The transmission T is almost constant and below 1% in the excitation field range from 0.6 to 4 mT and then strongly increases from T = 0.78% at b_{e} = 4 mT to T = 51.5% at b_{e} = 13 mT due to the strong nonlinear shift and the steep slope of the transmission curve vs. frequency (compare to Fig. 5). A high contrast of around 18 dB between the output states is observed. This is reflected in the fact that by increasing the input spinwave intensity by a factor of about 10, the transmitted intensity (absolute spinwave output power) is increased by a factor of around 700. This strong nonlinearity can be achieved, as energy is stored in the ring at resonance. The contrast, threshold, and maximum transmission level can be tuned by adjusting the radius of the ring R, the transmission coefficient τ, and the loss coefficient β. As a further example for the use of the ring resonator for data processing, Fig. 6b shows a functionality that can be considered as a kind of “passivation function,” meaning that the transmitted intensity decreases with increasing input intensity. Due to the foldover effect shown in the transmission curve in the highpower region (see the magenta line in Fig. 5), the transmission T at the frequency of f = 2.638 GHz drops down from 68.6 to 14.2% by slightly increasing the excitation field from 10 to 12 mT. This functionality can be used to filter out the highpower spin waves or normalize the output spinwave power, i.e., the absolute spinwave output power could be made independent of the input spinwave power in a certain power range. It is worth noting that the precession angle is only around 5° even for the high excitation field of 13 mT, which reveals the fact that the energy consumption is very low in the magnon domain.
In conclusion, a nanoscale nonlinear magnonic ring resonator is proposed and its functionality is demonstrated using micromagnetic simulations. The transmission curve of the ring resonator in the linear region is of a notch filter type due to the resonant critical coupling effect. Spin waves at resonance frequencies are stored in the ring and cannot pass through it, whereas spin waves of a frequency in between the resonances pass the ring resonator with only a small loss. Importantly, the nonlinear shift of the spinwave resonance frequency and, consequently, of the spinwave phase accumulation leads to a strong power dependence of the magnonic ring transmission curves. In this nonlinear regime, the resonance frequencies are shifted, the transmission curves become asymmetric, and, at largeenough input power, exhibit a bistability. The transmission at the linear resonance frequency shows a thresholdlike behavior: a lowinput spinwave power is stored in the ring structure and the ring only generates an output if the input power exceeds a threshold. This functionality is useful for magnonic logic applications, for instance, in the field of neuromorphic computing. Very different transmission curves can be realized at frequencies not coinciding with the linear resonances. In addition, the transmission functions can be reconfigured by changing the alignment of the magnetization in the ring and the adjacent waveguide. The obtained results are supported by the developed analytical theory, which allows to calculate the ring resonator characteristics in both the linear and nonlinear regimes.
Methods
Spinwave dispersion in a ring
The dispersion of spin waves in a magnonic ring can be calculated similarly to those of a vortexstate magnetic disk^{46,47}. It is noteworthy that the ring dispersion relation in the considered system is continuous, ω = ω_{k}, and spin waves with continuous wavenumbers k can propagate in it, as the ring resonator is not isolated. In our case, the width of the ring is sufficiently small^{40}, leading to an almost uniform (unpinned) spinwave profile across the ring width, which greatly simplifies the calculations. In this approximation, the dispersion relation is given by
where ω_{M} = γμ_{0}M_{s}, M_{s} is the saturation magnetization, γ is the gyromagnetic ratio, λ is the exchange length, and \(\widehat {\mathbf{F}}_{k} = {\int} {{\int} {\widehat {\mathbf{G}}\left( {{\mathbf{r}}{\mathrm{,}}\,{\mathbf{r}}^\prime } \right)\exp \left[ {{\mathrm{i}}{k}R\left( {\phi  \phi ^\prime } \right)} \right]d{r}d{r}^\prime /\left( {2{\uppi}Rw} \right)} }\) is the effective dynamic demagnetization tensor with \(\widehat {\mathbf{G}}\left( {{\mathbf{r}},{\mathbf{r}}^\prime } \right)\) being the magnetostatic Green’s function in the polar coordinate system^{48} and the integration going over the ring surface. For an arbitrary wavenumber, the calculation of \(\widehat {\mathbf{F}}_{k}\) is complicated. However, for k_{n} = n/R, which are the wavenumbers corresponding the resonant modes of an isolated ring, it is greatly simplified and yields
where \(f\left( {{k}h} \right) = 1  \left( {1  \exp \left( {  {k}h} \right)} \right)/\left( {{k}h} \right)\), and we use the notation
with the Bessel functions J_{n}. The function I_{n}(k) can be expressed via a combination of hypergeometric functions or calculated numerically. The complete continuous spinwave dispersion ω_{k} can be numerically found by interpolation of the dispersion relations of the ring resonance frequencies \(\omega_{k_n}\). The spinwave group velocity is found via v_{gr} = dω_{k}/dk. The spinwave damping rate is calculated using the following general formalism^{49}:
Coupling between waveguide and ring
The dynamics of spinwave amplitudes a_{1}(x) and a_{2}(x) in coupled waveguides is described by the following system of equations^{24}:
where ω_{c} is the coupling frequency, which has the meaning of a splitting between the symmetric and antisymmetric collective modes in the coupled waveguide. The difference in dispersion relations (and, consequently, in v_{gr}) in the waveguide and ring leads to only a small (second order) correction and is neglected here. The coupling frequency is given by
and it depends on the coordinate x via the dependence of the distance between centers of straight and ring waveguides \(d( x ) = d_0 + ( {R + \sqrt {R^2  x^2} } )\) with d_{0} = δ + w. The positiondependent angle between the waveguide and ring, and consequently between their static magnetizations, is not accounted for, as in the region that contributes most to the overall coupling, this angle is negligible. Here and below the tensors \(\widehat {\mathbf{{\Omega}}}\), \(\widehat {\mathbf{F}}\) and \(\widehat {\mathbf{N}}_{k}\) are defined as in ref. ^{24}.
From the solution of Eq. (9), one finds the coupling and transmission coefficients, which enter into Eq. (1):
where \(\overline \omega _{\mathrm {c}} = \left( {1/2R} \right){\int}_{  R}^R {\omega _{\mathrm {c}}\left( x \right)dx}\) is the “averaged” coupling frequency. This equation can be used for any shape of the coupling area, e.g., if the ring is changed to a polygon. For the ring structure, the calculation of \(\overline \omega _{\mathrm {c}}\) is greatly simplified (note that \(\widehat {\mathbf{F}}\left( d \right)\) is an integral itself), noting that the coupling frequency decays fast with the separation d, so we can use the approximation d(x) ≈ d_{0} + x^{2}/(2R) and change the integration limits to (−∞,∞). Then,
where
Micromagnetic simulations
The micromagnetic simulations were performed using the software package FastMag developed at the University of California, San Diego^{50}. This software uses a finite element method to solve the LLG equation and can use the power of modern Graphics Processing Units, which leads to the capability to handle ultracomplex geometries at a high speed^{50}. The finite element method is especially useful if nonrectangular systems such as the presented ring are simulated. The simulated structure of a magnonic ring resonator is shown in Fig. 1a. The parameters of the nanometerthick YIG are obtained from the experiment as following^{36}: saturation magnetization M_{s} = 1.4 × 10^{5} A m^{−1}, exchange constant A = 3.5 pJ m^{−1}, and Gilbert damping for most of the structure α = 2 × 10^{−4}, except for the ring structure. The Gilbert damping in the ring structure is increased to 2 × 10^{−3} to meet the critical coupling condition and the damping at the ends of the simulated structure is set to exponentially increase to 0.2, to prevent spinwave reflection. The high damping region can be realized in the experiment by placing another magnetic material or a metal on top of YIG. The averaged cell size was set to 10 × 10 × 10 nm^{3}, which is smaller than the exchange length of YIG (~16 nm) and the studied wavelength (~220 nm). To excite a propagating spin wave, a sinusoidal magnetic field b = b_{e}sin(2πft) was applied over an area of 40 nm in length, with a varying oscillation amplitude b_{e} and microwave frequency f. The magnetization M_{z}(x,y,t) was obtained over a period of 250 ns, which is long enough to reach a stable dynamic equilibrium. The spinwave intensity is calculated by performing a Fourier transform from 200 to 250 ns, which is long enough to resemble the condition of a dynamic equilibrium. The transmission T is defined by calculating the ratio of the spinwave intensities in the straight waveguide behind and in front of the ring.
Data availability
The data that support the plots presented in this paper are available from the corresponding authors upon reasonable request.
Change history
23 January 2021
A Correction to this paper has been published: https://doi.org/10.1038/s41524021004976.
01 February 2021
A Correction to this paper has been published: https://doi.org/10.1038/s41524021004976
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Acknowledgements
The project was funded by the European Research Council (ERC) Starting Grant 678309 MagnonCircuits and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–TRR 173–268565370 (“Spin + X”, Project B01), the Nachwuchsring of the TU Kaiserslautern. R.V. acknowledges support of National Research Foundation of Ukraine (grant number 2020.02/0261).
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Q.W. conceived the idea and designed the structure. P.P. and A.V.C led this project. A.H., V.L., and Q.W. carried out the micromagnetic simulations. R.V. developed the analytical theory. Q.W. wrote the manuscript with the help of all the coauthors. All authors contributed to the scientific discussion and commented on the manuscript.
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Wang, Q., Hamadeh, A., Verba, R. et al. A nonlinear magnonic nanoring resonator. npj Comput Mater 6, 192 (2020). https://doi.org/10.1038/s41524020004656
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