Abstract
Magnons are elementary excitations in magnetic materials and undergo nonlinear multimode scattering processes at large input powers. In experiments and simulations, we show that the interaction between magnon modes of a confined magnetic vortex can be harnessed for pattern recognition. We study the magnetic response to signals comprising sine wave pulses with frequencies corresponding to radial mode excitations. Threemagnon scattering results in the excitation of different azimuthal modes, whose amplitudes depend strongly on the input sequences. We show that recognition rates as high as 99.4% can be attained for foursymbol sequences using the scattered modes, with strong performance maintained with the presence of amplitude noise in the inputs.
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Introduction
A key challenge in modern electronics is to develop lowpower solutions for information processing tasks such as pattern recognition on noisy or incomplete data. One promising approach is physical reservoir computing, which exploits the nonlinearity and recurrence of dynamical systems (the reservoir) as a computational resource^{1,2,3,4}. Examples include a diverse range of materials such as water^{5}, optoelectronic systems^{6,7,8,9}, silicon photonics^{10}, microcavity lasers^{11}, organic electrochemical transistors^{12}, dynamic memristors^{13}, nanowire networks^{14}, and magnetic devices^{15,16,17,18,19,20,21}.
The physical reservoir embodies a recurrent neural network. A natural implementation comprises interconnected nonlinear elements in space (spatial multiplexing, Fig. 1a), where information is fed into the system via input nodes representing distinct spatial elements, and the dynamical state is read out through another set of output nodes^{5,12,14,22}. Another approach involves mapping the network onto a set of virtual nodes in time by using delayedfeedback dynamics on a single nonlinear node (temporal multiplexing, Fig. 1b)^{6,7,8,9,13,18}, which reduces the complexity in spatial connectivity at the expense of more intricate timedependent signal processing.
Here, we study an alternative paradigm in which we exploit instead the dynamics in the modal space of a magnetic element. This scheme relies on magnon interactions in magnetic materials whereby inputs and outputs correspond to particular eigenmodes of a micromagnetic state. Micrometersized magnetic structures can exhibit hundreds of modes in the GHz range^{23}. Processes such as threemagnonscattering interconnect the modes with each other and, with that, provide the nonlinearity and recurrence required for computing. We refer to this approach as modal multiplexing with signals evolving in reciprocal space, in which the actual computation is performed. This is distinct from other wavebased schemes where information is processed with wave propagation and interference in real space^{11,16,19,24,25}, and differs from temporal multiplexing where virtual nodes are constructed with delayed feedback^{6,7,8,9,13,18}. The latter also includes reservoirs based on optical cavities where multimode dynamics (such as frequency combs) are exploited but the output spaces are still constructed by temporal multiplexing^{26,27,28}.
We illustrate the concept of modal multiplexing with a pattern recognition task using a magnonscattering reservoir (MSR). The patterns comprise a sequence of symbols “A” and “B” represented by radiofrequency (rf) signals, which consist of sine wave pulses with two distinct frequencies, f_{A} and f_{B}, and amplitudes b_{rf,A} and b_{rf,b} as shown in Fig. 1d. An example of the power spectrum of the input sequence is given in Fig. 1e. The rf pulses generate oscillating magnetic fields along the z direction through an Ωshaped antenna, which surrounds a 5.1 μm wide, 50nm thick Ni_{81}Fe_{19} disk which hosts a magnetic vortex as a ground state (Fig. 1f). f_{A} and f_{B} are chosen to coincide with the frequencies of primary radial eigenmodes of the vortex, which, when excited above a given threshold, result in the excitation of secondary azimuthal eigenmodes through threemagnonscattering processes^{23}. In our previous work^{29}, we have shown that individual threemagnon splitting channels, e.g. exciting only f_{A}, can be stimulated below their threshold power, and their temporal evolution is significantly modified by additionally exciting one of the secondary modes. In ref. ^{29}, this stimulation was achieved by magnons propagating in a waveguide adjacent to the vortex disk. Here, as a logical extension, the role of the stimulating magnon is provided by the secondary modes of another (active) threemagnon channel f_{B}, a process that we refer to as crossstimulated threemagnon splitting. The operation of our MSR strongly relies on the fact that the crossstimulation between f_{B} and f_{A} is not reciprocal due to the involved nonlinear transients. In other words, the effect of channel f_{A} on the channel f_{B} via crossstimulation differs from the feedback of f_{B} on f_{A}.
The power spectrum of excited magnons is obtained experimentally through microfocused Brillouin light scattering spectroscopy (μBLS), where a portion of the disk is probed (see Methods and Supplementary Fig. 1a). It is important to note that in the linear response regime neither the input spectrum (Fig. 1e) nor the directlyexcited magnon spectrum (Fig. 1g) gives any information about the actual sequence of “A” and “B” (e.g., “AB” and “BA” are equivalent). This means that no linear classifier can be employed. However, when nonlinear processes are at play, magnonscattering, and associated transient processes result in distinct spectral signatures that can be used to distinguish between different input sequences (Fig. 1h).
Results
Figure 2 illustrates the role of threemagnon splitting (3MS), the primary nonlinear process at play for the MSR, in which a stronglyexcited primary magnon splits into two secondary magnons under the conservation of energy and momentum. In experiments, we choose 20ns pulses of f_{A} = 8.9 GHz (20 dBm) and f_{B} = 7.4 GHz (24 dBm), which excite different radial modes of the vortex above their respective power threshold for 3MS, to represent the symbols “A” and “B”, respectively (Fig. 2a). The magnon intensity is probed as a function of frequency and time using timeresolved (TR) μBLS (see Methods) and is colorcoded in Fig. 2b. We measure not only the directly excited primary magnons at f_{A} and f_{B}, but also magnons at frequencies around half the respective excitation frequencies which result from the nonlinearity of spontaneous 3MS (see Fig. 2c)^{23,29}. Here, only the scattering channel with the lowest power threshold is active while other allowed scattering channels remain silent (depicted by dotted lines in Fig. 2c).
Crossstimulation occurs when signals “A” and “B” overlap in time, as shown in Fig. 2d. Two different primary magnons that share a common secondary mode, as is depicted in Fig. 2f, can result in two 3MS channels that mutually crossstimulate each other, even below their intrinsic thresholds and along silent channels^{29}. Thus, within the overlap interval, the pumped secondary magnon of the first symbol influences the primary mode scattering of the second symbol, and vice versa, leading to the primary mode scattering into multiple pairs of secondary modes (Fig. 2e).
Because crossstimulation strongly depends on the temporal order of the primary excitation (Fig. 2f), it provides an important physical resource for processing the temporal sequence of our “AB” signals. This is shown by the experimental results plotted in Fig. 2g, where we compare the timeaveraged power spectra for the “AB” and “BA” sequences. These spectra are computed by integrating the temporal data in Fig. 2e. When only signal “A” or only signal “B” is applied, we measure conventional spontaneous 3MS of the respective primary modes with the secondary modes already discussed above in context of Fig. 2b. Within the overlap interval, the mutual crossstimulation leads to additional peaks in the scattered mode spectrum. As highlighted by shaded areas in Fig. 2e, g, the frequencies and amplitudes of these additional scattered modes strongly depend on the temporal order of the two input signals. Consequently, the average spectra of “AB” and “BA” are different from each other, and neither can be constructed from a simple superposition of the average spectra of “A” and “B” individually (Fig. 2g). This is the key principle that underpins how the MSR processes temporal signals.
To highlight the significance of the transient times, we vary the overlap Δt of the symbols “A” and “B” in experiments and determine the frequencyaveraged difference between the timeaveraged spectra of “AB” and “BA” (Fig. 2h). This difference is zero when the two input pulses do not overlap since no crossstimulation takes place. With increasing overlap, however, crossstimulation between the two pulses becomes more significant and leads to a difference in the output of the reservoir. This difference vanishes again when the input pulses fully overlap and, thus, arrive at the same time.
In order to explore the capabilities of the presented MSR, the complexity of the input signals was further increased experimentally. Figure 3a shows the nonlinear response to the foursymbol pulse pattern “ABAB” measured by TRμBLS. In contrast to a reference spectrum composed by a simple linear superposition of two consecutive “AB” patterns, shown in Fig. 3b, the real spectrum of the foursymbol response contains additional features which are generated by crossstimulated scattering when two pulses overlap. The differences are highlighted by the shaded areas in Fig. 3a and circled areas in Fig. 3b, respectively. This behavior illustrates that crossstimulation can result in distinct features that allow distinguishing also longer patterns. This is further exemplified in Fig. 3c, which shows the timeaveraged BLS spectra of the six foursymbol combinations comprising two “A” and two “B”. Like the data in Fig. 2, transient processes from crossstimulation generate distinct power spectra for the six combinations, which would be indistinguishable in the linear response regime.
Since the experimental data discussed so far requires the integration of thousands of pulse cycles, we rely on micromagnetic simulations to quantify the capacity of the MSR for recognizing all possible combinations of foursymbol sequences composed from “A” and “B” (see Methods). Thereby, we are able to analyze individual pulse sequences and study the influence of thermal noise and amplitude fluctuations on the recognition rate of the MSR. Figure 4a shows a simulated power spectrum (at T = 300 K) for the input pattern “ABAB” with f_{A} = 8.9 GHz (b_{rf,A} = 3 mT) and f_{B} = 7.4 GHz (b_{rf,B} = 3.5 mT), with the field strengths chosen to be above the respective power threshold for 3MS. The output spaces of the reservoir are defined by subdividing the timeaveraged power spectrum into frequency bins of different widths. To emphasize the importance of the scattering (interconnection) between the different magnon modes, we study the performance of the MSR for two separate output spaces (Fig. 4a). One output space for the scattered modes is constructed over a 4GHz window below f_{A} and f_{B}, where the different frequency bins result in an output vector with 16–80 components depending on the bin size (see Supplementary Note 1). For comparison, a twodimensional output space corresponding to the directlyexcited modes is constructed by averaging within bins centered around f_{A} and f_{B}. Note that, here, analyzing the directlyexcited modes does not correspond to a linear classifier, as these modes themselves experience nonlinear feedback (amplitude losses, frequency shift, etc.) above their power threshold for 3MS.
For each foursymbol sequence, 200 micromagnetic simulations were executed with different realizations of the thermal field in order to generate distinct output states. Supervised learning using logistic regression was then performed on this data set to construct trained models of the output states based on either the directlyexcited or scattered modes. The accuracy of these models for different combinations of input frequencies f_{A} = 8.9 GHz, f_{B} = 7.2 GHz, f_{C} = 6.5 GHz, f_{D} = 10.7 GHz (and corresponding input strengths b_{rf,i}) is shown in Fig. 4b as a function of bin size. We find that the MSR performs comparably well when choosing different input frequencies (different radial modes) to represent the input symbols. Hence, an extension of the input space to more than two frequencies/symbols (“ABC”, “BDC”, “ABCD”, etc.), or even to more broadband signals, is straightforward. To this end, Supplementary Note 2 contains the measured and simulated distinct nonlinear responses for different permutations of the three and fourfrequency sequences “ABC” and “ABCD”.
Overall, the accuracy depends weakly on the bin size. The recognition rate slightly increases with increasing bin size for the directlyexited modes whereas it decreases marginally for the scattered modes. This can be understood from the fact that smaller bin sizes capture more features of the power spectrum of the scattered modes, while for the directlyexcited modes, the larger bin sizes contain more information about potential nonlinear frequency shifts, which helps to separate the inputs. We observe that outputs based on the directlyexcited modes can yield an accuracy of ~84%, while scattered modes provide a notable improvement in performance, with an accuracy reaching 99.4% for the case considered in Fig. 4a.
In general, the scattered modes provide higher accuracy for pattern recognition compared with the directlyexcited modes. The difference in accuracy becomes even more pronounced when amplitude fluctuations are present. Figure 4c illustrates how the accuracy evolves with the fluctuation strength, which represents the width of the normal distributions (in %), centered around the nominal values of b_{rf,A} and b_{rf,B}, from which the field strengths are drawn, as shown in the inset for b_{rf,A} = 3 mT and b_{rf,B} = 3.5 mT. The performance of the MSR based on the directlyexcited modes drops significantly with increasing fluctuation strength (42% accuracy at 10% fluctuation). However, recognition based on the scattered modes is much more resilient, with a decrease to only between ~74% and ~92% accuracy (depending on the bin size).
Figure 4d, e show confusion matrices for the scattered and directlyexcited modes, respectively, both for the same set of parameters. They highlight the robustness of the MSR which is based on the scattered modes since it mainly fails to distinguish “AABA” from “ABAA” and “BBAB” from “BBAB” in ~12% of the cases. The MSR based on the directlyexcited modes, on the other hand, fails to recognize almost all of the patterns, except for the trivial cases of “AAAA” and “BBBB” for which there is practically no ambiguity in the inputs. These trends do not depend on the type of supervised learning used and highlight the important role of crossstimulated 3MS in the MSR for the pattern recognition of noisy radiofrequency signals.
Discussion
Our findings demonstrate the possibility of performing reservoir computing in modal space utilizing the intrinsic nonlinear properties of a magnetic system, namely the scattering processes between magnons in a magnetic vortex disk. Temporal patterns encoded using two different inputfrequency pulses can be distinguished with high accuracy. The results also indicate that input patterns can be extended to more broadband signals. We note that the technical design of the physical reservoir is extremely simple and requires very little prepossessing, while the complexity of the data handling arises mostly from the intrinsic nonlinear dynamics of the magnon system. Additionally, recent findings have shown that the magnon interactions in micrometersized disks can be modified significantly by small static magnetic fields^{30}, providing effective means to enhance the complexity of the magnonscattering reservoir further. Although our current readout scheme is based on optical methods, magnetoresistive sensors hold promising possibilities for an allelectric readout.
Methods
Sample preparation and characterization
The magnetic disk housing the magnonscattering reservoir for our experiments was manufactured in a twostep procedure: In a first step, using electronbeam evaporation and subsequent lift off, a magnetic disk with a diameter of 5.1 μm was patterned from a Ti(2)/Ni_{81}Fe_{19}(50)/Ti(5) film deposited on a SiO_{2} substrate which had been capped with a 5nm thick aluminum layer. All thicknesses are given in nanometers. In a second step, an Ωshaped antenna used to excite magnon dynamics in the reservoir was patterned around the disk from a Ti(2)/Au(200), also using electronbeam evaporation and subsequent lift off. The inner and outer diameter of the antenna are 8.3 μm and 11.1 μm, respectively. An image of the sample, obtained with scanning electron microscopy, can be seen in Supplementary Fig. 1a.
Signal generation
The radiofrequency (rf) pulses were generated by two separate rfsources set to a fixed frequency corresponding to pulse “A” and pulse “B”, respectively (see Supplementary Fig. 1b). In order to synchronize the two sources, a pattern generator (Pulsestreamer by Swabian Instruments) was used to create a pattern of arbitrary shape gating the rfsources. The two generated microwave signals were combined and fed onto the Ωshaped antenna using picoprobes.
Timeresolved Brillouinlightscattering microscopy
All experimental measurements were carried out at room temperature. Magnon spectra were obtained by means of Brillouinlightscattering microscopy as schematically shown in Supplementary Fig. 1b^{31}. A monochromatic 532nm laser (CW) was focused onto the sample surface using a 100x microscope lens with a numerical aperture of 0.7. The backscattered light was then directed into a Tandem FabryPérot interferometer (TFPI) using a beam splitter (BS) in order to measure the frequency shift caused by inelastic scattering of photons and magnons. Control signals that encode the current state of the interferometer, signals of the photon counter inside the TFPI and a clock signal from the pattern generator were acquired continuously by a timetodigital converter (Timetagger 20 by Swabian Instruments). From these data, the temporal evolution of the magnon spectra with respect to the stroboscopic rf excitation was reconstructed. During the experiments, the investigated structure was imaged using a red LED and a CCD camera (red beam path in Supplementary Fig. 1b). Displacements and drifts of the sample were tracked by an image recognition algorithm and compensated by the positioning system (XMS linear stages by Newport). The laser and imaging beam path were separated by the dichroic mirror as shown in Supplementary Fig. 1b. In order to ensure that all stationary magnon modes were measured, the signal was averaged over 10 positions across half the disk as seen in Supplementary Fig. 1a.
Micromagnetic simulations
Simulations of the vortex dynamics were performed using the opensource finitedifference micromagnetics code MuMax3^{32}, which performs a time integration of the LandauLifshitzGilbert equation of motion of the magnetization m(r, t),
Here, m(r, t) = M(r, t)/M_{s} is a unit vector representing the orientation of the magnetization field M(r, t) with M_{s} being the saturation magnetization, γ = gμ_{B}/ℏ is the gyromagnetic constant, and α is the dimensionless Gilbertdamping constant. The effective field, B_{eff} = − δU/δM, represents a variational derivative of the total magnetic energy U with respect to the magnetization, where U contains contributions from the Zeeman, nearestneighbor Heisenberg exchange, and dipoledipole interactions. The term b_{th} represents a stochastic field with zero mean, \(\langle {b}_{{{{{{{{\rm{th}}}}}}}}}^{i}({{{{{{{\bf{r}}}}}}}},t)\rangle=0\) and spectral properties satisfying^{33}
with amplitudes drawn from a Gaussian distribution. Here, k_{B} is Boltzmann’s constant, T is the temperature, and V denotes the volume of the finitedifference cell. This stochastic term models the effect of thermal fluctuations acting on the magnetization dynamics. An adaptive timestep algorithm based on a sixthorder RungeKuttaFehlberg method was used to perform the time integration^{34}.
We model our 50nm thick, 5.1μm diameter disk using 512 × 512 × 1 finitedifference cells with γ/2π = 29.6 GHz/T, M_{s} = 810 kA/m, an exchange constant of A_{ex} = 13 pJ/m, and α = 0.008. Previous work has shown that these simulation parameters provide excellent agreement with previous experimental results^{23}.
For the magnon dynamics shown in Fig. 4, we first obtain the magnetic ground state of the disk by initializing with a vortex state and subsequently relaxing the magnetization by minimizing the total magnetic energy in the absence of any static external applied fields. Magnons are then excited under a finite temperature of 300 K. First, we let the system evolve for 5 ns under the action of thermal fluctuations alone. A spatially uniform oscillating magnetic field b_{rf}(t) = b_{rf}(t)e_{z} is then applied along the z direction, perpendicular to the film plane. Following the experimental work, the 4symbol pulse patterns are encoded into b_{rf}(t) as a combination of two inputfrequency signals,
W_{A}(t) and W_{B}(t) represent windowing functions where each “A” or “B” pattern lasts 20ns with a 5ns overlap between patterns. These windowing functions are illustrated in Supplementary Fig. 2 for the 16 4symbol pulse patterns considered. The excitation field amplitude b_{rf,i} and frequency f_{i} for each pattern is given in the main text. After the end of the last pattern, the transient dynamics is computed for an additional 10 ns. The dynamics is simulated for a total duration of 80 ns for each 4symbol pattern.
The power spectral density of the magnon excitations is computed by using a coarsegraining procedure (Supplementary Fig. 3).
The simulation geometry is further subdivided using a triangle mesh (Supplementary Fig. 3a) in the film plane whereby we record the spatial average of the magnetization vector as a function of time, i.e., for a mesh element j,
where V_{j} is the volume of the mesh element. With V = ∑_{j}V_{j} representing the total volume, the total power spectral density \({{{{{{{\mathcal{S}}}}}}}}(\omega )=(1/V){\sum }_{j}{{{{{{{{\mathcal{S}}}}}}}}}_{j}(\omega ){V}_{j}\) is then constructed from the discrete Fourier transform of the z component of the averaged magnetization for each element, \({{{{{{{{\mathcal{S}}}}}}}}}_{j}(\omega )={{{{{{{{\mathcal{M}}}}}}}}}_{j}(\omega ){}^{2}\), where
Δt = 20 ps is the timestep, and N = 8000 is the total number of time steps. Supplementary Fig. 3a and b illustrate the power spectrum for individual regions, i.e., for Region 1 and Region 2 in Supplementary Fig. 3a, respectively. Even at the level of a single mesh region, we can clearly identify the directly excited modes at f_{A} and f_{B}, along with a number of scattered modes. Averaging over a quadrant of the disk gives the power spectrum in Supplementary Fig. 3d, where we can see a muchimproved signaltonoise ratio of the excited and scattered modes. Supplementary Fig. 3e shows the power spectrum averaged over all the mesh regions of the disk, which is qualitatively very similar to the quadrantaveraged result in Supplementary Fig. 3d. For this reason, we only used the quadrantaveraged spectra for the pattern recognition tasks in the interest of minimizing computation time without loss of generality. The construction of the output spaces from the obtained spectra is described in Supplementary Note 1.
Data availability
The numerical and experimental data used in this study are available in the RODARE database under https://doi.org/10.14278/rodare.2344.
Code availability
The software package used for micromagnetic simulations is found at http://mumax.github.io/api.html.
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Acknowledgements
The authors are thankful to D. Rontani and K. Knobloch for providing feedback on the manuscript and fruitful discussions. This study was supported by the German Research Foundation (DFG) within programs SCHU 2922/11 (H.S., T.H., C.H.), KA 5069/11, and KA 5069/31 (L.K.), as well as by the French Research Agency (ANR) under contract No. ANR20CE240012 (MARIN) (S.T., J.V.K.). The project has received funding from the EU Research and Innovation Programme Horizon Europe under grant agreement no. 101070290 (NIMFEIA) (K.S.). Support by the Nanofabrication Facilities Rossendorf (NanoFaRo) at the IBC is gratefully acknowledged.
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Conceptualization: H.S., J.V.K., K.S. Investigation: C.H. Simulation: J.V.K., S.T. Visualization: L.K., T.H., J.V.K., H.S. Funding acquisition: J.V.K., H.S., J.F. Project administration: H.S., K.S. Writing–original draft: L.K. Writing—review and editing: L.K., C.H., T.H., J.V.K., H.S., J.F., K.S.
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Körber, L., Heins, C., Hula, T. et al. Pattern recognition in reciprocal space with a magnonscattering reservoir. Nat Commun 14, 3954 (2023). https://doi.org/10.1038/s4146702339452y
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DOI: https://doi.org/10.1038/s4146702339452y
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