Abstract
Nonlinear processes are a key feature in the emerging field of spinwave based information processing and allow to convert uniform spinwave excitations into propagating modes at different frequencies. Recently, the existence of nonlinear magnons at halfinteger multiples of the driving frequency has been predicted for Ni_{80}Fe_{20} at low bias fields. However, it is an open question under which conditions such nonlinear spin waves emerge coherently and how they may be used in device structures. Usually nonlinear processes are explored in the small modulation regime and result in the well known three and four magnon scattering processes. Here we demonstrate and image a class of spin waves oscillating at halfinteger harmonics that have only recently been proposed for the strong modulation regime. The direct imaging of these parametrically generated magnons in Ni_{80}Fe_{20} elements allows to visualize their wave vectors. In addition, we demonstrate the presence of two degenerate phase states that may be selected by external phaselocking. These results open new possibilities for applications such as spinwave sources, amplifiers and phaseencoded information processing with magnons.
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Introduction
In contrast to other common types of waves (e.g., electromagnetic or sound) the interaction of spin waves in magnetic materials is intrinsically nonlinear due to dipolar coupling effects. Nonlinear processes even dominate the response of magnetic materials for large excitation amplitudes^{1,2} and can be utilized to generate and amplify coherent collective excitations of the spin system—known as magnons. These nonlinear processes are usually described as magnonmagnon scattering processes^{3} and act as additional decay channels for the homogeneous mode^{4}.
Three and fourmagnon scattering processes are ubiquitous effects due to the strong nonlinearity of the equation of motion. For example, in threemagnon scattering^{5,6} typically the excitation of a uniform magnon leads to scattering into a magnon pair at half of the driving frequency with opposing wave vectors^{7,8,9}. In addition, four magnon scattering processes^{4,10,11,12} may be used for harmonic generation^{13,14,15,16} or the amplification and stabilization of propagating spin waves^{17}. Besides frequency conversion effects, nonlinear magnonscattering processes are potentially also suitable for short wavelength spinwave generation. Such sources are required for highlyintegrated magnonic devices^{18,19,20}. For spinwave based information processing well controlled propagation, as well as phaselocking, are required. Recently, nonreciprocal propagation and frequency selection of spin waves was demonstrated in magnetic hybrid structures ^{21,22}. Phaselocking of multiple spinwave sources has been achieved via propagating spin waves using spinHall^{23,24} or spin–torque nanooscillators^{25,26,27,28}. More generally, phaselocking in spin systems can be achieved by a variety of coupling mechanisms such as cavity photons^{29}, acoustic phonons^{30}, exchange spin waves^{22}, or currents in a superconductor^{31}. In addition, nonlinear phenomena were discussed as control mechanism for coherent information processing^{32} in the emerging field of cavity magnonics^{33,34,35}.
The theoretical description of these nonlinear spinwave (NLSW) phenomena, also known as spinwave instabilities^{36}, was established by Suhl^{2,37} and L’vov^{3}. Recently, Bauer et al.^{38} presented a model for the description of NLSWs, considering the inherent frequency modulation of the excited magnons at low magnetic bias fields due to the anisotropic spinwave dispersion. Here, the limit of large magnetization modulation can be treated and gives rise to a novel class of nonlinear excitations. Specifically, strong amplitudephaseoscillations result in parametrically generated spin waves oscillating at odd halfinteger multiples of the driving frequency. Such excitations are not described by the conventional nonlinear spinwave theory. In soft ferromagnets, such as Ni_{80}Fe_{20}, these NLSWs are predicted to dominate at low bias fields with threshold driving fields below the ones of conventional nonlinear processes^{38,39}. So far, no direct evidence for the existence of such spin waves has been provided.
In this work, we demonstrate the direct phaseresolved imaging of this novel class of parametric spinwave excitations in magnetic microstructures by using superNyquist sampling magnetooptical Kerr microscopy. In doing so, we identify the predicted large wave vector nonlinear magnons oscillating at halfinteger harmonics as the dominant excitation. Our experimental results are in agreement with theoretical predictions from our analytical model as well as with micromagnetic simulations. Moreover, by exploiting the phase sensitivity of our technique we reveal different regimes of phase stability of the parametrically generated spinwave excitations which we link to two distinct phase states. Finally, we demonstrate that phaselocking of the parametric spin waves can be achieved revealing the potential of this phenomenon for phaseencoded information processing in magnonbased devices, such as spinwave emitters.
Results
Nonlinear spinwave excitations
In the experiments, we utilize superNyquist sampling Kerr microscopy (SNSMOKE)^{40} to investigate coherent nonlinear spinwave excitations in 20 nm thick Ni_{80}Fe_{20} elements placed on top of a coplanar wave guide (CPW), as depicted in Fig. 1a. At low bias fields, the rfmagnetic field generated by the CPW causes magnetization precession with large ellipticity. For low rfdriving field amplitudes the imaginary part of the dynamic susceptibility results in a Lorentzian resonance line shape as a function of the external bias field, as demonstrated in Fig. 1b for the case of a 30 μm × 15 μm elliptical Ni_{80}Fe_{20} element. As the excitation amplitude is increased, the inplane excursion angle of the magnetization becomes very large (of the order of 20°)^{41}. The elliptical precession trajectory caused by the inplane shape anisotropy results in an inherent frequency modulation due to the anisotropic nature of the spinwave dispersion. A good example where these conditions are satisfied is a 20 nm thick Ni_{80}Fe_{20} layer, as predicted in ref. 38. For large driving amplitudes the phase of the response changes as soon as a threshold rffield of ≈0.30 mT is exceeded (inset in Fig. 1b). At the same time the resonance condition shifts towards lower magnetic bias fields^{41}.
In order to capture the physics of nonlinear spin waves excited at low bias fields and large driving amplitudes, it is necessary to treat the magnetization dynamics explicitly within the limit of large modulation. For this, the Landau–Lifshitz–Gilbert equation can be considered in kspace. Using algebraic transformations, it assumes the form of wellknown Mathieu or Hill equations. The advantage of this procedure is that the critical spinwave modes oscillating at halfinteger multiples of the pumping frequency can be quickly and reliably identified numerically^{38}. The obtained solutions for the critical spin waves are characterized by strong amplitudephaseoscillations. These amplitudephaseoscillations are connected with frequency components at several odd halfinteger multiples of the driving frequency. Our aim is now to directly image these spin waves and to resolve their frequency components. For this, a phasestable nonlinear response is required. In comparison to more common magnetooptical imaging approaches which are typically sensitive to the magnitude of the magnetic excitation, the SNSMOKE technique provides a unique combination of phaseresolved lockin detection while offering an arbitrary highfrequency resolution^{40}. This feature allows obtaining the real and imaginary parts of the dynamic susceptibility at arbitrary frequency components simultaneously, such as 3/2 of the driving frequency as shown in Fig. 1c (see methods for further details). However, when this method is applied to a continuous magnetic layer, spin waves oscillating at 3/2 of the driving frequency are expected, but no corresponding signal is observed by means of SNSMOKE (see Supplementary Fig. S1). The most likely explanation is that in this case the parametric spin waves are not phase stable during the measurement. To promote phase stability, an elliptical magnetic element is prepared. Here the fixed boundary conditions facilitate the formation of a standing spinwave mode pattern that is fixed in space. Indeed, above a threshold rfamplitude, a coherent signal starts to appear (cf. Fig. 1c). Simultaneously, we detect additional coherent nonlinear signals at other integer and halfinteger multiples as a direct consequence of the strongly anharmonic precession of the magnetization, as demonstrated in Fig. 2. This behavior is a hallmark of the amplitudephaseoscillations expected for nonlinear processes at low bias fields^{38}. While the integer multiples^{13,16} cover a field range as broad as the ferromagnetic resonance line shape, the odd halfinteger harmonics are restricted to a very narrow field window in which they emerge coherently. Using additional micromagnetic simulations for a comparable sample geometry we find, that the additional halfinteger multiples with integer index n indeed arise from the precessional motion, spanning a frequency comb n⋅f_{rf}/2 with amplitudes matching the experimental results, as shown Fig. 2i.
For the nonlinear spinwave process shown in Fig. 2c, one clearly observes a threshold behavior which is in good agreement with the prediction from our analytical kspace model^{38} (see Supplementary Fig. S2). However, as already discussed, the detection of coherent NLSWs at halfinteger harmonics strongly depends on the sample geometry. Thus, the values obtained by SNSMOKE might be larger compared to the simulations due to the lack of phase stability near the threshold. While one can observe phasestable NLSW generation in larger elliptical elements, even smaller structures, such as a 5 μm × 4 μm rectangle, shows no phasestable NLSW response in our measurements. We attribute this effect to rapid thermal fluctuations of the phase of the nonlinear response.
Phase stability
The fact that a direct observation of NLSWs is only possible within a narrow field window and in certain element geometries requires further attention. Especially since indirect evidence for the onset of these nonlinearities is observed for all investigated structures and in agreement with micromagnetic simulations. For conventional parametric spinwave processes it is known that the resulting nonlinear magnon pair populates one of two energetically equivalent phase states either inphase or outofphase with respect to the rfexcitation^{42}. In the case of threemagnon scattering in parallel pumping geometry randomly occurring phase transitions between the 0state and the πstate appear stochastically with equal probability. In our experiments, we identify such degenerate phase states for the transverse pumping geometry, as depicted in Fig. 3a and reveal their stochastic switching behavior in phaseresolved measurement of the 3/2 f_{rf} nonlinearity as randomly emerging sign reversals of the signal. Note that for SNSMOKE experiments phase stability of the excited NLSWs with respect to the excitation on time scales larger than the measurement bandwidth is essential to obtain a coherent signal. For this purpose, we investigate how experimental parameters such as the excitation amplitude and the bias field influence phase stability.
Interestingly, near the threshold condition, the NLSW signals appear to be less stable in phase over time, resulting in randomly occurring phase flips between the two distinct states most likely caused by thermal fluctuations. A further increase of the rfdriving power stabilizes the NLSWs over a broader field range allowing for phasestable spatiallyresolved imaging, as demonstrated in Fig. 3e. However, even in this stable regime, the initial phase of the NLSWs with respect to the excitation is set randomly to one of the degenerate states. This is demonstrated in Fig. 3b, c where the NLSW is prepared in the stable regime while a brief interruption of the rfpower results either in a 180° phase flip or leaves the phase unchanged with equal probability. This observation illustrates the existence of two degenerate phase states from which one is randomly chosen as soon the nonlinearity at 3/2 f_{rf} sets in. In fact, whenever halfinteger harmonics of the driving signal are generated a twofold degeneracy of the possible phase state occurs (cf. Fig. 3a). Thus, in contrast to SNSMOKE, conventional TRMOKE experiments with rfamplitude or phase modulation would be unable to detect such NLSW signals due to a periodic resetting to a random phase state. In the next step, we aim for a better understanding of the stability criteria of the NLSWs. Therefore, we slowly sweep the external magnetic field and detect the phase stability for each step on the time scale of 50 s (cf. Fig. 3f–h). Here, we reveal a regime of enhanced stability where phase flips are less likely (gray area). For slightly larger (or smaller) bias fields the stability is reduced as indicated by the occurrence of switching events on the timescale of a few seconds, as recently reported for threemagnon scattering processes^{42}. In Fig. 3f–h we demonstrate how the increasing number of switching events in the vicinity of the stable regime results in a decreased average amplitude and an enhanced standard deviation of the measured signal. In comparison to this small field range of enhanced stability micromagnetic simulations indicate that the field range in which 3/2 f_{rf} nonlinear spin waves are generated is as broad as the line width of the ferromagnetic resonance (see Supplementary Fig. S3). Moreover, the formation of this stable regime, where phase flips are mostly prevented, strongly depends on the chosen element geometry. In SNSMOKE experiments, therefore only a small fraction of the entire NLSW spectrum is observable as random phase flips due to thermal fluctuations occur on a time scales faster than the measurement bandwidth, resulting in a loss of signal.
Spinwave imaging
The experimental observation of phasestable NLSWs at 3/2 f_{rf} implies that the predicted amplitudephaseoscillations^{38} occur coherently with respect to the driving frequency in a certain bias field window. In the phasestable regime (cf. Fig. 3), SNSMOKE allows for spatiallyresolved imaging of the NLSWs and directly reveals the inplane wave vector components of the parametrically generated magnons at 3/2 f_{rf}. For this type of measurement, all experimental parameters are fixed while real and imaginary parts of the polar magnetic response are recorded in a pointwise fashion. Here, a mostly uniform signal is obtained at the driving frequency f_{rf} = 2.3 GHz, as shown in Fig. 4a, while above the threshold condition a phasestable pattern of NLSWs is detected, oscillating at a frequency of 3/2 f_{rf} = 3.45 GHz (see Fig. 4b). At the same time, additional nonlinear signal components due to the amplitudephaseoscillations are detected at other halfinteger multiples, as demonstrated in Supplementary Fig. S4. Corresponding micromagnetic simulations of the nonlinear magnetization dynamics are carried out and compared to the experimental results, as shown in Fig. 4d, e, respectively. The direct comparison of the excited wave vectors in the standing spinwave patterns is achieved by performing a twodimensional fast Fourier transform (2DFFT) for experimental data and the simulation. For both cases, the 2DFFT pattern shows the predicted NLSW pairs at 3/2 f_{rf}^{38}. Furthermore, we find a good agreement between our results and the analytical model as we compare the obtained nonlinear wave vector components from spatiallyresolved imaging with our calculations as a function of the rffrequency (see Supplementary Fig. S5). The nonlinear effects we observe convert the uniform mode into shortwavelength dipoleexchange spin waves within the ferromagnetic material without the need for patterning while the wavelength can be tuned by external parameters. This is an advantage compared to recent concepts of generating shortwavelength spin waves^{19,20} using periodic metallic coupling structures.
We would like to point out that the observed spinwave patterns emerge in a frequencylocked regime^{38}, additional signal components are visible and follow the isofrequency line of the nonlinear dispersion at 3/2 f_{rf} obtained from the analytical model (red line in Fig. 4). We explain this population of different modes by elastic magnon–magnon scattering processes facilitated by slight spatial variations of magnetic properties as it can be expected for a polycrystalline Ni_{80}Fe_{20} sample at low bias fields. To support this assumption, the impact of spatially varying magnetic properties is demonstrated using micromagnetic simulations, as shown in Supplementary Fig. S6. In the following, we aim to achieve phase control by stabilizing the NLSWs in one of the degenerate phase states.
Phaselocking of nonlinear spin waves
The integration of NLSWs into magnonbased devices requires direct control of propagation direction, amplitude or phase. Furthermore, the synchronization of excited nonlinear magnons to an external reference frequency has enormous potential for magnonbased computing^{23,25,26,27}. Phase control in such a nonlinear system would allow to precisely set the NLSW’s initial phase and prevent the phase state from random switching events by increasing its robustness against thermal fluctuations. For this, we modify the excitation scheme by integrating a second rfsource^{43} operating at 3/2 f_{rf}, as shown in Fig. 5a. This allows to seed the magnetic system with an additional frequency component at the desired phase. Physically, phaselocking is achieved due to the coupling between the homogeneous rffield and the NLSWs for the following reasons: (i) As NLSWs are generated within a confined element their standing spinwave pattern is fixed in space even if the phase state randomly switches. As a result of these boundary conditions only two degenerate phase states exist. (ii) The interplay of a uniform rffield with the boundaries of the elements results in nonuniform dynamic demagnetization fields^{44}. Here, the nonuniformity of the internal fields provides a spectrum of wave vectors in the range of the NLSWs with an relative amplitude of a few percent with respect to the uniform contribution. The presence of this rffield breaks the symmetry of the initially degenerate states and allows to select only one of them. As soon as coherent magnons with the corresponding wave vectors are provided by the seed rffield, the NLSWs will parametrically amplify this specific mode with the phase of the seed source.
The applied power level of the seed frequency component is typically two orders of magnitude smaller compared to the pump excitation. This modified excitation scheme is now applied to one of the elements which do not show a coherent NLSW signal without a seed excitation, namely a 5 μm × 4 μm rectangular element (cf. Supplementary Fig. S1). In the following, we analyze the signal amplitude at 3/2 f_{rf} while the power level of rfpump and rfseed were varied in a stepwise fashion. Figure 5b shows the obtained 2D map. In agreement with our analysis above, a regime exists where the coherent excitation of NLSWs only occurs when an additional seed amplitude (at f_{seed} = 3/2 f_{rf}) is applied. We would like to point out, that due to the amplitudephaseoscillations, the introduced stabilization process can also be performed at other odd halfinteger seed frequencies, as demonstrated in Supplementary Fig. S10. Here, the separation of seed and detection frequency reduces the influence of direct excitations at the detection frequency, and thus provides a pure NLSW signal. Clearly, the seed frequency sets and stabilizes the phase of the obtained NLSW signal.
Next, we study the phase control of the excited modes for the 5 μm × 4 μm rectangular magnetic element. In a discrete structure, one can expect that only two phase states with respect to the excitation are possible for a standing spin wave. For this purpose spatiallyresolved imaging is performed with both power levels set to the stable regime above threshold, while the phase of the seed (at 3/2 f_{rf}) was varied with respect to the phase of 3/2 of the pump frequency for the images shown in Fig. 5c–e. Here, the amplitude of the NLSW signal is strongly suppressed for a relative phase Δθ = 180° (cf. Fig. 5d), while the sign of the signal changes from c to e corresponding to the two distinct phase states. Measuring the magnitude of the signal in Fig. 5f as a function of Δθ in the center of this device results in a 180° periodicity. At the same time the phase of the signal ϕ shifts by 180°, as shown in g. We link the rapid phase jump in g to the transition from one of the two possible phase states to the other while the magnitude returns to the same level as the phase of the seed frequency is varied. For the transition region the two phase states are equally probable and the signal vanishes. The observation of these two possible phase states and their accessibility via phase shifts of the seed frequency is further corroborated by micromagnetic simulations (cf. Fig. S11). Note that the observed phaselocking effect strongly depends on the applied pump (f_{rf}) and seed (3/2 f_{rf}) power levels. For example, as the pump power is slightly decreased, the transition between the states becomes less sharp and at the same time, the magnitude is reduced. However, the applied scheme allows to initialize the NLSW in a defined and stable phase state, and provides a direct control for setting the phase state.
Discussion
In the experiments, we found that a series of signals at halfinteger multiples of the driving frequency arises and their spatial distribution can be imaged under phasestable conditions. The agreement between the experimental data and kspace modeling as well as micromagnetic simulations is excellent. Moreover, by utilizing the phase sensitivity of our imaging technique we revealed different regimes of stability. Within the stable regime, we found that the excitation tends to stochastically populate one of two degenerate phase states. As we demonstrated, these phase states of the NLSWs can be easily manipulated by external rffields illustrating their potential for phaseencoded information processing. A possible application of our results is that externally stabilized nonlinear spin waves are integrated into magnonbased devices, e.g., as phaselocked large wave vector spinwave emitters. We point out that the required very low bias fields may be generated using electrical currents flowing in microscopic wire structures. This should also enable magnonic devices, such as spinwave amplifiers that are crucial for realizing magnonic computing applications.
Methods
Sample preparation
The metallic structures were prepared by electron beam lithography, thermal evaporation and liftoff processes on top of galliumarsenide (GaAs) substrates. First, a 100 nm thick gold CPW was deposited. The 50 Ω CPW has a gap width of 25 μm while the signal line is 50 μm wide. On top of the signal line of the CPW 20 nm thick Ni_{80}Fe_{20} elements were structured by electron beam lithography and liftoff processes. The sample geometry provides mostly uniform inplane excitation in the center of the waveguide where the elements are located. Various different geometries were prepared on the sample: Elliptical and rectangular elements with lateral dimensions of 40 μm × 20 μm, 30 μm × 15 μm and 20 μm × 10 μm, 5 μm × 4 μm, respectively. A second sample, used for initial measurements shown in Supplementary Fig. S1, was prepared in a similar fashion but here the entire signal line of the CPW is covered with a 20 nm thick Ni_{80}Fe_{20} film.
SNSMOKE
In the presented experiments, a 520 nm laser with a pulse length of 200 fs is focused with a high numerical aperture objective lens onto the sample. The dynamics within the Ni_{80}Fe_{20} elements are excited by the uniform inplane rfmagnetic field generated by the CPW. In SNSMOKE^{40} measurements the femtosecond laser is utilized to sample the magnetization dynamics in a phaseresolved fashion. Phasestable measurements become possible when laser, rfsource, and lockin amplifier (used for the detection) are stabilized to the same master clock. The train of ultrafast laser pulses with a repetition rate f_{rep} = 80 MHz corresponds to a frequency comb with a spacing f_{rep} of the comb lines and converts the GHz precessional motion of the spins down to the intermediate frequency ε derived from the offset of the rffrequency f_{rf} to the nearest laser comb line n⋅f_{rep}. This process leads to an inherent modulation of the dynamics and allows to detect the precessional motion without the loss of phase or amplitude information at kHz or MHz alias frequencies by direct demodulation at this frequency component ε. This technique allows for simultaneous detection of the real and imaginary part of the dynamic susceptibility due to the polar magnetooptical Kerr effect. The obtained signal components allow to reconstruct the magnitude and phase of the M_{z} component for arbitrary detection frequencies. As an example one may use an excitation frequency of f_{rf} = 2.001 GHz, generating the lowest alias frequency at ε = ∣f_{rf} − 25⋅f_{rep}∣ = 1 MHz. For the detection of a signal component at 3/2 f_{rf} = 3.0015 GHz, the signal needs to be demodulated at ε_{nonlinear} = 1.5 MHz. For simultaneous measurements at different frequencies, a multifrequency lockin amplifier is used. The rfpumping signal is amplified up to 23 dBm using a broadband rfamplifier to access the nonlinear regime. For the phase locking experiments a power combiner is used to join the amplified f_{rf} component and the seed frequency component. Both frequency sources are are mutually isolated by rfcirculators (see Fig. 5a).
Analytical model
To describe the observed phenomena, we utilize a model introduced by Bauer et al.^{38} to obtain the spinwave life times and the nonlinear dispersion in kspace within the limit of strong modulation. Therefore, the LLG equation is transformed into the Mathieu equation by substituting the magnetization component with f as described in ref. 38. This results in
where, \(a={({\omega }_{{{{{{{{\rm{k}}}}}}}}}/{\omega }_{{{{{{{{\rm{mod}}}}}}}}})}^{2}\) includes the spinwave frequency ω_{k} as well as the modulation frequency \({\omega }_{{{{{{{{\rm{mod}}}}}}}}}\), while q depicts the modulation strength. The Mathieu equation can be solved by the ansatz
where ν(a, q) is the complex Mathieu exponent and P denotes a periodic function. The real part of ν(a, q) results in the (nonlinear) spinwave dispersion as a function of k_{x} and k_{y}. The imaginary part yields the spinwave life time and allows to determine threshold values when nonlinear spin waves become critical as a function of the rfdriving field amplitude for fixed rffrequency and bias field. For this threshold condition, the wavevector components of the 3/2 f_{rf} spin waves can be accessed in a frequencyresolved manner. For the calculation, we used a Ni_{80}Fe_{20} thickness of 20 nm, the saturation magnetization M_{s} = 800 kA/m, the exchange constant A_{exch} = 13 pJ/m, gyromagnetic ratio γ = 1.76 × 10^{11}(T^{−1}s^{−1}), and a damping constant of α = 0.008.
Micromagnetic simulations
The micromagnetic simulations complementing our experimental findings, are performed using the GPUaccelerated software package mumax3^{45}. The time and spacedependent effective magnetic field is defined by the functional derivative of the free energy density \({{{{{{{\mathcal{F}}}}}}}}[{{{{{{{\bf{m}}}}}}}}]\) with respect to the unit vector field of the magnetization m(r, t)
It is composed of the external field \({{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{ext}}}}}}}}}(t)\), including the static bias field and the oscillating excitation field contribution; the exchange interaction field \({{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{exch}}}}}}}}}=(2{A}_{{{{{{{{\rm{exch}}}}}}}}}/{M}_{{{{{{{{\rm{s}}}}}}}}}){{\Delta }}{{{{{{{{\bf{m}}}}}}}}}_{i}\), with the exchange stiffness A_{exch} and the saturation magnetization M_{s}, the demagnetizing field \({{{{{{{{\bf{B}}}}}}}}}_{i}^{{{{{{{{\rm{d}}}}}}}}}={M}_{{{{{{{{\rm{s}}}}}}}}}{\hat{{{{{{{{\bf{K}}}}}}}}}}_{ij}*{{{{{{{{\bf{m}}}}}}}}}_{j}\), where we refer to ref. 45 for details of the calculation of the demagnetizing kernel \(\hat{{{{{{{{\bf{K}}}}}}}}}\). The magnetization dynamics is described by the Landau Lifshitz Gilbert equation (LLG). The effective magnetic field enters the LLG equation as
This equation is solved for each simulation cell i of the discretized magnetization vector field m_{i}. In the simulations we used the same material parameters as in the analytical modelling.
Data availability
All primary data that support our findings of this study, as well as the code employed in simulations, are available at Zenodo^{46}.
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Acknowledgements
Financial support from the German Research Foundation (DFG) through the collaborative research center CRC/TRR 227 (Project ID 328545488, Projects B02 G.W., B06 J.B.) and Priority Program SPP 2137 (Skyrmionics, G.W.) is gratefully acknowledged.
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G.W. conceived the study. R.D. carried out the experiments. N.L. fabricated the samples. A.F. S., H.G. B., and R.D. performed the simulations. G.W. and R.D. wrote the manuscript. All authors analyzed the data, discussed the results, and commented on the manuscript.
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Dreyer, R., Schäffer, A.F., Bauer, H.G. et al. Imaging and phaselocking of nonlinear spin waves. Nat Commun 13, 4939 (2022). https://doi.org/10.1038/s41467022322240
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DOI: https://doi.org/10.1038/s41467022322240
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